Research ArticleTackling Complexity in Green Contractor Selection for MegaInfrastructure Projects A Hesitant Fuzzy Linguistic MADMApproach with considering Group Attitudinal Character andAttributesrsquo Interdependency

Junling Zhang 1 Xiaowen Qi 2 and Changyong Liang3

1School of Economics and Management Zhejiang Normal University Jinhua 321004 China2School of Business Administration Zhejiang University of Finance amp Economics Hangzhou 310018 Zhejiang China3School of Management Hefei University of Technology Hefei 230009 China

Correspondence should be addressed to Junling Zhang zhangjunlingzjnucn

Received 25 May 2018 Revised 3 October 2018 Accepted 18 October 2018 Published 2 December 2018

Academic Editor Changzhi Wu

Copyright copy 2018 Junling Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Continuous environmental concerns regarding construction industry have been driving general constructors of mega infras-tructure projects to incorporate green contractors Although conventional multiple attributes decision-making (MADM)methodologies have provided feasible ways to select contractor high complexity in scenarios of megaprojects still challengesexisting MADMmethods in concurrently accommodating three key issues of decision hesitancy attributes interdependency andgroup attitudinal character To elicit decision-makersrsquo hesitant fuzzy assessments more objectively and comprehensively we definean expression tool called interval-valued dual hesitant fuzzy uncertain unbalanced linguistic set (IVDHF UUBLS) and developaggregation operators through its operations To exploit attributes interdependencywe establish a synthesized attributesrsquo weightingmodel to fuse an attributes interdependency-based weighting vector and an argument-dependent weighting vector which arerespectively derived throughDecision-Making andTrial Evaluation Laboratory (DEMATEL) technique andmaximizing deviationmethod To effectively utilize decision-makersrsquo group attitudinal characters we also develop a TOPSIS-based method to rationallytransform group ideal attitudes into order-inducing vectors On the strength of the above methods an integratedMADMapproachis then constructed Finally illustrative case study and experiments are conducted to validate our approach

1 Introduction

In comparison with other industries the industry of con-structionhas been observed as exertingmajor contribution toenvironmental pollution [1] including those related to noiseair solid waste and water pollution [2 3] The problems ofconstruction pollution and wastes have been more critical incountries that are undertaking accelerated urbanization con-struction with high growth rate of economics As can be seenthe environmental concerns especially stand out for megainfrastructure construction projects because those megapro-jects are basically one-off and sustaining in a relatively verylong period [4 5] As a matter of fact mega infrastructureconstruction projects are assumed to take responsibility

to minimize their negative impact on environments whenmaximizing contribution to economics and society To thatend not only high level of projectmanagement sophisticationis needed but a nexus of green contractors also should beincorporated in project organization [6] As a result greencontractor selection plays a strategically important role inachieving success of megaprojects

Due to self-evident complexity in large-scale projectslike mega infrastructure understanding scenarios of workpackages and selecting appropriate contractors are intrin-sically complicate [7 8] To confront the complexity incontractor selection Safa et al [5] suggested utilizing resultsof competitive intelligence (CI) systems as favorable inputs

HindawiComplexityVolume 2018 Article ID 4903572 31 pageshttpsdoiorg10115520184903572

2 Complexity

tomultiple attributes decision-making (MADM)methodolo-gies However during contractor selection activities at thefront end planning phase [9] decision-makersrsquo assessmentsinevitably characterize with vagueness and uncertainty dueto limitedness in their experience knowledge and expertise[10 11] To help decision-makers decrease complexity underdifferent scenarios of green contractor selection pioneeringresearches have introduced effective MADM approaches thatintegrate uncertainty expression tools such as fuzzy sets[12] and linguistic sets [13ndash15] BuiThiNuong et al [16] putforward a fuzzy AHP method to accommodate contractorselection with imprecise information Kusi-Sarpong et al[17] developed a joint rough sets and fuzzy TOPSIS methodZavadskas et al [18] presented a weighted aggregated sumproduct assessment method which uses grey values for con-tractor selection When facing more complicate contractorselection problems with ill-structured definition Vahdani etal [19] devised an ideal solutions-based MADM approachto contractor selection by employing linguistic variablesto describe uncertain assessments Alhumaidi [20] utilizedlinguistic variables to construct a fuzzy-logic based multipleattributes group decision-making method Most recently inviewing of common phenomena of decision hesitancy incomplex decision-making processes [21 22] Borujeni andGitinavard [23] studied a group decision-making approachbased on hesitant fuzzy preference relations

Actually advocating practicality of hesitant fuzzy set(HFS) [21] one of popular research directions in fuzzyMADM literature has also focused on tackling complicatesituations with decision hesitancy based on HFS [40] Suchas the MADM approach based on Shapley-Choquet integraloperators to consider interdependencies among attributes[41] the hesitant fuzzy TOPSIS method [42] the MADMapproach based onhesitant fuzzyHamacher aggregation [43]the hesitant fuzzy QUALIFLEX approach [44] among othersFollowing the idea of HFS Zhu et al [45] extendedHFS to thedual hesitant fuzzy set (DHFS) by adding nonmembershipdegrees in depicting hesitancy Since then DHFS has beenstudied with a deep extent such as those listed in [46ndash50] Regarding decision-making situations where linguisticvariables gain feasibility and adaptability on the basic ideas ofHFS Rodrıguez et al [51] firstly introduced the hesitant fuzzylinguistic term sets to allow decision hesitancy of possiblelinguistic terms However in practical cases decision-makersare capable of reaching the most favorite linguistic labelbut generally holding decision hesitancy to that label orunder the decision-making situations where voting andmajority rule apply group opinions will locate at a linguisticlabel or a linguistic interval [52] but obviously there existsdecision hesitancy to the voted label As a result compoundexpression tools of hesitant fuzzy linguistic sets and theirMADMapproaches have attracted widespread interests Suchas hesitant fuzzy linguistic set [38 53] and dual hesitantfuzzy linguistic set [39 54] among others Comparativelycompound hesitant fuzzy linguistic expression tools attainmore adaptability and feasibility in practical complex deci-sion environments

Although researches in both contractor selection litera-ture and MADM literature have presented rich approaches

for application to green contractor selection problems thereis still a gap with three facets to be further investigatedconcurrently First most of existing hesitant fuzzy linguisticexpression tools were based on rigidly uniform or symmet-rical linguistic label sets [55 56] however practical studies[57 58] have revealed that decision-makers are inclinedto express their complicate assessments more precisely andobjectively by use of non-uniform or asymmetric linguisticterm set that is the unbalanced linguistic term set (ULTS)[59] Second under complicate decision-making environ-ments interdependency among attributes is a serious issuethat needs to be addressed [60] However Choquet integralincorporated by Lin et al [38] can only capture interactionsbetween adjacent coalitions while the prioritized aggregationoperator in [53 54] requires linear ordering of evaluativeattributes that is often difficult for decision-makers to deduceup front in complex practices Third in order to deal withreal-world complex problems various degrees of optimism(degree of orness) of decision-makers should be coordinatedas complex attitudinal characters and taken into accountduring decision-making [61]Merigo andCasanovas [61] sug-gested to use order-inducing variables for reflecting complexattitudinal characters and integrate them into informationaggregation operators in which the order-inducing variablesare usually not directly defined and need to be derivedappropriately [62] But to our best knowledge thus farthere is scarcely any attention has been paid to the complexattitudinal characters when tackling complicate contractorselection problems

Therefore to bridge the above-discussed gap we firstintroduce an interval-valued dual hesitant fuzzy uncertainunbalanced linguistic set (IVDHF UUBLS) whose elementshold a hybrid structure of ⟨119909 119904 ℎ(119909) 119892(119909)⟩ in which twointerval-valued fuzzy sets ℎ(119909) and 119892(119909) denote possiblemembership and nonmembership degrees to the uncertainunbalanced linguistic term 119904 IVDHF UUBLS can depictfuzzy properties of an object more comprehensively andflexibly We also develop a series of generalized aggregationoperators for IVDHF UUBLS In order to exploit interdepen-dency relations among evaluative attributes more effectivelywe next employ the Decision-Making and Trial EvaluationLaboratory (DEMATEL) [63 64] to capture both direct andindirect influences between attributes thereby constructinga more rational attributes weighting model To includedecision-makerrsquos complex attitudinal characters instead ofassuming subjective values we propose a robust TOPSIS-based method to identify the order-inducing variables basedon positive and negative ideal attitudes On the strengthof above methods we subsequently propose an effectiveMADM approach for resolving the complex problems ofgreen contractor selection

The rest of this paper is organized as follows In Sec-tion 2 research problem is described Section 3 definesthe new hybrid hesitant fuzzy linguistic expression toolie IVDHF UUBLS and studies basic operations and dis-tance measure for IVDHF UUBLS Section 4 investigatesseveral fundamental generalized aggregation operators for

Complexity 3

IVDHF UUBLS In Section 5 We detail the DEMATEL-based model for weighting evaluative attributes and theTOPSIS-based method for deriving order-inducing variablesbased on which we then construct an effective approach forcomplex MADM In Section 6 illustrative case study andexperiments are further conducted to verify the proposedMADM approach Finally conclusions and future researchare presented in Section 7

2 Description of Research Problem

The significant impacts of construction activities on theenvironment have triggered serious alarms and governmentsworldwide have introduced various policies regulations andindustrial evaluation systems for controlling them [65] forinstance the cleaner production promotion law and thepollution prevention law in China and the green buildingrating systems in the US UK and China [66] To meetthe requirements of environmental concerns constructionprojects have to select contractors in operations by simultane-ously considering potential contractorsrsquo green characteristics[6] Actually the practical needs of selecting green con-tractors are especially indispensable for mega infrastructureconstruction projects as they generally are one-off and nearlyever-lasting in the environment

21 Evaluative Attributes for Green Contractor SelectionTypically when construction organizations seek to developappropriate approaches to contractor selection the organiza-tionsrsquo specific requirements are firstly introduced Thereforedifferent sets of evaluative attributes for contractor selectionwith different scenarios are needed During last two decadesmany efforts have been paid to identify selection attributesand construct comprehensive contractor selection methodswith different applications [8 10 18 19 23 24 26 28 67 68]

One basic observation from existing literature for consid-ering evaluative attributes is that being analogous to selectingsuppliers in supply chain management [69] decision-makersare supposed to not only consider competitive capabilitythat distinguishes contractors from each other by businessoperations but also examine contractorrsquos cooperation capac-ity with other partner contractors that indeed influencesthe project success [8 70] More importantly since theconstruction industry has great impact on environment andthe governments worldwide have introduced various policiesand regulations for controlling them [6] green practices thushas become a crucial facet for evaluating green contractors[68 71] As a result three aspects of business competivenesscooperation and green practices thus become indispen-sible to derive evaluative attributes for green contractorselection

Many earlier studies put their emphasis on competitiveattributes to evaluate comprehensive performance of contrac-tors Hatush and Skitmore [26] suggested five competitiveattributes to assess contractors including financial sound-ness technical ability management capability health andsafety and reputation Fong and Choi [24] studied contractorselection problem for Hong Kong scenario and derived a set

of eight evaluative attributes according to a questionnairesurvey among which seven are competitive attributes includ-ing price financial capability past performance past expe-rience resource current workload and safety performanceIn their developed computer-aided decision support systemShen et al [28] employed a parameter system to evaluatecontractorrsquos competitiveness in the Chinese constructionindustry in which six competitive attributes were includedas social influence technical ability financing ability andaccounting status marketing ability management skills andorganizational structure and operations Darvish et al [25]developed a graph theory-based decision-making method tocope with contractor prequalification problem under Iranianscenarios they introduced the Iranian domestic prequalifi-cation criteria system that included nine main indicatorswork experience technology amp equipments managementexperience and knowledge of the operation team finan-cial stability quality being familiar with the area or beingdomestic reputation and creativity and innovation Nieto-Morote and Ruz-Vila [10] also investigated a fuzzy decision-making model based on TOPSIS to accommodate construc-tion contractor prequalification problem they summarizedthe most common factors for comprehensively consideringcontractors during the prequalification that fall into followingmain aspects technical capacity experience managementcapability financial stability past performance past relation-ship reputation and occupational health amp safety Focusingon contractor selection problems in Lithuania constructionindustry Zavadskas et al [18] proposed an effective multipleattributes decision-making approach called WASPAS-G inwhich main attributes are identified to cover bid amountcapability amp skill occupational health and security technicalcapacity managerial capacity past performance and pastexperience As seen from the above representative literaturequotation technical strength and resource strength are thethreemain factors which arewidely accepted and adopted wethus include these factors as main evaluative attributes in thispaper Another finding from the above reviewed literature isthat nearly all of them emphasized the inclusion of indicatorsto examine credibility of contractors in their managingexternal and internal challenges [72 73] such as financialcapability [24] safety performance [24] and reputation [1025 26] therefore we introduce the lsquoCredibilityrsquo as anothermain attribute in this paper

Indeed cooperation attributes must be taken into con-sideration because cooperation among selected contractorinfluences the project success [74] Actually some of ear-lier pioneering studies also noticed and took considera-tion of cooperation attribute in their parameter systemssuch as lsquoclientcontractor relationshiprsquo in references [10 24]Recently increasing attentions have been paid to investigateappropriate evaluative cooperation attributes for contractorselection Representatively based on the conceptual modelof partnering and alliancing in construction project man-agement Liang et al [8] elaborated a set of cooperationattributes for selecting joint venture contractors in large-scale infrastructure projects including compatible culturecontract communication collaboration cooperation abilityand cooperation satisfaction To provide a deeper insights

4 Complexity

on factors that affect contractorsrsquo cooperation under inter-national construction joint ventures in construction projectsHwang et al [27] conducted a survey and reported lsquosharingof project risksrsquo as the top attractive factor and lsquodifferences inculture and working stylersquo as the top negative factor amongothers Furthermore aiming at finding risks that most affectcontractorsrsquo cooperation within project networks Hwangand Han [29] conducted a survey from the viewpoint ofcontractors and sub-contractors in Singapore constructionenvironment they identified ten top critical network riskssuch as lsquodifferent cultural normsrsquo lsquoinaccurate informationdeliveryrsquo lsquooccurrence of disputersquo and lsquolack of risk manage-ment knowledgersquo Obviously Hwang et al [27] and Hwangand Han [29] contributed a feasible way to deduce morereasonable evaluative attributes on contractorrsquos cooperationcapability however to our best knowledge there is stilla lack of thorough investigation on a consensus set ofevaluative attributes on contractorrsquos cooperation capabilityfor referencing Therefore in the light of Liang et al [8]Hwang et al [27] and Hwang and Han [29] we here takelsquoCooperation management capabilityrsquo as one of the mainevaluative attributes to assess contractors

To reduce the significant environmental footprint [6]internal and external factors (such as government regulationsand managerial concerns) are driving the entire construc-tion industry to enforce green practices all over the world[6] Green practices can be considered as an outcome ofstrategic processes through cooperation within the nexus ofcontractors [75] Therefore it is intrinsically crucial to takeinto account evaluative attributes on green practices in con-tractor selection especially for those nearly ever-lasting megainfrastructure construction projects Since environmentalregulations of different scales (ie domestic governmentaland international [76]) have increasingly become compulsoryin various industrial markets contractors should have thecapability of manage and keep track of the compliance [35]And the widely accepted way [77] to attain the compliancecapability by building environment management systems[32 34 78] whose most-cited components [77] compriseof ISO-14001 certification [32 34 36 37] eco-labeling [3132 37] environment policies [33] environment planning[33] and environmental management information system(continuous monitoring and regulatory compliance) [31 32]Due to the reason that environmental impacts occur at everystage of the construction cycle Rwelamila et al [79] stronglysuggested to contractors should implement green design ampprocurement to improve their green practices Construc-tion life-cycle analysis is critical in both green design forenvironment [33] and pollution reduction through greenprocurement [31 32 34] Measurements of green design forenvironment include tracking all material and reverse flowof a project from the retrieval of raw materials out of theenvironment to the disposal of the product back into theenvironment generally including recycle [31] reuse [32 33]remanufacture [32 33] disassembly and disposal [33] In thegreen procurement aspect Tan et al [68] suggested to reduceenvironmental footprints throughout the whole constructionsupply chain by addressing issues such as waste reduction [3234] environment material substitution [31 33] hazardous

material minimization [35] and clean technology availability[33] Furthermore from the stance of strategic manage-ment Fergusson and Langford [80] pointed out environ-mental performance of green practices positively contributesto comprehensive competitive advantages of contractorsEnvironmental performance has thus been recognized asone of the indispensible evaluative attributes to contractorselection [34 78] Sharma and Vredenburg [81] referred toenvironmental performance as the environmental effects thatcorporationrsquos activities have on the natural milieu To makeenvironmental performance more measurable many effortshave been paid to establish effective assessing approachessuch as the approach to examination on waste flows andcontrol on construction sites [3 82] As for indicators ofenvironmental performance according to the thorough liter-ature review by Govindan et al [77] the commonly adoptedones include [32ndash34 36] solid waste chemical waste airemission waste water disposal and energy In sum we hereinclude lsquoDesign amp procurementrsquo lsquoCompliance with greenlegislationrsquo and lsquoEnvironmental performancersquo as three othermain evaluative attributes to comprehensively assess greenpractices of alternative contractors

For more clarity based on the above literature review welisted all eight main evaluative attributes and present theirdefinitionsprinciples And the optional subdimensions ofthese main attributes have been also listed in Table 1 forvarious comprehensive considerations according to specificnature in practical problems

(A1) Quotation Project value of reviewed The lowesttender price tends to attract a clientrsquos interest as superior toother criteria

(A2) Credibility Comprehensive evaluation on trustwor-thiness of contractors in developmental dynamics and risksfrom both internal and external environments focusing onfinancial soundness health and safety past performancework experience etc

(A3) Technical strength Comprehensive evaluation oncontractorsrsquo capability of technology and innovation thattackling forthcoming complicate tasks generally based onquality rank technical ability experience and knowledge ofoperation team creativity amp innovation etc

(A4) Resource strength Comprehensive evaluation oncontractorsrsquo competitive strength on productive resourcesthat indispensable for construction needs generally based ontechnical human resource current workload constructionmachinery amp equipment fixed assets amp liquidity etc

(A5) Cooperation management capacity Comprehensiveevaluation on contractorsrsquo cooperation practices with partici-pators in projects focusing on clientcontractor relationshiporganizational structure and operations compatible culturecommunication and information delivery knowledge shar-ing of risk management among others

(A6) Design amp procurement Comprehensive evaluationon contractorsrsquo practices that improve projectrsquos whole lifevalue by using green design environmental friendly mate-rials and green production processes which promote bestpractices of green construction procurement throughout thesupply chain

Complexity 5

Table1Mainevaluativ

eattributesandtheiro

ptionalsub

dimensio

nsforg

reen

contractor

selection

No

Attributes

Sub-dimensio

nsRe

ferences

A1

Quo

tatio

n(i)

Tend

erprice(

advancep

aymentcapitalbidrou

tine

maintenance

andmajor

repairs

)[81824]

Liangetal[8]Fon

gandCh

oi[24]Z

avadskasetal[18]

A2

Credibility

(i)pastperfo

rmance

[101824]

(ii)w

orkexperie

nce[25]

(iii)repu

tatio

nampcreditratin

g[1025ndash27]

(iv)h

ealth

ampsafety

[10182426]

(v)fi

nancialstability[1024ndash2628]

Nieto-M

orotea

ndRu

z-Vila[10]Z

avadskasetal[18]H

atushandSkitm

ore

[26]Fon

gandCh

oi[24]Shenetal[28]Darvishetal[2527]

A3

Technical

strength

(i)technicalability[810182628]

(ii)e

xperienceamp

know

ledgeo

foperatio

nteam

[1025]

(iii)qu

ality

rank

[25]

(iv)c

reativity

ampinno

vatio

n[25]

Liangetal[8]N

ieto-M

orotea

ndRu

z-Vila[10]Z

avadskasetal[18]H

atush

andSkitm

ore[

26]Sh

enetal[28]D

arvishetal[25]

A4

Resource

strength

(i)technicalhum

anresource

[2427]

(ii)c

urrent

workload

[24]

(iii)constructio

nmachinery

ampequipm

ent[2527]

(iv)fi

xedassetsampliq

uidity

[27]

Fong

andCh

oi[24]D

arvishetal[25]H

wangetal[27]

A5

Coo

peratio

nmanagem

ent

capability

(i)clientcon

tractorrelationship[102429]

(ii)o

rganizationalstructureampop

erations

[828]

(iii)compatib

lecultu

re[827]

(iv)c

ommun

ication[829]

(v)k

nowledges

harin

gof

riskmanagem

ent[2729]

(vi)inform

ationdelivery[29]

(vii)

contractmanagem

ent[8]

Liangetal[8]N

ieto-M

orotea

ndRu

z-Vila[10]H

atushandSkitm

ore[

26]

Fong

andCh

oi[24]Shenetal[28]Darvishetal[25]H

wangetal[27]

HwangandHan

[29]A

nvuu

rAaron

andKu

maraswam

yMoh

an[30]

A6

Designamp

Procurem

ent

(i)recycle[31]

(ii)reuse

[3233]

(iii)remanufacture[3233]

(iv)d

isassem

blyanddisposal[33]

(v)w

asteredu

ction[3234]

(vi)environm

entm

aterialsub

stitutio

n[3133]

(vii)

hazardou

smaterialm

inim

ization[35]

(viii)c

lean

techno

logy

availability[33]

Lin[35]H

umph

reysetal[34]C

hiou

etal[32]Leeetal[31]H

umph

reys

etal[33]

A7

Com

pliance

with

green

legisla

tion

(i)ISO-140

01certificatio

n[32343637]

(ii)e

co-la

belin

g[313237]

(iii)environm

entp

olicies[33]

(iv)e

nviro

nmentp

lann

ing[33]

(v)e

nviro

nmentalm

anagem

entinformationsyste

m[3132]

Hum

phreysetal[34]C

hiou

etal[32]YangandWu[36]Liu

andLiu[37]

Leeetal[31]Hum

phreysetal[33]

A8

Environm

ental

perfo

rmance

(i)solid

waste[333436]

(ii)c

hemicalwaste[333436]

(iii)aire

mission[333436]

(iv)w

astewater

disposal[32ndash3436]

(v)e

nergy[

3334]

Hum

phreysetal[34]C

hiou

etal[32]YangandWu[36]Hum

phreysetal

[33]

6 Complexity

(A7) Compliance with green legislation Comprehensiveevaluation on the extent to which contractorsrsquo practicessatisfy different governmental green or sustainability legis-lations according to the aspects of ISO-14001 certificationeco-labeling environment policies environment planningenvironmental management information system etc

(A8) Environmental performance Comprehensive evalu-ation on environmental effects that the corporationrsquos activitieshave on the natural milieu Environmental performance cancommonly bemeasured throughoperative performance indi-cators (ie energyresource utilization emission reductionand waste disposal)

22 Problem Definition The contractor selection as well asmany other multicriteria decisions impacting the overallproject should bemade during the front end planning stage ofa project the point at which a group of designated decision-makers have the power to accept or reject a contractorfor a specific project or its work packages [5 9] WhenusingMADMmechanism to copewith complicate contractorselection problems [5] normally few nominated contractorswill be ready for the decision-makers to vote on Due tocomplexity of the problems and limitedness of knowledgedecision-makers usually feel confident in expressing theiropinions by use of interval numbers [83] or uncertain linguis-tic terms [52] Although widely-accepted majority rule willquickly help the group of decision-makers arrive at a decisionon an uncertain linguistic term (eg [s4 s5]) to whichobviously there exists decision hesitancy because differentopinions existTherefore in this paper we define the interval-valued dual hesitant fuzzy uncertain unbalanced linguisticset (IVDHF UUBLS) to help the decision-making panel elicittheir assessments more objectively and completely

From another point of view various attitudinal charac-ters (degree of orness) commonly exist because individualexpert holds specific backgrounds and decision processthat involves the attitudinal character of group decision-makers must coordinate those various attitudinal charactersinto one complex attitudinal character [61] Therefore weadopt the concept of order-inducing vector [62] to reflectgroup complex attitudinal character and develop a TOPSIS-based method to rationally determine the order-inducingvector Besides as Tan and Chen [84] pointed out forreal-world sophisticated MADM problems the indepen-dency axiom [85] cannot generally satisfied For exampleupgrading in green performance will raise the quotationand intrinsically result in requirements for high-standardcollaboration between contractors and general constructorIn viewing of this common phenomenon we take attributesrsquointerdependency as a third indispensible characteristic intackling complexity in green contractor selection in megainfrastructure projects In sum we take three characteristicsof complexity to model the practical problems of complicategreen contractor selection ie (i) compound structure ofhesitant fuzzy linguistic assessments (ii) group attitudinalcharacters and (iii) attributesrsquo interdependency To producegreater clarity Figure 1 demonstrates the conceptual MADMmodel for green contractor selection

Now we can give the symbolized description of thetargeted complex green contractor problems Given a megainfrastructure project there are a set of alternative greencontractors ie 119883 = 1199091 1199092 119909119899 for its subprojectsLet 119860 = 1198601 1198602 119860119898 be the evaluative attributesaccording to which decision-makers consider each greencontractor Due to high complexity in the sophisticatedproblem scenarios there exists interdependency relationsamong the evaluative attribute A panel of decision-makers119864 = 1198641 1198642 119864119905 have been organized to give theirassessments to each alternative contractor 119909119894 (119894 = 1 119899)under every attribute 119860119895 (119895 = 1 119898) In order to reflect thecomplicate group assessments of all alternative contactorsthe hybrid expression tool of IVDHF UUBLS that will bedetailed in Section 3 is adopted to depict the assessmentsmore effectively and comprehensively As a result a specificdecision matrix 119877 = (119903119894119895)119898times119899 whose elements are in theform of IVDHF UUBLS is obtained According to Merigoand Casanovas [61] suppose that an order-inducing vec-tor 120576 for denoting group attitudinal characters has beenreasonably obtained Then effective MADM approachesmust be developed to determine the most appropriate greencontractor(s)

3 Interval-Valued Dual Hesitant FuzzyUncertain Unbalanced Linguistic Set

As demonstrated in Figure 1 after the panel of decision-makers votes on an alternative contractor under certainattribute by use of specific uncertain unbalanced linguisticterm set the uncertain linguistic term [s4 s5] stands outbecause of the majority rule while different opinions shouldalso be included and considered inMADMprocesses To thatend based on interval-valued dual hesitant fuzzy set (IVD-HFS) [47] and the unbalanced linguistic term set (ULTS)[59] we here first introduce an interval-valued dual hesitantfuzzy uncertain unbalanced linguistic set (IVDHF UUBLS)which incorporate different opinions of decision-makersas membership degrees or nonmembership degrees to themajority-voted [s4 s5] Then we develop operational rules aswell as distance measure for the IVDHF UUBLS Regardingdefinitions of the IVDHFS and ULTS one can refer toAppendix A

31 Definition of IVDHF UUBLS

Definition 1 Let119883 be a fixed set and 119878 be a finite and continu-ous unbalanced linguistic label set Then an IVDHF UUBLS119878119863 on119883 is defined as

119878119863 = ⟨119909 119904120599(119909) ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (1)

where 119904120579(119909) = [119904120572 119904120573] represents judgment to object 119909119904120572 and 119904120573 are two unbalanced linguistic variables frompredefined unbalanced linguistic label set 119878 which repre-sents decision-makersrsquo judgments to an evaluated object119883 ℎ(119909) = ⋃120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and 119892(119909) =⋃] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two sets of closed intervals

Complexity 7

e panelof decision

makers Alternativegreen

contractors

Evaluativeattributes

Comprehensiveevaluation

Attributesrsquo interdependency

+

alternativecontractor

vote

Voting results on

MajorityRule

radic

Specificdecision

matrix

AppropriateMADM

approach

Decisionresults

Section 3 Defines an effectivehybrid expression tool

Group attitudinal characters (ie complex degrees of orness)

s

⟨Contractori [s4 s5] ℎ (x) g(x)⟩

i-th

1 2 3 4 5 6 7

j-th attribute

Figure 1 Conceptual MADMmodel for tackling complexity in green contractor selection

in [0 1] ℎ(119909) denotes possible membership degrees that 119909belongs to 119904120579(119909) and119892(119909) represents possible nonmembershipdegrees of 119909 to 119904120579(119909) ℎ(119909) and 119892(119909) hold conditions 120583 ] isin[0 1] and 0 le (120583119880)+ + (]119880)+ le 1 where(120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880 for all 119909 isin 119883When119883 = 1199091 1199092 119909119899has only one element 119878119863 reduces to (s120599 ℎ 119892) which is calledan interval-valued dual hesitant fuzzy uncertain unbalancedlinguistic (IVDHF UUBL) number (IVDHF UUBLN)

32 Operational Rules for IVDHF UUBLS On the strength ofoperational rules for uncertain linguistic set [52] unbalancedlinguistic set [59] and interval-valued dual hesitant fuzzy set[47] we get the following operations for IVDHF UUBLS

Definition 2 Let 119904119889 = (119904120599 ℎ 119892) = ([119904120572 119904120573] ℎ 119892) 1199041198891 =(1199041205991 ℎ1 1198921) = ([1199041205721 1199041205731] ℎ1 1198921) and 1199041198892 = (1199041205992 ℎ2 1198922) =([1199041205722 1199041205732] ℎ2 1198922) be any three IVDHF UUBLNs 120582 isin [0 1]operations on these IVDHF UUBLNs are defined as

(1) 120582119904119889 = ⋃(119904120599ℎ119892)isin119904119889

([119904120582Δminus11199050 (119879119865

11990511989611199050(120595(119904120572)))

119904120582Δminus11199050 (119879119865

11990511989621199050(120595(119904120573)))

] ⋃

[120583119871120583119880]isinℎ[]119871]119880]isin119892

[1 minus (1 minus 120583119871)120582 1 minus (1 minus 120583119880)120582] [(]119871)120582 (]119880)120582]) (2) 119904119889120582 = ⋃

(119904120599 ℎ119892)isin119904119889

([119904(Δminus11199050 (119879119865

11990511989611199050(120595(119904120572))))

120582 119904(Δminus11199050 (119879119865

11990511989621199050(120595(119904120573))))

120582])

⋃[120583119871120583119880]isinℎ[]119871]119880]isin119892

[(120583119871)120582 (120583119880)120582] [1 minus (1 minus ]119871)120582 1 minus (1 minus ]119880)120582] )(3) 1199041198891 oplus 1199041198892 = ⋃

(1199041205991 ℎ11198921)isin1199041198891(1199041205992 ℎ21198922)isin1199041198892

([119904Δminus11199050 (119879119865

11990511989411199050(120595(1199041205721 )))+Δ

minus11199050(11987911986511990511989511199050(120595(1199041205722 )))

119904Δminus11199050 (119879119865

11990511989421199050(120595(1199041205731 )))+Δ

minus11199050(11987911986511990511989521199050(120595(1199041205732 )))

] ⋃

[1205831198711 1205831198801 ]isinℎ1[120583

1198712 1205831198802 ]isinℎ2[]

1198711 ]1198801 ]isin1198921[]

1198712 ]1198802 ]isin1198922

[1205831198711 + 1205831198712 minus 12058311987111205831198712 1205831198801 + 1205831198802 minus 1205831198801 1205831198802 ] []1198711]1198712 ]1198801 ]1198802 ]) (4) 1199041198891 otimes 1199041198892 = ⋃

(1199041205991 ℎ11198921)isin1199041198891(1199041205992 ℎ21198922)isin1199041198892

([119904Δminus11199050 (119879119865

11990511989411199050(120595(1199041205721 )))timesΔ

minus11199050(11987911986511990511989511199050(120595(1199041205722 )))

119904Δminus11199050 (119879119865

11990511989421199050(120595(1199041205731 )))timesΔ

minus11199050(11987911986511990511989521199050(120595(1199041205732 )))

] ⋃

[1205831198711 1205831198801 ]isinℎ1[120583

1198712 1205831198802 ]isinℎ2[]

1198711 ]1198801 ]isin1198921[]

1198712 ]1198802 ]isin1198922

[12058311987111205831198712 1205831198801 1205831198802 ] []1198711 + ]1198712 minus ]1198711]1198712 ]1198801 + ]1198802 minus ]1198801 ]

1198802 ])

(2)

8 Complexity

In above details about the transformation functionsof 119879119865 and Δ linguistic hierarchies (119871119867) as well as thetransformation procedures for unbalanced linguistic termsets are shown in Appendix A

eorem 3 Letting 119904119889 = (119904120599 ℎ 119892) 1199041198891 = (1199041205991 ℎ1 1198921) and1199041198892 = (1199041205992 ℎ2 1198922) be any three IVDHF UUBLNs 120582 1205821 1205822 isin[0 1]then following properties are true

(1) 1199041198891 oplus 1199041198892 = 1199041198892 oplus 1199041198891 (2) 1199041198891 otimes 1199041198892 = 1199041198892 otimes 1199041198891 (3)120582(1199041198891 oplus 1199041198892) = 1205821199041198891 oplus 1205821199041198892(4) 1199041198891120582 otimes 1199041198892120582 = (1199041198891 otimes 1199041198892)120582 (5) 1205821119904119889 oplus 1205822119904119889 = (1205821 +1205822)119904119889 (6) 1199041198891205821 otimes 1199041198891205822 = 1199041198891205821+1205822

Proof Omitted

In Definition 2 and Theorem 3 1199051198961 1199051198962 1199051198941 1199051198942 1199051198951 and1199051198952 are corresponding levels of 119904120572 119904120573 1199041205721 1199041205731 1199041205722 and 1199041205722in 119871119867 respectively 1199050 is the maximum level of 119904120572 119904120573 1199041205721 1199041205731 1199041205722 and 1199041205722 in 119871119867 Furthermore to compare any twoIVDHF UUBLNs we also have following definitions

Definition 4 Let 119904119889 = (119904120599 ℎ 119892) be an IVDHF UUBLN119904120599 = [119904120572 119904120573] then score function 119878(119904119889) and accuracy function119875(119904119889) can be represented by

119878 (119904119889) = Δminus11199050 (11987911986511990511989611199050 (120595 (119904120572))) + Δminus11199050 (11987911986511990511989621199050 (120595 (119904120573)))2 times 12 ( 1119897 (ℎ) sum

[120583119871120583119880]isinℎ

120583119871 minus 1119897 (119892) sum[]119871]119880]isin119892

]119871

+ 1119897 (ℎ) sum[120583119871120583119880]isinℎ

120583119880 minus 1119897 (119892) sum[]119871]119880]isin119892

]119880)(3)

119875 (119904119889)= Δminus11199050 (11987911986511990511989611199050 (120595 (119904120572))) + Δminus11199050 (11987911986511990511989621199050 (120595 (119904120573)))2

times 12 ( 1119897 (ℎ) sum[120583119871120583119880]isinℎ

120583119871 + 1119897 (119892) sum[]119871]119880]isin119892

]119871

+ 1119897 (ℎ) sum[120583119871120583119880]isinℎ

120583119880 + 1119897 (119892) sum[]119871]119880]isin119892

]119880)

(4)

where 119897(ℎ) and 119897(119892) are numbers of values in ℎ and 119892respectively 1199051198961 and 1199051198962 are the corresponding levels of 119904120572 and119904120573 in 119871119867 and 1199050 is the maximum level of 119905119896 in 119871119867

Definition 5 Given any IVDHF UUBLNs 1199041198891 = (1199041205991 ℎ1 1198921)1199041198892 = (1199041205992 ℎ2 1198922) then(1) If 119878(1199041198891) lt 119878(1199041198892) then 1199041198891 lt 1199041198892(2) If 119878(1199041198891) = 119878(1199041198892) then

(a) If 119875(1199041198891) = 119875(1199041198892) then 1199041198891 = 1199041198892(b) If 119875(1199041198891) lt 119875(1199041198892) then 1199041198891 lt 1199041198892

33 Distance Measure for IVDHF UUBLS When 119897(ℎ) or119897(119892) of two IVDHF UUBLNs are unequal the complement-ing method [86] is normally adopted to design distancemeasures Note that artificially adding values to shorterones in the complementing method will cause informationdistortion To avoid this limitation we provide the followingdistance measure

Definition 6 Let two IVDHF UUBLNs 1199041198891 = (1199041205991 ℎ1 1198921)and 1199041198892 = (1199041205992 ℎ2 1198922) where 1199041205991 = [1199041205721 1199041205731] 1199041205992 = [1199041205722 1199041205732]119897ℎ1 119897ℎ2 1198971198921 and 1198971198922 are lengths of ℎ1 ℎ2 1198921 and 1198922 respectivelydenoting number of elements in ℎ1 ℎ2 1198921 and 1198922 Suppose1198681 = (1(119899(1199051198941) minus 1))Δminus11199050 (11987911986511990511989411199050 (120595(1199041205721))) 1198682 = (1(119899(1199051198942) minus1))Δminus11199050 (11987911986511990511989421199050 (120595(1199041205731))) 1198683 = (1(119899(1199051198951)minus1))Δminus11199050 (11987911986511990511989511199050 (120595(1199041205722)))1198684 = (1(119899(1199051198952) minus 1))Δminus11199050 (11987911986511990511989521199050 (120595(1199041205732))) where 1199051198941 1199051198942 1199051198951 and1199051198952 are the corresponding levels of unbalanced linguistic terms1199041205721 1199041205731 1199041205722 and 1199041205732 in the linguistic hierarchy 119871119867 and 1199050 is themaximum level of 1199041205721 1199041205731 1199041205722 and 1199041205732 in 119871119867 Then by use ofnormalized Euclidean distance we define a distance measure119889 for IVDHF UUBLNs as follows

Situation 7 When 119897ℎ1 = 119897ℎ2 = 1198971 and 1198971198921 = 1198971198922 = 1198972 then119889 (1199041198891 1199041198892) = (12 ( 11198971sdot 1198971sum119896=1

(10038161003816100381610038161003816100381610038161198681120583119871119895ℎ1 minus 1198683120583119871119896ℎ2 10038161003816100381610038161003816100381610038162 + 10038161003816100381610038161003816100381610038161198682120583119880119895ℎ1 minus 1198684120583119880119896ℎ2 10038161003816100381610038161003816100381610038162) + 11198972sdot 1198972sum119896=1

(1003816100381610038161003816100381610038161198681]1198711198951198921 minus 1198683]1198711198961198922 1003816100381610038161003816100381610038162 + 1003816100381610038161003816100381610038161198682]1198801198951198921 minus 1198684]1198801198961198922 1003816100381610038161003816100381610038162)))12

(5)

Situation 8 When 119897ℎ1 = 119897ℎ2 or 1198971198921 = 1198971198922 then119889 (1199041198891 1199041198892) = (12 ( 1119897ℎ1 119897ℎ2sdot 119897ℎ1sum119895=1

119897ℎ2sum119896=1

(10038161003816100381610038161003816100381610038161198681120583119871119895ℎ1 minus 1198683120583119871119896ℎ2 10038161003816100381610038161003816100381610038162 + 10038161003816100381610038161003816100381610038161198682120583119880119895ℎ1 minus 1198684120583119880119896ℎ2 10038161003816100381610038161003816100381610038162)+ 11198971198921 1198971198922sdot 1198971198921sum119895=1

1198971198922sum119896=1

(1003816100381610038161003816100381610038161198681]1198711198951198921 minus 1198683]1198711198961198922 1003816100381610038161003816100381610038162 + 1003816100381610038161003816100381610038161198682]1198801198951198921 minus 1198684]1198801198961198922 1003816100381610038161003816100381610038162))12

(6)

eorem 9 The distance measure 119889 defined in Definition 6satisfies following properties

(1) 0 le 119889(1199041198891 1199041198892) le 1

Complexity 9

(2) 119889(1199041198891 1199041198892) = 0 if and only if 1199041198891 and 1199041198892 are perfectlyconsistent

(3) 119889(1199041198891 1199041198892) = 119889(1199041198892 1199041198891)4 Generalized Aggregation Operators forIVDHF_UUBLS

41 Definitions of Operators Based on the generalized oper-ators firstly introduced by Yager [87] we here develop some

fundamental generalized aggregation operators for the newlydefined IVDHF UUBLS

Definition 10 Given a collection of IVDHF UUBLNs 119904119889119895(119895 =1 119899) their weighting vector 120596 = (1205961 1205962 120596119899)119879120596119895 isin[0 1]sum119899119895=1 120596119895 = 1 120582 be a parameter 120582 isin (0 +infin)(1) Generalized IVDHFUUBLWeighted Average(GIVDHFUUBLWA) Operator

119866119868119881119863119867119865119880119880119861119871119882119860120596120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(120596119895119904119889119895120582))1120582

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120582)1120582 119904(sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120582)1120582]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[(1 minus

119899prod119895=1

(1 minus (120583119871119895 )120582)120596119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880119895 )120582)120596119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871119895)120582)120596119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880119895 )120582)120596119895)1120582]])

(7)

(2) Generalized IVDHFUUBLWeighted Geometric(GIVDHFUUBLWG)Operator

119866119868119881119863119867119865119880119880119861119871119882119866120596120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889119895)120596119895)= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871119895 )120582)120596119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880119895 )120582)120596119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871119895 )120582)120596119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880119895 )120582)120596119895)1120582]])

(8)

Definition 11 For a collection of IVDHF UUBLNs 119904119889119895(119895 = 1 119899) 119904119889120590(119895) be the 119895th largest 119908 = (1199081 1199082 119908119899)119879 be the aggregation-associated weighting vector 119908119895 isin [0 1]sum119899119895=1119908119895= 1 120582 is a parameter such that 120582 isin (0 +infin) 119878119899 997888rarr 119878 Then

10 Complexity

(1) Generalized IVDHFUUBL Ordered Weighted Average(GIVDHFUUBLOWA) Operator

119866119868119881119863119867119865119880119880119861119871119874119882119860120596120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(119908119895119904119889120590(119895)120582))1120582

= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

120582)1120582]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895) []119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120590(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120590(119895))120582)119908119895)1120582]])

(9)

(2) Generalized IVDHFUUBL Ordered Weighted Geometric(GIVDHFUUBLOWG)Operator

119866119868119881119863119867119865119880119880119861119871119874119882119866119908120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889120590(119895))119908119895)= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120590(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120590(119895))120582)120596119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120590(119895))120582)120596119895)1120582]])

(10)

Definition 12 For a collection of IVDHF UUBLNs 119904119889119895(119895 =1 119899) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector120596119895 isin [0 1] sum119899119895=1 120596119895 = 1 119899 is a balancing coefficient119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weight-ing vector with 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1

(1) Generalized IVDHFUUBL Hybrid Average (GIVDHFU-UBLHA) Operator

119866119868119881119863119867119865119880119880119861119871119867119860120596119908120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(119908119895119904119889120590(119895)120582))1120582

= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895)

))))120582)1120582]

Complexity 11

⋃[120583119871120590(119895) 120583

119880120590(119895)]isinℎ120590(119895) []

119871120590(119895) ]119880120590(119895)]isin119892120590(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120590(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120590(119895))120582)119908119895)1120582]])

(11)

where 119904119889120590(119895) is the 119895th largest and

119904119889119895 = 119899120596119895119904119889119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904119899120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904119899120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

] ⋃

[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[1 minus (1 minus 120583119871119895 )119899120596119895 1 minus (1 minus 120583119880119895 )119899120596119895] [(]119871119895)119899120596119895 (]119880119895 )119899120596119895]) (12)

(2) Generalized IVDHFUUBL Hybrid Geometric(GIVDHFUUBLHG) Operator

119866119868119881119863119867119865119880119880119861119871119867119866119908120596120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889120590(119895))119908119895)= ⋃(119904120599120590(119895)

ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120590(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895)

))))119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895) []119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120590(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120590(119895))120582)119908119895)1120582]])

(13)

where 119904119889120590(119895) is the 119895th largest and

119904119889119895 = 119904119889119895119899120596119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(Δminus11199050 (119879119865

11990511989511199050(120595(119904120572119895 ))))

119899120596119895 119904(Δminus11199050 (119879119865

11990511989521199050(120595(119904120573119895 ))))

119899120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[(120583119871119895 )119899120596119895 (120583119880119895 )119899120596119895] [1 minus (1 minus ]119871119895 )119899120596119895 1 minus (1 minus ]119880119895 )119899120596119895]) (14)

12 Complexity

When confronted with ill-structured situations wheredecision-makerrsquos complex attitudinal characters need to beincluded order-inducing vectors provide an effective way[88ndash90]Thus we further define following induced operatorsfor IVDHF UUBLNs

Definition 13 For a collection of IVDHF UUBLNs 119904119889119895 (119895 =1 119899) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector

120596119895 isin [0 1] sum119899119895=1 120596119895 = 1 119899 is a balancing coefficient119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weight-ing vector 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1 120576119895 denote a set oforder inducing vectors

(1) Induced Generalized IVDHFUUBL Hybrid Average (I-GIVDHFUUBLHA) Operator

119868-119866119868119881119863119867119865119880119880119861119871119867119860120596119908120582 (lt 1205761 1199041198891 gt lt 1205762 1199041198892 gt lt 120576119899 119904119889119899 gt) = ( 119899⨁119895=1

(119908119895119904119889120587(119895)120582))1120582

= ⋃(119904120599120587(119895) ℎ120587(119895) 119892120587(119895))isin119904119889120587(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120587(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587(119895)

))))120582)1120582]

⋃[120583119871120587(119895) 120583

119880120587(119895)]isinℎ120587(119895) []

119871120587(119895) ]119880120587(119895)]isin119892120587(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120587(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120587(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120587(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120587(119895))120582)119908119895)1120582]])

(15)

where (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) is decreasing order of (11990411988911199041198892 119904119889119899) according to the order inducing vector 120576119895 and119904119889119895 = 119899120596119895119904119889119895 = ⋃

(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904119899120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904119899120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

] ⋃

[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[1 minus (1 minus 120583119871119895 )119899120596119895 1 minus (1 minus 120583119880119895 )119899120596119895] [(]119871119895)119899120596119895 (]119880119895 )119899120596119895]) (16)

(2) Induced Generalized IVDHFUUBL Hybrid Geometric(I-GIVDHFUUBLHG) Operator

119868-119866119868119881119863119867119865119880119880119861119871119867119866119908120596120582 (lt 1205761 1199041198891 gt lt 1205762 1199041198892 gt lt 120576119899 119904119889119899 gt) = 1120582 ( 119899⨂119895=1

(120582119904119889120587(119895))119908119895)= ⋃(119904120599120587(119895)

ℎ120587(119895) 119892120587(119895))isin119904119889120587(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120587(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587(119895)

))))119908119895]

⋃[120583119871120587(119895)120583119880120587(119895)]isinℎ120587(119895) []119871120587(119895) ]

119880120587(119895)]isin119892120587(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120587(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120587(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120587(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120587(119895))120582)119908119895)1120582]])

(17)

Complexity 13

Weighted Aggregation

Geometric Arithmetic

GIVDHFUUBLHG GIVDHFUUBLHA

GIVDHFUUBLWG GIVDHFUUBLOWG

IVDHFUUBLWG IVDHFUUBLOWG

GIVDHFUUBLWA GIVDHFUUBLOWA

IVDHFUUBLWA IVDHFUUBLOWA

= 1 w =1

n = 1 =

1

n = 1 =1

n = 1 =

1

n

= 1 = 1 = 1 = 1

Figure 2

where (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) is decreasing order of (11990411988911199041198892 119904119889119899) according to the order inducing variables 120576119895 and

119904119889119895 = 119904119889119895119899120596119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(Δminus11199050 (119879119865

11990511989511199050(120595(119904120572119895 ))))

119899120596119895 119904(Δminus11199050 (119879119865

11990511989521199050(120595(119904120573119895 ))))

119899120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[(120583119871119895 )119899120596119895 (120583119880119895 )119899120596119895] [1 minus (1 minus ]119871119895 )119899120596119895 1 minus (1 minus ]119880119895 )119899120596119895]) (18)

42 Properties of the Proposed GeneralizedAggregation Operators

eorem 14 With special values of 120582 119908 and 120596 the operatorsGIVDHFUUBLHA and GIVDHFUUBLHG can include aseries of traditional aggregation operators as special cases andtheir relationship can be depicted in Figure 2

Proof See Appendix B

As for the induced hybrid aggregation operators we alsohave following theorem

eorem 15 If (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) = (119904119889120590(1) 119904119889120590(2) 119904119889120590(119899)) I-GIVDHFUUBLHA reduces toGIVDHFUUBLHAIf (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) = (119904119889120590(1) 119904119889120590(2) 119904119889120590(119899)) thenI-GIVDHFUUBLHG reduces to the GIVDHFUUBLHG oper-ator

eorem 16 All the proposed generalized operatorsGIVDHFUUBLWAGIVDHFUUBLWGGIVDHFUUBLOWAGIVDHFUUBLOWGGIVDHFUUBLHAGIVDHFUUBLHGI-GIVDHFUUBLHA and I-GIVDHFUUBLHG hold thefollowing properties (1) Commutativity (2) Idempotency (3)Boundedness

Based on above theorems following properties can alsobe derived

eorem 17 For a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weight vector of 119904119889119895with 120596119895 isin [0 1] and sum119899119895=1 120596119895 = 1 120582 gt 0 then we have

(1) 119868119881119863119867119865119880119880119861119871119882119866120596(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119882119860120596(1199041198891 1199041198892 119904119889119899)(2) 119868119881119863119867119865119880119880119861119871119882119866120596(1199041198891 1199041198892 119904119889119899) le119866119868119881119863119867119865119880119880119861119871119882119860120596120582(1199041198891 1199041198892 119904119889119899)(3) 119866119868119881119863119867119865119880119880119861119871119882119866120596120582(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119882119860120596(1199041198891 1199041198892 119904119889119899)

Proof Omitted for concision

eorem 18 For a collection of IVDHF UUBLNs 119904119889119895 = (119904120599119895 ℎ119895 119892119895) 119908 = (1199081 1199082 119908119899)119879 is the weighting vector of 119904119889119895with 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1 120582 gt 0 Then

(1) 119868119881119863119867119865119880119880119861119871119874119882119866119908(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119874119882119860119908(1199041198891 1199041198892 119904119889119899)(2) 119868119881119863119867119865119880119880119861119871119874119882119866119908(1199041198891 1199041198892 119904119889119899) le119866119868119881119863119867119865119880119880119861119871119874119882119860119908120582(1199041198891 1199041198892 119904119889119899)(3) 119866119868119881119863119867119865119880119880119861119871119874119882119866119908120582(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119874119882119860119908(1199041198891 1199041198892 119904119889119899)

Proof Omitted for concision

14 Complexity

5 An Integrated MADM Approach forTackling Complex Green ContractorSelection Problems

As described in Section 2 we take the green contractorselection as a special type of complicate MADM problemsthat synthesizes three characteristics of decision hesitancy[21 22] attributes interdependency [84] and group atti-tudinal characters [61] Therefore in this section we con-struct an integrated MADM approach to tackle the com-plex green contractor selection problems Suppose 119883 =1199091 1199092 119909119898 is the set of alternative green contractors and119860 = 1198601 1198602 119860119899 is the set of evaluative attributes 120596 =(1205961 1205962 120596119899)119879 is weighting vector for the attributes 120596119894 ge 0sum119899119894=1 120596119894 = 1 Let = (119903119894119895)119899times119898 denote the decision matrix inwhich 119903119894119895 = ([119904120572119894119895 119904120573119894119895] ℎ119894119895 119892119894119895) is an IVDHF UUBLN givenby decision-makers for alternative contractor 119909119895 with respectto attribute 119860 119894 According to the mechanism of pair-wisecomparisons among attributes in the DEMATELmethod [6364] the interdependency among attributes can be obtained asamatrix119885 = (119911119894119896)119899times119899 where 119911119894119896 indicates the degree towhich119860 119894 affects119860119896 Subsequently based on the IVDHFUUBLS andits operations we now present detailed steps of our MADMapproach as shown in following Algorithm I

Algorithm I Hesitant fuzzy linguistic MADMwith attributesinterdependency and decision-makersrsquo group attitudinalcharacters

Step 1 Determine argument-dependent weighting vector120596119860119863 according to attribute values by programming modeldeveloped in the following Section 51

Step 2 Obtain the attribute-interdependences based weight-ing vector 120596119860119868 by use of DEMATEL method described in thefollowing Section 52

Step 3 Calculate synthesized attribute weighting vectoraccording to

120596 = 120572120596119860119868 + 120573120596119860119863 (19)

where 120572 and 120573 are parameters to reflect decision character-istics of decision organizations 0 le 120572 120573 le 1 120572 + 120573 =1Step 4 Check requirements for order inducing If no addi-tional order inducing required then go to Step 5 otherwisego to Step 6

Step 5 Utilize generalized aggregation operators to get theoverall IVDHF UUBLNs 119903119895(119895 = 1 119898) for each alternative119909119895 Here we take GIVDHFUUBLHA operator for examplebecause it can include other traditional operators as its specialcases Therefore we have

119866119868119881119863119867119865119880119880119861119871119867119860(1199031119895 1199032119895 119903119899119895) = ( 119899⨁119894=1

(119908119894 (119903120590119894119895)120582))1120582= ⋃(119904120590119894119895ℎ120590119894119895 119892120590119894119895

)isin119903120590119894119895

([119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590119894119895

))))120582)1120582 119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590119894119895

))))120582)1120582]

⋃[120583119871120590119894119895120583

119880120590119894119895]isinℎ120590119894119895[]

119871120590119894119895]119880120590119894119895]isin119892120590119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120590119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120590119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120590119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120590119894119895)120582)119908119894)1120582]])

(20)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of 119903119894119895119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weightvector 119903120590119894119895 is the 119894th largest of 119903119894119895 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))

119904119899120596119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119894119895))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (21)

Complexity 15

Then go to Step 8

Step 6 By use of the TOPSIS-based method developed in thefollowing Section 53 we determine order-inducing vectors 120576to reflect decision-makersrsquo group attitudinal characters

Step 7 Utilizing induced generalized aggregation operatorsto get the overall IVDHF UUBLNs 119903119895(119895 = 1 2 119898) foreach alternative 119909119895 we have

119868-119866119868119881119863119867119865119880119871119867119860(1199031119895 1199032119895 119903119899119898) = ( 119899⨁119894=1

(119908119894 (119903120587119894119895)120582))1120582

= ⋃(119904120587119894119895ℎ120587119894119895119892120587119894119895)isin119903120587119894119895

([119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120587119894119895

))))120582)1120582 119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587119894119895

))))120582)1120582]

⋃[120583119871120587119894119895 120583

119880120587119894119895]isinℎ120587119894119895 []

119871120587119894119895]119880120587119894119895]isin119892120587119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120587119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120587119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120587119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120587119894119895)120582)119908119894)1120582]])

(22)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of119903119894119895 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 119903120587119894119895 is the 119894th largest of 119903119894119895 reordered by theorder-inducing vectors 120576 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))119899120596119894120572119894119895

119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120573119894119895 ))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (23)

Step 8 Calculate scores 119878(119903119895) and accuracy degrees 119875(119903119895) for119903119895(119895 = 1 2 119898)Step 9 According to Definition 5 output ranking order ofalternative contractors

In Algorithm I to objectively derive unknown weight-ing vector for evaluative attributes a synthesized attributeweighting vector is devised by fusing two parts of concern(i) argument-dependent weighting vector and (ii) attribute-interdependences based weighting vector The former is toreflect effects of attribute values on attribute weights whilethe latter is to exploit effects of interdependences amongattributes on their weights For clarity processing flowchartof Algorithm I is depicted in Figure 3

51 Determining Argument-Dependent Attribute WeightingVector 120596119860119863 In order to derive attribute weighting vec-tor appropriately from assessments maximizing deviationmethod [91] is adopted here for its distinguishing ability

and objectivity [92] According to Wang [91] if performancevalues of each alternative have little differences under anattribute it implicates such an attribute plays a less importantrole otherwise it lives a more important role in decision-making Thus if one attribute has similar values across alter-natives it should be assigned a smaller weight contrariwiseit should be a bigger weight

In light of the idea in maximizing deviation methodwe develop a programming model to determine argument-dependent weighting vector 120596119860119863 For attribute 119860 119894 the stan-dard deviation of alternative 119909119895 to all the other alternatives isdenoted as

119865119894119895 (120596119860119863) = 119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (24)

Let119865119894(120596119860119863) represent deviation value of all alternatives toother alternatives according to attribute119860 119894 and be formulatedas follows

16 Complexity

Attribute weighting

Hesitant fuzzy linguistic MADM with attributes interdependency and decision makersrsquo group attitudinal characters

Step 9 Output ranking order of alternatives

interdependency among attributes by DEMATEL in Section 52

attributes values by standard deviation method in Section 51

Step 6 Ensure TOPSIS order inducing variables proposed in Section 53

Step 7 Utilizing proposed induced generalized hybrid aggregation operators developed in Section

41 to get overall value of each alternative

Step 8 Calculate score degrees and accuracy degrees

Step 4 Order inducingYes

Step 5 Choose proposed generalized operators in Section 41 to get overall value of each alternative

No

Collect decision matrix in the form of IVDHF_UUBLS

Complex problems of green contractor selection

Step 1 Weighting vector AD derived from Step 2 Weighting vector AI derived from

Step 3 Obtain synthesized weighting vector of attributes =AI + AD

Figure 3 Processing flowchart of Algorithm I

119865119894 (120596119860119863) = 119898sum119895=1

119865119894119895 (120596119860119863) = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (25)

Based on the analysis above we can get followingprogramming model (M-1) for obtaining optimal attributeweighting vector 120596119860119863

(119872 minus 1)max 119865 (120596119860119863) = 119899sum

119894=1119865119894 (120596119860119863) = 119899sum

119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894st 119899sum

119894=1(120596119860119863119894 )2 = 1 120596119860119863119894 ge 0 119894 = 1 2 119899 (26)

where 119889(119903119894119895 119903119894119897) can be calculated by Definition 6To solve model (M-1) we can construct Lagrange func-

tion

119871 (120596119860119863119894 120577) = 119899sum119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894+ 1205772 ( 119899sum

119894=1

(120596119860119863119894 )2 minus 1) (27)

where 120577 is the Lagrange multiplierDifferentiate with respect to 120596119860119863119894 (119894 = 1 2 119898) and 120577

and set these partial derivatives equal to zero the followingset of equations can be obtained

120597119871120597120596119860119863119894 = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) + 120577120596119860119863119894 = 0120597119871120597120577 = 119899sum

119894=1

(120596119860119863119894 )2 minus 1 = 0 (28)

By normalizing the solutions of above equations we canget a simple and exact formula for determining the attributeweights as follows

120596119860119863119894 = sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897)sum119899119894=1sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897) (29)

52 Deriving Attributes Interdependency-Based AttributeWeighting Vector 120596119860119868 Some efforts [53 60 84 93ndash95]

Complexity 17

have been made to address interdependencies betweenattributes in MADM However they unanimously cannotreveal the interdependences in a more complete way WhileDEMATAEL technique [63 96] gathers collective knowledgeto capture influences between attributes more precisely andcompletely so that weighting vector for attributes can berationally determined according to strength measure ofthose influences [64 97] Therefore in tackling the complexMADMwith interdependent attributes under IVDHF UUBLenvironments we present a DEMATEL-based method as thefollowing Algorithm II to derive attribute weighting vector120596119860119868 from the viewpoint of attributes interdependency

Algorithm II DEMATEL-basedmethod for ensuringweight-ing vector 120596119860119868Step 1 Construct initial direct-influence matrix 119885 = (119911119894119896)119899times119899

For attributes 119860 = 1198601 1198602 119860119899 decision-makersare asked to give direct effect that attribute 119860 119894 has onattribute 119860119896 using a five-point scale with scores representedby natural language (1) 0 mdash lsquoabsolutely no influencersquo (2)1 mdash lsquolow influencersquo (3) 2 mdash lsquomedium influencersquo (4) 3 mdashlsquohigh influencersquo (5) 4mdash lsquovery high influencersquoThen the initialdirect-influence matrix 119885 = (119911119894119896)119899times119899 can be established witheach element being the mean of same factors in differentdirect matrices

Step 2 Calculate normalized direct-influence matrix 119883 =(119909119894119896)119899times119899 Using matrix 119885 the normalized direct-influencematrix 119883 = (119909119894119896)119899times119899 is calculated according to

119909119894119896 = 119911119894119896119904 (30)

where 119904 = maxmax1le119894le119899sum119899119896=1 119911119894119896max1le119896le119899sum119899119895=1 119911119894119896 Allelements in matrix 119883 are complying with 0 le 119909119894119896 le 10 le sum119899119896=1 119909119894119896 le 1 and 0 le sum119899119894=1 119909119894119896 le 1 and at least onerow or column of the summation equals 1

Step 3 Derive total-influence matrix 119879 = (119905119894119896)119899times119899 Basedon matrix 119883 the total-influence matrix 119879 = (119905119894119896)119899times119899 can bederived by summing the direct effects and all of the indirecteffects

119879 = 119883 + 1198832 + 1198833 + sdot sdot sdot + 119883ℎ = 119883 (119868 minus 119883)minus1 (31)

where 119868 represents identity matrix When ℎ 997888rarr infin 119883ℎ =(0)119899times119899Step 4 Compute influencing and influenced degrees of eachattribute The sum of rows and the sum of columns withinthe total-influence matrix 119879 are respectively expressed as thevectors 119903 and 119888

119903 = (119903119894)119899times1 = ( 119899sum119896=1

119905119894119896)119899times1

119888 = (119888119896)119899times1 = ( 119899sum

119895=1

119905119894119896)119879

1times119899

(32)

where 119903119894 denotes the sumof the 119894th row inmatrix119879 and showsthe sum of direct and indirect effects that attribute 119860 119894 hason other attributes Similarly 119888119896 shows the sum of direct andindirect effects that attribute 119860119896 has receivedStep 5 Estimate weights of attributes based on influencedegrees according to

120596119868lowast119894 = [(119903119894 + 119888119894)2 + (119903119894 minus 119888119894)2]12 (33)

Then the weighting vector 120596119860119868 can be obtained by normaliz-ing (33)

120596119860119868119894 = 120596119868lowast119894sum119899119894=1 120596119868lowast119894 (34)

53 Obtaining Order Inducing Variables 120576 by TOPSIS-BasedMethod For ill-defined MADM problems induced aggre-gation operators [89 90 98] provide an effective channelfor experts to express complex attitudinal characters Thekey step in induced aggregation operators is to obtainorder-inducing vectors 120576 which however are presumptivelyassigned in most papers [62] Therefore in this sectionaiming at deducing the order-inducing vectors 120576 rationallywe here devise a TOPSIS-based method in Algorithm IIIObviously the Algorithm III is of generality for MADMwithdifferent forms of linguistic expression

Algorithm III TOPSIS-based method for obtaining orderinducing vectors 120576Step 1 Let the decision maker define IVDHF UUBL idealsolution(s) 119903+119894 and IVDHF UUBL negative ideal solution(s)119903minus119894 corresponding to each attribute in which

119903+119894 = (119903+1198941 119903+1198942 119903+119894119901) 119903minus119894 = (119903minus1198941 119903minus1198942 119903minus119894119902) (35)

where 119901 and 119902 denotes the number of 119903+119894 and 119903minus119894 respectively

Step 2 Calculate distances from each element in decisionmatrix to ideal solution(s) and negative solution(s) respec-tively

119889+119894119895 = max119897=12119901

119889 (119903119894119895 119903+119894119897 ) 119889minus119894119895 = min

119897=12119902119889 (119903119894119895 119903minus119894119897 ) (36)

where 119889(119903119894119895 119903+119894119897 ) and 119889(119903119894119895 119903minus119894119897 ) are calculated by the distancemeasure in Definition 6 If 119901 gt 1 we choose the onewith maximum distance if 119902 gt 1 we choose the one withminimum distance

Step 3 Calculate coefficients according to

119888119894119895 = 119889minus119894119895119889+119894119895 + 119889minus119894119895 119894 = 1 2 119899 119895 = 1 2 119898 (37)

18 Complexity

Table 2 Decision matrix in the form of IVDHF UUBLS1199091 1199092 11990931198601([AHVH] [02 03][02 04] [03 04]) ([HT] [04 05][05 06] [02 03] [02 04]) ([ALAH] [01 02][01 03] [06 07])1198602([MH] [05 06][02 03]) ([AHT] [06 07][01 02] ) ([MVH] [02 03][05 06] [06 07])1198603([VLAL] [03 04][04 05] [05 06]) ([MVH] [06 07] [07 08][01 02]) ([VLL] [04 05][02 03])1198604([MAH] [02 04][05 06]) ([LAL] [03 05][02 03]) ([ALM] [06 07][01 02] [02 03])1198605([ALM] [04 05][01 02] [04 05]) ([AHH] [06 07][01 02]) ([HVH] [04 05][06 07] [01 03])1198606([LM] [01 02][03 05] [03 05]) ([VLL] [02 04] [05 06][02 03]) ([AHH] [03 04][04 05])1198607([VLL] [04 05][05 06] [02 03] [02 04]) ([MAH] [05 06] [07 08][01 02]) ([ALM] [04 05][05 06] [03 04])1198608([ALAH] [01 03][04 06]) ([AHH] [05 07][01 02] [02 03]) ([AHH] [04 06][03 04])

Table 3 Direct influence matrix Z1198601 1198602 1198603 1198604 1198605 1198606 1198607 11986081198601 0 3 4 1 3 1 2 41198602 1 0 2 4 4 2 3 11198603 4 2 0 1 3 2 4 11198604 2 3 1 0 3 4 1 21198605 4 2 1 3 0 1 4 31198606 2 4 3 4 4 0 4 21198607 4 1 2 3 3 1 0 41198608 1 2 4 3 3 2 4 0

Step 4 Derive order inducing variable 120576119894119895 by descendingorder of 119888119894119895 The greater the value 119888119894119895 the bigger the 1205761198941198956 Illustrative Example and Experiments

61 Illustrative Case Study Suppose a mega infrastructureproject is selecting green contractor for one of its subprojectsA panel of decision-makers including managers professorsand competitive intelligence experts has been organized tocomprehensively evaluate three alternative contractors (ie1199091 1199092 and 1199093) under the eight attributes (ie 1198601ndash1198608) listedin Section 2

To elicit true decision preferences we apply theIVDHF UUBLS to assess the three alternatives (1199091ndash1199093)Firstly decision-makers are asked to vote on 119909119895 under eachattribute119860 119894 using different unbalanced linguistic term sets 1198781 1198782 and 1198783 where 1198781 = 119873 119871119860119871119872119860119867119867119876119867119881119867119860119879 1198791198782 = 119873119860119873119881119871 119876119871 119871 119860119871119872119860119867119881119867 119879 1198783 = 119873119881119871119871119860119872119872119876119872119860119867119867 119879 In Figure 4 we show the adoptedlinguistic hierarchies and relationship between 1198781 1198782 and1198783 Attributes 1198601 1198602 1198605 and 1198608 are evaluated by use of 1198781attributes11986031198604 and1198607 are evaluated by 1198782 and attribute1198606is evaluated by 1198783 The highest-voted uncertain unbalancedlinguistic term [119904119898 119904119899] are identified Secondly the panel of

decision-makers is asked to further express their opinionson membership degrees ℎ(119909) and nonmembership degrees119892(119909) to [119904119898 119904119899] Then we can derive compound assessments(ie ⟨119909 119904120599(119909) ℎ(119909) 119892(119909)⟩) in the form of IVDHF UUBLSand obtain the decision matrix in Table 2 Additionallyin order to examine the interdependency among attributesdecision-makers are also required to assess the influentialrelations among the eight attributes using DEMATEL-basedAlgorithm II The direct influence matrix is collected andshown in Table 3

Next we apply the proposed Algorithm I to solve thisgreen contractor selection problem To derive the synthe-sized attribute weighting vector in Algorithm I we assign120572 = 120573 = 05 For scenarios with no group attitudinalcharacters operator GIVDHFUUBLHA in Definition 12 ischosen In contrast for scenarios with group attitudinalcharacters operator I-GIVDHFUUBLHA in Definition 13will be chosen Further the panel of decision-makers is askedto determine their positive and negative ideal alternativesthat are collected in Table 4 where corresponding order-inducing vectors that are derived from Algorithm III arealso listed (see the rightmost column) Thereafter operatorI-GIVDHFUUBLHA in Definition 13 is utilized for aggrega-tion For GIVDHFUUBLHA and I-GIVDHFUUBLHA we

Complexity 19

Table 4 Ideal alternatives (119903+ and 119903minus) and derived order-inducing vectors 120576119903+ 119903minus 1205761199091 1199092 11990931198601 ([VHT] 1 0) ([NL] 1 0) 5 6 61198602 ([MAH] [07 08] [01 02]) ([NL] 1 0) 1 1 41198603 (T 1 0) (L 1 0) 8 7 81198604 ([ALAH] 09 01) (T 1 0) 2 2 11198605 ([HT] [09 1] 0) (N 1 0) 7 5 51198606 (M 1 0) (N 1 0) 6 3 31198607 (M 1 0) ([NL] 1 0) 3 8 71198608 ([MAH] 1 0) (N 1 0) 4 4 2

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 1 2 3 4

0 1 2

N L AH H QH VH AT TM

Level 1 l(13)

Level 2 l(25)

Level 3 l(39)

Level 4 l(417)

Unbalanced linguistic term set S1

N AH VH TM Unbalanced linguistic term set S2VL L ALAN QL

N H TM Unbalanced linguistic term set S3

L AM AHQM

AL

VL

Figure 4 Unbalanced linguistic term sets (1198781 1198782 and 1198783) and their mapping in linguistic hierarchies

assume 120582=2 and 119908 = (0 0 015 025 025 025 01 0)119879Note that 119908 is a position weighting vector which is derivedby the fuzzy semantic quantitative operator [99] Aftercompleting all steps in Algorithm I (see Appendix C) weobtain the ranking results and present them in Table 5 Forthe scores derived from Step 5 we find the ranking order is1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the best solution Step 7gives the ranking order of 1199092 ≻ 1199091 ≻ 1199093 indicating 1199092 is stillthe most appropriate one while 1199091 is better than 1199093 under theinfluence of 12057662 Experiments on Ranking Results along with VariousParameter Configurations In order to inspect the impactsof various parameter configurations on ranking results inthis section more numerical examples are carried out Alleight operators that defined in Section 41 are applied to theabove illustrative case where the parameter 120582 is configured asintegers ranging from 1 to 20 and parameter 120572 is configuredto a value set of 0 03 05 08 09 1

Firstly the four basic aggregation operators of GIVDH-FUUBLWA GIVDHFUUBLWG GIVDHFUUBLOWA and

Table 5 Ranking results of operators GIVDHFUUBLHA and I-GIVDHFUUBLHA

Aggregation operator Ranking orderGIVDHFUUBLHA 1199092 ≻ 1199093 ≻ 1199091I-GIVDHFUUBLHA 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG are applied to the former case Thesame configuration 120572 = 120573 = 05 is chosen while 120582 is testedas integers ranging from 1 to 20 The ranking results derivedfrom different parameter combinations have been collectedin Table 6 As can be seen four operators unanimouslyidentify the contractor 1199092 is the best one no matter how120582 changes Regarding priority relation between 1199091 and 1199093GIVDHFUUBLWA and GIVDHFUUBLOWA generate thesame result of1199093 ≻ 1199091 while results fromGIVDHFUUBLWGandGIVDHFUUBLOWGgot a change to 1199091 ≻ 1199093 at a certainvalue of parameter 120582

Actually GIVDHFUUBLWA and GIVDHFUUBLOWAare generalized version of weighted arithmetic aggregationoperators that producemore favorable results by emphasizing

20 Complexity

Table 6 Ranking orders obtained by GIVDHFUUBLWA GIVDHFUUBLOWA GIVDHFUUBLWG and GIVDHFUUBLOWG along with120582Operators 120582 (Integer) Ranking orderGIVDHFUUBLWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLOWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLWG [1 2] 1199092 ≻ 1199093 ≻ 1199091[3 20] 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG [1 8] 1199092 ≻ 1199093 ≻ 1199091[9 20] 1199092 ≻ 1199091 ≻ 1199093

Table 7 Ranking results obtained by GIVDHFUUBLHA and I-GIVDHFUUBLHA along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHA I-GIVDHFUUBLHA120572=0 120573=1 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=03 120573=07 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=05 120573=05 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=08 120573=02 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=09 120573=01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=1 120573=0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093

Table 8 Ranking results obtained by GIVDHFUUBLHG and I-GIVDHFUUBLHG along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHG I-GIVDHFUUBLHG

120572 = 0 120573 = 1 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 03 120573 = 07 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 05 120573 = 05 [1 8] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[9 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 08 120573 = 02 [1 17] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[18 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 09 120573 = 01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572 = 1 120573 = 0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093membership degrees to a uncertain linguistic term thus canbe considered as optimistic operators while GIVDHFU-UBLWG and GIVDHFUUBLOWG are generalized geomet-ric aggregation operators that producemore favorable resultsby emphasizing nonmembership degrees thus can be consid-ered as pessimistic operators [100 101]Therefore the rankingresults in Table 6 could be perceived as for optimism-baseddecision-making alternative 1199093 always stands out better than1199091 when 120582 gradually amplifies degree of optimism but forpessimism-based decision-making 1199091 turns out better than1199093 when 120582 gradually amplifies degree of pessimism 120582 canbe considered as the parameter for amplifying degree ofoptimism or pessimism [101]

Next to inspect the impacts of various combinationsof 120596119860119868 and 120596119860119863 on ranking results another four aggrega-tion operators of GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG also have beenapplied to the above illustrative case with varying 120572 120573 and 120582

All results have been collected in Tables 7 and 8 for compari-son As can be seen fromTable 7 ranking results generated byoperator GIVDHFUUBLHA keep the same as 1199092 ≻ 1199093 ≻ 1199091for all test combinations of parameters However as shownin Table 8 although GIVDHFUUBLHG is also capable ofidentifying 1199092 as the best contractor the priority relationsbetween 1199091 and 1199093 hold similar changing patterns in rankingresults in Table 6 that were output byGIVDHFUUBLWGandGIVDHFUUBLOWG Regarding decision situations whereobjective attributeweighting vector120596119860119863 is only considered (120572= 0) or its dominant importance (such as 120572 = 03) is accepted1199091 ≻ 1199093 is a clear observation when parameter 120582 is set toan integer in interval [8 20] Contrariwise under decisionscenarios where subjective attribute weighting vector 120596119860119868 isonly needed (120572 = 1) or its dominant importance (such as 120572 =09) is acknowledged 1199093 ≻ 1199091 holds all the way 120582 varies

In order to more clearly show the score trajectories ofalternative contractors along variations of parameter 120582 we

Complexity 21

HAInducedHA

InducedHA-2HA-2

HA-3

InducedHA-1InducedHA-3

= =

HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(a) 120572 = 0

HAInducedHA

= 03 = 07

InducedHA-2

HA-2HA-3

InducedHA-1 InducedHA-3HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(b) 120572 = 03

HAInducedHA

= 5 = 05

InducedHA-2HA-2

HA-3

InducedHA-1 InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(c) 120572 = 05

HAInducedHA

= 08 = 02

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572 = 08

HAInducedHA

= 9 = 01

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(e) 120572 = 09

HAInducedHA

246 2462 2464 2466 2468 247

= 1 = 0InducedHA-2HA-2

HA-3

HA- 3

HA-3

InducedHA-1

InducedHA-3

HA-1

HA-1

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

1085

1086

1087

(f) 120572 = 1

Figure 5 Score trajectories obtained by aggregation operators HA and InducedHA with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

further configure parameter 120582 to the value set of [0 01100]and conduct computational experiments to collect scorevalues and ranking results (as shown in Figures 5 and 6) byfour aggregation operators of GIVDHFUUBLHA GIVDH-FUUBLHG I-GIVDHFUUBLHAand I-GIVDHFUUBLHGFor more clarity in Figures 5 and 6 please note that weuse HA HG Induced-HA and Induced HG to respec-tively denote GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG In Figure 5it can be observed that when dominant importance of 120596119860119863

is emphasized (120572 = 0 120572 = 03 120572 = 05) GIVDHFUUBLHAis capable of differentiating the three alternative contractorsas same ranking order of 1199092 ≻ 1199093 ≻ 1199091 for all 120582 but withincreases in 120572 the margins between scores of 1199091 and 1199093 nar-rows and the rank order changes to 1199091 ≻ 1199093when 120572 gets 09and 1 Interestingly in Figure 6 although GIVDHFUUBLHGis able to significantly differentiate alternative contractors 1199091and 1199093 when dominant importance of 120596119860119863 is emphasized (120572= 0 120572 = 03 120572 = 05) rank order of 1199091 and 1199093 obviouslychanges once from 1199093 ≻ 1199091 to 1199091 ≻ 1199093 when 120582 is bigger

22 Complexity

HGInducedHG

= = 1

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(a) 120572= 0

HGInducedHG

= 3 = 07

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(b) 120572= 03

HGInducedHG

= 05 = 05

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(c) 120572= 05

HGInducedHG

= 8 = 02

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572= 08

HGInducedHG

292 293 294 295 296 297minus08

minus0795

minus079

= 09 = 01

HA-1

HA-3

InducedHG-2

HG-1

HG-1

HG-2

HG-3

HG-3 InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4

minus3

minus2

minus1

0

1

2

scor

e val

ues

(e) 120572= 09

HGInducedHG

50 60 70 80 90 100minus095minus09

minus085

= 1 = 0

HG-x3

HG-x1

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(f) 120572= 1

Figure 6 Score trajectories obtained by aggregation operators HG and InducedHG with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

than 8 When 120572 increases to 1 GIVDHFUUBLHGmaintainsthe rank order of 1199093 ≻ 1199091 but their score trajectorieswalk very close Comparatively I-GIVDHFUUBLHA andI-GIVDHFUUBLHG generate consistent ranking results of1199092 ≻ 1199091 ≻ 1199093 no matter 120582 varies in all the above experiments

In sum when parameters are configured with typical val-ues (ie 120572 = 05 and 120582=1) and no additional attitudinal infor-mation is taken into consideration the proposed aggregationoperators unanimously derive the same rank order 1199092 ≻1199093 ≻ 1199091 of for the three alternative contractors However

with the amplification effect of parameter 120582 analysis resultsfrom linear combination of 120596119860119868 and 120596119860119863 reveal alternative1199092 which consistently is the best choice but rank order of1199091 and 1199093 changes because their score trajectories get closeand crossed which probably indicates that only decision-making informationderived from the decisionmatrix is inad-equate to significantly differentiate alternatives 1199091 and 1199093While by integrating additional decision information aboutattitudinal characters of group experts I-GIVDHFUUBLHAand I-GIVDHFUUBLHG manage to generate the identical

Complexity 23

Table 9 Order-inducing vectors obtained by our TOPSIS-based method for problem in [38]

1198661 1198662 1198663 11986641198601 02522 01629 02282 027141198602 01 05243 01364 046891198603 04081 01756 05244 037791198604 04507 01409 01064 013221198605 00697 02203 03753 01161

Table 10 Order-inducing vectors obtained by our TOPSIS-based method for problem in [39]

1198661 1198662 1198663 11986641198601 04548 04969 04923 047321198602 05092 047 04768 046451198603 07192 04888 04953 049151198604 06722 06163 05706 049291198605 06104 04942 06463 04946

Table 11 Ranking results obtained by applying different order-inducing vectors to [38 39]

Problems Approaches Ranking results

Problem in [38] IHFLOWA [38] + order-inducing vectors in [38] 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198605 ≻ 1198604

IHFLOWA [38] + order-inducing vectors in Table 9 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198604 ≻ 1198605

Problem in [39] I-IVDHFLOWA [39] + order-inducing vectors in [39] 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198602 ≻ 1198601

I-IVDHFLOWA [39] + order-inducing vectors in Table 10 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602

permutation of 1199092 ≻ 1199091 ≻ 1199093 which verifies that ourproposed TOPSIS-based order-inducing approach is prac-tical and useful in helping work out decision results morereasonably and robustly

63 Application Studies of Proposed TOPSIS-Based Order-Inducing Method When confronting complex decision-making situations that need additional attitudinal charactersof decision-makers Yager and Filev [88] and Yager [102]introduced and encouraged to employ order-inducing vectorto represent those attitudinal characters Since then theidea of order-inducing vector has been extended to variousdecision-making environments [98 103] such as those basedon intuitionistic fuzzy information [104 105] linguistic infor-mation [106] hesitant fuzzy linguistic information [38] andinterval-valued dual hesitant fuzzy uncertain linguistic infor-mation [39] Generally in existing literature order-inducingvectors rely on experts to directly give [38 39 104ndash106]

In order to examine the adaptability of our TOPSIS-based order-inducing method to linguistic decision-makingenvironments we respectively apply the method to resolvethe same problems in references [38 39] Due to fact thatthere is no descriptions about how the order-inducing vectorswere derived in both references [38 39] we here applyextremums into our TOPSIS-based method to obtain order-inducing vectors that is (1199047 1 0) and (1199041 0 1) for[38] ([1199046 1199046] 1 0) and ([1199040 1199040] 0 1) for [39] Thenwe obtain our order-inducing vectors listed in Table 9 forthe problem adopted in [38] and our order-inducing vectorslisted in Table 10 for [39]

Now we apply the order-inducing vectors as shownin Tables 9 and 10 to resolve the problems in [38] and[39] respectively Regarding the problem in [38] the scorevalues of all alternatives are calculated as 119878(1198601)=02602119878(1198602)=02417 119878(1198603)=03577 119878(1198604)=0221 and 119878(1198605)=0185so the corresponding ranking result is 1198603 ≻ 1198601 ≻ 1198602 ≻1198604 ≻ 1198605 As for the problem in [39] the score values ofall alternatives are computed as 119878(1198601)=39954 119878(1198602)=3783119878(1198603)=70827 119878(1198604)=71792 and 119878(1198605)=5813 we thus obtainthe ranking result 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602 Formore clarity brief descriptions of comparative approachesand obtained ranking results also have been put together inTable 11

According to Table 11 the approaches that employ ourorder-inducing vectors derive the same ranking order ofthe better three alternatives for both comparative problemsfrom [38 39] As seen our proposed TOPSIS-based order-inducing method exhibits effective and more understand-able Furthermore when decision-makers are incapable ofdirectly put forward all desired order-inducing vectors due tocomplexity in ill-structured problems our method providesa feasible directions to obtain order-inducing vectors byfiguring out ideal solutions Comparatively our proposedTOPSIS-based order-inducing method behaves effective andmore operational in practical use

64 Comparison with Decision-Making Approach Based onDegraded form of IVDHFULS Note that our approachis intrinsically adaptive to decision situations based ondegraded forms of IVDHF UUBLS such as the dual hesitant

24 Complexity

fuzzy linguistic sets introduced by Yang and Ju [54] Tofurther inspect feasibility of our approach we here conductcomparative study on the same case adopted by Yang and Ju[54] We firstly convert their original decision matrix intothe form of IVDHF UUBLS crisp membership and non-membership degrees are equivalently changed to intervalswith same upper and lower limits (eg 02 turns out to be[02 02]) and linguistic variables are rewritten as uncertainlinguistic variables (eg 1199042 turns out to be [11990421199042])

Since the case only got three attributes among whichinterdependency could be not significant under generalassumption weighting vector 120596119860119868 is not calculated Andfor straight comparison order inducing vector is notincluded either Then the argument-dependent attributeweighting vector can be obtained from Eq(29) as 120596119860119863 =(03192 03132 03676)119879 By use of Algorithm Iwith operatorIVDHFUUBLWA (ie GIVDHFUUBLWA in case of 120582 = 1see Appendix B) we have the scores of alternatives 119904(1198601) =03412 119904(1198602) = 01354 119904(1198603) = 03932 119904(1198604) = 07352Therefore the ordering of alternatives is1198604 ≻ 1198603 ≻ 1198601 ≻ 1198602which is mainly in alignment with the ordering of1198604 ≻ 1198601 ≻1198603 ≻ 1198602 in Yang and Ju [54] It is worth noticing that 120596119860119863used in our approach is derived objectively from assessmentswhile attribute weights 120596lowast = (035 025 040)119879used by Yangand Ju [54] were assigned empirically Differences between120596119860119863 and 120596lowast lead to different ordering of1198601 and1198603 Compar-atively speaking our approach holds more adaptability andobjectivity

In general our proposed MADM approach in this paperexhibits an effective way to tackle complex green contractorselection problems in which subjective attributes weightingvector (such as120596119860119868) and objective attributes weighting vector(such as 120596119860119863) exist To comprehensively utilize these twofundamental facets of attributes weighting information ourcase study and computational experiments suggest that linearcombination method (ie 120596 = 120572120596119860119868 + 120573120596119860119863) is effectiveto reflect effects of 120596119860119868 and 120596119860119863 by configuring parameter 120572or 120573 When there is no specific expert opinions to specify orhard to obtain we suggest that parameter 120572 chooses the valueset of 0 02 05 08 1 Furthermore generalized aggrega-tion operators are suggested to inspect score trajectoriesof alternative contractors along increasing of the operatorsrsquoamplification parameter 120582 If some alternative contractorsrsquoscore values get very close or crossed so that they cannot bedifferentiated from each other significantly we suggest to askthe panel of experts to derive opinions of ideal solutions andutilize our TOPSIS-based order-inducing method to includethose expert opinions as attitudinal decision informationthereby obtaining more rational and robust decision results

7 Conclusions

Aiming at tackling complex decision-making problems ofgreen contractor selection that often hold simultaneouslythe characteristics of decision hesitancy attributes inter-dependency and group complex attitudinal characters wehave developed an effective IVDHF UUBLS-based approach

We utilize IVDHF UUBLS to elicit complicate decision-makersrsquo assessments more objectively and comprehensivelywe employ DEMATEL for addressing attribute interdepen-dency more precisely and we determine group attitudinalcharacters as order-inducing vectors based on ideal attitudesof experts rather than by empirical assignment The mainadvantages of our approach are (i) IVDHF UUBLS offers anadequate way to depict decision hesitancy (ii) the designeddistance measure for IVDHF UUBLS avoids informationdistortion of traditional ones which artificially add mis-matching membership or nonmembership degrees (iii) thedeveloped generalized operators include traditional ones asspecial cases so as to provide flexibility in MADM basedon IVDHF UUBLS (iv) the devised synthesized attributeweighting scheme provides an adaptive way to consider twonecessary aspects attributes interdependency-based weightsand argument-dependent weights (v) the TOPSIS-basedmethod can deduce reasonable order-inducing vectors ratherthan empirical ones and holds generality in other linguisticdecision environments Results of case studies and experi-ments have verified the validity of the proposed approach

When multiple decision organizations are involvedgroup decision-making based on IVDHF UUBLS is animportant research direction Besides inducing preferencesfor decision-making problems of high uncertainty interactivemechanisms should also be studied to extract preferences Inaddition empirical studies of the proposed approach can becarried out to various areas such as investment evaluationproject management and vehicle selection in fleet operations

Appendix

A Basic Notions for IVDHFS and UnbalancedLinguistic Term Set

Definition A1 (see [47]) Letting 119883 be a fixed set then anIVDHFS on 119883 is defined as119863 = ⟨119909 ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (A1)

where ℎ(119909) = ⋃[120583119871120583119880]isinℎ(119909)120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and119892(119909) = ⋃[]119871]119880]isin119892(119909)] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two setsof interval values in [0 1] denoting possible membershipand non-membership degrees of element 119909 isin 119883 to theset 119863 respectively with conditions 120583 ] isin [0 1] and0 le (120583119880)+ + (]119880)+ le 1 and for all 119909 isin 119883 (120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880

For convenience normally 119889 = ℎ 119892 is called an interval-valued dual hesitant fuzzy element (IVDHFE) and 119863 is theset of all IVDHFEs

Definition A2 (see [107]) Suppose 119878 = 119904119895 | 119895 = 0 1 119892 minus1 is a finite and totally ordered discrete linguistic term setwhere 119904119895 represent possible values for a linguistic variableand 119892 is an odd cardinality 119878 is uniformly and symmetricallydistributed if the following two conditions are satisfied (1)

Complexity 25

there exists a unique constant 120582 gt 0 such that 119873119878(119904119894) minus119873119878(119904119895) = 119897(119894 minus 119895) for all 119894 119895 = 0 1 119892 minus 1 (2) Let 119878119877 =119904 | 119904 isin 119878 119904 gt 119878119862 and 119878119871 = 119904 | 119904 isin 119878 119904 lt 119878119862 Let C(119878119877) andC(119878119871) be the cardinality of 119878119877 and 119878119871 then C(119878119877) = C(119878119871)If 119878 is uniformly and symmetrically distributed then 119878 iscalled a balanced linguistic term set Otherwise 119878 is calledan unbalanced linguistic term set

The following 2-tuple fuzzy linguistic representationmodel extends traditional linguistic term set to a continuouscase so as to facilitate computing with linguistic variables

Definition A3 (see [59]) Let 119878 = 1199040 1199041 119904119892minus1 be alinguistic term set and 120573 isin [0 119892] Then a 2-tuple fuzzylinguistic variable expresses the equivalent information to 120573is defined as

Δ [0 119892] 997888rarr 119878 times [minus05 05) (A2)

Δ (120573) = (119904119894 120572) with

119904119894 119894 = 119903119900119906119899119889 (120573)120572 = 120573 minus 119894 120572 isin [minus05 05)

(A3)

Δminus1 (119904119894 120572) = 119894 + 120572 = 120573 (A4)

where 119903119900119906119899119889() is the usual rounding operation and120572 is calledsymbolic translation

To obtain 2-tuple fuzzy linguistic representations ofunbalanced linguistic terms the concept of linguistic hierar-chies ie 119871119867 = ⋃119905 119897(119905 119899(119905)) is used 119897(119905 119899(119905)) is a linguistichierarchy with 119905 indicating the level of hierarchy and 119899(119905)denotes the granularity of the linguistic term set of 119905 Herreraet al [59] defined the following transformation functionsbetween labels from different levels in multigranular linguis-tic information contexts without loss of information

Definition A4 (see [59]) In linguistic hierarchies 119871119867 =⋃119905 119897(119905 119899(119905)) whose linguistic term sets are represented by119878119899(119905) = 119904119899(119905)0 119904119899(119905)119899(119905)minus1 the transformation function froma linguistic label in level 119905 to a label in consecutive level 1199051015840 isdefined as 1198791198651199051199051015840 119897(119905 119899(119905)) 997888rarr 119897(1199051015840 119899(1199051015840)) such that

1198791198651199051199051015840 (119904119899(119905)119894 120572119899(119905))= Δ 1199051015840 (Δminus1119905 (119904119899(119905)119894 120572119899(119905)) (119899 (1199051015840) minus 1)119899 (119905) minus 1 ) (A5)

By use of the above transformation function any 2-tuplelinguistic representation can be transformed into a term in119871119867 Detailed transformation procedures are listed as follows

(1) Representation in linguistic hierarchy To transformthe unbalanced terms of 119878 into the corresponding terms inthe 119871119867 transformation function 120595 is employed to associateeach unbalanced linguistic 2-tuple (119904119894 120572) with its linguistic 2-tuple in 119871119867(119878) ie

120595 119878 997888rarr 119871119867(119878) (A6)

so that 120595(119904119894 120572) = (119904119866(119894)119868(119894) 120582) for forall(119904119894 120572) isin 119878(2) Computational phase Firstly transform (119904119866(119894)119868(119894) 120582) intolinguistic 2-tuples denoted as (119904119899(1199051015840)

1198681015840(119894) 1205821015840) in 119878119899(1199051015840) where

(119904119899(1199051015840)1198681015840(119894)

1205821015840) = 119879119865 (119904119866(119894)119868(119894) 120582)= Δ(Δminus1 (119904119866(119894)119868(119894) 120582) sdot (119899 (1199051015840) minus 1)119866 (119894) minus 1 ) (A7)

Then a computational model is used over 119878119899(1199051015840) with aresult denoted as (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)(3) Retranslation process The result (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)is transformed into the unbalanced term in 119878 by using thetransformation function 120595minus1 ie

120595minus1 119871119867 (119878) 997888rarr 119878 (A8)

so that 120595minus1(119904119899(1199051015840)119903 120582119903) = (119904119903119890119904119906119897119905 120582119903119890119904119906119897119905) isin 119878B Details of Theorem 14

Proof Given a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector for119904119889119895 and 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 120582 gt 0 then(1) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLWA and GIVDHFUUBLWG operators respectively(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLOWA and GIVDHFUUBLOWG operators respectively

(2) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to theIVDHF UUBL weighted average (IVDHFUUBLWA)and IVDHF UUBL weighted geometric (IVDHFUUBLWG)operators respectively where

26 Complexity

119868119881119863119867119865119880119880119861119871119882119860120596 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

120596119895119904119889119895= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus

119899prod119895=1

(1 minus 120583119871119895 )120596119895 1 minus 119899prod119895=1

(1 minus 120583119880119895 )120596119895]] [[

119899prod119895=1

(]119871119895)120596119895 119899prod119895=1

(]119880119895 )120596119895]])

(B1)

119868119881119863119867119865119880119880119861119871119882119866120596 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889119895120596119895

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[

119899prod119895=1

(120583119871119895 )120596119895 119899prod119895=1

(120583119880119895 )120596119895]] [[1 minus

119899prod119895=1

(1 minus ]119871119895)120596119895 1 minus 119899prod119895=1

(1 minus ]119880119895 )120596119895]])

(B2)

(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHA andGIVDHFUUBLHG operators reduce to the IVDHF UUBL

orderedweighted average (IVDHFUUBLOWA) and IVDHFUUBL ordered weighted geometric (IVDHFUUBLOWG)operators respectively where

119868119881119863119867119865119880119880119861119871119874119882119860119908 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

119908119895119904119889120590(119895)= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) )))

119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) )))

]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus

119899prod119895=1

(1 minus 120583119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus 120583119880120590(119895))119908119895]] [[

119899prod119895=1

(]119871120590(119895))119908119895 119899prod119895=1

(]119880120590(119895))119908119895]])

(B3)

119868119881119863119867119865119880119880119861119871119874119882119866119908 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889120590(119895)119908119895

= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[

119899prod119895=1

(120583119871120590(119895))119908119895 119899prod119895=1

(120583119880120590(119895))119908119895]] [[1 minus

119899prod119895=1

(1 minus ]119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus ]119880120590(119895))119908119895]])

(B4)

Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

References

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[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

[11] X Qi C Liang and J Zhang ldquoGeneralized cross-entropy basedgroup decision making with unknown expert and attributeweights under interval-valued intuitionistic fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 79 pp 52ndash642015

[12] H Zimmermann Fuzzy Set Theorymdashand Its ApplicationsKluwer Academic Boston Mass USA 2001

[13] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 pp 199ndash249 1975

[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

[16] B T Nuong K-W Kim L Prathumratana et al ldquoSustainabledevelopment in the mining sector and its evaluation usingfuzzy AHP (Analytic Hierarchy Process) approachrdquo GeosystemEngineering vol 14 no 1 pp 43ndash50 2011

[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

[18] E K Zavadskas Z Turskis and J Antucheviciene ldquoSelectinga contractor by using a novel method for multiple attributeanalysis Weighted aggregated sum product assessment withgrey values (WASPAS-G)rdquo Studies in Informatics and Controlvol 24 no 2 pp 141ndash150 2015

[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

[21] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

[25] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009

[26] Z Hatush and M Skitmore ldquoContractor Selection UsingMulticriteria Utility Theory An Additive Modelrdquo Building andEnvironment vol 33 no 2-3 pp 105ndash115 1998

[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

[30] AM Anvuur andMM Kumaraswamy ldquoConceptualmodel ofpartnering and alliancingrdquo Journal of Construction Engineeringand Management vol 133 no 3 pp 225ndash234 2007

[31] A H I Lee H Kang C F Hsu and H Hung ldquoA green supplierselection model for high-tech industryrdquo Expert Systems withApplications vol 36 no 4 pp 7917ndash7927 2009

[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

[33] P Humphreys R McIvor and F Chan ldquoUsing case-basedreasoning to evaluate supplier environmental managementperformancerdquo Expert Systems with Applications vol 25 no 2pp 141ndash153 2003

[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

[45] Bin Zhu Zeshui Xu and Meimei Xia ldquoDual Hesitant FuzzySetsrdquo Journal of Applied Mathematics vol 2012 Article ID879629 13 pages 2012

[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

[51] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[52] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

[55] Z Tao X Liu H Chen and L Zhou ldquoUsing new versionof extended t-norms and s-norms for aggregating intervallinguistic labelsrdquo IEEETransactions on SystemsManCyberneticsSystems vol vol p 15 2016

[56] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[57] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[58] L Martınez M Espinilla J Liu L G Perez and P J SanchezldquoAn evaluation model with unbalanced linguistic informationapplied to olive oil sensory evaluationrdquo Journal of Multiple-Valued Logic and Soft Computing vol 15 no 2-3 pp 229ndash2512009

[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

[62] L Marin A Valls D Isern A Moreno and J M MerigoldquoInduced unbalanced linguistic ordered weighted average andits application in multiperson decision makingrdquo The ScientificWorld Journal vol 2014 2014

[63] A Baykasoglu V Kaplanoglu Z D U Durmusoglu and CSahin ldquoIntegrating fuzzy DEMATEL and fuzzy hierarchicalTOPSIS methods for truck selectionrdquo Expert Systems withApplications vol 40 no 3 pp 899ndash907 2013

[64] S K Patil and R Kant ldquoA hybrid approach based on fuzzyDEMATEL and FMCDM to predict success of knowledge man-agement adoption in supply chainrdquoApplied Soft Computing vol18 pp 126ndash135 2014

[65] S Liyin and Y Hong ldquoImproving environmental performanceby means of empowerment of contractorsrdquo Management ofEnvironmental Quality An International Journal vol 17 no 3pp 242ndash257 2006

[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

[69] Y Kazancoglu I Kazancoglu and M Sagnak ldquoA new holisticconceptual framework for green supply chain management

performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

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Complexity 3

IVDHF UUBLS In Section 5 We detail the DEMATEL-based model for weighting evaluative attributes and theTOPSIS-based method for deriving order-inducing variablesbased on which we then construct an effective approach forcomplex MADM In Section 6 illustrative case study andexperiments are further conducted to verify the proposedMADM approach Finally conclusions and future researchare presented in Section 7

2 Description of Research Problem

The significant impacts of construction activities on theenvironment have triggered serious alarms and governmentsworldwide have introduced various policies regulations andindustrial evaluation systems for controlling them [65] forinstance the cleaner production promotion law and thepollution prevention law in China and the green buildingrating systems in the US UK and China [66] To meetthe requirements of environmental concerns constructionprojects have to select contractors in operations by simultane-ously considering potential contractorsrsquo green characteristics[6] Actually the practical needs of selecting green con-tractors are especially indispensable for mega infrastructureconstruction projects as they generally are one-off and nearlyever-lasting in the environment

21 Evaluative Attributes for Green Contractor SelectionTypically when construction organizations seek to developappropriate approaches to contractor selection the organiza-tionsrsquo specific requirements are firstly introduced Thereforedifferent sets of evaluative attributes for contractor selectionwith different scenarios are needed During last two decadesmany efforts have been paid to identify selection attributesand construct comprehensive contractor selection methodswith different applications [8 10 18 19 23 24 26 28 67 68]

One basic observation from existing literature for consid-ering evaluative attributes is that being analogous to selectingsuppliers in supply chain management [69] decision-makersare supposed to not only consider competitive capabilitythat distinguishes contractors from each other by businessoperations but also examine contractorrsquos cooperation capac-ity with other partner contractors that indeed influencesthe project success [8 70] More importantly since theconstruction industry has great impact on environment andthe governments worldwide have introduced various policiesand regulations for controlling them [6] green practices thushas become a crucial facet for evaluating green contractors[68 71] As a result three aspects of business competivenesscooperation and green practices thus become indispen-sible to derive evaluative attributes for green contractorselection

Many earlier studies put their emphasis on competitiveattributes to evaluate comprehensive performance of contrac-tors Hatush and Skitmore [26] suggested five competitiveattributes to assess contractors including financial sound-ness technical ability management capability health andsafety and reputation Fong and Choi [24] studied contractorselection problem for Hong Kong scenario and derived a set

of eight evaluative attributes according to a questionnairesurvey among which seven are competitive attributes includ-ing price financial capability past performance past expe-rience resource current workload and safety performanceIn their developed computer-aided decision support systemShen et al [28] employed a parameter system to evaluatecontractorrsquos competitiveness in the Chinese constructionindustry in which six competitive attributes were includedas social influence technical ability financing ability andaccounting status marketing ability management skills andorganizational structure and operations Darvish et al [25]developed a graph theory-based decision-making method tocope with contractor prequalification problem under Iranianscenarios they introduced the Iranian domestic prequalifi-cation criteria system that included nine main indicatorswork experience technology amp equipments managementexperience and knowledge of the operation team finan-cial stability quality being familiar with the area or beingdomestic reputation and creativity and innovation Nieto-Morote and Ruz-Vila [10] also investigated a fuzzy decision-making model based on TOPSIS to accommodate construc-tion contractor prequalification problem they summarizedthe most common factors for comprehensively consideringcontractors during the prequalification that fall into followingmain aspects technical capacity experience managementcapability financial stability past performance past relation-ship reputation and occupational health amp safety Focusingon contractor selection problems in Lithuania constructionindustry Zavadskas et al [18] proposed an effective multipleattributes decision-making approach called WASPAS-G inwhich main attributes are identified to cover bid amountcapability amp skill occupational health and security technicalcapacity managerial capacity past performance and pastexperience As seen from the above representative literaturequotation technical strength and resource strength are thethreemain factors which arewidely accepted and adopted wethus include these factors as main evaluative attributes in thispaper Another finding from the above reviewed literature isthat nearly all of them emphasized the inclusion of indicatorsto examine credibility of contractors in their managingexternal and internal challenges [72 73] such as financialcapability [24] safety performance [24] and reputation [1025 26] therefore we introduce the lsquoCredibilityrsquo as anothermain attribute in this paper

Indeed cooperation attributes must be taken into con-sideration because cooperation among selected contractorinfluences the project success [74] Actually some of ear-lier pioneering studies also noticed and took considera-tion of cooperation attribute in their parameter systemssuch as lsquoclientcontractor relationshiprsquo in references [10 24]Recently increasing attentions have been paid to investigateappropriate evaluative cooperation attributes for contractorselection Representatively based on the conceptual modelof partnering and alliancing in construction project man-agement Liang et al [8] elaborated a set of cooperationattributes for selecting joint venture contractors in large-scale infrastructure projects including compatible culturecontract communication collaboration cooperation abilityand cooperation satisfaction To provide a deeper insights

4 Complexity

on factors that affect contractorsrsquo cooperation under inter-national construction joint ventures in construction projectsHwang et al [27] conducted a survey and reported lsquosharingof project risksrsquo as the top attractive factor and lsquodifferences inculture and working stylersquo as the top negative factor amongothers Furthermore aiming at finding risks that most affectcontractorsrsquo cooperation within project networks Hwangand Han [29] conducted a survey from the viewpoint ofcontractors and sub-contractors in Singapore constructionenvironment they identified ten top critical network riskssuch as lsquodifferent cultural normsrsquo lsquoinaccurate informationdeliveryrsquo lsquooccurrence of disputersquo and lsquolack of risk manage-ment knowledgersquo Obviously Hwang et al [27] and Hwangand Han [29] contributed a feasible way to deduce morereasonable evaluative attributes on contractorrsquos cooperationcapability however to our best knowledge there is stilla lack of thorough investigation on a consensus set ofevaluative attributes on contractorrsquos cooperation capabilityfor referencing Therefore in the light of Liang et al [8]Hwang et al [27] and Hwang and Han [29] we here takelsquoCooperation management capabilityrsquo as one of the mainevaluative attributes to assess contractors

To reduce the significant environmental footprint [6]internal and external factors (such as government regulationsand managerial concerns) are driving the entire construc-tion industry to enforce green practices all over the world[6] Green practices can be considered as an outcome ofstrategic processes through cooperation within the nexus ofcontractors [75] Therefore it is intrinsically crucial to takeinto account evaluative attributes on green practices in con-tractor selection especially for those nearly ever-lasting megainfrastructure construction projects Since environmentalregulations of different scales (ie domestic governmentaland international [76]) have increasingly become compulsoryin various industrial markets contractors should have thecapability of manage and keep track of the compliance [35]And the widely accepted way [77] to attain the compliancecapability by building environment management systems[32 34 78] whose most-cited components [77] compriseof ISO-14001 certification [32 34 36 37] eco-labeling [3132 37] environment policies [33] environment planning[33] and environmental management information system(continuous monitoring and regulatory compliance) [31 32]Due to the reason that environmental impacts occur at everystage of the construction cycle Rwelamila et al [79] stronglysuggested to contractors should implement green design ampprocurement to improve their green practices Construc-tion life-cycle analysis is critical in both green design forenvironment [33] and pollution reduction through greenprocurement [31 32 34] Measurements of green design forenvironment include tracking all material and reverse flowof a project from the retrieval of raw materials out of theenvironment to the disposal of the product back into theenvironment generally including recycle [31] reuse [32 33]remanufacture [32 33] disassembly and disposal [33] In thegreen procurement aspect Tan et al [68] suggested to reduceenvironmental footprints throughout the whole constructionsupply chain by addressing issues such as waste reduction [3234] environment material substitution [31 33] hazardous

material minimization [35] and clean technology availability[33] Furthermore from the stance of strategic manage-ment Fergusson and Langford [80] pointed out environ-mental performance of green practices positively contributesto comprehensive competitive advantages of contractorsEnvironmental performance has thus been recognized asone of the indispensible evaluative attributes to contractorselection [34 78] Sharma and Vredenburg [81] referred toenvironmental performance as the environmental effects thatcorporationrsquos activities have on the natural milieu To makeenvironmental performance more measurable many effortshave been paid to establish effective assessing approachessuch as the approach to examination on waste flows andcontrol on construction sites [3 82] As for indicators ofenvironmental performance according to the thorough liter-ature review by Govindan et al [77] the commonly adoptedones include [32ndash34 36] solid waste chemical waste airemission waste water disposal and energy In sum we hereinclude lsquoDesign amp procurementrsquo lsquoCompliance with greenlegislationrsquo and lsquoEnvironmental performancersquo as three othermain evaluative attributes to comprehensively assess greenpractices of alternative contractors

For more clarity based on the above literature review welisted all eight main evaluative attributes and present theirdefinitionsprinciples And the optional subdimensions ofthese main attributes have been also listed in Table 1 forvarious comprehensive considerations according to specificnature in practical problems

(A1) Quotation Project value of reviewed The lowesttender price tends to attract a clientrsquos interest as superior toother criteria

(A2) Credibility Comprehensive evaluation on trustwor-thiness of contractors in developmental dynamics and risksfrom both internal and external environments focusing onfinancial soundness health and safety past performancework experience etc

(A3) Technical strength Comprehensive evaluation oncontractorsrsquo capability of technology and innovation thattackling forthcoming complicate tasks generally based onquality rank technical ability experience and knowledge ofoperation team creativity amp innovation etc

(A4) Resource strength Comprehensive evaluation oncontractorsrsquo competitive strength on productive resourcesthat indispensable for construction needs generally based ontechnical human resource current workload constructionmachinery amp equipment fixed assets amp liquidity etc

(A5) Cooperation management capacity Comprehensiveevaluation on contractorsrsquo cooperation practices with partici-pators in projects focusing on clientcontractor relationshiporganizational structure and operations compatible culturecommunication and information delivery knowledge shar-ing of risk management among others

(A6) Design amp procurement Comprehensive evaluationon contractorsrsquo practices that improve projectrsquos whole lifevalue by using green design environmental friendly mate-rials and green production processes which promote bestpractices of green construction procurement throughout thesupply chain

Complexity 5

Table1Mainevaluativ

eattributesandtheiro

ptionalsub

dimensio

nsforg

reen

contractor

selection

No

Attributes

Sub-dimensio

nsRe

ferences

A1

Quo

tatio

n(i)

Tend

erprice(

advancep

aymentcapitalbidrou

tine

maintenance

andmajor

repairs

)[81824]

Liangetal[8]Fon

gandCh

oi[24]Z

avadskasetal[18]

A2

Credibility

(i)pastperfo

rmance

[101824]

(ii)w

orkexperie

nce[25]

(iii)repu

tatio

nampcreditratin

g[1025ndash27]

(iv)h

ealth

ampsafety

[10182426]

(v)fi

nancialstability[1024ndash2628]

Nieto-M

orotea

ndRu

z-Vila[10]Z

avadskasetal[18]H

atushandSkitm

ore

[26]Fon

gandCh

oi[24]Shenetal[28]Darvishetal[2527]

A3

Technical

strength

(i)technicalability[810182628]

(ii)e

xperienceamp

know

ledgeo

foperatio

nteam

[1025]

(iii)qu

ality

rank

[25]

(iv)c

reativity

ampinno

vatio

n[25]

Liangetal[8]N

ieto-M

orotea

ndRu

z-Vila[10]Z

avadskasetal[18]H

atush

andSkitm

ore[

26]Sh

enetal[28]D

arvishetal[25]

A4

Resource

strength

(i)technicalhum

anresource

[2427]

(ii)c

urrent

workload

[24]

(iii)constructio

nmachinery

ampequipm

ent[2527]

(iv)fi

xedassetsampliq

uidity

[27]

Fong

andCh

oi[24]D

arvishetal[25]H

wangetal[27]

A5

Coo

peratio

nmanagem

ent

capability

(i)clientcon

tractorrelationship[102429]

(ii)o

rganizationalstructureampop

erations

[828]

(iii)compatib

lecultu

re[827]

(iv)c

ommun

ication[829]

(v)k

nowledges

harin

gof

riskmanagem

ent[2729]

(vi)inform

ationdelivery[29]

(vii)

contractmanagem

ent[8]

Liangetal[8]N

ieto-M

orotea

ndRu

z-Vila[10]H

atushandSkitm

ore[

26]

Fong

andCh

oi[24]Shenetal[28]Darvishetal[25]H

wangetal[27]

HwangandHan

[29]A

nvuu

rAaron

andKu

maraswam

yMoh

an[30]

A6

Designamp

Procurem

ent

(i)recycle[31]

(ii)reuse

[3233]

(iii)remanufacture[3233]

(iv)d

isassem

blyanddisposal[33]

(v)w

asteredu

ction[3234]

(vi)environm

entm

aterialsub

stitutio

n[3133]

(vii)

hazardou

smaterialm

inim

ization[35]

(viii)c

lean

techno

logy

availability[33]

Lin[35]H

umph

reysetal[34]C

hiou

etal[32]Leeetal[31]H

umph

reys

etal[33]

A7

Com

pliance

with

green

legisla

tion

(i)ISO-140

01certificatio

n[32343637]

(ii)e

co-la

belin

g[313237]

(iii)environm

entp

olicies[33]

(iv)e

nviro

nmentp

lann

ing[33]

(v)e

nviro

nmentalm

anagem

entinformationsyste

m[3132]

Hum

phreysetal[34]C

hiou

etal[32]YangandWu[36]Liu

andLiu[37]

Leeetal[31]Hum

phreysetal[33]

A8

Environm

ental

perfo

rmance

(i)solid

waste[333436]

(ii)c

hemicalwaste[333436]

(iii)aire

mission[333436]

(iv)w

astewater

disposal[32ndash3436]

(v)e

nergy[

3334]

Hum

phreysetal[34]C

hiou

etal[32]YangandWu[36]Hum

phreysetal

[33]

6 Complexity

(A7) Compliance with green legislation Comprehensiveevaluation on the extent to which contractorsrsquo practicessatisfy different governmental green or sustainability legis-lations according to the aspects of ISO-14001 certificationeco-labeling environment policies environment planningenvironmental management information system etc

(A8) Environmental performance Comprehensive evalu-ation on environmental effects that the corporationrsquos activitieshave on the natural milieu Environmental performance cancommonly bemeasured throughoperative performance indi-cators (ie energyresource utilization emission reductionand waste disposal)

22 Problem Definition The contractor selection as well asmany other multicriteria decisions impacting the overallproject should bemade during the front end planning stage ofa project the point at which a group of designated decision-makers have the power to accept or reject a contractorfor a specific project or its work packages [5 9] WhenusingMADMmechanism to copewith complicate contractorselection problems [5] normally few nominated contractorswill be ready for the decision-makers to vote on Due tocomplexity of the problems and limitedness of knowledgedecision-makers usually feel confident in expressing theiropinions by use of interval numbers [83] or uncertain linguis-tic terms [52] Although widely-accepted majority rule willquickly help the group of decision-makers arrive at a decisionon an uncertain linguistic term (eg [s4 s5]) to whichobviously there exists decision hesitancy because differentopinions existTherefore in this paper we define the interval-valued dual hesitant fuzzy uncertain unbalanced linguisticset (IVDHF UUBLS) to help the decision-making panel elicittheir assessments more objectively and completely

From another point of view various attitudinal charac-ters (degree of orness) commonly exist because individualexpert holds specific backgrounds and decision processthat involves the attitudinal character of group decision-makers must coordinate those various attitudinal charactersinto one complex attitudinal character [61] Therefore weadopt the concept of order-inducing vector [62] to reflectgroup complex attitudinal character and develop a TOPSIS-based method to rationally determine the order-inducingvector Besides as Tan and Chen [84] pointed out forreal-world sophisticated MADM problems the indepen-dency axiom [85] cannot generally satisfied For exampleupgrading in green performance will raise the quotationand intrinsically result in requirements for high-standardcollaboration between contractors and general constructorIn viewing of this common phenomenon we take attributesrsquointerdependency as a third indispensible characteristic intackling complexity in green contractor selection in megainfrastructure projects In sum we take three characteristicsof complexity to model the practical problems of complicategreen contractor selection ie (i) compound structure ofhesitant fuzzy linguistic assessments (ii) group attitudinalcharacters and (iii) attributesrsquo interdependency To producegreater clarity Figure 1 demonstrates the conceptual MADMmodel for green contractor selection

Now we can give the symbolized description of thetargeted complex green contractor problems Given a megainfrastructure project there are a set of alternative greencontractors ie 119883 = 1199091 1199092 119909119899 for its subprojectsLet 119860 = 1198601 1198602 119860119898 be the evaluative attributesaccording to which decision-makers consider each greencontractor Due to high complexity in the sophisticatedproblem scenarios there exists interdependency relationsamong the evaluative attribute A panel of decision-makers119864 = 1198641 1198642 119864119905 have been organized to give theirassessments to each alternative contractor 119909119894 (119894 = 1 119899)under every attribute 119860119895 (119895 = 1 119898) In order to reflect thecomplicate group assessments of all alternative contactorsthe hybrid expression tool of IVDHF UUBLS that will bedetailed in Section 3 is adopted to depict the assessmentsmore effectively and comprehensively As a result a specificdecision matrix 119877 = (119903119894119895)119898times119899 whose elements are in theform of IVDHF UUBLS is obtained According to Merigoand Casanovas [61] suppose that an order-inducing vec-tor 120576 for denoting group attitudinal characters has beenreasonably obtained Then effective MADM approachesmust be developed to determine the most appropriate greencontractor(s)

3 Interval-Valued Dual Hesitant FuzzyUncertain Unbalanced Linguistic Set

As demonstrated in Figure 1 after the panel of decision-makers votes on an alternative contractor under certainattribute by use of specific uncertain unbalanced linguisticterm set the uncertain linguistic term [s4 s5] stands outbecause of the majority rule while different opinions shouldalso be included and considered inMADMprocesses To thatend based on interval-valued dual hesitant fuzzy set (IVD-HFS) [47] and the unbalanced linguistic term set (ULTS)[59] we here first introduce an interval-valued dual hesitantfuzzy uncertain unbalanced linguistic set (IVDHF UUBLS)which incorporate different opinions of decision-makersas membership degrees or nonmembership degrees to themajority-voted [s4 s5] Then we develop operational rules aswell as distance measure for the IVDHF UUBLS Regardingdefinitions of the IVDHFS and ULTS one can refer toAppendix A

31 Definition of IVDHF UUBLS

Definition 1 Let119883 be a fixed set and 119878 be a finite and continu-ous unbalanced linguistic label set Then an IVDHF UUBLS119878119863 on119883 is defined as

119878119863 = ⟨119909 119904120599(119909) ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (1)

where 119904120579(119909) = [119904120572 119904120573] represents judgment to object 119909119904120572 and 119904120573 are two unbalanced linguistic variables frompredefined unbalanced linguistic label set 119878 which repre-sents decision-makersrsquo judgments to an evaluated object119883 ℎ(119909) = ⋃120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and 119892(119909) =⋃] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two sets of closed intervals

Complexity 7

e panelof decision

makers Alternativegreen

contractors

Evaluativeattributes

Comprehensiveevaluation

Attributesrsquo interdependency

+

alternativecontractor

vote

Voting results on

MajorityRule

radic

Specificdecision

matrix

AppropriateMADM

approach

Decisionresults

Section 3 Defines an effectivehybrid expression tool

Group attitudinal characters (ie complex degrees of orness)

s

⟨Contractori [s4 s5] ℎ (x) g(x)⟩

i-th

1 2 3 4 5 6 7

j-th attribute

Figure 1 Conceptual MADMmodel for tackling complexity in green contractor selection

in [0 1] ℎ(119909) denotes possible membership degrees that 119909belongs to 119904120579(119909) and119892(119909) represents possible nonmembershipdegrees of 119909 to 119904120579(119909) ℎ(119909) and 119892(119909) hold conditions 120583 ] isin[0 1] and 0 le (120583119880)+ + (]119880)+ le 1 where(120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880 for all 119909 isin 119883When119883 = 1199091 1199092 119909119899has only one element 119878119863 reduces to (s120599 ℎ 119892) which is calledan interval-valued dual hesitant fuzzy uncertain unbalancedlinguistic (IVDHF UUBL) number (IVDHF UUBLN)

32 Operational Rules for IVDHF UUBLS On the strength ofoperational rules for uncertain linguistic set [52] unbalancedlinguistic set [59] and interval-valued dual hesitant fuzzy set[47] we get the following operations for IVDHF UUBLS

Definition 2 Let 119904119889 = (119904120599 ℎ 119892) = ([119904120572 119904120573] ℎ 119892) 1199041198891 =(1199041205991 ℎ1 1198921) = ([1199041205721 1199041205731] ℎ1 1198921) and 1199041198892 = (1199041205992 ℎ2 1198922) =([1199041205722 1199041205732] ℎ2 1198922) be any three IVDHF UUBLNs 120582 isin [0 1]operations on these IVDHF UUBLNs are defined as

(1) 120582119904119889 = ⋃(119904120599ℎ119892)isin119904119889

([119904120582Δminus11199050 (119879119865

11990511989611199050(120595(119904120572)))

119904120582Δminus11199050 (119879119865

11990511989621199050(120595(119904120573)))

] ⋃

[120583119871120583119880]isinℎ[]119871]119880]isin119892

[1 minus (1 minus 120583119871)120582 1 minus (1 minus 120583119880)120582] [(]119871)120582 (]119880)120582]) (2) 119904119889120582 = ⋃

(119904120599 ℎ119892)isin119904119889

([119904(Δminus11199050 (119879119865

11990511989611199050(120595(119904120572))))

120582 119904(Δminus11199050 (119879119865

11990511989621199050(120595(119904120573))))

120582])

⋃[120583119871120583119880]isinℎ[]119871]119880]isin119892

[(120583119871)120582 (120583119880)120582] [1 minus (1 minus ]119871)120582 1 minus (1 minus ]119880)120582] )(3) 1199041198891 oplus 1199041198892 = ⋃

(1199041205991 ℎ11198921)isin1199041198891(1199041205992 ℎ21198922)isin1199041198892

([119904Δminus11199050 (119879119865

11990511989411199050(120595(1199041205721 )))+Δ

minus11199050(11987911986511990511989511199050(120595(1199041205722 )))

119904Δminus11199050 (119879119865

11990511989421199050(120595(1199041205731 )))+Δ

minus11199050(11987911986511990511989521199050(120595(1199041205732 )))

] ⋃

[1205831198711 1205831198801 ]isinℎ1[120583

1198712 1205831198802 ]isinℎ2[]

1198711 ]1198801 ]isin1198921[]

1198712 ]1198802 ]isin1198922

[1205831198711 + 1205831198712 minus 12058311987111205831198712 1205831198801 + 1205831198802 minus 1205831198801 1205831198802 ] []1198711]1198712 ]1198801 ]1198802 ]) (4) 1199041198891 otimes 1199041198892 = ⋃

(1199041205991 ℎ11198921)isin1199041198891(1199041205992 ℎ21198922)isin1199041198892

([119904Δminus11199050 (119879119865

11990511989411199050(120595(1199041205721 )))timesΔ

minus11199050(11987911986511990511989511199050(120595(1199041205722 )))

119904Δminus11199050 (119879119865

11990511989421199050(120595(1199041205731 )))timesΔ

minus11199050(11987911986511990511989521199050(120595(1199041205732 )))

] ⋃

[1205831198711 1205831198801 ]isinℎ1[120583

1198712 1205831198802 ]isinℎ2[]

1198711 ]1198801 ]isin1198921[]

1198712 ]1198802 ]isin1198922

[12058311987111205831198712 1205831198801 1205831198802 ] []1198711 + ]1198712 minus ]1198711]1198712 ]1198801 + ]1198802 minus ]1198801 ]

1198802 ])

(2)

8 Complexity

In above details about the transformation functionsof 119879119865 and Δ linguistic hierarchies (119871119867) as well as thetransformation procedures for unbalanced linguistic termsets are shown in Appendix A

eorem 3 Letting 119904119889 = (119904120599 ℎ 119892) 1199041198891 = (1199041205991 ℎ1 1198921) and1199041198892 = (1199041205992 ℎ2 1198922) be any three IVDHF UUBLNs 120582 1205821 1205822 isin[0 1]then following properties are true

(1) 1199041198891 oplus 1199041198892 = 1199041198892 oplus 1199041198891 (2) 1199041198891 otimes 1199041198892 = 1199041198892 otimes 1199041198891 (3)120582(1199041198891 oplus 1199041198892) = 1205821199041198891 oplus 1205821199041198892(4) 1199041198891120582 otimes 1199041198892120582 = (1199041198891 otimes 1199041198892)120582 (5) 1205821119904119889 oplus 1205822119904119889 = (1205821 +1205822)119904119889 (6) 1199041198891205821 otimes 1199041198891205822 = 1199041198891205821+1205822

Proof Omitted

In Definition 2 and Theorem 3 1199051198961 1199051198962 1199051198941 1199051198942 1199051198951 and1199051198952 are corresponding levels of 119904120572 119904120573 1199041205721 1199041205731 1199041205722 and 1199041205722in 119871119867 respectively 1199050 is the maximum level of 119904120572 119904120573 1199041205721 1199041205731 1199041205722 and 1199041205722 in 119871119867 Furthermore to compare any twoIVDHF UUBLNs we also have following definitions

Definition 4 Let 119904119889 = (119904120599 ℎ 119892) be an IVDHF UUBLN119904120599 = [119904120572 119904120573] then score function 119878(119904119889) and accuracy function119875(119904119889) can be represented by

119878 (119904119889) = Δminus11199050 (11987911986511990511989611199050 (120595 (119904120572))) + Δminus11199050 (11987911986511990511989621199050 (120595 (119904120573)))2 times 12 ( 1119897 (ℎ) sum

[120583119871120583119880]isinℎ

120583119871 minus 1119897 (119892) sum[]119871]119880]isin119892

]119871

+ 1119897 (ℎ) sum[120583119871120583119880]isinℎ

120583119880 minus 1119897 (119892) sum[]119871]119880]isin119892

]119880)(3)

119875 (119904119889)= Δminus11199050 (11987911986511990511989611199050 (120595 (119904120572))) + Δminus11199050 (11987911986511990511989621199050 (120595 (119904120573)))2

times 12 ( 1119897 (ℎ) sum[120583119871120583119880]isinℎ

120583119871 + 1119897 (119892) sum[]119871]119880]isin119892

]119871

+ 1119897 (ℎ) sum[120583119871120583119880]isinℎ

120583119880 + 1119897 (119892) sum[]119871]119880]isin119892

]119880)

(4)

where 119897(ℎ) and 119897(119892) are numbers of values in ℎ and 119892respectively 1199051198961 and 1199051198962 are the corresponding levels of 119904120572 and119904120573 in 119871119867 and 1199050 is the maximum level of 119905119896 in 119871119867

Definition 5 Given any IVDHF UUBLNs 1199041198891 = (1199041205991 ℎ1 1198921)1199041198892 = (1199041205992 ℎ2 1198922) then(1) If 119878(1199041198891) lt 119878(1199041198892) then 1199041198891 lt 1199041198892(2) If 119878(1199041198891) = 119878(1199041198892) then

(a) If 119875(1199041198891) = 119875(1199041198892) then 1199041198891 = 1199041198892(b) If 119875(1199041198891) lt 119875(1199041198892) then 1199041198891 lt 1199041198892

33 Distance Measure for IVDHF UUBLS When 119897(ℎ) or119897(119892) of two IVDHF UUBLNs are unequal the complement-ing method [86] is normally adopted to design distancemeasures Note that artificially adding values to shorterones in the complementing method will cause informationdistortion To avoid this limitation we provide the followingdistance measure

Definition 6 Let two IVDHF UUBLNs 1199041198891 = (1199041205991 ℎ1 1198921)and 1199041198892 = (1199041205992 ℎ2 1198922) where 1199041205991 = [1199041205721 1199041205731] 1199041205992 = [1199041205722 1199041205732]119897ℎ1 119897ℎ2 1198971198921 and 1198971198922 are lengths of ℎ1 ℎ2 1198921 and 1198922 respectivelydenoting number of elements in ℎ1 ℎ2 1198921 and 1198922 Suppose1198681 = (1(119899(1199051198941) minus 1))Δminus11199050 (11987911986511990511989411199050 (120595(1199041205721))) 1198682 = (1(119899(1199051198942) minus1))Δminus11199050 (11987911986511990511989421199050 (120595(1199041205731))) 1198683 = (1(119899(1199051198951)minus1))Δminus11199050 (11987911986511990511989511199050 (120595(1199041205722)))1198684 = (1(119899(1199051198952) minus 1))Δminus11199050 (11987911986511990511989521199050 (120595(1199041205732))) where 1199051198941 1199051198942 1199051198951 and1199051198952 are the corresponding levels of unbalanced linguistic terms1199041205721 1199041205731 1199041205722 and 1199041205732 in the linguistic hierarchy 119871119867 and 1199050 is themaximum level of 1199041205721 1199041205731 1199041205722 and 1199041205732 in 119871119867 Then by use ofnormalized Euclidean distance we define a distance measure119889 for IVDHF UUBLNs as follows

Situation 7 When 119897ℎ1 = 119897ℎ2 = 1198971 and 1198971198921 = 1198971198922 = 1198972 then119889 (1199041198891 1199041198892) = (12 ( 11198971sdot 1198971sum119896=1

(10038161003816100381610038161003816100381610038161198681120583119871119895ℎ1 minus 1198683120583119871119896ℎ2 10038161003816100381610038161003816100381610038162 + 10038161003816100381610038161003816100381610038161198682120583119880119895ℎ1 minus 1198684120583119880119896ℎ2 10038161003816100381610038161003816100381610038162) + 11198972sdot 1198972sum119896=1

(1003816100381610038161003816100381610038161198681]1198711198951198921 minus 1198683]1198711198961198922 1003816100381610038161003816100381610038162 + 1003816100381610038161003816100381610038161198682]1198801198951198921 minus 1198684]1198801198961198922 1003816100381610038161003816100381610038162)))12

(5)

Situation 8 When 119897ℎ1 = 119897ℎ2 or 1198971198921 = 1198971198922 then119889 (1199041198891 1199041198892) = (12 ( 1119897ℎ1 119897ℎ2sdot 119897ℎ1sum119895=1

119897ℎ2sum119896=1

(10038161003816100381610038161003816100381610038161198681120583119871119895ℎ1 minus 1198683120583119871119896ℎ2 10038161003816100381610038161003816100381610038162 + 10038161003816100381610038161003816100381610038161198682120583119880119895ℎ1 minus 1198684120583119880119896ℎ2 10038161003816100381610038161003816100381610038162)+ 11198971198921 1198971198922sdot 1198971198921sum119895=1

1198971198922sum119896=1

(1003816100381610038161003816100381610038161198681]1198711198951198921 minus 1198683]1198711198961198922 1003816100381610038161003816100381610038162 + 1003816100381610038161003816100381610038161198682]1198801198951198921 minus 1198684]1198801198961198922 1003816100381610038161003816100381610038162))12

(6)

eorem 9 The distance measure 119889 defined in Definition 6satisfies following properties

(1) 0 le 119889(1199041198891 1199041198892) le 1

Complexity 9

(2) 119889(1199041198891 1199041198892) = 0 if and only if 1199041198891 and 1199041198892 are perfectlyconsistent

(3) 119889(1199041198891 1199041198892) = 119889(1199041198892 1199041198891)4 Generalized Aggregation Operators forIVDHF_UUBLS

41 Definitions of Operators Based on the generalized oper-ators firstly introduced by Yager [87] we here develop some

fundamental generalized aggregation operators for the newlydefined IVDHF UUBLS

Definition 10 Given a collection of IVDHF UUBLNs 119904119889119895(119895 =1 119899) their weighting vector 120596 = (1205961 1205962 120596119899)119879120596119895 isin[0 1]sum119899119895=1 120596119895 = 1 120582 be a parameter 120582 isin (0 +infin)(1) Generalized IVDHFUUBLWeighted Average(GIVDHFUUBLWA) Operator

119866119868119881119863119867119865119880119880119861119871119882119860120596120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(120596119895119904119889119895120582))1120582

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120582)1120582 119904(sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120582)1120582]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[(1 minus

119899prod119895=1

(1 minus (120583119871119895 )120582)120596119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880119895 )120582)120596119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871119895)120582)120596119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880119895 )120582)120596119895)1120582]])

(7)

(2) Generalized IVDHFUUBLWeighted Geometric(GIVDHFUUBLWG)Operator

119866119868119881119863119867119865119880119880119861119871119882119866120596120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889119895)120596119895)= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871119895 )120582)120596119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880119895 )120582)120596119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871119895 )120582)120596119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880119895 )120582)120596119895)1120582]])

(8)

Definition 11 For a collection of IVDHF UUBLNs 119904119889119895(119895 = 1 119899) 119904119889120590(119895) be the 119895th largest 119908 = (1199081 1199082 119908119899)119879 be the aggregation-associated weighting vector 119908119895 isin [0 1]sum119899119895=1119908119895= 1 120582 is a parameter such that 120582 isin (0 +infin) 119878119899 997888rarr 119878 Then

10 Complexity

(1) Generalized IVDHFUUBL Ordered Weighted Average(GIVDHFUUBLOWA) Operator

119866119868119881119863119867119865119880119880119861119871119874119882119860120596120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(119908119895119904119889120590(119895)120582))1120582

= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

120582)1120582]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895) []119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120590(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120590(119895))120582)119908119895)1120582]])

(9)

(2) Generalized IVDHFUUBL Ordered Weighted Geometric(GIVDHFUUBLOWG)Operator

119866119868119881119863119867119865119880119880119861119871119874119882119866119908120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889120590(119895))119908119895)= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120590(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120590(119895))120582)120596119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120590(119895))120582)120596119895)1120582]])

(10)

Definition 12 For a collection of IVDHF UUBLNs 119904119889119895(119895 =1 119899) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector120596119895 isin [0 1] sum119899119895=1 120596119895 = 1 119899 is a balancing coefficient119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weight-ing vector with 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1

(1) Generalized IVDHFUUBL Hybrid Average (GIVDHFU-UBLHA) Operator

119866119868119881119863119867119865119880119880119861119871119867119860120596119908120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(119908119895119904119889120590(119895)120582))1120582

= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895)

))))120582)1120582]

Complexity 11

⋃[120583119871120590(119895) 120583

119880120590(119895)]isinℎ120590(119895) []

119871120590(119895) ]119880120590(119895)]isin119892120590(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120590(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120590(119895))120582)119908119895)1120582]])

(11)

where 119904119889120590(119895) is the 119895th largest and

119904119889119895 = 119899120596119895119904119889119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904119899120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904119899120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

] ⋃

[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[1 minus (1 minus 120583119871119895 )119899120596119895 1 minus (1 minus 120583119880119895 )119899120596119895] [(]119871119895)119899120596119895 (]119880119895 )119899120596119895]) (12)

(2) Generalized IVDHFUUBL Hybrid Geometric(GIVDHFUUBLHG) Operator

119866119868119881119863119867119865119880119880119861119871119867119866119908120596120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889120590(119895))119908119895)= ⋃(119904120599120590(119895)

ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120590(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895)

))))119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895) []119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120590(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120590(119895))120582)119908119895)1120582]])

(13)

where 119904119889120590(119895) is the 119895th largest and

119904119889119895 = 119904119889119895119899120596119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(Δminus11199050 (119879119865

11990511989511199050(120595(119904120572119895 ))))

119899120596119895 119904(Δminus11199050 (119879119865

11990511989521199050(120595(119904120573119895 ))))

119899120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[(120583119871119895 )119899120596119895 (120583119880119895 )119899120596119895] [1 minus (1 minus ]119871119895 )119899120596119895 1 minus (1 minus ]119880119895 )119899120596119895]) (14)

12 Complexity

When confronted with ill-structured situations wheredecision-makerrsquos complex attitudinal characters need to beincluded order-inducing vectors provide an effective way[88ndash90]Thus we further define following induced operatorsfor IVDHF UUBLNs

Definition 13 For a collection of IVDHF UUBLNs 119904119889119895 (119895 =1 119899) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector

120596119895 isin [0 1] sum119899119895=1 120596119895 = 1 119899 is a balancing coefficient119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weight-ing vector 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1 120576119895 denote a set oforder inducing vectors

(1) Induced Generalized IVDHFUUBL Hybrid Average (I-GIVDHFUUBLHA) Operator

119868-119866119868119881119863119867119865119880119880119861119871119867119860120596119908120582 (lt 1205761 1199041198891 gt lt 1205762 1199041198892 gt lt 120576119899 119904119889119899 gt) = ( 119899⨁119895=1

(119908119895119904119889120587(119895)120582))1120582

= ⋃(119904120599120587(119895) ℎ120587(119895) 119892120587(119895))isin119904119889120587(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120587(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587(119895)

))))120582)1120582]

⋃[120583119871120587(119895) 120583

119880120587(119895)]isinℎ120587(119895) []

119871120587(119895) ]119880120587(119895)]isin119892120587(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120587(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120587(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120587(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120587(119895))120582)119908119895)1120582]])

(15)

where (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) is decreasing order of (11990411988911199041198892 119904119889119899) according to the order inducing vector 120576119895 and119904119889119895 = 119899120596119895119904119889119895 = ⋃

(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904119899120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904119899120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

] ⋃

[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[1 minus (1 minus 120583119871119895 )119899120596119895 1 minus (1 minus 120583119880119895 )119899120596119895] [(]119871119895)119899120596119895 (]119880119895 )119899120596119895]) (16)

(2) Induced Generalized IVDHFUUBL Hybrid Geometric(I-GIVDHFUUBLHG) Operator

119868-119866119868119881119863119867119865119880119880119861119871119867119866119908120596120582 (lt 1205761 1199041198891 gt lt 1205762 1199041198892 gt lt 120576119899 119904119889119899 gt) = 1120582 ( 119899⨂119895=1

(120582119904119889120587(119895))119908119895)= ⋃(119904120599120587(119895)

ℎ120587(119895) 119892120587(119895))isin119904119889120587(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120587(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587(119895)

))))119908119895]

⋃[120583119871120587(119895)120583119880120587(119895)]isinℎ120587(119895) []119871120587(119895) ]

119880120587(119895)]isin119892120587(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120587(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120587(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120587(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120587(119895))120582)119908119895)1120582]])

(17)

Complexity 13

Weighted Aggregation

Geometric Arithmetic

GIVDHFUUBLHG GIVDHFUUBLHA

GIVDHFUUBLWG GIVDHFUUBLOWG

IVDHFUUBLWG IVDHFUUBLOWG

GIVDHFUUBLWA GIVDHFUUBLOWA

IVDHFUUBLWA IVDHFUUBLOWA

= 1 w =1

n = 1 =

1

n = 1 =1

n = 1 =

1

n

= 1 = 1 = 1 = 1

Figure 2

where (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) is decreasing order of (11990411988911199041198892 119904119889119899) according to the order inducing variables 120576119895 and

119904119889119895 = 119904119889119895119899120596119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(Δminus11199050 (119879119865

11990511989511199050(120595(119904120572119895 ))))

119899120596119895 119904(Δminus11199050 (119879119865

11990511989521199050(120595(119904120573119895 ))))

119899120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[(120583119871119895 )119899120596119895 (120583119880119895 )119899120596119895] [1 minus (1 minus ]119871119895 )119899120596119895 1 minus (1 minus ]119880119895 )119899120596119895]) (18)

42 Properties of the Proposed GeneralizedAggregation Operators

eorem 14 With special values of 120582 119908 and 120596 the operatorsGIVDHFUUBLHA and GIVDHFUUBLHG can include aseries of traditional aggregation operators as special cases andtheir relationship can be depicted in Figure 2

Proof See Appendix B

As for the induced hybrid aggregation operators we alsohave following theorem

eorem 15 If (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) = (119904119889120590(1) 119904119889120590(2) 119904119889120590(119899)) I-GIVDHFUUBLHA reduces toGIVDHFUUBLHAIf (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) = (119904119889120590(1) 119904119889120590(2) 119904119889120590(119899)) thenI-GIVDHFUUBLHG reduces to the GIVDHFUUBLHG oper-ator

eorem 16 All the proposed generalized operatorsGIVDHFUUBLWAGIVDHFUUBLWGGIVDHFUUBLOWAGIVDHFUUBLOWGGIVDHFUUBLHAGIVDHFUUBLHGI-GIVDHFUUBLHA and I-GIVDHFUUBLHG hold thefollowing properties (1) Commutativity (2) Idempotency (3)Boundedness

Based on above theorems following properties can alsobe derived

eorem 17 For a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weight vector of 119904119889119895with 120596119895 isin [0 1] and sum119899119895=1 120596119895 = 1 120582 gt 0 then we have

(1) 119868119881119863119867119865119880119880119861119871119882119866120596(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119882119860120596(1199041198891 1199041198892 119904119889119899)(2) 119868119881119863119867119865119880119880119861119871119882119866120596(1199041198891 1199041198892 119904119889119899) le119866119868119881119863119867119865119880119880119861119871119882119860120596120582(1199041198891 1199041198892 119904119889119899)(3) 119866119868119881119863119867119865119880119880119861119871119882119866120596120582(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119882119860120596(1199041198891 1199041198892 119904119889119899)

Proof Omitted for concision

eorem 18 For a collection of IVDHF UUBLNs 119904119889119895 = (119904120599119895 ℎ119895 119892119895) 119908 = (1199081 1199082 119908119899)119879 is the weighting vector of 119904119889119895with 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1 120582 gt 0 Then

(1) 119868119881119863119867119865119880119880119861119871119874119882119866119908(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119874119882119860119908(1199041198891 1199041198892 119904119889119899)(2) 119868119881119863119867119865119880119880119861119871119874119882119866119908(1199041198891 1199041198892 119904119889119899) le119866119868119881119863119867119865119880119880119861119871119874119882119860119908120582(1199041198891 1199041198892 119904119889119899)(3) 119866119868119881119863119867119865119880119880119861119871119874119882119866119908120582(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119874119882119860119908(1199041198891 1199041198892 119904119889119899)

Proof Omitted for concision

14 Complexity

5 An Integrated MADM Approach forTackling Complex Green ContractorSelection Problems

As described in Section 2 we take the green contractorselection as a special type of complicate MADM problemsthat synthesizes three characteristics of decision hesitancy[21 22] attributes interdependency [84] and group atti-tudinal characters [61] Therefore in this section we con-struct an integrated MADM approach to tackle the com-plex green contractor selection problems Suppose 119883 =1199091 1199092 119909119898 is the set of alternative green contractors and119860 = 1198601 1198602 119860119899 is the set of evaluative attributes 120596 =(1205961 1205962 120596119899)119879 is weighting vector for the attributes 120596119894 ge 0sum119899119894=1 120596119894 = 1 Let = (119903119894119895)119899times119898 denote the decision matrix inwhich 119903119894119895 = ([119904120572119894119895 119904120573119894119895] ℎ119894119895 119892119894119895) is an IVDHF UUBLN givenby decision-makers for alternative contractor 119909119895 with respectto attribute 119860 119894 According to the mechanism of pair-wisecomparisons among attributes in the DEMATELmethod [6364] the interdependency among attributes can be obtained asamatrix119885 = (119911119894119896)119899times119899 where 119911119894119896 indicates the degree towhich119860 119894 affects119860119896 Subsequently based on the IVDHFUUBLS andits operations we now present detailed steps of our MADMapproach as shown in following Algorithm I

Algorithm I Hesitant fuzzy linguistic MADMwith attributesinterdependency and decision-makersrsquo group attitudinalcharacters

Step 1 Determine argument-dependent weighting vector120596119860119863 according to attribute values by programming modeldeveloped in the following Section 51

Step 2 Obtain the attribute-interdependences based weight-ing vector 120596119860119868 by use of DEMATEL method described in thefollowing Section 52

Step 3 Calculate synthesized attribute weighting vectoraccording to

120596 = 120572120596119860119868 + 120573120596119860119863 (19)

where 120572 and 120573 are parameters to reflect decision character-istics of decision organizations 0 le 120572 120573 le 1 120572 + 120573 =1Step 4 Check requirements for order inducing If no addi-tional order inducing required then go to Step 5 otherwisego to Step 6

Step 5 Utilize generalized aggregation operators to get theoverall IVDHF UUBLNs 119903119895(119895 = 1 119898) for each alternative119909119895 Here we take GIVDHFUUBLHA operator for examplebecause it can include other traditional operators as its specialcases Therefore we have

119866119868119881119863119867119865119880119880119861119871119867119860(1199031119895 1199032119895 119903119899119895) = ( 119899⨁119894=1

(119908119894 (119903120590119894119895)120582))1120582= ⋃(119904120590119894119895ℎ120590119894119895 119892120590119894119895

)isin119903120590119894119895

([119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590119894119895

))))120582)1120582 119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590119894119895

))))120582)1120582]

⋃[120583119871120590119894119895120583

119880120590119894119895]isinℎ120590119894119895[]

119871120590119894119895]119880120590119894119895]isin119892120590119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120590119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120590119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120590119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120590119894119895)120582)119908119894)1120582]])

(20)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of 119903119894119895119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weightvector 119903120590119894119895 is the 119894th largest of 119903119894119895 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))

119904119899120596119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119894119895))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (21)

Complexity 15

Then go to Step 8

Step 6 By use of the TOPSIS-based method developed in thefollowing Section 53 we determine order-inducing vectors 120576to reflect decision-makersrsquo group attitudinal characters

Step 7 Utilizing induced generalized aggregation operatorsto get the overall IVDHF UUBLNs 119903119895(119895 = 1 2 119898) foreach alternative 119909119895 we have

119868-119866119868119881119863119867119865119880119871119867119860(1199031119895 1199032119895 119903119899119898) = ( 119899⨁119894=1

(119908119894 (119903120587119894119895)120582))1120582

= ⋃(119904120587119894119895ℎ120587119894119895119892120587119894119895)isin119903120587119894119895

([119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120587119894119895

))))120582)1120582 119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587119894119895

))))120582)1120582]

⋃[120583119871120587119894119895 120583

119880120587119894119895]isinℎ120587119894119895 []

119871120587119894119895]119880120587119894119895]isin119892120587119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120587119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120587119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120587119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120587119894119895)120582)119908119894)1120582]])

(22)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of119903119894119895 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 119903120587119894119895 is the 119894th largest of 119903119894119895 reordered by theorder-inducing vectors 120576 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))119899120596119894120572119894119895

119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120573119894119895 ))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (23)

Step 8 Calculate scores 119878(119903119895) and accuracy degrees 119875(119903119895) for119903119895(119895 = 1 2 119898)Step 9 According to Definition 5 output ranking order ofalternative contractors

In Algorithm I to objectively derive unknown weight-ing vector for evaluative attributes a synthesized attributeweighting vector is devised by fusing two parts of concern(i) argument-dependent weighting vector and (ii) attribute-interdependences based weighting vector The former is toreflect effects of attribute values on attribute weights whilethe latter is to exploit effects of interdependences amongattributes on their weights For clarity processing flowchartof Algorithm I is depicted in Figure 3

51 Determining Argument-Dependent Attribute WeightingVector 120596119860119863 In order to derive attribute weighting vec-tor appropriately from assessments maximizing deviationmethod [91] is adopted here for its distinguishing ability

and objectivity [92] According to Wang [91] if performancevalues of each alternative have little differences under anattribute it implicates such an attribute plays a less importantrole otherwise it lives a more important role in decision-making Thus if one attribute has similar values across alter-natives it should be assigned a smaller weight contrariwiseit should be a bigger weight

In light of the idea in maximizing deviation methodwe develop a programming model to determine argument-dependent weighting vector 120596119860119863 For attribute 119860 119894 the stan-dard deviation of alternative 119909119895 to all the other alternatives isdenoted as

119865119894119895 (120596119860119863) = 119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (24)

Let119865119894(120596119860119863) represent deviation value of all alternatives toother alternatives according to attribute119860 119894 and be formulatedas follows

16 Complexity

Attribute weighting

Hesitant fuzzy linguistic MADM with attributes interdependency and decision makersrsquo group attitudinal characters

Step 9 Output ranking order of alternatives

interdependency among attributes by DEMATEL in Section 52

attributes values by standard deviation method in Section 51

Step 6 Ensure TOPSIS order inducing variables proposed in Section 53

Step 7 Utilizing proposed induced generalized hybrid aggregation operators developed in Section

41 to get overall value of each alternative

Step 8 Calculate score degrees and accuracy degrees

Step 4 Order inducingYes

Step 5 Choose proposed generalized operators in Section 41 to get overall value of each alternative

No

Collect decision matrix in the form of IVDHF_UUBLS

Complex problems of green contractor selection

Step 1 Weighting vector AD derived from Step 2 Weighting vector AI derived from

Step 3 Obtain synthesized weighting vector of attributes =AI + AD

Figure 3 Processing flowchart of Algorithm I

119865119894 (120596119860119863) = 119898sum119895=1

119865119894119895 (120596119860119863) = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (25)

Based on the analysis above we can get followingprogramming model (M-1) for obtaining optimal attributeweighting vector 120596119860119863

(119872 minus 1)max 119865 (120596119860119863) = 119899sum

119894=1119865119894 (120596119860119863) = 119899sum

119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894st 119899sum

119894=1(120596119860119863119894 )2 = 1 120596119860119863119894 ge 0 119894 = 1 2 119899 (26)

where 119889(119903119894119895 119903119894119897) can be calculated by Definition 6To solve model (M-1) we can construct Lagrange func-

tion

119871 (120596119860119863119894 120577) = 119899sum119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894+ 1205772 ( 119899sum

119894=1

(120596119860119863119894 )2 minus 1) (27)

where 120577 is the Lagrange multiplierDifferentiate with respect to 120596119860119863119894 (119894 = 1 2 119898) and 120577

and set these partial derivatives equal to zero the followingset of equations can be obtained

120597119871120597120596119860119863119894 = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) + 120577120596119860119863119894 = 0120597119871120597120577 = 119899sum

119894=1

(120596119860119863119894 )2 minus 1 = 0 (28)

By normalizing the solutions of above equations we canget a simple and exact formula for determining the attributeweights as follows

120596119860119863119894 = sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897)sum119899119894=1sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897) (29)

52 Deriving Attributes Interdependency-Based AttributeWeighting Vector 120596119860119868 Some efforts [53 60 84 93ndash95]

Complexity 17

have been made to address interdependencies betweenattributes in MADM However they unanimously cannotreveal the interdependences in a more complete way WhileDEMATAEL technique [63 96] gathers collective knowledgeto capture influences between attributes more precisely andcompletely so that weighting vector for attributes can berationally determined according to strength measure ofthose influences [64 97] Therefore in tackling the complexMADMwith interdependent attributes under IVDHF UUBLenvironments we present a DEMATEL-based method as thefollowing Algorithm II to derive attribute weighting vector120596119860119868 from the viewpoint of attributes interdependency

Algorithm II DEMATEL-basedmethod for ensuringweight-ing vector 120596119860119868Step 1 Construct initial direct-influence matrix 119885 = (119911119894119896)119899times119899

For attributes 119860 = 1198601 1198602 119860119899 decision-makersare asked to give direct effect that attribute 119860 119894 has onattribute 119860119896 using a five-point scale with scores representedby natural language (1) 0 mdash lsquoabsolutely no influencersquo (2)1 mdash lsquolow influencersquo (3) 2 mdash lsquomedium influencersquo (4) 3 mdashlsquohigh influencersquo (5) 4mdash lsquovery high influencersquoThen the initialdirect-influence matrix 119885 = (119911119894119896)119899times119899 can be established witheach element being the mean of same factors in differentdirect matrices

Step 2 Calculate normalized direct-influence matrix 119883 =(119909119894119896)119899times119899 Using matrix 119885 the normalized direct-influencematrix 119883 = (119909119894119896)119899times119899 is calculated according to

119909119894119896 = 119911119894119896119904 (30)

where 119904 = maxmax1le119894le119899sum119899119896=1 119911119894119896max1le119896le119899sum119899119895=1 119911119894119896 Allelements in matrix 119883 are complying with 0 le 119909119894119896 le 10 le sum119899119896=1 119909119894119896 le 1 and 0 le sum119899119894=1 119909119894119896 le 1 and at least onerow or column of the summation equals 1

Step 3 Derive total-influence matrix 119879 = (119905119894119896)119899times119899 Basedon matrix 119883 the total-influence matrix 119879 = (119905119894119896)119899times119899 can bederived by summing the direct effects and all of the indirecteffects

119879 = 119883 + 1198832 + 1198833 + sdot sdot sdot + 119883ℎ = 119883 (119868 minus 119883)minus1 (31)

where 119868 represents identity matrix When ℎ 997888rarr infin 119883ℎ =(0)119899times119899Step 4 Compute influencing and influenced degrees of eachattribute The sum of rows and the sum of columns withinthe total-influence matrix 119879 are respectively expressed as thevectors 119903 and 119888

119903 = (119903119894)119899times1 = ( 119899sum119896=1

119905119894119896)119899times1

119888 = (119888119896)119899times1 = ( 119899sum

119895=1

119905119894119896)119879

1times119899

(32)

where 119903119894 denotes the sumof the 119894th row inmatrix119879 and showsthe sum of direct and indirect effects that attribute 119860 119894 hason other attributes Similarly 119888119896 shows the sum of direct andindirect effects that attribute 119860119896 has receivedStep 5 Estimate weights of attributes based on influencedegrees according to

120596119868lowast119894 = [(119903119894 + 119888119894)2 + (119903119894 minus 119888119894)2]12 (33)

Then the weighting vector 120596119860119868 can be obtained by normaliz-ing (33)

120596119860119868119894 = 120596119868lowast119894sum119899119894=1 120596119868lowast119894 (34)

53 Obtaining Order Inducing Variables 120576 by TOPSIS-BasedMethod For ill-defined MADM problems induced aggre-gation operators [89 90 98] provide an effective channelfor experts to express complex attitudinal characters Thekey step in induced aggregation operators is to obtainorder-inducing vectors 120576 which however are presumptivelyassigned in most papers [62] Therefore in this sectionaiming at deducing the order-inducing vectors 120576 rationallywe here devise a TOPSIS-based method in Algorithm IIIObviously the Algorithm III is of generality for MADMwithdifferent forms of linguistic expression

Algorithm III TOPSIS-based method for obtaining orderinducing vectors 120576Step 1 Let the decision maker define IVDHF UUBL idealsolution(s) 119903+119894 and IVDHF UUBL negative ideal solution(s)119903minus119894 corresponding to each attribute in which

119903+119894 = (119903+1198941 119903+1198942 119903+119894119901) 119903minus119894 = (119903minus1198941 119903minus1198942 119903minus119894119902) (35)

where 119901 and 119902 denotes the number of 119903+119894 and 119903minus119894 respectively

Step 2 Calculate distances from each element in decisionmatrix to ideal solution(s) and negative solution(s) respec-tively

119889+119894119895 = max119897=12119901

119889 (119903119894119895 119903+119894119897 ) 119889minus119894119895 = min

119897=12119902119889 (119903119894119895 119903minus119894119897 ) (36)

where 119889(119903119894119895 119903+119894119897 ) and 119889(119903119894119895 119903minus119894119897 ) are calculated by the distancemeasure in Definition 6 If 119901 gt 1 we choose the onewith maximum distance if 119902 gt 1 we choose the one withminimum distance

Step 3 Calculate coefficients according to

119888119894119895 = 119889minus119894119895119889+119894119895 + 119889minus119894119895 119894 = 1 2 119899 119895 = 1 2 119898 (37)

18 Complexity

Table 2 Decision matrix in the form of IVDHF UUBLS1199091 1199092 11990931198601([AHVH] [02 03][02 04] [03 04]) ([HT] [04 05][05 06] [02 03] [02 04]) ([ALAH] [01 02][01 03] [06 07])1198602([MH] [05 06][02 03]) ([AHT] [06 07][01 02] ) ([MVH] [02 03][05 06] [06 07])1198603([VLAL] [03 04][04 05] [05 06]) ([MVH] [06 07] [07 08][01 02]) ([VLL] [04 05][02 03])1198604([MAH] [02 04][05 06]) ([LAL] [03 05][02 03]) ([ALM] [06 07][01 02] [02 03])1198605([ALM] [04 05][01 02] [04 05]) ([AHH] [06 07][01 02]) ([HVH] [04 05][06 07] [01 03])1198606([LM] [01 02][03 05] [03 05]) ([VLL] [02 04] [05 06][02 03]) ([AHH] [03 04][04 05])1198607([VLL] [04 05][05 06] [02 03] [02 04]) ([MAH] [05 06] [07 08][01 02]) ([ALM] [04 05][05 06] [03 04])1198608([ALAH] [01 03][04 06]) ([AHH] [05 07][01 02] [02 03]) ([AHH] [04 06][03 04])

Table 3 Direct influence matrix Z1198601 1198602 1198603 1198604 1198605 1198606 1198607 11986081198601 0 3 4 1 3 1 2 41198602 1 0 2 4 4 2 3 11198603 4 2 0 1 3 2 4 11198604 2 3 1 0 3 4 1 21198605 4 2 1 3 0 1 4 31198606 2 4 3 4 4 0 4 21198607 4 1 2 3 3 1 0 41198608 1 2 4 3 3 2 4 0

Step 4 Derive order inducing variable 120576119894119895 by descendingorder of 119888119894119895 The greater the value 119888119894119895 the bigger the 1205761198941198956 Illustrative Example and Experiments

61 Illustrative Case Study Suppose a mega infrastructureproject is selecting green contractor for one of its subprojectsA panel of decision-makers including managers professorsand competitive intelligence experts has been organized tocomprehensively evaluate three alternative contractors (ie1199091 1199092 and 1199093) under the eight attributes (ie 1198601ndash1198608) listedin Section 2

To elicit true decision preferences we apply theIVDHF UUBLS to assess the three alternatives (1199091ndash1199093)Firstly decision-makers are asked to vote on 119909119895 under eachattribute119860 119894 using different unbalanced linguistic term sets 1198781 1198782 and 1198783 where 1198781 = 119873 119871119860119871119872119860119867119867119876119867119881119867119860119879 1198791198782 = 119873119860119873119881119871 119876119871 119871 119860119871119872119860119867119881119867 119879 1198783 = 119873119881119871119871119860119872119872119876119872119860119867119867 119879 In Figure 4 we show the adoptedlinguistic hierarchies and relationship between 1198781 1198782 and1198783 Attributes 1198601 1198602 1198605 and 1198608 are evaluated by use of 1198781attributes11986031198604 and1198607 are evaluated by 1198782 and attribute1198606is evaluated by 1198783 The highest-voted uncertain unbalancedlinguistic term [119904119898 119904119899] are identified Secondly the panel of

decision-makers is asked to further express their opinionson membership degrees ℎ(119909) and nonmembership degrees119892(119909) to [119904119898 119904119899] Then we can derive compound assessments(ie ⟨119909 119904120599(119909) ℎ(119909) 119892(119909)⟩) in the form of IVDHF UUBLSand obtain the decision matrix in Table 2 Additionallyin order to examine the interdependency among attributesdecision-makers are also required to assess the influentialrelations among the eight attributes using DEMATEL-basedAlgorithm II The direct influence matrix is collected andshown in Table 3

Next we apply the proposed Algorithm I to solve thisgreen contractor selection problem To derive the synthe-sized attribute weighting vector in Algorithm I we assign120572 = 120573 = 05 For scenarios with no group attitudinalcharacters operator GIVDHFUUBLHA in Definition 12 ischosen In contrast for scenarios with group attitudinalcharacters operator I-GIVDHFUUBLHA in Definition 13will be chosen Further the panel of decision-makers is askedto determine their positive and negative ideal alternativesthat are collected in Table 4 where corresponding order-inducing vectors that are derived from Algorithm III arealso listed (see the rightmost column) Thereafter operatorI-GIVDHFUUBLHA in Definition 13 is utilized for aggrega-tion For GIVDHFUUBLHA and I-GIVDHFUUBLHA we

Complexity 19

Table 4 Ideal alternatives (119903+ and 119903minus) and derived order-inducing vectors 120576119903+ 119903minus 1205761199091 1199092 11990931198601 ([VHT] 1 0) ([NL] 1 0) 5 6 61198602 ([MAH] [07 08] [01 02]) ([NL] 1 0) 1 1 41198603 (T 1 0) (L 1 0) 8 7 81198604 ([ALAH] 09 01) (T 1 0) 2 2 11198605 ([HT] [09 1] 0) (N 1 0) 7 5 51198606 (M 1 0) (N 1 0) 6 3 31198607 (M 1 0) ([NL] 1 0) 3 8 71198608 ([MAH] 1 0) (N 1 0) 4 4 2

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 1 2 3 4

0 1 2

N L AH H QH VH AT TM

Level 1 l(13)

Level 2 l(25)

Level 3 l(39)

Level 4 l(417)

Unbalanced linguistic term set S1

N AH VH TM Unbalanced linguistic term set S2VL L ALAN QL

N H TM Unbalanced linguistic term set S3

L AM AHQM

AL

VL

Figure 4 Unbalanced linguistic term sets (1198781 1198782 and 1198783) and their mapping in linguistic hierarchies

assume 120582=2 and 119908 = (0 0 015 025 025 025 01 0)119879Note that 119908 is a position weighting vector which is derivedby the fuzzy semantic quantitative operator [99] Aftercompleting all steps in Algorithm I (see Appendix C) weobtain the ranking results and present them in Table 5 Forthe scores derived from Step 5 we find the ranking order is1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the best solution Step 7gives the ranking order of 1199092 ≻ 1199091 ≻ 1199093 indicating 1199092 is stillthe most appropriate one while 1199091 is better than 1199093 under theinfluence of 12057662 Experiments on Ranking Results along with VariousParameter Configurations In order to inspect the impactsof various parameter configurations on ranking results inthis section more numerical examples are carried out Alleight operators that defined in Section 41 are applied to theabove illustrative case where the parameter 120582 is configured asintegers ranging from 1 to 20 and parameter 120572 is configuredto a value set of 0 03 05 08 09 1

Firstly the four basic aggregation operators of GIVDH-FUUBLWA GIVDHFUUBLWG GIVDHFUUBLOWA and

Table 5 Ranking results of operators GIVDHFUUBLHA and I-GIVDHFUUBLHA

Aggregation operator Ranking orderGIVDHFUUBLHA 1199092 ≻ 1199093 ≻ 1199091I-GIVDHFUUBLHA 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG are applied to the former case Thesame configuration 120572 = 120573 = 05 is chosen while 120582 is testedas integers ranging from 1 to 20 The ranking results derivedfrom different parameter combinations have been collectedin Table 6 As can be seen four operators unanimouslyidentify the contractor 1199092 is the best one no matter how120582 changes Regarding priority relation between 1199091 and 1199093GIVDHFUUBLWA and GIVDHFUUBLOWA generate thesame result of1199093 ≻ 1199091 while results fromGIVDHFUUBLWGandGIVDHFUUBLOWGgot a change to 1199091 ≻ 1199093 at a certainvalue of parameter 120582

Actually GIVDHFUUBLWA and GIVDHFUUBLOWAare generalized version of weighted arithmetic aggregationoperators that producemore favorable results by emphasizing

20 Complexity

Table 6 Ranking orders obtained by GIVDHFUUBLWA GIVDHFUUBLOWA GIVDHFUUBLWG and GIVDHFUUBLOWG along with120582Operators 120582 (Integer) Ranking orderGIVDHFUUBLWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLOWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLWG [1 2] 1199092 ≻ 1199093 ≻ 1199091[3 20] 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG [1 8] 1199092 ≻ 1199093 ≻ 1199091[9 20] 1199092 ≻ 1199091 ≻ 1199093

Table 7 Ranking results obtained by GIVDHFUUBLHA and I-GIVDHFUUBLHA along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHA I-GIVDHFUUBLHA120572=0 120573=1 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=03 120573=07 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=05 120573=05 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=08 120573=02 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=09 120573=01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=1 120573=0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093

Table 8 Ranking results obtained by GIVDHFUUBLHG and I-GIVDHFUUBLHG along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHG I-GIVDHFUUBLHG

120572 = 0 120573 = 1 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 03 120573 = 07 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 05 120573 = 05 [1 8] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[9 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 08 120573 = 02 [1 17] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[18 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 09 120573 = 01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572 = 1 120573 = 0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093membership degrees to a uncertain linguistic term thus canbe considered as optimistic operators while GIVDHFU-UBLWG and GIVDHFUUBLOWG are generalized geomet-ric aggregation operators that producemore favorable resultsby emphasizing nonmembership degrees thus can be consid-ered as pessimistic operators [100 101]Therefore the rankingresults in Table 6 could be perceived as for optimism-baseddecision-making alternative 1199093 always stands out better than1199091 when 120582 gradually amplifies degree of optimism but forpessimism-based decision-making 1199091 turns out better than1199093 when 120582 gradually amplifies degree of pessimism 120582 canbe considered as the parameter for amplifying degree ofoptimism or pessimism [101]

Next to inspect the impacts of various combinationsof 120596119860119868 and 120596119860119863 on ranking results another four aggrega-tion operators of GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG also have beenapplied to the above illustrative case with varying 120572 120573 and 120582

All results have been collected in Tables 7 and 8 for compari-son As can be seen fromTable 7 ranking results generated byoperator GIVDHFUUBLHA keep the same as 1199092 ≻ 1199093 ≻ 1199091for all test combinations of parameters However as shownin Table 8 although GIVDHFUUBLHG is also capable ofidentifying 1199092 as the best contractor the priority relationsbetween 1199091 and 1199093 hold similar changing patterns in rankingresults in Table 6 that were output byGIVDHFUUBLWGandGIVDHFUUBLOWG Regarding decision situations whereobjective attributeweighting vector120596119860119863 is only considered (120572= 0) or its dominant importance (such as 120572 = 03) is accepted1199091 ≻ 1199093 is a clear observation when parameter 120582 is set toan integer in interval [8 20] Contrariwise under decisionscenarios where subjective attribute weighting vector 120596119860119868 isonly needed (120572 = 1) or its dominant importance (such as 120572 =09) is acknowledged 1199093 ≻ 1199091 holds all the way 120582 varies

In order to more clearly show the score trajectories ofalternative contractors along variations of parameter 120582 we

Complexity 21

HAInducedHA

InducedHA-2HA-2

HA-3

InducedHA-1InducedHA-3

= =

HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(a) 120572 = 0

HAInducedHA

= 03 = 07

InducedHA-2

HA-2HA-3

InducedHA-1 InducedHA-3HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(b) 120572 = 03

HAInducedHA

= 5 = 05

InducedHA-2HA-2

HA-3

InducedHA-1 InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(c) 120572 = 05

HAInducedHA

= 08 = 02

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572 = 08

HAInducedHA

= 9 = 01

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(e) 120572 = 09

HAInducedHA

246 2462 2464 2466 2468 247

= 1 = 0InducedHA-2HA-2

HA-3

HA- 3

HA-3

InducedHA-1

InducedHA-3

HA-1

HA-1

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

1085

1086

1087

(f) 120572 = 1

Figure 5 Score trajectories obtained by aggregation operators HA and InducedHA with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

further configure parameter 120582 to the value set of [0 01100]and conduct computational experiments to collect scorevalues and ranking results (as shown in Figures 5 and 6) byfour aggregation operators of GIVDHFUUBLHA GIVDH-FUUBLHG I-GIVDHFUUBLHAand I-GIVDHFUUBLHGFor more clarity in Figures 5 and 6 please note that weuse HA HG Induced-HA and Induced HG to respec-tively denote GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG In Figure 5it can be observed that when dominant importance of 120596119860119863

is emphasized (120572 = 0 120572 = 03 120572 = 05) GIVDHFUUBLHAis capable of differentiating the three alternative contractorsas same ranking order of 1199092 ≻ 1199093 ≻ 1199091 for all 120582 but withincreases in 120572 the margins between scores of 1199091 and 1199093 nar-rows and the rank order changes to 1199091 ≻ 1199093when 120572 gets 09and 1 Interestingly in Figure 6 although GIVDHFUUBLHGis able to significantly differentiate alternative contractors 1199091and 1199093 when dominant importance of 120596119860119863 is emphasized (120572= 0 120572 = 03 120572 = 05) rank order of 1199091 and 1199093 obviouslychanges once from 1199093 ≻ 1199091 to 1199091 ≻ 1199093 when 120582 is bigger

22 Complexity

HGInducedHG

= = 1

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(a) 120572= 0

HGInducedHG

= 3 = 07

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(b) 120572= 03

HGInducedHG

= 05 = 05

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(c) 120572= 05

HGInducedHG

= 8 = 02

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572= 08

HGInducedHG

292 293 294 295 296 297minus08

minus0795

minus079

= 09 = 01

HA-1

HA-3

InducedHG-2

HG-1

HG-1

HG-2

HG-3

HG-3 InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4

minus3

minus2

minus1

0

1

2

scor

e val

ues

(e) 120572= 09

HGInducedHG

50 60 70 80 90 100minus095minus09

minus085

= 1 = 0

HG-x3

HG-x1

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(f) 120572= 1

Figure 6 Score trajectories obtained by aggregation operators HG and InducedHG with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

than 8 When 120572 increases to 1 GIVDHFUUBLHGmaintainsthe rank order of 1199093 ≻ 1199091 but their score trajectorieswalk very close Comparatively I-GIVDHFUUBLHA andI-GIVDHFUUBLHG generate consistent ranking results of1199092 ≻ 1199091 ≻ 1199093 no matter 120582 varies in all the above experiments

In sum when parameters are configured with typical val-ues (ie 120572 = 05 and 120582=1) and no additional attitudinal infor-mation is taken into consideration the proposed aggregationoperators unanimously derive the same rank order 1199092 ≻1199093 ≻ 1199091 of for the three alternative contractors However

with the amplification effect of parameter 120582 analysis resultsfrom linear combination of 120596119860119868 and 120596119860119863 reveal alternative1199092 which consistently is the best choice but rank order of1199091 and 1199093 changes because their score trajectories get closeand crossed which probably indicates that only decision-making informationderived from the decisionmatrix is inad-equate to significantly differentiate alternatives 1199091 and 1199093While by integrating additional decision information aboutattitudinal characters of group experts I-GIVDHFUUBLHAand I-GIVDHFUUBLHG manage to generate the identical

Complexity 23

Table 9 Order-inducing vectors obtained by our TOPSIS-based method for problem in [38]

1198661 1198662 1198663 11986641198601 02522 01629 02282 027141198602 01 05243 01364 046891198603 04081 01756 05244 037791198604 04507 01409 01064 013221198605 00697 02203 03753 01161

Table 10 Order-inducing vectors obtained by our TOPSIS-based method for problem in [39]

1198661 1198662 1198663 11986641198601 04548 04969 04923 047321198602 05092 047 04768 046451198603 07192 04888 04953 049151198604 06722 06163 05706 049291198605 06104 04942 06463 04946

Table 11 Ranking results obtained by applying different order-inducing vectors to [38 39]

Problems Approaches Ranking results

Problem in [38] IHFLOWA [38] + order-inducing vectors in [38] 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198605 ≻ 1198604

IHFLOWA [38] + order-inducing vectors in Table 9 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198604 ≻ 1198605

Problem in [39] I-IVDHFLOWA [39] + order-inducing vectors in [39] 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198602 ≻ 1198601

I-IVDHFLOWA [39] + order-inducing vectors in Table 10 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602

permutation of 1199092 ≻ 1199091 ≻ 1199093 which verifies that ourproposed TOPSIS-based order-inducing approach is prac-tical and useful in helping work out decision results morereasonably and robustly

63 Application Studies of Proposed TOPSIS-Based Order-Inducing Method When confronting complex decision-making situations that need additional attitudinal charactersof decision-makers Yager and Filev [88] and Yager [102]introduced and encouraged to employ order-inducing vectorto represent those attitudinal characters Since then theidea of order-inducing vector has been extended to variousdecision-making environments [98 103] such as those basedon intuitionistic fuzzy information [104 105] linguistic infor-mation [106] hesitant fuzzy linguistic information [38] andinterval-valued dual hesitant fuzzy uncertain linguistic infor-mation [39] Generally in existing literature order-inducingvectors rely on experts to directly give [38 39 104ndash106]

In order to examine the adaptability of our TOPSIS-based order-inducing method to linguistic decision-makingenvironments we respectively apply the method to resolvethe same problems in references [38 39] Due to fact thatthere is no descriptions about how the order-inducing vectorswere derived in both references [38 39] we here applyextremums into our TOPSIS-based method to obtain order-inducing vectors that is (1199047 1 0) and (1199041 0 1) for[38] ([1199046 1199046] 1 0) and ([1199040 1199040] 0 1) for [39] Thenwe obtain our order-inducing vectors listed in Table 9 forthe problem adopted in [38] and our order-inducing vectorslisted in Table 10 for [39]

Now we apply the order-inducing vectors as shownin Tables 9 and 10 to resolve the problems in [38] and[39] respectively Regarding the problem in [38] the scorevalues of all alternatives are calculated as 119878(1198601)=02602119878(1198602)=02417 119878(1198603)=03577 119878(1198604)=0221 and 119878(1198605)=0185so the corresponding ranking result is 1198603 ≻ 1198601 ≻ 1198602 ≻1198604 ≻ 1198605 As for the problem in [39] the score values ofall alternatives are computed as 119878(1198601)=39954 119878(1198602)=3783119878(1198603)=70827 119878(1198604)=71792 and 119878(1198605)=5813 we thus obtainthe ranking result 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602 Formore clarity brief descriptions of comparative approachesand obtained ranking results also have been put together inTable 11

According to Table 11 the approaches that employ ourorder-inducing vectors derive the same ranking order ofthe better three alternatives for both comparative problemsfrom [38 39] As seen our proposed TOPSIS-based order-inducing method exhibits effective and more understand-able Furthermore when decision-makers are incapable ofdirectly put forward all desired order-inducing vectors due tocomplexity in ill-structured problems our method providesa feasible directions to obtain order-inducing vectors byfiguring out ideal solutions Comparatively our proposedTOPSIS-based order-inducing method behaves effective andmore operational in practical use

64 Comparison with Decision-Making Approach Based onDegraded form of IVDHFULS Note that our approachis intrinsically adaptive to decision situations based ondegraded forms of IVDHF UUBLS such as the dual hesitant

24 Complexity

fuzzy linguistic sets introduced by Yang and Ju [54] Tofurther inspect feasibility of our approach we here conductcomparative study on the same case adopted by Yang and Ju[54] We firstly convert their original decision matrix intothe form of IVDHF UUBLS crisp membership and non-membership degrees are equivalently changed to intervalswith same upper and lower limits (eg 02 turns out to be[02 02]) and linguistic variables are rewritten as uncertainlinguistic variables (eg 1199042 turns out to be [11990421199042])

Since the case only got three attributes among whichinterdependency could be not significant under generalassumption weighting vector 120596119860119868 is not calculated Andfor straight comparison order inducing vector is notincluded either Then the argument-dependent attributeweighting vector can be obtained from Eq(29) as 120596119860119863 =(03192 03132 03676)119879 By use of Algorithm Iwith operatorIVDHFUUBLWA (ie GIVDHFUUBLWA in case of 120582 = 1see Appendix B) we have the scores of alternatives 119904(1198601) =03412 119904(1198602) = 01354 119904(1198603) = 03932 119904(1198604) = 07352Therefore the ordering of alternatives is1198604 ≻ 1198603 ≻ 1198601 ≻ 1198602which is mainly in alignment with the ordering of1198604 ≻ 1198601 ≻1198603 ≻ 1198602 in Yang and Ju [54] It is worth noticing that 120596119860119863used in our approach is derived objectively from assessmentswhile attribute weights 120596lowast = (035 025 040)119879used by Yangand Ju [54] were assigned empirically Differences between120596119860119863 and 120596lowast lead to different ordering of1198601 and1198603 Compar-atively speaking our approach holds more adaptability andobjectivity

In general our proposed MADM approach in this paperexhibits an effective way to tackle complex green contractorselection problems in which subjective attributes weightingvector (such as120596119860119868) and objective attributes weighting vector(such as 120596119860119863) exist To comprehensively utilize these twofundamental facets of attributes weighting information ourcase study and computational experiments suggest that linearcombination method (ie 120596 = 120572120596119860119868 + 120573120596119860119863) is effectiveto reflect effects of 120596119860119868 and 120596119860119863 by configuring parameter 120572or 120573 When there is no specific expert opinions to specify orhard to obtain we suggest that parameter 120572 chooses the valueset of 0 02 05 08 1 Furthermore generalized aggrega-tion operators are suggested to inspect score trajectoriesof alternative contractors along increasing of the operatorsrsquoamplification parameter 120582 If some alternative contractorsrsquoscore values get very close or crossed so that they cannot bedifferentiated from each other significantly we suggest to askthe panel of experts to derive opinions of ideal solutions andutilize our TOPSIS-based order-inducing method to includethose expert opinions as attitudinal decision informationthereby obtaining more rational and robust decision results

7 Conclusions

Aiming at tackling complex decision-making problems ofgreen contractor selection that often hold simultaneouslythe characteristics of decision hesitancy attributes inter-dependency and group complex attitudinal characters wehave developed an effective IVDHF UUBLS-based approach

We utilize IVDHF UUBLS to elicit complicate decision-makersrsquo assessments more objectively and comprehensivelywe employ DEMATEL for addressing attribute interdepen-dency more precisely and we determine group attitudinalcharacters as order-inducing vectors based on ideal attitudesof experts rather than by empirical assignment The mainadvantages of our approach are (i) IVDHF UUBLS offers anadequate way to depict decision hesitancy (ii) the designeddistance measure for IVDHF UUBLS avoids informationdistortion of traditional ones which artificially add mis-matching membership or nonmembership degrees (iii) thedeveloped generalized operators include traditional ones asspecial cases so as to provide flexibility in MADM basedon IVDHF UUBLS (iv) the devised synthesized attributeweighting scheme provides an adaptive way to consider twonecessary aspects attributes interdependency-based weightsand argument-dependent weights (v) the TOPSIS-basedmethod can deduce reasonable order-inducing vectors ratherthan empirical ones and holds generality in other linguisticdecision environments Results of case studies and experi-ments have verified the validity of the proposed approach

When multiple decision organizations are involvedgroup decision-making based on IVDHF UUBLS is animportant research direction Besides inducing preferencesfor decision-making problems of high uncertainty interactivemechanisms should also be studied to extract preferences Inaddition empirical studies of the proposed approach can becarried out to various areas such as investment evaluationproject management and vehicle selection in fleet operations

Appendix

A Basic Notions for IVDHFS and UnbalancedLinguistic Term Set

Definition A1 (see [47]) Letting 119883 be a fixed set then anIVDHFS on 119883 is defined as119863 = ⟨119909 ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (A1)

where ℎ(119909) = ⋃[120583119871120583119880]isinℎ(119909)120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and119892(119909) = ⋃[]119871]119880]isin119892(119909)] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two setsof interval values in [0 1] denoting possible membershipand non-membership degrees of element 119909 isin 119883 to theset 119863 respectively with conditions 120583 ] isin [0 1] and0 le (120583119880)+ + (]119880)+ le 1 and for all 119909 isin 119883 (120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880

For convenience normally 119889 = ℎ 119892 is called an interval-valued dual hesitant fuzzy element (IVDHFE) and 119863 is theset of all IVDHFEs

Definition A2 (see [107]) Suppose 119878 = 119904119895 | 119895 = 0 1 119892 minus1 is a finite and totally ordered discrete linguistic term setwhere 119904119895 represent possible values for a linguistic variableand 119892 is an odd cardinality 119878 is uniformly and symmetricallydistributed if the following two conditions are satisfied (1)

Complexity 25

there exists a unique constant 120582 gt 0 such that 119873119878(119904119894) minus119873119878(119904119895) = 119897(119894 minus 119895) for all 119894 119895 = 0 1 119892 minus 1 (2) Let 119878119877 =119904 | 119904 isin 119878 119904 gt 119878119862 and 119878119871 = 119904 | 119904 isin 119878 119904 lt 119878119862 Let C(119878119877) andC(119878119871) be the cardinality of 119878119877 and 119878119871 then C(119878119877) = C(119878119871)If 119878 is uniformly and symmetrically distributed then 119878 iscalled a balanced linguistic term set Otherwise 119878 is calledan unbalanced linguistic term set

The following 2-tuple fuzzy linguistic representationmodel extends traditional linguistic term set to a continuouscase so as to facilitate computing with linguistic variables

Definition A3 (see [59]) Let 119878 = 1199040 1199041 119904119892minus1 be alinguistic term set and 120573 isin [0 119892] Then a 2-tuple fuzzylinguistic variable expresses the equivalent information to 120573is defined as

Δ [0 119892] 997888rarr 119878 times [minus05 05) (A2)

Δ (120573) = (119904119894 120572) with

119904119894 119894 = 119903119900119906119899119889 (120573)120572 = 120573 minus 119894 120572 isin [minus05 05)

(A3)

Δminus1 (119904119894 120572) = 119894 + 120572 = 120573 (A4)

where 119903119900119906119899119889() is the usual rounding operation and120572 is calledsymbolic translation

To obtain 2-tuple fuzzy linguistic representations ofunbalanced linguistic terms the concept of linguistic hierar-chies ie 119871119867 = ⋃119905 119897(119905 119899(119905)) is used 119897(119905 119899(119905)) is a linguistichierarchy with 119905 indicating the level of hierarchy and 119899(119905)denotes the granularity of the linguistic term set of 119905 Herreraet al [59] defined the following transformation functionsbetween labels from different levels in multigranular linguis-tic information contexts without loss of information

Definition A4 (see [59]) In linguistic hierarchies 119871119867 =⋃119905 119897(119905 119899(119905)) whose linguistic term sets are represented by119878119899(119905) = 119904119899(119905)0 119904119899(119905)119899(119905)minus1 the transformation function froma linguistic label in level 119905 to a label in consecutive level 1199051015840 isdefined as 1198791198651199051199051015840 119897(119905 119899(119905)) 997888rarr 119897(1199051015840 119899(1199051015840)) such that

1198791198651199051199051015840 (119904119899(119905)119894 120572119899(119905))= Δ 1199051015840 (Δminus1119905 (119904119899(119905)119894 120572119899(119905)) (119899 (1199051015840) minus 1)119899 (119905) minus 1 ) (A5)

By use of the above transformation function any 2-tuplelinguistic representation can be transformed into a term in119871119867 Detailed transformation procedures are listed as follows

(1) Representation in linguistic hierarchy To transformthe unbalanced terms of 119878 into the corresponding terms inthe 119871119867 transformation function 120595 is employed to associateeach unbalanced linguistic 2-tuple (119904119894 120572) with its linguistic 2-tuple in 119871119867(119878) ie

120595 119878 997888rarr 119871119867(119878) (A6)

so that 120595(119904119894 120572) = (119904119866(119894)119868(119894) 120582) for forall(119904119894 120572) isin 119878(2) Computational phase Firstly transform (119904119866(119894)119868(119894) 120582) intolinguistic 2-tuples denoted as (119904119899(1199051015840)

1198681015840(119894) 1205821015840) in 119878119899(1199051015840) where

(119904119899(1199051015840)1198681015840(119894)

1205821015840) = 119879119865 (119904119866(119894)119868(119894) 120582)= Δ(Δminus1 (119904119866(119894)119868(119894) 120582) sdot (119899 (1199051015840) minus 1)119866 (119894) minus 1 ) (A7)

Then a computational model is used over 119878119899(1199051015840) with aresult denoted as (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)(3) Retranslation process The result (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)is transformed into the unbalanced term in 119878 by using thetransformation function 120595minus1 ie

120595minus1 119871119867 (119878) 997888rarr 119878 (A8)

so that 120595minus1(119904119899(1199051015840)119903 120582119903) = (119904119903119890119904119906119897119905 120582119903119890119904119906119897119905) isin 119878B Details of Theorem 14

Proof Given a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector for119904119889119895 and 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 120582 gt 0 then(1) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLWA and GIVDHFUUBLWG operators respectively(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLOWA and GIVDHFUUBLOWG operators respectively

(2) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to theIVDHF UUBL weighted average (IVDHFUUBLWA)and IVDHF UUBL weighted geometric (IVDHFUUBLWG)operators respectively where

26 Complexity

119868119881119863119867119865119880119880119861119871119882119860120596 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

120596119895119904119889119895= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus

119899prod119895=1

(1 minus 120583119871119895 )120596119895 1 minus 119899prod119895=1

(1 minus 120583119880119895 )120596119895]] [[

119899prod119895=1

(]119871119895)120596119895 119899prod119895=1

(]119880119895 )120596119895]])

(B1)

119868119881119863119867119865119880119880119861119871119882119866120596 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889119895120596119895

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[

119899prod119895=1

(120583119871119895 )120596119895 119899prod119895=1

(120583119880119895 )120596119895]] [[1 minus

119899prod119895=1

(1 minus ]119871119895)120596119895 1 minus 119899prod119895=1

(1 minus ]119880119895 )120596119895]])

(B2)

(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHA andGIVDHFUUBLHG operators reduce to the IVDHF UUBL

orderedweighted average (IVDHFUUBLOWA) and IVDHFUUBL ordered weighted geometric (IVDHFUUBLOWG)operators respectively where

119868119881119863119867119865119880119880119861119871119874119882119860119908 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

119908119895119904119889120590(119895)= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) )))

119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) )))

]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus

119899prod119895=1

(1 minus 120583119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus 120583119880120590(119895))119908119895]] [[

119899prod119895=1

(]119871120590(119895))119908119895 119899prod119895=1

(]119880120590(119895))119908119895]])

(B3)

119868119881119863119867119865119880119880119861119871119874119882119866119908 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889120590(119895)119908119895

= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[

119899prod119895=1

(120583119871120590(119895))119908119895 119899prod119895=1

(120583119880120590(119895))119908119895]] [[1 minus

119899prod119895=1

(1 minus ]119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus ]119880120590(119895))119908119895]])

(B4)

Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

References

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[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

[11] X Qi C Liang and J Zhang ldquoGeneralized cross-entropy basedgroup decision making with unknown expert and attributeweights under interval-valued intuitionistic fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 79 pp 52ndash642015

[12] H Zimmermann Fuzzy Set Theorymdashand Its ApplicationsKluwer Academic Boston Mass USA 2001

[13] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 pp 199ndash249 1975

[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

[16] B T Nuong K-W Kim L Prathumratana et al ldquoSustainabledevelopment in the mining sector and its evaluation usingfuzzy AHP (Analytic Hierarchy Process) approachrdquo GeosystemEngineering vol 14 no 1 pp 43ndash50 2011

[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

[18] E K Zavadskas Z Turskis and J Antucheviciene ldquoSelectinga contractor by using a novel method for multiple attributeanalysis Weighted aggregated sum product assessment withgrey values (WASPAS-G)rdquo Studies in Informatics and Controlvol 24 no 2 pp 141ndash150 2015

[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

[21] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

[25] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009

[26] Z Hatush and M Skitmore ldquoContractor Selection UsingMulticriteria Utility Theory An Additive Modelrdquo Building andEnvironment vol 33 no 2-3 pp 105ndash115 1998

[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

[30] AM Anvuur andMM Kumaraswamy ldquoConceptualmodel ofpartnering and alliancingrdquo Journal of Construction Engineeringand Management vol 133 no 3 pp 225ndash234 2007

[31] A H I Lee H Kang C F Hsu and H Hung ldquoA green supplierselection model for high-tech industryrdquo Expert Systems withApplications vol 36 no 4 pp 7917ndash7927 2009

[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

[33] P Humphreys R McIvor and F Chan ldquoUsing case-basedreasoning to evaluate supplier environmental managementperformancerdquo Expert Systems with Applications vol 25 no 2pp 141ndash153 2003

[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

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[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

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[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

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[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

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[64] S K Patil and R Kant ldquoA hybrid approach based on fuzzyDEMATEL and FMCDM to predict success of knowledge man-agement adoption in supply chainrdquoApplied Soft Computing vol18 pp 126ndash135 2014

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[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

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performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

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[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

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4 Complexity

on factors that affect contractorsrsquo cooperation under inter-national construction joint ventures in construction projectsHwang et al [27] conducted a survey and reported lsquosharingof project risksrsquo as the top attractive factor and lsquodifferences inculture and working stylersquo as the top negative factor amongothers Furthermore aiming at finding risks that most affectcontractorsrsquo cooperation within project networks Hwangand Han [29] conducted a survey from the viewpoint ofcontractors and sub-contractors in Singapore constructionenvironment they identified ten top critical network riskssuch as lsquodifferent cultural normsrsquo lsquoinaccurate informationdeliveryrsquo lsquooccurrence of disputersquo and lsquolack of risk manage-ment knowledgersquo Obviously Hwang et al [27] and Hwangand Han [29] contributed a feasible way to deduce morereasonable evaluative attributes on contractorrsquos cooperationcapability however to our best knowledge there is stilla lack of thorough investigation on a consensus set ofevaluative attributes on contractorrsquos cooperation capabilityfor referencing Therefore in the light of Liang et al [8]Hwang et al [27] and Hwang and Han [29] we here takelsquoCooperation management capabilityrsquo as one of the mainevaluative attributes to assess contractors

To reduce the significant environmental footprint [6]internal and external factors (such as government regulationsand managerial concerns) are driving the entire construc-tion industry to enforce green practices all over the world[6] Green practices can be considered as an outcome ofstrategic processes through cooperation within the nexus ofcontractors [75] Therefore it is intrinsically crucial to takeinto account evaluative attributes on green practices in con-tractor selection especially for those nearly ever-lasting megainfrastructure construction projects Since environmentalregulations of different scales (ie domestic governmentaland international [76]) have increasingly become compulsoryin various industrial markets contractors should have thecapability of manage and keep track of the compliance [35]And the widely accepted way [77] to attain the compliancecapability by building environment management systems[32 34 78] whose most-cited components [77] compriseof ISO-14001 certification [32 34 36 37] eco-labeling [3132 37] environment policies [33] environment planning[33] and environmental management information system(continuous monitoring and regulatory compliance) [31 32]Due to the reason that environmental impacts occur at everystage of the construction cycle Rwelamila et al [79] stronglysuggested to contractors should implement green design ampprocurement to improve their green practices Construc-tion life-cycle analysis is critical in both green design forenvironment [33] and pollution reduction through greenprocurement [31 32 34] Measurements of green design forenvironment include tracking all material and reverse flowof a project from the retrieval of raw materials out of theenvironment to the disposal of the product back into theenvironment generally including recycle [31] reuse [32 33]remanufacture [32 33] disassembly and disposal [33] In thegreen procurement aspect Tan et al [68] suggested to reduceenvironmental footprints throughout the whole constructionsupply chain by addressing issues such as waste reduction [3234] environment material substitution [31 33] hazardous

material minimization [35] and clean technology availability[33] Furthermore from the stance of strategic manage-ment Fergusson and Langford [80] pointed out environ-mental performance of green practices positively contributesto comprehensive competitive advantages of contractorsEnvironmental performance has thus been recognized asone of the indispensible evaluative attributes to contractorselection [34 78] Sharma and Vredenburg [81] referred toenvironmental performance as the environmental effects thatcorporationrsquos activities have on the natural milieu To makeenvironmental performance more measurable many effortshave been paid to establish effective assessing approachessuch as the approach to examination on waste flows andcontrol on construction sites [3 82] As for indicators ofenvironmental performance according to the thorough liter-ature review by Govindan et al [77] the commonly adoptedones include [32ndash34 36] solid waste chemical waste airemission waste water disposal and energy In sum we hereinclude lsquoDesign amp procurementrsquo lsquoCompliance with greenlegislationrsquo and lsquoEnvironmental performancersquo as three othermain evaluative attributes to comprehensively assess greenpractices of alternative contractors

For more clarity based on the above literature review welisted all eight main evaluative attributes and present theirdefinitionsprinciples And the optional subdimensions ofthese main attributes have been also listed in Table 1 forvarious comprehensive considerations according to specificnature in practical problems

(A1) Quotation Project value of reviewed The lowesttender price tends to attract a clientrsquos interest as superior toother criteria

(A2) Credibility Comprehensive evaluation on trustwor-thiness of contractors in developmental dynamics and risksfrom both internal and external environments focusing onfinancial soundness health and safety past performancework experience etc

(A3) Technical strength Comprehensive evaluation oncontractorsrsquo capability of technology and innovation thattackling forthcoming complicate tasks generally based onquality rank technical ability experience and knowledge ofoperation team creativity amp innovation etc

(A4) Resource strength Comprehensive evaluation oncontractorsrsquo competitive strength on productive resourcesthat indispensable for construction needs generally based ontechnical human resource current workload constructionmachinery amp equipment fixed assets amp liquidity etc

(A5) Cooperation management capacity Comprehensiveevaluation on contractorsrsquo cooperation practices with partici-pators in projects focusing on clientcontractor relationshiporganizational structure and operations compatible culturecommunication and information delivery knowledge shar-ing of risk management among others

(A6) Design amp procurement Comprehensive evaluationon contractorsrsquo practices that improve projectrsquos whole lifevalue by using green design environmental friendly mate-rials and green production processes which promote bestpractices of green construction procurement throughout thesupply chain

Complexity 5

Table1Mainevaluativ

eattributesandtheiro

ptionalsub

dimensio

nsforg

reen

contractor

selection

No

Attributes

Sub-dimensio

nsRe

ferences

A1

Quo

tatio

n(i)

Tend

erprice(

advancep

aymentcapitalbidrou

tine

maintenance

andmajor

repairs

)[81824]

Liangetal[8]Fon

gandCh

oi[24]Z

avadskasetal[18]

A2

Credibility

(i)pastperfo

rmance

[101824]

(ii)w

orkexperie

nce[25]

(iii)repu

tatio

nampcreditratin

g[1025ndash27]

(iv)h

ealth

ampsafety

[10182426]

(v)fi

nancialstability[1024ndash2628]

Nieto-M

orotea

ndRu

z-Vila[10]Z

avadskasetal[18]H

atushandSkitm

ore

[26]Fon

gandCh

oi[24]Shenetal[28]Darvishetal[2527]

A3

Technical

strength

(i)technicalability[810182628]

(ii)e

xperienceamp

know

ledgeo

foperatio

nteam

[1025]

(iii)qu

ality

rank

[25]

(iv)c

reativity

ampinno

vatio

n[25]

Liangetal[8]N

ieto-M

orotea

ndRu

z-Vila[10]Z

avadskasetal[18]H

atush

andSkitm

ore[

26]Sh

enetal[28]D

arvishetal[25]

A4

Resource

strength

(i)technicalhum

anresource

[2427]

(ii)c

urrent

workload

[24]

(iii)constructio

nmachinery

ampequipm

ent[2527]

(iv)fi

xedassetsampliq

uidity

[27]

Fong

andCh

oi[24]D

arvishetal[25]H

wangetal[27]

A5

Coo

peratio

nmanagem

ent

capability

(i)clientcon

tractorrelationship[102429]

(ii)o

rganizationalstructureampop

erations

[828]

(iii)compatib

lecultu

re[827]

(iv)c

ommun

ication[829]

(v)k

nowledges

harin

gof

riskmanagem

ent[2729]

(vi)inform

ationdelivery[29]

(vii)

contractmanagem

ent[8]

Liangetal[8]N

ieto-M

orotea

ndRu

z-Vila[10]H

atushandSkitm

ore[

26]

Fong

andCh

oi[24]Shenetal[28]Darvishetal[25]H

wangetal[27]

HwangandHan

[29]A

nvuu

rAaron

andKu

maraswam

yMoh

an[30]

A6

Designamp

Procurem

ent

(i)recycle[31]

(ii)reuse

[3233]

(iii)remanufacture[3233]

(iv)d

isassem

blyanddisposal[33]

(v)w

asteredu

ction[3234]

(vi)environm

entm

aterialsub

stitutio

n[3133]

(vii)

hazardou

smaterialm

inim

ization[35]

(viii)c

lean

techno

logy

availability[33]

Lin[35]H

umph

reysetal[34]C

hiou

etal[32]Leeetal[31]H

umph

reys

etal[33]

A7

Com

pliance

with

green

legisla

tion

(i)ISO-140

01certificatio

n[32343637]

(ii)e

co-la

belin

g[313237]

(iii)environm

entp

olicies[33]

(iv)e

nviro

nmentp

lann

ing[33]

(v)e

nviro

nmentalm

anagem

entinformationsyste

m[3132]

Hum

phreysetal[34]C

hiou

etal[32]YangandWu[36]Liu

andLiu[37]

Leeetal[31]Hum

phreysetal[33]

A8

Environm

ental

perfo

rmance

(i)solid

waste[333436]

(ii)c

hemicalwaste[333436]

(iii)aire

mission[333436]

(iv)w

astewater

disposal[32ndash3436]

(v)e

nergy[

3334]

Hum

phreysetal[34]C

hiou

etal[32]YangandWu[36]Hum

phreysetal

[33]

6 Complexity

(A7) Compliance with green legislation Comprehensiveevaluation on the extent to which contractorsrsquo practicessatisfy different governmental green or sustainability legis-lations according to the aspects of ISO-14001 certificationeco-labeling environment policies environment planningenvironmental management information system etc

(A8) Environmental performance Comprehensive evalu-ation on environmental effects that the corporationrsquos activitieshave on the natural milieu Environmental performance cancommonly bemeasured throughoperative performance indi-cators (ie energyresource utilization emission reductionand waste disposal)

22 Problem Definition The contractor selection as well asmany other multicriteria decisions impacting the overallproject should bemade during the front end planning stage ofa project the point at which a group of designated decision-makers have the power to accept or reject a contractorfor a specific project or its work packages [5 9] WhenusingMADMmechanism to copewith complicate contractorselection problems [5] normally few nominated contractorswill be ready for the decision-makers to vote on Due tocomplexity of the problems and limitedness of knowledgedecision-makers usually feel confident in expressing theiropinions by use of interval numbers [83] or uncertain linguis-tic terms [52] Although widely-accepted majority rule willquickly help the group of decision-makers arrive at a decisionon an uncertain linguistic term (eg [s4 s5]) to whichobviously there exists decision hesitancy because differentopinions existTherefore in this paper we define the interval-valued dual hesitant fuzzy uncertain unbalanced linguisticset (IVDHF UUBLS) to help the decision-making panel elicittheir assessments more objectively and completely

From another point of view various attitudinal charac-ters (degree of orness) commonly exist because individualexpert holds specific backgrounds and decision processthat involves the attitudinal character of group decision-makers must coordinate those various attitudinal charactersinto one complex attitudinal character [61] Therefore weadopt the concept of order-inducing vector [62] to reflectgroup complex attitudinal character and develop a TOPSIS-based method to rationally determine the order-inducingvector Besides as Tan and Chen [84] pointed out forreal-world sophisticated MADM problems the indepen-dency axiom [85] cannot generally satisfied For exampleupgrading in green performance will raise the quotationand intrinsically result in requirements for high-standardcollaboration between contractors and general constructorIn viewing of this common phenomenon we take attributesrsquointerdependency as a third indispensible characteristic intackling complexity in green contractor selection in megainfrastructure projects In sum we take three characteristicsof complexity to model the practical problems of complicategreen contractor selection ie (i) compound structure ofhesitant fuzzy linguistic assessments (ii) group attitudinalcharacters and (iii) attributesrsquo interdependency To producegreater clarity Figure 1 demonstrates the conceptual MADMmodel for green contractor selection

Now we can give the symbolized description of thetargeted complex green contractor problems Given a megainfrastructure project there are a set of alternative greencontractors ie 119883 = 1199091 1199092 119909119899 for its subprojectsLet 119860 = 1198601 1198602 119860119898 be the evaluative attributesaccording to which decision-makers consider each greencontractor Due to high complexity in the sophisticatedproblem scenarios there exists interdependency relationsamong the evaluative attribute A panel of decision-makers119864 = 1198641 1198642 119864119905 have been organized to give theirassessments to each alternative contractor 119909119894 (119894 = 1 119899)under every attribute 119860119895 (119895 = 1 119898) In order to reflect thecomplicate group assessments of all alternative contactorsthe hybrid expression tool of IVDHF UUBLS that will bedetailed in Section 3 is adopted to depict the assessmentsmore effectively and comprehensively As a result a specificdecision matrix 119877 = (119903119894119895)119898times119899 whose elements are in theform of IVDHF UUBLS is obtained According to Merigoand Casanovas [61] suppose that an order-inducing vec-tor 120576 for denoting group attitudinal characters has beenreasonably obtained Then effective MADM approachesmust be developed to determine the most appropriate greencontractor(s)

3 Interval-Valued Dual Hesitant FuzzyUncertain Unbalanced Linguistic Set

As demonstrated in Figure 1 after the panel of decision-makers votes on an alternative contractor under certainattribute by use of specific uncertain unbalanced linguisticterm set the uncertain linguistic term [s4 s5] stands outbecause of the majority rule while different opinions shouldalso be included and considered inMADMprocesses To thatend based on interval-valued dual hesitant fuzzy set (IVD-HFS) [47] and the unbalanced linguistic term set (ULTS)[59] we here first introduce an interval-valued dual hesitantfuzzy uncertain unbalanced linguistic set (IVDHF UUBLS)which incorporate different opinions of decision-makersas membership degrees or nonmembership degrees to themajority-voted [s4 s5] Then we develop operational rules aswell as distance measure for the IVDHF UUBLS Regardingdefinitions of the IVDHFS and ULTS one can refer toAppendix A

31 Definition of IVDHF UUBLS

Definition 1 Let119883 be a fixed set and 119878 be a finite and continu-ous unbalanced linguistic label set Then an IVDHF UUBLS119878119863 on119883 is defined as

119878119863 = ⟨119909 119904120599(119909) ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (1)

where 119904120579(119909) = [119904120572 119904120573] represents judgment to object 119909119904120572 and 119904120573 are two unbalanced linguistic variables frompredefined unbalanced linguistic label set 119878 which repre-sents decision-makersrsquo judgments to an evaluated object119883 ℎ(119909) = ⋃120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and 119892(119909) =⋃] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two sets of closed intervals

Complexity 7

e panelof decision

makers Alternativegreen

contractors

Evaluativeattributes

Comprehensiveevaluation

Attributesrsquo interdependency

+

alternativecontractor

vote

Voting results on

MajorityRule

radic

Specificdecision

matrix

AppropriateMADM

approach

Decisionresults

Section 3 Defines an effectivehybrid expression tool

Group attitudinal characters (ie complex degrees of orness)

s

⟨Contractori [s4 s5] ℎ (x) g(x)⟩

i-th

1 2 3 4 5 6 7

j-th attribute

Figure 1 Conceptual MADMmodel for tackling complexity in green contractor selection

in [0 1] ℎ(119909) denotes possible membership degrees that 119909belongs to 119904120579(119909) and119892(119909) represents possible nonmembershipdegrees of 119909 to 119904120579(119909) ℎ(119909) and 119892(119909) hold conditions 120583 ] isin[0 1] and 0 le (120583119880)+ + (]119880)+ le 1 where(120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880 for all 119909 isin 119883When119883 = 1199091 1199092 119909119899has only one element 119878119863 reduces to (s120599 ℎ 119892) which is calledan interval-valued dual hesitant fuzzy uncertain unbalancedlinguistic (IVDHF UUBL) number (IVDHF UUBLN)

32 Operational Rules for IVDHF UUBLS On the strength ofoperational rules for uncertain linguistic set [52] unbalancedlinguistic set [59] and interval-valued dual hesitant fuzzy set[47] we get the following operations for IVDHF UUBLS

Definition 2 Let 119904119889 = (119904120599 ℎ 119892) = ([119904120572 119904120573] ℎ 119892) 1199041198891 =(1199041205991 ℎ1 1198921) = ([1199041205721 1199041205731] ℎ1 1198921) and 1199041198892 = (1199041205992 ℎ2 1198922) =([1199041205722 1199041205732] ℎ2 1198922) be any three IVDHF UUBLNs 120582 isin [0 1]operations on these IVDHF UUBLNs are defined as

(1) 120582119904119889 = ⋃(119904120599ℎ119892)isin119904119889

([119904120582Δminus11199050 (119879119865

11990511989611199050(120595(119904120572)))

119904120582Δminus11199050 (119879119865

11990511989621199050(120595(119904120573)))

] ⋃

[120583119871120583119880]isinℎ[]119871]119880]isin119892

[1 minus (1 minus 120583119871)120582 1 minus (1 minus 120583119880)120582] [(]119871)120582 (]119880)120582]) (2) 119904119889120582 = ⋃

(119904120599 ℎ119892)isin119904119889

([119904(Δminus11199050 (119879119865

11990511989611199050(120595(119904120572))))

120582 119904(Δminus11199050 (119879119865

11990511989621199050(120595(119904120573))))

120582])

⋃[120583119871120583119880]isinℎ[]119871]119880]isin119892

[(120583119871)120582 (120583119880)120582] [1 minus (1 minus ]119871)120582 1 minus (1 minus ]119880)120582] )(3) 1199041198891 oplus 1199041198892 = ⋃

(1199041205991 ℎ11198921)isin1199041198891(1199041205992 ℎ21198922)isin1199041198892

([119904Δminus11199050 (119879119865

11990511989411199050(120595(1199041205721 )))+Δ

minus11199050(11987911986511990511989511199050(120595(1199041205722 )))

119904Δminus11199050 (119879119865

11990511989421199050(120595(1199041205731 )))+Δ

minus11199050(11987911986511990511989521199050(120595(1199041205732 )))

] ⋃

[1205831198711 1205831198801 ]isinℎ1[120583

1198712 1205831198802 ]isinℎ2[]

1198711 ]1198801 ]isin1198921[]

1198712 ]1198802 ]isin1198922

[1205831198711 + 1205831198712 minus 12058311987111205831198712 1205831198801 + 1205831198802 minus 1205831198801 1205831198802 ] []1198711]1198712 ]1198801 ]1198802 ]) (4) 1199041198891 otimes 1199041198892 = ⋃

(1199041205991 ℎ11198921)isin1199041198891(1199041205992 ℎ21198922)isin1199041198892

([119904Δminus11199050 (119879119865

11990511989411199050(120595(1199041205721 )))timesΔ

minus11199050(11987911986511990511989511199050(120595(1199041205722 )))

119904Δminus11199050 (119879119865

11990511989421199050(120595(1199041205731 )))timesΔ

minus11199050(11987911986511990511989521199050(120595(1199041205732 )))

] ⋃

[1205831198711 1205831198801 ]isinℎ1[120583

1198712 1205831198802 ]isinℎ2[]

1198711 ]1198801 ]isin1198921[]

1198712 ]1198802 ]isin1198922

[12058311987111205831198712 1205831198801 1205831198802 ] []1198711 + ]1198712 minus ]1198711]1198712 ]1198801 + ]1198802 minus ]1198801 ]

1198802 ])

(2)

8 Complexity

In above details about the transformation functionsof 119879119865 and Δ linguistic hierarchies (119871119867) as well as thetransformation procedures for unbalanced linguistic termsets are shown in Appendix A

eorem 3 Letting 119904119889 = (119904120599 ℎ 119892) 1199041198891 = (1199041205991 ℎ1 1198921) and1199041198892 = (1199041205992 ℎ2 1198922) be any three IVDHF UUBLNs 120582 1205821 1205822 isin[0 1]then following properties are true

(1) 1199041198891 oplus 1199041198892 = 1199041198892 oplus 1199041198891 (2) 1199041198891 otimes 1199041198892 = 1199041198892 otimes 1199041198891 (3)120582(1199041198891 oplus 1199041198892) = 1205821199041198891 oplus 1205821199041198892(4) 1199041198891120582 otimes 1199041198892120582 = (1199041198891 otimes 1199041198892)120582 (5) 1205821119904119889 oplus 1205822119904119889 = (1205821 +1205822)119904119889 (6) 1199041198891205821 otimes 1199041198891205822 = 1199041198891205821+1205822

Proof Omitted

In Definition 2 and Theorem 3 1199051198961 1199051198962 1199051198941 1199051198942 1199051198951 and1199051198952 are corresponding levels of 119904120572 119904120573 1199041205721 1199041205731 1199041205722 and 1199041205722in 119871119867 respectively 1199050 is the maximum level of 119904120572 119904120573 1199041205721 1199041205731 1199041205722 and 1199041205722 in 119871119867 Furthermore to compare any twoIVDHF UUBLNs we also have following definitions

Definition 4 Let 119904119889 = (119904120599 ℎ 119892) be an IVDHF UUBLN119904120599 = [119904120572 119904120573] then score function 119878(119904119889) and accuracy function119875(119904119889) can be represented by

119878 (119904119889) = Δminus11199050 (11987911986511990511989611199050 (120595 (119904120572))) + Δminus11199050 (11987911986511990511989621199050 (120595 (119904120573)))2 times 12 ( 1119897 (ℎ) sum

[120583119871120583119880]isinℎ

120583119871 minus 1119897 (119892) sum[]119871]119880]isin119892

]119871

+ 1119897 (ℎ) sum[120583119871120583119880]isinℎ

120583119880 minus 1119897 (119892) sum[]119871]119880]isin119892

]119880)(3)

119875 (119904119889)= Δminus11199050 (11987911986511990511989611199050 (120595 (119904120572))) + Δminus11199050 (11987911986511990511989621199050 (120595 (119904120573)))2

times 12 ( 1119897 (ℎ) sum[120583119871120583119880]isinℎ

120583119871 + 1119897 (119892) sum[]119871]119880]isin119892

]119871

+ 1119897 (ℎ) sum[120583119871120583119880]isinℎ

120583119880 + 1119897 (119892) sum[]119871]119880]isin119892

]119880)

(4)

where 119897(ℎ) and 119897(119892) are numbers of values in ℎ and 119892respectively 1199051198961 and 1199051198962 are the corresponding levels of 119904120572 and119904120573 in 119871119867 and 1199050 is the maximum level of 119905119896 in 119871119867

Definition 5 Given any IVDHF UUBLNs 1199041198891 = (1199041205991 ℎ1 1198921)1199041198892 = (1199041205992 ℎ2 1198922) then(1) If 119878(1199041198891) lt 119878(1199041198892) then 1199041198891 lt 1199041198892(2) If 119878(1199041198891) = 119878(1199041198892) then

(a) If 119875(1199041198891) = 119875(1199041198892) then 1199041198891 = 1199041198892(b) If 119875(1199041198891) lt 119875(1199041198892) then 1199041198891 lt 1199041198892

33 Distance Measure for IVDHF UUBLS When 119897(ℎ) or119897(119892) of two IVDHF UUBLNs are unequal the complement-ing method [86] is normally adopted to design distancemeasures Note that artificially adding values to shorterones in the complementing method will cause informationdistortion To avoid this limitation we provide the followingdistance measure

Definition 6 Let two IVDHF UUBLNs 1199041198891 = (1199041205991 ℎ1 1198921)and 1199041198892 = (1199041205992 ℎ2 1198922) where 1199041205991 = [1199041205721 1199041205731] 1199041205992 = [1199041205722 1199041205732]119897ℎ1 119897ℎ2 1198971198921 and 1198971198922 are lengths of ℎ1 ℎ2 1198921 and 1198922 respectivelydenoting number of elements in ℎ1 ℎ2 1198921 and 1198922 Suppose1198681 = (1(119899(1199051198941) minus 1))Δminus11199050 (11987911986511990511989411199050 (120595(1199041205721))) 1198682 = (1(119899(1199051198942) minus1))Δminus11199050 (11987911986511990511989421199050 (120595(1199041205731))) 1198683 = (1(119899(1199051198951)minus1))Δminus11199050 (11987911986511990511989511199050 (120595(1199041205722)))1198684 = (1(119899(1199051198952) minus 1))Δminus11199050 (11987911986511990511989521199050 (120595(1199041205732))) where 1199051198941 1199051198942 1199051198951 and1199051198952 are the corresponding levels of unbalanced linguistic terms1199041205721 1199041205731 1199041205722 and 1199041205732 in the linguistic hierarchy 119871119867 and 1199050 is themaximum level of 1199041205721 1199041205731 1199041205722 and 1199041205732 in 119871119867 Then by use ofnormalized Euclidean distance we define a distance measure119889 for IVDHF UUBLNs as follows

Situation 7 When 119897ℎ1 = 119897ℎ2 = 1198971 and 1198971198921 = 1198971198922 = 1198972 then119889 (1199041198891 1199041198892) = (12 ( 11198971sdot 1198971sum119896=1

(10038161003816100381610038161003816100381610038161198681120583119871119895ℎ1 minus 1198683120583119871119896ℎ2 10038161003816100381610038161003816100381610038162 + 10038161003816100381610038161003816100381610038161198682120583119880119895ℎ1 minus 1198684120583119880119896ℎ2 10038161003816100381610038161003816100381610038162) + 11198972sdot 1198972sum119896=1

(1003816100381610038161003816100381610038161198681]1198711198951198921 minus 1198683]1198711198961198922 1003816100381610038161003816100381610038162 + 1003816100381610038161003816100381610038161198682]1198801198951198921 minus 1198684]1198801198961198922 1003816100381610038161003816100381610038162)))12

(5)

Situation 8 When 119897ℎ1 = 119897ℎ2 or 1198971198921 = 1198971198922 then119889 (1199041198891 1199041198892) = (12 ( 1119897ℎ1 119897ℎ2sdot 119897ℎ1sum119895=1

119897ℎ2sum119896=1

(10038161003816100381610038161003816100381610038161198681120583119871119895ℎ1 minus 1198683120583119871119896ℎ2 10038161003816100381610038161003816100381610038162 + 10038161003816100381610038161003816100381610038161198682120583119880119895ℎ1 minus 1198684120583119880119896ℎ2 10038161003816100381610038161003816100381610038162)+ 11198971198921 1198971198922sdot 1198971198921sum119895=1

1198971198922sum119896=1

(1003816100381610038161003816100381610038161198681]1198711198951198921 minus 1198683]1198711198961198922 1003816100381610038161003816100381610038162 + 1003816100381610038161003816100381610038161198682]1198801198951198921 minus 1198684]1198801198961198922 1003816100381610038161003816100381610038162))12

(6)

eorem 9 The distance measure 119889 defined in Definition 6satisfies following properties

(1) 0 le 119889(1199041198891 1199041198892) le 1

Complexity 9

(2) 119889(1199041198891 1199041198892) = 0 if and only if 1199041198891 and 1199041198892 are perfectlyconsistent

(3) 119889(1199041198891 1199041198892) = 119889(1199041198892 1199041198891)4 Generalized Aggregation Operators forIVDHF_UUBLS

41 Definitions of Operators Based on the generalized oper-ators firstly introduced by Yager [87] we here develop some

fundamental generalized aggregation operators for the newlydefined IVDHF UUBLS

Definition 10 Given a collection of IVDHF UUBLNs 119904119889119895(119895 =1 119899) their weighting vector 120596 = (1205961 1205962 120596119899)119879120596119895 isin[0 1]sum119899119895=1 120596119895 = 1 120582 be a parameter 120582 isin (0 +infin)(1) Generalized IVDHFUUBLWeighted Average(GIVDHFUUBLWA) Operator

119866119868119881119863119867119865119880119880119861119871119882119860120596120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(120596119895119904119889119895120582))1120582

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120582)1120582 119904(sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120582)1120582]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[(1 minus

119899prod119895=1

(1 minus (120583119871119895 )120582)120596119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880119895 )120582)120596119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871119895)120582)120596119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880119895 )120582)120596119895)1120582]])

(7)

(2) Generalized IVDHFUUBLWeighted Geometric(GIVDHFUUBLWG)Operator

119866119868119881119863119867119865119880119880119861119871119882119866120596120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889119895)120596119895)= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871119895 )120582)120596119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880119895 )120582)120596119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871119895 )120582)120596119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880119895 )120582)120596119895)1120582]])

(8)

Definition 11 For a collection of IVDHF UUBLNs 119904119889119895(119895 = 1 119899) 119904119889120590(119895) be the 119895th largest 119908 = (1199081 1199082 119908119899)119879 be the aggregation-associated weighting vector 119908119895 isin [0 1]sum119899119895=1119908119895= 1 120582 is a parameter such that 120582 isin (0 +infin) 119878119899 997888rarr 119878 Then

10 Complexity

(1) Generalized IVDHFUUBL Ordered Weighted Average(GIVDHFUUBLOWA) Operator

119866119868119881119863119867119865119880119880119861119871119874119882119860120596120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(119908119895119904119889120590(119895)120582))1120582

= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

120582)1120582]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895) []119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120590(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120590(119895))120582)119908119895)1120582]])

(9)

(2) Generalized IVDHFUUBL Ordered Weighted Geometric(GIVDHFUUBLOWG)Operator

119866119868119881119863119867119865119880119880119861119871119874119882119866119908120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889120590(119895))119908119895)= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120590(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120590(119895))120582)120596119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120590(119895))120582)120596119895)1120582]])

(10)

Definition 12 For a collection of IVDHF UUBLNs 119904119889119895(119895 =1 119899) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector120596119895 isin [0 1] sum119899119895=1 120596119895 = 1 119899 is a balancing coefficient119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weight-ing vector with 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1

(1) Generalized IVDHFUUBL Hybrid Average (GIVDHFU-UBLHA) Operator

119866119868119881119863119867119865119880119880119861119871119867119860120596119908120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(119908119895119904119889120590(119895)120582))1120582

= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895)

))))120582)1120582]

Complexity 11

⋃[120583119871120590(119895) 120583

119880120590(119895)]isinℎ120590(119895) []

119871120590(119895) ]119880120590(119895)]isin119892120590(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120590(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120590(119895))120582)119908119895)1120582]])

(11)

where 119904119889120590(119895) is the 119895th largest and

119904119889119895 = 119899120596119895119904119889119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904119899120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904119899120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

] ⋃

[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[1 minus (1 minus 120583119871119895 )119899120596119895 1 minus (1 minus 120583119880119895 )119899120596119895] [(]119871119895)119899120596119895 (]119880119895 )119899120596119895]) (12)

(2) Generalized IVDHFUUBL Hybrid Geometric(GIVDHFUUBLHG) Operator

119866119868119881119863119867119865119880119880119861119871119867119866119908120596120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889120590(119895))119908119895)= ⋃(119904120599120590(119895)

ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120590(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895)

))))119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895) []119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120590(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120590(119895))120582)119908119895)1120582]])

(13)

where 119904119889120590(119895) is the 119895th largest and

119904119889119895 = 119904119889119895119899120596119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(Δminus11199050 (119879119865

11990511989511199050(120595(119904120572119895 ))))

119899120596119895 119904(Δminus11199050 (119879119865

11990511989521199050(120595(119904120573119895 ))))

119899120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[(120583119871119895 )119899120596119895 (120583119880119895 )119899120596119895] [1 minus (1 minus ]119871119895 )119899120596119895 1 minus (1 minus ]119880119895 )119899120596119895]) (14)

12 Complexity

When confronted with ill-structured situations wheredecision-makerrsquos complex attitudinal characters need to beincluded order-inducing vectors provide an effective way[88ndash90]Thus we further define following induced operatorsfor IVDHF UUBLNs

Definition 13 For a collection of IVDHF UUBLNs 119904119889119895 (119895 =1 119899) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector

120596119895 isin [0 1] sum119899119895=1 120596119895 = 1 119899 is a balancing coefficient119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weight-ing vector 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1 120576119895 denote a set oforder inducing vectors

(1) Induced Generalized IVDHFUUBL Hybrid Average (I-GIVDHFUUBLHA) Operator

119868-119866119868119881119863119867119865119880119880119861119871119867119860120596119908120582 (lt 1205761 1199041198891 gt lt 1205762 1199041198892 gt lt 120576119899 119904119889119899 gt) = ( 119899⨁119895=1

(119908119895119904119889120587(119895)120582))1120582

= ⋃(119904120599120587(119895) ℎ120587(119895) 119892120587(119895))isin119904119889120587(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120587(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587(119895)

))))120582)1120582]

⋃[120583119871120587(119895) 120583

119880120587(119895)]isinℎ120587(119895) []

119871120587(119895) ]119880120587(119895)]isin119892120587(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120587(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120587(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120587(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120587(119895))120582)119908119895)1120582]])

(15)

where (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) is decreasing order of (11990411988911199041198892 119904119889119899) according to the order inducing vector 120576119895 and119904119889119895 = 119899120596119895119904119889119895 = ⋃

(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904119899120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904119899120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

] ⋃

[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[1 minus (1 minus 120583119871119895 )119899120596119895 1 minus (1 minus 120583119880119895 )119899120596119895] [(]119871119895)119899120596119895 (]119880119895 )119899120596119895]) (16)

(2) Induced Generalized IVDHFUUBL Hybrid Geometric(I-GIVDHFUUBLHG) Operator

119868-119866119868119881119863119867119865119880119880119861119871119867119866119908120596120582 (lt 1205761 1199041198891 gt lt 1205762 1199041198892 gt lt 120576119899 119904119889119899 gt) = 1120582 ( 119899⨂119895=1

(120582119904119889120587(119895))119908119895)= ⋃(119904120599120587(119895)

ℎ120587(119895) 119892120587(119895))isin119904119889120587(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120587(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587(119895)

))))119908119895]

⋃[120583119871120587(119895)120583119880120587(119895)]isinℎ120587(119895) []119871120587(119895) ]

119880120587(119895)]isin119892120587(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120587(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120587(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120587(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120587(119895))120582)119908119895)1120582]])

(17)

Complexity 13

Weighted Aggregation

Geometric Arithmetic

GIVDHFUUBLHG GIVDHFUUBLHA

GIVDHFUUBLWG GIVDHFUUBLOWG

IVDHFUUBLWG IVDHFUUBLOWG

GIVDHFUUBLWA GIVDHFUUBLOWA

IVDHFUUBLWA IVDHFUUBLOWA

= 1 w =1

n = 1 =

1

n = 1 =1

n = 1 =

1

n

= 1 = 1 = 1 = 1

Figure 2

where (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) is decreasing order of (11990411988911199041198892 119904119889119899) according to the order inducing variables 120576119895 and

119904119889119895 = 119904119889119895119899120596119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(Δminus11199050 (119879119865

11990511989511199050(120595(119904120572119895 ))))

119899120596119895 119904(Δminus11199050 (119879119865

11990511989521199050(120595(119904120573119895 ))))

119899120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[(120583119871119895 )119899120596119895 (120583119880119895 )119899120596119895] [1 minus (1 minus ]119871119895 )119899120596119895 1 minus (1 minus ]119880119895 )119899120596119895]) (18)

42 Properties of the Proposed GeneralizedAggregation Operators

eorem 14 With special values of 120582 119908 and 120596 the operatorsGIVDHFUUBLHA and GIVDHFUUBLHG can include aseries of traditional aggregation operators as special cases andtheir relationship can be depicted in Figure 2

Proof See Appendix B

As for the induced hybrid aggregation operators we alsohave following theorem

eorem 15 If (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) = (119904119889120590(1) 119904119889120590(2) 119904119889120590(119899)) I-GIVDHFUUBLHA reduces toGIVDHFUUBLHAIf (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) = (119904119889120590(1) 119904119889120590(2) 119904119889120590(119899)) thenI-GIVDHFUUBLHG reduces to the GIVDHFUUBLHG oper-ator

eorem 16 All the proposed generalized operatorsGIVDHFUUBLWAGIVDHFUUBLWGGIVDHFUUBLOWAGIVDHFUUBLOWGGIVDHFUUBLHAGIVDHFUUBLHGI-GIVDHFUUBLHA and I-GIVDHFUUBLHG hold thefollowing properties (1) Commutativity (2) Idempotency (3)Boundedness

Based on above theorems following properties can alsobe derived

eorem 17 For a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weight vector of 119904119889119895with 120596119895 isin [0 1] and sum119899119895=1 120596119895 = 1 120582 gt 0 then we have

(1) 119868119881119863119867119865119880119880119861119871119882119866120596(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119882119860120596(1199041198891 1199041198892 119904119889119899)(2) 119868119881119863119867119865119880119880119861119871119882119866120596(1199041198891 1199041198892 119904119889119899) le119866119868119881119863119867119865119880119880119861119871119882119860120596120582(1199041198891 1199041198892 119904119889119899)(3) 119866119868119881119863119867119865119880119880119861119871119882119866120596120582(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119882119860120596(1199041198891 1199041198892 119904119889119899)

Proof Omitted for concision

eorem 18 For a collection of IVDHF UUBLNs 119904119889119895 = (119904120599119895 ℎ119895 119892119895) 119908 = (1199081 1199082 119908119899)119879 is the weighting vector of 119904119889119895with 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1 120582 gt 0 Then

(1) 119868119881119863119867119865119880119880119861119871119874119882119866119908(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119874119882119860119908(1199041198891 1199041198892 119904119889119899)(2) 119868119881119863119867119865119880119880119861119871119874119882119866119908(1199041198891 1199041198892 119904119889119899) le119866119868119881119863119867119865119880119880119861119871119874119882119860119908120582(1199041198891 1199041198892 119904119889119899)(3) 119866119868119881119863119867119865119880119880119861119871119874119882119866119908120582(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119874119882119860119908(1199041198891 1199041198892 119904119889119899)

Proof Omitted for concision

14 Complexity

5 An Integrated MADM Approach forTackling Complex Green ContractorSelection Problems

As described in Section 2 we take the green contractorselection as a special type of complicate MADM problemsthat synthesizes three characteristics of decision hesitancy[21 22] attributes interdependency [84] and group atti-tudinal characters [61] Therefore in this section we con-struct an integrated MADM approach to tackle the com-plex green contractor selection problems Suppose 119883 =1199091 1199092 119909119898 is the set of alternative green contractors and119860 = 1198601 1198602 119860119899 is the set of evaluative attributes 120596 =(1205961 1205962 120596119899)119879 is weighting vector for the attributes 120596119894 ge 0sum119899119894=1 120596119894 = 1 Let = (119903119894119895)119899times119898 denote the decision matrix inwhich 119903119894119895 = ([119904120572119894119895 119904120573119894119895] ℎ119894119895 119892119894119895) is an IVDHF UUBLN givenby decision-makers for alternative contractor 119909119895 with respectto attribute 119860 119894 According to the mechanism of pair-wisecomparisons among attributes in the DEMATELmethod [6364] the interdependency among attributes can be obtained asamatrix119885 = (119911119894119896)119899times119899 where 119911119894119896 indicates the degree towhich119860 119894 affects119860119896 Subsequently based on the IVDHFUUBLS andits operations we now present detailed steps of our MADMapproach as shown in following Algorithm I

Algorithm I Hesitant fuzzy linguistic MADMwith attributesinterdependency and decision-makersrsquo group attitudinalcharacters

Step 1 Determine argument-dependent weighting vector120596119860119863 according to attribute values by programming modeldeveloped in the following Section 51

Step 2 Obtain the attribute-interdependences based weight-ing vector 120596119860119868 by use of DEMATEL method described in thefollowing Section 52

Step 3 Calculate synthesized attribute weighting vectoraccording to

120596 = 120572120596119860119868 + 120573120596119860119863 (19)

where 120572 and 120573 are parameters to reflect decision character-istics of decision organizations 0 le 120572 120573 le 1 120572 + 120573 =1Step 4 Check requirements for order inducing If no addi-tional order inducing required then go to Step 5 otherwisego to Step 6

Step 5 Utilize generalized aggregation operators to get theoverall IVDHF UUBLNs 119903119895(119895 = 1 119898) for each alternative119909119895 Here we take GIVDHFUUBLHA operator for examplebecause it can include other traditional operators as its specialcases Therefore we have

119866119868119881119863119867119865119880119880119861119871119867119860(1199031119895 1199032119895 119903119899119895) = ( 119899⨁119894=1

(119908119894 (119903120590119894119895)120582))1120582= ⋃(119904120590119894119895ℎ120590119894119895 119892120590119894119895

)isin119903120590119894119895

([119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590119894119895

))))120582)1120582 119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590119894119895

))))120582)1120582]

⋃[120583119871120590119894119895120583

119880120590119894119895]isinℎ120590119894119895[]

119871120590119894119895]119880120590119894119895]isin119892120590119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120590119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120590119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120590119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120590119894119895)120582)119908119894)1120582]])

(20)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of 119903119894119895119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weightvector 119903120590119894119895 is the 119894th largest of 119903119894119895 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))

119904119899120596119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119894119895))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (21)

Complexity 15

Then go to Step 8

Step 6 By use of the TOPSIS-based method developed in thefollowing Section 53 we determine order-inducing vectors 120576to reflect decision-makersrsquo group attitudinal characters

Step 7 Utilizing induced generalized aggregation operatorsto get the overall IVDHF UUBLNs 119903119895(119895 = 1 2 119898) foreach alternative 119909119895 we have

119868-119866119868119881119863119867119865119880119871119867119860(1199031119895 1199032119895 119903119899119898) = ( 119899⨁119894=1

(119908119894 (119903120587119894119895)120582))1120582

= ⋃(119904120587119894119895ℎ120587119894119895119892120587119894119895)isin119903120587119894119895

([119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120587119894119895

))))120582)1120582 119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587119894119895

))))120582)1120582]

⋃[120583119871120587119894119895 120583

119880120587119894119895]isinℎ120587119894119895 []

119871120587119894119895]119880120587119894119895]isin119892120587119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120587119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120587119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120587119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120587119894119895)120582)119908119894)1120582]])

(22)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of119903119894119895 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 119903120587119894119895 is the 119894th largest of 119903119894119895 reordered by theorder-inducing vectors 120576 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))119899120596119894120572119894119895

119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120573119894119895 ))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (23)

Step 8 Calculate scores 119878(119903119895) and accuracy degrees 119875(119903119895) for119903119895(119895 = 1 2 119898)Step 9 According to Definition 5 output ranking order ofalternative contractors

In Algorithm I to objectively derive unknown weight-ing vector for evaluative attributes a synthesized attributeweighting vector is devised by fusing two parts of concern(i) argument-dependent weighting vector and (ii) attribute-interdependences based weighting vector The former is toreflect effects of attribute values on attribute weights whilethe latter is to exploit effects of interdependences amongattributes on their weights For clarity processing flowchartof Algorithm I is depicted in Figure 3

51 Determining Argument-Dependent Attribute WeightingVector 120596119860119863 In order to derive attribute weighting vec-tor appropriately from assessments maximizing deviationmethod [91] is adopted here for its distinguishing ability

and objectivity [92] According to Wang [91] if performancevalues of each alternative have little differences under anattribute it implicates such an attribute plays a less importantrole otherwise it lives a more important role in decision-making Thus if one attribute has similar values across alter-natives it should be assigned a smaller weight contrariwiseit should be a bigger weight

In light of the idea in maximizing deviation methodwe develop a programming model to determine argument-dependent weighting vector 120596119860119863 For attribute 119860 119894 the stan-dard deviation of alternative 119909119895 to all the other alternatives isdenoted as

119865119894119895 (120596119860119863) = 119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (24)

Let119865119894(120596119860119863) represent deviation value of all alternatives toother alternatives according to attribute119860 119894 and be formulatedas follows

16 Complexity

Attribute weighting

Hesitant fuzzy linguistic MADM with attributes interdependency and decision makersrsquo group attitudinal characters

Step 9 Output ranking order of alternatives

interdependency among attributes by DEMATEL in Section 52

attributes values by standard deviation method in Section 51

Step 6 Ensure TOPSIS order inducing variables proposed in Section 53

Step 7 Utilizing proposed induced generalized hybrid aggregation operators developed in Section

41 to get overall value of each alternative

Step 8 Calculate score degrees and accuracy degrees

Step 4 Order inducingYes

Step 5 Choose proposed generalized operators in Section 41 to get overall value of each alternative

No

Collect decision matrix in the form of IVDHF_UUBLS

Complex problems of green contractor selection

Step 1 Weighting vector AD derived from Step 2 Weighting vector AI derived from

Step 3 Obtain synthesized weighting vector of attributes =AI + AD

Figure 3 Processing flowchart of Algorithm I

119865119894 (120596119860119863) = 119898sum119895=1

119865119894119895 (120596119860119863) = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (25)

Based on the analysis above we can get followingprogramming model (M-1) for obtaining optimal attributeweighting vector 120596119860119863

(119872 minus 1)max 119865 (120596119860119863) = 119899sum

119894=1119865119894 (120596119860119863) = 119899sum

119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894st 119899sum

119894=1(120596119860119863119894 )2 = 1 120596119860119863119894 ge 0 119894 = 1 2 119899 (26)

where 119889(119903119894119895 119903119894119897) can be calculated by Definition 6To solve model (M-1) we can construct Lagrange func-

tion

119871 (120596119860119863119894 120577) = 119899sum119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894+ 1205772 ( 119899sum

119894=1

(120596119860119863119894 )2 minus 1) (27)

where 120577 is the Lagrange multiplierDifferentiate with respect to 120596119860119863119894 (119894 = 1 2 119898) and 120577

and set these partial derivatives equal to zero the followingset of equations can be obtained

120597119871120597120596119860119863119894 = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) + 120577120596119860119863119894 = 0120597119871120597120577 = 119899sum

119894=1

(120596119860119863119894 )2 minus 1 = 0 (28)

By normalizing the solutions of above equations we canget a simple and exact formula for determining the attributeweights as follows

120596119860119863119894 = sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897)sum119899119894=1sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897) (29)

52 Deriving Attributes Interdependency-Based AttributeWeighting Vector 120596119860119868 Some efforts [53 60 84 93ndash95]

Complexity 17

have been made to address interdependencies betweenattributes in MADM However they unanimously cannotreveal the interdependences in a more complete way WhileDEMATAEL technique [63 96] gathers collective knowledgeto capture influences between attributes more precisely andcompletely so that weighting vector for attributes can berationally determined according to strength measure ofthose influences [64 97] Therefore in tackling the complexMADMwith interdependent attributes under IVDHF UUBLenvironments we present a DEMATEL-based method as thefollowing Algorithm II to derive attribute weighting vector120596119860119868 from the viewpoint of attributes interdependency

Algorithm II DEMATEL-basedmethod for ensuringweight-ing vector 120596119860119868Step 1 Construct initial direct-influence matrix 119885 = (119911119894119896)119899times119899

For attributes 119860 = 1198601 1198602 119860119899 decision-makersare asked to give direct effect that attribute 119860 119894 has onattribute 119860119896 using a five-point scale with scores representedby natural language (1) 0 mdash lsquoabsolutely no influencersquo (2)1 mdash lsquolow influencersquo (3) 2 mdash lsquomedium influencersquo (4) 3 mdashlsquohigh influencersquo (5) 4mdash lsquovery high influencersquoThen the initialdirect-influence matrix 119885 = (119911119894119896)119899times119899 can be established witheach element being the mean of same factors in differentdirect matrices

Step 2 Calculate normalized direct-influence matrix 119883 =(119909119894119896)119899times119899 Using matrix 119885 the normalized direct-influencematrix 119883 = (119909119894119896)119899times119899 is calculated according to

119909119894119896 = 119911119894119896119904 (30)

where 119904 = maxmax1le119894le119899sum119899119896=1 119911119894119896max1le119896le119899sum119899119895=1 119911119894119896 Allelements in matrix 119883 are complying with 0 le 119909119894119896 le 10 le sum119899119896=1 119909119894119896 le 1 and 0 le sum119899119894=1 119909119894119896 le 1 and at least onerow or column of the summation equals 1

Step 3 Derive total-influence matrix 119879 = (119905119894119896)119899times119899 Basedon matrix 119883 the total-influence matrix 119879 = (119905119894119896)119899times119899 can bederived by summing the direct effects and all of the indirecteffects

119879 = 119883 + 1198832 + 1198833 + sdot sdot sdot + 119883ℎ = 119883 (119868 minus 119883)minus1 (31)

where 119868 represents identity matrix When ℎ 997888rarr infin 119883ℎ =(0)119899times119899Step 4 Compute influencing and influenced degrees of eachattribute The sum of rows and the sum of columns withinthe total-influence matrix 119879 are respectively expressed as thevectors 119903 and 119888

119903 = (119903119894)119899times1 = ( 119899sum119896=1

119905119894119896)119899times1

119888 = (119888119896)119899times1 = ( 119899sum

119895=1

119905119894119896)119879

1times119899

(32)

where 119903119894 denotes the sumof the 119894th row inmatrix119879 and showsthe sum of direct and indirect effects that attribute 119860 119894 hason other attributes Similarly 119888119896 shows the sum of direct andindirect effects that attribute 119860119896 has receivedStep 5 Estimate weights of attributes based on influencedegrees according to

120596119868lowast119894 = [(119903119894 + 119888119894)2 + (119903119894 minus 119888119894)2]12 (33)

Then the weighting vector 120596119860119868 can be obtained by normaliz-ing (33)

120596119860119868119894 = 120596119868lowast119894sum119899119894=1 120596119868lowast119894 (34)

53 Obtaining Order Inducing Variables 120576 by TOPSIS-BasedMethod For ill-defined MADM problems induced aggre-gation operators [89 90 98] provide an effective channelfor experts to express complex attitudinal characters Thekey step in induced aggregation operators is to obtainorder-inducing vectors 120576 which however are presumptivelyassigned in most papers [62] Therefore in this sectionaiming at deducing the order-inducing vectors 120576 rationallywe here devise a TOPSIS-based method in Algorithm IIIObviously the Algorithm III is of generality for MADMwithdifferent forms of linguistic expression

Algorithm III TOPSIS-based method for obtaining orderinducing vectors 120576Step 1 Let the decision maker define IVDHF UUBL idealsolution(s) 119903+119894 and IVDHF UUBL negative ideal solution(s)119903minus119894 corresponding to each attribute in which

119903+119894 = (119903+1198941 119903+1198942 119903+119894119901) 119903minus119894 = (119903minus1198941 119903minus1198942 119903minus119894119902) (35)

where 119901 and 119902 denotes the number of 119903+119894 and 119903minus119894 respectively

Step 2 Calculate distances from each element in decisionmatrix to ideal solution(s) and negative solution(s) respec-tively

119889+119894119895 = max119897=12119901

119889 (119903119894119895 119903+119894119897 ) 119889minus119894119895 = min

119897=12119902119889 (119903119894119895 119903minus119894119897 ) (36)

where 119889(119903119894119895 119903+119894119897 ) and 119889(119903119894119895 119903minus119894119897 ) are calculated by the distancemeasure in Definition 6 If 119901 gt 1 we choose the onewith maximum distance if 119902 gt 1 we choose the one withminimum distance

Step 3 Calculate coefficients according to

119888119894119895 = 119889minus119894119895119889+119894119895 + 119889minus119894119895 119894 = 1 2 119899 119895 = 1 2 119898 (37)

18 Complexity

Table 2 Decision matrix in the form of IVDHF UUBLS1199091 1199092 11990931198601([AHVH] [02 03][02 04] [03 04]) ([HT] [04 05][05 06] [02 03] [02 04]) ([ALAH] [01 02][01 03] [06 07])1198602([MH] [05 06][02 03]) ([AHT] [06 07][01 02] ) ([MVH] [02 03][05 06] [06 07])1198603([VLAL] [03 04][04 05] [05 06]) ([MVH] [06 07] [07 08][01 02]) ([VLL] [04 05][02 03])1198604([MAH] [02 04][05 06]) ([LAL] [03 05][02 03]) ([ALM] [06 07][01 02] [02 03])1198605([ALM] [04 05][01 02] [04 05]) ([AHH] [06 07][01 02]) ([HVH] [04 05][06 07] [01 03])1198606([LM] [01 02][03 05] [03 05]) ([VLL] [02 04] [05 06][02 03]) ([AHH] [03 04][04 05])1198607([VLL] [04 05][05 06] [02 03] [02 04]) ([MAH] [05 06] [07 08][01 02]) ([ALM] [04 05][05 06] [03 04])1198608([ALAH] [01 03][04 06]) ([AHH] [05 07][01 02] [02 03]) ([AHH] [04 06][03 04])

Table 3 Direct influence matrix Z1198601 1198602 1198603 1198604 1198605 1198606 1198607 11986081198601 0 3 4 1 3 1 2 41198602 1 0 2 4 4 2 3 11198603 4 2 0 1 3 2 4 11198604 2 3 1 0 3 4 1 21198605 4 2 1 3 0 1 4 31198606 2 4 3 4 4 0 4 21198607 4 1 2 3 3 1 0 41198608 1 2 4 3 3 2 4 0

Step 4 Derive order inducing variable 120576119894119895 by descendingorder of 119888119894119895 The greater the value 119888119894119895 the bigger the 1205761198941198956 Illustrative Example and Experiments

61 Illustrative Case Study Suppose a mega infrastructureproject is selecting green contractor for one of its subprojectsA panel of decision-makers including managers professorsand competitive intelligence experts has been organized tocomprehensively evaluate three alternative contractors (ie1199091 1199092 and 1199093) under the eight attributes (ie 1198601ndash1198608) listedin Section 2

To elicit true decision preferences we apply theIVDHF UUBLS to assess the three alternatives (1199091ndash1199093)Firstly decision-makers are asked to vote on 119909119895 under eachattribute119860 119894 using different unbalanced linguistic term sets 1198781 1198782 and 1198783 where 1198781 = 119873 119871119860119871119872119860119867119867119876119867119881119867119860119879 1198791198782 = 119873119860119873119881119871 119876119871 119871 119860119871119872119860119867119881119867 119879 1198783 = 119873119881119871119871119860119872119872119876119872119860119867119867 119879 In Figure 4 we show the adoptedlinguistic hierarchies and relationship between 1198781 1198782 and1198783 Attributes 1198601 1198602 1198605 and 1198608 are evaluated by use of 1198781attributes11986031198604 and1198607 are evaluated by 1198782 and attribute1198606is evaluated by 1198783 The highest-voted uncertain unbalancedlinguistic term [119904119898 119904119899] are identified Secondly the panel of

decision-makers is asked to further express their opinionson membership degrees ℎ(119909) and nonmembership degrees119892(119909) to [119904119898 119904119899] Then we can derive compound assessments(ie ⟨119909 119904120599(119909) ℎ(119909) 119892(119909)⟩) in the form of IVDHF UUBLSand obtain the decision matrix in Table 2 Additionallyin order to examine the interdependency among attributesdecision-makers are also required to assess the influentialrelations among the eight attributes using DEMATEL-basedAlgorithm II The direct influence matrix is collected andshown in Table 3

Next we apply the proposed Algorithm I to solve thisgreen contractor selection problem To derive the synthe-sized attribute weighting vector in Algorithm I we assign120572 = 120573 = 05 For scenarios with no group attitudinalcharacters operator GIVDHFUUBLHA in Definition 12 ischosen In contrast for scenarios with group attitudinalcharacters operator I-GIVDHFUUBLHA in Definition 13will be chosen Further the panel of decision-makers is askedto determine their positive and negative ideal alternativesthat are collected in Table 4 where corresponding order-inducing vectors that are derived from Algorithm III arealso listed (see the rightmost column) Thereafter operatorI-GIVDHFUUBLHA in Definition 13 is utilized for aggrega-tion For GIVDHFUUBLHA and I-GIVDHFUUBLHA we

Complexity 19

Table 4 Ideal alternatives (119903+ and 119903minus) and derived order-inducing vectors 120576119903+ 119903minus 1205761199091 1199092 11990931198601 ([VHT] 1 0) ([NL] 1 0) 5 6 61198602 ([MAH] [07 08] [01 02]) ([NL] 1 0) 1 1 41198603 (T 1 0) (L 1 0) 8 7 81198604 ([ALAH] 09 01) (T 1 0) 2 2 11198605 ([HT] [09 1] 0) (N 1 0) 7 5 51198606 (M 1 0) (N 1 0) 6 3 31198607 (M 1 0) ([NL] 1 0) 3 8 71198608 ([MAH] 1 0) (N 1 0) 4 4 2

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 1 2 3 4

0 1 2

N L AH H QH VH AT TM

Level 1 l(13)

Level 2 l(25)

Level 3 l(39)

Level 4 l(417)

Unbalanced linguistic term set S1

N AH VH TM Unbalanced linguistic term set S2VL L ALAN QL

N H TM Unbalanced linguistic term set S3

L AM AHQM

AL

VL

Figure 4 Unbalanced linguistic term sets (1198781 1198782 and 1198783) and their mapping in linguistic hierarchies

assume 120582=2 and 119908 = (0 0 015 025 025 025 01 0)119879Note that 119908 is a position weighting vector which is derivedby the fuzzy semantic quantitative operator [99] Aftercompleting all steps in Algorithm I (see Appendix C) weobtain the ranking results and present them in Table 5 Forthe scores derived from Step 5 we find the ranking order is1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the best solution Step 7gives the ranking order of 1199092 ≻ 1199091 ≻ 1199093 indicating 1199092 is stillthe most appropriate one while 1199091 is better than 1199093 under theinfluence of 12057662 Experiments on Ranking Results along with VariousParameter Configurations In order to inspect the impactsof various parameter configurations on ranking results inthis section more numerical examples are carried out Alleight operators that defined in Section 41 are applied to theabove illustrative case where the parameter 120582 is configured asintegers ranging from 1 to 20 and parameter 120572 is configuredto a value set of 0 03 05 08 09 1

Firstly the four basic aggregation operators of GIVDH-FUUBLWA GIVDHFUUBLWG GIVDHFUUBLOWA and

Table 5 Ranking results of operators GIVDHFUUBLHA and I-GIVDHFUUBLHA

Aggregation operator Ranking orderGIVDHFUUBLHA 1199092 ≻ 1199093 ≻ 1199091I-GIVDHFUUBLHA 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG are applied to the former case Thesame configuration 120572 = 120573 = 05 is chosen while 120582 is testedas integers ranging from 1 to 20 The ranking results derivedfrom different parameter combinations have been collectedin Table 6 As can be seen four operators unanimouslyidentify the contractor 1199092 is the best one no matter how120582 changes Regarding priority relation between 1199091 and 1199093GIVDHFUUBLWA and GIVDHFUUBLOWA generate thesame result of1199093 ≻ 1199091 while results fromGIVDHFUUBLWGandGIVDHFUUBLOWGgot a change to 1199091 ≻ 1199093 at a certainvalue of parameter 120582

Actually GIVDHFUUBLWA and GIVDHFUUBLOWAare generalized version of weighted arithmetic aggregationoperators that producemore favorable results by emphasizing

20 Complexity

Table 6 Ranking orders obtained by GIVDHFUUBLWA GIVDHFUUBLOWA GIVDHFUUBLWG and GIVDHFUUBLOWG along with120582Operators 120582 (Integer) Ranking orderGIVDHFUUBLWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLOWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLWG [1 2] 1199092 ≻ 1199093 ≻ 1199091[3 20] 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG [1 8] 1199092 ≻ 1199093 ≻ 1199091[9 20] 1199092 ≻ 1199091 ≻ 1199093

Table 7 Ranking results obtained by GIVDHFUUBLHA and I-GIVDHFUUBLHA along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHA I-GIVDHFUUBLHA120572=0 120573=1 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=03 120573=07 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=05 120573=05 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=08 120573=02 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=09 120573=01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=1 120573=0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093

Table 8 Ranking results obtained by GIVDHFUUBLHG and I-GIVDHFUUBLHG along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHG I-GIVDHFUUBLHG

120572 = 0 120573 = 1 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 03 120573 = 07 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 05 120573 = 05 [1 8] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[9 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 08 120573 = 02 [1 17] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[18 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 09 120573 = 01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572 = 1 120573 = 0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093membership degrees to a uncertain linguistic term thus canbe considered as optimistic operators while GIVDHFU-UBLWG and GIVDHFUUBLOWG are generalized geomet-ric aggregation operators that producemore favorable resultsby emphasizing nonmembership degrees thus can be consid-ered as pessimistic operators [100 101]Therefore the rankingresults in Table 6 could be perceived as for optimism-baseddecision-making alternative 1199093 always stands out better than1199091 when 120582 gradually amplifies degree of optimism but forpessimism-based decision-making 1199091 turns out better than1199093 when 120582 gradually amplifies degree of pessimism 120582 canbe considered as the parameter for amplifying degree ofoptimism or pessimism [101]

Next to inspect the impacts of various combinationsof 120596119860119868 and 120596119860119863 on ranking results another four aggrega-tion operators of GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG also have beenapplied to the above illustrative case with varying 120572 120573 and 120582

All results have been collected in Tables 7 and 8 for compari-son As can be seen fromTable 7 ranking results generated byoperator GIVDHFUUBLHA keep the same as 1199092 ≻ 1199093 ≻ 1199091for all test combinations of parameters However as shownin Table 8 although GIVDHFUUBLHG is also capable ofidentifying 1199092 as the best contractor the priority relationsbetween 1199091 and 1199093 hold similar changing patterns in rankingresults in Table 6 that were output byGIVDHFUUBLWGandGIVDHFUUBLOWG Regarding decision situations whereobjective attributeweighting vector120596119860119863 is only considered (120572= 0) or its dominant importance (such as 120572 = 03) is accepted1199091 ≻ 1199093 is a clear observation when parameter 120582 is set toan integer in interval [8 20] Contrariwise under decisionscenarios where subjective attribute weighting vector 120596119860119868 isonly needed (120572 = 1) or its dominant importance (such as 120572 =09) is acknowledged 1199093 ≻ 1199091 holds all the way 120582 varies

In order to more clearly show the score trajectories ofalternative contractors along variations of parameter 120582 we

Complexity 21

HAInducedHA

InducedHA-2HA-2

HA-3

InducedHA-1InducedHA-3

= =

HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(a) 120572 = 0

HAInducedHA

= 03 = 07

InducedHA-2

HA-2HA-3

InducedHA-1 InducedHA-3HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(b) 120572 = 03

HAInducedHA

= 5 = 05

InducedHA-2HA-2

HA-3

InducedHA-1 InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(c) 120572 = 05

HAInducedHA

= 08 = 02

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572 = 08

HAInducedHA

= 9 = 01

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(e) 120572 = 09

HAInducedHA

246 2462 2464 2466 2468 247

= 1 = 0InducedHA-2HA-2

HA-3

HA- 3

HA-3

InducedHA-1

InducedHA-3

HA-1

HA-1

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

1085

1086

1087

(f) 120572 = 1

Figure 5 Score trajectories obtained by aggregation operators HA and InducedHA with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

further configure parameter 120582 to the value set of [0 01100]and conduct computational experiments to collect scorevalues and ranking results (as shown in Figures 5 and 6) byfour aggregation operators of GIVDHFUUBLHA GIVDH-FUUBLHG I-GIVDHFUUBLHAand I-GIVDHFUUBLHGFor more clarity in Figures 5 and 6 please note that weuse HA HG Induced-HA and Induced HG to respec-tively denote GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG In Figure 5it can be observed that when dominant importance of 120596119860119863

is emphasized (120572 = 0 120572 = 03 120572 = 05) GIVDHFUUBLHAis capable of differentiating the three alternative contractorsas same ranking order of 1199092 ≻ 1199093 ≻ 1199091 for all 120582 but withincreases in 120572 the margins between scores of 1199091 and 1199093 nar-rows and the rank order changes to 1199091 ≻ 1199093when 120572 gets 09and 1 Interestingly in Figure 6 although GIVDHFUUBLHGis able to significantly differentiate alternative contractors 1199091and 1199093 when dominant importance of 120596119860119863 is emphasized (120572= 0 120572 = 03 120572 = 05) rank order of 1199091 and 1199093 obviouslychanges once from 1199093 ≻ 1199091 to 1199091 ≻ 1199093 when 120582 is bigger

22 Complexity

HGInducedHG

= = 1

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(a) 120572= 0

HGInducedHG

= 3 = 07

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(b) 120572= 03

HGInducedHG

= 05 = 05

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(c) 120572= 05

HGInducedHG

= 8 = 02

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572= 08

HGInducedHG

292 293 294 295 296 297minus08

minus0795

minus079

= 09 = 01

HA-1

HA-3

InducedHG-2

HG-1

HG-1

HG-2

HG-3

HG-3 InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4

minus3

minus2

minus1

0

1

2

scor

e val

ues

(e) 120572= 09

HGInducedHG

50 60 70 80 90 100minus095minus09

minus085

= 1 = 0

HG-x3

HG-x1

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(f) 120572= 1

Figure 6 Score trajectories obtained by aggregation operators HG and InducedHG with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

than 8 When 120572 increases to 1 GIVDHFUUBLHGmaintainsthe rank order of 1199093 ≻ 1199091 but their score trajectorieswalk very close Comparatively I-GIVDHFUUBLHA andI-GIVDHFUUBLHG generate consistent ranking results of1199092 ≻ 1199091 ≻ 1199093 no matter 120582 varies in all the above experiments

In sum when parameters are configured with typical val-ues (ie 120572 = 05 and 120582=1) and no additional attitudinal infor-mation is taken into consideration the proposed aggregationoperators unanimously derive the same rank order 1199092 ≻1199093 ≻ 1199091 of for the three alternative contractors However

with the amplification effect of parameter 120582 analysis resultsfrom linear combination of 120596119860119868 and 120596119860119863 reveal alternative1199092 which consistently is the best choice but rank order of1199091 and 1199093 changes because their score trajectories get closeand crossed which probably indicates that only decision-making informationderived from the decisionmatrix is inad-equate to significantly differentiate alternatives 1199091 and 1199093While by integrating additional decision information aboutattitudinal characters of group experts I-GIVDHFUUBLHAand I-GIVDHFUUBLHG manage to generate the identical

Complexity 23

Table 9 Order-inducing vectors obtained by our TOPSIS-based method for problem in [38]

1198661 1198662 1198663 11986641198601 02522 01629 02282 027141198602 01 05243 01364 046891198603 04081 01756 05244 037791198604 04507 01409 01064 013221198605 00697 02203 03753 01161

Table 10 Order-inducing vectors obtained by our TOPSIS-based method for problem in [39]

1198661 1198662 1198663 11986641198601 04548 04969 04923 047321198602 05092 047 04768 046451198603 07192 04888 04953 049151198604 06722 06163 05706 049291198605 06104 04942 06463 04946

Table 11 Ranking results obtained by applying different order-inducing vectors to [38 39]

Problems Approaches Ranking results

Problem in [38] IHFLOWA [38] + order-inducing vectors in [38] 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198605 ≻ 1198604

IHFLOWA [38] + order-inducing vectors in Table 9 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198604 ≻ 1198605

Problem in [39] I-IVDHFLOWA [39] + order-inducing vectors in [39] 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198602 ≻ 1198601

I-IVDHFLOWA [39] + order-inducing vectors in Table 10 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602

permutation of 1199092 ≻ 1199091 ≻ 1199093 which verifies that ourproposed TOPSIS-based order-inducing approach is prac-tical and useful in helping work out decision results morereasonably and robustly

63 Application Studies of Proposed TOPSIS-Based Order-Inducing Method When confronting complex decision-making situations that need additional attitudinal charactersof decision-makers Yager and Filev [88] and Yager [102]introduced and encouraged to employ order-inducing vectorto represent those attitudinal characters Since then theidea of order-inducing vector has been extended to variousdecision-making environments [98 103] such as those basedon intuitionistic fuzzy information [104 105] linguistic infor-mation [106] hesitant fuzzy linguistic information [38] andinterval-valued dual hesitant fuzzy uncertain linguistic infor-mation [39] Generally in existing literature order-inducingvectors rely on experts to directly give [38 39 104ndash106]

In order to examine the adaptability of our TOPSIS-based order-inducing method to linguistic decision-makingenvironments we respectively apply the method to resolvethe same problems in references [38 39] Due to fact thatthere is no descriptions about how the order-inducing vectorswere derived in both references [38 39] we here applyextremums into our TOPSIS-based method to obtain order-inducing vectors that is (1199047 1 0) and (1199041 0 1) for[38] ([1199046 1199046] 1 0) and ([1199040 1199040] 0 1) for [39] Thenwe obtain our order-inducing vectors listed in Table 9 forthe problem adopted in [38] and our order-inducing vectorslisted in Table 10 for [39]

Now we apply the order-inducing vectors as shownin Tables 9 and 10 to resolve the problems in [38] and[39] respectively Regarding the problem in [38] the scorevalues of all alternatives are calculated as 119878(1198601)=02602119878(1198602)=02417 119878(1198603)=03577 119878(1198604)=0221 and 119878(1198605)=0185so the corresponding ranking result is 1198603 ≻ 1198601 ≻ 1198602 ≻1198604 ≻ 1198605 As for the problem in [39] the score values ofall alternatives are computed as 119878(1198601)=39954 119878(1198602)=3783119878(1198603)=70827 119878(1198604)=71792 and 119878(1198605)=5813 we thus obtainthe ranking result 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602 Formore clarity brief descriptions of comparative approachesand obtained ranking results also have been put together inTable 11

According to Table 11 the approaches that employ ourorder-inducing vectors derive the same ranking order ofthe better three alternatives for both comparative problemsfrom [38 39] As seen our proposed TOPSIS-based order-inducing method exhibits effective and more understand-able Furthermore when decision-makers are incapable ofdirectly put forward all desired order-inducing vectors due tocomplexity in ill-structured problems our method providesa feasible directions to obtain order-inducing vectors byfiguring out ideal solutions Comparatively our proposedTOPSIS-based order-inducing method behaves effective andmore operational in practical use

64 Comparison with Decision-Making Approach Based onDegraded form of IVDHFULS Note that our approachis intrinsically adaptive to decision situations based ondegraded forms of IVDHF UUBLS such as the dual hesitant

24 Complexity

fuzzy linguistic sets introduced by Yang and Ju [54] Tofurther inspect feasibility of our approach we here conductcomparative study on the same case adopted by Yang and Ju[54] We firstly convert their original decision matrix intothe form of IVDHF UUBLS crisp membership and non-membership degrees are equivalently changed to intervalswith same upper and lower limits (eg 02 turns out to be[02 02]) and linguistic variables are rewritten as uncertainlinguistic variables (eg 1199042 turns out to be [11990421199042])

Since the case only got three attributes among whichinterdependency could be not significant under generalassumption weighting vector 120596119860119868 is not calculated Andfor straight comparison order inducing vector is notincluded either Then the argument-dependent attributeweighting vector can be obtained from Eq(29) as 120596119860119863 =(03192 03132 03676)119879 By use of Algorithm Iwith operatorIVDHFUUBLWA (ie GIVDHFUUBLWA in case of 120582 = 1see Appendix B) we have the scores of alternatives 119904(1198601) =03412 119904(1198602) = 01354 119904(1198603) = 03932 119904(1198604) = 07352Therefore the ordering of alternatives is1198604 ≻ 1198603 ≻ 1198601 ≻ 1198602which is mainly in alignment with the ordering of1198604 ≻ 1198601 ≻1198603 ≻ 1198602 in Yang and Ju [54] It is worth noticing that 120596119860119863used in our approach is derived objectively from assessmentswhile attribute weights 120596lowast = (035 025 040)119879used by Yangand Ju [54] were assigned empirically Differences between120596119860119863 and 120596lowast lead to different ordering of1198601 and1198603 Compar-atively speaking our approach holds more adaptability andobjectivity

In general our proposed MADM approach in this paperexhibits an effective way to tackle complex green contractorselection problems in which subjective attributes weightingvector (such as120596119860119868) and objective attributes weighting vector(such as 120596119860119863) exist To comprehensively utilize these twofundamental facets of attributes weighting information ourcase study and computational experiments suggest that linearcombination method (ie 120596 = 120572120596119860119868 + 120573120596119860119863) is effectiveto reflect effects of 120596119860119868 and 120596119860119863 by configuring parameter 120572or 120573 When there is no specific expert opinions to specify orhard to obtain we suggest that parameter 120572 chooses the valueset of 0 02 05 08 1 Furthermore generalized aggrega-tion operators are suggested to inspect score trajectoriesof alternative contractors along increasing of the operatorsrsquoamplification parameter 120582 If some alternative contractorsrsquoscore values get very close or crossed so that they cannot bedifferentiated from each other significantly we suggest to askthe panel of experts to derive opinions of ideal solutions andutilize our TOPSIS-based order-inducing method to includethose expert opinions as attitudinal decision informationthereby obtaining more rational and robust decision results

7 Conclusions

Aiming at tackling complex decision-making problems ofgreen contractor selection that often hold simultaneouslythe characteristics of decision hesitancy attributes inter-dependency and group complex attitudinal characters wehave developed an effective IVDHF UUBLS-based approach

We utilize IVDHF UUBLS to elicit complicate decision-makersrsquo assessments more objectively and comprehensivelywe employ DEMATEL for addressing attribute interdepen-dency more precisely and we determine group attitudinalcharacters as order-inducing vectors based on ideal attitudesof experts rather than by empirical assignment The mainadvantages of our approach are (i) IVDHF UUBLS offers anadequate way to depict decision hesitancy (ii) the designeddistance measure for IVDHF UUBLS avoids informationdistortion of traditional ones which artificially add mis-matching membership or nonmembership degrees (iii) thedeveloped generalized operators include traditional ones asspecial cases so as to provide flexibility in MADM basedon IVDHF UUBLS (iv) the devised synthesized attributeweighting scheme provides an adaptive way to consider twonecessary aspects attributes interdependency-based weightsand argument-dependent weights (v) the TOPSIS-basedmethod can deduce reasonable order-inducing vectors ratherthan empirical ones and holds generality in other linguisticdecision environments Results of case studies and experi-ments have verified the validity of the proposed approach

When multiple decision organizations are involvedgroup decision-making based on IVDHF UUBLS is animportant research direction Besides inducing preferencesfor decision-making problems of high uncertainty interactivemechanisms should also be studied to extract preferences Inaddition empirical studies of the proposed approach can becarried out to various areas such as investment evaluationproject management and vehicle selection in fleet operations

Appendix

A Basic Notions for IVDHFS and UnbalancedLinguistic Term Set

Definition A1 (see [47]) Letting 119883 be a fixed set then anIVDHFS on 119883 is defined as119863 = ⟨119909 ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (A1)

where ℎ(119909) = ⋃[120583119871120583119880]isinℎ(119909)120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and119892(119909) = ⋃[]119871]119880]isin119892(119909)] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two setsof interval values in [0 1] denoting possible membershipand non-membership degrees of element 119909 isin 119883 to theset 119863 respectively with conditions 120583 ] isin [0 1] and0 le (120583119880)+ + (]119880)+ le 1 and for all 119909 isin 119883 (120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880

For convenience normally 119889 = ℎ 119892 is called an interval-valued dual hesitant fuzzy element (IVDHFE) and 119863 is theset of all IVDHFEs

Definition A2 (see [107]) Suppose 119878 = 119904119895 | 119895 = 0 1 119892 minus1 is a finite and totally ordered discrete linguistic term setwhere 119904119895 represent possible values for a linguistic variableand 119892 is an odd cardinality 119878 is uniformly and symmetricallydistributed if the following two conditions are satisfied (1)

Complexity 25

there exists a unique constant 120582 gt 0 such that 119873119878(119904119894) minus119873119878(119904119895) = 119897(119894 minus 119895) for all 119894 119895 = 0 1 119892 minus 1 (2) Let 119878119877 =119904 | 119904 isin 119878 119904 gt 119878119862 and 119878119871 = 119904 | 119904 isin 119878 119904 lt 119878119862 Let C(119878119877) andC(119878119871) be the cardinality of 119878119877 and 119878119871 then C(119878119877) = C(119878119871)If 119878 is uniformly and symmetrically distributed then 119878 iscalled a balanced linguistic term set Otherwise 119878 is calledan unbalanced linguistic term set

The following 2-tuple fuzzy linguistic representationmodel extends traditional linguistic term set to a continuouscase so as to facilitate computing with linguistic variables

Definition A3 (see [59]) Let 119878 = 1199040 1199041 119904119892minus1 be alinguistic term set and 120573 isin [0 119892] Then a 2-tuple fuzzylinguistic variable expresses the equivalent information to 120573is defined as

Δ [0 119892] 997888rarr 119878 times [minus05 05) (A2)

Δ (120573) = (119904119894 120572) with

119904119894 119894 = 119903119900119906119899119889 (120573)120572 = 120573 minus 119894 120572 isin [minus05 05)

(A3)

Δminus1 (119904119894 120572) = 119894 + 120572 = 120573 (A4)

where 119903119900119906119899119889() is the usual rounding operation and120572 is calledsymbolic translation

To obtain 2-tuple fuzzy linguistic representations ofunbalanced linguistic terms the concept of linguistic hierar-chies ie 119871119867 = ⋃119905 119897(119905 119899(119905)) is used 119897(119905 119899(119905)) is a linguistichierarchy with 119905 indicating the level of hierarchy and 119899(119905)denotes the granularity of the linguistic term set of 119905 Herreraet al [59] defined the following transformation functionsbetween labels from different levels in multigranular linguis-tic information contexts without loss of information

Definition A4 (see [59]) In linguistic hierarchies 119871119867 =⋃119905 119897(119905 119899(119905)) whose linguistic term sets are represented by119878119899(119905) = 119904119899(119905)0 119904119899(119905)119899(119905)minus1 the transformation function froma linguistic label in level 119905 to a label in consecutive level 1199051015840 isdefined as 1198791198651199051199051015840 119897(119905 119899(119905)) 997888rarr 119897(1199051015840 119899(1199051015840)) such that

1198791198651199051199051015840 (119904119899(119905)119894 120572119899(119905))= Δ 1199051015840 (Δminus1119905 (119904119899(119905)119894 120572119899(119905)) (119899 (1199051015840) minus 1)119899 (119905) minus 1 ) (A5)

By use of the above transformation function any 2-tuplelinguistic representation can be transformed into a term in119871119867 Detailed transformation procedures are listed as follows

(1) Representation in linguistic hierarchy To transformthe unbalanced terms of 119878 into the corresponding terms inthe 119871119867 transformation function 120595 is employed to associateeach unbalanced linguistic 2-tuple (119904119894 120572) with its linguistic 2-tuple in 119871119867(119878) ie

120595 119878 997888rarr 119871119867(119878) (A6)

so that 120595(119904119894 120572) = (119904119866(119894)119868(119894) 120582) for forall(119904119894 120572) isin 119878(2) Computational phase Firstly transform (119904119866(119894)119868(119894) 120582) intolinguistic 2-tuples denoted as (119904119899(1199051015840)

1198681015840(119894) 1205821015840) in 119878119899(1199051015840) where

(119904119899(1199051015840)1198681015840(119894)

1205821015840) = 119879119865 (119904119866(119894)119868(119894) 120582)= Δ(Δminus1 (119904119866(119894)119868(119894) 120582) sdot (119899 (1199051015840) minus 1)119866 (119894) minus 1 ) (A7)

Then a computational model is used over 119878119899(1199051015840) with aresult denoted as (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)(3) Retranslation process The result (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)is transformed into the unbalanced term in 119878 by using thetransformation function 120595minus1 ie

120595minus1 119871119867 (119878) 997888rarr 119878 (A8)

so that 120595minus1(119904119899(1199051015840)119903 120582119903) = (119904119903119890119904119906119897119905 120582119903119890119904119906119897119905) isin 119878B Details of Theorem 14

Proof Given a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector for119904119889119895 and 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 120582 gt 0 then(1) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLWA and GIVDHFUUBLWG operators respectively(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLOWA and GIVDHFUUBLOWG operators respectively

(2) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to theIVDHF UUBL weighted average (IVDHFUUBLWA)and IVDHF UUBL weighted geometric (IVDHFUUBLWG)operators respectively where

26 Complexity

119868119881119863119867119865119880119880119861119871119882119860120596 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

120596119895119904119889119895= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus

119899prod119895=1

(1 minus 120583119871119895 )120596119895 1 minus 119899prod119895=1

(1 minus 120583119880119895 )120596119895]] [[

119899prod119895=1

(]119871119895)120596119895 119899prod119895=1

(]119880119895 )120596119895]])

(B1)

119868119881119863119867119865119880119880119861119871119882119866120596 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889119895120596119895

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[

119899prod119895=1

(120583119871119895 )120596119895 119899prod119895=1

(120583119880119895 )120596119895]] [[1 minus

119899prod119895=1

(1 minus ]119871119895)120596119895 1 minus 119899prod119895=1

(1 minus ]119880119895 )120596119895]])

(B2)

(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHA andGIVDHFUUBLHG operators reduce to the IVDHF UUBL

orderedweighted average (IVDHFUUBLOWA) and IVDHFUUBL ordered weighted geometric (IVDHFUUBLOWG)operators respectively where

119868119881119863119867119865119880119880119861119871119874119882119860119908 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

119908119895119904119889120590(119895)= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) )))

119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) )))

]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus

119899prod119895=1

(1 minus 120583119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus 120583119880120590(119895))119908119895]] [[

119899prod119895=1

(]119871120590(119895))119908119895 119899prod119895=1

(]119880120590(119895))119908119895]])

(B3)

119868119881119863119867119865119880119880119861119871119874119882119866119908 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889120590(119895)119908119895

= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[

119899prod119895=1

(120583119871120590(119895))119908119895 119899prod119895=1

(120583119880120590(119895))119908119895]] [[1 minus

119899prod119895=1

(1 minus ]119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus ]119880120590(119895))119908119895]])

(B4)

Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

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[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

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[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

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[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

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[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

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[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

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[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

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[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

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[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

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[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

[51] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

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[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

[55] Z Tao X Liu H Chen and L Zhou ldquoUsing new versionof extended t-norms and s-norms for aggregating intervallinguistic labelsrdquo IEEETransactions on SystemsManCyberneticsSystems vol vol p 15 2016

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[57] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[58] L Martınez M Espinilla J Liu L G Perez and P J SanchezldquoAn evaluation model with unbalanced linguistic informationapplied to olive oil sensory evaluationrdquo Journal of Multiple-Valued Logic and Soft Computing vol 15 no 2-3 pp 229ndash2512009

[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

[62] L Marin A Valls D Isern A Moreno and J M MerigoldquoInduced unbalanced linguistic ordered weighted average andits application in multiperson decision makingrdquo The ScientificWorld Journal vol 2014 2014

[63] A Baykasoglu V Kaplanoglu Z D U Durmusoglu and CSahin ldquoIntegrating fuzzy DEMATEL and fuzzy hierarchicalTOPSIS methods for truck selectionrdquo Expert Systems withApplications vol 40 no 3 pp 899ndash907 2013

[64] S K Patil and R Kant ldquoA hybrid approach based on fuzzyDEMATEL and FMCDM to predict success of knowledge man-agement adoption in supply chainrdquoApplied Soft Computing vol18 pp 126ndash135 2014

[65] S Liyin and Y Hong ldquoImproving environmental performanceby means of empowerment of contractorsrdquo Management ofEnvironmental Quality An International Journal vol 17 no 3pp 242ndash257 2006

[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

[69] Y Kazancoglu I Kazancoglu and M Sagnak ldquoA new holisticconceptual framework for green supply chain management

performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

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Complexity 5

Table1Mainevaluativ

eattributesandtheiro

ptionalsub

dimensio

nsforg

reen

contractor

selection

No

Attributes

Sub-dimensio

nsRe

ferences

A1

Quo

tatio

n(i)

Tend

erprice(

advancep

aymentcapitalbidrou

tine

maintenance

andmajor

repairs

)[81824]

Liangetal[8]Fon

gandCh

oi[24]Z

avadskasetal[18]

A2

Credibility

(i)pastperfo

rmance

[101824]

(ii)w

orkexperie

nce[25]

(iii)repu

tatio

nampcreditratin

g[1025ndash27]

(iv)h

ealth

ampsafety

[10182426]

(v)fi

nancialstability[1024ndash2628]

Nieto-M

orotea

ndRu

z-Vila[10]Z

avadskasetal[18]H

atushandSkitm

ore

[26]Fon

gandCh

oi[24]Shenetal[28]Darvishetal[2527]

A3

Technical

strength

(i)technicalability[810182628]

(ii)e

xperienceamp

know

ledgeo

foperatio

nteam

[1025]

(iii)qu

ality

rank

[25]

(iv)c

reativity

ampinno

vatio

n[25]

Liangetal[8]N

ieto-M

orotea

ndRu

z-Vila[10]Z

avadskasetal[18]H

atush

andSkitm

ore[

26]Sh

enetal[28]D

arvishetal[25]

A4

Resource

strength

(i)technicalhum

anresource

[2427]

(ii)c

urrent

workload

[24]

(iii)constructio

nmachinery

ampequipm

ent[2527]

(iv)fi

xedassetsampliq

uidity

[27]

Fong

andCh

oi[24]D

arvishetal[25]H

wangetal[27]

A5

Coo

peratio

nmanagem

ent

capability

(i)clientcon

tractorrelationship[102429]

(ii)o

rganizationalstructureampop

erations

[828]

(iii)compatib

lecultu

re[827]

(iv)c

ommun

ication[829]

(v)k

nowledges

harin

gof

riskmanagem

ent[2729]

(vi)inform

ationdelivery[29]

(vii)

contractmanagem

ent[8]

Liangetal[8]N

ieto-M

orotea

ndRu

z-Vila[10]H

atushandSkitm

ore[

26]

Fong

andCh

oi[24]Shenetal[28]Darvishetal[25]H

wangetal[27]

HwangandHan

[29]A

nvuu

rAaron

andKu

maraswam

yMoh

an[30]

A6

Designamp

Procurem

ent

(i)recycle[31]

(ii)reuse

[3233]

(iii)remanufacture[3233]

(iv)d

isassem

blyanddisposal[33]

(v)w

asteredu

ction[3234]

(vi)environm

entm

aterialsub

stitutio

n[3133]

(vii)

hazardou

smaterialm

inim

ization[35]

(viii)c

lean

techno

logy

availability[33]

Lin[35]H

umph

reysetal[34]C

hiou

etal[32]Leeetal[31]H

umph

reys

etal[33]

A7

Com

pliance

with

green

legisla

tion

(i)ISO-140

01certificatio

n[32343637]

(ii)e

co-la

belin

g[313237]

(iii)environm

entp

olicies[33]

(iv)e

nviro

nmentp

lann

ing[33]

(v)e

nviro

nmentalm

anagem

entinformationsyste

m[3132]

Hum

phreysetal[34]C

hiou

etal[32]YangandWu[36]Liu

andLiu[37]

Leeetal[31]Hum

phreysetal[33]

A8

Environm

ental

perfo

rmance

(i)solid

waste[333436]

(ii)c

hemicalwaste[333436]

(iii)aire

mission[333436]

(iv)w

astewater

disposal[32ndash3436]

(v)e

nergy[

3334]

Hum

phreysetal[34]C

hiou

etal[32]YangandWu[36]Hum

phreysetal

[33]

6 Complexity

(A7) Compliance with green legislation Comprehensiveevaluation on the extent to which contractorsrsquo practicessatisfy different governmental green or sustainability legis-lations according to the aspects of ISO-14001 certificationeco-labeling environment policies environment planningenvironmental management information system etc

(A8) Environmental performance Comprehensive evalu-ation on environmental effects that the corporationrsquos activitieshave on the natural milieu Environmental performance cancommonly bemeasured throughoperative performance indi-cators (ie energyresource utilization emission reductionand waste disposal)

22 Problem Definition The contractor selection as well asmany other multicriteria decisions impacting the overallproject should bemade during the front end planning stage ofa project the point at which a group of designated decision-makers have the power to accept or reject a contractorfor a specific project or its work packages [5 9] WhenusingMADMmechanism to copewith complicate contractorselection problems [5] normally few nominated contractorswill be ready for the decision-makers to vote on Due tocomplexity of the problems and limitedness of knowledgedecision-makers usually feel confident in expressing theiropinions by use of interval numbers [83] or uncertain linguis-tic terms [52] Although widely-accepted majority rule willquickly help the group of decision-makers arrive at a decisionon an uncertain linguistic term (eg [s4 s5]) to whichobviously there exists decision hesitancy because differentopinions existTherefore in this paper we define the interval-valued dual hesitant fuzzy uncertain unbalanced linguisticset (IVDHF UUBLS) to help the decision-making panel elicittheir assessments more objectively and completely

From another point of view various attitudinal charac-ters (degree of orness) commonly exist because individualexpert holds specific backgrounds and decision processthat involves the attitudinal character of group decision-makers must coordinate those various attitudinal charactersinto one complex attitudinal character [61] Therefore weadopt the concept of order-inducing vector [62] to reflectgroup complex attitudinal character and develop a TOPSIS-based method to rationally determine the order-inducingvector Besides as Tan and Chen [84] pointed out forreal-world sophisticated MADM problems the indepen-dency axiom [85] cannot generally satisfied For exampleupgrading in green performance will raise the quotationand intrinsically result in requirements for high-standardcollaboration between contractors and general constructorIn viewing of this common phenomenon we take attributesrsquointerdependency as a third indispensible characteristic intackling complexity in green contractor selection in megainfrastructure projects In sum we take three characteristicsof complexity to model the practical problems of complicategreen contractor selection ie (i) compound structure ofhesitant fuzzy linguistic assessments (ii) group attitudinalcharacters and (iii) attributesrsquo interdependency To producegreater clarity Figure 1 demonstrates the conceptual MADMmodel for green contractor selection

Now we can give the symbolized description of thetargeted complex green contractor problems Given a megainfrastructure project there are a set of alternative greencontractors ie 119883 = 1199091 1199092 119909119899 for its subprojectsLet 119860 = 1198601 1198602 119860119898 be the evaluative attributesaccording to which decision-makers consider each greencontractor Due to high complexity in the sophisticatedproblem scenarios there exists interdependency relationsamong the evaluative attribute A panel of decision-makers119864 = 1198641 1198642 119864119905 have been organized to give theirassessments to each alternative contractor 119909119894 (119894 = 1 119899)under every attribute 119860119895 (119895 = 1 119898) In order to reflect thecomplicate group assessments of all alternative contactorsthe hybrid expression tool of IVDHF UUBLS that will bedetailed in Section 3 is adopted to depict the assessmentsmore effectively and comprehensively As a result a specificdecision matrix 119877 = (119903119894119895)119898times119899 whose elements are in theform of IVDHF UUBLS is obtained According to Merigoand Casanovas [61] suppose that an order-inducing vec-tor 120576 for denoting group attitudinal characters has beenreasonably obtained Then effective MADM approachesmust be developed to determine the most appropriate greencontractor(s)

3 Interval-Valued Dual Hesitant FuzzyUncertain Unbalanced Linguistic Set

As demonstrated in Figure 1 after the panel of decision-makers votes on an alternative contractor under certainattribute by use of specific uncertain unbalanced linguisticterm set the uncertain linguistic term [s4 s5] stands outbecause of the majority rule while different opinions shouldalso be included and considered inMADMprocesses To thatend based on interval-valued dual hesitant fuzzy set (IVD-HFS) [47] and the unbalanced linguistic term set (ULTS)[59] we here first introduce an interval-valued dual hesitantfuzzy uncertain unbalanced linguistic set (IVDHF UUBLS)which incorporate different opinions of decision-makersas membership degrees or nonmembership degrees to themajority-voted [s4 s5] Then we develop operational rules aswell as distance measure for the IVDHF UUBLS Regardingdefinitions of the IVDHFS and ULTS one can refer toAppendix A

31 Definition of IVDHF UUBLS

Definition 1 Let119883 be a fixed set and 119878 be a finite and continu-ous unbalanced linguistic label set Then an IVDHF UUBLS119878119863 on119883 is defined as

119878119863 = ⟨119909 119904120599(119909) ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (1)

where 119904120579(119909) = [119904120572 119904120573] represents judgment to object 119909119904120572 and 119904120573 are two unbalanced linguistic variables frompredefined unbalanced linguistic label set 119878 which repre-sents decision-makersrsquo judgments to an evaluated object119883 ℎ(119909) = ⋃120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and 119892(119909) =⋃] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two sets of closed intervals

Complexity 7

e panelof decision

makers Alternativegreen

contractors

Evaluativeattributes

Comprehensiveevaluation

Attributesrsquo interdependency

+

alternativecontractor

vote

Voting results on

MajorityRule

radic

Specificdecision

matrix

AppropriateMADM

approach

Decisionresults

Section 3 Defines an effectivehybrid expression tool

Group attitudinal characters (ie complex degrees of orness)

s

⟨Contractori [s4 s5] ℎ (x) g(x)⟩

i-th

1 2 3 4 5 6 7

j-th attribute

Figure 1 Conceptual MADMmodel for tackling complexity in green contractor selection

in [0 1] ℎ(119909) denotes possible membership degrees that 119909belongs to 119904120579(119909) and119892(119909) represents possible nonmembershipdegrees of 119909 to 119904120579(119909) ℎ(119909) and 119892(119909) hold conditions 120583 ] isin[0 1] and 0 le (120583119880)+ + (]119880)+ le 1 where(120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880 for all 119909 isin 119883When119883 = 1199091 1199092 119909119899has only one element 119878119863 reduces to (s120599 ℎ 119892) which is calledan interval-valued dual hesitant fuzzy uncertain unbalancedlinguistic (IVDHF UUBL) number (IVDHF UUBLN)

32 Operational Rules for IVDHF UUBLS On the strength ofoperational rules for uncertain linguistic set [52] unbalancedlinguistic set [59] and interval-valued dual hesitant fuzzy set[47] we get the following operations for IVDHF UUBLS

Definition 2 Let 119904119889 = (119904120599 ℎ 119892) = ([119904120572 119904120573] ℎ 119892) 1199041198891 =(1199041205991 ℎ1 1198921) = ([1199041205721 1199041205731] ℎ1 1198921) and 1199041198892 = (1199041205992 ℎ2 1198922) =([1199041205722 1199041205732] ℎ2 1198922) be any three IVDHF UUBLNs 120582 isin [0 1]operations on these IVDHF UUBLNs are defined as

(1) 120582119904119889 = ⋃(119904120599ℎ119892)isin119904119889

([119904120582Δminus11199050 (119879119865

11990511989611199050(120595(119904120572)))

119904120582Δminus11199050 (119879119865

11990511989621199050(120595(119904120573)))

] ⋃

[120583119871120583119880]isinℎ[]119871]119880]isin119892

[1 minus (1 minus 120583119871)120582 1 minus (1 minus 120583119880)120582] [(]119871)120582 (]119880)120582]) (2) 119904119889120582 = ⋃

(119904120599 ℎ119892)isin119904119889

([119904(Δminus11199050 (119879119865

11990511989611199050(120595(119904120572))))

120582 119904(Δminus11199050 (119879119865

11990511989621199050(120595(119904120573))))

120582])

⋃[120583119871120583119880]isinℎ[]119871]119880]isin119892

[(120583119871)120582 (120583119880)120582] [1 minus (1 minus ]119871)120582 1 minus (1 minus ]119880)120582] )(3) 1199041198891 oplus 1199041198892 = ⋃

(1199041205991 ℎ11198921)isin1199041198891(1199041205992 ℎ21198922)isin1199041198892

([119904Δminus11199050 (119879119865

11990511989411199050(120595(1199041205721 )))+Δ

minus11199050(11987911986511990511989511199050(120595(1199041205722 )))

119904Δminus11199050 (119879119865

11990511989421199050(120595(1199041205731 )))+Δ

minus11199050(11987911986511990511989521199050(120595(1199041205732 )))

] ⋃

[1205831198711 1205831198801 ]isinℎ1[120583

1198712 1205831198802 ]isinℎ2[]

1198711 ]1198801 ]isin1198921[]

1198712 ]1198802 ]isin1198922

[1205831198711 + 1205831198712 minus 12058311987111205831198712 1205831198801 + 1205831198802 minus 1205831198801 1205831198802 ] []1198711]1198712 ]1198801 ]1198802 ]) (4) 1199041198891 otimes 1199041198892 = ⋃

(1199041205991 ℎ11198921)isin1199041198891(1199041205992 ℎ21198922)isin1199041198892

([119904Δminus11199050 (119879119865

11990511989411199050(120595(1199041205721 )))timesΔ

minus11199050(11987911986511990511989511199050(120595(1199041205722 )))

119904Δminus11199050 (119879119865

11990511989421199050(120595(1199041205731 )))timesΔ

minus11199050(11987911986511990511989521199050(120595(1199041205732 )))

] ⋃

[1205831198711 1205831198801 ]isinℎ1[120583

1198712 1205831198802 ]isinℎ2[]

1198711 ]1198801 ]isin1198921[]

1198712 ]1198802 ]isin1198922

[12058311987111205831198712 1205831198801 1205831198802 ] []1198711 + ]1198712 minus ]1198711]1198712 ]1198801 + ]1198802 minus ]1198801 ]

1198802 ])

(2)

8 Complexity

In above details about the transformation functionsof 119879119865 and Δ linguistic hierarchies (119871119867) as well as thetransformation procedures for unbalanced linguistic termsets are shown in Appendix A

eorem 3 Letting 119904119889 = (119904120599 ℎ 119892) 1199041198891 = (1199041205991 ℎ1 1198921) and1199041198892 = (1199041205992 ℎ2 1198922) be any three IVDHF UUBLNs 120582 1205821 1205822 isin[0 1]then following properties are true

(1) 1199041198891 oplus 1199041198892 = 1199041198892 oplus 1199041198891 (2) 1199041198891 otimes 1199041198892 = 1199041198892 otimes 1199041198891 (3)120582(1199041198891 oplus 1199041198892) = 1205821199041198891 oplus 1205821199041198892(4) 1199041198891120582 otimes 1199041198892120582 = (1199041198891 otimes 1199041198892)120582 (5) 1205821119904119889 oplus 1205822119904119889 = (1205821 +1205822)119904119889 (6) 1199041198891205821 otimes 1199041198891205822 = 1199041198891205821+1205822

Proof Omitted

In Definition 2 and Theorem 3 1199051198961 1199051198962 1199051198941 1199051198942 1199051198951 and1199051198952 are corresponding levels of 119904120572 119904120573 1199041205721 1199041205731 1199041205722 and 1199041205722in 119871119867 respectively 1199050 is the maximum level of 119904120572 119904120573 1199041205721 1199041205731 1199041205722 and 1199041205722 in 119871119867 Furthermore to compare any twoIVDHF UUBLNs we also have following definitions

Definition 4 Let 119904119889 = (119904120599 ℎ 119892) be an IVDHF UUBLN119904120599 = [119904120572 119904120573] then score function 119878(119904119889) and accuracy function119875(119904119889) can be represented by

119878 (119904119889) = Δminus11199050 (11987911986511990511989611199050 (120595 (119904120572))) + Δminus11199050 (11987911986511990511989621199050 (120595 (119904120573)))2 times 12 ( 1119897 (ℎ) sum

[120583119871120583119880]isinℎ

120583119871 minus 1119897 (119892) sum[]119871]119880]isin119892

]119871

+ 1119897 (ℎ) sum[120583119871120583119880]isinℎ

120583119880 minus 1119897 (119892) sum[]119871]119880]isin119892

]119880)(3)

119875 (119904119889)= Δminus11199050 (11987911986511990511989611199050 (120595 (119904120572))) + Δminus11199050 (11987911986511990511989621199050 (120595 (119904120573)))2

times 12 ( 1119897 (ℎ) sum[120583119871120583119880]isinℎ

120583119871 + 1119897 (119892) sum[]119871]119880]isin119892

]119871

+ 1119897 (ℎ) sum[120583119871120583119880]isinℎ

120583119880 + 1119897 (119892) sum[]119871]119880]isin119892

]119880)

(4)

where 119897(ℎ) and 119897(119892) are numbers of values in ℎ and 119892respectively 1199051198961 and 1199051198962 are the corresponding levels of 119904120572 and119904120573 in 119871119867 and 1199050 is the maximum level of 119905119896 in 119871119867

Definition 5 Given any IVDHF UUBLNs 1199041198891 = (1199041205991 ℎ1 1198921)1199041198892 = (1199041205992 ℎ2 1198922) then(1) If 119878(1199041198891) lt 119878(1199041198892) then 1199041198891 lt 1199041198892(2) If 119878(1199041198891) = 119878(1199041198892) then

(a) If 119875(1199041198891) = 119875(1199041198892) then 1199041198891 = 1199041198892(b) If 119875(1199041198891) lt 119875(1199041198892) then 1199041198891 lt 1199041198892

33 Distance Measure for IVDHF UUBLS When 119897(ℎ) or119897(119892) of two IVDHF UUBLNs are unequal the complement-ing method [86] is normally adopted to design distancemeasures Note that artificially adding values to shorterones in the complementing method will cause informationdistortion To avoid this limitation we provide the followingdistance measure

Definition 6 Let two IVDHF UUBLNs 1199041198891 = (1199041205991 ℎ1 1198921)and 1199041198892 = (1199041205992 ℎ2 1198922) where 1199041205991 = [1199041205721 1199041205731] 1199041205992 = [1199041205722 1199041205732]119897ℎ1 119897ℎ2 1198971198921 and 1198971198922 are lengths of ℎ1 ℎ2 1198921 and 1198922 respectivelydenoting number of elements in ℎ1 ℎ2 1198921 and 1198922 Suppose1198681 = (1(119899(1199051198941) minus 1))Δminus11199050 (11987911986511990511989411199050 (120595(1199041205721))) 1198682 = (1(119899(1199051198942) minus1))Δminus11199050 (11987911986511990511989421199050 (120595(1199041205731))) 1198683 = (1(119899(1199051198951)minus1))Δminus11199050 (11987911986511990511989511199050 (120595(1199041205722)))1198684 = (1(119899(1199051198952) minus 1))Δminus11199050 (11987911986511990511989521199050 (120595(1199041205732))) where 1199051198941 1199051198942 1199051198951 and1199051198952 are the corresponding levels of unbalanced linguistic terms1199041205721 1199041205731 1199041205722 and 1199041205732 in the linguistic hierarchy 119871119867 and 1199050 is themaximum level of 1199041205721 1199041205731 1199041205722 and 1199041205732 in 119871119867 Then by use ofnormalized Euclidean distance we define a distance measure119889 for IVDHF UUBLNs as follows

Situation 7 When 119897ℎ1 = 119897ℎ2 = 1198971 and 1198971198921 = 1198971198922 = 1198972 then119889 (1199041198891 1199041198892) = (12 ( 11198971sdot 1198971sum119896=1

(10038161003816100381610038161003816100381610038161198681120583119871119895ℎ1 minus 1198683120583119871119896ℎ2 10038161003816100381610038161003816100381610038162 + 10038161003816100381610038161003816100381610038161198682120583119880119895ℎ1 minus 1198684120583119880119896ℎ2 10038161003816100381610038161003816100381610038162) + 11198972sdot 1198972sum119896=1

(1003816100381610038161003816100381610038161198681]1198711198951198921 minus 1198683]1198711198961198922 1003816100381610038161003816100381610038162 + 1003816100381610038161003816100381610038161198682]1198801198951198921 minus 1198684]1198801198961198922 1003816100381610038161003816100381610038162)))12

(5)

Situation 8 When 119897ℎ1 = 119897ℎ2 or 1198971198921 = 1198971198922 then119889 (1199041198891 1199041198892) = (12 ( 1119897ℎ1 119897ℎ2sdot 119897ℎ1sum119895=1

119897ℎ2sum119896=1

(10038161003816100381610038161003816100381610038161198681120583119871119895ℎ1 minus 1198683120583119871119896ℎ2 10038161003816100381610038161003816100381610038162 + 10038161003816100381610038161003816100381610038161198682120583119880119895ℎ1 minus 1198684120583119880119896ℎ2 10038161003816100381610038161003816100381610038162)+ 11198971198921 1198971198922sdot 1198971198921sum119895=1

1198971198922sum119896=1

(1003816100381610038161003816100381610038161198681]1198711198951198921 minus 1198683]1198711198961198922 1003816100381610038161003816100381610038162 + 1003816100381610038161003816100381610038161198682]1198801198951198921 minus 1198684]1198801198961198922 1003816100381610038161003816100381610038162))12

(6)

eorem 9 The distance measure 119889 defined in Definition 6satisfies following properties

(1) 0 le 119889(1199041198891 1199041198892) le 1

Complexity 9

(2) 119889(1199041198891 1199041198892) = 0 if and only if 1199041198891 and 1199041198892 are perfectlyconsistent

(3) 119889(1199041198891 1199041198892) = 119889(1199041198892 1199041198891)4 Generalized Aggregation Operators forIVDHF_UUBLS

41 Definitions of Operators Based on the generalized oper-ators firstly introduced by Yager [87] we here develop some

fundamental generalized aggregation operators for the newlydefined IVDHF UUBLS

Definition 10 Given a collection of IVDHF UUBLNs 119904119889119895(119895 =1 119899) their weighting vector 120596 = (1205961 1205962 120596119899)119879120596119895 isin[0 1]sum119899119895=1 120596119895 = 1 120582 be a parameter 120582 isin (0 +infin)(1) Generalized IVDHFUUBLWeighted Average(GIVDHFUUBLWA) Operator

119866119868119881119863119867119865119880119880119861119871119882119860120596120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(120596119895119904119889119895120582))1120582

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120582)1120582 119904(sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120582)1120582]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[(1 minus

119899prod119895=1

(1 minus (120583119871119895 )120582)120596119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880119895 )120582)120596119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871119895)120582)120596119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880119895 )120582)120596119895)1120582]])

(7)

(2) Generalized IVDHFUUBLWeighted Geometric(GIVDHFUUBLWG)Operator

119866119868119881119863119867119865119880119880119861119871119882119866120596120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889119895)120596119895)= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871119895 )120582)120596119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880119895 )120582)120596119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871119895 )120582)120596119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880119895 )120582)120596119895)1120582]])

(8)

Definition 11 For a collection of IVDHF UUBLNs 119904119889119895(119895 = 1 119899) 119904119889120590(119895) be the 119895th largest 119908 = (1199081 1199082 119908119899)119879 be the aggregation-associated weighting vector 119908119895 isin [0 1]sum119899119895=1119908119895= 1 120582 is a parameter such that 120582 isin (0 +infin) 119878119899 997888rarr 119878 Then

10 Complexity

(1) Generalized IVDHFUUBL Ordered Weighted Average(GIVDHFUUBLOWA) Operator

119866119868119881119863119867119865119880119880119861119871119874119882119860120596120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(119908119895119904119889120590(119895)120582))1120582

= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

120582)1120582]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895) []119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120590(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120590(119895))120582)119908119895)1120582]])

(9)

(2) Generalized IVDHFUUBL Ordered Weighted Geometric(GIVDHFUUBLOWG)Operator

119866119868119881119863119867119865119880119880119861119871119874119882119866119908120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889120590(119895))119908119895)= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120590(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120590(119895))120582)120596119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120590(119895))120582)120596119895)1120582]])

(10)

Definition 12 For a collection of IVDHF UUBLNs 119904119889119895(119895 =1 119899) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector120596119895 isin [0 1] sum119899119895=1 120596119895 = 1 119899 is a balancing coefficient119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weight-ing vector with 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1

(1) Generalized IVDHFUUBL Hybrid Average (GIVDHFU-UBLHA) Operator

119866119868119881119863119867119865119880119880119861119871119867119860120596119908120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(119908119895119904119889120590(119895)120582))1120582

= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895)

))))120582)1120582]

Complexity 11

⋃[120583119871120590(119895) 120583

119880120590(119895)]isinℎ120590(119895) []

119871120590(119895) ]119880120590(119895)]isin119892120590(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120590(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120590(119895))120582)119908119895)1120582]])

(11)

where 119904119889120590(119895) is the 119895th largest and

119904119889119895 = 119899120596119895119904119889119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904119899120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904119899120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

] ⋃

[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[1 minus (1 minus 120583119871119895 )119899120596119895 1 minus (1 minus 120583119880119895 )119899120596119895] [(]119871119895)119899120596119895 (]119880119895 )119899120596119895]) (12)

(2) Generalized IVDHFUUBL Hybrid Geometric(GIVDHFUUBLHG) Operator

119866119868119881119863119867119865119880119880119861119871119867119866119908120596120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889120590(119895))119908119895)= ⋃(119904120599120590(119895)

ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120590(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895)

))))119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895) []119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120590(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120590(119895))120582)119908119895)1120582]])

(13)

where 119904119889120590(119895) is the 119895th largest and

119904119889119895 = 119904119889119895119899120596119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(Δminus11199050 (119879119865

11990511989511199050(120595(119904120572119895 ))))

119899120596119895 119904(Δminus11199050 (119879119865

11990511989521199050(120595(119904120573119895 ))))

119899120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[(120583119871119895 )119899120596119895 (120583119880119895 )119899120596119895] [1 minus (1 minus ]119871119895 )119899120596119895 1 minus (1 minus ]119880119895 )119899120596119895]) (14)

12 Complexity

When confronted with ill-structured situations wheredecision-makerrsquos complex attitudinal characters need to beincluded order-inducing vectors provide an effective way[88ndash90]Thus we further define following induced operatorsfor IVDHF UUBLNs

Definition 13 For a collection of IVDHF UUBLNs 119904119889119895 (119895 =1 119899) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector

120596119895 isin [0 1] sum119899119895=1 120596119895 = 1 119899 is a balancing coefficient119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weight-ing vector 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1 120576119895 denote a set oforder inducing vectors

(1) Induced Generalized IVDHFUUBL Hybrid Average (I-GIVDHFUUBLHA) Operator

119868-119866119868119881119863119867119865119880119880119861119871119867119860120596119908120582 (lt 1205761 1199041198891 gt lt 1205762 1199041198892 gt lt 120576119899 119904119889119899 gt) = ( 119899⨁119895=1

(119908119895119904119889120587(119895)120582))1120582

= ⋃(119904120599120587(119895) ℎ120587(119895) 119892120587(119895))isin119904119889120587(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120587(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587(119895)

))))120582)1120582]

⋃[120583119871120587(119895) 120583

119880120587(119895)]isinℎ120587(119895) []

119871120587(119895) ]119880120587(119895)]isin119892120587(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120587(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120587(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120587(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120587(119895))120582)119908119895)1120582]])

(15)

where (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) is decreasing order of (11990411988911199041198892 119904119889119899) according to the order inducing vector 120576119895 and119904119889119895 = 119899120596119895119904119889119895 = ⋃

(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904119899120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904119899120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

] ⋃

[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[1 minus (1 minus 120583119871119895 )119899120596119895 1 minus (1 minus 120583119880119895 )119899120596119895] [(]119871119895)119899120596119895 (]119880119895 )119899120596119895]) (16)

(2) Induced Generalized IVDHFUUBL Hybrid Geometric(I-GIVDHFUUBLHG) Operator

119868-119866119868119881119863119867119865119880119880119861119871119867119866119908120596120582 (lt 1205761 1199041198891 gt lt 1205762 1199041198892 gt lt 120576119899 119904119889119899 gt) = 1120582 ( 119899⨂119895=1

(120582119904119889120587(119895))119908119895)= ⋃(119904120599120587(119895)

ℎ120587(119895) 119892120587(119895))isin119904119889120587(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120587(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587(119895)

))))119908119895]

⋃[120583119871120587(119895)120583119880120587(119895)]isinℎ120587(119895) []119871120587(119895) ]

119880120587(119895)]isin119892120587(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120587(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120587(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120587(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120587(119895))120582)119908119895)1120582]])

(17)

Complexity 13

Weighted Aggregation

Geometric Arithmetic

GIVDHFUUBLHG GIVDHFUUBLHA

GIVDHFUUBLWG GIVDHFUUBLOWG

IVDHFUUBLWG IVDHFUUBLOWG

GIVDHFUUBLWA GIVDHFUUBLOWA

IVDHFUUBLWA IVDHFUUBLOWA

= 1 w =1

n = 1 =

1

n = 1 =1

n = 1 =

1

n

= 1 = 1 = 1 = 1

Figure 2

where (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) is decreasing order of (11990411988911199041198892 119904119889119899) according to the order inducing variables 120576119895 and

119904119889119895 = 119904119889119895119899120596119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(Δminus11199050 (119879119865

11990511989511199050(120595(119904120572119895 ))))

119899120596119895 119904(Δminus11199050 (119879119865

11990511989521199050(120595(119904120573119895 ))))

119899120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[(120583119871119895 )119899120596119895 (120583119880119895 )119899120596119895] [1 minus (1 minus ]119871119895 )119899120596119895 1 minus (1 minus ]119880119895 )119899120596119895]) (18)

42 Properties of the Proposed GeneralizedAggregation Operators

eorem 14 With special values of 120582 119908 and 120596 the operatorsGIVDHFUUBLHA and GIVDHFUUBLHG can include aseries of traditional aggregation operators as special cases andtheir relationship can be depicted in Figure 2

Proof See Appendix B

As for the induced hybrid aggregation operators we alsohave following theorem

eorem 15 If (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) = (119904119889120590(1) 119904119889120590(2) 119904119889120590(119899)) I-GIVDHFUUBLHA reduces toGIVDHFUUBLHAIf (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) = (119904119889120590(1) 119904119889120590(2) 119904119889120590(119899)) thenI-GIVDHFUUBLHG reduces to the GIVDHFUUBLHG oper-ator

eorem 16 All the proposed generalized operatorsGIVDHFUUBLWAGIVDHFUUBLWGGIVDHFUUBLOWAGIVDHFUUBLOWGGIVDHFUUBLHAGIVDHFUUBLHGI-GIVDHFUUBLHA and I-GIVDHFUUBLHG hold thefollowing properties (1) Commutativity (2) Idempotency (3)Boundedness

Based on above theorems following properties can alsobe derived

eorem 17 For a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weight vector of 119904119889119895with 120596119895 isin [0 1] and sum119899119895=1 120596119895 = 1 120582 gt 0 then we have

(1) 119868119881119863119867119865119880119880119861119871119882119866120596(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119882119860120596(1199041198891 1199041198892 119904119889119899)(2) 119868119881119863119867119865119880119880119861119871119882119866120596(1199041198891 1199041198892 119904119889119899) le119866119868119881119863119867119865119880119880119861119871119882119860120596120582(1199041198891 1199041198892 119904119889119899)(3) 119866119868119881119863119867119865119880119880119861119871119882119866120596120582(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119882119860120596(1199041198891 1199041198892 119904119889119899)

Proof Omitted for concision

eorem 18 For a collection of IVDHF UUBLNs 119904119889119895 = (119904120599119895 ℎ119895 119892119895) 119908 = (1199081 1199082 119908119899)119879 is the weighting vector of 119904119889119895with 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1 120582 gt 0 Then

(1) 119868119881119863119867119865119880119880119861119871119874119882119866119908(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119874119882119860119908(1199041198891 1199041198892 119904119889119899)(2) 119868119881119863119867119865119880119880119861119871119874119882119866119908(1199041198891 1199041198892 119904119889119899) le119866119868119881119863119867119865119880119880119861119871119874119882119860119908120582(1199041198891 1199041198892 119904119889119899)(3) 119866119868119881119863119867119865119880119880119861119871119874119882119866119908120582(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119874119882119860119908(1199041198891 1199041198892 119904119889119899)

Proof Omitted for concision

14 Complexity

5 An Integrated MADM Approach forTackling Complex Green ContractorSelection Problems

As described in Section 2 we take the green contractorselection as a special type of complicate MADM problemsthat synthesizes three characteristics of decision hesitancy[21 22] attributes interdependency [84] and group atti-tudinal characters [61] Therefore in this section we con-struct an integrated MADM approach to tackle the com-plex green contractor selection problems Suppose 119883 =1199091 1199092 119909119898 is the set of alternative green contractors and119860 = 1198601 1198602 119860119899 is the set of evaluative attributes 120596 =(1205961 1205962 120596119899)119879 is weighting vector for the attributes 120596119894 ge 0sum119899119894=1 120596119894 = 1 Let = (119903119894119895)119899times119898 denote the decision matrix inwhich 119903119894119895 = ([119904120572119894119895 119904120573119894119895] ℎ119894119895 119892119894119895) is an IVDHF UUBLN givenby decision-makers for alternative contractor 119909119895 with respectto attribute 119860 119894 According to the mechanism of pair-wisecomparisons among attributes in the DEMATELmethod [6364] the interdependency among attributes can be obtained asamatrix119885 = (119911119894119896)119899times119899 where 119911119894119896 indicates the degree towhich119860 119894 affects119860119896 Subsequently based on the IVDHFUUBLS andits operations we now present detailed steps of our MADMapproach as shown in following Algorithm I

Algorithm I Hesitant fuzzy linguistic MADMwith attributesinterdependency and decision-makersrsquo group attitudinalcharacters

Step 1 Determine argument-dependent weighting vector120596119860119863 according to attribute values by programming modeldeveloped in the following Section 51

Step 2 Obtain the attribute-interdependences based weight-ing vector 120596119860119868 by use of DEMATEL method described in thefollowing Section 52

Step 3 Calculate synthesized attribute weighting vectoraccording to

120596 = 120572120596119860119868 + 120573120596119860119863 (19)

where 120572 and 120573 are parameters to reflect decision character-istics of decision organizations 0 le 120572 120573 le 1 120572 + 120573 =1Step 4 Check requirements for order inducing If no addi-tional order inducing required then go to Step 5 otherwisego to Step 6

Step 5 Utilize generalized aggregation operators to get theoverall IVDHF UUBLNs 119903119895(119895 = 1 119898) for each alternative119909119895 Here we take GIVDHFUUBLHA operator for examplebecause it can include other traditional operators as its specialcases Therefore we have

119866119868119881119863119867119865119880119880119861119871119867119860(1199031119895 1199032119895 119903119899119895) = ( 119899⨁119894=1

(119908119894 (119903120590119894119895)120582))1120582= ⋃(119904120590119894119895ℎ120590119894119895 119892120590119894119895

)isin119903120590119894119895

([119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590119894119895

))))120582)1120582 119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590119894119895

))))120582)1120582]

⋃[120583119871120590119894119895120583

119880120590119894119895]isinℎ120590119894119895[]

119871120590119894119895]119880120590119894119895]isin119892120590119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120590119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120590119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120590119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120590119894119895)120582)119908119894)1120582]])

(20)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of 119903119894119895119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weightvector 119903120590119894119895 is the 119894th largest of 119903119894119895 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))

119904119899120596119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119894119895))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (21)

Complexity 15

Then go to Step 8

Step 6 By use of the TOPSIS-based method developed in thefollowing Section 53 we determine order-inducing vectors 120576to reflect decision-makersrsquo group attitudinal characters

Step 7 Utilizing induced generalized aggregation operatorsto get the overall IVDHF UUBLNs 119903119895(119895 = 1 2 119898) foreach alternative 119909119895 we have

119868-119866119868119881119863119867119865119880119871119867119860(1199031119895 1199032119895 119903119899119898) = ( 119899⨁119894=1

(119908119894 (119903120587119894119895)120582))1120582

= ⋃(119904120587119894119895ℎ120587119894119895119892120587119894119895)isin119903120587119894119895

([119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120587119894119895

))))120582)1120582 119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587119894119895

))))120582)1120582]

⋃[120583119871120587119894119895 120583

119880120587119894119895]isinℎ120587119894119895 []

119871120587119894119895]119880120587119894119895]isin119892120587119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120587119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120587119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120587119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120587119894119895)120582)119908119894)1120582]])

(22)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of119903119894119895 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 119903120587119894119895 is the 119894th largest of 119903119894119895 reordered by theorder-inducing vectors 120576 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))119899120596119894120572119894119895

119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120573119894119895 ))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (23)

Step 8 Calculate scores 119878(119903119895) and accuracy degrees 119875(119903119895) for119903119895(119895 = 1 2 119898)Step 9 According to Definition 5 output ranking order ofalternative contractors

In Algorithm I to objectively derive unknown weight-ing vector for evaluative attributes a synthesized attributeweighting vector is devised by fusing two parts of concern(i) argument-dependent weighting vector and (ii) attribute-interdependences based weighting vector The former is toreflect effects of attribute values on attribute weights whilethe latter is to exploit effects of interdependences amongattributes on their weights For clarity processing flowchartof Algorithm I is depicted in Figure 3

51 Determining Argument-Dependent Attribute WeightingVector 120596119860119863 In order to derive attribute weighting vec-tor appropriately from assessments maximizing deviationmethod [91] is adopted here for its distinguishing ability

and objectivity [92] According to Wang [91] if performancevalues of each alternative have little differences under anattribute it implicates such an attribute plays a less importantrole otherwise it lives a more important role in decision-making Thus if one attribute has similar values across alter-natives it should be assigned a smaller weight contrariwiseit should be a bigger weight

In light of the idea in maximizing deviation methodwe develop a programming model to determine argument-dependent weighting vector 120596119860119863 For attribute 119860 119894 the stan-dard deviation of alternative 119909119895 to all the other alternatives isdenoted as

119865119894119895 (120596119860119863) = 119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (24)

Let119865119894(120596119860119863) represent deviation value of all alternatives toother alternatives according to attribute119860 119894 and be formulatedas follows

16 Complexity

Attribute weighting

Hesitant fuzzy linguistic MADM with attributes interdependency and decision makersrsquo group attitudinal characters

Step 9 Output ranking order of alternatives

interdependency among attributes by DEMATEL in Section 52

attributes values by standard deviation method in Section 51

Step 6 Ensure TOPSIS order inducing variables proposed in Section 53

Step 7 Utilizing proposed induced generalized hybrid aggregation operators developed in Section

41 to get overall value of each alternative

Step 8 Calculate score degrees and accuracy degrees

Step 4 Order inducingYes

Step 5 Choose proposed generalized operators in Section 41 to get overall value of each alternative

No

Collect decision matrix in the form of IVDHF_UUBLS

Complex problems of green contractor selection

Step 1 Weighting vector AD derived from Step 2 Weighting vector AI derived from

Step 3 Obtain synthesized weighting vector of attributes =AI + AD

Figure 3 Processing flowchart of Algorithm I

119865119894 (120596119860119863) = 119898sum119895=1

119865119894119895 (120596119860119863) = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (25)

Based on the analysis above we can get followingprogramming model (M-1) for obtaining optimal attributeweighting vector 120596119860119863

(119872 minus 1)max 119865 (120596119860119863) = 119899sum

119894=1119865119894 (120596119860119863) = 119899sum

119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894st 119899sum

119894=1(120596119860119863119894 )2 = 1 120596119860119863119894 ge 0 119894 = 1 2 119899 (26)

where 119889(119903119894119895 119903119894119897) can be calculated by Definition 6To solve model (M-1) we can construct Lagrange func-

tion

119871 (120596119860119863119894 120577) = 119899sum119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894+ 1205772 ( 119899sum

119894=1

(120596119860119863119894 )2 minus 1) (27)

where 120577 is the Lagrange multiplierDifferentiate with respect to 120596119860119863119894 (119894 = 1 2 119898) and 120577

and set these partial derivatives equal to zero the followingset of equations can be obtained

120597119871120597120596119860119863119894 = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) + 120577120596119860119863119894 = 0120597119871120597120577 = 119899sum

119894=1

(120596119860119863119894 )2 minus 1 = 0 (28)

By normalizing the solutions of above equations we canget a simple and exact formula for determining the attributeweights as follows

120596119860119863119894 = sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897)sum119899119894=1sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897) (29)

52 Deriving Attributes Interdependency-Based AttributeWeighting Vector 120596119860119868 Some efforts [53 60 84 93ndash95]

Complexity 17

have been made to address interdependencies betweenattributes in MADM However they unanimously cannotreveal the interdependences in a more complete way WhileDEMATAEL technique [63 96] gathers collective knowledgeto capture influences between attributes more precisely andcompletely so that weighting vector for attributes can berationally determined according to strength measure ofthose influences [64 97] Therefore in tackling the complexMADMwith interdependent attributes under IVDHF UUBLenvironments we present a DEMATEL-based method as thefollowing Algorithm II to derive attribute weighting vector120596119860119868 from the viewpoint of attributes interdependency

Algorithm II DEMATEL-basedmethod for ensuringweight-ing vector 120596119860119868Step 1 Construct initial direct-influence matrix 119885 = (119911119894119896)119899times119899

For attributes 119860 = 1198601 1198602 119860119899 decision-makersare asked to give direct effect that attribute 119860 119894 has onattribute 119860119896 using a five-point scale with scores representedby natural language (1) 0 mdash lsquoabsolutely no influencersquo (2)1 mdash lsquolow influencersquo (3) 2 mdash lsquomedium influencersquo (4) 3 mdashlsquohigh influencersquo (5) 4mdash lsquovery high influencersquoThen the initialdirect-influence matrix 119885 = (119911119894119896)119899times119899 can be established witheach element being the mean of same factors in differentdirect matrices

Step 2 Calculate normalized direct-influence matrix 119883 =(119909119894119896)119899times119899 Using matrix 119885 the normalized direct-influencematrix 119883 = (119909119894119896)119899times119899 is calculated according to

119909119894119896 = 119911119894119896119904 (30)

where 119904 = maxmax1le119894le119899sum119899119896=1 119911119894119896max1le119896le119899sum119899119895=1 119911119894119896 Allelements in matrix 119883 are complying with 0 le 119909119894119896 le 10 le sum119899119896=1 119909119894119896 le 1 and 0 le sum119899119894=1 119909119894119896 le 1 and at least onerow or column of the summation equals 1

Step 3 Derive total-influence matrix 119879 = (119905119894119896)119899times119899 Basedon matrix 119883 the total-influence matrix 119879 = (119905119894119896)119899times119899 can bederived by summing the direct effects and all of the indirecteffects

119879 = 119883 + 1198832 + 1198833 + sdot sdot sdot + 119883ℎ = 119883 (119868 minus 119883)minus1 (31)

where 119868 represents identity matrix When ℎ 997888rarr infin 119883ℎ =(0)119899times119899Step 4 Compute influencing and influenced degrees of eachattribute The sum of rows and the sum of columns withinthe total-influence matrix 119879 are respectively expressed as thevectors 119903 and 119888

119903 = (119903119894)119899times1 = ( 119899sum119896=1

119905119894119896)119899times1

119888 = (119888119896)119899times1 = ( 119899sum

119895=1

119905119894119896)119879

1times119899

(32)

where 119903119894 denotes the sumof the 119894th row inmatrix119879 and showsthe sum of direct and indirect effects that attribute 119860 119894 hason other attributes Similarly 119888119896 shows the sum of direct andindirect effects that attribute 119860119896 has receivedStep 5 Estimate weights of attributes based on influencedegrees according to

120596119868lowast119894 = [(119903119894 + 119888119894)2 + (119903119894 minus 119888119894)2]12 (33)

Then the weighting vector 120596119860119868 can be obtained by normaliz-ing (33)

120596119860119868119894 = 120596119868lowast119894sum119899119894=1 120596119868lowast119894 (34)

53 Obtaining Order Inducing Variables 120576 by TOPSIS-BasedMethod For ill-defined MADM problems induced aggre-gation operators [89 90 98] provide an effective channelfor experts to express complex attitudinal characters Thekey step in induced aggregation operators is to obtainorder-inducing vectors 120576 which however are presumptivelyassigned in most papers [62] Therefore in this sectionaiming at deducing the order-inducing vectors 120576 rationallywe here devise a TOPSIS-based method in Algorithm IIIObviously the Algorithm III is of generality for MADMwithdifferent forms of linguistic expression

Algorithm III TOPSIS-based method for obtaining orderinducing vectors 120576Step 1 Let the decision maker define IVDHF UUBL idealsolution(s) 119903+119894 and IVDHF UUBL negative ideal solution(s)119903minus119894 corresponding to each attribute in which

119903+119894 = (119903+1198941 119903+1198942 119903+119894119901) 119903minus119894 = (119903minus1198941 119903minus1198942 119903minus119894119902) (35)

where 119901 and 119902 denotes the number of 119903+119894 and 119903minus119894 respectively

Step 2 Calculate distances from each element in decisionmatrix to ideal solution(s) and negative solution(s) respec-tively

119889+119894119895 = max119897=12119901

119889 (119903119894119895 119903+119894119897 ) 119889minus119894119895 = min

119897=12119902119889 (119903119894119895 119903minus119894119897 ) (36)

where 119889(119903119894119895 119903+119894119897 ) and 119889(119903119894119895 119903minus119894119897 ) are calculated by the distancemeasure in Definition 6 If 119901 gt 1 we choose the onewith maximum distance if 119902 gt 1 we choose the one withminimum distance

Step 3 Calculate coefficients according to

119888119894119895 = 119889minus119894119895119889+119894119895 + 119889minus119894119895 119894 = 1 2 119899 119895 = 1 2 119898 (37)

18 Complexity

Table 2 Decision matrix in the form of IVDHF UUBLS1199091 1199092 11990931198601([AHVH] [02 03][02 04] [03 04]) ([HT] [04 05][05 06] [02 03] [02 04]) ([ALAH] [01 02][01 03] [06 07])1198602([MH] [05 06][02 03]) ([AHT] [06 07][01 02] ) ([MVH] [02 03][05 06] [06 07])1198603([VLAL] [03 04][04 05] [05 06]) ([MVH] [06 07] [07 08][01 02]) ([VLL] [04 05][02 03])1198604([MAH] [02 04][05 06]) ([LAL] [03 05][02 03]) ([ALM] [06 07][01 02] [02 03])1198605([ALM] [04 05][01 02] [04 05]) ([AHH] [06 07][01 02]) ([HVH] [04 05][06 07] [01 03])1198606([LM] [01 02][03 05] [03 05]) ([VLL] [02 04] [05 06][02 03]) ([AHH] [03 04][04 05])1198607([VLL] [04 05][05 06] [02 03] [02 04]) ([MAH] [05 06] [07 08][01 02]) ([ALM] [04 05][05 06] [03 04])1198608([ALAH] [01 03][04 06]) ([AHH] [05 07][01 02] [02 03]) ([AHH] [04 06][03 04])

Table 3 Direct influence matrix Z1198601 1198602 1198603 1198604 1198605 1198606 1198607 11986081198601 0 3 4 1 3 1 2 41198602 1 0 2 4 4 2 3 11198603 4 2 0 1 3 2 4 11198604 2 3 1 0 3 4 1 21198605 4 2 1 3 0 1 4 31198606 2 4 3 4 4 0 4 21198607 4 1 2 3 3 1 0 41198608 1 2 4 3 3 2 4 0

Step 4 Derive order inducing variable 120576119894119895 by descendingorder of 119888119894119895 The greater the value 119888119894119895 the bigger the 1205761198941198956 Illustrative Example and Experiments

61 Illustrative Case Study Suppose a mega infrastructureproject is selecting green contractor for one of its subprojectsA panel of decision-makers including managers professorsand competitive intelligence experts has been organized tocomprehensively evaluate three alternative contractors (ie1199091 1199092 and 1199093) under the eight attributes (ie 1198601ndash1198608) listedin Section 2

To elicit true decision preferences we apply theIVDHF UUBLS to assess the three alternatives (1199091ndash1199093)Firstly decision-makers are asked to vote on 119909119895 under eachattribute119860 119894 using different unbalanced linguistic term sets 1198781 1198782 and 1198783 where 1198781 = 119873 119871119860119871119872119860119867119867119876119867119881119867119860119879 1198791198782 = 119873119860119873119881119871 119876119871 119871 119860119871119872119860119867119881119867 119879 1198783 = 119873119881119871119871119860119872119872119876119872119860119867119867 119879 In Figure 4 we show the adoptedlinguistic hierarchies and relationship between 1198781 1198782 and1198783 Attributes 1198601 1198602 1198605 and 1198608 are evaluated by use of 1198781attributes11986031198604 and1198607 are evaluated by 1198782 and attribute1198606is evaluated by 1198783 The highest-voted uncertain unbalancedlinguistic term [119904119898 119904119899] are identified Secondly the panel of

decision-makers is asked to further express their opinionson membership degrees ℎ(119909) and nonmembership degrees119892(119909) to [119904119898 119904119899] Then we can derive compound assessments(ie ⟨119909 119904120599(119909) ℎ(119909) 119892(119909)⟩) in the form of IVDHF UUBLSand obtain the decision matrix in Table 2 Additionallyin order to examine the interdependency among attributesdecision-makers are also required to assess the influentialrelations among the eight attributes using DEMATEL-basedAlgorithm II The direct influence matrix is collected andshown in Table 3

Next we apply the proposed Algorithm I to solve thisgreen contractor selection problem To derive the synthe-sized attribute weighting vector in Algorithm I we assign120572 = 120573 = 05 For scenarios with no group attitudinalcharacters operator GIVDHFUUBLHA in Definition 12 ischosen In contrast for scenarios with group attitudinalcharacters operator I-GIVDHFUUBLHA in Definition 13will be chosen Further the panel of decision-makers is askedto determine their positive and negative ideal alternativesthat are collected in Table 4 where corresponding order-inducing vectors that are derived from Algorithm III arealso listed (see the rightmost column) Thereafter operatorI-GIVDHFUUBLHA in Definition 13 is utilized for aggrega-tion For GIVDHFUUBLHA and I-GIVDHFUUBLHA we

Complexity 19

Table 4 Ideal alternatives (119903+ and 119903minus) and derived order-inducing vectors 120576119903+ 119903minus 1205761199091 1199092 11990931198601 ([VHT] 1 0) ([NL] 1 0) 5 6 61198602 ([MAH] [07 08] [01 02]) ([NL] 1 0) 1 1 41198603 (T 1 0) (L 1 0) 8 7 81198604 ([ALAH] 09 01) (T 1 0) 2 2 11198605 ([HT] [09 1] 0) (N 1 0) 7 5 51198606 (M 1 0) (N 1 0) 6 3 31198607 (M 1 0) ([NL] 1 0) 3 8 71198608 ([MAH] 1 0) (N 1 0) 4 4 2

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 1 2 3 4

0 1 2

N L AH H QH VH AT TM

Level 1 l(13)

Level 2 l(25)

Level 3 l(39)

Level 4 l(417)

Unbalanced linguistic term set S1

N AH VH TM Unbalanced linguistic term set S2VL L ALAN QL

N H TM Unbalanced linguistic term set S3

L AM AHQM

AL

VL

Figure 4 Unbalanced linguistic term sets (1198781 1198782 and 1198783) and their mapping in linguistic hierarchies

assume 120582=2 and 119908 = (0 0 015 025 025 025 01 0)119879Note that 119908 is a position weighting vector which is derivedby the fuzzy semantic quantitative operator [99] Aftercompleting all steps in Algorithm I (see Appendix C) weobtain the ranking results and present them in Table 5 Forthe scores derived from Step 5 we find the ranking order is1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the best solution Step 7gives the ranking order of 1199092 ≻ 1199091 ≻ 1199093 indicating 1199092 is stillthe most appropriate one while 1199091 is better than 1199093 under theinfluence of 12057662 Experiments on Ranking Results along with VariousParameter Configurations In order to inspect the impactsof various parameter configurations on ranking results inthis section more numerical examples are carried out Alleight operators that defined in Section 41 are applied to theabove illustrative case where the parameter 120582 is configured asintegers ranging from 1 to 20 and parameter 120572 is configuredto a value set of 0 03 05 08 09 1

Firstly the four basic aggregation operators of GIVDH-FUUBLWA GIVDHFUUBLWG GIVDHFUUBLOWA and

Table 5 Ranking results of operators GIVDHFUUBLHA and I-GIVDHFUUBLHA

Aggregation operator Ranking orderGIVDHFUUBLHA 1199092 ≻ 1199093 ≻ 1199091I-GIVDHFUUBLHA 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG are applied to the former case Thesame configuration 120572 = 120573 = 05 is chosen while 120582 is testedas integers ranging from 1 to 20 The ranking results derivedfrom different parameter combinations have been collectedin Table 6 As can be seen four operators unanimouslyidentify the contractor 1199092 is the best one no matter how120582 changes Regarding priority relation between 1199091 and 1199093GIVDHFUUBLWA and GIVDHFUUBLOWA generate thesame result of1199093 ≻ 1199091 while results fromGIVDHFUUBLWGandGIVDHFUUBLOWGgot a change to 1199091 ≻ 1199093 at a certainvalue of parameter 120582

Actually GIVDHFUUBLWA and GIVDHFUUBLOWAare generalized version of weighted arithmetic aggregationoperators that producemore favorable results by emphasizing

20 Complexity

Table 6 Ranking orders obtained by GIVDHFUUBLWA GIVDHFUUBLOWA GIVDHFUUBLWG and GIVDHFUUBLOWG along with120582Operators 120582 (Integer) Ranking orderGIVDHFUUBLWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLOWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLWG [1 2] 1199092 ≻ 1199093 ≻ 1199091[3 20] 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG [1 8] 1199092 ≻ 1199093 ≻ 1199091[9 20] 1199092 ≻ 1199091 ≻ 1199093

Table 7 Ranking results obtained by GIVDHFUUBLHA and I-GIVDHFUUBLHA along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHA I-GIVDHFUUBLHA120572=0 120573=1 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=03 120573=07 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=05 120573=05 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=08 120573=02 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=09 120573=01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=1 120573=0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093

Table 8 Ranking results obtained by GIVDHFUUBLHG and I-GIVDHFUUBLHG along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHG I-GIVDHFUUBLHG

120572 = 0 120573 = 1 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 03 120573 = 07 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 05 120573 = 05 [1 8] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[9 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 08 120573 = 02 [1 17] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[18 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 09 120573 = 01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572 = 1 120573 = 0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093membership degrees to a uncertain linguistic term thus canbe considered as optimistic operators while GIVDHFU-UBLWG and GIVDHFUUBLOWG are generalized geomet-ric aggregation operators that producemore favorable resultsby emphasizing nonmembership degrees thus can be consid-ered as pessimistic operators [100 101]Therefore the rankingresults in Table 6 could be perceived as for optimism-baseddecision-making alternative 1199093 always stands out better than1199091 when 120582 gradually amplifies degree of optimism but forpessimism-based decision-making 1199091 turns out better than1199093 when 120582 gradually amplifies degree of pessimism 120582 canbe considered as the parameter for amplifying degree ofoptimism or pessimism [101]

Next to inspect the impacts of various combinationsof 120596119860119868 and 120596119860119863 on ranking results another four aggrega-tion operators of GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG also have beenapplied to the above illustrative case with varying 120572 120573 and 120582

All results have been collected in Tables 7 and 8 for compari-son As can be seen fromTable 7 ranking results generated byoperator GIVDHFUUBLHA keep the same as 1199092 ≻ 1199093 ≻ 1199091for all test combinations of parameters However as shownin Table 8 although GIVDHFUUBLHG is also capable ofidentifying 1199092 as the best contractor the priority relationsbetween 1199091 and 1199093 hold similar changing patterns in rankingresults in Table 6 that were output byGIVDHFUUBLWGandGIVDHFUUBLOWG Regarding decision situations whereobjective attributeweighting vector120596119860119863 is only considered (120572= 0) or its dominant importance (such as 120572 = 03) is accepted1199091 ≻ 1199093 is a clear observation when parameter 120582 is set toan integer in interval [8 20] Contrariwise under decisionscenarios where subjective attribute weighting vector 120596119860119868 isonly needed (120572 = 1) or its dominant importance (such as 120572 =09) is acknowledged 1199093 ≻ 1199091 holds all the way 120582 varies

In order to more clearly show the score trajectories ofalternative contractors along variations of parameter 120582 we

Complexity 21

HAInducedHA

InducedHA-2HA-2

HA-3

InducedHA-1InducedHA-3

= =

HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(a) 120572 = 0

HAInducedHA

= 03 = 07

InducedHA-2

HA-2HA-3

InducedHA-1 InducedHA-3HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(b) 120572 = 03

HAInducedHA

= 5 = 05

InducedHA-2HA-2

HA-3

InducedHA-1 InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(c) 120572 = 05

HAInducedHA

= 08 = 02

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572 = 08

HAInducedHA

= 9 = 01

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(e) 120572 = 09

HAInducedHA

246 2462 2464 2466 2468 247

= 1 = 0InducedHA-2HA-2

HA-3

HA- 3

HA-3

InducedHA-1

InducedHA-3

HA-1

HA-1

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

1085

1086

1087

(f) 120572 = 1

Figure 5 Score trajectories obtained by aggregation operators HA and InducedHA with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

further configure parameter 120582 to the value set of [0 01100]and conduct computational experiments to collect scorevalues and ranking results (as shown in Figures 5 and 6) byfour aggregation operators of GIVDHFUUBLHA GIVDH-FUUBLHG I-GIVDHFUUBLHAand I-GIVDHFUUBLHGFor more clarity in Figures 5 and 6 please note that weuse HA HG Induced-HA and Induced HG to respec-tively denote GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG In Figure 5it can be observed that when dominant importance of 120596119860119863

is emphasized (120572 = 0 120572 = 03 120572 = 05) GIVDHFUUBLHAis capable of differentiating the three alternative contractorsas same ranking order of 1199092 ≻ 1199093 ≻ 1199091 for all 120582 but withincreases in 120572 the margins between scores of 1199091 and 1199093 nar-rows and the rank order changes to 1199091 ≻ 1199093when 120572 gets 09and 1 Interestingly in Figure 6 although GIVDHFUUBLHGis able to significantly differentiate alternative contractors 1199091and 1199093 when dominant importance of 120596119860119863 is emphasized (120572= 0 120572 = 03 120572 = 05) rank order of 1199091 and 1199093 obviouslychanges once from 1199093 ≻ 1199091 to 1199091 ≻ 1199093 when 120582 is bigger

22 Complexity

HGInducedHG

= = 1

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(a) 120572= 0

HGInducedHG

= 3 = 07

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(b) 120572= 03

HGInducedHG

= 05 = 05

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(c) 120572= 05

HGInducedHG

= 8 = 02

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572= 08

HGInducedHG

292 293 294 295 296 297minus08

minus0795

minus079

= 09 = 01

HA-1

HA-3

InducedHG-2

HG-1

HG-1

HG-2

HG-3

HG-3 InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4

minus3

minus2

minus1

0

1

2

scor

e val

ues

(e) 120572= 09

HGInducedHG

50 60 70 80 90 100minus095minus09

minus085

= 1 = 0

HG-x3

HG-x1

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(f) 120572= 1

Figure 6 Score trajectories obtained by aggregation operators HG and InducedHG with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

than 8 When 120572 increases to 1 GIVDHFUUBLHGmaintainsthe rank order of 1199093 ≻ 1199091 but their score trajectorieswalk very close Comparatively I-GIVDHFUUBLHA andI-GIVDHFUUBLHG generate consistent ranking results of1199092 ≻ 1199091 ≻ 1199093 no matter 120582 varies in all the above experiments

In sum when parameters are configured with typical val-ues (ie 120572 = 05 and 120582=1) and no additional attitudinal infor-mation is taken into consideration the proposed aggregationoperators unanimously derive the same rank order 1199092 ≻1199093 ≻ 1199091 of for the three alternative contractors However

with the amplification effect of parameter 120582 analysis resultsfrom linear combination of 120596119860119868 and 120596119860119863 reveal alternative1199092 which consistently is the best choice but rank order of1199091 and 1199093 changes because their score trajectories get closeand crossed which probably indicates that only decision-making informationderived from the decisionmatrix is inad-equate to significantly differentiate alternatives 1199091 and 1199093While by integrating additional decision information aboutattitudinal characters of group experts I-GIVDHFUUBLHAand I-GIVDHFUUBLHG manage to generate the identical

Complexity 23

Table 9 Order-inducing vectors obtained by our TOPSIS-based method for problem in [38]

1198661 1198662 1198663 11986641198601 02522 01629 02282 027141198602 01 05243 01364 046891198603 04081 01756 05244 037791198604 04507 01409 01064 013221198605 00697 02203 03753 01161

Table 10 Order-inducing vectors obtained by our TOPSIS-based method for problem in [39]

1198661 1198662 1198663 11986641198601 04548 04969 04923 047321198602 05092 047 04768 046451198603 07192 04888 04953 049151198604 06722 06163 05706 049291198605 06104 04942 06463 04946

Table 11 Ranking results obtained by applying different order-inducing vectors to [38 39]

Problems Approaches Ranking results

Problem in [38] IHFLOWA [38] + order-inducing vectors in [38] 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198605 ≻ 1198604

IHFLOWA [38] + order-inducing vectors in Table 9 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198604 ≻ 1198605

Problem in [39] I-IVDHFLOWA [39] + order-inducing vectors in [39] 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198602 ≻ 1198601

I-IVDHFLOWA [39] + order-inducing vectors in Table 10 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602

permutation of 1199092 ≻ 1199091 ≻ 1199093 which verifies that ourproposed TOPSIS-based order-inducing approach is prac-tical and useful in helping work out decision results morereasonably and robustly

63 Application Studies of Proposed TOPSIS-Based Order-Inducing Method When confronting complex decision-making situations that need additional attitudinal charactersof decision-makers Yager and Filev [88] and Yager [102]introduced and encouraged to employ order-inducing vectorto represent those attitudinal characters Since then theidea of order-inducing vector has been extended to variousdecision-making environments [98 103] such as those basedon intuitionistic fuzzy information [104 105] linguistic infor-mation [106] hesitant fuzzy linguistic information [38] andinterval-valued dual hesitant fuzzy uncertain linguistic infor-mation [39] Generally in existing literature order-inducingvectors rely on experts to directly give [38 39 104ndash106]

In order to examine the adaptability of our TOPSIS-based order-inducing method to linguistic decision-makingenvironments we respectively apply the method to resolvethe same problems in references [38 39] Due to fact thatthere is no descriptions about how the order-inducing vectorswere derived in both references [38 39] we here applyextremums into our TOPSIS-based method to obtain order-inducing vectors that is (1199047 1 0) and (1199041 0 1) for[38] ([1199046 1199046] 1 0) and ([1199040 1199040] 0 1) for [39] Thenwe obtain our order-inducing vectors listed in Table 9 forthe problem adopted in [38] and our order-inducing vectorslisted in Table 10 for [39]

Now we apply the order-inducing vectors as shownin Tables 9 and 10 to resolve the problems in [38] and[39] respectively Regarding the problem in [38] the scorevalues of all alternatives are calculated as 119878(1198601)=02602119878(1198602)=02417 119878(1198603)=03577 119878(1198604)=0221 and 119878(1198605)=0185so the corresponding ranking result is 1198603 ≻ 1198601 ≻ 1198602 ≻1198604 ≻ 1198605 As for the problem in [39] the score values ofall alternatives are computed as 119878(1198601)=39954 119878(1198602)=3783119878(1198603)=70827 119878(1198604)=71792 and 119878(1198605)=5813 we thus obtainthe ranking result 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602 Formore clarity brief descriptions of comparative approachesand obtained ranking results also have been put together inTable 11

According to Table 11 the approaches that employ ourorder-inducing vectors derive the same ranking order ofthe better three alternatives for both comparative problemsfrom [38 39] As seen our proposed TOPSIS-based order-inducing method exhibits effective and more understand-able Furthermore when decision-makers are incapable ofdirectly put forward all desired order-inducing vectors due tocomplexity in ill-structured problems our method providesa feasible directions to obtain order-inducing vectors byfiguring out ideal solutions Comparatively our proposedTOPSIS-based order-inducing method behaves effective andmore operational in practical use

64 Comparison with Decision-Making Approach Based onDegraded form of IVDHFULS Note that our approachis intrinsically adaptive to decision situations based ondegraded forms of IVDHF UUBLS such as the dual hesitant

24 Complexity

fuzzy linguistic sets introduced by Yang and Ju [54] Tofurther inspect feasibility of our approach we here conductcomparative study on the same case adopted by Yang and Ju[54] We firstly convert their original decision matrix intothe form of IVDHF UUBLS crisp membership and non-membership degrees are equivalently changed to intervalswith same upper and lower limits (eg 02 turns out to be[02 02]) and linguistic variables are rewritten as uncertainlinguistic variables (eg 1199042 turns out to be [11990421199042])

Since the case only got three attributes among whichinterdependency could be not significant under generalassumption weighting vector 120596119860119868 is not calculated Andfor straight comparison order inducing vector is notincluded either Then the argument-dependent attributeweighting vector can be obtained from Eq(29) as 120596119860119863 =(03192 03132 03676)119879 By use of Algorithm Iwith operatorIVDHFUUBLWA (ie GIVDHFUUBLWA in case of 120582 = 1see Appendix B) we have the scores of alternatives 119904(1198601) =03412 119904(1198602) = 01354 119904(1198603) = 03932 119904(1198604) = 07352Therefore the ordering of alternatives is1198604 ≻ 1198603 ≻ 1198601 ≻ 1198602which is mainly in alignment with the ordering of1198604 ≻ 1198601 ≻1198603 ≻ 1198602 in Yang and Ju [54] It is worth noticing that 120596119860119863used in our approach is derived objectively from assessmentswhile attribute weights 120596lowast = (035 025 040)119879used by Yangand Ju [54] were assigned empirically Differences between120596119860119863 and 120596lowast lead to different ordering of1198601 and1198603 Compar-atively speaking our approach holds more adaptability andobjectivity

In general our proposed MADM approach in this paperexhibits an effective way to tackle complex green contractorselection problems in which subjective attributes weightingvector (such as120596119860119868) and objective attributes weighting vector(such as 120596119860119863) exist To comprehensively utilize these twofundamental facets of attributes weighting information ourcase study and computational experiments suggest that linearcombination method (ie 120596 = 120572120596119860119868 + 120573120596119860119863) is effectiveto reflect effects of 120596119860119868 and 120596119860119863 by configuring parameter 120572or 120573 When there is no specific expert opinions to specify orhard to obtain we suggest that parameter 120572 chooses the valueset of 0 02 05 08 1 Furthermore generalized aggrega-tion operators are suggested to inspect score trajectoriesof alternative contractors along increasing of the operatorsrsquoamplification parameter 120582 If some alternative contractorsrsquoscore values get very close or crossed so that they cannot bedifferentiated from each other significantly we suggest to askthe panel of experts to derive opinions of ideal solutions andutilize our TOPSIS-based order-inducing method to includethose expert opinions as attitudinal decision informationthereby obtaining more rational and robust decision results

7 Conclusions

Aiming at tackling complex decision-making problems ofgreen contractor selection that often hold simultaneouslythe characteristics of decision hesitancy attributes inter-dependency and group complex attitudinal characters wehave developed an effective IVDHF UUBLS-based approach

We utilize IVDHF UUBLS to elicit complicate decision-makersrsquo assessments more objectively and comprehensivelywe employ DEMATEL for addressing attribute interdepen-dency more precisely and we determine group attitudinalcharacters as order-inducing vectors based on ideal attitudesof experts rather than by empirical assignment The mainadvantages of our approach are (i) IVDHF UUBLS offers anadequate way to depict decision hesitancy (ii) the designeddistance measure for IVDHF UUBLS avoids informationdistortion of traditional ones which artificially add mis-matching membership or nonmembership degrees (iii) thedeveloped generalized operators include traditional ones asspecial cases so as to provide flexibility in MADM basedon IVDHF UUBLS (iv) the devised synthesized attributeweighting scheme provides an adaptive way to consider twonecessary aspects attributes interdependency-based weightsand argument-dependent weights (v) the TOPSIS-basedmethod can deduce reasonable order-inducing vectors ratherthan empirical ones and holds generality in other linguisticdecision environments Results of case studies and experi-ments have verified the validity of the proposed approach

When multiple decision organizations are involvedgroup decision-making based on IVDHF UUBLS is animportant research direction Besides inducing preferencesfor decision-making problems of high uncertainty interactivemechanisms should also be studied to extract preferences Inaddition empirical studies of the proposed approach can becarried out to various areas such as investment evaluationproject management and vehicle selection in fleet operations

Appendix

A Basic Notions for IVDHFS and UnbalancedLinguistic Term Set

Definition A1 (see [47]) Letting 119883 be a fixed set then anIVDHFS on 119883 is defined as119863 = ⟨119909 ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (A1)

where ℎ(119909) = ⋃[120583119871120583119880]isinℎ(119909)120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and119892(119909) = ⋃[]119871]119880]isin119892(119909)] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two setsof interval values in [0 1] denoting possible membershipand non-membership degrees of element 119909 isin 119883 to theset 119863 respectively with conditions 120583 ] isin [0 1] and0 le (120583119880)+ + (]119880)+ le 1 and for all 119909 isin 119883 (120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880

For convenience normally 119889 = ℎ 119892 is called an interval-valued dual hesitant fuzzy element (IVDHFE) and 119863 is theset of all IVDHFEs

Definition A2 (see [107]) Suppose 119878 = 119904119895 | 119895 = 0 1 119892 minus1 is a finite and totally ordered discrete linguistic term setwhere 119904119895 represent possible values for a linguistic variableand 119892 is an odd cardinality 119878 is uniformly and symmetricallydistributed if the following two conditions are satisfied (1)

Complexity 25

there exists a unique constant 120582 gt 0 such that 119873119878(119904119894) minus119873119878(119904119895) = 119897(119894 minus 119895) for all 119894 119895 = 0 1 119892 minus 1 (2) Let 119878119877 =119904 | 119904 isin 119878 119904 gt 119878119862 and 119878119871 = 119904 | 119904 isin 119878 119904 lt 119878119862 Let C(119878119877) andC(119878119871) be the cardinality of 119878119877 and 119878119871 then C(119878119877) = C(119878119871)If 119878 is uniformly and symmetrically distributed then 119878 iscalled a balanced linguistic term set Otherwise 119878 is calledan unbalanced linguistic term set

The following 2-tuple fuzzy linguistic representationmodel extends traditional linguistic term set to a continuouscase so as to facilitate computing with linguistic variables

Definition A3 (see [59]) Let 119878 = 1199040 1199041 119904119892minus1 be alinguistic term set and 120573 isin [0 119892] Then a 2-tuple fuzzylinguistic variable expresses the equivalent information to 120573is defined as

Δ [0 119892] 997888rarr 119878 times [minus05 05) (A2)

Δ (120573) = (119904119894 120572) with

119904119894 119894 = 119903119900119906119899119889 (120573)120572 = 120573 minus 119894 120572 isin [minus05 05)

(A3)

Δminus1 (119904119894 120572) = 119894 + 120572 = 120573 (A4)

where 119903119900119906119899119889() is the usual rounding operation and120572 is calledsymbolic translation

To obtain 2-tuple fuzzy linguistic representations ofunbalanced linguistic terms the concept of linguistic hierar-chies ie 119871119867 = ⋃119905 119897(119905 119899(119905)) is used 119897(119905 119899(119905)) is a linguistichierarchy with 119905 indicating the level of hierarchy and 119899(119905)denotes the granularity of the linguistic term set of 119905 Herreraet al [59] defined the following transformation functionsbetween labels from different levels in multigranular linguis-tic information contexts without loss of information

Definition A4 (see [59]) In linguistic hierarchies 119871119867 =⋃119905 119897(119905 119899(119905)) whose linguistic term sets are represented by119878119899(119905) = 119904119899(119905)0 119904119899(119905)119899(119905)minus1 the transformation function froma linguistic label in level 119905 to a label in consecutive level 1199051015840 isdefined as 1198791198651199051199051015840 119897(119905 119899(119905)) 997888rarr 119897(1199051015840 119899(1199051015840)) such that

1198791198651199051199051015840 (119904119899(119905)119894 120572119899(119905))= Δ 1199051015840 (Δminus1119905 (119904119899(119905)119894 120572119899(119905)) (119899 (1199051015840) minus 1)119899 (119905) minus 1 ) (A5)

By use of the above transformation function any 2-tuplelinguistic representation can be transformed into a term in119871119867 Detailed transformation procedures are listed as follows

(1) Representation in linguistic hierarchy To transformthe unbalanced terms of 119878 into the corresponding terms inthe 119871119867 transformation function 120595 is employed to associateeach unbalanced linguistic 2-tuple (119904119894 120572) with its linguistic 2-tuple in 119871119867(119878) ie

120595 119878 997888rarr 119871119867(119878) (A6)

so that 120595(119904119894 120572) = (119904119866(119894)119868(119894) 120582) for forall(119904119894 120572) isin 119878(2) Computational phase Firstly transform (119904119866(119894)119868(119894) 120582) intolinguistic 2-tuples denoted as (119904119899(1199051015840)

1198681015840(119894) 1205821015840) in 119878119899(1199051015840) where

(119904119899(1199051015840)1198681015840(119894)

1205821015840) = 119879119865 (119904119866(119894)119868(119894) 120582)= Δ(Δminus1 (119904119866(119894)119868(119894) 120582) sdot (119899 (1199051015840) minus 1)119866 (119894) minus 1 ) (A7)

Then a computational model is used over 119878119899(1199051015840) with aresult denoted as (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)(3) Retranslation process The result (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)is transformed into the unbalanced term in 119878 by using thetransformation function 120595minus1 ie

120595minus1 119871119867 (119878) 997888rarr 119878 (A8)

so that 120595minus1(119904119899(1199051015840)119903 120582119903) = (119904119903119890119904119906119897119905 120582119903119890119904119906119897119905) isin 119878B Details of Theorem 14

Proof Given a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector for119904119889119895 and 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 120582 gt 0 then(1) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLWA and GIVDHFUUBLWG operators respectively(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLOWA and GIVDHFUUBLOWG operators respectively

(2) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to theIVDHF UUBL weighted average (IVDHFUUBLWA)and IVDHF UUBL weighted geometric (IVDHFUUBLWG)operators respectively where

26 Complexity

119868119881119863119867119865119880119880119861119871119882119860120596 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

120596119895119904119889119895= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus

119899prod119895=1

(1 minus 120583119871119895 )120596119895 1 minus 119899prod119895=1

(1 minus 120583119880119895 )120596119895]] [[

119899prod119895=1

(]119871119895)120596119895 119899prod119895=1

(]119880119895 )120596119895]])

(B1)

119868119881119863119867119865119880119880119861119871119882119866120596 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889119895120596119895

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[

119899prod119895=1

(120583119871119895 )120596119895 119899prod119895=1

(120583119880119895 )120596119895]] [[1 minus

119899prod119895=1

(1 minus ]119871119895)120596119895 1 minus 119899prod119895=1

(1 minus ]119880119895 )120596119895]])

(B2)

(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHA andGIVDHFUUBLHG operators reduce to the IVDHF UUBL

orderedweighted average (IVDHFUUBLOWA) and IVDHFUUBL ordered weighted geometric (IVDHFUUBLOWG)operators respectively where

119868119881119863119867119865119880119880119861119871119874119882119860119908 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

119908119895119904119889120590(119895)= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) )))

119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) )))

]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus

119899prod119895=1

(1 minus 120583119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus 120583119880120590(119895))119908119895]] [[

119899prod119895=1

(]119871120590(119895))119908119895 119899prod119895=1

(]119880120590(119895))119908119895]])

(B3)

119868119881119863119867119865119880119880119861119871119874119882119866119908 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889120590(119895)119908119895

= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[

119899prod119895=1

(120583119871120590(119895))119908119895 119899prod119895=1

(120583119880120590(119895))119908119895]] [[1 minus

119899prod119895=1

(1 minus ]119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus ]119880120590(119895))119908119895]])

(B4)

Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

References

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[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

[11] X Qi C Liang and J Zhang ldquoGeneralized cross-entropy basedgroup decision making with unknown expert and attributeweights under interval-valued intuitionistic fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 79 pp 52ndash642015

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[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

[16] B T Nuong K-W Kim L Prathumratana et al ldquoSustainabledevelopment in the mining sector and its evaluation usingfuzzy AHP (Analytic Hierarchy Process) approachrdquo GeosystemEngineering vol 14 no 1 pp 43ndash50 2011

[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

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[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

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[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

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[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

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[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

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[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

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[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

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[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

[51] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[52] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

[55] Z Tao X Liu H Chen and L Zhou ldquoUsing new versionof extended t-norms and s-norms for aggregating intervallinguistic labelsrdquo IEEETransactions on SystemsManCyberneticsSystems vol vol p 15 2016

[56] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[57] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[58] L Martınez M Espinilla J Liu L G Perez and P J SanchezldquoAn evaluation model with unbalanced linguistic informationapplied to olive oil sensory evaluationrdquo Journal of Multiple-Valued Logic and Soft Computing vol 15 no 2-3 pp 229ndash2512009

[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

[62] L Marin A Valls D Isern A Moreno and J M MerigoldquoInduced unbalanced linguistic ordered weighted average andits application in multiperson decision makingrdquo The ScientificWorld Journal vol 2014 2014

[63] A Baykasoglu V Kaplanoglu Z D U Durmusoglu and CSahin ldquoIntegrating fuzzy DEMATEL and fuzzy hierarchicalTOPSIS methods for truck selectionrdquo Expert Systems withApplications vol 40 no 3 pp 899ndash907 2013

[64] S K Patil and R Kant ldquoA hybrid approach based on fuzzyDEMATEL and FMCDM to predict success of knowledge man-agement adoption in supply chainrdquoApplied Soft Computing vol18 pp 126ndash135 2014

[65] S Liyin and Y Hong ldquoImproving environmental performanceby means of empowerment of contractorsrdquo Management ofEnvironmental Quality An International Journal vol 17 no 3pp 242ndash257 2006

[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

[69] Y Kazancoglu I Kazancoglu and M Sagnak ldquoA new holisticconceptual framework for green supply chain management

performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

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6 Complexity

(A7) Compliance with green legislation Comprehensiveevaluation on the extent to which contractorsrsquo practicessatisfy different governmental green or sustainability legis-lations according to the aspects of ISO-14001 certificationeco-labeling environment policies environment planningenvironmental management information system etc

(A8) Environmental performance Comprehensive evalu-ation on environmental effects that the corporationrsquos activitieshave on the natural milieu Environmental performance cancommonly bemeasured throughoperative performance indi-cators (ie energyresource utilization emission reductionand waste disposal)

22 Problem Definition The contractor selection as well asmany other multicriteria decisions impacting the overallproject should bemade during the front end planning stage ofa project the point at which a group of designated decision-makers have the power to accept or reject a contractorfor a specific project or its work packages [5 9] WhenusingMADMmechanism to copewith complicate contractorselection problems [5] normally few nominated contractorswill be ready for the decision-makers to vote on Due tocomplexity of the problems and limitedness of knowledgedecision-makers usually feel confident in expressing theiropinions by use of interval numbers [83] or uncertain linguis-tic terms [52] Although widely-accepted majority rule willquickly help the group of decision-makers arrive at a decisionon an uncertain linguistic term (eg [s4 s5]) to whichobviously there exists decision hesitancy because differentopinions existTherefore in this paper we define the interval-valued dual hesitant fuzzy uncertain unbalanced linguisticset (IVDHF UUBLS) to help the decision-making panel elicittheir assessments more objectively and completely

From another point of view various attitudinal charac-ters (degree of orness) commonly exist because individualexpert holds specific backgrounds and decision processthat involves the attitudinal character of group decision-makers must coordinate those various attitudinal charactersinto one complex attitudinal character [61] Therefore weadopt the concept of order-inducing vector [62] to reflectgroup complex attitudinal character and develop a TOPSIS-based method to rationally determine the order-inducingvector Besides as Tan and Chen [84] pointed out forreal-world sophisticated MADM problems the indepen-dency axiom [85] cannot generally satisfied For exampleupgrading in green performance will raise the quotationand intrinsically result in requirements for high-standardcollaboration between contractors and general constructorIn viewing of this common phenomenon we take attributesrsquointerdependency as a third indispensible characteristic intackling complexity in green contractor selection in megainfrastructure projects In sum we take three characteristicsof complexity to model the practical problems of complicategreen contractor selection ie (i) compound structure ofhesitant fuzzy linguistic assessments (ii) group attitudinalcharacters and (iii) attributesrsquo interdependency To producegreater clarity Figure 1 demonstrates the conceptual MADMmodel for green contractor selection

Now we can give the symbolized description of thetargeted complex green contractor problems Given a megainfrastructure project there are a set of alternative greencontractors ie 119883 = 1199091 1199092 119909119899 for its subprojectsLet 119860 = 1198601 1198602 119860119898 be the evaluative attributesaccording to which decision-makers consider each greencontractor Due to high complexity in the sophisticatedproblem scenarios there exists interdependency relationsamong the evaluative attribute A panel of decision-makers119864 = 1198641 1198642 119864119905 have been organized to give theirassessments to each alternative contractor 119909119894 (119894 = 1 119899)under every attribute 119860119895 (119895 = 1 119898) In order to reflect thecomplicate group assessments of all alternative contactorsthe hybrid expression tool of IVDHF UUBLS that will bedetailed in Section 3 is adopted to depict the assessmentsmore effectively and comprehensively As a result a specificdecision matrix 119877 = (119903119894119895)119898times119899 whose elements are in theform of IVDHF UUBLS is obtained According to Merigoand Casanovas [61] suppose that an order-inducing vec-tor 120576 for denoting group attitudinal characters has beenreasonably obtained Then effective MADM approachesmust be developed to determine the most appropriate greencontractor(s)

3 Interval-Valued Dual Hesitant FuzzyUncertain Unbalanced Linguistic Set

As demonstrated in Figure 1 after the panel of decision-makers votes on an alternative contractor under certainattribute by use of specific uncertain unbalanced linguisticterm set the uncertain linguistic term [s4 s5] stands outbecause of the majority rule while different opinions shouldalso be included and considered inMADMprocesses To thatend based on interval-valued dual hesitant fuzzy set (IVD-HFS) [47] and the unbalanced linguistic term set (ULTS)[59] we here first introduce an interval-valued dual hesitantfuzzy uncertain unbalanced linguistic set (IVDHF UUBLS)which incorporate different opinions of decision-makersas membership degrees or nonmembership degrees to themajority-voted [s4 s5] Then we develop operational rules aswell as distance measure for the IVDHF UUBLS Regardingdefinitions of the IVDHFS and ULTS one can refer toAppendix A

31 Definition of IVDHF UUBLS

Definition 1 Let119883 be a fixed set and 119878 be a finite and continu-ous unbalanced linguistic label set Then an IVDHF UUBLS119878119863 on119883 is defined as

119878119863 = ⟨119909 119904120599(119909) ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (1)

where 119904120579(119909) = [119904120572 119904120573] represents judgment to object 119909119904120572 and 119904120573 are two unbalanced linguistic variables frompredefined unbalanced linguistic label set 119878 which repre-sents decision-makersrsquo judgments to an evaluated object119883 ℎ(119909) = ⋃120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and 119892(119909) =⋃] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two sets of closed intervals

Complexity 7

e panelof decision

makers Alternativegreen

contractors

Evaluativeattributes

Comprehensiveevaluation

Attributesrsquo interdependency

+

alternativecontractor

vote

Voting results on

MajorityRule

radic

Specificdecision

matrix

AppropriateMADM

approach

Decisionresults

Section 3 Defines an effectivehybrid expression tool

Group attitudinal characters (ie complex degrees of orness)

s

⟨Contractori [s4 s5] ℎ (x) g(x)⟩

i-th

1 2 3 4 5 6 7

j-th attribute

Figure 1 Conceptual MADMmodel for tackling complexity in green contractor selection

in [0 1] ℎ(119909) denotes possible membership degrees that 119909belongs to 119904120579(119909) and119892(119909) represents possible nonmembershipdegrees of 119909 to 119904120579(119909) ℎ(119909) and 119892(119909) hold conditions 120583 ] isin[0 1] and 0 le (120583119880)+ + (]119880)+ le 1 where(120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880 for all 119909 isin 119883When119883 = 1199091 1199092 119909119899has only one element 119878119863 reduces to (s120599 ℎ 119892) which is calledan interval-valued dual hesitant fuzzy uncertain unbalancedlinguistic (IVDHF UUBL) number (IVDHF UUBLN)

32 Operational Rules for IVDHF UUBLS On the strength ofoperational rules for uncertain linguistic set [52] unbalancedlinguistic set [59] and interval-valued dual hesitant fuzzy set[47] we get the following operations for IVDHF UUBLS

Definition 2 Let 119904119889 = (119904120599 ℎ 119892) = ([119904120572 119904120573] ℎ 119892) 1199041198891 =(1199041205991 ℎ1 1198921) = ([1199041205721 1199041205731] ℎ1 1198921) and 1199041198892 = (1199041205992 ℎ2 1198922) =([1199041205722 1199041205732] ℎ2 1198922) be any three IVDHF UUBLNs 120582 isin [0 1]operations on these IVDHF UUBLNs are defined as

(1) 120582119904119889 = ⋃(119904120599ℎ119892)isin119904119889

([119904120582Δminus11199050 (119879119865

11990511989611199050(120595(119904120572)))

119904120582Δminus11199050 (119879119865

11990511989621199050(120595(119904120573)))

] ⋃

[120583119871120583119880]isinℎ[]119871]119880]isin119892

[1 minus (1 minus 120583119871)120582 1 minus (1 minus 120583119880)120582] [(]119871)120582 (]119880)120582]) (2) 119904119889120582 = ⋃

(119904120599 ℎ119892)isin119904119889

([119904(Δminus11199050 (119879119865

11990511989611199050(120595(119904120572))))

120582 119904(Δminus11199050 (119879119865

11990511989621199050(120595(119904120573))))

120582])

⋃[120583119871120583119880]isinℎ[]119871]119880]isin119892

[(120583119871)120582 (120583119880)120582] [1 minus (1 minus ]119871)120582 1 minus (1 minus ]119880)120582] )(3) 1199041198891 oplus 1199041198892 = ⋃

(1199041205991 ℎ11198921)isin1199041198891(1199041205992 ℎ21198922)isin1199041198892

([119904Δminus11199050 (119879119865

11990511989411199050(120595(1199041205721 )))+Δ

minus11199050(11987911986511990511989511199050(120595(1199041205722 )))

119904Δminus11199050 (119879119865

11990511989421199050(120595(1199041205731 )))+Δ

minus11199050(11987911986511990511989521199050(120595(1199041205732 )))

] ⋃

[1205831198711 1205831198801 ]isinℎ1[120583

1198712 1205831198802 ]isinℎ2[]

1198711 ]1198801 ]isin1198921[]

1198712 ]1198802 ]isin1198922

[1205831198711 + 1205831198712 minus 12058311987111205831198712 1205831198801 + 1205831198802 minus 1205831198801 1205831198802 ] []1198711]1198712 ]1198801 ]1198802 ]) (4) 1199041198891 otimes 1199041198892 = ⋃

(1199041205991 ℎ11198921)isin1199041198891(1199041205992 ℎ21198922)isin1199041198892

([119904Δminus11199050 (119879119865

11990511989411199050(120595(1199041205721 )))timesΔ

minus11199050(11987911986511990511989511199050(120595(1199041205722 )))

119904Δminus11199050 (119879119865

11990511989421199050(120595(1199041205731 )))timesΔ

minus11199050(11987911986511990511989521199050(120595(1199041205732 )))

] ⋃

[1205831198711 1205831198801 ]isinℎ1[120583

1198712 1205831198802 ]isinℎ2[]

1198711 ]1198801 ]isin1198921[]

1198712 ]1198802 ]isin1198922

[12058311987111205831198712 1205831198801 1205831198802 ] []1198711 + ]1198712 minus ]1198711]1198712 ]1198801 + ]1198802 minus ]1198801 ]

1198802 ])

(2)

8 Complexity

In above details about the transformation functionsof 119879119865 and Δ linguistic hierarchies (119871119867) as well as thetransformation procedures for unbalanced linguistic termsets are shown in Appendix A

eorem 3 Letting 119904119889 = (119904120599 ℎ 119892) 1199041198891 = (1199041205991 ℎ1 1198921) and1199041198892 = (1199041205992 ℎ2 1198922) be any three IVDHF UUBLNs 120582 1205821 1205822 isin[0 1]then following properties are true

(1) 1199041198891 oplus 1199041198892 = 1199041198892 oplus 1199041198891 (2) 1199041198891 otimes 1199041198892 = 1199041198892 otimes 1199041198891 (3)120582(1199041198891 oplus 1199041198892) = 1205821199041198891 oplus 1205821199041198892(4) 1199041198891120582 otimes 1199041198892120582 = (1199041198891 otimes 1199041198892)120582 (5) 1205821119904119889 oplus 1205822119904119889 = (1205821 +1205822)119904119889 (6) 1199041198891205821 otimes 1199041198891205822 = 1199041198891205821+1205822

Proof Omitted

In Definition 2 and Theorem 3 1199051198961 1199051198962 1199051198941 1199051198942 1199051198951 and1199051198952 are corresponding levels of 119904120572 119904120573 1199041205721 1199041205731 1199041205722 and 1199041205722in 119871119867 respectively 1199050 is the maximum level of 119904120572 119904120573 1199041205721 1199041205731 1199041205722 and 1199041205722 in 119871119867 Furthermore to compare any twoIVDHF UUBLNs we also have following definitions

Definition 4 Let 119904119889 = (119904120599 ℎ 119892) be an IVDHF UUBLN119904120599 = [119904120572 119904120573] then score function 119878(119904119889) and accuracy function119875(119904119889) can be represented by

119878 (119904119889) = Δminus11199050 (11987911986511990511989611199050 (120595 (119904120572))) + Δminus11199050 (11987911986511990511989621199050 (120595 (119904120573)))2 times 12 ( 1119897 (ℎ) sum

[120583119871120583119880]isinℎ

120583119871 minus 1119897 (119892) sum[]119871]119880]isin119892

]119871

+ 1119897 (ℎ) sum[120583119871120583119880]isinℎ

120583119880 minus 1119897 (119892) sum[]119871]119880]isin119892

]119880)(3)

119875 (119904119889)= Δminus11199050 (11987911986511990511989611199050 (120595 (119904120572))) + Δminus11199050 (11987911986511990511989621199050 (120595 (119904120573)))2

times 12 ( 1119897 (ℎ) sum[120583119871120583119880]isinℎ

120583119871 + 1119897 (119892) sum[]119871]119880]isin119892

]119871

+ 1119897 (ℎ) sum[120583119871120583119880]isinℎ

120583119880 + 1119897 (119892) sum[]119871]119880]isin119892

]119880)

(4)

where 119897(ℎ) and 119897(119892) are numbers of values in ℎ and 119892respectively 1199051198961 and 1199051198962 are the corresponding levels of 119904120572 and119904120573 in 119871119867 and 1199050 is the maximum level of 119905119896 in 119871119867

Definition 5 Given any IVDHF UUBLNs 1199041198891 = (1199041205991 ℎ1 1198921)1199041198892 = (1199041205992 ℎ2 1198922) then(1) If 119878(1199041198891) lt 119878(1199041198892) then 1199041198891 lt 1199041198892(2) If 119878(1199041198891) = 119878(1199041198892) then

(a) If 119875(1199041198891) = 119875(1199041198892) then 1199041198891 = 1199041198892(b) If 119875(1199041198891) lt 119875(1199041198892) then 1199041198891 lt 1199041198892

33 Distance Measure for IVDHF UUBLS When 119897(ℎ) or119897(119892) of two IVDHF UUBLNs are unequal the complement-ing method [86] is normally adopted to design distancemeasures Note that artificially adding values to shorterones in the complementing method will cause informationdistortion To avoid this limitation we provide the followingdistance measure

Definition 6 Let two IVDHF UUBLNs 1199041198891 = (1199041205991 ℎ1 1198921)and 1199041198892 = (1199041205992 ℎ2 1198922) where 1199041205991 = [1199041205721 1199041205731] 1199041205992 = [1199041205722 1199041205732]119897ℎ1 119897ℎ2 1198971198921 and 1198971198922 are lengths of ℎ1 ℎ2 1198921 and 1198922 respectivelydenoting number of elements in ℎ1 ℎ2 1198921 and 1198922 Suppose1198681 = (1(119899(1199051198941) minus 1))Δminus11199050 (11987911986511990511989411199050 (120595(1199041205721))) 1198682 = (1(119899(1199051198942) minus1))Δminus11199050 (11987911986511990511989421199050 (120595(1199041205731))) 1198683 = (1(119899(1199051198951)minus1))Δminus11199050 (11987911986511990511989511199050 (120595(1199041205722)))1198684 = (1(119899(1199051198952) minus 1))Δminus11199050 (11987911986511990511989521199050 (120595(1199041205732))) where 1199051198941 1199051198942 1199051198951 and1199051198952 are the corresponding levels of unbalanced linguistic terms1199041205721 1199041205731 1199041205722 and 1199041205732 in the linguistic hierarchy 119871119867 and 1199050 is themaximum level of 1199041205721 1199041205731 1199041205722 and 1199041205732 in 119871119867 Then by use ofnormalized Euclidean distance we define a distance measure119889 for IVDHF UUBLNs as follows

Situation 7 When 119897ℎ1 = 119897ℎ2 = 1198971 and 1198971198921 = 1198971198922 = 1198972 then119889 (1199041198891 1199041198892) = (12 ( 11198971sdot 1198971sum119896=1

(10038161003816100381610038161003816100381610038161198681120583119871119895ℎ1 minus 1198683120583119871119896ℎ2 10038161003816100381610038161003816100381610038162 + 10038161003816100381610038161003816100381610038161198682120583119880119895ℎ1 minus 1198684120583119880119896ℎ2 10038161003816100381610038161003816100381610038162) + 11198972sdot 1198972sum119896=1

(1003816100381610038161003816100381610038161198681]1198711198951198921 minus 1198683]1198711198961198922 1003816100381610038161003816100381610038162 + 1003816100381610038161003816100381610038161198682]1198801198951198921 minus 1198684]1198801198961198922 1003816100381610038161003816100381610038162)))12

(5)

Situation 8 When 119897ℎ1 = 119897ℎ2 or 1198971198921 = 1198971198922 then119889 (1199041198891 1199041198892) = (12 ( 1119897ℎ1 119897ℎ2sdot 119897ℎ1sum119895=1

119897ℎ2sum119896=1

(10038161003816100381610038161003816100381610038161198681120583119871119895ℎ1 minus 1198683120583119871119896ℎ2 10038161003816100381610038161003816100381610038162 + 10038161003816100381610038161003816100381610038161198682120583119880119895ℎ1 minus 1198684120583119880119896ℎ2 10038161003816100381610038161003816100381610038162)+ 11198971198921 1198971198922sdot 1198971198921sum119895=1

1198971198922sum119896=1

(1003816100381610038161003816100381610038161198681]1198711198951198921 minus 1198683]1198711198961198922 1003816100381610038161003816100381610038162 + 1003816100381610038161003816100381610038161198682]1198801198951198921 minus 1198684]1198801198961198922 1003816100381610038161003816100381610038162))12

(6)

eorem 9 The distance measure 119889 defined in Definition 6satisfies following properties

(1) 0 le 119889(1199041198891 1199041198892) le 1

Complexity 9

(2) 119889(1199041198891 1199041198892) = 0 if and only if 1199041198891 and 1199041198892 are perfectlyconsistent

(3) 119889(1199041198891 1199041198892) = 119889(1199041198892 1199041198891)4 Generalized Aggregation Operators forIVDHF_UUBLS

41 Definitions of Operators Based on the generalized oper-ators firstly introduced by Yager [87] we here develop some

fundamental generalized aggregation operators for the newlydefined IVDHF UUBLS

Definition 10 Given a collection of IVDHF UUBLNs 119904119889119895(119895 =1 119899) their weighting vector 120596 = (1205961 1205962 120596119899)119879120596119895 isin[0 1]sum119899119895=1 120596119895 = 1 120582 be a parameter 120582 isin (0 +infin)(1) Generalized IVDHFUUBLWeighted Average(GIVDHFUUBLWA) Operator

119866119868119881119863119867119865119880119880119861119871119882119860120596120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(120596119895119904119889119895120582))1120582

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120582)1120582 119904(sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120582)1120582]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[(1 minus

119899prod119895=1

(1 minus (120583119871119895 )120582)120596119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880119895 )120582)120596119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871119895)120582)120596119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880119895 )120582)120596119895)1120582]])

(7)

(2) Generalized IVDHFUUBLWeighted Geometric(GIVDHFUUBLWG)Operator

119866119868119881119863119867119865119880119880119861119871119882119866120596120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889119895)120596119895)= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871119895 )120582)120596119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880119895 )120582)120596119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871119895 )120582)120596119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880119895 )120582)120596119895)1120582]])

(8)

Definition 11 For a collection of IVDHF UUBLNs 119904119889119895(119895 = 1 119899) 119904119889120590(119895) be the 119895th largest 119908 = (1199081 1199082 119908119899)119879 be the aggregation-associated weighting vector 119908119895 isin [0 1]sum119899119895=1119908119895= 1 120582 is a parameter such that 120582 isin (0 +infin) 119878119899 997888rarr 119878 Then

10 Complexity

(1) Generalized IVDHFUUBL Ordered Weighted Average(GIVDHFUUBLOWA) Operator

119866119868119881119863119867119865119880119880119861119871119874119882119860120596120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(119908119895119904119889120590(119895)120582))1120582

= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

120582)1120582]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895) []119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120590(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120590(119895))120582)119908119895)1120582]])

(9)

(2) Generalized IVDHFUUBL Ordered Weighted Geometric(GIVDHFUUBLOWG)Operator

119866119868119881119863119867119865119880119880119861119871119874119882119866119908120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889120590(119895))119908119895)= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120590(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120590(119895))120582)120596119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120590(119895))120582)120596119895)1120582]])

(10)

Definition 12 For a collection of IVDHF UUBLNs 119904119889119895(119895 =1 119899) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector120596119895 isin [0 1] sum119899119895=1 120596119895 = 1 119899 is a balancing coefficient119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weight-ing vector with 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1

(1) Generalized IVDHFUUBL Hybrid Average (GIVDHFU-UBLHA) Operator

119866119868119881119863119867119865119880119880119861119871119867119860120596119908120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(119908119895119904119889120590(119895)120582))1120582

= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895)

))))120582)1120582]

Complexity 11

⋃[120583119871120590(119895) 120583

119880120590(119895)]isinℎ120590(119895) []

119871120590(119895) ]119880120590(119895)]isin119892120590(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120590(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120590(119895))120582)119908119895)1120582]])

(11)

where 119904119889120590(119895) is the 119895th largest and

119904119889119895 = 119899120596119895119904119889119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904119899120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904119899120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

] ⋃

[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[1 minus (1 minus 120583119871119895 )119899120596119895 1 minus (1 minus 120583119880119895 )119899120596119895] [(]119871119895)119899120596119895 (]119880119895 )119899120596119895]) (12)

(2) Generalized IVDHFUUBL Hybrid Geometric(GIVDHFUUBLHG) Operator

119866119868119881119863119867119865119880119880119861119871119867119866119908120596120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889120590(119895))119908119895)= ⋃(119904120599120590(119895)

ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120590(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895)

))))119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895) []119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120590(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120590(119895))120582)119908119895)1120582]])

(13)

where 119904119889120590(119895) is the 119895th largest and

119904119889119895 = 119904119889119895119899120596119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(Δminus11199050 (119879119865

11990511989511199050(120595(119904120572119895 ))))

119899120596119895 119904(Δminus11199050 (119879119865

11990511989521199050(120595(119904120573119895 ))))

119899120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[(120583119871119895 )119899120596119895 (120583119880119895 )119899120596119895] [1 minus (1 minus ]119871119895 )119899120596119895 1 minus (1 minus ]119880119895 )119899120596119895]) (14)

12 Complexity

When confronted with ill-structured situations wheredecision-makerrsquos complex attitudinal characters need to beincluded order-inducing vectors provide an effective way[88ndash90]Thus we further define following induced operatorsfor IVDHF UUBLNs

Definition 13 For a collection of IVDHF UUBLNs 119904119889119895 (119895 =1 119899) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector

120596119895 isin [0 1] sum119899119895=1 120596119895 = 1 119899 is a balancing coefficient119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weight-ing vector 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1 120576119895 denote a set oforder inducing vectors

(1) Induced Generalized IVDHFUUBL Hybrid Average (I-GIVDHFUUBLHA) Operator

119868-119866119868119881119863119867119865119880119880119861119871119867119860120596119908120582 (lt 1205761 1199041198891 gt lt 1205762 1199041198892 gt lt 120576119899 119904119889119899 gt) = ( 119899⨁119895=1

(119908119895119904119889120587(119895)120582))1120582

= ⋃(119904120599120587(119895) ℎ120587(119895) 119892120587(119895))isin119904119889120587(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120587(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587(119895)

))))120582)1120582]

⋃[120583119871120587(119895) 120583

119880120587(119895)]isinℎ120587(119895) []

119871120587(119895) ]119880120587(119895)]isin119892120587(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120587(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120587(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120587(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120587(119895))120582)119908119895)1120582]])

(15)

where (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) is decreasing order of (11990411988911199041198892 119904119889119899) according to the order inducing vector 120576119895 and119904119889119895 = 119899120596119895119904119889119895 = ⋃

(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904119899120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904119899120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

] ⋃

[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[1 minus (1 minus 120583119871119895 )119899120596119895 1 minus (1 minus 120583119880119895 )119899120596119895] [(]119871119895)119899120596119895 (]119880119895 )119899120596119895]) (16)

(2) Induced Generalized IVDHFUUBL Hybrid Geometric(I-GIVDHFUUBLHG) Operator

119868-119866119868119881119863119867119865119880119880119861119871119867119866119908120596120582 (lt 1205761 1199041198891 gt lt 1205762 1199041198892 gt lt 120576119899 119904119889119899 gt) = 1120582 ( 119899⨂119895=1

(120582119904119889120587(119895))119908119895)= ⋃(119904120599120587(119895)

ℎ120587(119895) 119892120587(119895))isin119904119889120587(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120587(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587(119895)

))))119908119895]

⋃[120583119871120587(119895)120583119880120587(119895)]isinℎ120587(119895) []119871120587(119895) ]

119880120587(119895)]isin119892120587(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120587(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120587(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120587(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120587(119895))120582)119908119895)1120582]])

(17)

Complexity 13

Weighted Aggregation

Geometric Arithmetic

GIVDHFUUBLHG GIVDHFUUBLHA

GIVDHFUUBLWG GIVDHFUUBLOWG

IVDHFUUBLWG IVDHFUUBLOWG

GIVDHFUUBLWA GIVDHFUUBLOWA

IVDHFUUBLWA IVDHFUUBLOWA

= 1 w =1

n = 1 =

1

n = 1 =1

n = 1 =

1

n

= 1 = 1 = 1 = 1

Figure 2

where (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) is decreasing order of (11990411988911199041198892 119904119889119899) according to the order inducing variables 120576119895 and

119904119889119895 = 119904119889119895119899120596119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(Δminus11199050 (119879119865

11990511989511199050(120595(119904120572119895 ))))

119899120596119895 119904(Δminus11199050 (119879119865

11990511989521199050(120595(119904120573119895 ))))

119899120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[(120583119871119895 )119899120596119895 (120583119880119895 )119899120596119895] [1 minus (1 minus ]119871119895 )119899120596119895 1 minus (1 minus ]119880119895 )119899120596119895]) (18)

42 Properties of the Proposed GeneralizedAggregation Operators

eorem 14 With special values of 120582 119908 and 120596 the operatorsGIVDHFUUBLHA and GIVDHFUUBLHG can include aseries of traditional aggregation operators as special cases andtheir relationship can be depicted in Figure 2

Proof See Appendix B

As for the induced hybrid aggregation operators we alsohave following theorem

eorem 15 If (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) = (119904119889120590(1) 119904119889120590(2) 119904119889120590(119899)) I-GIVDHFUUBLHA reduces toGIVDHFUUBLHAIf (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) = (119904119889120590(1) 119904119889120590(2) 119904119889120590(119899)) thenI-GIVDHFUUBLHG reduces to the GIVDHFUUBLHG oper-ator

eorem 16 All the proposed generalized operatorsGIVDHFUUBLWAGIVDHFUUBLWGGIVDHFUUBLOWAGIVDHFUUBLOWGGIVDHFUUBLHAGIVDHFUUBLHGI-GIVDHFUUBLHA and I-GIVDHFUUBLHG hold thefollowing properties (1) Commutativity (2) Idempotency (3)Boundedness

Based on above theorems following properties can alsobe derived

eorem 17 For a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weight vector of 119904119889119895with 120596119895 isin [0 1] and sum119899119895=1 120596119895 = 1 120582 gt 0 then we have

(1) 119868119881119863119867119865119880119880119861119871119882119866120596(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119882119860120596(1199041198891 1199041198892 119904119889119899)(2) 119868119881119863119867119865119880119880119861119871119882119866120596(1199041198891 1199041198892 119904119889119899) le119866119868119881119863119867119865119880119880119861119871119882119860120596120582(1199041198891 1199041198892 119904119889119899)(3) 119866119868119881119863119867119865119880119880119861119871119882119866120596120582(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119882119860120596(1199041198891 1199041198892 119904119889119899)

Proof Omitted for concision

eorem 18 For a collection of IVDHF UUBLNs 119904119889119895 = (119904120599119895 ℎ119895 119892119895) 119908 = (1199081 1199082 119908119899)119879 is the weighting vector of 119904119889119895with 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1 120582 gt 0 Then

(1) 119868119881119863119867119865119880119880119861119871119874119882119866119908(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119874119882119860119908(1199041198891 1199041198892 119904119889119899)(2) 119868119881119863119867119865119880119880119861119871119874119882119866119908(1199041198891 1199041198892 119904119889119899) le119866119868119881119863119867119865119880119880119861119871119874119882119860119908120582(1199041198891 1199041198892 119904119889119899)(3) 119866119868119881119863119867119865119880119880119861119871119874119882119866119908120582(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119874119882119860119908(1199041198891 1199041198892 119904119889119899)

Proof Omitted for concision

14 Complexity

5 An Integrated MADM Approach forTackling Complex Green ContractorSelection Problems

As described in Section 2 we take the green contractorselection as a special type of complicate MADM problemsthat synthesizes three characteristics of decision hesitancy[21 22] attributes interdependency [84] and group atti-tudinal characters [61] Therefore in this section we con-struct an integrated MADM approach to tackle the com-plex green contractor selection problems Suppose 119883 =1199091 1199092 119909119898 is the set of alternative green contractors and119860 = 1198601 1198602 119860119899 is the set of evaluative attributes 120596 =(1205961 1205962 120596119899)119879 is weighting vector for the attributes 120596119894 ge 0sum119899119894=1 120596119894 = 1 Let = (119903119894119895)119899times119898 denote the decision matrix inwhich 119903119894119895 = ([119904120572119894119895 119904120573119894119895] ℎ119894119895 119892119894119895) is an IVDHF UUBLN givenby decision-makers for alternative contractor 119909119895 with respectto attribute 119860 119894 According to the mechanism of pair-wisecomparisons among attributes in the DEMATELmethod [6364] the interdependency among attributes can be obtained asamatrix119885 = (119911119894119896)119899times119899 where 119911119894119896 indicates the degree towhich119860 119894 affects119860119896 Subsequently based on the IVDHFUUBLS andits operations we now present detailed steps of our MADMapproach as shown in following Algorithm I

Algorithm I Hesitant fuzzy linguistic MADMwith attributesinterdependency and decision-makersrsquo group attitudinalcharacters

Step 1 Determine argument-dependent weighting vector120596119860119863 according to attribute values by programming modeldeveloped in the following Section 51

Step 2 Obtain the attribute-interdependences based weight-ing vector 120596119860119868 by use of DEMATEL method described in thefollowing Section 52

Step 3 Calculate synthesized attribute weighting vectoraccording to

120596 = 120572120596119860119868 + 120573120596119860119863 (19)

where 120572 and 120573 are parameters to reflect decision character-istics of decision organizations 0 le 120572 120573 le 1 120572 + 120573 =1Step 4 Check requirements for order inducing If no addi-tional order inducing required then go to Step 5 otherwisego to Step 6

Step 5 Utilize generalized aggregation operators to get theoverall IVDHF UUBLNs 119903119895(119895 = 1 119898) for each alternative119909119895 Here we take GIVDHFUUBLHA operator for examplebecause it can include other traditional operators as its specialcases Therefore we have

119866119868119881119863119867119865119880119880119861119871119867119860(1199031119895 1199032119895 119903119899119895) = ( 119899⨁119894=1

(119908119894 (119903120590119894119895)120582))1120582= ⋃(119904120590119894119895ℎ120590119894119895 119892120590119894119895

)isin119903120590119894119895

([119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590119894119895

))))120582)1120582 119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590119894119895

))))120582)1120582]

⋃[120583119871120590119894119895120583

119880120590119894119895]isinℎ120590119894119895[]

119871120590119894119895]119880120590119894119895]isin119892120590119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120590119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120590119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120590119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120590119894119895)120582)119908119894)1120582]])

(20)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of 119903119894119895119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weightvector 119903120590119894119895 is the 119894th largest of 119903119894119895 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))

119904119899120596119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119894119895))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (21)

Complexity 15

Then go to Step 8

Step 6 By use of the TOPSIS-based method developed in thefollowing Section 53 we determine order-inducing vectors 120576to reflect decision-makersrsquo group attitudinal characters

Step 7 Utilizing induced generalized aggregation operatorsto get the overall IVDHF UUBLNs 119903119895(119895 = 1 2 119898) foreach alternative 119909119895 we have

119868-119866119868119881119863119867119865119880119871119867119860(1199031119895 1199032119895 119903119899119898) = ( 119899⨁119894=1

(119908119894 (119903120587119894119895)120582))1120582

= ⋃(119904120587119894119895ℎ120587119894119895119892120587119894119895)isin119903120587119894119895

([119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120587119894119895

))))120582)1120582 119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587119894119895

))))120582)1120582]

⋃[120583119871120587119894119895 120583

119880120587119894119895]isinℎ120587119894119895 []

119871120587119894119895]119880120587119894119895]isin119892120587119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120587119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120587119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120587119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120587119894119895)120582)119908119894)1120582]])

(22)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of119903119894119895 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 119903120587119894119895 is the 119894th largest of 119903119894119895 reordered by theorder-inducing vectors 120576 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))119899120596119894120572119894119895

119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120573119894119895 ))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (23)

Step 8 Calculate scores 119878(119903119895) and accuracy degrees 119875(119903119895) for119903119895(119895 = 1 2 119898)Step 9 According to Definition 5 output ranking order ofalternative contractors

In Algorithm I to objectively derive unknown weight-ing vector for evaluative attributes a synthesized attributeweighting vector is devised by fusing two parts of concern(i) argument-dependent weighting vector and (ii) attribute-interdependences based weighting vector The former is toreflect effects of attribute values on attribute weights whilethe latter is to exploit effects of interdependences amongattributes on their weights For clarity processing flowchartof Algorithm I is depicted in Figure 3

51 Determining Argument-Dependent Attribute WeightingVector 120596119860119863 In order to derive attribute weighting vec-tor appropriately from assessments maximizing deviationmethod [91] is adopted here for its distinguishing ability

and objectivity [92] According to Wang [91] if performancevalues of each alternative have little differences under anattribute it implicates such an attribute plays a less importantrole otherwise it lives a more important role in decision-making Thus if one attribute has similar values across alter-natives it should be assigned a smaller weight contrariwiseit should be a bigger weight

In light of the idea in maximizing deviation methodwe develop a programming model to determine argument-dependent weighting vector 120596119860119863 For attribute 119860 119894 the stan-dard deviation of alternative 119909119895 to all the other alternatives isdenoted as

119865119894119895 (120596119860119863) = 119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (24)

Let119865119894(120596119860119863) represent deviation value of all alternatives toother alternatives according to attribute119860 119894 and be formulatedas follows

16 Complexity

Attribute weighting

Hesitant fuzzy linguistic MADM with attributes interdependency and decision makersrsquo group attitudinal characters

Step 9 Output ranking order of alternatives

interdependency among attributes by DEMATEL in Section 52

attributes values by standard deviation method in Section 51

Step 6 Ensure TOPSIS order inducing variables proposed in Section 53

Step 7 Utilizing proposed induced generalized hybrid aggregation operators developed in Section

41 to get overall value of each alternative

Step 8 Calculate score degrees and accuracy degrees

Step 4 Order inducingYes

Step 5 Choose proposed generalized operators in Section 41 to get overall value of each alternative

No

Collect decision matrix in the form of IVDHF_UUBLS

Complex problems of green contractor selection

Step 1 Weighting vector AD derived from Step 2 Weighting vector AI derived from

Step 3 Obtain synthesized weighting vector of attributes =AI + AD

Figure 3 Processing flowchart of Algorithm I

119865119894 (120596119860119863) = 119898sum119895=1

119865119894119895 (120596119860119863) = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (25)

Based on the analysis above we can get followingprogramming model (M-1) for obtaining optimal attributeweighting vector 120596119860119863

(119872 minus 1)max 119865 (120596119860119863) = 119899sum

119894=1119865119894 (120596119860119863) = 119899sum

119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894st 119899sum

119894=1(120596119860119863119894 )2 = 1 120596119860119863119894 ge 0 119894 = 1 2 119899 (26)

where 119889(119903119894119895 119903119894119897) can be calculated by Definition 6To solve model (M-1) we can construct Lagrange func-

tion

119871 (120596119860119863119894 120577) = 119899sum119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894+ 1205772 ( 119899sum

119894=1

(120596119860119863119894 )2 minus 1) (27)

where 120577 is the Lagrange multiplierDifferentiate with respect to 120596119860119863119894 (119894 = 1 2 119898) and 120577

and set these partial derivatives equal to zero the followingset of equations can be obtained

120597119871120597120596119860119863119894 = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) + 120577120596119860119863119894 = 0120597119871120597120577 = 119899sum

119894=1

(120596119860119863119894 )2 minus 1 = 0 (28)

By normalizing the solutions of above equations we canget a simple and exact formula for determining the attributeweights as follows

120596119860119863119894 = sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897)sum119899119894=1sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897) (29)

52 Deriving Attributes Interdependency-Based AttributeWeighting Vector 120596119860119868 Some efforts [53 60 84 93ndash95]

Complexity 17

have been made to address interdependencies betweenattributes in MADM However they unanimously cannotreveal the interdependences in a more complete way WhileDEMATAEL technique [63 96] gathers collective knowledgeto capture influences between attributes more precisely andcompletely so that weighting vector for attributes can berationally determined according to strength measure ofthose influences [64 97] Therefore in tackling the complexMADMwith interdependent attributes under IVDHF UUBLenvironments we present a DEMATEL-based method as thefollowing Algorithm II to derive attribute weighting vector120596119860119868 from the viewpoint of attributes interdependency

Algorithm II DEMATEL-basedmethod for ensuringweight-ing vector 120596119860119868Step 1 Construct initial direct-influence matrix 119885 = (119911119894119896)119899times119899

For attributes 119860 = 1198601 1198602 119860119899 decision-makersare asked to give direct effect that attribute 119860 119894 has onattribute 119860119896 using a five-point scale with scores representedby natural language (1) 0 mdash lsquoabsolutely no influencersquo (2)1 mdash lsquolow influencersquo (3) 2 mdash lsquomedium influencersquo (4) 3 mdashlsquohigh influencersquo (5) 4mdash lsquovery high influencersquoThen the initialdirect-influence matrix 119885 = (119911119894119896)119899times119899 can be established witheach element being the mean of same factors in differentdirect matrices

Step 2 Calculate normalized direct-influence matrix 119883 =(119909119894119896)119899times119899 Using matrix 119885 the normalized direct-influencematrix 119883 = (119909119894119896)119899times119899 is calculated according to

119909119894119896 = 119911119894119896119904 (30)

where 119904 = maxmax1le119894le119899sum119899119896=1 119911119894119896max1le119896le119899sum119899119895=1 119911119894119896 Allelements in matrix 119883 are complying with 0 le 119909119894119896 le 10 le sum119899119896=1 119909119894119896 le 1 and 0 le sum119899119894=1 119909119894119896 le 1 and at least onerow or column of the summation equals 1

Step 3 Derive total-influence matrix 119879 = (119905119894119896)119899times119899 Basedon matrix 119883 the total-influence matrix 119879 = (119905119894119896)119899times119899 can bederived by summing the direct effects and all of the indirecteffects

119879 = 119883 + 1198832 + 1198833 + sdot sdot sdot + 119883ℎ = 119883 (119868 minus 119883)minus1 (31)

where 119868 represents identity matrix When ℎ 997888rarr infin 119883ℎ =(0)119899times119899Step 4 Compute influencing and influenced degrees of eachattribute The sum of rows and the sum of columns withinthe total-influence matrix 119879 are respectively expressed as thevectors 119903 and 119888

119903 = (119903119894)119899times1 = ( 119899sum119896=1

119905119894119896)119899times1

119888 = (119888119896)119899times1 = ( 119899sum

119895=1

119905119894119896)119879

1times119899

(32)

where 119903119894 denotes the sumof the 119894th row inmatrix119879 and showsthe sum of direct and indirect effects that attribute 119860 119894 hason other attributes Similarly 119888119896 shows the sum of direct andindirect effects that attribute 119860119896 has receivedStep 5 Estimate weights of attributes based on influencedegrees according to

120596119868lowast119894 = [(119903119894 + 119888119894)2 + (119903119894 minus 119888119894)2]12 (33)

Then the weighting vector 120596119860119868 can be obtained by normaliz-ing (33)

120596119860119868119894 = 120596119868lowast119894sum119899119894=1 120596119868lowast119894 (34)

53 Obtaining Order Inducing Variables 120576 by TOPSIS-BasedMethod For ill-defined MADM problems induced aggre-gation operators [89 90 98] provide an effective channelfor experts to express complex attitudinal characters Thekey step in induced aggregation operators is to obtainorder-inducing vectors 120576 which however are presumptivelyassigned in most papers [62] Therefore in this sectionaiming at deducing the order-inducing vectors 120576 rationallywe here devise a TOPSIS-based method in Algorithm IIIObviously the Algorithm III is of generality for MADMwithdifferent forms of linguistic expression

Algorithm III TOPSIS-based method for obtaining orderinducing vectors 120576Step 1 Let the decision maker define IVDHF UUBL idealsolution(s) 119903+119894 and IVDHF UUBL negative ideal solution(s)119903minus119894 corresponding to each attribute in which

119903+119894 = (119903+1198941 119903+1198942 119903+119894119901) 119903minus119894 = (119903minus1198941 119903minus1198942 119903minus119894119902) (35)

where 119901 and 119902 denotes the number of 119903+119894 and 119903minus119894 respectively

Step 2 Calculate distances from each element in decisionmatrix to ideal solution(s) and negative solution(s) respec-tively

119889+119894119895 = max119897=12119901

119889 (119903119894119895 119903+119894119897 ) 119889minus119894119895 = min

119897=12119902119889 (119903119894119895 119903minus119894119897 ) (36)

where 119889(119903119894119895 119903+119894119897 ) and 119889(119903119894119895 119903minus119894119897 ) are calculated by the distancemeasure in Definition 6 If 119901 gt 1 we choose the onewith maximum distance if 119902 gt 1 we choose the one withminimum distance

Step 3 Calculate coefficients according to

119888119894119895 = 119889minus119894119895119889+119894119895 + 119889minus119894119895 119894 = 1 2 119899 119895 = 1 2 119898 (37)

18 Complexity

Table 2 Decision matrix in the form of IVDHF UUBLS1199091 1199092 11990931198601([AHVH] [02 03][02 04] [03 04]) ([HT] [04 05][05 06] [02 03] [02 04]) ([ALAH] [01 02][01 03] [06 07])1198602([MH] [05 06][02 03]) ([AHT] [06 07][01 02] ) ([MVH] [02 03][05 06] [06 07])1198603([VLAL] [03 04][04 05] [05 06]) ([MVH] [06 07] [07 08][01 02]) ([VLL] [04 05][02 03])1198604([MAH] [02 04][05 06]) ([LAL] [03 05][02 03]) ([ALM] [06 07][01 02] [02 03])1198605([ALM] [04 05][01 02] [04 05]) ([AHH] [06 07][01 02]) ([HVH] [04 05][06 07] [01 03])1198606([LM] [01 02][03 05] [03 05]) ([VLL] [02 04] [05 06][02 03]) ([AHH] [03 04][04 05])1198607([VLL] [04 05][05 06] [02 03] [02 04]) ([MAH] [05 06] [07 08][01 02]) ([ALM] [04 05][05 06] [03 04])1198608([ALAH] [01 03][04 06]) ([AHH] [05 07][01 02] [02 03]) ([AHH] [04 06][03 04])

Table 3 Direct influence matrix Z1198601 1198602 1198603 1198604 1198605 1198606 1198607 11986081198601 0 3 4 1 3 1 2 41198602 1 0 2 4 4 2 3 11198603 4 2 0 1 3 2 4 11198604 2 3 1 0 3 4 1 21198605 4 2 1 3 0 1 4 31198606 2 4 3 4 4 0 4 21198607 4 1 2 3 3 1 0 41198608 1 2 4 3 3 2 4 0

Step 4 Derive order inducing variable 120576119894119895 by descendingorder of 119888119894119895 The greater the value 119888119894119895 the bigger the 1205761198941198956 Illustrative Example and Experiments

61 Illustrative Case Study Suppose a mega infrastructureproject is selecting green contractor for one of its subprojectsA panel of decision-makers including managers professorsand competitive intelligence experts has been organized tocomprehensively evaluate three alternative contractors (ie1199091 1199092 and 1199093) under the eight attributes (ie 1198601ndash1198608) listedin Section 2

To elicit true decision preferences we apply theIVDHF UUBLS to assess the three alternatives (1199091ndash1199093)Firstly decision-makers are asked to vote on 119909119895 under eachattribute119860 119894 using different unbalanced linguistic term sets 1198781 1198782 and 1198783 where 1198781 = 119873 119871119860119871119872119860119867119867119876119867119881119867119860119879 1198791198782 = 119873119860119873119881119871 119876119871 119871 119860119871119872119860119867119881119867 119879 1198783 = 119873119881119871119871119860119872119872119876119872119860119867119867 119879 In Figure 4 we show the adoptedlinguistic hierarchies and relationship between 1198781 1198782 and1198783 Attributes 1198601 1198602 1198605 and 1198608 are evaluated by use of 1198781attributes11986031198604 and1198607 are evaluated by 1198782 and attribute1198606is evaluated by 1198783 The highest-voted uncertain unbalancedlinguistic term [119904119898 119904119899] are identified Secondly the panel of

decision-makers is asked to further express their opinionson membership degrees ℎ(119909) and nonmembership degrees119892(119909) to [119904119898 119904119899] Then we can derive compound assessments(ie ⟨119909 119904120599(119909) ℎ(119909) 119892(119909)⟩) in the form of IVDHF UUBLSand obtain the decision matrix in Table 2 Additionallyin order to examine the interdependency among attributesdecision-makers are also required to assess the influentialrelations among the eight attributes using DEMATEL-basedAlgorithm II The direct influence matrix is collected andshown in Table 3

Next we apply the proposed Algorithm I to solve thisgreen contractor selection problem To derive the synthe-sized attribute weighting vector in Algorithm I we assign120572 = 120573 = 05 For scenarios with no group attitudinalcharacters operator GIVDHFUUBLHA in Definition 12 ischosen In contrast for scenarios with group attitudinalcharacters operator I-GIVDHFUUBLHA in Definition 13will be chosen Further the panel of decision-makers is askedto determine their positive and negative ideal alternativesthat are collected in Table 4 where corresponding order-inducing vectors that are derived from Algorithm III arealso listed (see the rightmost column) Thereafter operatorI-GIVDHFUUBLHA in Definition 13 is utilized for aggrega-tion For GIVDHFUUBLHA and I-GIVDHFUUBLHA we

Complexity 19

Table 4 Ideal alternatives (119903+ and 119903minus) and derived order-inducing vectors 120576119903+ 119903minus 1205761199091 1199092 11990931198601 ([VHT] 1 0) ([NL] 1 0) 5 6 61198602 ([MAH] [07 08] [01 02]) ([NL] 1 0) 1 1 41198603 (T 1 0) (L 1 0) 8 7 81198604 ([ALAH] 09 01) (T 1 0) 2 2 11198605 ([HT] [09 1] 0) (N 1 0) 7 5 51198606 (M 1 0) (N 1 0) 6 3 31198607 (M 1 0) ([NL] 1 0) 3 8 71198608 ([MAH] 1 0) (N 1 0) 4 4 2

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 1 2 3 4

0 1 2

N L AH H QH VH AT TM

Level 1 l(13)

Level 2 l(25)

Level 3 l(39)

Level 4 l(417)

Unbalanced linguistic term set S1

N AH VH TM Unbalanced linguistic term set S2VL L ALAN QL

N H TM Unbalanced linguistic term set S3

L AM AHQM

AL

VL

Figure 4 Unbalanced linguistic term sets (1198781 1198782 and 1198783) and their mapping in linguistic hierarchies

assume 120582=2 and 119908 = (0 0 015 025 025 025 01 0)119879Note that 119908 is a position weighting vector which is derivedby the fuzzy semantic quantitative operator [99] Aftercompleting all steps in Algorithm I (see Appendix C) weobtain the ranking results and present them in Table 5 Forthe scores derived from Step 5 we find the ranking order is1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the best solution Step 7gives the ranking order of 1199092 ≻ 1199091 ≻ 1199093 indicating 1199092 is stillthe most appropriate one while 1199091 is better than 1199093 under theinfluence of 12057662 Experiments on Ranking Results along with VariousParameter Configurations In order to inspect the impactsof various parameter configurations on ranking results inthis section more numerical examples are carried out Alleight operators that defined in Section 41 are applied to theabove illustrative case where the parameter 120582 is configured asintegers ranging from 1 to 20 and parameter 120572 is configuredto a value set of 0 03 05 08 09 1

Firstly the four basic aggregation operators of GIVDH-FUUBLWA GIVDHFUUBLWG GIVDHFUUBLOWA and

Table 5 Ranking results of operators GIVDHFUUBLHA and I-GIVDHFUUBLHA

Aggregation operator Ranking orderGIVDHFUUBLHA 1199092 ≻ 1199093 ≻ 1199091I-GIVDHFUUBLHA 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG are applied to the former case Thesame configuration 120572 = 120573 = 05 is chosen while 120582 is testedas integers ranging from 1 to 20 The ranking results derivedfrom different parameter combinations have been collectedin Table 6 As can be seen four operators unanimouslyidentify the contractor 1199092 is the best one no matter how120582 changes Regarding priority relation between 1199091 and 1199093GIVDHFUUBLWA and GIVDHFUUBLOWA generate thesame result of1199093 ≻ 1199091 while results fromGIVDHFUUBLWGandGIVDHFUUBLOWGgot a change to 1199091 ≻ 1199093 at a certainvalue of parameter 120582

Actually GIVDHFUUBLWA and GIVDHFUUBLOWAare generalized version of weighted arithmetic aggregationoperators that producemore favorable results by emphasizing

20 Complexity

Table 6 Ranking orders obtained by GIVDHFUUBLWA GIVDHFUUBLOWA GIVDHFUUBLWG and GIVDHFUUBLOWG along with120582Operators 120582 (Integer) Ranking orderGIVDHFUUBLWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLOWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLWG [1 2] 1199092 ≻ 1199093 ≻ 1199091[3 20] 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG [1 8] 1199092 ≻ 1199093 ≻ 1199091[9 20] 1199092 ≻ 1199091 ≻ 1199093

Table 7 Ranking results obtained by GIVDHFUUBLHA and I-GIVDHFUUBLHA along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHA I-GIVDHFUUBLHA120572=0 120573=1 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=03 120573=07 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=05 120573=05 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=08 120573=02 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=09 120573=01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=1 120573=0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093

Table 8 Ranking results obtained by GIVDHFUUBLHG and I-GIVDHFUUBLHG along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHG I-GIVDHFUUBLHG

120572 = 0 120573 = 1 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 03 120573 = 07 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 05 120573 = 05 [1 8] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[9 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 08 120573 = 02 [1 17] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[18 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 09 120573 = 01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572 = 1 120573 = 0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093membership degrees to a uncertain linguistic term thus canbe considered as optimistic operators while GIVDHFU-UBLWG and GIVDHFUUBLOWG are generalized geomet-ric aggregation operators that producemore favorable resultsby emphasizing nonmembership degrees thus can be consid-ered as pessimistic operators [100 101]Therefore the rankingresults in Table 6 could be perceived as for optimism-baseddecision-making alternative 1199093 always stands out better than1199091 when 120582 gradually amplifies degree of optimism but forpessimism-based decision-making 1199091 turns out better than1199093 when 120582 gradually amplifies degree of pessimism 120582 canbe considered as the parameter for amplifying degree ofoptimism or pessimism [101]

Next to inspect the impacts of various combinationsof 120596119860119868 and 120596119860119863 on ranking results another four aggrega-tion operators of GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG also have beenapplied to the above illustrative case with varying 120572 120573 and 120582

All results have been collected in Tables 7 and 8 for compari-son As can be seen fromTable 7 ranking results generated byoperator GIVDHFUUBLHA keep the same as 1199092 ≻ 1199093 ≻ 1199091for all test combinations of parameters However as shownin Table 8 although GIVDHFUUBLHG is also capable ofidentifying 1199092 as the best contractor the priority relationsbetween 1199091 and 1199093 hold similar changing patterns in rankingresults in Table 6 that were output byGIVDHFUUBLWGandGIVDHFUUBLOWG Regarding decision situations whereobjective attributeweighting vector120596119860119863 is only considered (120572= 0) or its dominant importance (such as 120572 = 03) is accepted1199091 ≻ 1199093 is a clear observation when parameter 120582 is set toan integer in interval [8 20] Contrariwise under decisionscenarios where subjective attribute weighting vector 120596119860119868 isonly needed (120572 = 1) or its dominant importance (such as 120572 =09) is acknowledged 1199093 ≻ 1199091 holds all the way 120582 varies

In order to more clearly show the score trajectories ofalternative contractors along variations of parameter 120582 we

Complexity 21

HAInducedHA

InducedHA-2HA-2

HA-3

InducedHA-1InducedHA-3

= =

HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(a) 120572 = 0

HAInducedHA

= 03 = 07

InducedHA-2

HA-2HA-3

InducedHA-1 InducedHA-3HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(b) 120572 = 03

HAInducedHA

= 5 = 05

InducedHA-2HA-2

HA-3

InducedHA-1 InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(c) 120572 = 05

HAInducedHA

= 08 = 02

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572 = 08

HAInducedHA

= 9 = 01

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(e) 120572 = 09

HAInducedHA

246 2462 2464 2466 2468 247

= 1 = 0InducedHA-2HA-2

HA-3

HA- 3

HA-3

InducedHA-1

InducedHA-3

HA-1

HA-1

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

1085

1086

1087

(f) 120572 = 1

Figure 5 Score trajectories obtained by aggregation operators HA and InducedHA with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

further configure parameter 120582 to the value set of [0 01100]and conduct computational experiments to collect scorevalues and ranking results (as shown in Figures 5 and 6) byfour aggregation operators of GIVDHFUUBLHA GIVDH-FUUBLHG I-GIVDHFUUBLHAand I-GIVDHFUUBLHGFor more clarity in Figures 5 and 6 please note that weuse HA HG Induced-HA and Induced HG to respec-tively denote GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG In Figure 5it can be observed that when dominant importance of 120596119860119863

is emphasized (120572 = 0 120572 = 03 120572 = 05) GIVDHFUUBLHAis capable of differentiating the three alternative contractorsas same ranking order of 1199092 ≻ 1199093 ≻ 1199091 for all 120582 but withincreases in 120572 the margins between scores of 1199091 and 1199093 nar-rows and the rank order changes to 1199091 ≻ 1199093when 120572 gets 09and 1 Interestingly in Figure 6 although GIVDHFUUBLHGis able to significantly differentiate alternative contractors 1199091and 1199093 when dominant importance of 120596119860119863 is emphasized (120572= 0 120572 = 03 120572 = 05) rank order of 1199091 and 1199093 obviouslychanges once from 1199093 ≻ 1199091 to 1199091 ≻ 1199093 when 120582 is bigger

22 Complexity

HGInducedHG

= = 1

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(a) 120572= 0

HGInducedHG

= 3 = 07

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(b) 120572= 03

HGInducedHG

= 05 = 05

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(c) 120572= 05

HGInducedHG

= 8 = 02

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572= 08

HGInducedHG

292 293 294 295 296 297minus08

minus0795

minus079

= 09 = 01

HA-1

HA-3

InducedHG-2

HG-1

HG-1

HG-2

HG-3

HG-3 InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4

minus3

minus2

minus1

0

1

2

scor

e val

ues

(e) 120572= 09

HGInducedHG

50 60 70 80 90 100minus095minus09

minus085

= 1 = 0

HG-x3

HG-x1

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(f) 120572= 1

Figure 6 Score trajectories obtained by aggregation operators HG and InducedHG with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

than 8 When 120572 increases to 1 GIVDHFUUBLHGmaintainsthe rank order of 1199093 ≻ 1199091 but their score trajectorieswalk very close Comparatively I-GIVDHFUUBLHA andI-GIVDHFUUBLHG generate consistent ranking results of1199092 ≻ 1199091 ≻ 1199093 no matter 120582 varies in all the above experiments

In sum when parameters are configured with typical val-ues (ie 120572 = 05 and 120582=1) and no additional attitudinal infor-mation is taken into consideration the proposed aggregationoperators unanimously derive the same rank order 1199092 ≻1199093 ≻ 1199091 of for the three alternative contractors However

with the amplification effect of parameter 120582 analysis resultsfrom linear combination of 120596119860119868 and 120596119860119863 reveal alternative1199092 which consistently is the best choice but rank order of1199091 and 1199093 changes because their score trajectories get closeand crossed which probably indicates that only decision-making informationderived from the decisionmatrix is inad-equate to significantly differentiate alternatives 1199091 and 1199093While by integrating additional decision information aboutattitudinal characters of group experts I-GIVDHFUUBLHAand I-GIVDHFUUBLHG manage to generate the identical

Complexity 23

Table 9 Order-inducing vectors obtained by our TOPSIS-based method for problem in [38]

1198661 1198662 1198663 11986641198601 02522 01629 02282 027141198602 01 05243 01364 046891198603 04081 01756 05244 037791198604 04507 01409 01064 013221198605 00697 02203 03753 01161

Table 10 Order-inducing vectors obtained by our TOPSIS-based method for problem in [39]

1198661 1198662 1198663 11986641198601 04548 04969 04923 047321198602 05092 047 04768 046451198603 07192 04888 04953 049151198604 06722 06163 05706 049291198605 06104 04942 06463 04946

Table 11 Ranking results obtained by applying different order-inducing vectors to [38 39]

Problems Approaches Ranking results

Problem in [38] IHFLOWA [38] + order-inducing vectors in [38] 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198605 ≻ 1198604

IHFLOWA [38] + order-inducing vectors in Table 9 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198604 ≻ 1198605

Problem in [39] I-IVDHFLOWA [39] + order-inducing vectors in [39] 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198602 ≻ 1198601

I-IVDHFLOWA [39] + order-inducing vectors in Table 10 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602

permutation of 1199092 ≻ 1199091 ≻ 1199093 which verifies that ourproposed TOPSIS-based order-inducing approach is prac-tical and useful in helping work out decision results morereasonably and robustly

63 Application Studies of Proposed TOPSIS-Based Order-Inducing Method When confronting complex decision-making situations that need additional attitudinal charactersof decision-makers Yager and Filev [88] and Yager [102]introduced and encouraged to employ order-inducing vectorto represent those attitudinal characters Since then theidea of order-inducing vector has been extended to variousdecision-making environments [98 103] such as those basedon intuitionistic fuzzy information [104 105] linguistic infor-mation [106] hesitant fuzzy linguistic information [38] andinterval-valued dual hesitant fuzzy uncertain linguistic infor-mation [39] Generally in existing literature order-inducingvectors rely on experts to directly give [38 39 104ndash106]

In order to examine the adaptability of our TOPSIS-based order-inducing method to linguistic decision-makingenvironments we respectively apply the method to resolvethe same problems in references [38 39] Due to fact thatthere is no descriptions about how the order-inducing vectorswere derived in both references [38 39] we here applyextremums into our TOPSIS-based method to obtain order-inducing vectors that is (1199047 1 0) and (1199041 0 1) for[38] ([1199046 1199046] 1 0) and ([1199040 1199040] 0 1) for [39] Thenwe obtain our order-inducing vectors listed in Table 9 forthe problem adopted in [38] and our order-inducing vectorslisted in Table 10 for [39]

Now we apply the order-inducing vectors as shownin Tables 9 and 10 to resolve the problems in [38] and[39] respectively Regarding the problem in [38] the scorevalues of all alternatives are calculated as 119878(1198601)=02602119878(1198602)=02417 119878(1198603)=03577 119878(1198604)=0221 and 119878(1198605)=0185so the corresponding ranking result is 1198603 ≻ 1198601 ≻ 1198602 ≻1198604 ≻ 1198605 As for the problem in [39] the score values ofall alternatives are computed as 119878(1198601)=39954 119878(1198602)=3783119878(1198603)=70827 119878(1198604)=71792 and 119878(1198605)=5813 we thus obtainthe ranking result 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602 Formore clarity brief descriptions of comparative approachesand obtained ranking results also have been put together inTable 11

According to Table 11 the approaches that employ ourorder-inducing vectors derive the same ranking order ofthe better three alternatives for both comparative problemsfrom [38 39] As seen our proposed TOPSIS-based order-inducing method exhibits effective and more understand-able Furthermore when decision-makers are incapable ofdirectly put forward all desired order-inducing vectors due tocomplexity in ill-structured problems our method providesa feasible directions to obtain order-inducing vectors byfiguring out ideal solutions Comparatively our proposedTOPSIS-based order-inducing method behaves effective andmore operational in practical use

64 Comparison with Decision-Making Approach Based onDegraded form of IVDHFULS Note that our approachis intrinsically adaptive to decision situations based ondegraded forms of IVDHF UUBLS such as the dual hesitant

24 Complexity

fuzzy linguistic sets introduced by Yang and Ju [54] Tofurther inspect feasibility of our approach we here conductcomparative study on the same case adopted by Yang and Ju[54] We firstly convert their original decision matrix intothe form of IVDHF UUBLS crisp membership and non-membership degrees are equivalently changed to intervalswith same upper and lower limits (eg 02 turns out to be[02 02]) and linguistic variables are rewritten as uncertainlinguistic variables (eg 1199042 turns out to be [11990421199042])

Since the case only got three attributes among whichinterdependency could be not significant under generalassumption weighting vector 120596119860119868 is not calculated Andfor straight comparison order inducing vector is notincluded either Then the argument-dependent attributeweighting vector can be obtained from Eq(29) as 120596119860119863 =(03192 03132 03676)119879 By use of Algorithm Iwith operatorIVDHFUUBLWA (ie GIVDHFUUBLWA in case of 120582 = 1see Appendix B) we have the scores of alternatives 119904(1198601) =03412 119904(1198602) = 01354 119904(1198603) = 03932 119904(1198604) = 07352Therefore the ordering of alternatives is1198604 ≻ 1198603 ≻ 1198601 ≻ 1198602which is mainly in alignment with the ordering of1198604 ≻ 1198601 ≻1198603 ≻ 1198602 in Yang and Ju [54] It is worth noticing that 120596119860119863used in our approach is derived objectively from assessmentswhile attribute weights 120596lowast = (035 025 040)119879used by Yangand Ju [54] were assigned empirically Differences between120596119860119863 and 120596lowast lead to different ordering of1198601 and1198603 Compar-atively speaking our approach holds more adaptability andobjectivity

In general our proposed MADM approach in this paperexhibits an effective way to tackle complex green contractorselection problems in which subjective attributes weightingvector (such as120596119860119868) and objective attributes weighting vector(such as 120596119860119863) exist To comprehensively utilize these twofundamental facets of attributes weighting information ourcase study and computational experiments suggest that linearcombination method (ie 120596 = 120572120596119860119868 + 120573120596119860119863) is effectiveto reflect effects of 120596119860119868 and 120596119860119863 by configuring parameter 120572or 120573 When there is no specific expert opinions to specify orhard to obtain we suggest that parameter 120572 chooses the valueset of 0 02 05 08 1 Furthermore generalized aggrega-tion operators are suggested to inspect score trajectoriesof alternative contractors along increasing of the operatorsrsquoamplification parameter 120582 If some alternative contractorsrsquoscore values get very close or crossed so that they cannot bedifferentiated from each other significantly we suggest to askthe panel of experts to derive opinions of ideal solutions andutilize our TOPSIS-based order-inducing method to includethose expert opinions as attitudinal decision informationthereby obtaining more rational and robust decision results

7 Conclusions

Aiming at tackling complex decision-making problems ofgreen contractor selection that often hold simultaneouslythe characteristics of decision hesitancy attributes inter-dependency and group complex attitudinal characters wehave developed an effective IVDHF UUBLS-based approach

We utilize IVDHF UUBLS to elicit complicate decision-makersrsquo assessments more objectively and comprehensivelywe employ DEMATEL for addressing attribute interdepen-dency more precisely and we determine group attitudinalcharacters as order-inducing vectors based on ideal attitudesof experts rather than by empirical assignment The mainadvantages of our approach are (i) IVDHF UUBLS offers anadequate way to depict decision hesitancy (ii) the designeddistance measure for IVDHF UUBLS avoids informationdistortion of traditional ones which artificially add mis-matching membership or nonmembership degrees (iii) thedeveloped generalized operators include traditional ones asspecial cases so as to provide flexibility in MADM basedon IVDHF UUBLS (iv) the devised synthesized attributeweighting scheme provides an adaptive way to consider twonecessary aspects attributes interdependency-based weightsand argument-dependent weights (v) the TOPSIS-basedmethod can deduce reasonable order-inducing vectors ratherthan empirical ones and holds generality in other linguisticdecision environments Results of case studies and experi-ments have verified the validity of the proposed approach

When multiple decision organizations are involvedgroup decision-making based on IVDHF UUBLS is animportant research direction Besides inducing preferencesfor decision-making problems of high uncertainty interactivemechanisms should also be studied to extract preferences Inaddition empirical studies of the proposed approach can becarried out to various areas such as investment evaluationproject management and vehicle selection in fleet operations

Appendix

A Basic Notions for IVDHFS and UnbalancedLinguistic Term Set

Definition A1 (see [47]) Letting 119883 be a fixed set then anIVDHFS on 119883 is defined as119863 = ⟨119909 ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (A1)

where ℎ(119909) = ⋃[120583119871120583119880]isinℎ(119909)120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and119892(119909) = ⋃[]119871]119880]isin119892(119909)] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two setsof interval values in [0 1] denoting possible membershipand non-membership degrees of element 119909 isin 119883 to theset 119863 respectively with conditions 120583 ] isin [0 1] and0 le (120583119880)+ + (]119880)+ le 1 and for all 119909 isin 119883 (120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880

For convenience normally 119889 = ℎ 119892 is called an interval-valued dual hesitant fuzzy element (IVDHFE) and 119863 is theset of all IVDHFEs

Definition A2 (see [107]) Suppose 119878 = 119904119895 | 119895 = 0 1 119892 minus1 is a finite and totally ordered discrete linguistic term setwhere 119904119895 represent possible values for a linguistic variableand 119892 is an odd cardinality 119878 is uniformly and symmetricallydistributed if the following two conditions are satisfied (1)

Complexity 25

there exists a unique constant 120582 gt 0 such that 119873119878(119904119894) minus119873119878(119904119895) = 119897(119894 minus 119895) for all 119894 119895 = 0 1 119892 minus 1 (2) Let 119878119877 =119904 | 119904 isin 119878 119904 gt 119878119862 and 119878119871 = 119904 | 119904 isin 119878 119904 lt 119878119862 Let C(119878119877) andC(119878119871) be the cardinality of 119878119877 and 119878119871 then C(119878119877) = C(119878119871)If 119878 is uniformly and symmetrically distributed then 119878 iscalled a balanced linguistic term set Otherwise 119878 is calledan unbalanced linguistic term set

The following 2-tuple fuzzy linguistic representationmodel extends traditional linguistic term set to a continuouscase so as to facilitate computing with linguistic variables

Definition A3 (see [59]) Let 119878 = 1199040 1199041 119904119892minus1 be alinguistic term set and 120573 isin [0 119892] Then a 2-tuple fuzzylinguistic variable expresses the equivalent information to 120573is defined as

Δ [0 119892] 997888rarr 119878 times [minus05 05) (A2)

Δ (120573) = (119904119894 120572) with

119904119894 119894 = 119903119900119906119899119889 (120573)120572 = 120573 minus 119894 120572 isin [minus05 05)

(A3)

Δminus1 (119904119894 120572) = 119894 + 120572 = 120573 (A4)

where 119903119900119906119899119889() is the usual rounding operation and120572 is calledsymbolic translation

To obtain 2-tuple fuzzy linguistic representations ofunbalanced linguistic terms the concept of linguistic hierar-chies ie 119871119867 = ⋃119905 119897(119905 119899(119905)) is used 119897(119905 119899(119905)) is a linguistichierarchy with 119905 indicating the level of hierarchy and 119899(119905)denotes the granularity of the linguistic term set of 119905 Herreraet al [59] defined the following transformation functionsbetween labels from different levels in multigranular linguis-tic information contexts without loss of information

Definition A4 (see [59]) In linguistic hierarchies 119871119867 =⋃119905 119897(119905 119899(119905)) whose linguistic term sets are represented by119878119899(119905) = 119904119899(119905)0 119904119899(119905)119899(119905)minus1 the transformation function froma linguistic label in level 119905 to a label in consecutive level 1199051015840 isdefined as 1198791198651199051199051015840 119897(119905 119899(119905)) 997888rarr 119897(1199051015840 119899(1199051015840)) such that

1198791198651199051199051015840 (119904119899(119905)119894 120572119899(119905))= Δ 1199051015840 (Δminus1119905 (119904119899(119905)119894 120572119899(119905)) (119899 (1199051015840) minus 1)119899 (119905) minus 1 ) (A5)

By use of the above transformation function any 2-tuplelinguistic representation can be transformed into a term in119871119867 Detailed transformation procedures are listed as follows

(1) Representation in linguistic hierarchy To transformthe unbalanced terms of 119878 into the corresponding terms inthe 119871119867 transformation function 120595 is employed to associateeach unbalanced linguistic 2-tuple (119904119894 120572) with its linguistic 2-tuple in 119871119867(119878) ie

120595 119878 997888rarr 119871119867(119878) (A6)

so that 120595(119904119894 120572) = (119904119866(119894)119868(119894) 120582) for forall(119904119894 120572) isin 119878(2) Computational phase Firstly transform (119904119866(119894)119868(119894) 120582) intolinguistic 2-tuples denoted as (119904119899(1199051015840)

1198681015840(119894) 1205821015840) in 119878119899(1199051015840) where

(119904119899(1199051015840)1198681015840(119894)

1205821015840) = 119879119865 (119904119866(119894)119868(119894) 120582)= Δ(Δminus1 (119904119866(119894)119868(119894) 120582) sdot (119899 (1199051015840) minus 1)119866 (119894) minus 1 ) (A7)

Then a computational model is used over 119878119899(1199051015840) with aresult denoted as (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)(3) Retranslation process The result (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)is transformed into the unbalanced term in 119878 by using thetransformation function 120595minus1 ie

120595minus1 119871119867 (119878) 997888rarr 119878 (A8)

so that 120595minus1(119904119899(1199051015840)119903 120582119903) = (119904119903119890119904119906119897119905 120582119903119890119904119906119897119905) isin 119878B Details of Theorem 14

Proof Given a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector for119904119889119895 and 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 120582 gt 0 then(1) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLWA and GIVDHFUUBLWG operators respectively(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLOWA and GIVDHFUUBLOWG operators respectively

(2) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to theIVDHF UUBL weighted average (IVDHFUUBLWA)and IVDHF UUBL weighted geometric (IVDHFUUBLWG)operators respectively where

26 Complexity

119868119881119863119867119865119880119880119861119871119882119860120596 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

120596119895119904119889119895= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus

119899prod119895=1

(1 minus 120583119871119895 )120596119895 1 minus 119899prod119895=1

(1 minus 120583119880119895 )120596119895]] [[

119899prod119895=1

(]119871119895)120596119895 119899prod119895=1

(]119880119895 )120596119895]])

(B1)

119868119881119863119867119865119880119880119861119871119882119866120596 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889119895120596119895

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[

119899prod119895=1

(120583119871119895 )120596119895 119899prod119895=1

(120583119880119895 )120596119895]] [[1 minus

119899prod119895=1

(1 minus ]119871119895)120596119895 1 minus 119899prod119895=1

(1 minus ]119880119895 )120596119895]])

(B2)

(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHA andGIVDHFUUBLHG operators reduce to the IVDHF UUBL

orderedweighted average (IVDHFUUBLOWA) and IVDHFUUBL ordered weighted geometric (IVDHFUUBLOWG)operators respectively where

119868119881119863119867119865119880119880119861119871119874119882119860119908 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

119908119895119904119889120590(119895)= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) )))

119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) )))

]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus

119899prod119895=1

(1 minus 120583119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus 120583119880120590(119895))119908119895]] [[

119899prod119895=1

(]119871120590(119895))119908119895 119899prod119895=1

(]119880120590(119895))119908119895]])

(B3)

119868119881119863119867119865119880119880119861119871119874119882119866119908 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889120590(119895)119908119895

= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[

119899prod119895=1

(120583119871120590(119895))119908119895 119899prod119895=1

(120583119880120590(119895))119908119895]] [[1 minus

119899prod119895=1

(1 minus ]119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus ]119880120590(119895))119908119895]])

(B4)

Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

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[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

[11] X Qi C Liang and J Zhang ldquoGeneralized cross-entropy basedgroup decision making with unknown expert and attributeweights under interval-valued intuitionistic fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 79 pp 52ndash642015

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[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

[16] B T Nuong K-W Kim L Prathumratana et al ldquoSustainabledevelopment in the mining sector and its evaluation usingfuzzy AHP (Analytic Hierarchy Process) approachrdquo GeosystemEngineering vol 14 no 1 pp 43ndash50 2011

[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

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[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

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[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

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[26] Z Hatush and M Skitmore ldquoContractor Selection UsingMulticriteria Utility Theory An Additive Modelrdquo Building andEnvironment vol 33 no 2-3 pp 105ndash115 1998

[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

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[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

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[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

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[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

[51] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

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[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

[55] Z Tao X Liu H Chen and L Zhou ldquoUsing new versionof extended t-norms and s-norms for aggregating intervallinguistic labelsrdquo IEEETransactions on SystemsManCyberneticsSystems vol vol p 15 2016

[56] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[57] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[58] L Martınez M Espinilla J Liu L G Perez and P J SanchezldquoAn evaluation model with unbalanced linguistic informationapplied to olive oil sensory evaluationrdquo Journal of Multiple-Valued Logic and Soft Computing vol 15 no 2-3 pp 229ndash2512009

[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

[62] L Marin A Valls D Isern A Moreno and J M MerigoldquoInduced unbalanced linguistic ordered weighted average andits application in multiperson decision makingrdquo The ScientificWorld Journal vol 2014 2014

[63] A Baykasoglu V Kaplanoglu Z D U Durmusoglu and CSahin ldquoIntegrating fuzzy DEMATEL and fuzzy hierarchicalTOPSIS methods for truck selectionrdquo Expert Systems withApplications vol 40 no 3 pp 899ndash907 2013

[64] S K Patil and R Kant ldquoA hybrid approach based on fuzzyDEMATEL and FMCDM to predict success of knowledge man-agement adoption in supply chainrdquoApplied Soft Computing vol18 pp 126ndash135 2014

[65] S Liyin and Y Hong ldquoImproving environmental performanceby means of empowerment of contractorsrdquo Management ofEnvironmental Quality An International Journal vol 17 no 3pp 242ndash257 2006

[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

[69] Y Kazancoglu I Kazancoglu and M Sagnak ldquoA new holisticconceptual framework for green supply chain management

performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

Hindawiwwwhindawicom Volume 2018

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Submit your manuscripts atwwwhindawicom

Complexity 7

e panelof decision

makers Alternativegreen

contractors

Evaluativeattributes

Comprehensiveevaluation

Attributesrsquo interdependency

+

alternativecontractor

vote

Voting results on

MajorityRule

radic

Specificdecision

matrix

AppropriateMADM

approach

Decisionresults

Section 3 Defines an effectivehybrid expression tool

Group attitudinal characters (ie complex degrees of orness)

s

⟨Contractori [s4 s5] ℎ (x) g(x)⟩

i-th

1 2 3 4 5 6 7

j-th attribute

Figure 1 Conceptual MADMmodel for tackling complexity in green contractor selection

in [0 1] ℎ(119909) denotes possible membership degrees that 119909belongs to 119904120579(119909) and119892(119909) represents possible nonmembershipdegrees of 119909 to 119904120579(119909) ℎ(119909) and 119892(119909) hold conditions 120583 ] isin[0 1] and 0 le (120583119880)+ + (]119880)+ le 1 where(120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880 for all 119909 isin 119883When119883 = 1199091 1199092 119909119899has only one element 119878119863 reduces to (s120599 ℎ 119892) which is calledan interval-valued dual hesitant fuzzy uncertain unbalancedlinguistic (IVDHF UUBL) number (IVDHF UUBLN)

32 Operational Rules for IVDHF UUBLS On the strength ofoperational rules for uncertain linguistic set [52] unbalancedlinguistic set [59] and interval-valued dual hesitant fuzzy set[47] we get the following operations for IVDHF UUBLS

Definition 2 Let 119904119889 = (119904120599 ℎ 119892) = ([119904120572 119904120573] ℎ 119892) 1199041198891 =(1199041205991 ℎ1 1198921) = ([1199041205721 1199041205731] ℎ1 1198921) and 1199041198892 = (1199041205992 ℎ2 1198922) =([1199041205722 1199041205732] ℎ2 1198922) be any three IVDHF UUBLNs 120582 isin [0 1]operations on these IVDHF UUBLNs are defined as

(1) 120582119904119889 = ⋃(119904120599ℎ119892)isin119904119889

([119904120582Δminus11199050 (119879119865

11990511989611199050(120595(119904120572)))

119904120582Δminus11199050 (119879119865

11990511989621199050(120595(119904120573)))

] ⋃

[120583119871120583119880]isinℎ[]119871]119880]isin119892

[1 minus (1 minus 120583119871)120582 1 minus (1 minus 120583119880)120582] [(]119871)120582 (]119880)120582]) (2) 119904119889120582 = ⋃

(119904120599 ℎ119892)isin119904119889

([119904(Δminus11199050 (119879119865

11990511989611199050(120595(119904120572))))

120582 119904(Δminus11199050 (119879119865

11990511989621199050(120595(119904120573))))

120582])

⋃[120583119871120583119880]isinℎ[]119871]119880]isin119892

[(120583119871)120582 (120583119880)120582] [1 minus (1 minus ]119871)120582 1 minus (1 minus ]119880)120582] )(3) 1199041198891 oplus 1199041198892 = ⋃

(1199041205991 ℎ11198921)isin1199041198891(1199041205992 ℎ21198922)isin1199041198892

([119904Δminus11199050 (119879119865

11990511989411199050(120595(1199041205721 )))+Δ

minus11199050(11987911986511990511989511199050(120595(1199041205722 )))

119904Δminus11199050 (119879119865

11990511989421199050(120595(1199041205731 )))+Δ

minus11199050(11987911986511990511989521199050(120595(1199041205732 )))

] ⋃

[1205831198711 1205831198801 ]isinℎ1[120583

1198712 1205831198802 ]isinℎ2[]

1198711 ]1198801 ]isin1198921[]

1198712 ]1198802 ]isin1198922

[1205831198711 + 1205831198712 minus 12058311987111205831198712 1205831198801 + 1205831198802 minus 1205831198801 1205831198802 ] []1198711]1198712 ]1198801 ]1198802 ]) (4) 1199041198891 otimes 1199041198892 = ⋃

(1199041205991 ℎ11198921)isin1199041198891(1199041205992 ℎ21198922)isin1199041198892

([119904Δminus11199050 (119879119865

11990511989411199050(120595(1199041205721 )))timesΔ

minus11199050(11987911986511990511989511199050(120595(1199041205722 )))

119904Δminus11199050 (119879119865

11990511989421199050(120595(1199041205731 )))timesΔ

minus11199050(11987911986511990511989521199050(120595(1199041205732 )))

] ⋃

[1205831198711 1205831198801 ]isinℎ1[120583

1198712 1205831198802 ]isinℎ2[]

1198711 ]1198801 ]isin1198921[]

1198712 ]1198802 ]isin1198922

[12058311987111205831198712 1205831198801 1205831198802 ] []1198711 + ]1198712 minus ]1198711]1198712 ]1198801 + ]1198802 minus ]1198801 ]

1198802 ])

(2)

8 Complexity

In above details about the transformation functionsof 119879119865 and Δ linguistic hierarchies (119871119867) as well as thetransformation procedures for unbalanced linguistic termsets are shown in Appendix A

eorem 3 Letting 119904119889 = (119904120599 ℎ 119892) 1199041198891 = (1199041205991 ℎ1 1198921) and1199041198892 = (1199041205992 ℎ2 1198922) be any three IVDHF UUBLNs 120582 1205821 1205822 isin[0 1]then following properties are true

(1) 1199041198891 oplus 1199041198892 = 1199041198892 oplus 1199041198891 (2) 1199041198891 otimes 1199041198892 = 1199041198892 otimes 1199041198891 (3)120582(1199041198891 oplus 1199041198892) = 1205821199041198891 oplus 1205821199041198892(4) 1199041198891120582 otimes 1199041198892120582 = (1199041198891 otimes 1199041198892)120582 (5) 1205821119904119889 oplus 1205822119904119889 = (1205821 +1205822)119904119889 (6) 1199041198891205821 otimes 1199041198891205822 = 1199041198891205821+1205822

Proof Omitted

In Definition 2 and Theorem 3 1199051198961 1199051198962 1199051198941 1199051198942 1199051198951 and1199051198952 are corresponding levels of 119904120572 119904120573 1199041205721 1199041205731 1199041205722 and 1199041205722in 119871119867 respectively 1199050 is the maximum level of 119904120572 119904120573 1199041205721 1199041205731 1199041205722 and 1199041205722 in 119871119867 Furthermore to compare any twoIVDHF UUBLNs we also have following definitions

Definition 4 Let 119904119889 = (119904120599 ℎ 119892) be an IVDHF UUBLN119904120599 = [119904120572 119904120573] then score function 119878(119904119889) and accuracy function119875(119904119889) can be represented by

119878 (119904119889) = Δminus11199050 (11987911986511990511989611199050 (120595 (119904120572))) + Δminus11199050 (11987911986511990511989621199050 (120595 (119904120573)))2 times 12 ( 1119897 (ℎ) sum

[120583119871120583119880]isinℎ

120583119871 minus 1119897 (119892) sum[]119871]119880]isin119892

]119871

+ 1119897 (ℎ) sum[120583119871120583119880]isinℎ

120583119880 minus 1119897 (119892) sum[]119871]119880]isin119892

]119880)(3)

119875 (119904119889)= Δminus11199050 (11987911986511990511989611199050 (120595 (119904120572))) + Δminus11199050 (11987911986511990511989621199050 (120595 (119904120573)))2

times 12 ( 1119897 (ℎ) sum[120583119871120583119880]isinℎ

120583119871 + 1119897 (119892) sum[]119871]119880]isin119892

]119871

+ 1119897 (ℎ) sum[120583119871120583119880]isinℎ

120583119880 + 1119897 (119892) sum[]119871]119880]isin119892

]119880)

(4)

where 119897(ℎ) and 119897(119892) are numbers of values in ℎ and 119892respectively 1199051198961 and 1199051198962 are the corresponding levels of 119904120572 and119904120573 in 119871119867 and 1199050 is the maximum level of 119905119896 in 119871119867

Definition 5 Given any IVDHF UUBLNs 1199041198891 = (1199041205991 ℎ1 1198921)1199041198892 = (1199041205992 ℎ2 1198922) then(1) If 119878(1199041198891) lt 119878(1199041198892) then 1199041198891 lt 1199041198892(2) If 119878(1199041198891) = 119878(1199041198892) then

(a) If 119875(1199041198891) = 119875(1199041198892) then 1199041198891 = 1199041198892(b) If 119875(1199041198891) lt 119875(1199041198892) then 1199041198891 lt 1199041198892

33 Distance Measure for IVDHF UUBLS When 119897(ℎ) or119897(119892) of two IVDHF UUBLNs are unequal the complement-ing method [86] is normally adopted to design distancemeasures Note that artificially adding values to shorterones in the complementing method will cause informationdistortion To avoid this limitation we provide the followingdistance measure

Definition 6 Let two IVDHF UUBLNs 1199041198891 = (1199041205991 ℎ1 1198921)and 1199041198892 = (1199041205992 ℎ2 1198922) where 1199041205991 = [1199041205721 1199041205731] 1199041205992 = [1199041205722 1199041205732]119897ℎ1 119897ℎ2 1198971198921 and 1198971198922 are lengths of ℎ1 ℎ2 1198921 and 1198922 respectivelydenoting number of elements in ℎ1 ℎ2 1198921 and 1198922 Suppose1198681 = (1(119899(1199051198941) minus 1))Δminus11199050 (11987911986511990511989411199050 (120595(1199041205721))) 1198682 = (1(119899(1199051198942) minus1))Δminus11199050 (11987911986511990511989421199050 (120595(1199041205731))) 1198683 = (1(119899(1199051198951)minus1))Δminus11199050 (11987911986511990511989511199050 (120595(1199041205722)))1198684 = (1(119899(1199051198952) minus 1))Δminus11199050 (11987911986511990511989521199050 (120595(1199041205732))) where 1199051198941 1199051198942 1199051198951 and1199051198952 are the corresponding levels of unbalanced linguistic terms1199041205721 1199041205731 1199041205722 and 1199041205732 in the linguistic hierarchy 119871119867 and 1199050 is themaximum level of 1199041205721 1199041205731 1199041205722 and 1199041205732 in 119871119867 Then by use ofnormalized Euclidean distance we define a distance measure119889 for IVDHF UUBLNs as follows

Situation 7 When 119897ℎ1 = 119897ℎ2 = 1198971 and 1198971198921 = 1198971198922 = 1198972 then119889 (1199041198891 1199041198892) = (12 ( 11198971sdot 1198971sum119896=1

(10038161003816100381610038161003816100381610038161198681120583119871119895ℎ1 minus 1198683120583119871119896ℎ2 10038161003816100381610038161003816100381610038162 + 10038161003816100381610038161003816100381610038161198682120583119880119895ℎ1 minus 1198684120583119880119896ℎ2 10038161003816100381610038161003816100381610038162) + 11198972sdot 1198972sum119896=1

(1003816100381610038161003816100381610038161198681]1198711198951198921 minus 1198683]1198711198961198922 1003816100381610038161003816100381610038162 + 1003816100381610038161003816100381610038161198682]1198801198951198921 minus 1198684]1198801198961198922 1003816100381610038161003816100381610038162)))12

(5)

Situation 8 When 119897ℎ1 = 119897ℎ2 or 1198971198921 = 1198971198922 then119889 (1199041198891 1199041198892) = (12 ( 1119897ℎ1 119897ℎ2sdot 119897ℎ1sum119895=1

119897ℎ2sum119896=1

(10038161003816100381610038161003816100381610038161198681120583119871119895ℎ1 minus 1198683120583119871119896ℎ2 10038161003816100381610038161003816100381610038162 + 10038161003816100381610038161003816100381610038161198682120583119880119895ℎ1 minus 1198684120583119880119896ℎ2 10038161003816100381610038161003816100381610038162)+ 11198971198921 1198971198922sdot 1198971198921sum119895=1

1198971198922sum119896=1

(1003816100381610038161003816100381610038161198681]1198711198951198921 minus 1198683]1198711198961198922 1003816100381610038161003816100381610038162 + 1003816100381610038161003816100381610038161198682]1198801198951198921 minus 1198684]1198801198961198922 1003816100381610038161003816100381610038162))12

(6)

eorem 9 The distance measure 119889 defined in Definition 6satisfies following properties

(1) 0 le 119889(1199041198891 1199041198892) le 1

Complexity 9

(2) 119889(1199041198891 1199041198892) = 0 if and only if 1199041198891 and 1199041198892 are perfectlyconsistent

(3) 119889(1199041198891 1199041198892) = 119889(1199041198892 1199041198891)4 Generalized Aggregation Operators forIVDHF_UUBLS

41 Definitions of Operators Based on the generalized oper-ators firstly introduced by Yager [87] we here develop some

fundamental generalized aggregation operators for the newlydefined IVDHF UUBLS

Definition 10 Given a collection of IVDHF UUBLNs 119904119889119895(119895 =1 119899) their weighting vector 120596 = (1205961 1205962 120596119899)119879120596119895 isin[0 1]sum119899119895=1 120596119895 = 1 120582 be a parameter 120582 isin (0 +infin)(1) Generalized IVDHFUUBLWeighted Average(GIVDHFUUBLWA) Operator

119866119868119881119863119867119865119880119880119861119871119882119860120596120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(120596119895119904119889119895120582))1120582

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120582)1120582 119904(sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120582)1120582]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[(1 minus

119899prod119895=1

(1 minus (120583119871119895 )120582)120596119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880119895 )120582)120596119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871119895)120582)120596119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880119895 )120582)120596119895)1120582]])

(7)

(2) Generalized IVDHFUUBLWeighted Geometric(GIVDHFUUBLWG)Operator

119866119868119881119863119867119865119880119880119861119871119882119866120596120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889119895)120596119895)= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871119895 )120582)120596119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880119895 )120582)120596119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871119895 )120582)120596119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880119895 )120582)120596119895)1120582]])

(8)

Definition 11 For a collection of IVDHF UUBLNs 119904119889119895(119895 = 1 119899) 119904119889120590(119895) be the 119895th largest 119908 = (1199081 1199082 119908119899)119879 be the aggregation-associated weighting vector 119908119895 isin [0 1]sum119899119895=1119908119895= 1 120582 is a parameter such that 120582 isin (0 +infin) 119878119899 997888rarr 119878 Then

10 Complexity

(1) Generalized IVDHFUUBL Ordered Weighted Average(GIVDHFUUBLOWA) Operator

119866119868119881119863119867119865119880119880119861119871119874119882119860120596120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(119908119895119904119889120590(119895)120582))1120582

= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

120582)1120582]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895) []119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120590(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120590(119895))120582)119908119895)1120582]])

(9)

(2) Generalized IVDHFUUBL Ordered Weighted Geometric(GIVDHFUUBLOWG)Operator

119866119868119881119863119867119865119880119880119861119871119874119882119866119908120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889120590(119895))119908119895)= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120590(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120590(119895))120582)120596119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120590(119895))120582)120596119895)1120582]])

(10)

Definition 12 For a collection of IVDHF UUBLNs 119904119889119895(119895 =1 119899) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector120596119895 isin [0 1] sum119899119895=1 120596119895 = 1 119899 is a balancing coefficient119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weight-ing vector with 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1

(1) Generalized IVDHFUUBL Hybrid Average (GIVDHFU-UBLHA) Operator

119866119868119881119863119867119865119880119880119861119871119867119860120596119908120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(119908119895119904119889120590(119895)120582))1120582

= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895)

))))120582)1120582]

Complexity 11

⋃[120583119871120590(119895) 120583

119880120590(119895)]isinℎ120590(119895) []

119871120590(119895) ]119880120590(119895)]isin119892120590(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120590(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120590(119895))120582)119908119895)1120582]])

(11)

where 119904119889120590(119895) is the 119895th largest and

119904119889119895 = 119899120596119895119904119889119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904119899120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904119899120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

] ⋃

[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[1 minus (1 minus 120583119871119895 )119899120596119895 1 minus (1 minus 120583119880119895 )119899120596119895] [(]119871119895)119899120596119895 (]119880119895 )119899120596119895]) (12)

(2) Generalized IVDHFUUBL Hybrid Geometric(GIVDHFUUBLHG) Operator

119866119868119881119863119867119865119880119880119861119871119867119866119908120596120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889120590(119895))119908119895)= ⋃(119904120599120590(119895)

ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120590(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895)

))))119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895) []119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120590(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120590(119895))120582)119908119895)1120582]])

(13)

where 119904119889120590(119895) is the 119895th largest and

119904119889119895 = 119904119889119895119899120596119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(Δminus11199050 (119879119865

11990511989511199050(120595(119904120572119895 ))))

119899120596119895 119904(Δminus11199050 (119879119865

11990511989521199050(120595(119904120573119895 ))))

119899120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[(120583119871119895 )119899120596119895 (120583119880119895 )119899120596119895] [1 minus (1 minus ]119871119895 )119899120596119895 1 minus (1 minus ]119880119895 )119899120596119895]) (14)

12 Complexity

When confronted with ill-structured situations wheredecision-makerrsquos complex attitudinal characters need to beincluded order-inducing vectors provide an effective way[88ndash90]Thus we further define following induced operatorsfor IVDHF UUBLNs

Definition 13 For a collection of IVDHF UUBLNs 119904119889119895 (119895 =1 119899) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector

120596119895 isin [0 1] sum119899119895=1 120596119895 = 1 119899 is a balancing coefficient119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weight-ing vector 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1 120576119895 denote a set oforder inducing vectors

(1) Induced Generalized IVDHFUUBL Hybrid Average (I-GIVDHFUUBLHA) Operator

119868-119866119868119881119863119867119865119880119880119861119871119867119860120596119908120582 (lt 1205761 1199041198891 gt lt 1205762 1199041198892 gt lt 120576119899 119904119889119899 gt) = ( 119899⨁119895=1

(119908119895119904119889120587(119895)120582))1120582

= ⋃(119904120599120587(119895) ℎ120587(119895) 119892120587(119895))isin119904119889120587(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120587(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587(119895)

))))120582)1120582]

⋃[120583119871120587(119895) 120583

119880120587(119895)]isinℎ120587(119895) []

119871120587(119895) ]119880120587(119895)]isin119892120587(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120587(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120587(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120587(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120587(119895))120582)119908119895)1120582]])

(15)

where (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) is decreasing order of (11990411988911199041198892 119904119889119899) according to the order inducing vector 120576119895 and119904119889119895 = 119899120596119895119904119889119895 = ⋃

(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904119899120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904119899120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

] ⋃

[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[1 minus (1 minus 120583119871119895 )119899120596119895 1 minus (1 minus 120583119880119895 )119899120596119895] [(]119871119895)119899120596119895 (]119880119895 )119899120596119895]) (16)

(2) Induced Generalized IVDHFUUBL Hybrid Geometric(I-GIVDHFUUBLHG) Operator

119868-119866119868119881119863119867119865119880119880119861119871119867119866119908120596120582 (lt 1205761 1199041198891 gt lt 1205762 1199041198892 gt lt 120576119899 119904119889119899 gt) = 1120582 ( 119899⨂119895=1

(120582119904119889120587(119895))119908119895)= ⋃(119904120599120587(119895)

ℎ120587(119895) 119892120587(119895))isin119904119889120587(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120587(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587(119895)

))))119908119895]

⋃[120583119871120587(119895)120583119880120587(119895)]isinℎ120587(119895) []119871120587(119895) ]

119880120587(119895)]isin119892120587(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120587(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120587(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120587(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120587(119895))120582)119908119895)1120582]])

(17)

Complexity 13

Weighted Aggregation

Geometric Arithmetic

GIVDHFUUBLHG GIVDHFUUBLHA

GIVDHFUUBLWG GIVDHFUUBLOWG

IVDHFUUBLWG IVDHFUUBLOWG

GIVDHFUUBLWA GIVDHFUUBLOWA

IVDHFUUBLWA IVDHFUUBLOWA

= 1 w =1

n = 1 =

1

n = 1 =1

n = 1 =

1

n

= 1 = 1 = 1 = 1

Figure 2

where (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) is decreasing order of (11990411988911199041198892 119904119889119899) according to the order inducing variables 120576119895 and

119904119889119895 = 119904119889119895119899120596119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(Δminus11199050 (119879119865

11990511989511199050(120595(119904120572119895 ))))

119899120596119895 119904(Δminus11199050 (119879119865

11990511989521199050(120595(119904120573119895 ))))

119899120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[(120583119871119895 )119899120596119895 (120583119880119895 )119899120596119895] [1 minus (1 minus ]119871119895 )119899120596119895 1 minus (1 minus ]119880119895 )119899120596119895]) (18)

42 Properties of the Proposed GeneralizedAggregation Operators

eorem 14 With special values of 120582 119908 and 120596 the operatorsGIVDHFUUBLHA and GIVDHFUUBLHG can include aseries of traditional aggregation operators as special cases andtheir relationship can be depicted in Figure 2

Proof See Appendix B

As for the induced hybrid aggregation operators we alsohave following theorem

eorem 15 If (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) = (119904119889120590(1) 119904119889120590(2) 119904119889120590(119899)) I-GIVDHFUUBLHA reduces toGIVDHFUUBLHAIf (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) = (119904119889120590(1) 119904119889120590(2) 119904119889120590(119899)) thenI-GIVDHFUUBLHG reduces to the GIVDHFUUBLHG oper-ator

eorem 16 All the proposed generalized operatorsGIVDHFUUBLWAGIVDHFUUBLWGGIVDHFUUBLOWAGIVDHFUUBLOWGGIVDHFUUBLHAGIVDHFUUBLHGI-GIVDHFUUBLHA and I-GIVDHFUUBLHG hold thefollowing properties (1) Commutativity (2) Idempotency (3)Boundedness

Based on above theorems following properties can alsobe derived

eorem 17 For a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weight vector of 119904119889119895with 120596119895 isin [0 1] and sum119899119895=1 120596119895 = 1 120582 gt 0 then we have

(1) 119868119881119863119867119865119880119880119861119871119882119866120596(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119882119860120596(1199041198891 1199041198892 119904119889119899)(2) 119868119881119863119867119865119880119880119861119871119882119866120596(1199041198891 1199041198892 119904119889119899) le119866119868119881119863119867119865119880119880119861119871119882119860120596120582(1199041198891 1199041198892 119904119889119899)(3) 119866119868119881119863119867119865119880119880119861119871119882119866120596120582(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119882119860120596(1199041198891 1199041198892 119904119889119899)

Proof Omitted for concision

eorem 18 For a collection of IVDHF UUBLNs 119904119889119895 = (119904120599119895 ℎ119895 119892119895) 119908 = (1199081 1199082 119908119899)119879 is the weighting vector of 119904119889119895with 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1 120582 gt 0 Then

(1) 119868119881119863119867119865119880119880119861119871119874119882119866119908(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119874119882119860119908(1199041198891 1199041198892 119904119889119899)(2) 119868119881119863119867119865119880119880119861119871119874119882119866119908(1199041198891 1199041198892 119904119889119899) le119866119868119881119863119867119865119880119880119861119871119874119882119860119908120582(1199041198891 1199041198892 119904119889119899)(3) 119866119868119881119863119867119865119880119880119861119871119874119882119866119908120582(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119874119882119860119908(1199041198891 1199041198892 119904119889119899)

Proof Omitted for concision

14 Complexity

5 An Integrated MADM Approach forTackling Complex Green ContractorSelection Problems

As described in Section 2 we take the green contractorselection as a special type of complicate MADM problemsthat synthesizes three characteristics of decision hesitancy[21 22] attributes interdependency [84] and group atti-tudinal characters [61] Therefore in this section we con-struct an integrated MADM approach to tackle the com-plex green contractor selection problems Suppose 119883 =1199091 1199092 119909119898 is the set of alternative green contractors and119860 = 1198601 1198602 119860119899 is the set of evaluative attributes 120596 =(1205961 1205962 120596119899)119879 is weighting vector for the attributes 120596119894 ge 0sum119899119894=1 120596119894 = 1 Let = (119903119894119895)119899times119898 denote the decision matrix inwhich 119903119894119895 = ([119904120572119894119895 119904120573119894119895] ℎ119894119895 119892119894119895) is an IVDHF UUBLN givenby decision-makers for alternative contractor 119909119895 with respectto attribute 119860 119894 According to the mechanism of pair-wisecomparisons among attributes in the DEMATELmethod [6364] the interdependency among attributes can be obtained asamatrix119885 = (119911119894119896)119899times119899 where 119911119894119896 indicates the degree towhich119860 119894 affects119860119896 Subsequently based on the IVDHFUUBLS andits operations we now present detailed steps of our MADMapproach as shown in following Algorithm I

Algorithm I Hesitant fuzzy linguistic MADMwith attributesinterdependency and decision-makersrsquo group attitudinalcharacters

Step 1 Determine argument-dependent weighting vector120596119860119863 according to attribute values by programming modeldeveloped in the following Section 51

Step 2 Obtain the attribute-interdependences based weight-ing vector 120596119860119868 by use of DEMATEL method described in thefollowing Section 52

Step 3 Calculate synthesized attribute weighting vectoraccording to

120596 = 120572120596119860119868 + 120573120596119860119863 (19)

where 120572 and 120573 are parameters to reflect decision character-istics of decision organizations 0 le 120572 120573 le 1 120572 + 120573 =1Step 4 Check requirements for order inducing If no addi-tional order inducing required then go to Step 5 otherwisego to Step 6

Step 5 Utilize generalized aggregation operators to get theoverall IVDHF UUBLNs 119903119895(119895 = 1 119898) for each alternative119909119895 Here we take GIVDHFUUBLHA operator for examplebecause it can include other traditional operators as its specialcases Therefore we have

119866119868119881119863119867119865119880119880119861119871119867119860(1199031119895 1199032119895 119903119899119895) = ( 119899⨁119894=1

(119908119894 (119903120590119894119895)120582))1120582= ⋃(119904120590119894119895ℎ120590119894119895 119892120590119894119895

)isin119903120590119894119895

([119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590119894119895

))))120582)1120582 119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590119894119895

))))120582)1120582]

⋃[120583119871120590119894119895120583

119880120590119894119895]isinℎ120590119894119895[]

119871120590119894119895]119880120590119894119895]isin119892120590119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120590119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120590119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120590119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120590119894119895)120582)119908119894)1120582]])

(20)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of 119903119894119895119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weightvector 119903120590119894119895 is the 119894th largest of 119903119894119895 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))

119904119899120596119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119894119895))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (21)

Complexity 15

Then go to Step 8

Step 6 By use of the TOPSIS-based method developed in thefollowing Section 53 we determine order-inducing vectors 120576to reflect decision-makersrsquo group attitudinal characters

Step 7 Utilizing induced generalized aggregation operatorsto get the overall IVDHF UUBLNs 119903119895(119895 = 1 2 119898) foreach alternative 119909119895 we have

119868-119866119868119881119863119867119865119880119871119867119860(1199031119895 1199032119895 119903119899119898) = ( 119899⨁119894=1

(119908119894 (119903120587119894119895)120582))1120582

= ⋃(119904120587119894119895ℎ120587119894119895119892120587119894119895)isin119903120587119894119895

([119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120587119894119895

))))120582)1120582 119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587119894119895

))))120582)1120582]

⋃[120583119871120587119894119895 120583

119880120587119894119895]isinℎ120587119894119895 []

119871120587119894119895]119880120587119894119895]isin119892120587119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120587119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120587119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120587119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120587119894119895)120582)119908119894)1120582]])

(22)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of119903119894119895 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 119903120587119894119895 is the 119894th largest of 119903119894119895 reordered by theorder-inducing vectors 120576 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))119899120596119894120572119894119895

119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120573119894119895 ))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (23)

Step 8 Calculate scores 119878(119903119895) and accuracy degrees 119875(119903119895) for119903119895(119895 = 1 2 119898)Step 9 According to Definition 5 output ranking order ofalternative contractors

In Algorithm I to objectively derive unknown weight-ing vector for evaluative attributes a synthesized attributeweighting vector is devised by fusing two parts of concern(i) argument-dependent weighting vector and (ii) attribute-interdependences based weighting vector The former is toreflect effects of attribute values on attribute weights whilethe latter is to exploit effects of interdependences amongattributes on their weights For clarity processing flowchartof Algorithm I is depicted in Figure 3

51 Determining Argument-Dependent Attribute WeightingVector 120596119860119863 In order to derive attribute weighting vec-tor appropriately from assessments maximizing deviationmethod [91] is adopted here for its distinguishing ability

and objectivity [92] According to Wang [91] if performancevalues of each alternative have little differences under anattribute it implicates such an attribute plays a less importantrole otherwise it lives a more important role in decision-making Thus if one attribute has similar values across alter-natives it should be assigned a smaller weight contrariwiseit should be a bigger weight

In light of the idea in maximizing deviation methodwe develop a programming model to determine argument-dependent weighting vector 120596119860119863 For attribute 119860 119894 the stan-dard deviation of alternative 119909119895 to all the other alternatives isdenoted as

119865119894119895 (120596119860119863) = 119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (24)

Let119865119894(120596119860119863) represent deviation value of all alternatives toother alternatives according to attribute119860 119894 and be formulatedas follows

16 Complexity

Attribute weighting

Hesitant fuzzy linguistic MADM with attributes interdependency and decision makersrsquo group attitudinal characters

Step 9 Output ranking order of alternatives

interdependency among attributes by DEMATEL in Section 52

attributes values by standard deviation method in Section 51

Step 6 Ensure TOPSIS order inducing variables proposed in Section 53

Step 7 Utilizing proposed induced generalized hybrid aggregation operators developed in Section

41 to get overall value of each alternative

Step 8 Calculate score degrees and accuracy degrees

Step 4 Order inducingYes

Step 5 Choose proposed generalized operators in Section 41 to get overall value of each alternative

No

Collect decision matrix in the form of IVDHF_UUBLS

Complex problems of green contractor selection

Step 1 Weighting vector AD derived from Step 2 Weighting vector AI derived from

Step 3 Obtain synthesized weighting vector of attributes =AI + AD

Figure 3 Processing flowchart of Algorithm I

119865119894 (120596119860119863) = 119898sum119895=1

119865119894119895 (120596119860119863) = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (25)

Based on the analysis above we can get followingprogramming model (M-1) for obtaining optimal attributeweighting vector 120596119860119863

(119872 minus 1)max 119865 (120596119860119863) = 119899sum

119894=1119865119894 (120596119860119863) = 119899sum

119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894st 119899sum

119894=1(120596119860119863119894 )2 = 1 120596119860119863119894 ge 0 119894 = 1 2 119899 (26)

where 119889(119903119894119895 119903119894119897) can be calculated by Definition 6To solve model (M-1) we can construct Lagrange func-

tion

119871 (120596119860119863119894 120577) = 119899sum119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894+ 1205772 ( 119899sum

119894=1

(120596119860119863119894 )2 minus 1) (27)

where 120577 is the Lagrange multiplierDifferentiate with respect to 120596119860119863119894 (119894 = 1 2 119898) and 120577

and set these partial derivatives equal to zero the followingset of equations can be obtained

120597119871120597120596119860119863119894 = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) + 120577120596119860119863119894 = 0120597119871120597120577 = 119899sum

119894=1

(120596119860119863119894 )2 minus 1 = 0 (28)

By normalizing the solutions of above equations we canget a simple and exact formula for determining the attributeweights as follows

120596119860119863119894 = sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897)sum119899119894=1sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897) (29)

52 Deriving Attributes Interdependency-Based AttributeWeighting Vector 120596119860119868 Some efforts [53 60 84 93ndash95]

Complexity 17

have been made to address interdependencies betweenattributes in MADM However they unanimously cannotreveal the interdependences in a more complete way WhileDEMATAEL technique [63 96] gathers collective knowledgeto capture influences between attributes more precisely andcompletely so that weighting vector for attributes can berationally determined according to strength measure ofthose influences [64 97] Therefore in tackling the complexMADMwith interdependent attributes under IVDHF UUBLenvironments we present a DEMATEL-based method as thefollowing Algorithm II to derive attribute weighting vector120596119860119868 from the viewpoint of attributes interdependency

Algorithm II DEMATEL-basedmethod for ensuringweight-ing vector 120596119860119868Step 1 Construct initial direct-influence matrix 119885 = (119911119894119896)119899times119899

For attributes 119860 = 1198601 1198602 119860119899 decision-makersare asked to give direct effect that attribute 119860 119894 has onattribute 119860119896 using a five-point scale with scores representedby natural language (1) 0 mdash lsquoabsolutely no influencersquo (2)1 mdash lsquolow influencersquo (3) 2 mdash lsquomedium influencersquo (4) 3 mdashlsquohigh influencersquo (5) 4mdash lsquovery high influencersquoThen the initialdirect-influence matrix 119885 = (119911119894119896)119899times119899 can be established witheach element being the mean of same factors in differentdirect matrices

Step 2 Calculate normalized direct-influence matrix 119883 =(119909119894119896)119899times119899 Using matrix 119885 the normalized direct-influencematrix 119883 = (119909119894119896)119899times119899 is calculated according to

119909119894119896 = 119911119894119896119904 (30)

where 119904 = maxmax1le119894le119899sum119899119896=1 119911119894119896max1le119896le119899sum119899119895=1 119911119894119896 Allelements in matrix 119883 are complying with 0 le 119909119894119896 le 10 le sum119899119896=1 119909119894119896 le 1 and 0 le sum119899119894=1 119909119894119896 le 1 and at least onerow or column of the summation equals 1

Step 3 Derive total-influence matrix 119879 = (119905119894119896)119899times119899 Basedon matrix 119883 the total-influence matrix 119879 = (119905119894119896)119899times119899 can bederived by summing the direct effects and all of the indirecteffects

119879 = 119883 + 1198832 + 1198833 + sdot sdot sdot + 119883ℎ = 119883 (119868 minus 119883)minus1 (31)

where 119868 represents identity matrix When ℎ 997888rarr infin 119883ℎ =(0)119899times119899Step 4 Compute influencing and influenced degrees of eachattribute The sum of rows and the sum of columns withinthe total-influence matrix 119879 are respectively expressed as thevectors 119903 and 119888

119903 = (119903119894)119899times1 = ( 119899sum119896=1

119905119894119896)119899times1

119888 = (119888119896)119899times1 = ( 119899sum

119895=1

119905119894119896)119879

1times119899

(32)

where 119903119894 denotes the sumof the 119894th row inmatrix119879 and showsthe sum of direct and indirect effects that attribute 119860 119894 hason other attributes Similarly 119888119896 shows the sum of direct andindirect effects that attribute 119860119896 has receivedStep 5 Estimate weights of attributes based on influencedegrees according to

120596119868lowast119894 = [(119903119894 + 119888119894)2 + (119903119894 minus 119888119894)2]12 (33)

Then the weighting vector 120596119860119868 can be obtained by normaliz-ing (33)

120596119860119868119894 = 120596119868lowast119894sum119899119894=1 120596119868lowast119894 (34)

53 Obtaining Order Inducing Variables 120576 by TOPSIS-BasedMethod For ill-defined MADM problems induced aggre-gation operators [89 90 98] provide an effective channelfor experts to express complex attitudinal characters Thekey step in induced aggregation operators is to obtainorder-inducing vectors 120576 which however are presumptivelyassigned in most papers [62] Therefore in this sectionaiming at deducing the order-inducing vectors 120576 rationallywe here devise a TOPSIS-based method in Algorithm IIIObviously the Algorithm III is of generality for MADMwithdifferent forms of linguistic expression

Algorithm III TOPSIS-based method for obtaining orderinducing vectors 120576Step 1 Let the decision maker define IVDHF UUBL idealsolution(s) 119903+119894 and IVDHF UUBL negative ideal solution(s)119903minus119894 corresponding to each attribute in which

119903+119894 = (119903+1198941 119903+1198942 119903+119894119901) 119903minus119894 = (119903minus1198941 119903minus1198942 119903minus119894119902) (35)

where 119901 and 119902 denotes the number of 119903+119894 and 119903minus119894 respectively

Step 2 Calculate distances from each element in decisionmatrix to ideal solution(s) and negative solution(s) respec-tively

119889+119894119895 = max119897=12119901

119889 (119903119894119895 119903+119894119897 ) 119889minus119894119895 = min

119897=12119902119889 (119903119894119895 119903minus119894119897 ) (36)

where 119889(119903119894119895 119903+119894119897 ) and 119889(119903119894119895 119903minus119894119897 ) are calculated by the distancemeasure in Definition 6 If 119901 gt 1 we choose the onewith maximum distance if 119902 gt 1 we choose the one withminimum distance

Step 3 Calculate coefficients according to

119888119894119895 = 119889minus119894119895119889+119894119895 + 119889minus119894119895 119894 = 1 2 119899 119895 = 1 2 119898 (37)

18 Complexity

Table 2 Decision matrix in the form of IVDHF UUBLS1199091 1199092 11990931198601([AHVH] [02 03][02 04] [03 04]) ([HT] [04 05][05 06] [02 03] [02 04]) ([ALAH] [01 02][01 03] [06 07])1198602([MH] [05 06][02 03]) ([AHT] [06 07][01 02] ) ([MVH] [02 03][05 06] [06 07])1198603([VLAL] [03 04][04 05] [05 06]) ([MVH] [06 07] [07 08][01 02]) ([VLL] [04 05][02 03])1198604([MAH] [02 04][05 06]) ([LAL] [03 05][02 03]) ([ALM] [06 07][01 02] [02 03])1198605([ALM] [04 05][01 02] [04 05]) ([AHH] [06 07][01 02]) ([HVH] [04 05][06 07] [01 03])1198606([LM] [01 02][03 05] [03 05]) ([VLL] [02 04] [05 06][02 03]) ([AHH] [03 04][04 05])1198607([VLL] [04 05][05 06] [02 03] [02 04]) ([MAH] [05 06] [07 08][01 02]) ([ALM] [04 05][05 06] [03 04])1198608([ALAH] [01 03][04 06]) ([AHH] [05 07][01 02] [02 03]) ([AHH] [04 06][03 04])

Table 3 Direct influence matrix Z1198601 1198602 1198603 1198604 1198605 1198606 1198607 11986081198601 0 3 4 1 3 1 2 41198602 1 0 2 4 4 2 3 11198603 4 2 0 1 3 2 4 11198604 2 3 1 0 3 4 1 21198605 4 2 1 3 0 1 4 31198606 2 4 3 4 4 0 4 21198607 4 1 2 3 3 1 0 41198608 1 2 4 3 3 2 4 0

Step 4 Derive order inducing variable 120576119894119895 by descendingorder of 119888119894119895 The greater the value 119888119894119895 the bigger the 1205761198941198956 Illustrative Example and Experiments

61 Illustrative Case Study Suppose a mega infrastructureproject is selecting green contractor for one of its subprojectsA panel of decision-makers including managers professorsand competitive intelligence experts has been organized tocomprehensively evaluate three alternative contractors (ie1199091 1199092 and 1199093) under the eight attributes (ie 1198601ndash1198608) listedin Section 2

To elicit true decision preferences we apply theIVDHF UUBLS to assess the three alternatives (1199091ndash1199093)Firstly decision-makers are asked to vote on 119909119895 under eachattribute119860 119894 using different unbalanced linguistic term sets 1198781 1198782 and 1198783 where 1198781 = 119873 119871119860119871119872119860119867119867119876119867119881119867119860119879 1198791198782 = 119873119860119873119881119871 119876119871 119871 119860119871119872119860119867119881119867 119879 1198783 = 119873119881119871119871119860119872119872119876119872119860119867119867 119879 In Figure 4 we show the adoptedlinguistic hierarchies and relationship between 1198781 1198782 and1198783 Attributes 1198601 1198602 1198605 and 1198608 are evaluated by use of 1198781attributes11986031198604 and1198607 are evaluated by 1198782 and attribute1198606is evaluated by 1198783 The highest-voted uncertain unbalancedlinguistic term [119904119898 119904119899] are identified Secondly the panel of

decision-makers is asked to further express their opinionson membership degrees ℎ(119909) and nonmembership degrees119892(119909) to [119904119898 119904119899] Then we can derive compound assessments(ie ⟨119909 119904120599(119909) ℎ(119909) 119892(119909)⟩) in the form of IVDHF UUBLSand obtain the decision matrix in Table 2 Additionallyin order to examine the interdependency among attributesdecision-makers are also required to assess the influentialrelations among the eight attributes using DEMATEL-basedAlgorithm II The direct influence matrix is collected andshown in Table 3

Next we apply the proposed Algorithm I to solve thisgreen contractor selection problem To derive the synthe-sized attribute weighting vector in Algorithm I we assign120572 = 120573 = 05 For scenarios with no group attitudinalcharacters operator GIVDHFUUBLHA in Definition 12 ischosen In contrast for scenarios with group attitudinalcharacters operator I-GIVDHFUUBLHA in Definition 13will be chosen Further the panel of decision-makers is askedto determine their positive and negative ideal alternativesthat are collected in Table 4 where corresponding order-inducing vectors that are derived from Algorithm III arealso listed (see the rightmost column) Thereafter operatorI-GIVDHFUUBLHA in Definition 13 is utilized for aggrega-tion For GIVDHFUUBLHA and I-GIVDHFUUBLHA we

Complexity 19

Table 4 Ideal alternatives (119903+ and 119903minus) and derived order-inducing vectors 120576119903+ 119903minus 1205761199091 1199092 11990931198601 ([VHT] 1 0) ([NL] 1 0) 5 6 61198602 ([MAH] [07 08] [01 02]) ([NL] 1 0) 1 1 41198603 (T 1 0) (L 1 0) 8 7 81198604 ([ALAH] 09 01) (T 1 0) 2 2 11198605 ([HT] [09 1] 0) (N 1 0) 7 5 51198606 (M 1 0) (N 1 0) 6 3 31198607 (M 1 0) ([NL] 1 0) 3 8 71198608 ([MAH] 1 0) (N 1 0) 4 4 2

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 1 2 3 4

0 1 2

N L AH H QH VH AT TM

Level 1 l(13)

Level 2 l(25)

Level 3 l(39)

Level 4 l(417)

Unbalanced linguistic term set S1

N AH VH TM Unbalanced linguistic term set S2VL L ALAN QL

N H TM Unbalanced linguistic term set S3

L AM AHQM

AL

VL

Figure 4 Unbalanced linguistic term sets (1198781 1198782 and 1198783) and their mapping in linguistic hierarchies

assume 120582=2 and 119908 = (0 0 015 025 025 025 01 0)119879Note that 119908 is a position weighting vector which is derivedby the fuzzy semantic quantitative operator [99] Aftercompleting all steps in Algorithm I (see Appendix C) weobtain the ranking results and present them in Table 5 Forthe scores derived from Step 5 we find the ranking order is1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the best solution Step 7gives the ranking order of 1199092 ≻ 1199091 ≻ 1199093 indicating 1199092 is stillthe most appropriate one while 1199091 is better than 1199093 under theinfluence of 12057662 Experiments on Ranking Results along with VariousParameter Configurations In order to inspect the impactsof various parameter configurations on ranking results inthis section more numerical examples are carried out Alleight operators that defined in Section 41 are applied to theabove illustrative case where the parameter 120582 is configured asintegers ranging from 1 to 20 and parameter 120572 is configuredto a value set of 0 03 05 08 09 1

Firstly the four basic aggregation operators of GIVDH-FUUBLWA GIVDHFUUBLWG GIVDHFUUBLOWA and

Table 5 Ranking results of operators GIVDHFUUBLHA and I-GIVDHFUUBLHA

Aggregation operator Ranking orderGIVDHFUUBLHA 1199092 ≻ 1199093 ≻ 1199091I-GIVDHFUUBLHA 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG are applied to the former case Thesame configuration 120572 = 120573 = 05 is chosen while 120582 is testedas integers ranging from 1 to 20 The ranking results derivedfrom different parameter combinations have been collectedin Table 6 As can be seen four operators unanimouslyidentify the contractor 1199092 is the best one no matter how120582 changes Regarding priority relation between 1199091 and 1199093GIVDHFUUBLWA and GIVDHFUUBLOWA generate thesame result of1199093 ≻ 1199091 while results fromGIVDHFUUBLWGandGIVDHFUUBLOWGgot a change to 1199091 ≻ 1199093 at a certainvalue of parameter 120582

Actually GIVDHFUUBLWA and GIVDHFUUBLOWAare generalized version of weighted arithmetic aggregationoperators that producemore favorable results by emphasizing

20 Complexity

Table 6 Ranking orders obtained by GIVDHFUUBLWA GIVDHFUUBLOWA GIVDHFUUBLWG and GIVDHFUUBLOWG along with120582Operators 120582 (Integer) Ranking orderGIVDHFUUBLWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLOWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLWG [1 2] 1199092 ≻ 1199093 ≻ 1199091[3 20] 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG [1 8] 1199092 ≻ 1199093 ≻ 1199091[9 20] 1199092 ≻ 1199091 ≻ 1199093

Table 7 Ranking results obtained by GIVDHFUUBLHA and I-GIVDHFUUBLHA along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHA I-GIVDHFUUBLHA120572=0 120573=1 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=03 120573=07 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=05 120573=05 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=08 120573=02 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=09 120573=01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=1 120573=0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093

Table 8 Ranking results obtained by GIVDHFUUBLHG and I-GIVDHFUUBLHG along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHG I-GIVDHFUUBLHG

120572 = 0 120573 = 1 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 03 120573 = 07 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 05 120573 = 05 [1 8] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[9 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 08 120573 = 02 [1 17] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[18 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 09 120573 = 01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572 = 1 120573 = 0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093membership degrees to a uncertain linguistic term thus canbe considered as optimistic operators while GIVDHFU-UBLWG and GIVDHFUUBLOWG are generalized geomet-ric aggregation operators that producemore favorable resultsby emphasizing nonmembership degrees thus can be consid-ered as pessimistic operators [100 101]Therefore the rankingresults in Table 6 could be perceived as for optimism-baseddecision-making alternative 1199093 always stands out better than1199091 when 120582 gradually amplifies degree of optimism but forpessimism-based decision-making 1199091 turns out better than1199093 when 120582 gradually amplifies degree of pessimism 120582 canbe considered as the parameter for amplifying degree ofoptimism or pessimism [101]

Next to inspect the impacts of various combinationsof 120596119860119868 and 120596119860119863 on ranking results another four aggrega-tion operators of GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG also have beenapplied to the above illustrative case with varying 120572 120573 and 120582

All results have been collected in Tables 7 and 8 for compari-son As can be seen fromTable 7 ranking results generated byoperator GIVDHFUUBLHA keep the same as 1199092 ≻ 1199093 ≻ 1199091for all test combinations of parameters However as shownin Table 8 although GIVDHFUUBLHG is also capable ofidentifying 1199092 as the best contractor the priority relationsbetween 1199091 and 1199093 hold similar changing patterns in rankingresults in Table 6 that were output byGIVDHFUUBLWGandGIVDHFUUBLOWG Regarding decision situations whereobjective attributeweighting vector120596119860119863 is only considered (120572= 0) or its dominant importance (such as 120572 = 03) is accepted1199091 ≻ 1199093 is a clear observation when parameter 120582 is set toan integer in interval [8 20] Contrariwise under decisionscenarios where subjective attribute weighting vector 120596119860119868 isonly needed (120572 = 1) or its dominant importance (such as 120572 =09) is acknowledged 1199093 ≻ 1199091 holds all the way 120582 varies

In order to more clearly show the score trajectories ofalternative contractors along variations of parameter 120582 we

Complexity 21

HAInducedHA

InducedHA-2HA-2

HA-3

InducedHA-1InducedHA-3

= =

HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(a) 120572 = 0

HAInducedHA

= 03 = 07

InducedHA-2

HA-2HA-3

InducedHA-1 InducedHA-3HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(b) 120572 = 03

HAInducedHA

= 5 = 05

InducedHA-2HA-2

HA-3

InducedHA-1 InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(c) 120572 = 05

HAInducedHA

= 08 = 02

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572 = 08

HAInducedHA

= 9 = 01

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(e) 120572 = 09

HAInducedHA

246 2462 2464 2466 2468 247

= 1 = 0InducedHA-2HA-2

HA-3

HA- 3

HA-3

InducedHA-1

InducedHA-3

HA-1

HA-1

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

1085

1086

1087

(f) 120572 = 1

Figure 5 Score trajectories obtained by aggregation operators HA and InducedHA with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

further configure parameter 120582 to the value set of [0 01100]and conduct computational experiments to collect scorevalues and ranking results (as shown in Figures 5 and 6) byfour aggregation operators of GIVDHFUUBLHA GIVDH-FUUBLHG I-GIVDHFUUBLHAand I-GIVDHFUUBLHGFor more clarity in Figures 5 and 6 please note that weuse HA HG Induced-HA and Induced HG to respec-tively denote GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG In Figure 5it can be observed that when dominant importance of 120596119860119863

is emphasized (120572 = 0 120572 = 03 120572 = 05) GIVDHFUUBLHAis capable of differentiating the three alternative contractorsas same ranking order of 1199092 ≻ 1199093 ≻ 1199091 for all 120582 but withincreases in 120572 the margins between scores of 1199091 and 1199093 nar-rows and the rank order changes to 1199091 ≻ 1199093when 120572 gets 09and 1 Interestingly in Figure 6 although GIVDHFUUBLHGis able to significantly differentiate alternative contractors 1199091and 1199093 when dominant importance of 120596119860119863 is emphasized (120572= 0 120572 = 03 120572 = 05) rank order of 1199091 and 1199093 obviouslychanges once from 1199093 ≻ 1199091 to 1199091 ≻ 1199093 when 120582 is bigger

22 Complexity

HGInducedHG

= = 1

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(a) 120572= 0

HGInducedHG

= 3 = 07

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(b) 120572= 03

HGInducedHG

= 05 = 05

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(c) 120572= 05

HGInducedHG

= 8 = 02

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572= 08

HGInducedHG

292 293 294 295 296 297minus08

minus0795

minus079

= 09 = 01

HA-1

HA-3

InducedHG-2

HG-1

HG-1

HG-2

HG-3

HG-3 InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4

minus3

minus2

minus1

0

1

2

scor

e val

ues

(e) 120572= 09

HGInducedHG

50 60 70 80 90 100minus095minus09

minus085

= 1 = 0

HG-x3

HG-x1

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(f) 120572= 1

Figure 6 Score trajectories obtained by aggregation operators HG and InducedHG with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

than 8 When 120572 increases to 1 GIVDHFUUBLHGmaintainsthe rank order of 1199093 ≻ 1199091 but their score trajectorieswalk very close Comparatively I-GIVDHFUUBLHA andI-GIVDHFUUBLHG generate consistent ranking results of1199092 ≻ 1199091 ≻ 1199093 no matter 120582 varies in all the above experiments

In sum when parameters are configured with typical val-ues (ie 120572 = 05 and 120582=1) and no additional attitudinal infor-mation is taken into consideration the proposed aggregationoperators unanimously derive the same rank order 1199092 ≻1199093 ≻ 1199091 of for the three alternative contractors However

with the amplification effect of parameter 120582 analysis resultsfrom linear combination of 120596119860119868 and 120596119860119863 reveal alternative1199092 which consistently is the best choice but rank order of1199091 and 1199093 changes because their score trajectories get closeand crossed which probably indicates that only decision-making informationderived from the decisionmatrix is inad-equate to significantly differentiate alternatives 1199091 and 1199093While by integrating additional decision information aboutattitudinal characters of group experts I-GIVDHFUUBLHAand I-GIVDHFUUBLHG manage to generate the identical

Complexity 23

Table 9 Order-inducing vectors obtained by our TOPSIS-based method for problem in [38]

1198661 1198662 1198663 11986641198601 02522 01629 02282 027141198602 01 05243 01364 046891198603 04081 01756 05244 037791198604 04507 01409 01064 013221198605 00697 02203 03753 01161

Table 10 Order-inducing vectors obtained by our TOPSIS-based method for problem in [39]

1198661 1198662 1198663 11986641198601 04548 04969 04923 047321198602 05092 047 04768 046451198603 07192 04888 04953 049151198604 06722 06163 05706 049291198605 06104 04942 06463 04946

Table 11 Ranking results obtained by applying different order-inducing vectors to [38 39]

Problems Approaches Ranking results

Problem in [38] IHFLOWA [38] + order-inducing vectors in [38] 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198605 ≻ 1198604

IHFLOWA [38] + order-inducing vectors in Table 9 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198604 ≻ 1198605

Problem in [39] I-IVDHFLOWA [39] + order-inducing vectors in [39] 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198602 ≻ 1198601

I-IVDHFLOWA [39] + order-inducing vectors in Table 10 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602

permutation of 1199092 ≻ 1199091 ≻ 1199093 which verifies that ourproposed TOPSIS-based order-inducing approach is prac-tical and useful in helping work out decision results morereasonably and robustly

63 Application Studies of Proposed TOPSIS-Based Order-Inducing Method When confronting complex decision-making situations that need additional attitudinal charactersof decision-makers Yager and Filev [88] and Yager [102]introduced and encouraged to employ order-inducing vectorto represent those attitudinal characters Since then theidea of order-inducing vector has been extended to variousdecision-making environments [98 103] such as those basedon intuitionistic fuzzy information [104 105] linguistic infor-mation [106] hesitant fuzzy linguistic information [38] andinterval-valued dual hesitant fuzzy uncertain linguistic infor-mation [39] Generally in existing literature order-inducingvectors rely on experts to directly give [38 39 104ndash106]

In order to examine the adaptability of our TOPSIS-based order-inducing method to linguistic decision-makingenvironments we respectively apply the method to resolvethe same problems in references [38 39] Due to fact thatthere is no descriptions about how the order-inducing vectorswere derived in both references [38 39] we here applyextremums into our TOPSIS-based method to obtain order-inducing vectors that is (1199047 1 0) and (1199041 0 1) for[38] ([1199046 1199046] 1 0) and ([1199040 1199040] 0 1) for [39] Thenwe obtain our order-inducing vectors listed in Table 9 forthe problem adopted in [38] and our order-inducing vectorslisted in Table 10 for [39]

Now we apply the order-inducing vectors as shownin Tables 9 and 10 to resolve the problems in [38] and[39] respectively Regarding the problem in [38] the scorevalues of all alternatives are calculated as 119878(1198601)=02602119878(1198602)=02417 119878(1198603)=03577 119878(1198604)=0221 and 119878(1198605)=0185so the corresponding ranking result is 1198603 ≻ 1198601 ≻ 1198602 ≻1198604 ≻ 1198605 As for the problem in [39] the score values ofall alternatives are computed as 119878(1198601)=39954 119878(1198602)=3783119878(1198603)=70827 119878(1198604)=71792 and 119878(1198605)=5813 we thus obtainthe ranking result 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602 Formore clarity brief descriptions of comparative approachesand obtained ranking results also have been put together inTable 11

According to Table 11 the approaches that employ ourorder-inducing vectors derive the same ranking order ofthe better three alternatives for both comparative problemsfrom [38 39] As seen our proposed TOPSIS-based order-inducing method exhibits effective and more understand-able Furthermore when decision-makers are incapable ofdirectly put forward all desired order-inducing vectors due tocomplexity in ill-structured problems our method providesa feasible directions to obtain order-inducing vectors byfiguring out ideal solutions Comparatively our proposedTOPSIS-based order-inducing method behaves effective andmore operational in practical use

64 Comparison with Decision-Making Approach Based onDegraded form of IVDHFULS Note that our approachis intrinsically adaptive to decision situations based ondegraded forms of IVDHF UUBLS such as the dual hesitant

24 Complexity

fuzzy linguistic sets introduced by Yang and Ju [54] Tofurther inspect feasibility of our approach we here conductcomparative study on the same case adopted by Yang and Ju[54] We firstly convert their original decision matrix intothe form of IVDHF UUBLS crisp membership and non-membership degrees are equivalently changed to intervalswith same upper and lower limits (eg 02 turns out to be[02 02]) and linguistic variables are rewritten as uncertainlinguistic variables (eg 1199042 turns out to be [11990421199042])

Since the case only got three attributes among whichinterdependency could be not significant under generalassumption weighting vector 120596119860119868 is not calculated Andfor straight comparison order inducing vector is notincluded either Then the argument-dependent attributeweighting vector can be obtained from Eq(29) as 120596119860119863 =(03192 03132 03676)119879 By use of Algorithm Iwith operatorIVDHFUUBLWA (ie GIVDHFUUBLWA in case of 120582 = 1see Appendix B) we have the scores of alternatives 119904(1198601) =03412 119904(1198602) = 01354 119904(1198603) = 03932 119904(1198604) = 07352Therefore the ordering of alternatives is1198604 ≻ 1198603 ≻ 1198601 ≻ 1198602which is mainly in alignment with the ordering of1198604 ≻ 1198601 ≻1198603 ≻ 1198602 in Yang and Ju [54] It is worth noticing that 120596119860119863used in our approach is derived objectively from assessmentswhile attribute weights 120596lowast = (035 025 040)119879used by Yangand Ju [54] were assigned empirically Differences between120596119860119863 and 120596lowast lead to different ordering of1198601 and1198603 Compar-atively speaking our approach holds more adaptability andobjectivity

In general our proposed MADM approach in this paperexhibits an effective way to tackle complex green contractorselection problems in which subjective attributes weightingvector (such as120596119860119868) and objective attributes weighting vector(such as 120596119860119863) exist To comprehensively utilize these twofundamental facets of attributes weighting information ourcase study and computational experiments suggest that linearcombination method (ie 120596 = 120572120596119860119868 + 120573120596119860119863) is effectiveto reflect effects of 120596119860119868 and 120596119860119863 by configuring parameter 120572or 120573 When there is no specific expert opinions to specify orhard to obtain we suggest that parameter 120572 chooses the valueset of 0 02 05 08 1 Furthermore generalized aggrega-tion operators are suggested to inspect score trajectoriesof alternative contractors along increasing of the operatorsrsquoamplification parameter 120582 If some alternative contractorsrsquoscore values get very close or crossed so that they cannot bedifferentiated from each other significantly we suggest to askthe panel of experts to derive opinions of ideal solutions andutilize our TOPSIS-based order-inducing method to includethose expert opinions as attitudinal decision informationthereby obtaining more rational and robust decision results

7 Conclusions

Aiming at tackling complex decision-making problems ofgreen contractor selection that often hold simultaneouslythe characteristics of decision hesitancy attributes inter-dependency and group complex attitudinal characters wehave developed an effective IVDHF UUBLS-based approach

We utilize IVDHF UUBLS to elicit complicate decision-makersrsquo assessments more objectively and comprehensivelywe employ DEMATEL for addressing attribute interdepen-dency more precisely and we determine group attitudinalcharacters as order-inducing vectors based on ideal attitudesof experts rather than by empirical assignment The mainadvantages of our approach are (i) IVDHF UUBLS offers anadequate way to depict decision hesitancy (ii) the designeddistance measure for IVDHF UUBLS avoids informationdistortion of traditional ones which artificially add mis-matching membership or nonmembership degrees (iii) thedeveloped generalized operators include traditional ones asspecial cases so as to provide flexibility in MADM basedon IVDHF UUBLS (iv) the devised synthesized attributeweighting scheme provides an adaptive way to consider twonecessary aspects attributes interdependency-based weightsand argument-dependent weights (v) the TOPSIS-basedmethod can deduce reasonable order-inducing vectors ratherthan empirical ones and holds generality in other linguisticdecision environments Results of case studies and experi-ments have verified the validity of the proposed approach

When multiple decision organizations are involvedgroup decision-making based on IVDHF UUBLS is animportant research direction Besides inducing preferencesfor decision-making problems of high uncertainty interactivemechanisms should also be studied to extract preferences Inaddition empirical studies of the proposed approach can becarried out to various areas such as investment evaluationproject management and vehicle selection in fleet operations

Appendix

A Basic Notions for IVDHFS and UnbalancedLinguistic Term Set

Definition A1 (see [47]) Letting 119883 be a fixed set then anIVDHFS on 119883 is defined as119863 = ⟨119909 ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (A1)

where ℎ(119909) = ⋃[120583119871120583119880]isinℎ(119909)120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and119892(119909) = ⋃[]119871]119880]isin119892(119909)] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two setsof interval values in [0 1] denoting possible membershipand non-membership degrees of element 119909 isin 119883 to theset 119863 respectively with conditions 120583 ] isin [0 1] and0 le (120583119880)+ + (]119880)+ le 1 and for all 119909 isin 119883 (120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880

For convenience normally 119889 = ℎ 119892 is called an interval-valued dual hesitant fuzzy element (IVDHFE) and 119863 is theset of all IVDHFEs

Definition A2 (see [107]) Suppose 119878 = 119904119895 | 119895 = 0 1 119892 minus1 is a finite and totally ordered discrete linguistic term setwhere 119904119895 represent possible values for a linguistic variableand 119892 is an odd cardinality 119878 is uniformly and symmetricallydistributed if the following two conditions are satisfied (1)

Complexity 25

there exists a unique constant 120582 gt 0 such that 119873119878(119904119894) minus119873119878(119904119895) = 119897(119894 minus 119895) for all 119894 119895 = 0 1 119892 minus 1 (2) Let 119878119877 =119904 | 119904 isin 119878 119904 gt 119878119862 and 119878119871 = 119904 | 119904 isin 119878 119904 lt 119878119862 Let C(119878119877) andC(119878119871) be the cardinality of 119878119877 and 119878119871 then C(119878119877) = C(119878119871)If 119878 is uniformly and symmetrically distributed then 119878 iscalled a balanced linguistic term set Otherwise 119878 is calledan unbalanced linguistic term set

The following 2-tuple fuzzy linguistic representationmodel extends traditional linguistic term set to a continuouscase so as to facilitate computing with linguistic variables

Definition A3 (see [59]) Let 119878 = 1199040 1199041 119904119892minus1 be alinguistic term set and 120573 isin [0 119892] Then a 2-tuple fuzzylinguistic variable expresses the equivalent information to 120573is defined as

Δ [0 119892] 997888rarr 119878 times [minus05 05) (A2)

Δ (120573) = (119904119894 120572) with

119904119894 119894 = 119903119900119906119899119889 (120573)120572 = 120573 minus 119894 120572 isin [minus05 05)

(A3)

Δminus1 (119904119894 120572) = 119894 + 120572 = 120573 (A4)

where 119903119900119906119899119889() is the usual rounding operation and120572 is calledsymbolic translation

To obtain 2-tuple fuzzy linguistic representations ofunbalanced linguistic terms the concept of linguistic hierar-chies ie 119871119867 = ⋃119905 119897(119905 119899(119905)) is used 119897(119905 119899(119905)) is a linguistichierarchy with 119905 indicating the level of hierarchy and 119899(119905)denotes the granularity of the linguistic term set of 119905 Herreraet al [59] defined the following transformation functionsbetween labels from different levels in multigranular linguis-tic information contexts without loss of information

Definition A4 (see [59]) In linguistic hierarchies 119871119867 =⋃119905 119897(119905 119899(119905)) whose linguistic term sets are represented by119878119899(119905) = 119904119899(119905)0 119904119899(119905)119899(119905)minus1 the transformation function froma linguistic label in level 119905 to a label in consecutive level 1199051015840 isdefined as 1198791198651199051199051015840 119897(119905 119899(119905)) 997888rarr 119897(1199051015840 119899(1199051015840)) such that

1198791198651199051199051015840 (119904119899(119905)119894 120572119899(119905))= Δ 1199051015840 (Δminus1119905 (119904119899(119905)119894 120572119899(119905)) (119899 (1199051015840) minus 1)119899 (119905) minus 1 ) (A5)

By use of the above transformation function any 2-tuplelinguistic representation can be transformed into a term in119871119867 Detailed transformation procedures are listed as follows

(1) Representation in linguistic hierarchy To transformthe unbalanced terms of 119878 into the corresponding terms inthe 119871119867 transformation function 120595 is employed to associateeach unbalanced linguistic 2-tuple (119904119894 120572) with its linguistic 2-tuple in 119871119867(119878) ie

120595 119878 997888rarr 119871119867(119878) (A6)

so that 120595(119904119894 120572) = (119904119866(119894)119868(119894) 120582) for forall(119904119894 120572) isin 119878(2) Computational phase Firstly transform (119904119866(119894)119868(119894) 120582) intolinguistic 2-tuples denoted as (119904119899(1199051015840)

1198681015840(119894) 1205821015840) in 119878119899(1199051015840) where

(119904119899(1199051015840)1198681015840(119894)

1205821015840) = 119879119865 (119904119866(119894)119868(119894) 120582)= Δ(Δminus1 (119904119866(119894)119868(119894) 120582) sdot (119899 (1199051015840) minus 1)119866 (119894) minus 1 ) (A7)

Then a computational model is used over 119878119899(1199051015840) with aresult denoted as (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)(3) Retranslation process The result (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)is transformed into the unbalanced term in 119878 by using thetransformation function 120595minus1 ie

120595minus1 119871119867 (119878) 997888rarr 119878 (A8)

so that 120595minus1(119904119899(1199051015840)119903 120582119903) = (119904119903119890119904119906119897119905 120582119903119890119904119906119897119905) isin 119878B Details of Theorem 14

Proof Given a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector for119904119889119895 and 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 120582 gt 0 then(1) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLWA and GIVDHFUUBLWG operators respectively(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLOWA and GIVDHFUUBLOWG operators respectively

(2) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to theIVDHF UUBL weighted average (IVDHFUUBLWA)and IVDHF UUBL weighted geometric (IVDHFUUBLWG)operators respectively where

26 Complexity

119868119881119863119867119865119880119880119861119871119882119860120596 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

120596119895119904119889119895= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus

119899prod119895=1

(1 minus 120583119871119895 )120596119895 1 minus 119899prod119895=1

(1 minus 120583119880119895 )120596119895]] [[

119899prod119895=1

(]119871119895)120596119895 119899prod119895=1

(]119880119895 )120596119895]])

(B1)

119868119881119863119867119865119880119880119861119871119882119866120596 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889119895120596119895

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[

119899prod119895=1

(120583119871119895 )120596119895 119899prod119895=1

(120583119880119895 )120596119895]] [[1 minus

119899prod119895=1

(1 minus ]119871119895)120596119895 1 minus 119899prod119895=1

(1 minus ]119880119895 )120596119895]])

(B2)

(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHA andGIVDHFUUBLHG operators reduce to the IVDHF UUBL

orderedweighted average (IVDHFUUBLOWA) and IVDHFUUBL ordered weighted geometric (IVDHFUUBLOWG)operators respectively where

119868119881119863119867119865119880119880119861119871119874119882119860119908 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

119908119895119904119889120590(119895)= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) )))

119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) )))

]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus

119899prod119895=1

(1 minus 120583119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus 120583119880120590(119895))119908119895]] [[

119899prod119895=1

(]119871120590(119895))119908119895 119899prod119895=1

(]119880120590(119895))119908119895]])

(B3)

119868119881119863119867119865119880119880119861119871119874119882119866119908 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889120590(119895)119908119895

= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[

119899prod119895=1

(120583119871120590(119895))119908119895 119899prod119895=1

(120583119880120590(119895))119908119895]] [[1 minus

119899prod119895=1

(1 minus ]119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus ]119880120590(119895))119908119895]])

(B4)

Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

References

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[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

[11] X Qi C Liang and J Zhang ldquoGeneralized cross-entropy basedgroup decision making with unknown expert and attributeweights under interval-valued intuitionistic fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 79 pp 52ndash642015

[12] H Zimmermann Fuzzy Set Theorymdashand Its ApplicationsKluwer Academic Boston Mass USA 2001

[13] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 pp 199ndash249 1975

[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

[16] B T Nuong K-W Kim L Prathumratana et al ldquoSustainabledevelopment in the mining sector and its evaluation usingfuzzy AHP (Analytic Hierarchy Process) approachrdquo GeosystemEngineering vol 14 no 1 pp 43ndash50 2011

[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

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[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

[21] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

[25] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009

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[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

[30] AM Anvuur andMM Kumaraswamy ldquoConceptualmodel ofpartnering and alliancingrdquo Journal of Construction Engineeringand Management vol 133 no 3 pp 225ndash234 2007

[31] A H I Lee H Kang C F Hsu and H Hung ldquoA green supplierselection model for high-tech industryrdquo Expert Systems withApplications vol 36 no 4 pp 7917ndash7927 2009

[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

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[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

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[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

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[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

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[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

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[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

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performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

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[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

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[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

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8 Complexity

In above details about the transformation functionsof 119879119865 and Δ linguistic hierarchies (119871119867) as well as thetransformation procedures for unbalanced linguistic termsets are shown in Appendix A

eorem 3 Letting 119904119889 = (119904120599 ℎ 119892) 1199041198891 = (1199041205991 ℎ1 1198921) and1199041198892 = (1199041205992 ℎ2 1198922) be any three IVDHF UUBLNs 120582 1205821 1205822 isin[0 1]then following properties are true

(1) 1199041198891 oplus 1199041198892 = 1199041198892 oplus 1199041198891 (2) 1199041198891 otimes 1199041198892 = 1199041198892 otimes 1199041198891 (3)120582(1199041198891 oplus 1199041198892) = 1205821199041198891 oplus 1205821199041198892(4) 1199041198891120582 otimes 1199041198892120582 = (1199041198891 otimes 1199041198892)120582 (5) 1205821119904119889 oplus 1205822119904119889 = (1205821 +1205822)119904119889 (6) 1199041198891205821 otimes 1199041198891205822 = 1199041198891205821+1205822

Proof Omitted

In Definition 2 and Theorem 3 1199051198961 1199051198962 1199051198941 1199051198942 1199051198951 and1199051198952 are corresponding levels of 119904120572 119904120573 1199041205721 1199041205731 1199041205722 and 1199041205722in 119871119867 respectively 1199050 is the maximum level of 119904120572 119904120573 1199041205721 1199041205731 1199041205722 and 1199041205722 in 119871119867 Furthermore to compare any twoIVDHF UUBLNs we also have following definitions

Definition 4 Let 119904119889 = (119904120599 ℎ 119892) be an IVDHF UUBLN119904120599 = [119904120572 119904120573] then score function 119878(119904119889) and accuracy function119875(119904119889) can be represented by

119878 (119904119889) = Δminus11199050 (11987911986511990511989611199050 (120595 (119904120572))) + Δminus11199050 (11987911986511990511989621199050 (120595 (119904120573)))2 times 12 ( 1119897 (ℎ) sum

[120583119871120583119880]isinℎ

120583119871 minus 1119897 (119892) sum[]119871]119880]isin119892

]119871

+ 1119897 (ℎ) sum[120583119871120583119880]isinℎ

120583119880 minus 1119897 (119892) sum[]119871]119880]isin119892

]119880)(3)

119875 (119904119889)= Δminus11199050 (11987911986511990511989611199050 (120595 (119904120572))) + Δminus11199050 (11987911986511990511989621199050 (120595 (119904120573)))2

times 12 ( 1119897 (ℎ) sum[120583119871120583119880]isinℎ

120583119871 + 1119897 (119892) sum[]119871]119880]isin119892

]119871

+ 1119897 (ℎ) sum[120583119871120583119880]isinℎ

120583119880 + 1119897 (119892) sum[]119871]119880]isin119892

]119880)

(4)

where 119897(ℎ) and 119897(119892) are numbers of values in ℎ and 119892respectively 1199051198961 and 1199051198962 are the corresponding levels of 119904120572 and119904120573 in 119871119867 and 1199050 is the maximum level of 119905119896 in 119871119867

Definition 5 Given any IVDHF UUBLNs 1199041198891 = (1199041205991 ℎ1 1198921)1199041198892 = (1199041205992 ℎ2 1198922) then(1) If 119878(1199041198891) lt 119878(1199041198892) then 1199041198891 lt 1199041198892(2) If 119878(1199041198891) = 119878(1199041198892) then

(a) If 119875(1199041198891) = 119875(1199041198892) then 1199041198891 = 1199041198892(b) If 119875(1199041198891) lt 119875(1199041198892) then 1199041198891 lt 1199041198892

33 Distance Measure for IVDHF UUBLS When 119897(ℎ) or119897(119892) of two IVDHF UUBLNs are unequal the complement-ing method [86] is normally adopted to design distancemeasures Note that artificially adding values to shorterones in the complementing method will cause informationdistortion To avoid this limitation we provide the followingdistance measure

Definition 6 Let two IVDHF UUBLNs 1199041198891 = (1199041205991 ℎ1 1198921)and 1199041198892 = (1199041205992 ℎ2 1198922) where 1199041205991 = [1199041205721 1199041205731] 1199041205992 = [1199041205722 1199041205732]119897ℎ1 119897ℎ2 1198971198921 and 1198971198922 are lengths of ℎ1 ℎ2 1198921 and 1198922 respectivelydenoting number of elements in ℎ1 ℎ2 1198921 and 1198922 Suppose1198681 = (1(119899(1199051198941) minus 1))Δminus11199050 (11987911986511990511989411199050 (120595(1199041205721))) 1198682 = (1(119899(1199051198942) minus1))Δminus11199050 (11987911986511990511989421199050 (120595(1199041205731))) 1198683 = (1(119899(1199051198951)minus1))Δminus11199050 (11987911986511990511989511199050 (120595(1199041205722)))1198684 = (1(119899(1199051198952) minus 1))Δminus11199050 (11987911986511990511989521199050 (120595(1199041205732))) where 1199051198941 1199051198942 1199051198951 and1199051198952 are the corresponding levels of unbalanced linguistic terms1199041205721 1199041205731 1199041205722 and 1199041205732 in the linguistic hierarchy 119871119867 and 1199050 is themaximum level of 1199041205721 1199041205731 1199041205722 and 1199041205732 in 119871119867 Then by use ofnormalized Euclidean distance we define a distance measure119889 for IVDHF UUBLNs as follows

Situation 7 When 119897ℎ1 = 119897ℎ2 = 1198971 and 1198971198921 = 1198971198922 = 1198972 then119889 (1199041198891 1199041198892) = (12 ( 11198971sdot 1198971sum119896=1

(10038161003816100381610038161003816100381610038161198681120583119871119895ℎ1 minus 1198683120583119871119896ℎ2 10038161003816100381610038161003816100381610038162 + 10038161003816100381610038161003816100381610038161198682120583119880119895ℎ1 minus 1198684120583119880119896ℎ2 10038161003816100381610038161003816100381610038162) + 11198972sdot 1198972sum119896=1

(1003816100381610038161003816100381610038161198681]1198711198951198921 minus 1198683]1198711198961198922 1003816100381610038161003816100381610038162 + 1003816100381610038161003816100381610038161198682]1198801198951198921 minus 1198684]1198801198961198922 1003816100381610038161003816100381610038162)))12

(5)

Situation 8 When 119897ℎ1 = 119897ℎ2 or 1198971198921 = 1198971198922 then119889 (1199041198891 1199041198892) = (12 ( 1119897ℎ1 119897ℎ2sdot 119897ℎ1sum119895=1

119897ℎ2sum119896=1

(10038161003816100381610038161003816100381610038161198681120583119871119895ℎ1 minus 1198683120583119871119896ℎ2 10038161003816100381610038161003816100381610038162 + 10038161003816100381610038161003816100381610038161198682120583119880119895ℎ1 minus 1198684120583119880119896ℎ2 10038161003816100381610038161003816100381610038162)+ 11198971198921 1198971198922sdot 1198971198921sum119895=1

1198971198922sum119896=1

(1003816100381610038161003816100381610038161198681]1198711198951198921 minus 1198683]1198711198961198922 1003816100381610038161003816100381610038162 + 1003816100381610038161003816100381610038161198682]1198801198951198921 minus 1198684]1198801198961198922 1003816100381610038161003816100381610038162))12

(6)

eorem 9 The distance measure 119889 defined in Definition 6satisfies following properties

(1) 0 le 119889(1199041198891 1199041198892) le 1

Complexity 9

(2) 119889(1199041198891 1199041198892) = 0 if and only if 1199041198891 and 1199041198892 are perfectlyconsistent

(3) 119889(1199041198891 1199041198892) = 119889(1199041198892 1199041198891)4 Generalized Aggregation Operators forIVDHF_UUBLS

41 Definitions of Operators Based on the generalized oper-ators firstly introduced by Yager [87] we here develop some

fundamental generalized aggregation operators for the newlydefined IVDHF UUBLS

Definition 10 Given a collection of IVDHF UUBLNs 119904119889119895(119895 =1 119899) their weighting vector 120596 = (1205961 1205962 120596119899)119879120596119895 isin[0 1]sum119899119895=1 120596119895 = 1 120582 be a parameter 120582 isin (0 +infin)(1) Generalized IVDHFUUBLWeighted Average(GIVDHFUUBLWA) Operator

119866119868119881119863119867119865119880119880119861119871119882119860120596120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(120596119895119904119889119895120582))1120582

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120582)1120582 119904(sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120582)1120582]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[(1 minus

119899prod119895=1

(1 minus (120583119871119895 )120582)120596119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880119895 )120582)120596119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871119895)120582)120596119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880119895 )120582)120596119895)1120582]])

(7)

(2) Generalized IVDHFUUBLWeighted Geometric(GIVDHFUUBLWG)Operator

119866119868119881119863119867119865119880119880119861119871119882119866120596120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889119895)120596119895)= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871119895 )120582)120596119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880119895 )120582)120596119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871119895 )120582)120596119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880119895 )120582)120596119895)1120582]])

(8)

Definition 11 For a collection of IVDHF UUBLNs 119904119889119895(119895 = 1 119899) 119904119889120590(119895) be the 119895th largest 119908 = (1199081 1199082 119908119899)119879 be the aggregation-associated weighting vector 119908119895 isin [0 1]sum119899119895=1119908119895= 1 120582 is a parameter such that 120582 isin (0 +infin) 119878119899 997888rarr 119878 Then

10 Complexity

(1) Generalized IVDHFUUBL Ordered Weighted Average(GIVDHFUUBLOWA) Operator

119866119868119881119863119867119865119880119880119861119871119874119882119860120596120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(119908119895119904119889120590(119895)120582))1120582

= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

120582)1120582]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895) []119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120590(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120590(119895))120582)119908119895)1120582]])

(9)

(2) Generalized IVDHFUUBL Ordered Weighted Geometric(GIVDHFUUBLOWG)Operator

119866119868119881119863119867119865119880119880119861119871119874119882119866119908120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889120590(119895))119908119895)= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120590(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120590(119895))120582)120596119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120590(119895))120582)120596119895)1120582]])

(10)

Definition 12 For a collection of IVDHF UUBLNs 119904119889119895(119895 =1 119899) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector120596119895 isin [0 1] sum119899119895=1 120596119895 = 1 119899 is a balancing coefficient119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weight-ing vector with 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1

(1) Generalized IVDHFUUBL Hybrid Average (GIVDHFU-UBLHA) Operator

119866119868119881119863119867119865119880119880119861119871119867119860120596119908120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(119908119895119904119889120590(119895)120582))1120582

= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895)

))))120582)1120582]

Complexity 11

⋃[120583119871120590(119895) 120583

119880120590(119895)]isinℎ120590(119895) []

119871120590(119895) ]119880120590(119895)]isin119892120590(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120590(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120590(119895))120582)119908119895)1120582]])

(11)

where 119904119889120590(119895) is the 119895th largest and

119904119889119895 = 119899120596119895119904119889119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904119899120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904119899120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

] ⋃

[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[1 minus (1 minus 120583119871119895 )119899120596119895 1 minus (1 minus 120583119880119895 )119899120596119895] [(]119871119895)119899120596119895 (]119880119895 )119899120596119895]) (12)

(2) Generalized IVDHFUUBL Hybrid Geometric(GIVDHFUUBLHG) Operator

119866119868119881119863119867119865119880119880119861119871119867119866119908120596120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889120590(119895))119908119895)= ⋃(119904120599120590(119895)

ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120590(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895)

))))119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895) []119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120590(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120590(119895))120582)119908119895)1120582]])

(13)

where 119904119889120590(119895) is the 119895th largest and

119904119889119895 = 119904119889119895119899120596119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(Δminus11199050 (119879119865

11990511989511199050(120595(119904120572119895 ))))

119899120596119895 119904(Δminus11199050 (119879119865

11990511989521199050(120595(119904120573119895 ))))

119899120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[(120583119871119895 )119899120596119895 (120583119880119895 )119899120596119895] [1 minus (1 minus ]119871119895 )119899120596119895 1 minus (1 minus ]119880119895 )119899120596119895]) (14)

12 Complexity

When confronted with ill-structured situations wheredecision-makerrsquos complex attitudinal characters need to beincluded order-inducing vectors provide an effective way[88ndash90]Thus we further define following induced operatorsfor IVDHF UUBLNs

Definition 13 For a collection of IVDHF UUBLNs 119904119889119895 (119895 =1 119899) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector

120596119895 isin [0 1] sum119899119895=1 120596119895 = 1 119899 is a balancing coefficient119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weight-ing vector 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1 120576119895 denote a set oforder inducing vectors

(1) Induced Generalized IVDHFUUBL Hybrid Average (I-GIVDHFUUBLHA) Operator

119868-119866119868119881119863119867119865119880119880119861119871119867119860120596119908120582 (lt 1205761 1199041198891 gt lt 1205762 1199041198892 gt lt 120576119899 119904119889119899 gt) = ( 119899⨁119895=1

(119908119895119904119889120587(119895)120582))1120582

= ⋃(119904120599120587(119895) ℎ120587(119895) 119892120587(119895))isin119904119889120587(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120587(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587(119895)

))))120582)1120582]

⋃[120583119871120587(119895) 120583

119880120587(119895)]isinℎ120587(119895) []

119871120587(119895) ]119880120587(119895)]isin119892120587(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120587(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120587(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120587(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120587(119895))120582)119908119895)1120582]])

(15)

where (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) is decreasing order of (11990411988911199041198892 119904119889119899) according to the order inducing vector 120576119895 and119904119889119895 = 119899120596119895119904119889119895 = ⋃

(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904119899120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904119899120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

] ⋃

[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[1 minus (1 minus 120583119871119895 )119899120596119895 1 minus (1 minus 120583119880119895 )119899120596119895] [(]119871119895)119899120596119895 (]119880119895 )119899120596119895]) (16)

(2) Induced Generalized IVDHFUUBL Hybrid Geometric(I-GIVDHFUUBLHG) Operator

119868-119866119868119881119863119867119865119880119880119861119871119867119866119908120596120582 (lt 1205761 1199041198891 gt lt 1205762 1199041198892 gt lt 120576119899 119904119889119899 gt) = 1120582 ( 119899⨂119895=1

(120582119904119889120587(119895))119908119895)= ⋃(119904120599120587(119895)

ℎ120587(119895) 119892120587(119895))isin119904119889120587(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120587(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587(119895)

))))119908119895]

⋃[120583119871120587(119895)120583119880120587(119895)]isinℎ120587(119895) []119871120587(119895) ]

119880120587(119895)]isin119892120587(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120587(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120587(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120587(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120587(119895))120582)119908119895)1120582]])

(17)

Complexity 13

Weighted Aggregation

Geometric Arithmetic

GIVDHFUUBLHG GIVDHFUUBLHA

GIVDHFUUBLWG GIVDHFUUBLOWG

IVDHFUUBLWG IVDHFUUBLOWG

GIVDHFUUBLWA GIVDHFUUBLOWA

IVDHFUUBLWA IVDHFUUBLOWA

= 1 w =1

n = 1 =

1

n = 1 =1

n = 1 =

1

n

= 1 = 1 = 1 = 1

Figure 2

where (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) is decreasing order of (11990411988911199041198892 119904119889119899) according to the order inducing variables 120576119895 and

119904119889119895 = 119904119889119895119899120596119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(Δminus11199050 (119879119865

11990511989511199050(120595(119904120572119895 ))))

119899120596119895 119904(Δminus11199050 (119879119865

11990511989521199050(120595(119904120573119895 ))))

119899120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[(120583119871119895 )119899120596119895 (120583119880119895 )119899120596119895] [1 minus (1 minus ]119871119895 )119899120596119895 1 minus (1 minus ]119880119895 )119899120596119895]) (18)

42 Properties of the Proposed GeneralizedAggregation Operators

eorem 14 With special values of 120582 119908 and 120596 the operatorsGIVDHFUUBLHA and GIVDHFUUBLHG can include aseries of traditional aggregation operators as special cases andtheir relationship can be depicted in Figure 2

Proof See Appendix B

As for the induced hybrid aggregation operators we alsohave following theorem

eorem 15 If (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) = (119904119889120590(1) 119904119889120590(2) 119904119889120590(119899)) I-GIVDHFUUBLHA reduces toGIVDHFUUBLHAIf (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) = (119904119889120590(1) 119904119889120590(2) 119904119889120590(119899)) thenI-GIVDHFUUBLHG reduces to the GIVDHFUUBLHG oper-ator

eorem 16 All the proposed generalized operatorsGIVDHFUUBLWAGIVDHFUUBLWGGIVDHFUUBLOWAGIVDHFUUBLOWGGIVDHFUUBLHAGIVDHFUUBLHGI-GIVDHFUUBLHA and I-GIVDHFUUBLHG hold thefollowing properties (1) Commutativity (2) Idempotency (3)Boundedness

Based on above theorems following properties can alsobe derived

eorem 17 For a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weight vector of 119904119889119895with 120596119895 isin [0 1] and sum119899119895=1 120596119895 = 1 120582 gt 0 then we have

(1) 119868119881119863119867119865119880119880119861119871119882119866120596(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119882119860120596(1199041198891 1199041198892 119904119889119899)(2) 119868119881119863119867119865119880119880119861119871119882119866120596(1199041198891 1199041198892 119904119889119899) le119866119868119881119863119867119865119880119880119861119871119882119860120596120582(1199041198891 1199041198892 119904119889119899)(3) 119866119868119881119863119867119865119880119880119861119871119882119866120596120582(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119882119860120596(1199041198891 1199041198892 119904119889119899)

Proof Omitted for concision

eorem 18 For a collection of IVDHF UUBLNs 119904119889119895 = (119904120599119895 ℎ119895 119892119895) 119908 = (1199081 1199082 119908119899)119879 is the weighting vector of 119904119889119895with 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1 120582 gt 0 Then

(1) 119868119881119863119867119865119880119880119861119871119874119882119866119908(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119874119882119860119908(1199041198891 1199041198892 119904119889119899)(2) 119868119881119863119867119865119880119880119861119871119874119882119866119908(1199041198891 1199041198892 119904119889119899) le119866119868119881119863119867119865119880119880119861119871119874119882119860119908120582(1199041198891 1199041198892 119904119889119899)(3) 119866119868119881119863119867119865119880119880119861119871119874119882119866119908120582(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119874119882119860119908(1199041198891 1199041198892 119904119889119899)

Proof Omitted for concision

14 Complexity

5 An Integrated MADM Approach forTackling Complex Green ContractorSelection Problems

As described in Section 2 we take the green contractorselection as a special type of complicate MADM problemsthat synthesizes three characteristics of decision hesitancy[21 22] attributes interdependency [84] and group atti-tudinal characters [61] Therefore in this section we con-struct an integrated MADM approach to tackle the com-plex green contractor selection problems Suppose 119883 =1199091 1199092 119909119898 is the set of alternative green contractors and119860 = 1198601 1198602 119860119899 is the set of evaluative attributes 120596 =(1205961 1205962 120596119899)119879 is weighting vector for the attributes 120596119894 ge 0sum119899119894=1 120596119894 = 1 Let = (119903119894119895)119899times119898 denote the decision matrix inwhich 119903119894119895 = ([119904120572119894119895 119904120573119894119895] ℎ119894119895 119892119894119895) is an IVDHF UUBLN givenby decision-makers for alternative contractor 119909119895 with respectto attribute 119860 119894 According to the mechanism of pair-wisecomparisons among attributes in the DEMATELmethod [6364] the interdependency among attributes can be obtained asamatrix119885 = (119911119894119896)119899times119899 where 119911119894119896 indicates the degree towhich119860 119894 affects119860119896 Subsequently based on the IVDHFUUBLS andits operations we now present detailed steps of our MADMapproach as shown in following Algorithm I

Algorithm I Hesitant fuzzy linguistic MADMwith attributesinterdependency and decision-makersrsquo group attitudinalcharacters

Step 1 Determine argument-dependent weighting vector120596119860119863 according to attribute values by programming modeldeveloped in the following Section 51

Step 2 Obtain the attribute-interdependences based weight-ing vector 120596119860119868 by use of DEMATEL method described in thefollowing Section 52

Step 3 Calculate synthesized attribute weighting vectoraccording to

120596 = 120572120596119860119868 + 120573120596119860119863 (19)

where 120572 and 120573 are parameters to reflect decision character-istics of decision organizations 0 le 120572 120573 le 1 120572 + 120573 =1Step 4 Check requirements for order inducing If no addi-tional order inducing required then go to Step 5 otherwisego to Step 6

Step 5 Utilize generalized aggregation operators to get theoverall IVDHF UUBLNs 119903119895(119895 = 1 119898) for each alternative119909119895 Here we take GIVDHFUUBLHA operator for examplebecause it can include other traditional operators as its specialcases Therefore we have

119866119868119881119863119867119865119880119880119861119871119867119860(1199031119895 1199032119895 119903119899119895) = ( 119899⨁119894=1

(119908119894 (119903120590119894119895)120582))1120582= ⋃(119904120590119894119895ℎ120590119894119895 119892120590119894119895

)isin119903120590119894119895

([119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590119894119895

))))120582)1120582 119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590119894119895

))))120582)1120582]

⋃[120583119871120590119894119895120583

119880120590119894119895]isinℎ120590119894119895[]

119871120590119894119895]119880120590119894119895]isin119892120590119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120590119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120590119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120590119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120590119894119895)120582)119908119894)1120582]])

(20)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of 119903119894119895119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weightvector 119903120590119894119895 is the 119894th largest of 119903119894119895 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))

119904119899120596119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119894119895))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (21)

Complexity 15

Then go to Step 8

Step 6 By use of the TOPSIS-based method developed in thefollowing Section 53 we determine order-inducing vectors 120576to reflect decision-makersrsquo group attitudinal characters

Step 7 Utilizing induced generalized aggregation operatorsto get the overall IVDHF UUBLNs 119903119895(119895 = 1 2 119898) foreach alternative 119909119895 we have

119868-119866119868119881119863119867119865119880119871119867119860(1199031119895 1199032119895 119903119899119898) = ( 119899⨁119894=1

(119908119894 (119903120587119894119895)120582))1120582

= ⋃(119904120587119894119895ℎ120587119894119895119892120587119894119895)isin119903120587119894119895

([119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120587119894119895

))))120582)1120582 119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587119894119895

))))120582)1120582]

⋃[120583119871120587119894119895 120583

119880120587119894119895]isinℎ120587119894119895 []

119871120587119894119895]119880120587119894119895]isin119892120587119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120587119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120587119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120587119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120587119894119895)120582)119908119894)1120582]])

(22)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of119903119894119895 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 119903120587119894119895 is the 119894th largest of 119903119894119895 reordered by theorder-inducing vectors 120576 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))119899120596119894120572119894119895

119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120573119894119895 ))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (23)

Step 8 Calculate scores 119878(119903119895) and accuracy degrees 119875(119903119895) for119903119895(119895 = 1 2 119898)Step 9 According to Definition 5 output ranking order ofalternative contractors

In Algorithm I to objectively derive unknown weight-ing vector for evaluative attributes a synthesized attributeweighting vector is devised by fusing two parts of concern(i) argument-dependent weighting vector and (ii) attribute-interdependences based weighting vector The former is toreflect effects of attribute values on attribute weights whilethe latter is to exploit effects of interdependences amongattributes on their weights For clarity processing flowchartof Algorithm I is depicted in Figure 3

51 Determining Argument-Dependent Attribute WeightingVector 120596119860119863 In order to derive attribute weighting vec-tor appropriately from assessments maximizing deviationmethod [91] is adopted here for its distinguishing ability

and objectivity [92] According to Wang [91] if performancevalues of each alternative have little differences under anattribute it implicates such an attribute plays a less importantrole otherwise it lives a more important role in decision-making Thus if one attribute has similar values across alter-natives it should be assigned a smaller weight contrariwiseit should be a bigger weight

In light of the idea in maximizing deviation methodwe develop a programming model to determine argument-dependent weighting vector 120596119860119863 For attribute 119860 119894 the stan-dard deviation of alternative 119909119895 to all the other alternatives isdenoted as

119865119894119895 (120596119860119863) = 119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (24)

Let119865119894(120596119860119863) represent deviation value of all alternatives toother alternatives according to attribute119860 119894 and be formulatedas follows

16 Complexity

Attribute weighting

Hesitant fuzzy linguistic MADM with attributes interdependency and decision makersrsquo group attitudinal characters

Step 9 Output ranking order of alternatives

interdependency among attributes by DEMATEL in Section 52

attributes values by standard deviation method in Section 51

Step 6 Ensure TOPSIS order inducing variables proposed in Section 53

Step 7 Utilizing proposed induced generalized hybrid aggregation operators developed in Section

41 to get overall value of each alternative

Step 8 Calculate score degrees and accuracy degrees

Step 4 Order inducingYes

Step 5 Choose proposed generalized operators in Section 41 to get overall value of each alternative

No

Collect decision matrix in the form of IVDHF_UUBLS

Complex problems of green contractor selection

Step 1 Weighting vector AD derived from Step 2 Weighting vector AI derived from

Step 3 Obtain synthesized weighting vector of attributes =AI + AD

Figure 3 Processing flowchart of Algorithm I

119865119894 (120596119860119863) = 119898sum119895=1

119865119894119895 (120596119860119863) = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (25)

Based on the analysis above we can get followingprogramming model (M-1) for obtaining optimal attributeweighting vector 120596119860119863

(119872 minus 1)max 119865 (120596119860119863) = 119899sum

119894=1119865119894 (120596119860119863) = 119899sum

119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894st 119899sum

119894=1(120596119860119863119894 )2 = 1 120596119860119863119894 ge 0 119894 = 1 2 119899 (26)

where 119889(119903119894119895 119903119894119897) can be calculated by Definition 6To solve model (M-1) we can construct Lagrange func-

tion

119871 (120596119860119863119894 120577) = 119899sum119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894+ 1205772 ( 119899sum

119894=1

(120596119860119863119894 )2 minus 1) (27)

where 120577 is the Lagrange multiplierDifferentiate with respect to 120596119860119863119894 (119894 = 1 2 119898) and 120577

and set these partial derivatives equal to zero the followingset of equations can be obtained

120597119871120597120596119860119863119894 = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) + 120577120596119860119863119894 = 0120597119871120597120577 = 119899sum

119894=1

(120596119860119863119894 )2 minus 1 = 0 (28)

By normalizing the solutions of above equations we canget a simple and exact formula for determining the attributeweights as follows

120596119860119863119894 = sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897)sum119899119894=1sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897) (29)

52 Deriving Attributes Interdependency-Based AttributeWeighting Vector 120596119860119868 Some efforts [53 60 84 93ndash95]

Complexity 17

have been made to address interdependencies betweenattributes in MADM However they unanimously cannotreveal the interdependences in a more complete way WhileDEMATAEL technique [63 96] gathers collective knowledgeto capture influences between attributes more precisely andcompletely so that weighting vector for attributes can berationally determined according to strength measure ofthose influences [64 97] Therefore in tackling the complexMADMwith interdependent attributes under IVDHF UUBLenvironments we present a DEMATEL-based method as thefollowing Algorithm II to derive attribute weighting vector120596119860119868 from the viewpoint of attributes interdependency

Algorithm II DEMATEL-basedmethod for ensuringweight-ing vector 120596119860119868Step 1 Construct initial direct-influence matrix 119885 = (119911119894119896)119899times119899

For attributes 119860 = 1198601 1198602 119860119899 decision-makersare asked to give direct effect that attribute 119860 119894 has onattribute 119860119896 using a five-point scale with scores representedby natural language (1) 0 mdash lsquoabsolutely no influencersquo (2)1 mdash lsquolow influencersquo (3) 2 mdash lsquomedium influencersquo (4) 3 mdashlsquohigh influencersquo (5) 4mdash lsquovery high influencersquoThen the initialdirect-influence matrix 119885 = (119911119894119896)119899times119899 can be established witheach element being the mean of same factors in differentdirect matrices

Step 2 Calculate normalized direct-influence matrix 119883 =(119909119894119896)119899times119899 Using matrix 119885 the normalized direct-influencematrix 119883 = (119909119894119896)119899times119899 is calculated according to

119909119894119896 = 119911119894119896119904 (30)

where 119904 = maxmax1le119894le119899sum119899119896=1 119911119894119896max1le119896le119899sum119899119895=1 119911119894119896 Allelements in matrix 119883 are complying with 0 le 119909119894119896 le 10 le sum119899119896=1 119909119894119896 le 1 and 0 le sum119899119894=1 119909119894119896 le 1 and at least onerow or column of the summation equals 1

Step 3 Derive total-influence matrix 119879 = (119905119894119896)119899times119899 Basedon matrix 119883 the total-influence matrix 119879 = (119905119894119896)119899times119899 can bederived by summing the direct effects and all of the indirecteffects

119879 = 119883 + 1198832 + 1198833 + sdot sdot sdot + 119883ℎ = 119883 (119868 minus 119883)minus1 (31)

where 119868 represents identity matrix When ℎ 997888rarr infin 119883ℎ =(0)119899times119899Step 4 Compute influencing and influenced degrees of eachattribute The sum of rows and the sum of columns withinthe total-influence matrix 119879 are respectively expressed as thevectors 119903 and 119888

119903 = (119903119894)119899times1 = ( 119899sum119896=1

119905119894119896)119899times1

119888 = (119888119896)119899times1 = ( 119899sum

119895=1

119905119894119896)119879

1times119899

(32)

where 119903119894 denotes the sumof the 119894th row inmatrix119879 and showsthe sum of direct and indirect effects that attribute 119860 119894 hason other attributes Similarly 119888119896 shows the sum of direct andindirect effects that attribute 119860119896 has receivedStep 5 Estimate weights of attributes based on influencedegrees according to

120596119868lowast119894 = [(119903119894 + 119888119894)2 + (119903119894 minus 119888119894)2]12 (33)

Then the weighting vector 120596119860119868 can be obtained by normaliz-ing (33)

120596119860119868119894 = 120596119868lowast119894sum119899119894=1 120596119868lowast119894 (34)

53 Obtaining Order Inducing Variables 120576 by TOPSIS-BasedMethod For ill-defined MADM problems induced aggre-gation operators [89 90 98] provide an effective channelfor experts to express complex attitudinal characters Thekey step in induced aggregation operators is to obtainorder-inducing vectors 120576 which however are presumptivelyassigned in most papers [62] Therefore in this sectionaiming at deducing the order-inducing vectors 120576 rationallywe here devise a TOPSIS-based method in Algorithm IIIObviously the Algorithm III is of generality for MADMwithdifferent forms of linguistic expression

Algorithm III TOPSIS-based method for obtaining orderinducing vectors 120576Step 1 Let the decision maker define IVDHF UUBL idealsolution(s) 119903+119894 and IVDHF UUBL negative ideal solution(s)119903minus119894 corresponding to each attribute in which

119903+119894 = (119903+1198941 119903+1198942 119903+119894119901) 119903minus119894 = (119903minus1198941 119903minus1198942 119903minus119894119902) (35)

where 119901 and 119902 denotes the number of 119903+119894 and 119903minus119894 respectively

Step 2 Calculate distances from each element in decisionmatrix to ideal solution(s) and negative solution(s) respec-tively

119889+119894119895 = max119897=12119901

119889 (119903119894119895 119903+119894119897 ) 119889minus119894119895 = min

119897=12119902119889 (119903119894119895 119903minus119894119897 ) (36)

where 119889(119903119894119895 119903+119894119897 ) and 119889(119903119894119895 119903minus119894119897 ) are calculated by the distancemeasure in Definition 6 If 119901 gt 1 we choose the onewith maximum distance if 119902 gt 1 we choose the one withminimum distance

Step 3 Calculate coefficients according to

119888119894119895 = 119889minus119894119895119889+119894119895 + 119889minus119894119895 119894 = 1 2 119899 119895 = 1 2 119898 (37)

18 Complexity

Table 2 Decision matrix in the form of IVDHF UUBLS1199091 1199092 11990931198601([AHVH] [02 03][02 04] [03 04]) ([HT] [04 05][05 06] [02 03] [02 04]) ([ALAH] [01 02][01 03] [06 07])1198602([MH] [05 06][02 03]) ([AHT] [06 07][01 02] ) ([MVH] [02 03][05 06] [06 07])1198603([VLAL] [03 04][04 05] [05 06]) ([MVH] [06 07] [07 08][01 02]) ([VLL] [04 05][02 03])1198604([MAH] [02 04][05 06]) ([LAL] [03 05][02 03]) ([ALM] [06 07][01 02] [02 03])1198605([ALM] [04 05][01 02] [04 05]) ([AHH] [06 07][01 02]) ([HVH] [04 05][06 07] [01 03])1198606([LM] [01 02][03 05] [03 05]) ([VLL] [02 04] [05 06][02 03]) ([AHH] [03 04][04 05])1198607([VLL] [04 05][05 06] [02 03] [02 04]) ([MAH] [05 06] [07 08][01 02]) ([ALM] [04 05][05 06] [03 04])1198608([ALAH] [01 03][04 06]) ([AHH] [05 07][01 02] [02 03]) ([AHH] [04 06][03 04])

Table 3 Direct influence matrix Z1198601 1198602 1198603 1198604 1198605 1198606 1198607 11986081198601 0 3 4 1 3 1 2 41198602 1 0 2 4 4 2 3 11198603 4 2 0 1 3 2 4 11198604 2 3 1 0 3 4 1 21198605 4 2 1 3 0 1 4 31198606 2 4 3 4 4 0 4 21198607 4 1 2 3 3 1 0 41198608 1 2 4 3 3 2 4 0

Step 4 Derive order inducing variable 120576119894119895 by descendingorder of 119888119894119895 The greater the value 119888119894119895 the bigger the 1205761198941198956 Illustrative Example and Experiments

61 Illustrative Case Study Suppose a mega infrastructureproject is selecting green contractor for one of its subprojectsA panel of decision-makers including managers professorsand competitive intelligence experts has been organized tocomprehensively evaluate three alternative contractors (ie1199091 1199092 and 1199093) under the eight attributes (ie 1198601ndash1198608) listedin Section 2

To elicit true decision preferences we apply theIVDHF UUBLS to assess the three alternatives (1199091ndash1199093)Firstly decision-makers are asked to vote on 119909119895 under eachattribute119860 119894 using different unbalanced linguistic term sets 1198781 1198782 and 1198783 where 1198781 = 119873 119871119860119871119872119860119867119867119876119867119881119867119860119879 1198791198782 = 119873119860119873119881119871 119876119871 119871 119860119871119872119860119867119881119867 119879 1198783 = 119873119881119871119871119860119872119872119876119872119860119867119867 119879 In Figure 4 we show the adoptedlinguistic hierarchies and relationship between 1198781 1198782 and1198783 Attributes 1198601 1198602 1198605 and 1198608 are evaluated by use of 1198781attributes11986031198604 and1198607 are evaluated by 1198782 and attribute1198606is evaluated by 1198783 The highest-voted uncertain unbalancedlinguistic term [119904119898 119904119899] are identified Secondly the panel of

decision-makers is asked to further express their opinionson membership degrees ℎ(119909) and nonmembership degrees119892(119909) to [119904119898 119904119899] Then we can derive compound assessments(ie ⟨119909 119904120599(119909) ℎ(119909) 119892(119909)⟩) in the form of IVDHF UUBLSand obtain the decision matrix in Table 2 Additionallyin order to examine the interdependency among attributesdecision-makers are also required to assess the influentialrelations among the eight attributes using DEMATEL-basedAlgorithm II The direct influence matrix is collected andshown in Table 3

Next we apply the proposed Algorithm I to solve thisgreen contractor selection problem To derive the synthe-sized attribute weighting vector in Algorithm I we assign120572 = 120573 = 05 For scenarios with no group attitudinalcharacters operator GIVDHFUUBLHA in Definition 12 ischosen In contrast for scenarios with group attitudinalcharacters operator I-GIVDHFUUBLHA in Definition 13will be chosen Further the panel of decision-makers is askedto determine their positive and negative ideal alternativesthat are collected in Table 4 where corresponding order-inducing vectors that are derived from Algorithm III arealso listed (see the rightmost column) Thereafter operatorI-GIVDHFUUBLHA in Definition 13 is utilized for aggrega-tion For GIVDHFUUBLHA and I-GIVDHFUUBLHA we

Complexity 19

Table 4 Ideal alternatives (119903+ and 119903minus) and derived order-inducing vectors 120576119903+ 119903minus 1205761199091 1199092 11990931198601 ([VHT] 1 0) ([NL] 1 0) 5 6 61198602 ([MAH] [07 08] [01 02]) ([NL] 1 0) 1 1 41198603 (T 1 0) (L 1 0) 8 7 81198604 ([ALAH] 09 01) (T 1 0) 2 2 11198605 ([HT] [09 1] 0) (N 1 0) 7 5 51198606 (M 1 0) (N 1 0) 6 3 31198607 (M 1 0) ([NL] 1 0) 3 8 71198608 ([MAH] 1 0) (N 1 0) 4 4 2

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 1 2 3 4

0 1 2

N L AH H QH VH AT TM

Level 1 l(13)

Level 2 l(25)

Level 3 l(39)

Level 4 l(417)

Unbalanced linguistic term set S1

N AH VH TM Unbalanced linguistic term set S2VL L ALAN QL

N H TM Unbalanced linguistic term set S3

L AM AHQM

AL

VL

Figure 4 Unbalanced linguistic term sets (1198781 1198782 and 1198783) and their mapping in linguistic hierarchies

assume 120582=2 and 119908 = (0 0 015 025 025 025 01 0)119879Note that 119908 is a position weighting vector which is derivedby the fuzzy semantic quantitative operator [99] Aftercompleting all steps in Algorithm I (see Appendix C) weobtain the ranking results and present them in Table 5 Forthe scores derived from Step 5 we find the ranking order is1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the best solution Step 7gives the ranking order of 1199092 ≻ 1199091 ≻ 1199093 indicating 1199092 is stillthe most appropriate one while 1199091 is better than 1199093 under theinfluence of 12057662 Experiments on Ranking Results along with VariousParameter Configurations In order to inspect the impactsof various parameter configurations on ranking results inthis section more numerical examples are carried out Alleight operators that defined in Section 41 are applied to theabove illustrative case where the parameter 120582 is configured asintegers ranging from 1 to 20 and parameter 120572 is configuredto a value set of 0 03 05 08 09 1

Firstly the four basic aggregation operators of GIVDH-FUUBLWA GIVDHFUUBLWG GIVDHFUUBLOWA and

Table 5 Ranking results of operators GIVDHFUUBLHA and I-GIVDHFUUBLHA

Aggregation operator Ranking orderGIVDHFUUBLHA 1199092 ≻ 1199093 ≻ 1199091I-GIVDHFUUBLHA 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG are applied to the former case Thesame configuration 120572 = 120573 = 05 is chosen while 120582 is testedas integers ranging from 1 to 20 The ranking results derivedfrom different parameter combinations have been collectedin Table 6 As can be seen four operators unanimouslyidentify the contractor 1199092 is the best one no matter how120582 changes Regarding priority relation between 1199091 and 1199093GIVDHFUUBLWA and GIVDHFUUBLOWA generate thesame result of1199093 ≻ 1199091 while results fromGIVDHFUUBLWGandGIVDHFUUBLOWGgot a change to 1199091 ≻ 1199093 at a certainvalue of parameter 120582

Actually GIVDHFUUBLWA and GIVDHFUUBLOWAare generalized version of weighted arithmetic aggregationoperators that producemore favorable results by emphasizing

20 Complexity

Table 6 Ranking orders obtained by GIVDHFUUBLWA GIVDHFUUBLOWA GIVDHFUUBLWG and GIVDHFUUBLOWG along with120582Operators 120582 (Integer) Ranking orderGIVDHFUUBLWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLOWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLWG [1 2] 1199092 ≻ 1199093 ≻ 1199091[3 20] 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG [1 8] 1199092 ≻ 1199093 ≻ 1199091[9 20] 1199092 ≻ 1199091 ≻ 1199093

Table 7 Ranking results obtained by GIVDHFUUBLHA and I-GIVDHFUUBLHA along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHA I-GIVDHFUUBLHA120572=0 120573=1 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=03 120573=07 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=05 120573=05 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=08 120573=02 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=09 120573=01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=1 120573=0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093

Table 8 Ranking results obtained by GIVDHFUUBLHG and I-GIVDHFUUBLHG along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHG I-GIVDHFUUBLHG

120572 = 0 120573 = 1 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 03 120573 = 07 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 05 120573 = 05 [1 8] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[9 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 08 120573 = 02 [1 17] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[18 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 09 120573 = 01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572 = 1 120573 = 0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093membership degrees to a uncertain linguistic term thus canbe considered as optimistic operators while GIVDHFU-UBLWG and GIVDHFUUBLOWG are generalized geomet-ric aggregation operators that producemore favorable resultsby emphasizing nonmembership degrees thus can be consid-ered as pessimistic operators [100 101]Therefore the rankingresults in Table 6 could be perceived as for optimism-baseddecision-making alternative 1199093 always stands out better than1199091 when 120582 gradually amplifies degree of optimism but forpessimism-based decision-making 1199091 turns out better than1199093 when 120582 gradually amplifies degree of pessimism 120582 canbe considered as the parameter for amplifying degree ofoptimism or pessimism [101]

Next to inspect the impacts of various combinationsof 120596119860119868 and 120596119860119863 on ranking results another four aggrega-tion operators of GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG also have beenapplied to the above illustrative case with varying 120572 120573 and 120582

All results have been collected in Tables 7 and 8 for compari-son As can be seen fromTable 7 ranking results generated byoperator GIVDHFUUBLHA keep the same as 1199092 ≻ 1199093 ≻ 1199091for all test combinations of parameters However as shownin Table 8 although GIVDHFUUBLHG is also capable ofidentifying 1199092 as the best contractor the priority relationsbetween 1199091 and 1199093 hold similar changing patterns in rankingresults in Table 6 that were output byGIVDHFUUBLWGandGIVDHFUUBLOWG Regarding decision situations whereobjective attributeweighting vector120596119860119863 is only considered (120572= 0) or its dominant importance (such as 120572 = 03) is accepted1199091 ≻ 1199093 is a clear observation when parameter 120582 is set toan integer in interval [8 20] Contrariwise under decisionscenarios where subjective attribute weighting vector 120596119860119868 isonly needed (120572 = 1) or its dominant importance (such as 120572 =09) is acknowledged 1199093 ≻ 1199091 holds all the way 120582 varies

In order to more clearly show the score trajectories ofalternative contractors along variations of parameter 120582 we

Complexity 21

HAInducedHA

InducedHA-2HA-2

HA-3

InducedHA-1InducedHA-3

= =

HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(a) 120572 = 0

HAInducedHA

= 03 = 07

InducedHA-2

HA-2HA-3

InducedHA-1 InducedHA-3HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(b) 120572 = 03

HAInducedHA

= 5 = 05

InducedHA-2HA-2

HA-3

InducedHA-1 InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(c) 120572 = 05

HAInducedHA

= 08 = 02

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572 = 08

HAInducedHA

= 9 = 01

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(e) 120572 = 09

HAInducedHA

246 2462 2464 2466 2468 247

= 1 = 0InducedHA-2HA-2

HA-3

HA- 3

HA-3

InducedHA-1

InducedHA-3

HA-1

HA-1

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

1085

1086

1087

(f) 120572 = 1

Figure 5 Score trajectories obtained by aggregation operators HA and InducedHA with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

further configure parameter 120582 to the value set of [0 01100]and conduct computational experiments to collect scorevalues and ranking results (as shown in Figures 5 and 6) byfour aggregation operators of GIVDHFUUBLHA GIVDH-FUUBLHG I-GIVDHFUUBLHAand I-GIVDHFUUBLHGFor more clarity in Figures 5 and 6 please note that weuse HA HG Induced-HA and Induced HG to respec-tively denote GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG In Figure 5it can be observed that when dominant importance of 120596119860119863

is emphasized (120572 = 0 120572 = 03 120572 = 05) GIVDHFUUBLHAis capable of differentiating the three alternative contractorsas same ranking order of 1199092 ≻ 1199093 ≻ 1199091 for all 120582 but withincreases in 120572 the margins between scores of 1199091 and 1199093 nar-rows and the rank order changes to 1199091 ≻ 1199093when 120572 gets 09and 1 Interestingly in Figure 6 although GIVDHFUUBLHGis able to significantly differentiate alternative contractors 1199091and 1199093 when dominant importance of 120596119860119863 is emphasized (120572= 0 120572 = 03 120572 = 05) rank order of 1199091 and 1199093 obviouslychanges once from 1199093 ≻ 1199091 to 1199091 ≻ 1199093 when 120582 is bigger

22 Complexity

HGInducedHG

= = 1

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(a) 120572= 0

HGInducedHG

= 3 = 07

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(b) 120572= 03

HGInducedHG

= 05 = 05

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(c) 120572= 05

HGInducedHG

= 8 = 02

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572= 08

HGInducedHG

292 293 294 295 296 297minus08

minus0795

minus079

= 09 = 01

HA-1

HA-3

InducedHG-2

HG-1

HG-1

HG-2

HG-3

HG-3 InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4

minus3

minus2

minus1

0

1

2

scor

e val

ues

(e) 120572= 09

HGInducedHG

50 60 70 80 90 100minus095minus09

minus085

= 1 = 0

HG-x3

HG-x1

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(f) 120572= 1

Figure 6 Score trajectories obtained by aggregation operators HG and InducedHG with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

than 8 When 120572 increases to 1 GIVDHFUUBLHGmaintainsthe rank order of 1199093 ≻ 1199091 but their score trajectorieswalk very close Comparatively I-GIVDHFUUBLHA andI-GIVDHFUUBLHG generate consistent ranking results of1199092 ≻ 1199091 ≻ 1199093 no matter 120582 varies in all the above experiments

In sum when parameters are configured with typical val-ues (ie 120572 = 05 and 120582=1) and no additional attitudinal infor-mation is taken into consideration the proposed aggregationoperators unanimously derive the same rank order 1199092 ≻1199093 ≻ 1199091 of for the three alternative contractors However

with the amplification effect of parameter 120582 analysis resultsfrom linear combination of 120596119860119868 and 120596119860119863 reveal alternative1199092 which consistently is the best choice but rank order of1199091 and 1199093 changes because their score trajectories get closeand crossed which probably indicates that only decision-making informationderived from the decisionmatrix is inad-equate to significantly differentiate alternatives 1199091 and 1199093While by integrating additional decision information aboutattitudinal characters of group experts I-GIVDHFUUBLHAand I-GIVDHFUUBLHG manage to generate the identical

Complexity 23

Table 9 Order-inducing vectors obtained by our TOPSIS-based method for problem in [38]

1198661 1198662 1198663 11986641198601 02522 01629 02282 027141198602 01 05243 01364 046891198603 04081 01756 05244 037791198604 04507 01409 01064 013221198605 00697 02203 03753 01161

Table 10 Order-inducing vectors obtained by our TOPSIS-based method for problem in [39]

1198661 1198662 1198663 11986641198601 04548 04969 04923 047321198602 05092 047 04768 046451198603 07192 04888 04953 049151198604 06722 06163 05706 049291198605 06104 04942 06463 04946

Table 11 Ranking results obtained by applying different order-inducing vectors to [38 39]

Problems Approaches Ranking results

Problem in [38] IHFLOWA [38] + order-inducing vectors in [38] 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198605 ≻ 1198604

IHFLOWA [38] + order-inducing vectors in Table 9 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198604 ≻ 1198605

Problem in [39] I-IVDHFLOWA [39] + order-inducing vectors in [39] 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198602 ≻ 1198601

I-IVDHFLOWA [39] + order-inducing vectors in Table 10 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602

permutation of 1199092 ≻ 1199091 ≻ 1199093 which verifies that ourproposed TOPSIS-based order-inducing approach is prac-tical and useful in helping work out decision results morereasonably and robustly

63 Application Studies of Proposed TOPSIS-Based Order-Inducing Method When confronting complex decision-making situations that need additional attitudinal charactersof decision-makers Yager and Filev [88] and Yager [102]introduced and encouraged to employ order-inducing vectorto represent those attitudinal characters Since then theidea of order-inducing vector has been extended to variousdecision-making environments [98 103] such as those basedon intuitionistic fuzzy information [104 105] linguistic infor-mation [106] hesitant fuzzy linguistic information [38] andinterval-valued dual hesitant fuzzy uncertain linguistic infor-mation [39] Generally in existing literature order-inducingvectors rely on experts to directly give [38 39 104ndash106]

In order to examine the adaptability of our TOPSIS-based order-inducing method to linguistic decision-makingenvironments we respectively apply the method to resolvethe same problems in references [38 39] Due to fact thatthere is no descriptions about how the order-inducing vectorswere derived in both references [38 39] we here applyextremums into our TOPSIS-based method to obtain order-inducing vectors that is (1199047 1 0) and (1199041 0 1) for[38] ([1199046 1199046] 1 0) and ([1199040 1199040] 0 1) for [39] Thenwe obtain our order-inducing vectors listed in Table 9 forthe problem adopted in [38] and our order-inducing vectorslisted in Table 10 for [39]

Now we apply the order-inducing vectors as shownin Tables 9 and 10 to resolve the problems in [38] and[39] respectively Regarding the problem in [38] the scorevalues of all alternatives are calculated as 119878(1198601)=02602119878(1198602)=02417 119878(1198603)=03577 119878(1198604)=0221 and 119878(1198605)=0185so the corresponding ranking result is 1198603 ≻ 1198601 ≻ 1198602 ≻1198604 ≻ 1198605 As for the problem in [39] the score values ofall alternatives are computed as 119878(1198601)=39954 119878(1198602)=3783119878(1198603)=70827 119878(1198604)=71792 and 119878(1198605)=5813 we thus obtainthe ranking result 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602 Formore clarity brief descriptions of comparative approachesand obtained ranking results also have been put together inTable 11

According to Table 11 the approaches that employ ourorder-inducing vectors derive the same ranking order ofthe better three alternatives for both comparative problemsfrom [38 39] As seen our proposed TOPSIS-based order-inducing method exhibits effective and more understand-able Furthermore when decision-makers are incapable ofdirectly put forward all desired order-inducing vectors due tocomplexity in ill-structured problems our method providesa feasible directions to obtain order-inducing vectors byfiguring out ideal solutions Comparatively our proposedTOPSIS-based order-inducing method behaves effective andmore operational in practical use

64 Comparison with Decision-Making Approach Based onDegraded form of IVDHFULS Note that our approachis intrinsically adaptive to decision situations based ondegraded forms of IVDHF UUBLS such as the dual hesitant

24 Complexity

fuzzy linguistic sets introduced by Yang and Ju [54] Tofurther inspect feasibility of our approach we here conductcomparative study on the same case adopted by Yang and Ju[54] We firstly convert their original decision matrix intothe form of IVDHF UUBLS crisp membership and non-membership degrees are equivalently changed to intervalswith same upper and lower limits (eg 02 turns out to be[02 02]) and linguistic variables are rewritten as uncertainlinguistic variables (eg 1199042 turns out to be [11990421199042])

Since the case only got three attributes among whichinterdependency could be not significant under generalassumption weighting vector 120596119860119868 is not calculated Andfor straight comparison order inducing vector is notincluded either Then the argument-dependent attributeweighting vector can be obtained from Eq(29) as 120596119860119863 =(03192 03132 03676)119879 By use of Algorithm Iwith operatorIVDHFUUBLWA (ie GIVDHFUUBLWA in case of 120582 = 1see Appendix B) we have the scores of alternatives 119904(1198601) =03412 119904(1198602) = 01354 119904(1198603) = 03932 119904(1198604) = 07352Therefore the ordering of alternatives is1198604 ≻ 1198603 ≻ 1198601 ≻ 1198602which is mainly in alignment with the ordering of1198604 ≻ 1198601 ≻1198603 ≻ 1198602 in Yang and Ju [54] It is worth noticing that 120596119860119863used in our approach is derived objectively from assessmentswhile attribute weights 120596lowast = (035 025 040)119879used by Yangand Ju [54] were assigned empirically Differences between120596119860119863 and 120596lowast lead to different ordering of1198601 and1198603 Compar-atively speaking our approach holds more adaptability andobjectivity

In general our proposed MADM approach in this paperexhibits an effective way to tackle complex green contractorselection problems in which subjective attributes weightingvector (such as120596119860119868) and objective attributes weighting vector(such as 120596119860119863) exist To comprehensively utilize these twofundamental facets of attributes weighting information ourcase study and computational experiments suggest that linearcombination method (ie 120596 = 120572120596119860119868 + 120573120596119860119863) is effectiveto reflect effects of 120596119860119868 and 120596119860119863 by configuring parameter 120572or 120573 When there is no specific expert opinions to specify orhard to obtain we suggest that parameter 120572 chooses the valueset of 0 02 05 08 1 Furthermore generalized aggrega-tion operators are suggested to inspect score trajectoriesof alternative contractors along increasing of the operatorsrsquoamplification parameter 120582 If some alternative contractorsrsquoscore values get very close or crossed so that they cannot bedifferentiated from each other significantly we suggest to askthe panel of experts to derive opinions of ideal solutions andutilize our TOPSIS-based order-inducing method to includethose expert opinions as attitudinal decision informationthereby obtaining more rational and robust decision results

7 Conclusions

Aiming at tackling complex decision-making problems ofgreen contractor selection that often hold simultaneouslythe characteristics of decision hesitancy attributes inter-dependency and group complex attitudinal characters wehave developed an effective IVDHF UUBLS-based approach

We utilize IVDHF UUBLS to elicit complicate decision-makersrsquo assessments more objectively and comprehensivelywe employ DEMATEL for addressing attribute interdepen-dency more precisely and we determine group attitudinalcharacters as order-inducing vectors based on ideal attitudesof experts rather than by empirical assignment The mainadvantages of our approach are (i) IVDHF UUBLS offers anadequate way to depict decision hesitancy (ii) the designeddistance measure for IVDHF UUBLS avoids informationdistortion of traditional ones which artificially add mis-matching membership or nonmembership degrees (iii) thedeveloped generalized operators include traditional ones asspecial cases so as to provide flexibility in MADM basedon IVDHF UUBLS (iv) the devised synthesized attributeweighting scheme provides an adaptive way to consider twonecessary aspects attributes interdependency-based weightsand argument-dependent weights (v) the TOPSIS-basedmethod can deduce reasonable order-inducing vectors ratherthan empirical ones and holds generality in other linguisticdecision environments Results of case studies and experi-ments have verified the validity of the proposed approach

When multiple decision organizations are involvedgroup decision-making based on IVDHF UUBLS is animportant research direction Besides inducing preferencesfor decision-making problems of high uncertainty interactivemechanisms should also be studied to extract preferences Inaddition empirical studies of the proposed approach can becarried out to various areas such as investment evaluationproject management and vehicle selection in fleet operations

Appendix

A Basic Notions for IVDHFS and UnbalancedLinguistic Term Set

Definition A1 (see [47]) Letting 119883 be a fixed set then anIVDHFS on 119883 is defined as119863 = ⟨119909 ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (A1)

where ℎ(119909) = ⋃[120583119871120583119880]isinℎ(119909)120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and119892(119909) = ⋃[]119871]119880]isin119892(119909)] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two setsof interval values in [0 1] denoting possible membershipand non-membership degrees of element 119909 isin 119883 to theset 119863 respectively with conditions 120583 ] isin [0 1] and0 le (120583119880)+ + (]119880)+ le 1 and for all 119909 isin 119883 (120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880

For convenience normally 119889 = ℎ 119892 is called an interval-valued dual hesitant fuzzy element (IVDHFE) and 119863 is theset of all IVDHFEs

Definition A2 (see [107]) Suppose 119878 = 119904119895 | 119895 = 0 1 119892 minus1 is a finite and totally ordered discrete linguistic term setwhere 119904119895 represent possible values for a linguistic variableand 119892 is an odd cardinality 119878 is uniformly and symmetricallydistributed if the following two conditions are satisfied (1)

Complexity 25

there exists a unique constant 120582 gt 0 such that 119873119878(119904119894) minus119873119878(119904119895) = 119897(119894 minus 119895) for all 119894 119895 = 0 1 119892 minus 1 (2) Let 119878119877 =119904 | 119904 isin 119878 119904 gt 119878119862 and 119878119871 = 119904 | 119904 isin 119878 119904 lt 119878119862 Let C(119878119877) andC(119878119871) be the cardinality of 119878119877 and 119878119871 then C(119878119877) = C(119878119871)If 119878 is uniformly and symmetrically distributed then 119878 iscalled a balanced linguistic term set Otherwise 119878 is calledan unbalanced linguistic term set

The following 2-tuple fuzzy linguistic representationmodel extends traditional linguistic term set to a continuouscase so as to facilitate computing with linguistic variables

Definition A3 (see [59]) Let 119878 = 1199040 1199041 119904119892minus1 be alinguistic term set and 120573 isin [0 119892] Then a 2-tuple fuzzylinguistic variable expresses the equivalent information to 120573is defined as

Δ [0 119892] 997888rarr 119878 times [minus05 05) (A2)

Δ (120573) = (119904119894 120572) with

119904119894 119894 = 119903119900119906119899119889 (120573)120572 = 120573 minus 119894 120572 isin [minus05 05)

(A3)

Δminus1 (119904119894 120572) = 119894 + 120572 = 120573 (A4)

where 119903119900119906119899119889() is the usual rounding operation and120572 is calledsymbolic translation

To obtain 2-tuple fuzzy linguistic representations ofunbalanced linguistic terms the concept of linguistic hierar-chies ie 119871119867 = ⋃119905 119897(119905 119899(119905)) is used 119897(119905 119899(119905)) is a linguistichierarchy with 119905 indicating the level of hierarchy and 119899(119905)denotes the granularity of the linguistic term set of 119905 Herreraet al [59] defined the following transformation functionsbetween labels from different levels in multigranular linguis-tic information contexts without loss of information

Definition A4 (see [59]) In linguistic hierarchies 119871119867 =⋃119905 119897(119905 119899(119905)) whose linguistic term sets are represented by119878119899(119905) = 119904119899(119905)0 119904119899(119905)119899(119905)minus1 the transformation function froma linguistic label in level 119905 to a label in consecutive level 1199051015840 isdefined as 1198791198651199051199051015840 119897(119905 119899(119905)) 997888rarr 119897(1199051015840 119899(1199051015840)) such that

1198791198651199051199051015840 (119904119899(119905)119894 120572119899(119905))= Δ 1199051015840 (Δminus1119905 (119904119899(119905)119894 120572119899(119905)) (119899 (1199051015840) minus 1)119899 (119905) minus 1 ) (A5)

By use of the above transformation function any 2-tuplelinguistic representation can be transformed into a term in119871119867 Detailed transformation procedures are listed as follows

(1) Representation in linguistic hierarchy To transformthe unbalanced terms of 119878 into the corresponding terms inthe 119871119867 transformation function 120595 is employed to associateeach unbalanced linguistic 2-tuple (119904119894 120572) with its linguistic 2-tuple in 119871119867(119878) ie

120595 119878 997888rarr 119871119867(119878) (A6)

so that 120595(119904119894 120572) = (119904119866(119894)119868(119894) 120582) for forall(119904119894 120572) isin 119878(2) Computational phase Firstly transform (119904119866(119894)119868(119894) 120582) intolinguistic 2-tuples denoted as (119904119899(1199051015840)

1198681015840(119894) 1205821015840) in 119878119899(1199051015840) where

(119904119899(1199051015840)1198681015840(119894)

1205821015840) = 119879119865 (119904119866(119894)119868(119894) 120582)= Δ(Δminus1 (119904119866(119894)119868(119894) 120582) sdot (119899 (1199051015840) minus 1)119866 (119894) minus 1 ) (A7)

Then a computational model is used over 119878119899(1199051015840) with aresult denoted as (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)(3) Retranslation process The result (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)is transformed into the unbalanced term in 119878 by using thetransformation function 120595minus1 ie

120595minus1 119871119867 (119878) 997888rarr 119878 (A8)

so that 120595minus1(119904119899(1199051015840)119903 120582119903) = (119904119903119890119904119906119897119905 120582119903119890119904119906119897119905) isin 119878B Details of Theorem 14

Proof Given a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector for119904119889119895 and 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 120582 gt 0 then(1) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLWA and GIVDHFUUBLWG operators respectively(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLOWA and GIVDHFUUBLOWG operators respectively

(2) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to theIVDHF UUBL weighted average (IVDHFUUBLWA)and IVDHF UUBL weighted geometric (IVDHFUUBLWG)operators respectively where

26 Complexity

119868119881119863119867119865119880119880119861119871119882119860120596 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

120596119895119904119889119895= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus

119899prod119895=1

(1 minus 120583119871119895 )120596119895 1 minus 119899prod119895=1

(1 minus 120583119880119895 )120596119895]] [[

119899prod119895=1

(]119871119895)120596119895 119899prod119895=1

(]119880119895 )120596119895]])

(B1)

119868119881119863119867119865119880119880119861119871119882119866120596 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889119895120596119895

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[

119899prod119895=1

(120583119871119895 )120596119895 119899prod119895=1

(120583119880119895 )120596119895]] [[1 minus

119899prod119895=1

(1 minus ]119871119895)120596119895 1 minus 119899prod119895=1

(1 minus ]119880119895 )120596119895]])

(B2)

(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHA andGIVDHFUUBLHG operators reduce to the IVDHF UUBL

orderedweighted average (IVDHFUUBLOWA) and IVDHFUUBL ordered weighted geometric (IVDHFUUBLOWG)operators respectively where

119868119881119863119867119865119880119880119861119871119874119882119860119908 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

119908119895119904119889120590(119895)= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) )))

119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) )))

]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus

119899prod119895=1

(1 minus 120583119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus 120583119880120590(119895))119908119895]] [[

119899prod119895=1

(]119871120590(119895))119908119895 119899prod119895=1

(]119880120590(119895))119908119895]])

(B3)

119868119881119863119867119865119880119880119861119871119874119882119866119908 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889120590(119895)119908119895

= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[

119899prod119895=1

(120583119871120590(119895))119908119895 119899prod119895=1

(120583119880120590(119895))119908119895]] [[1 minus

119899prod119895=1

(1 minus ]119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus ]119880120590(119895))119908119895]])

(B4)

Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

References

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[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

[11] X Qi C Liang and J Zhang ldquoGeneralized cross-entropy basedgroup decision making with unknown expert and attributeweights under interval-valued intuitionistic fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 79 pp 52ndash642015

[12] H Zimmermann Fuzzy Set Theorymdashand Its ApplicationsKluwer Academic Boston Mass USA 2001

[13] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 pp 199ndash249 1975

[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

[16] B T Nuong K-W Kim L Prathumratana et al ldquoSustainabledevelopment in the mining sector and its evaluation usingfuzzy AHP (Analytic Hierarchy Process) approachrdquo GeosystemEngineering vol 14 no 1 pp 43ndash50 2011

[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

[18] E K Zavadskas Z Turskis and J Antucheviciene ldquoSelectinga contractor by using a novel method for multiple attributeanalysis Weighted aggregated sum product assessment withgrey values (WASPAS-G)rdquo Studies in Informatics and Controlvol 24 no 2 pp 141ndash150 2015

[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

[21] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

[25] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009

[26] Z Hatush and M Skitmore ldquoContractor Selection UsingMulticriteria Utility Theory An Additive Modelrdquo Building andEnvironment vol 33 no 2-3 pp 105ndash115 1998

[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

[30] AM Anvuur andMM Kumaraswamy ldquoConceptualmodel ofpartnering and alliancingrdquo Journal of Construction Engineeringand Management vol 133 no 3 pp 225ndash234 2007

[31] A H I Lee H Kang C F Hsu and H Hung ldquoA green supplierselection model for high-tech industryrdquo Expert Systems withApplications vol 36 no 4 pp 7917ndash7927 2009

[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

[33] P Humphreys R McIvor and F Chan ldquoUsing case-basedreasoning to evaluate supplier environmental managementperformancerdquo Expert Systems with Applications vol 25 no 2pp 141ndash153 2003

[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

[45] Bin Zhu Zeshui Xu and Meimei Xia ldquoDual Hesitant FuzzySetsrdquo Journal of Applied Mathematics vol 2012 Article ID879629 13 pages 2012

[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

[51] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[52] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

[55] Z Tao X Liu H Chen and L Zhou ldquoUsing new versionof extended t-norms and s-norms for aggregating intervallinguistic labelsrdquo IEEETransactions on SystemsManCyberneticsSystems vol vol p 15 2016

[56] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[57] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[58] L Martınez M Espinilla J Liu L G Perez and P J SanchezldquoAn evaluation model with unbalanced linguistic informationapplied to olive oil sensory evaluationrdquo Journal of Multiple-Valued Logic and Soft Computing vol 15 no 2-3 pp 229ndash2512009

[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

[62] L Marin A Valls D Isern A Moreno and J M MerigoldquoInduced unbalanced linguistic ordered weighted average andits application in multiperson decision makingrdquo The ScientificWorld Journal vol 2014 2014

[63] A Baykasoglu V Kaplanoglu Z D U Durmusoglu and CSahin ldquoIntegrating fuzzy DEMATEL and fuzzy hierarchicalTOPSIS methods for truck selectionrdquo Expert Systems withApplications vol 40 no 3 pp 899ndash907 2013

[64] S K Patil and R Kant ldquoA hybrid approach based on fuzzyDEMATEL and FMCDM to predict success of knowledge man-agement adoption in supply chainrdquoApplied Soft Computing vol18 pp 126ndash135 2014

[65] S Liyin and Y Hong ldquoImproving environmental performanceby means of empowerment of contractorsrdquo Management ofEnvironmental Quality An International Journal vol 17 no 3pp 242ndash257 2006

[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

[69] Y Kazancoglu I Kazancoglu and M Sagnak ldquoA new holisticconceptual framework for green supply chain management

performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

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Complexity 9

(2) 119889(1199041198891 1199041198892) = 0 if and only if 1199041198891 and 1199041198892 are perfectlyconsistent

(3) 119889(1199041198891 1199041198892) = 119889(1199041198892 1199041198891)4 Generalized Aggregation Operators forIVDHF_UUBLS

41 Definitions of Operators Based on the generalized oper-ators firstly introduced by Yager [87] we here develop some

fundamental generalized aggregation operators for the newlydefined IVDHF UUBLS

Definition 10 Given a collection of IVDHF UUBLNs 119904119889119895(119895 =1 119899) their weighting vector 120596 = (1205961 1205962 120596119899)119879120596119895 isin[0 1]sum119899119895=1 120596119895 = 1 120582 be a parameter 120582 isin (0 +infin)(1) Generalized IVDHFUUBLWeighted Average(GIVDHFUUBLWA) Operator

119866119868119881119863119867119865119880119880119861119871119882119860120596120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(120596119895119904119889119895120582))1120582

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120582)1120582 119904(sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120582)1120582]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[(1 minus

119899prod119895=1

(1 minus (120583119871119895 )120582)120596119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880119895 )120582)120596119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871119895)120582)120596119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880119895 )120582)120596119895)1120582]])

(7)

(2) Generalized IVDHFUUBLWeighted Geometric(GIVDHFUUBLWG)Operator

119866119868119881119863119867119865119880119880119861119871119882119866120596120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889119895)120596119895)= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871119895 )120582)120596119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880119895 )120582)120596119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871119895 )120582)120596119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880119895 )120582)120596119895)1120582]])

(8)

Definition 11 For a collection of IVDHF UUBLNs 119904119889119895(119895 = 1 119899) 119904119889120590(119895) be the 119895th largest 119908 = (1199081 1199082 119908119899)119879 be the aggregation-associated weighting vector 119908119895 isin [0 1]sum119899119895=1119908119895= 1 120582 is a parameter such that 120582 isin (0 +infin) 119878119899 997888rarr 119878 Then

10 Complexity

(1) Generalized IVDHFUUBL Ordered Weighted Average(GIVDHFUUBLOWA) Operator

119866119868119881119863119867119865119880119880119861119871119874119882119860120596120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(119908119895119904119889120590(119895)120582))1120582

= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

120582)1120582]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895) []119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120590(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120590(119895))120582)119908119895)1120582]])

(9)

(2) Generalized IVDHFUUBL Ordered Weighted Geometric(GIVDHFUUBLOWG)Operator

119866119868119881119863119867119865119880119880119861119871119874119882119866119908120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889120590(119895))119908119895)= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120590(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120590(119895))120582)120596119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120590(119895))120582)120596119895)1120582]])

(10)

Definition 12 For a collection of IVDHF UUBLNs 119904119889119895(119895 =1 119899) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector120596119895 isin [0 1] sum119899119895=1 120596119895 = 1 119899 is a balancing coefficient119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weight-ing vector with 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1

(1) Generalized IVDHFUUBL Hybrid Average (GIVDHFU-UBLHA) Operator

119866119868119881119863119867119865119880119880119861119871119867119860120596119908120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(119908119895119904119889120590(119895)120582))1120582

= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895)

))))120582)1120582]

Complexity 11

⋃[120583119871120590(119895) 120583

119880120590(119895)]isinℎ120590(119895) []

119871120590(119895) ]119880120590(119895)]isin119892120590(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120590(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120590(119895))120582)119908119895)1120582]])

(11)

where 119904119889120590(119895) is the 119895th largest and

119904119889119895 = 119899120596119895119904119889119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904119899120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904119899120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

] ⋃

[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[1 minus (1 minus 120583119871119895 )119899120596119895 1 minus (1 minus 120583119880119895 )119899120596119895] [(]119871119895)119899120596119895 (]119880119895 )119899120596119895]) (12)

(2) Generalized IVDHFUUBL Hybrid Geometric(GIVDHFUUBLHG) Operator

119866119868119881119863119867119865119880119880119861119871119867119866119908120596120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889120590(119895))119908119895)= ⋃(119904120599120590(119895)

ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120590(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895)

))))119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895) []119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120590(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120590(119895))120582)119908119895)1120582]])

(13)

where 119904119889120590(119895) is the 119895th largest and

119904119889119895 = 119904119889119895119899120596119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(Δminus11199050 (119879119865

11990511989511199050(120595(119904120572119895 ))))

119899120596119895 119904(Δminus11199050 (119879119865

11990511989521199050(120595(119904120573119895 ))))

119899120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[(120583119871119895 )119899120596119895 (120583119880119895 )119899120596119895] [1 minus (1 minus ]119871119895 )119899120596119895 1 minus (1 minus ]119880119895 )119899120596119895]) (14)

12 Complexity

When confronted with ill-structured situations wheredecision-makerrsquos complex attitudinal characters need to beincluded order-inducing vectors provide an effective way[88ndash90]Thus we further define following induced operatorsfor IVDHF UUBLNs

Definition 13 For a collection of IVDHF UUBLNs 119904119889119895 (119895 =1 119899) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector

120596119895 isin [0 1] sum119899119895=1 120596119895 = 1 119899 is a balancing coefficient119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weight-ing vector 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1 120576119895 denote a set oforder inducing vectors

(1) Induced Generalized IVDHFUUBL Hybrid Average (I-GIVDHFUUBLHA) Operator

119868-119866119868119881119863119867119865119880119880119861119871119867119860120596119908120582 (lt 1205761 1199041198891 gt lt 1205762 1199041198892 gt lt 120576119899 119904119889119899 gt) = ( 119899⨁119895=1

(119908119895119904119889120587(119895)120582))1120582

= ⋃(119904120599120587(119895) ℎ120587(119895) 119892120587(119895))isin119904119889120587(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120587(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587(119895)

))))120582)1120582]

⋃[120583119871120587(119895) 120583

119880120587(119895)]isinℎ120587(119895) []

119871120587(119895) ]119880120587(119895)]isin119892120587(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120587(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120587(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120587(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120587(119895))120582)119908119895)1120582]])

(15)

where (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) is decreasing order of (11990411988911199041198892 119904119889119899) according to the order inducing vector 120576119895 and119904119889119895 = 119899120596119895119904119889119895 = ⋃

(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904119899120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904119899120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

] ⋃

[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[1 minus (1 minus 120583119871119895 )119899120596119895 1 minus (1 minus 120583119880119895 )119899120596119895] [(]119871119895)119899120596119895 (]119880119895 )119899120596119895]) (16)

(2) Induced Generalized IVDHFUUBL Hybrid Geometric(I-GIVDHFUUBLHG) Operator

119868-119866119868119881119863119867119865119880119880119861119871119867119866119908120596120582 (lt 1205761 1199041198891 gt lt 1205762 1199041198892 gt lt 120576119899 119904119889119899 gt) = 1120582 ( 119899⨂119895=1

(120582119904119889120587(119895))119908119895)= ⋃(119904120599120587(119895)

ℎ120587(119895) 119892120587(119895))isin119904119889120587(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120587(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587(119895)

))))119908119895]

⋃[120583119871120587(119895)120583119880120587(119895)]isinℎ120587(119895) []119871120587(119895) ]

119880120587(119895)]isin119892120587(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120587(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120587(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120587(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120587(119895))120582)119908119895)1120582]])

(17)

Complexity 13

Weighted Aggregation

Geometric Arithmetic

GIVDHFUUBLHG GIVDHFUUBLHA

GIVDHFUUBLWG GIVDHFUUBLOWG

IVDHFUUBLWG IVDHFUUBLOWG

GIVDHFUUBLWA GIVDHFUUBLOWA

IVDHFUUBLWA IVDHFUUBLOWA

= 1 w =1

n = 1 =

1

n = 1 =1

n = 1 =

1

n

= 1 = 1 = 1 = 1

Figure 2

where (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) is decreasing order of (11990411988911199041198892 119904119889119899) according to the order inducing variables 120576119895 and

119904119889119895 = 119904119889119895119899120596119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(Δminus11199050 (119879119865

11990511989511199050(120595(119904120572119895 ))))

119899120596119895 119904(Δminus11199050 (119879119865

11990511989521199050(120595(119904120573119895 ))))

119899120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[(120583119871119895 )119899120596119895 (120583119880119895 )119899120596119895] [1 minus (1 minus ]119871119895 )119899120596119895 1 minus (1 minus ]119880119895 )119899120596119895]) (18)

42 Properties of the Proposed GeneralizedAggregation Operators

eorem 14 With special values of 120582 119908 and 120596 the operatorsGIVDHFUUBLHA and GIVDHFUUBLHG can include aseries of traditional aggregation operators as special cases andtheir relationship can be depicted in Figure 2

Proof See Appendix B

As for the induced hybrid aggregation operators we alsohave following theorem

eorem 15 If (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) = (119904119889120590(1) 119904119889120590(2) 119904119889120590(119899)) I-GIVDHFUUBLHA reduces toGIVDHFUUBLHAIf (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) = (119904119889120590(1) 119904119889120590(2) 119904119889120590(119899)) thenI-GIVDHFUUBLHG reduces to the GIVDHFUUBLHG oper-ator

eorem 16 All the proposed generalized operatorsGIVDHFUUBLWAGIVDHFUUBLWGGIVDHFUUBLOWAGIVDHFUUBLOWGGIVDHFUUBLHAGIVDHFUUBLHGI-GIVDHFUUBLHA and I-GIVDHFUUBLHG hold thefollowing properties (1) Commutativity (2) Idempotency (3)Boundedness

Based on above theorems following properties can alsobe derived

eorem 17 For a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weight vector of 119904119889119895with 120596119895 isin [0 1] and sum119899119895=1 120596119895 = 1 120582 gt 0 then we have

(1) 119868119881119863119867119865119880119880119861119871119882119866120596(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119882119860120596(1199041198891 1199041198892 119904119889119899)(2) 119868119881119863119867119865119880119880119861119871119882119866120596(1199041198891 1199041198892 119904119889119899) le119866119868119881119863119867119865119880119880119861119871119882119860120596120582(1199041198891 1199041198892 119904119889119899)(3) 119866119868119881119863119867119865119880119880119861119871119882119866120596120582(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119882119860120596(1199041198891 1199041198892 119904119889119899)

Proof Omitted for concision

eorem 18 For a collection of IVDHF UUBLNs 119904119889119895 = (119904120599119895 ℎ119895 119892119895) 119908 = (1199081 1199082 119908119899)119879 is the weighting vector of 119904119889119895with 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1 120582 gt 0 Then

(1) 119868119881119863119867119865119880119880119861119871119874119882119866119908(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119874119882119860119908(1199041198891 1199041198892 119904119889119899)(2) 119868119881119863119867119865119880119880119861119871119874119882119866119908(1199041198891 1199041198892 119904119889119899) le119866119868119881119863119867119865119880119880119861119871119874119882119860119908120582(1199041198891 1199041198892 119904119889119899)(3) 119866119868119881119863119867119865119880119880119861119871119874119882119866119908120582(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119874119882119860119908(1199041198891 1199041198892 119904119889119899)

Proof Omitted for concision

14 Complexity

5 An Integrated MADM Approach forTackling Complex Green ContractorSelection Problems

As described in Section 2 we take the green contractorselection as a special type of complicate MADM problemsthat synthesizes three characteristics of decision hesitancy[21 22] attributes interdependency [84] and group atti-tudinal characters [61] Therefore in this section we con-struct an integrated MADM approach to tackle the com-plex green contractor selection problems Suppose 119883 =1199091 1199092 119909119898 is the set of alternative green contractors and119860 = 1198601 1198602 119860119899 is the set of evaluative attributes 120596 =(1205961 1205962 120596119899)119879 is weighting vector for the attributes 120596119894 ge 0sum119899119894=1 120596119894 = 1 Let = (119903119894119895)119899times119898 denote the decision matrix inwhich 119903119894119895 = ([119904120572119894119895 119904120573119894119895] ℎ119894119895 119892119894119895) is an IVDHF UUBLN givenby decision-makers for alternative contractor 119909119895 with respectto attribute 119860 119894 According to the mechanism of pair-wisecomparisons among attributes in the DEMATELmethod [6364] the interdependency among attributes can be obtained asamatrix119885 = (119911119894119896)119899times119899 where 119911119894119896 indicates the degree towhich119860 119894 affects119860119896 Subsequently based on the IVDHFUUBLS andits operations we now present detailed steps of our MADMapproach as shown in following Algorithm I

Algorithm I Hesitant fuzzy linguistic MADMwith attributesinterdependency and decision-makersrsquo group attitudinalcharacters

Step 1 Determine argument-dependent weighting vector120596119860119863 according to attribute values by programming modeldeveloped in the following Section 51

Step 2 Obtain the attribute-interdependences based weight-ing vector 120596119860119868 by use of DEMATEL method described in thefollowing Section 52

Step 3 Calculate synthesized attribute weighting vectoraccording to

120596 = 120572120596119860119868 + 120573120596119860119863 (19)

where 120572 and 120573 are parameters to reflect decision character-istics of decision organizations 0 le 120572 120573 le 1 120572 + 120573 =1Step 4 Check requirements for order inducing If no addi-tional order inducing required then go to Step 5 otherwisego to Step 6

Step 5 Utilize generalized aggregation operators to get theoverall IVDHF UUBLNs 119903119895(119895 = 1 119898) for each alternative119909119895 Here we take GIVDHFUUBLHA operator for examplebecause it can include other traditional operators as its specialcases Therefore we have

119866119868119881119863119867119865119880119880119861119871119867119860(1199031119895 1199032119895 119903119899119895) = ( 119899⨁119894=1

(119908119894 (119903120590119894119895)120582))1120582= ⋃(119904120590119894119895ℎ120590119894119895 119892120590119894119895

)isin119903120590119894119895

([119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590119894119895

))))120582)1120582 119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590119894119895

))))120582)1120582]

⋃[120583119871120590119894119895120583

119880120590119894119895]isinℎ120590119894119895[]

119871120590119894119895]119880120590119894119895]isin119892120590119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120590119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120590119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120590119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120590119894119895)120582)119908119894)1120582]])

(20)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of 119903119894119895119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weightvector 119903120590119894119895 is the 119894th largest of 119903119894119895 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))

119904119899120596119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119894119895))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (21)

Complexity 15

Then go to Step 8

Step 6 By use of the TOPSIS-based method developed in thefollowing Section 53 we determine order-inducing vectors 120576to reflect decision-makersrsquo group attitudinal characters

Step 7 Utilizing induced generalized aggregation operatorsto get the overall IVDHF UUBLNs 119903119895(119895 = 1 2 119898) foreach alternative 119909119895 we have

119868-119866119868119881119863119867119865119880119871119867119860(1199031119895 1199032119895 119903119899119898) = ( 119899⨁119894=1

(119908119894 (119903120587119894119895)120582))1120582

= ⋃(119904120587119894119895ℎ120587119894119895119892120587119894119895)isin119903120587119894119895

([119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120587119894119895

))))120582)1120582 119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587119894119895

))))120582)1120582]

⋃[120583119871120587119894119895 120583

119880120587119894119895]isinℎ120587119894119895 []

119871120587119894119895]119880120587119894119895]isin119892120587119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120587119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120587119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120587119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120587119894119895)120582)119908119894)1120582]])

(22)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of119903119894119895 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 119903120587119894119895 is the 119894th largest of 119903119894119895 reordered by theorder-inducing vectors 120576 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))119899120596119894120572119894119895

119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120573119894119895 ))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (23)

Step 8 Calculate scores 119878(119903119895) and accuracy degrees 119875(119903119895) for119903119895(119895 = 1 2 119898)Step 9 According to Definition 5 output ranking order ofalternative contractors

In Algorithm I to objectively derive unknown weight-ing vector for evaluative attributes a synthesized attributeweighting vector is devised by fusing two parts of concern(i) argument-dependent weighting vector and (ii) attribute-interdependences based weighting vector The former is toreflect effects of attribute values on attribute weights whilethe latter is to exploit effects of interdependences amongattributes on their weights For clarity processing flowchartof Algorithm I is depicted in Figure 3

51 Determining Argument-Dependent Attribute WeightingVector 120596119860119863 In order to derive attribute weighting vec-tor appropriately from assessments maximizing deviationmethod [91] is adopted here for its distinguishing ability

and objectivity [92] According to Wang [91] if performancevalues of each alternative have little differences under anattribute it implicates such an attribute plays a less importantrole otherwise it lives a more important role in decision-making Thus if one attribute has similar values across alter-natives it should be assigned a smaller weight contrariwiseit should be a bigger weight

In light of the idea in maximizing deviation methodwe develop a programming model to determine argument-dependent weighting vector 120596119860119863 For attribute 119860 119894 the stan-dard deviation of alternative 119909119895 to all the other alternatives isdenoted as

119865119894119895 (120596119860119863) = 119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (24)

Let119865119894(120596119860119863) represent deviation value of all alternatives toother alternatives according to attribute119860 119894 and be formulatedas follows

16 Complexity

Attribute weighting

Hesitant fuzzy linguistic MADM with attributes interdependency and decision makersrsquo group attitudinal characters

Step 9 Output ranking order of alternatives

interdependency among attributes by DEMATEL in Section 52

attributes values by standard deviation method in Section 51

Step 6 Ensure TOPSIS order inducing variables proposed in Section 53

Step 7 Utilizing proposed induced generalized hybrid aggregation operators developed in Section

41 to get overall value of each alternative

Step 8 Calculate score degrees and accuracy degrees

Step 4 Order inducingYes

Step 5 Choose proposed generalized operators in Section 41 to get overall value of each alternative

No

Collect decision matrix in the form of IVDHF_UUBLS

Complex problems of green contractor selection

Step 1 Weighting vector AD derived from Step 2 Weighting vector AI derived from

Step 3 Obtain synthesized weighting vector of attributes =AI + AD

Figure 3 Processing flowchart of Algorithm I

119865119894 (120596119860119863) = 119898sum119895=1

119865119894119895 (120596119860119863) = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (25)

Based on the analysis above we can get followingprogramming model (M-1) for obtaining optimal attributeweighting vector 120596119860119863

(119872 minus 1)max 119865 (120596119860119863) = 119899sum

119894=1119865119894 (120596119860119863) = 119899sum

119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894st 119899sum

119894=1(120596119860119863119894 )2 = 1 120596119860119863119894 ge 0 119894 = 1 2 119899 (26)

where 119889(119903119894119895 119903119894119897) can be calculated by Definition 6To solve model (M-1) we can construct Lagrange func-

tion

119871 (120596119860119863119894 120577) = 119899sum119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894+ 1205772 ( 119899sum

119894=1

(120596119860119863119894 )2 minus 1) (27)

where 120577 is the Lagrange multiplierDifferentiate with respect to 120596119860119863119894 (119894 = 1 2 119898) and 120577

and set these partial derivatives equal to zero the followingset of equations can be obtained

120597119871120597120596119860119863119894 = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) + 120577120596119860119863119894 = 0120597119871120597120577 = 119899sum

119894=1

(120596119860119863119894 )2 minus 1 = 0 (28)

By normalizing the solutions of above equations we canget a simple and exact formula for determining the attributeweights as follows

120596119860119863119894 = sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897)sum119899119894=1sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897) (29)

52 Deriving Attributes Interdependency-Based AttributeWeighting Vector 120596119860119868 Some efforts [53 60 84 93ndash95]

Complexity 17

have been made to address interdependencies betweenattributes in MADM However they unanimously cannotreveal the interdependences in a more complete way WhileDEMATAEL technique [63 96] gathers collective knowledgeto capture influences between attributes more precisely andcompletely so that weighting vector for attributes can berationally determined according to strength measure ofthose influences [64 97] Therefore in tackling the complexMADMwith interdependent attributes under IVDHF UUBLenvironments we present a DEMATEL-based method as thefollowing Algorithm II to derive attribute weighting vector120596119860119868 from the viewpoint of attributes interdependency

Algorithm II DEMATEL-basedmethod for ensuringweight-ing vector 120596119860119868Step 1 Construct initial direct-influence matrix 119885 = (119911119894119896)119899times119899

For attributes 119860 = 1198601 1198602 119860119899 decision-makersare asked to give direct effect that attribute 119860 119894 has onattribute 119860119896 using a five-point scale with scores representedby natural language (1) 0 mdash lsquoabsolutely no influencersquo (2)1 mdash lsquolow influencersquo (3) 2 mdash lsquomedium influencersquo (4) 3 mdashlsquohigh influencersquo (5) 4mdash lsquovery high influencersquoThen the initialdirect-influence matrix 119885 = (119911119894119896)119899times119899 can be established witheach element being the mean of same factors in differentdirect matrices

Step 2 Calculate normalized direct-influence matrix 119883 =(119909119894119896)119899times119899 Using matrix 119885 the normalized direct-influencematrix 119883 = (119909119894119896)119899times119899 is calculated according to

119909119894119896 = 119911119894119896119904 (30)

where 119904 = maxmax1le119894le119899sum119899119896=1 119911119894119896max1le119896le119899sum119899119895=1 119911119894119896 Allelements in matrix 119883 are complying with 0 le 119909119894119896 le 10 le sum119899119896=1 119909119894119896 le 1 and 0 le sum119899119894=1 119909119894119896 le 1 and at least onerow or column of the summation equals 1

Step 3 Derive total-influence matrix 119879 = (119905119894119896)119899times119899 Basedon matrix 119883 the total-influence matrix 119879 = (119905119894119896)119899times119899 can bederived by summing the direct effects and all of the indirecteffects

119879 = 119883 + 1198832 + 1198833 + sdot sdot sdot + 119883ℎ = 119883 (119868 minus 119883)minus1 (31)

where 119868 represents identity matrix When ℎ 997888rarr infin 119883ℎ =(0)119899times119899Step 4 Compute influencing and influenced degrees of eachattribute The sum of rows and the sum of columns withinthe total-influence matrix 119879 are respectively expressed as thevectors 119903 and 119888

119903 = (119903119894)119899times1 = ( 119899sum119896=1

119905119894119896)119899times1

119888 = (119888119896)119899times1 = ( 119899sum

119895=1

119905119894119896)119879

1times119899

(32)

where 119903119894 denotes the sumof the 119894th row inmatrix119879 and showsthe sum of direct and indirect effects that attribute 119860 119894 hason other attributes Similarly 119888119896 shows the sum of direct andindirect effects that attribute 119860119896 has receivedStep 5 Estimate weights of attributes based on influencedegrees according to

120596119868lowast119894 = [(119903119894 + 119888119894)2 + (119903119894 minus 119888119894)2]12 (33)

Then the weighting vector 120596119860119868 can be obtained by normaliz-ing (33)

120596119860119868119894 = 120596119868lowast119894sum119899119894=1 120596119868lowast119894 (34)

53 Obtaining Order Inducing Variables 120576 by TOPSIS-BasedMethod For ill-defined MADM problems induced aggre-gation operators [89 90 98] provide an effective channelfor experts to express complex attitudinal characters Thekey step in induced aggregation operators is to obtainorder-inducing vectors 120576 which however are presumptivelyassigned in most papers [62] Therefore in this sectionaiming at deducing the order-inducing vectors 120576 rationallywe here devise a TOPSIS-based method in Algorithm IIIObviously the Algorithm III is of generality for MADMwithdifferent forms of linguistic expression

Algorithm III TOPSIS-based method for obtaining orderinducing vectors 120576Step 1 Let the decision maker define IVDHF UUBL idealsolution(s) 119903+119894 and IVDHF UUBL negative ideal solution(s)119903minus119894 corresponding to each attribute in which

119903+119894 = (119903+1198941 119903+1198942 119903+119894119901) 119903minus119894 = (119903minus1198941 119903minus1198942 119903minus119894119902) (35)

where 119901 and 119902 denotes the number of 119903+119894 and 119903minus119894 respectively

Step 2 Calculate distances from each element in decisionmatrix to ideal solution(s) and negative solution(s) respec-tively

119889+119894119895 = max119897=12119901

119889 (119903119894119895 119903+119894119897 ) 119889minus119894119895 = min

119897=12119902119889 (119903119894119895 119903minus119894119897 ) (36)

where 119889(119903119894119895 119903+119894119897 ) and 119889(119903119894119895 119903minus119894119897 ) are calculated by the distancemeasure in Definition 6 If 119901 gt 1 we choose the onewith maximum distance if 119902 gt 1 we choose the one withminimum distance

Step 3 Calculate coefficients according to

119888119894119895 = 119889minus119894119895119889+119894119895 + 119889minus119894119895 119894 = 1 2 119899 119895 = 1 2 119898 (37)

18 Complexity

Table 2 Decision matrix in the form of IVDHF UUBLS1199091 1199092 11990931198601([AHVH] [02 03][02 04] [03 04]) ([HT] [04 05][05 06] [02 03] [02 04]) ([ALAH] [01 02][01 03] [06 07])1198602([MH] [05 06][02 03]) ([AHT] [06 07][01 02] ) ([MVH] [02 03][05 06] [06 07])1198603([VLAL] [03 04][04 05] [05 06]) ([MVH] [06 07] [07 08][01 02]) ([VLL] [04 05][02 03])1198604([MAH] [02 04][05 06]) ([LAL] [03 05][02 03]) ([ALM] [06 07][01 02] [02 03])1198605([ALM] [04 05][01 02] [04 05]) ([AHH] [06 07][01 02]) ([HVH] [04 05][06 07] [01 03])1198606([LM] [01 02][03 05] [03 05]) ([VLL] [02 04] [05 06][02 03]) ([AHH] [03 04][04 05])1198607([VLL] [04 05][05 06] [02 03] [02 04]) ([MAH] [05 06] [07 08][01 02]) ([ALM] [04 05][05 06] [03 04])1198608([ALAH] [01 03][04 06]) ([AHH] [05 07][01 02] [02 03]) ([AHH] [04 06][03 04])

Table 3 Direct influence matrix Z1198601 1198602 1198603 1198604 1198605 1198606 1198607 11986081198601 0 3 4 1 3 1 2 41198602 1 0 2 4 4 2 3 11198603 4 2 0 1 3 2 4 11198604 2 3 1 0 3 4 1 21198605 4 2 1 3 0 1 4 31198606 2 4 3 4 4 0 4 21198607 4 1 2 3 3 1 0 41198608 1 2 4 3 3 2 4 0

Step 4 Derive order inducing variable 120576119894119895 by descendingorder of 119888119894119895 The greater the value 119888119894119895 the bigger the 1205761198941198956 Illustrative Example and Experiments

61 Illustrative Case Study Suppose a mega infrastructureproject is selecting green contractor for one of its subprojectsA panel of decision-makers including managers professorsand competitive intelligence experts has been organized tocomprehensively evaluate three alternative contractors (ie1199091 1199092 and 1199093) under the eight attributes (ie 1198601ndash1198608) listedin Section 2

To elicit true decision preferences we apply theIVDHF UUBLS to assess the three alternatives (1199091ndash1199093)Firstly decision-makers are asked to vote on 119909119895 under eachattribute119860 119894 using different unbalanced linguistic term sets 1198781 1198782 and 1198783 where 1198781 = 119873 119871119860119871119872119860119867119867119876119867119881119867119860119879 1198791198782 = 119873119860119873119881119871 119876119871 119871 119860119871119872119860119867119881119867 119879 1198783 = 119873119881119871119871119860119872119872119876119872119860119867119867 119879 In Figure 4 we show the adoptedlinguistic hierarchies and relationship between 1198781 1198782 and1198783 Attributes 1198601 1198602 1198605 and 1198608 are evaluated by use of 1198781attributes11986031198604 and1198607 are evaluated by 1198782 and attribute1198606is evaluated by 1198783 The highest-voted uncertain unbalancedlinguistic term [119904119898 119904119899] are identified Secondly the panel of

decision-makers is asked to further express their opinionson membership degrees ℎ(119909) and nonmembership degrees119892(119909) to [119904119898 119904119899] Then we can derive compound assessments(ie ⟨119909 119904120599(119909) ℎ(119909) 119892(119909)⟩) in the form of IVDHF UUBLSand obtain the decision matrix in Table 2 Additionallyin order to examine the interdependency among attributesdecision-makers are also required to assess the influentialrelations among the eight attributes using DEMATEL-basedAlgorithm II The direct influence matrix is collected andshown in Table 3

Next we apply the proposed Algorithm I to solve thisgreen contractor selection problem To derive the synthe-sized attribute weighting vector in Algorithm I we assign120572 = 120573 = 05 For scenarios with no group attitudinalcharacters operator GIVDHFUUBLHA in Definition 12 ischosen In contrast for scenarios with group attitudinalcharacters operator I-GIVDHFUUBLHA in Definition 13will be chosen Further the panel of decision-makers is askedto determine their positive and negative ideal alternativesthat are collected in Table 4 where corresponding order-inducing vectors that are derived from Algorithm III arealso listed (see the rightmost column) Thereafter operatorI-GIVDHFUUBLHA in Definition 13 is utilized for aggrega-tion For GIVDHFUUBLHA and I-GIVDHFUUBLHA we

Complexity 19

Table 4 Ideal alternatives (119903+ and 119903minus) and derived order-inducing vectors 120576119903+ 119903minus 1205761199091 1199092 11990931198601 ([VHT] 1 0) ([NL] 1 0) 5 6 61198602 ([MAH] [07 08] [01 02]) ([NL] 1 0) 1 1 41198603 (T 1 0) (L 1 0) 8 7 81198604 ([ALAH] 09 01) (T 1 0) 2 2 11198605 ([HT] [09 1] 0) (N 1 0) 7 5 51198606 (M 1 0) (N 1 0) 6 3 31198607 (M 1 0) ([NL] 1 0) 3 8 71198608 ([MAH] 1 0) (N 1 0) 4 4 2

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 1 2 3 4

0 1 2

N L AH H QH VH AT TM

Level 1 l(13)

Level 2 l(25)

Level 3 l(39)

Level 4 l(417)

Unbalanced linguistic term set S1

N AH VH TM Unbalanced linguistic term set S2VL L ALAN QL

N H TM Unbalanced linguistic term set S3

L AM AHQM

AL

VL

Figure 4 Unbalanced linguistic term sets (1198781 1198782 and 1198783) and their mapping in linguistic hierarchies

assume 120582=2 and 119908 = (0 0 015 025 025 025 01 0)119879Note that 119908 is a position weighting vector which is derivedby the fuzzy semantic quantitative operator [99] Aftercompleting all steps in Algorithm I (see Appendix C) weobtain the ranking results and present them in Table 5 Forthe scores derived from Step 5 we find the ranking order is1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the best solution Step 7gives the ranking order of 1199092 ≻ 1199091 ≻ 1199093 indicating 1199092 is stillthe most appropriate one while 1199091 is better than 1199093 under theinfluence of 12057662 Experiments on Ranking Results along with VariousParameter Configurations In order to inspect the impactsof various parameter configurations on ranking results inthis section more numerical examples are carried out Alleight operators that defined in Section 41 are applied to theabove illustrative case where the parameter 120582 is configured asintegers ranging from 1 to 20 and parameter 120572 is configuredto a value set of 0 03 05 08 09 1

Firstly the four basic aggregation operators of GIVDH-FUUBLWA GIVDHFUUBLWG GIVDHFUUBLOWA and

Table 5 Ranking results of operators GIVDHFUUBLHA and I-GIVDHFUUBLHA

Aggregation operator Ranking orderGIVDHFUUBLHA 1199092 ≻ 1199093 ≻ 1199091I-GIVDHFUUBLHA 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG are applied to the former case Thesame configuration 120572 = 120573 = 05 is chosen while 120582 is testedas integers ranging from 1 to 20 The ranking results derivedfrom different parameter combinations have been collectedin Table 6 As can be seen four operators unanimouslyidentify the contractor 1199092 is the best one no matter how120582 changes Regarding priority relation between 1199091 and 1199093GIVDHFUUBLWA and GIVDHFUUBLOWA generate thesame result of1199093 ≻ 1199091 while results fromGIVDHFUUBLWGandGIVDHFUUBLOWGgot a change to 1199091 ≻ 1199093 at a certainvalue of parameter 120582

Actually GIVDHFUUBLWA and GIVDHFUUBLOWAare generalized version of weighted arithmetic aggregationoperators that producemore favorable results by emphasizing

20 Complexity

Table 6 Ranking orders obtained by GIVDHFUUBLWA GIVDHFUUBLOWA GIVDHFUUBLWG and GIVDHFUUBLOWG along with120582Operators 120582 (Integer) Ranking orderGIVDHFUUBLWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLOWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLWG [1 2] 1199092 ≻ 1199093 ≻ 1199091[3 20] 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG [1 8] 1199092 ≻ 1199093 ≻ 1199091[9 20] 1199092 ≻ 1199091 ≻ 1199093

Table 7 Ranking results obtained by GIVDHFUUBLHA and I-GIVDHFUUBLHA along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHA I-GIVDHFUUBLHA120572=0 120573=1 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=03 120573=07 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=05 120573=05 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=08 120573=02 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=09 120573=01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=1 120573=0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093

Table 8 Ranking results obtained by GIVDHFUUBLHG and I-GIVDHFUUBLHG along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHG I-GIVDHFUUBLHG

120572 = 0 120573 = 1 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 03 120573 = 07 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 05 120573 = 05 [1 8] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[9 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 08 120573 = 02 [1 17] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[18 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 09 120573 = 01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572 = 1 120573 = 0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093membership degrees to a uncertain linguistic term thus canbe considered as optimistic operators while GIVDHFU-UBLWG and GIVDHFUUBLOWG are generalized geomet-ric aggregation operators that producemore favorable resultsby emphasizing nonmembership degrees thus can be consid-ered as pessimistic operators [100 101]Therefore the rankingresults in Table 6 could be perceived as for optimism-baseddecision-making alternative 1199093 always stands out better than1199091 when 120582 gradually amplifies degree of optimism but forpessimism-based decision-making 1199091 turns out better than1199093 when 120582 gradually amplifies degree of pessimism 120582 canbe considered as the parameter for amplifying degree ofoptimism or pessimism [101]

Next to inspect the impacts of various combinationsof 120596119860119868 and 120596119860119863 on ranking results another four aggrega-tion operators of GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG also have beenapplied to the above illustrative case with varying 120572 120573 and 120582

All results have been collected in Tables 7 and 8 for compari-son As can be seen fromTable 7 ranking results generated byoperator GIVDHFUUBLHA keep the same as 1199092 ≻ 1199093 ≻ 1199091for all test combinations of parameters However as shownin Table 8 although GIVDHFUUBLHG is also capable ofidentifying 1199092 as the best contractor the priority relationsbetween 1199091 and 1199093 hold similar changing patterns in rankingresults in Table 6 that were output byGIVDHFUUBLWGandGIVDHFUUBLOWG Regarding decision situations whereobjective attributeweighting vector120596119860119863 is only considered (120572= 0) or its dominant importance (such as 120572 = 03) is accepted1199091 ≻ 1199093 is a clear observation when parameter 120582 is set toan integer in interval [8 20] Contrariwise under decisionscenarios where subjective attribute weighting vector 120596119860119868 isonly needed (120572 = 1) or its dominant importance (such as 120572 =09) is acknowledged 1199093 ≻ 1199091 holds all the way 120582 varies

In order to more clearly show the score trajectories ofalternative contractors along variations of parameter 120582 we

Complexity 21

HAInducedHA

InducedHA-2HA-2

HA-3

InducedHA-1InducedHA-3

= =

HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(a) 120572 = 0

HAInducedHA

= 03 = 07

InducedHA-2

HA-2HA-3

InducedHA-1 InducedHA-3HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(b) 120572 = 03

HAInducedHA

= 5 = 05

InducedHA-2HA-2

HA-3

InducedHA-1 InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(c) 120572 = 05

HAInducedHA

= 08 = 02

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572 = 08

HAInducedHA

= 9 = 01

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(e) 120572 = 09

HAInducedHA

246 2462 2464 2466 2468 247

= 1 = 0InducedHA-2HA-2

HA-3

HA- 3

HA-3

InducedHA-1

InducedHA-3

HA-1

HA-1

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

1085

1086

1087

(f) 120572 = 1

Figure 5 Score trajectories obtained by aggregation operators HA and InducedHA with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

further configure parameter 120582 to the value set of [0 01100]and conduct computational experiments to collect scorevalues and ranking results (as shown in Figures 5 and 6) byfour aggregation operators of GIVDHFUUBLHA GIVDH-FUUBLHG I-GIVDHFUUBLHAand I-GIVDHFUUBLHGFor more clarity in Figures 5 and 6 please note that weuse HA HG Induced-HA and Induced HG to respec-tively denote GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG In Figure 5it can be observed that when dominant importance of 120596119860119863

is emphasized (120572 = 0 120572 = 03 120572 = 05) GIVDHFUUBLHAis capable of differentiating the three alternative contractorsas same ranking order of 1199092 ≻ 1199093 ≻ 1199091 for all 120582 but withincreases in 120572 the margins between scores of 1199091 and 1199093 nar-rows and the rank order changes to 1199091 ≻ 1199093when 120572 gets 09and 1 Interestingly in Figure 6 although GIVDHFUUBLHGis able to significantly differentiate alternative contractors 1199091and 1199093 when dominant importance of 120596119860119863 is emphasized (120572= 0 120572 = 03 120572 = 05) rank order of 1199091 and 1199093 obviouslychanges once from 1199093 ≻ 1199091 to 1199091 ≻ 1199093 when 120582 is bigger

22 Complexity

HGInducedHG

= = 1

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(a) 120572= 0

HGInducedHG

= 3 = 07

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(b) 120572= 03

HGInducedHG

= 05 = 05

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(c) 120572= 05

HGInducedHG

= 8 = 02

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572= 08

HGInducedHG

292 293 294 295 296 297minus08

minus0795

minus079

= 09 = 01

HA-1

HA-3

InducedHG-2

HG-1

HG-1

HG-2

HG-3

HG-3 InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4

minus3

minus2

minus1

0

1

2

scor

e val

ues

(e) 120572= 09

HGInducedHG

50 60 70 80 90 100minus095minus09

minus085

= 1 = 0

HG-x3

HG-x1

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(f) 120572= 1

Figure 6 Score trajectories obtained by aggregation operators HG and InducedHG with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

than 8 When 120572 increases to 1 GIVDHFUUBLHGmaintainsthe rank order of 1199093 ≻ 1199091 but their score trajectorieswalk very close Comparatively I-GIVDHFUUBLHA andI-GIVDHFUUBLHG generate consistent ranking results of1199092 ≻ 1199091 ≻ 1199093 no matter 120582 varies in all the above experiments

In sum when parameters are configured with typical val-ues (ie 120572 = 05 and 120582=1) and no additional attitudinal infor-mation is taken into consideration the proposed aggregationoperators unanimously derive the same rank order 1199092 ≻1199093 ≻ 1199091 of for the three alternative contractors However

with the amplification effect of parameter 120582 analysis resultsfrom linear combination of 120596119860119868 and 120596119860119863 reveal alternative1199092 which consistently is the best choice but rank order of1199091 and 1199093 changes because their score trajectories get closeand crossed which probably indicates that only decision-making informationderived from the decisionmatrix is inad-equate to significantly differentiate alternatives 1199091 and 1199093While by integrating additional decision information aboutattitudinal characters of group experts I-GIVDHFUUBLHAand I-GIVDHFUUBLHG manage to generate the identical

Complexity 23

Table 9 Order-inducing vectors obtained by our TOPSIS-based method for problem in [38]

1198661 1198662 1198663 11986641198601 02522 01629 02282 027141198602 01 05243 01364 046891198603 04081 01756 05244 037791198604 04507 01409 01064 013221198605 00697 02203 03753 01161

Table 10 Order-inducing vectors obtained by our TOPSIS-based method for problem in [39]

1198661 1198662 1198663 11986641198601 04548 04969 04923 047321198602 05092 047 04768 046451198603 07192 04888 04953 049151198604 06722 06163 05706 049291198605 06104 04942 06463 04946

Table 11 Ranking results obtained by applying different order-inducing vectors to [38 39]

Problems Approaches Ranking results

Problem in [38] IHFLOWA [38] + order-inducing vectors in [38] 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198605 ≻ 1198604

IHFLOWA [38] + order-inducing vectors in Table 9 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198604 ≻ 1198605

Problem in [39] I-IVDHFLOWA [39] + order-inducing vectors in [39] 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198602 ≻ 1198601

I-IVDHFLOWA [39] + order-inducing vectors in Table 10 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602

permutation of 1199092 ≻ 1199091 ≻ 1199093 which verifies that ourproposed TOPSIS-based order-inducing approach is prac-tical and useful in helping work out decision results morereasonably and robustly

63 Application Studies of Proposed TOPSIS-Based Order-Inducing Method When confronting complex decision-making situations that need additional attitudinal charactersof decision-makers Yager and Filev [88] and Yager [102]introduced and encouraged to employ order-inducing vectorto represent those attitudinal characters Since then theidea of order-inducing vector has been extended to variousdecision-making environments [98 103] such as those basedon intuitionistic fuzzy information [104 105] linguistic infor-mation [106] hesitant fuzzy linguistic information [38] andinterval-valued dual hesitant fuzzy uncertain linguistic infor-mation [39] Generally in existing literature order-inducingvectors rely on experts to directly give [38 39 104ndash106]

In order to examine the adaptability of our TOPSIS-based order-inducing method to linguistic decision-makingenvironments we respectively apply the method to resolvethe same problems in references [38 39] Due to fact thatthere is no descriptions about how the order-inducing vectorswere derived in both references [38 39] we here applyextremums into our TOPSIS-based method to obtain order-inducing vectors that is (1199047 1 0) and (1199041 0 1) for[38] ([1199046 1199046] 1 0) and ([1199040 1199040] 0 1) for [39] Thenwe obtain our order-inducing vectors listed in Table 9 forthe problem adopted in [38] and our order-inducing vectorslisted in Table 10 for [39]

Now we apply the order-inducing vectors as shownin Tables 9 and 10 to resolve the problems in [38] and[39] respectively Regarding the problem in [38] the scorevalues of all alternatives are calculated as 119878(1198601)=02602119878(1198602)=02417 119878(1198603)=03577 119878(1198604)=0221 and 119878(1198605)=0185so the corresponding ranking result is 1198603 ≻ 1198601 ≻ 1198602 ≻1198604 ≻ 1198605 As for the problem in [39] the score values ofall alternatives are computed as 119878(1198601)=39954 119878(1198602)=3783119878(1198603)=70827 119878(1198604)=71792 and 119878(1198605)=5813 we thus obtainthe ranking result 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602 Formore clarity brief descriptions of comparative approachesand obtained ranking results also have been put together inTable 11

According to Table 11 the approaches that employ ourorder-inducing vectors derive the same ranking order ofthe better three alternatives for both comparative problemsfrom [38 39] As seen our proposed TOPSIS-based order-inducing method exhibits effective and more understand-able Furthermore when decision-makers are incapable ofdirectly put forward all desired order-inducing vectors due tocomplexity in ill-structured problems our method providesa feasible directions to obtain order-inducing vectors byfiguring out ideal solutions Comparatively our proposedTOPSIS-based order-inducing method behaves effective andmore operational in practical use

64 Comparison with Decision-Making Approach Based onDegraded form of IVDHFULS Note that our approachis intrinsically adaptive to decision situations based ondegraded forms of IVDHF UUBLS such as the dual hesitant

24 Complexity

fuzzy linguistic sets introduced by Yang and Ju [54] Tofurther inspect feasibility of our approach we here conductcomparative study on the same case adopted by Yang and Ju[54] We firstly convert their original decision matrix intothe form of IVDHF UUBLS crisp membership and non-membership degrees are equivalently changed to intervalswith same upper and lower limits (eg 02 turns out to be[02 02]) and linguistic variables are rewritten as uncertainlinguistic variables (eg 1199042 turns out to be [11990421199042])

Since the case only got three attributes among whichinterdependency could be not significant under generalassumption weighting vector 120596119860119868 is not calculated Andfor straight comparison order inducing vector is notincluded either Then the argument-dependent attributeweighting vector can be obtained from Eq(29) as 120596119860119863 =(03192 03132 03676)119879 By use of Algorithm Iwith operatorIVDHFUUBLWA (ie GIVDHFUUBLWA in case of 120582 = 1see Appendix B) we have the scores of alternatives 119904(1198601) =03412 119904(1198602) = 01354 119904(1198603) = 03932 119904(1198604) = 07352Therefore the ordering of alternatives is1198604 ≻ 1198603 ≻ 1198601 ≻ 1198602which is mainly in alignment with the ordering of1198604 ≻ 1198601 ≻1198603 ≻ 1198602 in Yang and Ju [54] It is worth noticing that 120596119860119863used in our approach is derived objectively from assessmentswhile attribute weights 120596lowast = (035 025 040)119879used by Yangand Ju [54] were assigned empirically Differences between120596119860119863 and 120596lowast lead to different ordering of1198601 and1198603 Compar-atively speaking our approach holds more adaptability andobjectivity

In general our proposed MADM approach in this paperexhibits an effective way to tackle complex green contractorselection problems in which subjective attributes weightingvector (such as120596119860119868) and objective attributes weighting vector(such as 120596119860119863) exist To comprehensively utilize these twofundamental facets of attributes weighting information ourcase study and computational experiments suggest that linearcombination method (ie 120596 = 120572120596119860119868 + 120573120596119860119863) is effectiveto reflect effects of 120596119860119868 and 120596119860119863 by configuring parameter 120572or 120573 When there is no specific expert opinions to specify orhard to obtain we suggest that parameter 120572 chooses the valueset of 0 02 05 08 1 Furthermore generalized aggrega-tion operators are suggested to inspect score trajectoriesof alternative contractors along increasing of the operatorsrsquoamplification parameter 120582 If some alternative contractorsrsquoscore values get very close or crossed so that they cannot bedifferentiated from each other significantly we suggest to askthe panel of experts to derive opinions of ideal solutions andutilize our TOPSIS-based order-inducing method to includethose expert opinions as attitudinal decision informationthereby obtaining more rational and robust decision results

7 Conclusions

Aiming at tackling complex decision-making problems ofgreen contractor selection that often hold simultaneouslythe characteristics of decision hesitancy attributes inter-dependency and group complex attitudinal characters wehave developed an effective IVDHF UUBLS-based approach

We utilize IVDHF UUBLS to elicit complicate decision-makersrsquo assessments more objectively and comprehensivelywe employ DEMATEL for addressing attribute interdepen-dency more precisely and we determine group attitudinalcharacters as order-inducing vectors based on ideal attitudesof experts rather than by empirical assignment The mainadvantages of our approach are (i) IVDHF UUBLS offers anadequate way to depict decision hesitancy (ii) the designeddistance measure for IVDHF UUBLS avoids informationdistortion of traditional ones which artificially add mis-matching membership or nonmembership degrees (iii) thedeveloped generalized operators include traditional ones asspecial cases so as to provide flexibility in MADM basedon IVDHF UUBLS (iv) the devised synthesized attributeweighting scheme provides an adaptive way to consider twonecessary aspects attributes interdependency-based weightsand argument-dependent weights (v) the TOPSIS-basedmethod can deduce reasonable order-inducing vectors ratherthan empirical ones and holds generality in other linguisticdecision environments Results of case studies and experi-ments have verified the validity of the proposed approach

When multiple decision organizations are involvedgroup decision-making based on IVDHF UUBLS is animportant research direction Besides inducing preferencesfor decision-making problems of high uncertainty interactivemechanisms should also be studied to extract preferences Inaddition empirical studies of the proposed approach can becarried out to various areas such as investment evaluationproject management and vehicle selection in fleet operations

Appendix

A Basic Notions for IVDHFS and UnbalancedLinguistic Term Set

Definition A1 (see [47]) Letting 119883 be a fixed set then anIVDHFS on 119883 is defined as119863 = ⟨119909 ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (A1)

where ℎ(119909) = ⋃[120583119871120583119880]isinℎ(119909)120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and119892(119909) = ⋃[]119871]119880]isin119892(119909)] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two setsof interval values in [0 1] denoting possible membershipand non-membership degrees of element 119909 isin 119883 to theset 119863 respectively with conditions 120583 ] isin [0 1] and0 le (120583119880)+ + (]119880)+ le 1 and for all 119909 isin 119883 (120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880

For convenience normally 119889 = ℎ 119892 is called an interval-valued dual hesitant fuzzy element (IVDHFE) and 119863 is theset of all IVDHFEs

Definition A2 (see [107]) Suppose 119878 = 119904119895 | 119895 = 0 1 119892 minus1 is a finite and totally ordered discrete linguistic term setwhere 119904119895 represent possible values for a linguistic variableand 119892 is an odd cardinality 119878 is uniformly and symmetricallydistributed if the following two conditions are satisfied (1)

Complexity 25

there exists a unique constant 120582 gt 0 such that 119873119878(119904119894) minus119873119878(119904119895) = 119897(119894 minus 119895) for all 119894 119895 = 0 1 119892 minus 1 (2) Let 119878119877 =119904 | 119904 isin 119878 119904 gt 119878119862 and 119878119871 = 119904 | 119904 isin 119878 119904 lt 119878119862 Let C(119878119877) andC(119878119871) be the cardinality of 119878119877 and 119878119871 then C(119878119877) = C(119878119871)If 119878 is uniformly and symmetrically distributed then 119878 iscalled a balanced linguistic term set Otherwise 119878 is calledan unbalanced linguistic term set

The following 2-tuple fuzzy linguistic representationmodel extends traditional linguistic term set to a continuouscase so as to facilitate computing with linguistic variables

Definition A3 (see [59]) Let 119878 = 1199040 1199041 119904119892minus1 be alinguistic term set and 120573 isin [0 119892] Then a 2-tuple fuzzylinguistic variable expresses the equivalent information to 120573is defined as

Δ [0 119892] 997888rarr 119878 times [minus05 05) (A2)

Δ (120573) = (119904119894 120572) with

119904119894 119894 = 119903119900119906119899119889 (120573)120572 = 120573 minus 119894 120572 isin [minus05 05)

(A3)

Δminus1 (119904119894 120572) = 119894 + 120572 = 120573 (A4)

where 119903119900119906119899119889() is the usual rounding operation and120572 is calledsymbolic translation

To obtain 2-tuple fuzzy linguistic representations ofunbalanced linguistic terms the concept of linguistic hierar-chies ie 119871119867 = ⋃119905 119897(119905 119899(119905)) is used 119897(119905 119899(119905)) is a linguistichierarchy with 119905 indicating the level of hierarchy and 119899(119905)denotes the granularity of the linguistic term set of 119905 Herreraet al [59] defined the following transformation functionsbetween labels from different levels in multigranular linguis-tic information contexts without loss of information

Definition A4 (see [59]) In linguistic hierarchies 119871119867 =⋃119905 119897(119905 119899(119905)) whose linguistic term sets are represented by119878119899(119905) = 119904119899(119905)0 119904119899(119905)119899(119905)minus1 the transformation function froma linguistic label in level 119905 to a label in consecutive level 1199051015840 isdefined as 1198791198651199051199051015840 119897(119905 119899(119905)) 997888rarr 119897(1199051015840 119899(1199051015840)) such that

1198791198651199051199051015840 (119904119899(119905)119894 120572119899(119905))= Δ 1199051015840 (Δminus1119905 (119904119899(119905)119894 120572119899(119905)) (119899 (1199051015840) minus 1)119899 (119905) minus 1 ) (A5)

By use of the above transformation function any 2-tuplelinguistic representation can be transformed into a term in119871119867 Detailed transformation procedures are listed as follows

(1) Representation in linguistic hierarchy To transformthe unbalanced terms of 119878 into the corresponding terms inthe 119871119867 transformation function 120595 is employed to associateeach unbalanced linguistic 2-tuple (119904119894 120572) with its linguistic 2-tuple in 119871119867(119878) ie

120595 119878 997888rarr 119871119867(119878) (A6)

so that 120595(119904119894 120572) = (119904119866(119894)119868(119894) 120582) for forall(119904119894 120572) isin 119878(2) Computational phase Firstly transform (119904119866(119894)119868(119894) 120582) intolinguistic 2-tuples denoted as (119904119899(1199051015840)

1198681015840(119894) 1205821015840) in 119878119899(1199051015840) where

(119904119899(1199051015840)1198681015840(119894)

1205821015840) = 119879119865 (119904119866(119894)119868(119894) 120582)= Δ(Δminus1 (119904119866(119894)119868(119894) 120582) sdot (119899 (1199051015840) minus 1)119866 (119894) minus 1 ) (A7)

Then a computational model is used over 119878119899(1199051015840) with aresult denoted as (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)(3) Retranslation process The result (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)is transformed into the unbalanced term in 119878 by using thetransformation function 120595minus1 ie

120595minus1 119871119867 (119878) 997888rarr 119878 (A8)

so that 120595minus1(119904119899(1199051015840)119903 120582119903) = (119904119903119890119904119906119897119905 120582119903119890119904119906119897119905) isin 119878B Details of Theorem 14

Proof Given a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector for119904119889119895 and 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 120582 gt 0 then(1) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLWA and GIVDHFUUBLWG operators respectively(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLOWA and GIVDHFUUBLOWG operators respectively

(2) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to theIVDHF UUBL weighted average (IVDHFUUBLWA)and IVDHF UUBL weighted geometric (IVDHFUUBLWG)operators respectively where

26 Complexity

119868119881119863119867119865119880119880119861119871119882119860120596 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

120596119895119904119889119895= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus

119899prod119895=1

(1 minus 120583119871119895 )120596119895 1 minus 119899prod119895=1

(1 minus 120583119880119895 )120596119895]] [[

119899prod119895=1

(]119871119895)120596119895 119899prod119895=1

(]119880119895 )120596119895]])

(B1)

119868119881119863119867119865119880119880119861119871119882119866120596 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889119895120596119895

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[

119899prod119895=1

(120583119871119895 )120596119895 119899prod119895=1

(120583119880119895 )120596119895]] [[1 minus

119899prod119895=1

(1 minus ]119871119895)120596119895 1 minus 119899prod119895=1

(1 minus ]119880119895 )120596119895]])

(B2)

(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHA andGIVDHFUUBLHG operators reduce to the IVDHF UUBL

orderedweighted average (IVDHFUUBLOWA) and IVDHFUUBL ordered weighted geometric (IVDHFUUBLOWG)operators respectively where

119868119881119863119867119865119880119880119861119871119874119882119860119908 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

119908119895119904119889120590(119895)= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) )))

119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) )))

]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus

119899prod119895=1

(1 minus 120583119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus 120583119880120590(119895))119908119895]] [[

119899prod119895=1

(]119871120590(119895))119908119895 119899prod119895=1

(]119880120590(119895))119908119895]])

(B3)

119868119881119863119867119865119880119880119861119871119874119882119866119908 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889120590(119895)119908119895

= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[

119899prod119895=1

(120583119871120590(119895))119908119895 119899prod119895=1

(120583119880120590(119895))119908119895]] [[1 minus

119899prod119895=1

(1 minus ]119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus ]119880120590(119895))119908119895]])

(B4)

Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

References

[1] V W Y Tam and C M Tam ldquoWaste reduction throughincentivesA case studyrdquoBuilding Research and Information vol36 no 1 pp 37ndash43 2008

[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

[11] X Qi C Liang and J Zhang ldquoGeneralized cross-entropy basedgroup decision making with unknown expert and attributeweights under interval-valued intuitionistic fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 79 pp 52ndash642015

[12] H Zimmermann Fuzzy Set Theorymdashand Its ApplicationsKluwer Academic Boston Mass USA 2001

[13] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 pp 199ndash249 1975

[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

[16] B T Nuong K-W Kim L Prathumratana et al ldquoSustainabledevelopment in the mining sector and its evaluation usingfuzzy AHP (Analytic Hierarchy Process) approachrdquo GeosystemEngineering vol 14 no 1 pp 43ndash50 2011

[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

[18] E K Zavadskas Z Turskis and J Antucheviciene ldquoSelectinga contractor by using a novel method for multiple attributeanalysis Weighted aggregated sum product assessment withgrey values (WASPAS-G)rdquo Studies in Informatics and Controlvol 24 no 2 pp 141ndash150 2015

[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

[21] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

[25] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009

[26] Z Hatush and M Skitmore ldquoContractor Selection UsingMulticriteria Utility Theory An Additive Modelrdquo Building andEnvironment vol 33 no 2-3 pp 105ndash115 1998

[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

[30] AM Anvuur andMM Kumaraswamy ldquoConceptualmodel ofpartnering and alliancingrdquo Journal of Construction Engineeringand Management vol 133 no 3 pp 225ndash234 2007

[31] A H I Lee H Kang C F Hsu and H Hung ldquoA green supplierselection model for high-tech industryrdquo Expert Systems withApplications vol 36 no 4 pp 7917ndash7927 2009

[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

[33] P Humphreys R McIvor and F Chan ldquoUsing case-basedreasoning to evaluate supplier environmental managementperformancerdquo Expert Systems with Applications vol 25 no 2pp 141ndash153 2003

[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

[45] Bin Zhu Zeshui Xu and Meimei Xia ldquoDual Hesitant FuzzySetsrdquo Journal of Applied Mathematics vol 2012 Article ID879629 13 pages 2012

[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

[51] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[52] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

[55] Z Tao X Liu H Chen and L Zhou ldquoUsing new versionof extended t-norms and s-norms for aggregating intervallinguistic labelsrdquo IEEETransactions on SystemsManCyberneticsSystems vol vol p 15 2016

[56] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[57] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[58] L Martınez M Espinilla J Liu L G Perez and P J SanchezldquoAn evaluation model with unbalanced linguistic informationapplied to olive oil sensory evaluationrdquo Journal of Multiple-Valued Logic and Soft Computing vol 15 no 2-3 pp 229ndash2512009

[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

[62] L Marin A Valls D Isern A Moreno and J M MerigoldquoInduced unbalanced linguistic ordered weighted average andits application in multiperson decision makingrdquo The ScientificWorld Journal vol 2014 2014

[63] A Baykasoglu V Kaplanoglu Z D U Durmusoglu and CSahin ldquoIntegrating fuzzy DEMATEL and fuzzy hierarchicalTOPSIS methods for truck selectionrdquo Expert Systems withApplications vol 40 no 3 pp 899ndash907 2013

[64] S K Patil and R Kant ldquoA hybrid approach based on fuzzyDEMATEL and FMCDM to predict success of knowledge man-agement adoption in supply chainrdquoApplied Soft Computing vol18 pp 126ndash135 2014

[65] S Liyin and Y Hong ldquoImproving environmental performanceby means of empowerment of contractorsrdquo Management ofEnvironmental Quality An International Journal vol 17 no 3pp 242ndash257 2006

[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

[69] Y Kazancoglu I Kazancoglu and M Sagnak ldquoA new holisticconceptual framework for green supply chain management

performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

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10 Complexity

(1) Generalized IVDHFUUBL Ordered Weighted Average(GIVDHFUUBLOWA) Operator

119866119868119881119863119867119865119880119880119861119871119874119882119860120596120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(119908119895119904119889120590(119895)120582))1120582

= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

120582)1120582]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895) []119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120590(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120590(119895))120582)119908119895)1120582]])

(9)

(2) Generalized IVDHFUUBL Ordered Weighted Geometric(GIVDHFUUBLOWG)Operator

119866119868119881119863119867119865119880119880119861119871119874119882119866119908120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889120590(119895))119908119895)= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120590(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120590(119895))120582)120596119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120590(119895))120582)120596119895)1120582]])

(10)

Definition 12 For a collection of IVDHF UUBLNs 119904119889119895(119895 =1 119899) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector120596119895 isin [0 1] sum119899119895=1 120596119895 = 1 119899 is a balancing coefficient119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weight-ing vector with 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1

(1) Generalized IVDHFUUBL Hybrid Average (GIVDHFU-UBLHA) Operator

119866119868119881119863119867119865119880119880119861119871119867119860120596119908120582 (1199041198891 1199041198892 119904119889119899) = ( 119899⨁119895=1

(119908119895119904119889120590(119895)120582))1120582

= ⋃(119904120599120590(119895) ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895)

))))120582)1120582]

Complexity 11

⋃[120583119871120590(119895) 120583

119880120590(119895)]isinℎ120590(119895) []

119871120590(119895) ]119880120590(119895)]isin119892120590(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120590(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120590(119895))120582)119908119895)1120582]])

(11)

where 119904119889120590(119895) is the 119895th largest and

119904119889119895 = 119899120596119895119904119889119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904119899120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904119899120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

] ⋃

[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[1 minus (1 minus 120583119871119895 )119899120596119895 1 minus (1 minus 120583119880119895 )119899120596119895] [(]119871119895)119899120596119895 (]119880119895 )119899120596119895]) (12)

(2) Generalized IVDHFUUBL Hybrid Geometric(GIVDHFUUBLHG) Operator

119866119868119881119863119867119865119880119880119861119871119867119866119908120596120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889120590(119895))119908119895)= ⋃(119904120599120590(119895)

ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120590(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895)

))))119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895) []119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120590(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120590(119895))120582)119908119895)1120582]])

(13)

where 119904119889120590(119895) is the 119895th largest and

119904119889119895 = 119904119889119895119899120596119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(Δminus11199050 (119879119865

11990511989511199050(120595(119904120572119895 ))))

119899120596119895 119904(Δminus11199050 (119879119865

11990511989521199050(120595(119904120573119895 ))))

119899120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[(120583119871119895 )119899120596119895 (120583119880119895 )119899120596119895] [1 minus (1 minus ]119871119895 )119899120596119895 1 minus (1 minus ]119880119895 )119899120596119895]) (14)

12 Complexity

When confronted with ill-structured situations wheredecision-makerrsquos complex attitudinal characters need to beincluded order-inducing vectors provide an effective way[88ndash90]Thus we further define following induced operatorsfor IVDHF UUBLNs

Definition 13 For a collection of IVDHF UUBLNs 119904119889119895 (119895 =1 119899) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector

120596119895 isin [0 1] sum119899119895=1 120596119895 = 1 119899 is a balancing coefficient119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weight-ing vector 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1 120576119895 denote a set oforder inducing vectors

(1) Induced Generalized IVDHFUUBL Hybrid Average (I-GIVDHFUUBLHA) Operator

119868-119866119868119881119863119867119865119880119880119861119871119867119860120596119908120582 (lt 1205761 1199041198891 gt lt 1205762 1199041198892 gt lt 120576119899 119904119889119899 gt) = ( 119899⨁119895=1

(119908119895119904119889120587(119895)120582))1120582

= ⋃(119904120599120587(119895) ℎ120587(119895) 119892120587(119895))isin119904119889120587(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120587(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587(119895)

))))120582)1120582]

⋃[120583119871120587(119895) 120583

119880120587(119895)]isinℎ120587(119895) []

119871120587(119895) ]119880120587(119895)]isin119892120587(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120587(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120587(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120587(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120587(119895))120582)119908119895)1120582]])

(15)

where (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) is decreasing order of (11990411988911199041198892 119904119889119899) according to the order inducing vector 120576119895 and119904119889119895 = 119899120596119895119904119889119895 = ⋃

(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904119899120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904119899120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

] ⋃

[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[1 minus (1 minus 120583119871119895 )119899120596119895 1 minus (1 minus 120583119880119895 )119899120596119895] [(]119871119895)119899120596119895 (]119880119895 )119899120596119895]) (16)

(2) Induced Generalized IVDHFUUBL Hybrid Geometric(I-GIVDHFUUBLHG) Operator

119868-119866119868119881119863119867119865119880119880119861119871119867119866119908120596120582 (lt 1205761 1199041198891 gt lt 1205762 1199041198892 gt lt 120576119899 119904119889119899 gt) = 1120582 ( 119899⨂119895=1

(120582119904119889120587(119895))119908119895)= ⋃(119904120599120587(119895)

ℎ120587(119895) 119892120587(119895))isin119904119889120587(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120587(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587(119895)

))))119908119895]

⋃[120583119871120587(119895)120583119880120587(119895)]isinℎ120587(119895) []119871120587(119895) ]

119880120587(119895)]isin119892120587(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120587(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120587(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120587(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120587(119895))120582)119908119895)1120582]])

(17)

Complexity 13

Weighted Aggregation

Geometric Arithmetic

GIVDHFUUBLHG GIVDHFUUBLHA

GIVDHFUUBLWG GIVDHFUUBLOWG

IVDHFUUBLWG IVDHFUUBLOWG

GIVDHFUUBLWA GIVDHFUUBLOWA

IVDHFUUBLWA IVDHFUUBLOWA

= 1 w =1

n = 1 =

1

n = 1 =1

n = 1 =

1

n

= 1 = 1 = 1 = 1

Figure 2

where (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) is decreasing order of (11990411988911199041198892 119904119889119899) according to the order inducing variables 120576119895 and

119904119889119895 = 119904119889119895119899120596119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(Δminus11199050 (119879119865

11990511989511199050(120595(119904120572119895 ))))

119899120596119895 119904(Δminus11199050 (119879119865

11990511989521199050(120595(119904120573119895 ))))

119899120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[(120583119871119895 )119899120596119895 (120583119880119895 )119899120596119895] [1 minus (1 minus ]119871119895 )119899120596119895 1 minus (1 minus ]119880119895 )119899120596119895]) (18)

42 Properties of the Proposed GeneralizedAggregation Operators

eorem 14 With special values of 120582 119908 and 120596 the operatorsGIVDHFUUBLHA and GIVDHFUUBLHG can include aseries of traditional aggregation operators as special cases andtheir relationship can be depicted in Figure 2

Proof See Appendix B

As for the induced hybrid aggregation operators we alsohave following theorem

eorem 15 If (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) = (119904119889120590(1) 119904119889120590(2) 119904119889120590(119899)) I-GIVDHFUUBLHA reduces toGIVDHFUUBLHAIf (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) = (119904119889120590(1) 119904119889120590(2) 119904119889120590(119899)) thenI-GIVDHFUUBLHG reduces to the GIVDHFUUBLHG oper-ator

eorem 16 All the proposed generalized operatorsGIVDHFUUBLWAGIVDHFUUBLWGGIVDHFUUBLOWAGIVDHFUUBLOWGGIVDHFUUBLHAGIVDHFUUBLHGI-GIVDHFUUBLHA and I-GIVDHFUUBLHG hold thefollowing properties (1) Commutativity (2) Idempotency (3)Boundedness

Based on above theorems following properties can alsobe derived

eorem 17 For a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weight vector of 119904119889119895with 120596119895 isin [0 1] and sum119899119895=1 120596119895 = 1 120582 gt 0 then we have

(1) 119868119881119863119867119865119880119880119861119871119882119866120596(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119882119860120596(1199041198891 1199041198892 119904119889119899)(2) 119868119881119863119867119865119880119880119861119871119882119866120596(1199041198891 1199041198892 119904119889119899) le119866119868119881119863119867119865119880119880119861119871119882119860120596120582(1199041198891 1199041198892 119904119889119899)(3) 119866119868119881119863119867119865119880119880119861119871119882119866120596120582(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119882119860120596(1199041198891 1199041198892 119904119889119899)

Proof Omitted for concision

eorem 18 For a collection of IVDHF UUBLNs 119904119889119895 = (119904120599119895 ℎ119895 119892119895) 119908 = (1199081 1199082 119908119899)119879 is the weighting vector of 119904119889119895with 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1 120582 gt 0 Then

(1) 119868119881119863119867119865119880119880119861119871119874119882119866119908(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119874119882119860119908(1199041198891 1199041198892 119904119889119899)(2) 119868119881119863119867119865119880119880119861119871119874119882119866119908(1199041198891 1199041198892 119904119889119899) le119866119868119881119863119867119865119880119880119861119871119874119882119860119908120582(1199041198891 1199041198892 119904119889119899)(3) 119866119868119881119863119867119865119880119880119861119871119874119882119866119908120582(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119874119882119860119908(1199041198891 1199041198892 119904119889119899)

Proof Omitted for concision

14 Complexity

5 An Integrated MADM Approach forTackling Complex Green ContractorSelection Problems

As described in Section 2 we take the green contractorselection as a special type of complicate MADM problemsthat synthesizes three characteristics of decision hesitancy[21 22] attributes interdependency [84] and group atti-tudinal characters [61] Therefore in this section we con-struct an integrated MADM approach to tackle the com-plex green contractor selection problems Suppose 119883 =1199091 1199092 119909119898 is the set of alternative green contractors and119860 = 1198601 1198602 119860119899 is the set of evaluative attributes 120596 =(1205961 1205962 120596119899)119879 is weighting vector for the attributes 120596119894 ge 0sum119899119894=1 120596119894 = 1 Let = (119903119894119895)119899times119898 denote the decision matrix inwhich 119903119894119895 = ([119904120572119894119895 119904120573119894119895] ℎ119894119895 119892119894119895) is an IVDHF UUBLN givenby decision-makers for alternative contractor 119909119895 with respectto attribute 119860 119894 According to the mechanism of pair-wisecomparisons among attributes in the DEMATELmethod [6364] the interdependency among attributes can be obtained asamatrix119885 = (119911119894119896)119899times119899 where 119911119894119896 indicates the degree towhich119860 119894 affects119860119896 Subsequently based on the IVDHFUUBLS andits operations we now present detailed steps of our MADMapproach as shown in following Algorithm I

Algorithm I Hesitant fuzzy linguistic MADMwith attributesinterdependency and decision-makersrsquo group attitudinalcharacters

Step 1 Determine argument-dependent weighting vector120596119860119863 according to attribute values by programming modeldeveloped in the following Section 51

Step 2 Obtain the attribute-interdependences based weight-ing vector 120596119860119868 by use of DEMATEL method described in thefollowing Section 52

Step 3 Calculate synthesized attribute weighting vectoraccording to

120596 = 120572120596119860119868 + 120573120596119860119863 (19)

where 120572 and 120573 are parameters to reflect decision character-istics of decision organizations 0 le 120572 120573 le 1 120572 + 120573 =1Step 4 Check requirements for order inducing If no addi-tional order inducing required then go to Step 5 otherwisego to Step 6

Step 5 Utilize generalized aggregation operators to get theoverall IVDHF UUBLNs 119903119895(119895 = 1 119898) for each alternative119909119895 Here we take GIVDHFUUBLHA operator for examplebecause it can include other traditional operators as its specialcases Therefore we have

119866119868119881119863119867119865119880119880119861119871119867119860(1199031119895 1199032119895 119903119899119895) = ( 119899⨁119894=1

(119908119894 (119903120590119894119895)120582))1120582= ⋃(119904120590119894119895ℎ120590119894119895 119892120590119894119895

)isin119903120590119894119895

([119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590119894119895

))))120582)1120582 119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590119894119895

))))120582)1120582]

⋃[120583119871120590119894119895120583

119880120590119894119895]isinℎ120590119894119895[]

119871120590119894119895]119880120590119894119895]isin119892120590119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120590119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120590119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120590119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120590119894119895)120582)119908119894)1120582]])

(20)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of 119903119894119895119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weightvector 119903120590119894119895 is the 119894th largest of 119903119894119895 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))

119904119899120596119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119894119895))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (21)

Complexity 15

Then go to Step 8

Step 6 By use of the TOPSIS-based method developed in thefollowing Section 53 we determine order-inducing vectors 120576to reflect decision-makersrsquo group attitudinal characters

Step 7 Utilizing induced generalized aggregation operatorsto get the overall IVDHF UUBLNs 119903119895(119895 = 1 2 119898) foreach alternative 119909119895 we have

119868-119866119868119881119863119867119865119880119871119867119860(1199031119895 1199032119895 119903119899119898) = ( 119899⨁119894=1

(119908119894 (119903120587119894119895)120582))1120582

= ⋃(119904120587119894119895ℎ120587119894119895119892120587119894119895)isin119903120587119894119895

([119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120587119894119895

))))120582)1120582 119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587119894119895

))))120582)1120582]

⋃[120583119871120587119894119895 120583

119880120587119894119895]isinℎ120587119894119895 []

119871120587119894119895]119880120587119894119895]isin119892120587119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120587119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120587119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120587119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120587119894119895)120582)119908119894)1120582]])

(22)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of119903119894119895 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 119903120587119894119895 is the 119894th largest of 119903119894119895 reordered by theorder-inducing vectors 120576 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))119899120596119894120572119894119895

119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120573119894119895 ))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (23)

Step 8 Calculate scores 119878(119903119895) and accuracy degrees 119875(119903119895) for119903119895(119895 = 1 2 119898)Step 9 According to Definition 5 output ranking order ofalternative contractors

In Algorithm I to objectively derive unknown weight-ing vector for evaluative attributes a synthesized attributeweighting vector is devised by fusing two parts of concern(i) argument-dependent weighting vector and (ii) attribute-interdependences based weighting vector The former is toreflect effects of attribute values on attribute weights whilethe latter is to exploit effects of interdependences amongattributes on their weights For clarity processing flowchartof Algorithm I is depicted in Figure 3

51 Determining Argument-Dependent Attribute WeightingVector 120596119860119863 In order to derive attribute weighting vec-tor appropriately from assessments maximizing deviationmethod [91] is adopted here for its distinguishing ability

and objectivity [92] According to Wang [91] if performancevalues of each alternative have little differences under anattribute it implicates such an attribute plays a less importantrole otherwise it lives a more important role in decision-making Thus if one attribute has similar values across alter-natives it should be assigned a smaller weight contrariwiseit should be a bigger weight

In light of the idea in maximizing deviation methodwe develop a programming model to determine argument-dependent weighting vector 120596119860119863 For attribute 119860 119894 the stan-dard deviation of alternative 119909119895 to all the other alternatives isdenoted as

119865119894119895 (120596119860119863) = 119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (24)

Let119865119894(120596119860119863) represent deviation value of all alternatives toother alternatives according to attribute119860 119894 and be formulatedas follows

16 Complexity

Attribute weighting

Hesitant fuzzy linguistic MADM with attributes interdependency and decision makersrsquo group attitudinal characters

Step 9 Output ranking order of alternatives

interdependency among attributes by DEMATEL in Section 52

attributes values by standard deviation method in Section 51

Step 6 Ensure TOPSIS order inducing variables proposed in Section 53

Step 7 Utilizing proposed induced generalized hybrid aggregation operators developed in Section

41 to get overall value of each alternative

Step 8 Calculate score degrees and accuracy degrees

Step 4 Order inducingYes

Step 5 Choose proposed generalized operators in Section 41 to get overall value of each alternative

No

Collect decision matrix in the form of IVDHF_UUBLS

Complex problems of green contractor selection

Step 1 Weighting vector AD derived from Step 2 Weighting vector AI derived from

Step 3 Obtain synthesized weighting vector of attributes =AI + AD

Figure 3 Processing flowchart of Algorithm I

119865119894 (120596119860119863) = 119898sum119895=1

119865119894119895 (120596119860119863) = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (25)

Based on the analysis above we can get followingprogramming model (M-1) for obtaining optimal attributeweighting vector 120596119860119863

(119872 minus 1)max 119865 (120596119860119863) = 119899sum

119894=1119865119894 (120596119860119863) = 119899sum

119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894st 119899sum

119894=1(120596119860119863119894 )2 = 1 120596119860119863119894 ge 0 119894 = 1 2 119899 (26)

where 119889(119903119894119895 119903119894119897) can be calculated by Definition 6To solve model (M-1) we can construct Lagrange func-

tion

119871 (120596119860119863119894 120577) = 119899sum119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894+ 1205772 ( 119899sum

119894=1

(120596119860119863119894 )2 minus 1) (27)

where 120577 is the Lagrange multiplierDifferentiate with respect to 120596119860119863119894 (119894 = 1 2 119898) and 120577

and set these partial derivatives equal to zero the followingset of equations can be obtained

120597119871120597120596119860119863119894 = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) + 120577120596119860119863119894 = 0120597119871120597120577 = 119899sum

119894=1

(120596119860119863119894 )2 minus 1 = 0 (28)

By normalizing the solutions of above equations we canget a simple and exact formula for determining the attributeweights as follows

120596119860119863119894 = sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897)sum119899119894=1sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897) (29)

52 Deriving Attributes Interdependency-Based AttributeWeighting Vector 120596119860119868 Some efforts [53 60 84 93ndash95]

Complexity 17

have been made to address interdependencies betweenattributes in MADM However they unanimously cannotreveal the interdependences in a more complete way WhileDEMATAEL technique [63 96] gathers collective knowledgeto capture influences between attributes more precisely andcompletely so that weighting vector for attributes can berationally determined according to strength measure ofthose influences [64 97] Therefore in tackling the complexMADMwith interdependent attributes under IVDHF UUBLenvironments we present a DEMATEL-based method as thefollowing Algorithm II to derive attribute weighting vector120596119860119868 from the viewpoint of attributes interdependency

Algorithm II DEMATEL-basedmethod for ensuringweight-ing vector 120596119860119868Step 1 Construct initial direct-influence matrix 119885 = (119911119894119896)119899times119899

For attributes 119860 = 1198601 1198602 119860119899 decision-makersare asked to give direct effect that attribute 119860 119894 has onattribute 119860119896 using a five-point scale with scores representedby natural language (1) 0 mdash lsquoabsolutely no influencersquo (2)1 mdash lsquolow influencersquo (3) 2 mdash lsquomedium influencersquo (4) 3 mdashlsquohigh influencersquo (5) 4mdash lsquovery high influencersquoThen the initialdirect-influence matrix 119885 = (119911119894119896)119899times119899 can be established witheach element being the mean of same factors in differentdirect matrices

Step 2 Calculate normalized direct-influence matrix 119883 =(119909119894119896)119899times119899 Using matrix 119885 the normalized direct-influencematrix 119883 = (119909119894119896)119899times119899 is calculated according to

119909119894119896 = 119911119894119896119904 (30)

where 119904 = maxmax1le119894le119899sum119899119896=1 119911119894119896max1le119896le119899sum119899119895=1 119911119894119896 Allelements in matrix 119883 are complying with 0 le 119909119894119896 le 10 le sum119899119896=1 119909119894119896 le 1 and 0 le sum119899119894=1 119909119894119896 le 1 and at least onerow or column of the summation equals 1

Step 3 Derive total-influence matrix 119879 = (119905119894119896)119899times119899 Basedon matrix 119883 the total-influence matrix 119879 = (119905119894119896)119899times119899 can bederived by summing the direct effects and all of the indirecteffects

119879 = 119883 + 1198832 + 1198833 + sdot sdot sdot + 119883ℎ = 119883 (119868 minus 119883)minus1 (31)

where 119868 represents identity matrix When ℎ 997888rarr infin 119883ℎ =(0)119899times119899Step 4 Compute influencing and influenced degrees of eachattribute The sum of rows and the sum of columns withinthe total-influence matrix 119879 are respectively expressed as thevectors 119903 and 119888

119903 = (119903119894)119899times1 = ( 119899sum119896=1

119905119894119896)119899times1

119888 = (119888119896)119899times1 = ( 119899sum

119895=1

119905119894119896)119879

1times119899

(32)

where 119903119894 denotes the sumof the 119894th row inmatrix119879 and showsthe sum of direct and indirect effects that attribute 119860 119894 hason other attributes Similarly 119888119896 shows the sum of direct andindirect effects that attribute 119860119896 has receivedStep 5 Estimate weights of attributes based on influencedegrees according to

120596119868lowast119894 = [(119903119894 + 119888119894)2 + (119903119894 minus 119888119894)2]12 (33)

Then the weighting vector 120596119860119868 can be obtained by normaliz-ing (33)

120596119860119868119894 = 120596119868lowast119894sum119899119894=1 120596119868lowast119894 (34)

53 Obtaining Order Inducing Variables 120576 by TOPSIS-BasedMethod For ill-defined MADM problems induced aggre-gation operators [89 90 98] provide an effective channelfor experts to express complex attitudinal characters Thekey step in induced aggregation operators is to obtainorder-inducing vectors 120576 which however are presumptivelyassigned in most papers [62] Therefore in this sectionaiming at deducing the order-inducing vectors 120576 rationallywe here devise a TOPSIS-based method in Algorithm IIIObviously the Algorithm III is of generality for MADMwithdifferent forms of linguistic expression

Algorithm III TOPSIS-based method for obtaining orderinducing vectors 120576Step 1 Let the decision maker define IVDHF UUBL idealsolution(s) 119903+119894 and IVDHF UUBL negative ideal solution(s)119903minus119894 corresponding to each attribute in which

119903+119894 = (119903+1198941 119903+1198942 119903+119894119901) 119903minus119894 = (119903minus1198941 119903minus1198942 119903minus119894119902) (35)

where 119901 and 119902 denotes the number of 119903+119894 and 119903minus119894 respectively

Step 2 Calculate distances from each element in decisionmatrix to ideal solution(s) and negative solution(s) respec-tively

119889+119894119895 = max119897=12119901

119889 (119903119894119895 119903+119894119897 ) 119889minus119894119895 = min

119897=12119902119889 (119903119894119895 119903minus119894119897 ) (36)

where 119889(119903119894119895 119903+119894119897 ) and 119889(119903119894119895 119903minus119894119897 ) are calculated by the distancemeasure in Definition 6 If 119901 gt 1 we choose the onewith maximum distance if 119902 gt 1 we choose the one withminimum distance

Step 3 Calculate coefficients according to

119888119894119895 = 119889minus119894119895119889+119894119895 + 119889minus119894119895 119894 = 1 2 119899 119895 = 1 2 119898 (37)

18 Complexity

Table 2 Decision matrix in the form of IVDHF UUBLS1199091 1199092 11990931198601([AHVH] [02 03][02 04] [03 04]) ([HT] [04 05][05 06] [02 03] [02 04]) ([ALAH] [01 02][01 03] [06 07])1198602([MH] [05 06][02 03]) ([AHT] [06 07][01 02] ) ([MVH] [02 03][05 06] [06 07])1198603([VLAL] [03 04][04 05] [05 06]) ([MVH] [06 07] [07 08][01 02]) ([VLL] [04 05][02 03])1198604([MAH] [02 04][05 06]) ([LAL] [03 05][02 03]) ([ALM] [06 07][01 02] [02 03])1198605([ALM] [04 05][01 02] [04 05]) ([AHH] [06 07][01 02]) ([HVH] [04 05][06 07] [01 03])1198606([LM] [01 02][03 05] [03 05]) ([VLL] [02 04] [05 06][02 03]) ([AHH] [03 04][04 05])1198607([VLL] [04 05][05 06] [02 03] [02 04]) ([MAH] [05 06] [07 08][01 02]) ([ALM] [04 05][05 06] [03 04])1198608([ALAH] [01 03][04 06]) ([AHH] [05 07][01 02] [02 03]) ([AHH] [04 06][03 04])

Table 3 Direct influence matrix Z1198601 1198602 1198603 1198604 1198605 1198606 1198607 11986081198601 0 3 4 1 3 1 2 41198602 1 0 2 4 4 2 3 11198603 4 2 0 1 3 2 4 11198604 2 3 1 0 3 4 1 21198605 4 2 1 3 0 1 4 31198606 2 4 3 4 4 0 4 21198607 4 1 2 3 3 1 0 41198608 1 2 4 3 3 2 4 0

Step 4 Derive order inducing variable 120576119894119895 by descendingorder of 119888119894119895 The greater the value 119888119894119895 the bigger the 1205761198941198956 Illustrative Example and Experiments

61 Illustrative Case Study Suppose a mega infrastructureproject is selecting green contractor for one of its subprojectsA panel of decision-makers including managers professorsand competitive intelligence experts has been organized tocomprehensively evaluate three alternative contractors (ie1199091 1199092 and 1199093) under the eight attributes (ie 1198601ndash1198608) listedin Section 2

To elicit true decision preferences we apply theIVDHF UUBLS to assess the three alternatives (1199091ndash1199093)Firstly decision-makers are asked to vote on 119909119895 under eachattribute119860 119894 using different unbalanced linguistic term sets 1198781 1198782 and 1198783 where 1198781 = 119873 119871119860119871119872119860119867119867119876119867119881119867119860119879 1198791198782 = 119873119860119873119881119871 119876119871 119871 119860119871119872119860119867119881119867 119879 1198783 = 119873119881119871119871119860119872119872119876119872119860119867119867 119879 In Figure 4 we show the adoptedlinguistic hierarchies and relationship between 1198781 1198782 and1198783 Attributes 1198601 1198602 1198605 and 1198608 are evaluated by use of 1198781attributes11986031198604 and1198607 are evaluated by 1198782 and attribute1198606is evaluated by 1198783 The highest-voted uncertain unbalancedlinguistic term [119904119898 119904119899] are identified Secondly the panel of

decision-makers is asked to further express their opinionson membership degrees ℎ(119909) and nonmembership degrees119892(119909) to [119904119898 119904119899] Then we can derive compound assessments(ie ⟨119909 119904120599(119909) ℎ(119909) 119892(119909)⟩) in the form of IVDHF UUBLSand obtain the decision matrix in Table 2 Additionallyin order to examine the interdependency among attributesdecision-makers are also required to assess the influentialrelations among the eight attributes using DEMATEL-basedAlgorithm II The direct influence matrix is collected andshown in Table 3

Next we apply the proposed Algorithm I to solve thisgreen contractor selection problem To derive the synthe-sized attribute weighting vector in Algorithm I we assign120572 = 120573 = 05 For scenarios with no group attitudinalcharacters operator GIVDHFUUBLHA in Definition 12 ischosen In contrast for scenarios with group attitudinalcharacters operator I-GIVDHFUUBLHA in Definition 13will be chosen Further the panel of decision-makers is askedto determine their positive and negative ideal alternativesthat are collected in Table 4 where corresponding order-inducing vectors that are derived from Algorithm III arealso listed (see the rightmost column) Thereafter operatorI-GIVDHFUUBLHA in Definition 13 is utilized for aggrega-tion For GIVDHFUUBLHA and I-GIVDHFUUBLHA we

Complexity 19

Table 4 Ideal alternatives (119903+ and 119903minus) and derived order-inducing vectors 120576119903+ 119903minus 1205761199091 1199092 11990931198601 ([VHT] 1 0) ([NL] 1 0) 5 6 61198602 ([MAH] [07 08] [01 02]) ([NL] 1 0) 1 1 41198603 (T 1 0) (L 1 0) 8 7 81198604 ([ALAH] 09 01) (T 1 0) 2 2 11198605 ([HT] [09 1] 0) (N 1 0) 7 5 51198606 (M 1 0) (N 1 0) 6 3 31198607 (M 1 0) ([NL] 1 0) 3 8 71198608 ([MAH] 1 0) (N 1 0) 4 4 2

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 1 2 3 4

0 1 2

N L AH H QH VH AT TM

Level 1 l(13)

Level 2 l(25)

Level 3 l(39)

Level 4 l(417)

Unbalanced linguistic term set S1

N AH VH TM Unbalanced linguistic term set S2VL L ALAN QL

N H TM Unbalanced linguistic term set S3

L AM AHQM

AL

VL

Figure 4 Unbalanced linguistic term sets (1198781 1198782 and 1198783) and their mapping in linguistic hierarchies

assume 120582=2 and 119908 = (0 0 015 025 025 025 01 0)119879Note that 119908 is a position weighting vector which is derivedby the fuzzy semantic quantitative operator [99] Aftercompleting all steps in Algorithm I (see Appendix C) weobtain the ranking results and present them in Table 5 Forthe scores derived from Step 5 we find the ranking order is1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the best solution Step 7gives the ranking order of 1199092 ≻ 1199091 ≻ 1199093 indicating 1199092 is stillthe most appropriate one while 1199091 is better than 1199093 under theinfluence of 12057662 Experiments on Ranking Results along with VariousParameter Configurations In order to inspect the impactsof various parameter configurations on ranking results inthis section more numerical examples are carried out Alleight operators that defined in Section 41 are applied to theabove illustrative case where the parameter 120582 is configured asintegers ranging from 1 to 20 and parameter 120572 is configuredto a value set of 0 03 05 08 09 1

Firstly the four basic aggregation operators of GIVDH-FUUBLWA GIVDHFUUBLWG GIVDHFUUBLOWA and

Table 5 Ranking results of operators GIVDHFUUBLHA and I-GIVDHFUUBLHA

Aggregation operator Ranking orderGIVDHFUUBLHA 1199092 ≻ 1199093 ≻ 1199091I-GIVDHFUUBLHA 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG are applied to the former case Thesame configuration 120572 = 120573 = 05 is chosen while 120582 is testedas integers ranging from 1 to 20 The ranking results derivedfrom different parameter combinations have been collectedin Table 6 As can be seen four operators unanimouslyidentify the contractor 1199092 is the best one no matter how120582 changes Regarding priority relation between 1199091 and 1199093GIVDHFUUBLWA and GIVDHFUUBLOWA generate thesame result of1199093 ≻ 1199091 while results fromGIVDHFUUBLWGandGIVDHFUUBLOWGgot a change to 1199091 ≻ 1199093 at a certainvalue of parameter 120582

Actually GIVDHFUUBLWA and GIVDHFUUBLOWAare generalized version of weighted arithmetic aggregationoperators that producemore favorable results by emphasizing

20 Complexity

Table 6 Ranking orders obtained by GIVDHFUUBLWA GIVDHFUUBLOWA GIVDHFUUBLWG and GIVDHFUUBLOWG along with120582Operators 120582 (Integer) Ranking orderGIVDHFUUBLWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLOWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLWG [1 2] 1199092 ≻ 1199093 ≻ 1199091[3 20] 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG [1 8] 1199092 ≻ 1199093 ≻ 1199091[9 20] 1199092 ≻ 1199091 ≻ 1199093

Table 7 Ranking results obtained by GIVDHFUUBLHA and I-GIVDHFUUBLHA along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHA I-GIVDHFUUBLHA120572=0 120573=1 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=03 120573=07 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=05 120573=05 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=08 120573=02 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=09 120573=01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=1 120573=0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093

Table 8 Ranking results obtained by GIVDHFUUBLHG and I-GIVDHFUUBLHG along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHG I-GIVDHFUUBLHG

120572 = 0 120573 = 1 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 03 120573 = 07 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 05 120573 = 05 [1 8] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[9 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 08 120573 = 02 [1 17] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[18 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 09 120573 = 01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572 = 1 120573 = 0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093membership degrees to a uncertain linguistic term thus canbe considered as optimistic operators while GIVDHFU-UBLWG and GIVDHFUUBLOWG are generalized geomet-ric aggregation operators that producemore favorable resultsby emphasizing nonmembership degrees thus can be consid-ered as pessimistic operators [100 101]Therefore the rankingresults in Table 6 could be perceived as for optimism-baseddecision-making alternative 1199093 always stands out better than1199091 when 120582 gradually amplifies degree of optimism but forpessimism-based decision-making 1199091 turns out better than1199093 when 120582 gradually amplifies degree of pessimism 120582 canbe considered as the parameter for amplifying degree ofoptimism or pessimism [101]

Next to inspect the impacts of various combinationsof 120596119860119868 and 120596119860119863 on ranking results another four aggrega-tion operators of GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG also have beenapplied to the above illustrative case with varying 120572 120573 and 120582

All results have been collected in Tables 7 and 8 for compari-son As can be seen fromTable 7 ranking results generated byoperator GIVDHFUUBLHA keep the same as 1199092 ≻ 1199093 ≻ 1199091for all test combinations of parameters However as shownin Table 8 although GIVDHFUUBLHG is also capable ofidentifying 1199092 as the best contractor the priority relationsbetween 1199091 and 1199093 hold similar changing patterns in rankingresults in Table 6 that were output byGIVDHFUUBLWGandGIVDHFUUBLOWG Regarding decision situations whereobjective attributeweighting vector120596119860119863 is only considered (120572= 0) or its dominant importance (such as 120572 = 03) is accepted1199091 ≻ 1199093 is a clear observation when parameter 120582 is set toan integer in interval [8 20] Contrariwise under decisionscenarios where subjective attribute weighting vector 120596119860119868 isonly needed (120572 = 1) or its dominant importance (such as 120572 =09) is acknowledged 1199093 ≻ 1199091 holds all the way 120582 varies

In order to more clearly show the score trajectories ofalternative contractors along variations of parameter 120582 we

Complexity 21

HAInducedHA

InducedHA-2HA-2

HA-3

InducedHA-1InducedHA-3

= =

HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(a) 120572 = 0

HAInducedHA

= 03 = 07

InducedHA-2

HA-2HA-3

InducedHA-1 InducedHA-3HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(b) 120572 = 03

HAInducedHA

= 5 = 05

InducedHA-2HA-2

HA-3

InducedHA-1 InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(c) 120572 = 05

HAInducedHA

= 08 = 02

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572 = 08

HAInducedHA

= 9 = 01

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(e) 120572 = 09

HAInducedHA

246 2462 2464 2466 2468 247

= 1 = 0InducedHA-2HA-2

HA-3

HA- 3

HA-3

InducedHA-1

InducedHA-3

HA-1

HA-1

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

1085

1086

1087

(f) 120572 = 1

Figure 5 Score trajectories obtained by aggregation operators HA and InducedHA with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

further configure parameter 120582 to the value set of [0 01100]and conduct computational experiments to collect scorevalues and ranking results (as shown in Figures 5 and 6) byfour aggregation operators of GIVDHFUUBLHA GIVDH-FUUBLHG I-GIVDHFUUBLHAand I-GIVDHFUUBLHGFor more clarity in Figures 5 and 6 please note that weuse HA HG Induced-HA and Induced HG to respec-tively denote GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG In Figure 5it can be observed that when dominant importance of 120596119860119863

is emphasized (120572 = 0 120572 = 03 120572 = 05) GIVDHFUUBLHAis capable of differentiating the three alternative contractorsas same ranking order of 1199092 ≻ 1199093 ≻ 1199091 for all 120582 but withincreases in 120572 the margins between scores of 1199091 and 1199093 nar-rows and the rank order changes to 1199091 ≻ 1199093when 120572 gets 09and 1 Interestingly in Figure 6 although GIVDHFUUBLHGis able to significantly differentiate alternative contractors 1199091and 1199093 when dominant importance of 120596119860119863 is emphasized (120572= 0 120572 = 03 120572 = 05) rank order of 1199091 and 1199093 obviouslychanges once from 1199093 ≻ 1199091 to 1199091 ≻ 1199093 when 120582 is bigger

22 Complexity

HGInducedHG

= = 1

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(a) 120572= 0

HGInducedHG

= 3 = 07

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(b) 120572= 03

HGInducedHG

= 05 = 05

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(c) 120572= 05

HGInducedHG

= 8 = 02

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572= 08

HGInducedHG

292 293 294 295 296 297minus08

minus0795

minus079

= 09 = 01

HA-1

HA-3

InducedHG-2

HG-1

HG-1

HG-2

HG-3

HG-3 InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4

minus3

minus2

minus1

0

1

2

scor

e val

ues

(e) 120572= 09

HGInducedHG

50 60 70 80 90 100minus095minus09

minus085

= 1 = 0

HG-x3

HG-x1

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(f) 120572= 1

Figure 6 Score trajectories obtained by aggregation operators HG and InducedHG with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

than 8 When 120572 increases to 1 GIVDHFUUBLHGmaintainsthe rank order of 1199093 ≻ 1199091 but their score trajectorieswalk very close Comparatively I-GIVDHFUUBLHA andI-GIVDHFUUBLHG generate consistent ranking results of1199092 ≻ 1199091 ≻ 1199093 no matter 120582 varies in all the above experiments

In sum when parameters are configured with typical val-ues (ie 120572 = 05 and 120582=1) and no additional attitudinal infor-mation is taken into consideration the proposed aggregationoperators unanimously derive the same rank order 1199092 ≻1199093 ≻ 1199091 of for the three alternative contractors However

with the amplification effect of parameter 120582 analysis resultsfrom linear combination of 120596119860119868 and 120596119860119863 reveal alternative1199092 which consistently is the best choice but rank order of1199091 and 1199093 changes because their score trajectories get closeand crossed which probably indicates that only decision-making informationderived from the decisionmatrix is inad-equate to significantly differentiate alternatives 1199091 and 1199093While by integrating additional decision information aboutattitudinal characters of group experts I-GIVDHFUUBLHAand I-GIVDHFUUBLHG manage to generate the identical

Complexity 23

Table 9 Order-inducing vectors obtained by our TOPSIS-based method for problem in [38]

1198661 1198662 1198663 11986641198601 02522 01629 02282 027141198602 01 05243 01364 046891198603 04081 01756 05244 037791198604 04507 01409 01064 013221198605 00697 02203 03753 01161

Table 10 Order-inducing vectors obtained by our TOPSIS-based method for problem in [39]

1198661 1198662 1198663 11986641198601 04548 04969 04923 047321198602 05092 047 04768 046451198603 07192 04888 04953 049151198604 06722 06163 05706 049291198605 06104 04942 06463 04946

Table 11 Ranking results obtained by applying different order-inducing vectors to [38 39]

Problems Approaches Ranking results

Problem in [38] IHFLOWA [38] + order-inducing vectors in [38] 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198605 ≻ 1198604

IHFLOWA [38] + order-inducing vectors in Table 9 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198604 ≻ 1198605

Problem in [39] I-IVDHFLOWA [39] + order-inducing vectors in [39] 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198602 ≻ 1198601

I-IVDHFLOWA [39] + order-inducing vectors in Table 10 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602

permutation of 1199092 ≻ 1199091 ≻ 1199093 which verifies that ourproposed TOPSIS-based order-inducing approach is prac-tical and useful in helping work out decision results morereasonably and robustly

63 Application Studies of Proposed TOPSIS-Based Order-Inducing Method When confronting complex decision-making situations that need additional attitudinal charactersof decision-makers Yager and Filev [88] and Yager [102]introduced and encouraged to employ order-inducing vectorto represent those attitudinal characters Since then theidea of order-inducing vector has been extended to variousdecision-making environments [98 103] such as those basedon intuitionistic fuzzy information [104 105] linguistic infor-mation [106] hesitant fuzzy linguistic information [38] andinterval-valued dual hesitant fuzzy uncertain linguistic infor-mation [39] Generally in existing literature order-inducingvectors rely on experts to directly give [38 39 104ndash106]

In order to examine the adaptability of our TOPSIS-based order-inducing method to linguistic decision-makingenvironments we respectively apply the method to resolvethe same problems in references [38 39] Due to fact thatthere is no descriptions about how the order-inducing vectorswere derived in both references [38 39] we here applyextremums into our TOPSIS-based method to obtain order-inducing vectors that is (1199047 1 0) and (1199041 0 1) for[38] ([1199046 1199046] 1 0) and ([1199040 1199040] 0 1) for [39] Thenwe obtain our order-inducing vectors listed in Table 9 forthe problem adopted in [38] and our order-inducing vectorslisted in Table 10 for [39]

Now we apply the order-inducing vectors as shownin Tables 9 and 10 to resolve the problems in [38] and[39] respectively Regarding the problem in [38] the scorevalues of all alternatives are calculated as 119878(1198601)=02602119878(1198602)=02417 119878(1198603)=03577 119878(1198604)=0221 and 119878(1198605)=0185so the corresponding ranking result is 1198603 ≻ 1198601 ≻ 1198602 ≻1198604 ≻ 1198605 As for the problem in [39] the score values ofall alternatives are computed as 119878(1198601)=39954 119878(1198602)=3783119878(1198603)=70827 119878(1198604)=71792 and 119878(1198605)=5813 we thus obtainthe ranking result 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602 Formore clarity brief descriptions of comparative approachesand obtained ranking results also have been put together inTable 11

According to Table 11 the approaches that employ ourorder-inducing vectors derive the same ranking order ofthe better three alternatives for both comparative problemsfrom [38 39] As seen our proposed TOPSIS-based order-inducing method exhibits effective and more understand-able Furthermore when decision-makers are incapable ofdirectly put forward all desired order-inducing vectors due tocomplexity in ill-structured problems our method providesa feasible directions to obtain order-inducing vectors byfiguring out ideal solutions Comparatively our proposedTOPSIS-based order-inducing method behaves effective andmore operational in practical use

64 Comparison with Decision-Making Approach Based onDegraded form of IVDHFULS Note that our approachis intrinsically adaptive to decision situations based ondegraded forms of IVDHF UUBLS such as the dual hesitant

24 Complexity

fuzzy linguistic sets introduced by Yang and Ju [54] Tofurther inspect feasibility of our approach we here conductcomparative study on the same case adopted by Yang and Ju[54] We firstly convert their original decision matrix intothe form of IVDHF UUBLS crisp membership and non-membership degrees are equivalently changed to intervalswith same upper and lower limits (eg 02 turns out to be[02 02]) and linguistic variables are rewritten as uncertainlinguistic variables (eg 1199042 turns out to be [11990421199042])

Since the case only got three attributes among whichinterdependency could be not significant under generalassumption weighting vector 120596119860119868 is not calculated Andfor straight comparison order inducing vector is notincluded either Then the argument-dependent attributeweighting vector can be obtained from Eq(29) as 120596119860119863 =(03192 03132 03676)119879 By use of Algorithm Iwith operatorIVDHFUUBLWA (ie GIVDHFUUBLWA in case of 120582 = 1see Appendix B) we have the scores of alternatives 119904(1198601) =03412 119904(1198602) = 01354 119904(1198603) = 03932 119904(1198604) = 07352Therefore the ordering of alternatives is1198604 ≻ 1198603 ≻ 1198601 ≻ 1198602which is mainly in alignment with the ordering of1198604 ≻ 1198601 ≻1198603 ≻ 1198602 in Yang and Ju [54] It is worth noticing that 120596119860119863used in our approach is derived objectively from assessmentswhile attribute weights 120596lowast = (035 025 040)119879used by Yangand Ju [54] were assigned empirically Differences between120596119860119863 and 120596lowast lead to different ordering of1198601 and1198603 Compar-atively speaking our approach holds more adaptability andobjectivity

In general our proposed MADM approach in this paperexhibits an effective way to tackle complex green contractorselection problems in which subjective attributes weightingvector (such as120596119860119868) and objective attributes weighting vector(such as 120596119860119863) exist To comprehensively utilize these twofundamental facets of attributes weighting information ourcase study and computational experiments suggest that linearcombination method (ie 120596 = 120572120596119860119868 + 120573120596119860119863) is effectiveto reflect effects of 120596119860119868 and 120596119860119863 by configuring parameter 120572or 120573 When there is no specific expert opinions to specify orhard to obtain we suggest that parameter 120572 chooses the valueset of 0 02 05 08 1 Furthermore generalized aggrega-tion operators are suggested to inspect score trajectoriesof alternative contractors along increasing of the operatorsrsquoamplification parameter 120582 If some alternative contractorsrsquoscore values get very close or crossed so that they cannot bedifferentiated from each other significantly we suggest to askthe panel of experts to derive opinions of ideal solutions andutilize our TOPSIS-based order-inducing method to includethose expert opinions as attitudinal decision informationthereby obtaining more rational and robust decision results

7 Conclusions

Aiming at tackling complex decision-making problems ofgreen contractor selection that often hold simultaneouslythe characteristics of decision hesitancy attributes inter-dependency and group complex attitudinal characters wehave developed an effective IVDHF UUBLS-based approach

We utilize IVDHF UUBLS to elicit complicate decision-makersrsquo assessments more objectively and comprehensivelywe employ DEMATEL for addressing attribute interdepen-dency more precisely and we determine group attitudinalcharacters as order-inducing vectors based on ideal attitudesof experts rather than by empirical assignment The mainadvantages of our approach are (i) IVDHF UUBLS offers anadequate way to depict decision hesitancy (ii) the designeddistance measure for IVDHF UUBLS avoids informationdistortion of traditional ones which artificially add mis-matching membership or nonmembership degrees (iii) thedeveloped generalized operators include traditional ones asspecial cases so as to provide flexibility in MADM basedon IVDHF UUBLS (iv) the devised synthesized attributeweighting scheme provides an adaptive way to consider twonecessary aspects attributes interdependency-based weightsand argument-dependent weights (v) the TOPSIS-basedmethod can deduce reasonable order-inducing vectors ratherthan empirical ones and holds generality in other linguisticdecision environments Results of case studies and experi-ments have verified the validity of the proposed approach

When multiple decision organizations are involvedgroup decision-making based on IVDHF UUBLS is animportant research direction Besides inducing preferencesfor decision-making problems of high uncertainty interactivemechanisms should also be studied to extract preferences Inaddition empirical studies of the proposed approach can becarried out to various areas such as investment evaluationproject management and vehicle selection in fleet operations

Appendix

A Basic Notions for IVDHFS and UnbalancedLinguistic Term Set

Definition A1 (see [47]) Letting 119883 be a fixed set then anIVDHFS on 119883 is defined as119863 = ⟨119909 ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (A1)

where ℎ(119909) = ⋃[120583119871120583119880]isinℎ(119909)120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and119892(119909) = ⋃[]119871]119880]isin119892(119909)] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two setsof interval values in [0 1] denoting possible membershipand non-membership degrees of element 119909 isin 119883 to theset 119863 respectively with conditions 120583 ] isin [0 1] and0 le (120583119880)+ + (]119880)+ le 1 and for all 119909 isin 119883 (120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880

For convenience normally 119889 = ℎ 119892 is called an interval-valued dual hesitant fuzzy element (IVDHFE) and 119863 is theset of all IVDHFEs

Definition A2 (see [107]) Suppose 119878 = 119904119895 | 119895 = 0 1 119892 minus1 is a finite and totally ordered discrete linguistic term setwhere 119904119895 represent possible values for a linguistic variableand 119892 is an odd cardinality 119878 is uniformly and symmetricallydistributed if the following two conditions are satisfied (1)

Complexity 25

there exists a unique constant 120582 gt 0 such that 119873119878(119904119894) minus119873119878(119904119895) = 119897(119894 minus 119895) for all 119894 119895 = 0 1 119892 minus 1 (2) Let 119878119877 =119904 | 119904 isin 119878 119904 gt 119878119862 and 119878119871 = 119904 | 119904 isin 119878 119904 lt 119878119862 Let C(119878119877) andC(119878119871) be the cardinality of 119878119877 and 119878119871 then C(119878119877) = C(119878119871)If 119878 is uniformly and symmetrically distributed then 119878 iscalled a balanced linguistic term set Otherwise 119878 is calledan unbalanced linguistic term set

The following 2-tuple fuzzy linguistic representationmodel extends traditional linguistic term set to a continuouscase so as to facilitate computing with linguistic variables

Definition A3 (see [59]) Let 119878 = 1199040 1199041 119904119892minus1 be alinguistic term set and 120573 isin [0 119892] Then a 2-tuple fuzzylinguistic variable expresses the equivalent information to 120573is defined as

Δ [0 119892] 997888rarr 119878 times [minus05 05) (A2)

Δ (120573) = (119904119894 120572) with

119904119894 119894 = 119903119900119906119899119889 (120573)120572 = 120573 minus 119894 120572 isin [minus05 05)

(A3)

Δminus1 (119904119894 120572) = 119894 + 120572 = 120573 (A4)

where 119903119900119906119899119889() is the usual rounding operation and120572 is calledsymbolic translation

To obtain 2-tuple fuzzy linguistic representations ofunbalanced linguistic terms the concept of linguistic hierar-chies ie 119871119867 = ⋃119905 119897(119905 119899(119905)) is used 119897(119905 119899(119905)) is a linguistichierarchy with 119905 indicating the level of hierarchy and 119899(119905)denotes the granularity of the linguistic term set of 119905 Herreraet al [59] defined the following transformation functionsbetween labels from different levels in multigranular linguis-tic information contexts without loss of information

Definition A4 (see [59]) In linguistic hierarchies 119871119867 =⋃119905 119897(119905 119899(119905)) whose linguistic term sets are represented by119878119899(119905) = 119904119899(119905)0 119904119899(119905)119899(119905)minus1 the transformation function froma linguistic label in level 119905 to a label in consecutive level 1199051015840 isdefined as 1198791198651199051199051015840 119897(119905 119899(119905)) 997888rarr 119897(1199051015840 119899(1199051015840)) such that

1198791198651199051199051015840 (119904119899(119905)119894 120572119899(119905))= Δ 1199051015840 (Δminus1119905 (119904119899(119905)119894 120572119899(119905)) (119899 (1199051015840) minus 1)119899 (119905) minus 1 ) (A5)

By use of the above transformation function any 2-tuplelinguistic representation can be transformed into a term in119871119867 Detailed transformation procedures are listed as follows

(1) Representation in linguistic hierarchy To transformthe unbalanced terms of 119878 into the corresponding terms inthe 119871119867 transformation function 120595 is employed to associateeach unbalanced linguistic 2-tuple (119904119894 120572) with its linguistic 2-tuple in 119871119867(119878) ie

120595 119878 997888rarr 119871119867(119878) (A6)

so that 120595(119904119894 120572) = (119904119866(119894)119868(119894) 120582) for forall(119904119894 120572) isin 119878(2) Computational phase Firstly transform (119904119866(119894)119868(119894) 120582) intolinguistic 2-tuples denoted as (119904119899(1199051015840)

1198681015840(119894) 1205821015840) in 119878119899(1199051015840) where

(119904119899(1199051015840)1198681015840(119894)

1205821015840) = 119879119865 (119904119866(119894)119868(119894) 120582)= Δ(Δminus1 (119904119866(119894)119868(119894) 120582) sdot (119899 (1199051015840) minus 1)119866 (119894) minus 1 ) (A7)

Then a computational model is used over 119878119899(1199051015840) with aresult denoted as (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)(3) Retranslation process The result (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)is transformed into the unbalanced term in 119878 by using thetransformation function 120595minus1 ie

120595minus1 119871119867 (119878) 997888rarr 119878 (A8)

so that 120595minus1(119904119899(1199051015840)119903 120582119903) = (119904119903119890119904119906119897119905 120582119903119890119904119906119897119905) isin 119878B Details of Theorem 14

Proof Given a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector for119904119889119895 and 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 120582 gt 0 then(1) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLWA and GIVDHFUUBLWG operators respectively(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLOWA and GIVDHFUUBLOWG operators respectively

(2) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to theIVDHF UUBL weighted average (IVDHFUUBLWA)and IVDHF UUBL weighted geometric (IVDHFUUBLWG)operators respectively where

26 Complexity

119868119881119863119867119865119880119880119861119871119882119860120596 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

120596119895119904119889119895= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus

119899prod119895=1

(1 minus 120583119871119895 )120596119895 1 minus 119899prod119895=1

(1 minus 120583119880119895 )120596119895]] [[

119899prod119895=1

(]119871119895)120596119895 119899prod119895=1

(]119880119895 )120596119895]])

(B1)

119868119881119863119867119865119880119880119861119871119882119866120596 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889119895120596119895

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[

119899prod119895=1

(120583119871119895 )120596119895 119899prod119895=1

(120583119880119895 )120596119895]] [[1 minus

119899prod119895=1

(1 minus ]119871119895)120596119895 1 minus 119899prod119895=1

(1 minus ]119880119895 )120596119895]])

(B2)

(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHA andGIVDHFUUBLHG operators reduce to the IVDHF UUBL

orderedweighted average (IVDHFUUBLOWA) and IVDHFUUBL ordered weighted geometric (IVDHFUUBLOWG)operators respectively where

119868119881119863119867119865119880119880119861119871119874119882119860119908 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

119908119895119904119889120590(119895)= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) )))

119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) )))

]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus

119899prod119895=1

(1 minus 120583119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus 120583119880120590(119895))119908119895]] [[

119899prod119895=1

(]119871120590(119895))119908119895 119899prod119895=1

(]119880120590(119895))119908119895]])

(B3)

119868119881119863119867119865119880119880119861119871119874119882119866119908 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889120590(119895)119908119895

= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[

119899prod119895=1

(120583119871120590(119895))119908119895 119899prod119895=1

(120583119880120590(119895))119908119895]] [[1 minus

119899prod119895=1

(1 minus ]119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus ]119880120590(119895))119908119895]])

(B4)

Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

References

[1] V W Y Tam and C M Tam ldquoWaste reduction throughincentivesA case studyrdquoBuilding Research and Information vol36 no 1 pp 37ndash43 2008

[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

[11] X Qi C Liang and J Zhang ldquoGeneralized cross-entropy basedgroup decision making with unknown expert and attributeweights under interval-valued intuitionistic fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 79 pp 52ndash642015

[12] H Zimmermann Fuzzy Set Theorymdashand Its ApplicationsKluwer Academic Boston Mass USA 2001

[13] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 pp 199ndash249 1975

[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

[16] B T Nuong K-W Kim L Prathumratana et al ldquoSustainabledevelopment in the mining sector and its evaluation usingfuzzy AHP (Analytic Hierarchy Process) approachrdquo GeosystemEngineering vol 14 no 1 pp 43ndash50 2011

[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

[18] E K Zavadskas Z Turskis and J Antucheviciene ldquoSelectinga contractor by using a novel method for multiple attributeanalysis Weighted aggregated sum product assessment withgrey values (WASPAS-G)rdquo Studies in Informatics and Controlvol 24 no 2 pp 141ndash150 2015

[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

[21] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

[25] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009

[26] Z Hatush and M Skitmore ldquoContractor Selection UsingMulticriteria Utility Theory An Additive Modelrdquo Building andEnvironment vol 33 no 2-3 pp 105ndash115 1998

[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

[30] AM Anvuur andMM Kumaraswamy ldquoConceptualmodel ofpartnering and alliancingrdquo Journal of Construction Engineeringand Management vol 133 no 3 pp 225ndash234 2007

[31] A H I Lee H Kang C F Hsu and H Hung ldquoA green supplierselection model for high-tech industryrdquo Expert Systems withApplications vol 36 no 4 pp 7917ndash7927 2009

[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

[33] P Humphreys R McIvor and F Chan ldquoUsing case-basedreasoning to evaluate supplier environmental managementperformancerdquo Expert Systems with Applications vol 25 no 2pp 141ndash153 2003

[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

[45] Bin Zhu Zeshui Xu and Meimei Xia ldquoDual Hesitant FuzzySetsrdquo Journal of Applied Mathematics vol 2012 Article ID879629 13 pages 2012

[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

[51] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[52] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

[55] Z Tao X Liu H Chen and L Zhou ldquoUsing new versionof extended t-norms and s-norms for aggregating intervallinguistic labelsrdquo IEEETransactions on SystemsManCyberneticsSystems vol vol p 15 2016

[56] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[57] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[58] L Martınez M Espinilla J Liu L G Perez and P J SanchezldquoAn evaluation model with unbalanced linguistic informationapplied to olive oil sensory evaluationrdquo Journal of Multiple-Valued Logic and Soft Computing vol 15 no 2-3 pp 229ndash2512009

[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

[62] L Marin A Valls D Isern A Moreno and J M MerigoldquoInduced unbalanced linguistic ordered weighted average andits application in multiperson decision makingrdquo The ScientificWorld Journal vol 2014 2014

[63] A Baykasoglu V Kaplanoglu Z D U Durmusoglu and CSahin ldquoIntegrating fuzzy DEMATEL and fuzzy hierarchicalTOPSIS methods for truck selectionrdquo Expert Systems withApplications vol 40 no 3 pp 899ndash907 2013

[64] S K Patil and R Kant ldquoA hybrid approach based on fuzzyDEMATEL and FMCDM to predict success of knowledge man-agement adoption in supply chainrdquoApplied Soft Computing vol18 pp 126ndash135 2014

[65] S Liyin and Y Hong ldquoImproving environmental performanceby means of empowerment of contractorsrdquo Management ofEnvironmental Quality An International Journal vol 17 no 3pp 242ndash257 2006

[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

[69] Y Kazancoglu I Kazancoglu and M Sagnak ldquoA new holisticconceptual framework for green supply chain management

performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

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Complexity 11

⋃[120583119871120590(119895) 120583

119880120590(119895)]isinℎ120590(119895) []

119871120590(119895) ]119880120590(119895)]isin119892120590(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120590(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120590(119895))120582)119908119895)1120582]])

(11)

where 119904119889120590(119895) is the 119895th largest and

119904119889119895 = 119899120596119895119904119889119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904119899120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904119899120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

] ⋃

[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[1 minus (1 minus 120583119871119895 )119899120596119895 1 minus (1 minus 120583119880119895 )119899120596119895] [(]119871119895)119899120596119895 (]119880119895 )119899120596119895]) (12)

(2) Generalized IVDHFUUBL Hybrid Geometric(GIVDHFUUBLHG) Operator

119866119868119881119863119867119865119880119880119861119871119867119866119908120596120582 (1199041198891 1199041198892 119904119889119899) = 1120582 ( 119899⨂119895=1

(120582119904119889120590(119895))119908119895)= ⋃(119904120599120590(119895)

ℎ120590(119895) 119892120590(119895))isin119904119889120590(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120590(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895)

))))119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895) []119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120590(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120590(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120590(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120590(119895))120582)119908119895)1120582]])

(13)

where 119904119889120590(119895) is the 119895th largest and

119904119889119895 = 119904119889119895119899120596119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(Δminus11199050 (119879119865

11990511989511199050(120595(119904120572119895 ))))

119899120596119895 119904(Δminus11199050 (119879119865

11990511989521199050(120595(119904120573119895 ))))

119899120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[(120583119871119895 )119899120596119895 (120583119880119895 )119899120596119895] [1 minus (1 minus ]119871119895 )119899120596119895 1 minus (1 minus ]119880119895 )119899120596119895]) (14)

12 Complexity

When confronted with ill-structured situations wheredecision-makerrsquos complex attitudinal characters need to beincluded order-inducing vectors provide an effective way[88ndash90]Thus we further define following induced operatorsfor IVDHF UUBLNs

Definition 13 For a collection of IVDHF UUBLNs 119904119889119895 (119895 =1 119899) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector

120596119895 isin [0 1] sum119899119895=1 120596119895 = 1 119899 is a balancing coefficient119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weight-ing vector 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1 120576119895 denote a set oforder inducing vectors

(1) Induced Generalized IVDHFUUBL Hybrid Average (I-GIVDHFUUBLHA) Operator

119868-119866119868119881119863119867119865119880119880119861119871119867119860120596119908120582 (lt 1205761 1199041198891 gt lt 1205762 1199041198892 gt lt 120576119899 119904119889119899 gt) = ( 119899⨁119895=1

(119908119895119904119889120587(119895)120582))1120582

= ⋃(119904120599120587(119895) ℎ120587(119895) 119892120587(119895))isin119904119889120587(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120587(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587(119895)

))))120582)1120582]

⋃[120583119871120587(119895) 120583

119880120587(119895)]isinℎ120587(119895) []

119871120587(119895) ]119880120587(119895)]isin119892120587(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120587(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120587(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120587(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120587(119895))120582)119908119895)1120582]])

(15)

where (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) is decreasing order of (11990411988911199041198892 119904119889119899) according to the order inducing vector 120576119895 and119904119889119895 = 119899120596119895119904119889119895 = ⋃

(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904119899120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904119899120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

] ⋃

[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[1 minus (1 minus 120583119871119895 )119899120596119895 1 minus (1 minus 120583119880119895 )119899120596119895] [(]119871119895)119899120596119895 (]119880119895 )119899120596119895]) (16)

(2) Induced Generalized IVDHFUUBL Hybrid Geometric(I-GIVDHFUUBLHG) Operator

119868-119866119868119881119863119867119865119880119880119861119871119867119866119908120596120582 (lt 1205761 1199041198891 gt lt 1205762 1199041198892 gt lt 120576119899 119904119889119899 gt) = 1120582 ( 119899⨂119895=1

(120582119904119889120587(119895))119908119895)= ⋃(119904120599120587(119895)

ℎ120587(119895) 119892120587(119895))isin119904119889120587(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120587(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587(119895)

))))119908119895]

⋃[120583119871120587(119895)120583119880120587(119895)]isinℎ120587(119895) []119871120587(119895) ]

119880120587(119895)]isin119892120587(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120587(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120587(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120587(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120587(119895))120582)119908119895)1120582]])

(17)

Complexity 13

Weighted Aggregation

Geometric Arithmetic

GIVDHFUUBLHG GIVDHFUUBLHA

GIVDHFUUBLWG GIVDHFUUBLOWG

IVDHFUUBLWG IVDHFUUBLOWG

GIVDHFUUBLWA GIVDHFUUBLOWA

IVDHFUUBLWA IVDHFUUBLOWA

= 1 w =1

n = 1 =

1

n = 1 =1

n = 1 =

1

n

= 1 = 1 = 1 = 1

Figure 2

where (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) is decreasing order of (11990411988911199041198892 119904119889119899) according to the order inducing variables 120576119895 and

119904119889119895 = 119904119889119895119899120596119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(Δminus11199050 (119879119865

11990511989511199050(120595(119904120572119895 ))))

119899120596119895 119904(Δminus11199050 (119879119865

11990511989521199050(120595(119904120573119895 ))))

119899120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[(120583119871119895 )119899120596119895 (120583119880119895 )119899120596119895] [1 minus (1 minus ]119871119895 )119899120596119895 1 minus (1 minus ]119880119895 )119899120596119895]) (18)

42 Properties of the Proposed GeneralizedAggregation Operators

eorem 14 With special values of 120582 119908 and 120596 the operatorsGIVDHFUUBLHA and GIVDHFUUBLHG can include aseries of traditional aggregation operators as special cases andtheir relationship can be depicted in Figure 2

Proof See Appendix B

As for the induced hybrid aggregation operators we alsohave following theorem

eorem 15 If (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) = (119904119889120590(1) 119904119889120590(2) 119904119889120590(119899)) I-GIVDHFUUBLHA reduces toGIVDHFUUBLHAIf (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) = (119904119889120590(1) 119904119889120590(2) 119904119889120590(119899)) thenI-GIVDHFUUBLHG reduces to the GIVDHFUUBLHG oper-ator

eorem 16 All the proposed generalized operatorsGIVDHFUUBLWAGIVDHFUUBLWGGIVDHFUUBLOWAGIVDHFUUBLOWGGIVDHFUUBLHAGIVDHFUUBLHGI-GIVDHFUUBLHA and I-GIVDHFUUBLHG hold thefollowing properties (1) Commutativity (2) Idempotency (3)Boundedness

Based on above theorems following properties can alsobe derived

eorem 17 For a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weight vector of 119904119889119895with 120596119895 isin [0 1] and sum119899119895=1 120596119895 = 1 120582 gt 0 then we have

(1) 119868119881119863119867119865119880119880119861119871119882119866120596(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119882119860120596(1199041198891 1199041198892 119904119889119899)(2) 119868119881119863119867119865119880119880119861119871119882119866120596(1199041198891 1199041198892 119904119889119899) le119866119868119881119863119867119865119880119880119861119871119882119860120596120582(1199041198891 1199041198892 119904119889119899)(3) 119866119868119881119863119867119865119880119880119861119871119882119866120596120582(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119882119860120596(1199041198891 1199041198892 119904119889119899)

Proof Omitted for concision

eorem 18 For a collection of IVDHF UUBLNs 119904119889119895 = (119904120599119895 ℎ119895 119892119895) 119908 = (1199081 1199082 119908119899)119879 is the weighting vector of 119904119889119895with 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1 120582 gt 0 Then

(1) 119868119881119863119867119865119880119880119861119871119874119882119866119908(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119874119882119860119908(1199041198891 1199041198892 119904119889119899)(2) 119868119881119863119867119865119880119880119861119871119874119882119866119908(1199041198891 1199041198892 119904119889119899) le119866119868119881119863119867119865119880119880119861119871119874119882119860119908120582(1199041198891 1199041198892 119904119889119899)(3) 119866119868119881119863119867119865119880119880119861119871119874119882119866119908120582(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119874119882119860119908(1199041198891 1199041198892 119904119889119899)

Proof Omitted for concision

14 Complexity

5 An Integrated MADM Approach forTackling Complex Green ContractorSelection Problems

As described in Section 2 we take the green contractorselection as a special type of complicate MADM problemsthat synthesizes three characteristics of decision hesitancy[21 22] attributes interdependency [84] and group atti-tudinal characters [61] Therefore in this section we con-struct an integrated MADM approach to tackle the com-plex green contractor selection problems Suppose 119883 =1199091 1199092 119909119898 is the set of alternative green contractors and119860 = 1198601 1198602 119860119899 is the set of evaluative attributes 120596 =(1205961 1205962 120596119899)119879 is weighting vector for the attributes 120596119894 ge 0sum119899119894=1 120596119894 = 1 Let = (119903119894119895)119899times119898 denote the decision matrix inwhich 119903119894119895 = ([119904120572119894119895 119904120573119894119895] ℎ119894119895 119892119894119895) is an IVDHF UUBLN givenby decision-makers for alternative contractor 119909119895 with respectto attribute 119860 119894 According to the mechanism of pair-wisecomparisons among attributes in the DEMATELmethod [6364] the interdependency among attributes can be obtained asamatrix119885 = (119911119894119896)119899times119899 where 119911119894119896 indicates the degree towhich119860 119894 affects119860119896 Subsequently based on the IVDHFUUBLS andits operations we now present detailed steps of our MADMapproach as shown in following Algorithm I

Algorithm I Hesitant fuzzy linguistic MADMwith attributesinterdependency and decision-makersrsquo group attitudinalcharacters

Step 1 Determine argument-dependent weighting vector120596119860119863 according to attribute values by programming modeldeveloped in the following Section 51

Step 2 Obtain the attribute-interdependences based weight-ing vector 120596119860119868 by use of DEMATEL method described in thefollowing Section 52

Step 3 Calculate synthesized attribute weighting vectoraccording to

120596 = 120572120596119860119868 + 120573120596119860119863 (19)

where 120572 and 120573 are parameters to reflect decision character-istics of decision organizations 0 le 120572 120573 le 1 120572 + 120573 =1Step 4 Check requirements for order inducing If no addi-tional order inducing required then go to Step 5 otherwisego to Step 6

Step 5 Utilize generalized aggregation operators to get theoverall IVDHF UUBLNs 119903119895(119895 = 1 119898) for each alternative119909119895 Here we take GIVDHFUUBLHA operator for examplebecause it can include other traditional operators as its specialcases Therefore we have

119866119868119881119863119867119865119880119880119861119871119867119860(1199031119895 1199032119895 119903119899119895) = ( 119899⨁119894=1

(119908119894 (119903120590119894119895)120582))1120582= ⋃(119904120590119894119895ℎ120590119894119895 119892120590119894119895

)isin119903120590119894119895

([119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590119894119895

))))120582)1120582 119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590119894119895

))))120582)1120582]

⋃[120583119871120590119894119895120583

119880120590119894119895]isinℎ120590119894119895[]

119871120590119894119895]119880120590119894119895]isin119892120590119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120590119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120590119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120590119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120590119894119895)120582)119908119894)1120582]])

(20)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of 119903119894119895119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weightvector 119903120590119894119895 is the 119894th largest of 119903119894119895 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))

119904119899120596119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119894119895))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (21)

Complexity 15

Then go to Step 8

Step 6 By use of the TOPSIS-based method developed in thefollowing Section 53 we determine order-inducing vectors 120576to reflect decision-makersrsquo group attitudinal characters

Step 7 Utilizing induced generalized aggregation operatorsto get the overall IVDHF UUBLNs 119903119895(119895 = 1 2 119898) foreach alternative 119909119895 we have

119868-119866119868119881119863119867119865119880119871119867119860(1199031119895 1199032119895 119903119899119898) = ( 119899⨁119894=1

(119908119894 (119903120587119894119895)120582))1120582

= ⋃(119904120587119894119895ℎ120587119894119895119892120587119894119895)isin119903120587119894119895

([119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120587119894119895

))))120582)1120582 119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587119894119895

))))120582)1120582]

⋃[120583119871120587119894119895 120583

119880120587119894119895]isinℎ120587119894119895 []

119871120587119894119895]119880120587119894119895]isin119892120587119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120587119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120587119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120587119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120587119894119895)120582)119908119894)1120582]])

(22)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of119903119894119895 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 119903120587119894119895 is the 119894th largest of 119903119894119895 reordered by theorder-inducing vectors 120576 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))119899120596119894120572119894119895

119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120573119894119895 ))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (23)

Step 8 Calculate scores 119878(119903119895) and accuracy degrees 119875(119903119895) for119903119895(119895 = 1 2 119898)Step 9 According to Definition 5 output ranking order ofalternative contractors

In Algorithm I to objectively derive unknown weight-ing vector for evaluative attributes a synthesized attributeweighting vector is devised by fusing two parts of concern(i) argument-dependent weighting vector and (ii) attribute-interdependences based weighting vector The former is toreflect effects of attribute values on attribute weights whilethe latter is to exploit effects of interdependences amongattributes on their weights For clarity processing flowchartof Algorithm I is depicted in Figure 3

51 Determining Argument-Dependent Attribute WeightingVector 120596119860119863 In order to derive attribute weighting vec-tor appropriately from assessments maximizing deviationmethod [91] is adopted here for its distinguishing ability

and objectivity [92] According to Wang [91] if performancevalues of each alternative have little differences under anattribute it implicates such an attribute plays a less importantrole otherwise it lives a more important role in decision-making Thus if one attribute has similar values across alter-natives it should be assigned a smaller weight contrariwiseit should be a bigger weight

In light of the idea in maximizing deviation methodwe develop a programming model to determine argument-dependent weighting vector 120596119860119863 For attribute 119860 119894 the stan-dard deviation of alternative 119909119895 to all the other alternatives isdenoted as

119865119894119895 (120596119860119863) = 119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (24)

Let119865119894(120596119860119863) represent deviation value of all alternatives toother alternatives according to attribute119860 119894 and be formulatedas follows

16 Complexity

Attribute weighting

Hesitant fuzzy linguistic MADM with attributes interdependency and decision makersrsquo group attitudinal characters

Step 9 Output ranking order of alternatives

interdependency among attributes by DEMATEL in Section 52

attributes values by standard deviation method in Section 51

Step 6 Ensure TOPSIS order inducing variables proposed in Section 53

Step 7 Utilizing proposed induced generalized hybrid aggregation operators developed in Section

41 to get overall value of each alternative

Step 8 Calculate score degrees and accuracy degrees

Step 4 Order inducingYes

Step 5 Choose proposed generalized operators in Section 41 to get overall value of each alternative

No

Collect decision matrix in the form of IVDHF_UUBLS

Complex problems of green contractor selection

Step 1 Weighting vector AD derived from Step 2 Weighting vector AI derived from

Step 3 Obtain synthesized weighting vector of attributes =AI + AD

Figure 3 Processing flowchart of Algorithm I

119865119894 (120596119860119863) = 119898sum119895=1

119865119894119895 (120596119860119863) = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (25)

Based on the analysis above we can get followingprogramming model (M-1) for obtaining optimal attributeweighting vector 120596119860119863

(119872 minus 1)max 119865 (120596119860119863) = 119899sum

119894=1119865119894 (120596119860119863) = 119899sum

119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894st 119899sum

119894=1(120596119860119863119894 )2 = 1 120596119860119863119894 ge 0 119894 = 1 2 119899 (26)

where 119889(119903119894119895 119903119894119897) can be calculated by Definition 6To solve model (M-1) we can construct Lagrange func-

tion

119871 (120596119860119863119894 120577) = 119899sum119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894+ 1205772 ( 119899sum

119894=1

(120596119860119863119894 )2 minus 1) (27)

where 120577 is the Lagrange multiplierDifferentiate with respect to 120596119860119863119894 (119894 = 1 2 119898) and 120577

and set these partial derivatives equal to zero the followingset of equations can be obtained

120597119871120597120596119860119863119894 = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) + 120577120596119860119863119894 = 0120597119871120597120577 = 119899sum

119894=1

(120596119860119863119894 )2 minus 1 = 0 (28)

By normalizing the solutions of above equations we canget a simple and exact formula for determining the attributeweights as follows

120596119860119863119894 = sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897)sum119899119894=1sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897) (29)

52 Deriving Attributes Interdependency-Based AttributeWeighting Vector 120596119860119868 Some efforts [53 60 84 93ndash95]

Complexity 17

have been made to address interdependencies betweenattributes in MADM However they unanimously cannotreveal the interdependences in a more complete way WhileDEMATAEL technique [63 96] gathers collective knowledgeto capture influences between attributes more precisely andcompletely so that weighting vector for attributes can berationally determined according to strength measure ofthose influences [64 97] Therefore in tackling the complexMADMwith interdependent attributes under IVDHF UUBLenvironments we present a DEMATEL-based method as thefollowing Algorithm II to derive attribute weighting vector120596119860119868 from the viewpoint of attributes interdependency

Algorithm II DEMATEL-basedmethod for ensuringweight-ing vector 120596119860119868Step 1 Construct initial direct-influence matrix 119885 = (119911119894119896)119899times119899

For attributes 119860 = 1198601 1198602 119860119899 decision-makersare asked to give direct effect that attribute 119860 119894 has onattribute 119860119896 using a five-point scale with scores representedby natural language (1) 0 mdash lsquoabsolutely no influencersquo (2)1 mdash lsquolow influencersquo (3) 2 mdash lsquomedium influencersquo (4) 3 mdashlsquohigh influencersquo (5) 4mdash lsquovery high influencersquoThen the initialdirect-influence matrix 119885 = (119911119894119896)119899times119899 can be established witheach element being the mean of same factors in differentdirect matrices

Step 2 Calculate normalized direct-influence matrix 119883 =(119909119894119896)119899times119899 Using matrix 119885 the normalized direct-influencematrix 119883 = (119909119894119896)119899times119899 is calculated according to

119909119894119896 = 119911119894119896119904 (30)

where 119904 = maxmax1le119894le119899sum119899119896=1 119911119894119896max1le119896le119899sum119899119895=1 119911119894119896 Allelements in matrix 119883 are complying with 0 le 119909119894119896 le 10 le sum119899119896=1 119909119894119896 le 1 and 0 le sum119899119894=1 119909119894119896 le 1 and at least onerow or column of the summation equals 1

Step 3 Derive total-influence matrix 119879 = (119905119894119896)119899times119899 Basedon matrix 119883 the total-influence matrix 119879 = (119905119894119896)119899times119899 can bederived by summing the direct effects and all of the indirecteffects

119879 = 119883 + 1198832 + 1198833 + sdot sdot sdot + 119883ℎ = 119883 (119868 minus 119883)minus1 (31)

where 119868 represents identity matrix When ℎ 997888rarr infin 119883ℎ =(0)119899times119899Step 4 Compute influencing and influenced degrees of eachattribute The sum of rows and the sum of columns withinthe total-influence matrix 119879 are respectively expressed as thevectors 119903 and 119888

119903 = (119903119894)119899times1 = ( 119899sum119896=1

119905119894119896)119899times1

119888 = (119888119896)119899times1 = ( 119899sum

119895=1

119905119894119896)119879

1times119899

(32)

where 119903119894 denotes the sumof the 119894th row inmatrix119879 and showsthe sum of direct and indirect effects that attribute 119860 119894 hason other attributes Similarly 119888119896 shows the sum of direct andindirect effects that attribute 119860119896 has receivedStep 5 Estimate weights of attributes based on influencedegrees according to

120596119868lowast119894 = [(119903119894 + 119888119894)2 + (119903119894 minus 119888119894)2]12 (33)

Then the weighting vector 120596119860119868 can be obtained by normaliz-ing (33)

120596119860119868119894 = 120596119868lowast119894sum119899119894=1 120596119868lowast119894 (34)

53 Obtaining Order Inducing Variables 120576 by TOPSIS-BasedMethod For ill-defined MADM problems induced aggre-gation operators [89 90 98] provide an effective channelfor experts to express complex attitudinal characters Thekey step in induced aggregation operators is to obtainorder-inducing vectors 120576 which however are presumptivelyassigned in most papers [62] Therefore in this sectionaiming at deducing the order-inducing vectors 120576 rationallywe here devise a TOPSIS-based method in Algorithm IIIObviously the Algorithm III is of generality for MADMwithdifferent forms of linguistic expression

Algorithm III TOPSIS-based method for obtaining orderinducing vectors 120576Step 1 Let the decision maker define IVDHF UUBL idealsolution(s) 119903+119894 and IVDHF UUBL negative ideal solution(s)119903minus119894 corresponding to each attribute in which

119903+119894 = (119903+1198941 119903+1198942 119903+119894119901) 119903minus119894 = (119903minus1198941 119903minus1198942 119903minus119894119902) (35)

where 119901 and 119902 denotes the number of 119903+119894 and 119903minus119894 respectively

Step 2 Calculate distances from each element in decisionmatrix to ideal solution(s) and negative solution(s) respec-tively

119889+119894119895 = max119897=12119901

119889 (119903119894119895 119903+119894119897 ) 119889minus119894119895 = min

119897=12119902119889 (119903119894119895 119903minus119894119897 ) (36)

where 119889(119903119894119895 119903+119894119897 ) and 119889(119903119894119895 119903minus119894119897 ) are calculated by the distancemeasure in Definition 6 If 119901 gt 1 we choose the onewith maximum distance if 119902 gt 1 we choose the one withminimum distance

Step 3 Calculate coefficients according to

119888119894119895 = 119889minus119894119895119889+119894119895 + 119889minus119894119895 119894 = 1 2 119899 119895 = 1 2 119898 (37)

18 Complexity

Table 2 Decision matrix in the form of IVDHF UUBLS1199091 1199092 11990931198601([AHVH] [02 03][02 04] [03 04]) ([HT] [04 05][05 06] [02 03] [02 04]) ([ALAH] [01 02][01 03] [06 07])1198602([MH] [05 06][02 03]) ([AHT] [06 07][01 02] ) ([MVH] [02 03][05 06] [06 07])1198603([VLAL] [03 04][04 05] [05 06]) ([MVH] [06 07] [07 08][01 02]) ([VLL] [04 05][02 03])1198604([MAH] [02 04][05 06]) ([LAL] [03 05][02 03]) ([ALM] [06 07][01 02] [02 03])1198605([ALM] [04 05][01 02] [04 05]) ([AHH] [06 07][01 02]) ([HVH] [04 05][06 07] [01 03])1198606([LM] [01 02][03 05] [03 05]) ([VLL] [02 04] [05 06][02 03]) ([AHH] [03 04][04 05])1198607([VLL] [04 05][05 06] [02 03] [02 04]) ([MAH] [05 06] [07 08][01 02]) ([ALM] [04 05][05 06] [03 04])1198608([ALAH] [01 03][04 06]) ([AHH] [05 07][01 02] [02 03]) ([AHH] [04 06][03 04])

Table 3 Direct influence matrix Z1198601 1198602 1198603 1198604 1198605 1198606 1198607 11986081198601 0 3 4 1 3 1 2 41198602 1 0 2 4 4 2 3 11198603 4 2 0 1 3 2 4 11198604 2 3 1 0 3 4 1 21198605 4 2 1 3 0 1 4 31198606 2 4 3 4 4 0 4 21198607 4 1 2 3 3 1 0 41198608 1 2 4 3 3 2 4 0

Step 4 Derive order inducing variable 120576119894119895 by descendingorder of 119888119894119895 The greater the value 119888119894119895 the bigger the 1205761198941198956 Illustrative Example and Experiments

61 Illustrative Case Study Suppose a mega infrastructureproject is selecting green contractor for one of its subprojectsA panel of decision-makers including managers professorsand competitive intelligence experts has been organized tocomprehensively evaluate three alternative contractors (ie1199091 1199092 and 1199093) under the eight attributes (ie 1198601ndash1198608) listedin Section 2

To elicit true decision preferences we apply theIVDHF UUBLS to assess the three alternatives (1199091ndash1199093)Firstly decision-makers are asked to vote on 119909119895 under eachattribute119860 119894 using different unbalanced linguistic term sets 1198781 1198782 and 1198783 where 1198781 = 119873 119871119860119871119872119860119867119867119876119867119881119867119860119879 1198791198782 = 119873119860119873119881119871 119876119871 119871 119860119871119872119860119867119881119867 119879 1198783 = 119873119881119871119871119860119872119872119876119872119860119867119867 119879 In Figure 4 we show the adoptedlinguistic hierarchies and relationship between 1198781 1198782 and1198783 Attributes 1198601 1198602 1198605 and 1198608 are evaluated by use of 1198781attributes11986031198604 and1198607 are evaluated by 1198782 and attribute1198606is evaluated by 1198783 The highest-voted uncertain unbalancedlinguistic term [119904119898 119904119899] are identified Secondly the panel of

decision-makers is asked to further express their opinionson membership degrees ℎ(119909) and nonmembership degrees119892(119909) to [119904119898 119904119899] Then we can derive compound assessments(ie ⟨119909 119904120599(119909) ℎ(119909) 119892(119909)⟩) in the form of IVDHF UUBLSand obtain the decision matrix in Table 2 Additionallyin order to examine the interdependency among attributesdecision-makers are also required to assess the influentialrelations among the eight attributes using DEMATEL-basedAlgorithm II The direct influence matrix is collected andshown in Table 3

Next we apply the proposed Algorithm I to solve thisgreen contractor selection problem To derive the synthe-sized attribute weighting vector in Algorithm I we assign120572 = 120573 = 05 For scenarios with no group attitudinalcharacters operator GIVDHFUUBLHA in Definition 12 ischosen In contrast for scenarios with group attitudinalcharacters operator I-GIVDHFUUBLHA in Definition 13will be chosen Further the panel of decision-makers is askedto determine their positive and negative ideal alternativesthat are collected in Table 4 where corresponding order-inducing vectors that are derived from Algorithm III arealso listed (see the rightmost column) Thereafter operatorI-GIVDHFUUBLHA in Definition 13 is utilized for aggrega-tion For GIVDHFUUBLHA and I-GIVDHFUUBLHA we

Complexity 19

Table 4 Ideal alternatives (119903+ and 119903minus) and derived order-inducing vectors 120576119903+ 119903minus 1205761199091 1199092 11990931198601 ([VHT] 1 0) ([NL] 1 0) 5 6 61198602 ([MAH] [07 08] [01 02]) ([NL] 1 0) 1 1 41198603 (T 1 0) (L 1 0) 8 7 81198604 ([ALAH] 09 01) (T 1 0) 2 2 11198605 ([HT] [09 1] 0) (N 1 0) 7 5 51198606 (M 1 0) (N 1 0) 6 3 31198607 (M 1 0) ([NL] 1 0) 3 8 71198608 ([MAH] 1 0) (N 1 0) 4 4 2

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 1 2 3 4

0 1 2

N L AH H QH VH AT TM

Level 1 l(13)

Level 2 l(25)

Level 3 l(39)

Level 4 l(417)

Unbalanced linguistic term set S1

N AH VH TM Unbalanced linguistic term set S2VL L ALAN QL

N H TM Unbalanced linguistic term set S3

L AM AHQM

AL

VL

Figure 4 Unbalanced linguistic term sets (1198781 1198782 and 1198783) and their mapping in linguistic hierarchies

assume 120582=2 and 119908 = (0 0 015 025 025 025 01 0)119879Note that 119908 is a position weighting vector which is derivedby the fuzzy semantic quantitative operator [99] Aftercompleting all steps in Algorithm I (see Appendix C) weobtain the ranking results and present them in Table 5 Forthe scores derived from Step 5 we find the ranking order is1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the best solution Step 7gives the ranking order of 1199092 ≻ 1199091 ≻ 1199093 indicating 1199092 is stillthe most appropriate one while 1199091 is better than 1199093 under theinfluence of 12057662 Experiments on Ranking Results along with VariousParameter Configurations In order to inspect the impactsof various parameter configurations on ranking results inthis section more numerical examples are carried out Alleight operators that defined in Section 41 are applied to theabove illustrative case where the parameter 120582 is configured asintegers ranging from 1 to 20 and parameter 120572 is configuredto a value set of 0 03 05 08 09 1

Firstly the four basic aggregation operators of GIVDH-FUUBLWA GIVDHFUUBLWG GIVDHFUUBLOWA and

Table 5 Ranking results of operators GIVDHFUUBLHA and I-GIVDHFUUBLHA

Aggregation operator Ranking orderGIVDHFUUBLHA 1199092 ≻ 1199093 ≻ 1199091I-GIVDHFUUBLHA 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG are applied to the former case Thesame configuration 120572 = 120573 = 05 is chosen while 120582 is testedas integers ranging from 1 to 20 The ranking results derivedfrom different parameter combinations have been collectedin Table 6 As can be seen four operators unanimouslyidentify the contractor 1199092 is the best one no matter how120582 changes Regarding priority relation between 1199091 and 1199093GIVDHFUUBLWA and GIVDHFUUBLOWA generate thesame result of1199093 ≻ 1199091 while results fromGIVDHFUUBLWGandGIVDHFUUBLOWGgot a change to 1199091 ≻ 1199093 at a certainvalue of parameter 120582

Actually GIVDHFUUBLWA and GIVDHFUUBLOWAare generalized version of weighted arithmetic aggregationoperators that producemore favorable results by emphasizing

20 Complexity

Table 6 Ranking orders obtained by GIVDHFUUBLWA GIVDHFUUBLOWA GIVDHFUUBLWG and GIVDHFUUBLOWG along with120582Operators 120582 (Integer) Ranking orderGIVDHFUUBLWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLOWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLWG [1 2] 1199092 ≻ 1199093 ≻ 1199091[3 20] 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG [1 8] 1199092 ≻ 1199093 ≻ 1199091[9 20] 1199092 ≻ 1199091 ≻ 1199093

Table 7 Ranking results obtained by GIVDHFUUBLHA and I-GIVDHFUUBLHA along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHA I-GIVDHFUUBLHA120572=0 120573=1 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=03 120573=07 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=05 120573=05 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=08 120573=02 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=09 120573=01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=1 120573=0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093

Table 8 Ranking results obtained by GIVDHFUUBLHG and I-GIVDHFUUBLHG along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHG I-GIVDHFUUBLHG

120572 = 0 120573 = 1 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 03 120573 = 07 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 05 120573 = 05 [1 8] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[9 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 08 120573 = 02 [1 17] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[18 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 09 120573 = 01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572 = 1 120573 = 0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093membership degrees to a uncertain linguistic term thus canbe considered as optimistic operators while GIVDHFU-UBLWG and GIVDHFUUBLOWG are generalized geomet-ric aggregation operators that producemore favorable resultsby emphasizing nonmembership degrees thus can be consid-ered as pessimistic operators [100 101]Therefore the rankingresults in Table 6 could be perceived as for optimism-baseddecision-making alternative 1199093 always stands out better than1199091 when 120582 gradually amplifies degree of optimism but forpessimism-based decision-making 1199091 turns out better than1199093 when 120582 gradually amplifies degree of pessimism 120582 canbe considered as the parameter for amplifying degree ofoptimism or pessimism [101]

Next to inspect the impacts of various combinationsof 120596119860119868 and 120596119860119863 on ranking results another four aggrega-tion operators of GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG also have beenapplied to the above illustrative case with varying 120572 120573 and 120582

All results have been collected in Tables 7 and 8 for compari-son As can be seen fromTable 7 ranking results generated byoperator GIVDHFUUBLHA keep the same as 1199092 ≻ 1199093 ≻ 1199091for all test combinations of parameters However as shownin Table 8 although GIVDHFUUBLHG is also capable ofidentifying 1199092 as the best contractor the priority relationsbetween 1199091 and 1199093 hold similar changing patterns in rankingresults in Table 6 that were output byGIVDHFUUBLWGandGIVDHFUUBLOWG Regarding decision situations whereobjective attributeweighting vector120596119860119863 is only considered (120572= 0) or its dominant importance (such as 120572 = 03) is accepted1199091 ≻ 1199093 is a clear observation when parameter 120582 is set toan integer in interval [8 20] Contrariwise under decisionscenarios where subjective attribute weighting vector 120596119860119868 isonly needed (120572 = 1) or its dominant importance (such as 120572 =09) is acknowledged 1199093 ≻ 1199091 holds all the way 120582 varies

In order to more clearly show the score trajectories ofalternative contractors along variations of parameter 120582 we

Complexity 21

HAInducedHA

InducedHA-2HA-2

HA-3

InducedHA-1InducedHA-3

= =

HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(a) 120572 = 0

HAInducedHA

= 03 = 07

InducedHA-2

HA-2HA-3

InducedHA-1 InducedHA-3HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(b) 120572 = 03

HAInducedHA

= 5 = 05

InducedHA-2HA-2

HA-3

InducedHA-1 InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(c) 120572 = 05

HAInducedHA

= 08 = 02

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572 = 08

HAInducedHA

= 9 = 01

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(e) 120572 = 09

HAInducedHA

246 2462 2464 2466 2468 247

= 1 = 0InducedHA-2HA-2

HA-3

HA- 3

HA-3

InducedHA-1

InducedHA-3

HA-1

HA-1

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

1085

1086

1087

(f) 120572 = 1

Figure 5 Score trajectories obtained by aggregation operators HA and InducedHA with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

further configure parameter 120582 to the value set of [0 01100]and conduct computational experiments to collect scorevalues and ranking results (as shown in Figures 5 and 6) byfour aggregation operators of GIVDHFUUBLHA GIVDH-FUUBLHG I-GIVDHFUUBLHAand I-GIVDHFUUBLHGFor more clarity in Figures 5 and 6 please note that weuse HA HG Induced-HA and Induced HG to respec-tively denote GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG In Figure 5it can be observed that when dominant importance of 120596119860119863

is emphasized (120572 = 0 120572 = 03 120572 = 05) GIVDHFUUBLHAis capable of differentiating the three alternative contractorsas same ranking order of 1199092 ≻ 1199093 ≻ 1199091 for all 120582 but withincreases in 120572 the margins between scores of 1199091 and 1199093 nar-rows and the rank order changes to 1199091 ≻ 1199093when 120572 gets 09and 1 Interestingly in Figure 6 although GIVDHFUUBLHGis able to significantly differentiate alternative contractors 1199091and 1199093 when dominant importance of 120596119860119863 is emphasized (120572= 0 120572 = 03 120572 = 05) rank order of 1199091 and 1199093 obviouslychanges once from 1199093 ≻ 1199091 to 1199091 ≻ 1199093 when 120582 is bigger

22 Complexity

HGInducedHG

= = 1

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(a) 120572= 0

HGInducedHG

= 3 = 07

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(b) 120572= 03

HGInducedHG

= 05 = 05

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(c) 120572= 05

HGInducedHG

= 8 = 02

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572= 08

HGInducedHG

292 293 294 295 296 297minus08

minus0795

minus079

= 09 = 01

HA-1

HA-3

InducedHG-2

HG-1

HG-1

HG-2

HG-3

HG-3 InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4

minus3

minus2

minus1

0

1

2

scor

e val

ues

(e) 120572= 09

HGInducedHG

50 60 70 80 90 100minus095minus09

minus085

= 1 = 0

HG-x3

HG-x1

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(f) 120572= 1

Figure 6 Score trajectories obtained by aggregation operators HG and InducedHG with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

than 8 When 120572 increases to 1 GIVDHFUUBLHGmaintainsthe rank order of 1199093 ≻ 1199091 but their score trajectorieswalk very close Comparatively I-GIVDHFUUBLHA andI-GIVDHFUUBLHG generate consistent ranking results of1199092 ≻ 1199091 ≻ 1199093 no matter 120582 varies in all the above experiments

In sum when parameters are configured with typical val-ues (ie 120572 = 05 and 120582=1) and no additional attitudinal infor-mation is taken into consideration the proposed aggregationoperators unanimously derive the same rank order 1199092 ≻1199093 ≻ 1199091 of for the three alternative contractors However

with the amplification effect of parameter 120582 analysis resultsfrom linear combination of 120596119860119868 and 120596119860119863 reveal alternative1199092 which consistently is the best choice but rank order of1199091 and 1199093 changes because their score trajectories get closeand crossed which probably indicates that only decision-making informationderived from the decisionmatrix is inad-equate to significantly differentiate alternatives 1199091 and 1199093While by integrating additional decision information aboutattitudinal characters of group experts I-GIVDHFUUBLHAand I-GIVDHFUUBLHG manage to generate the identical

Complexity 23

Table 9 Order-inducing vectors obtained by our TOPSIS-based method for problem in [38]

1198661 1198662 1198663 11986641198601 02522 01629 02282 027141198602 01 05243 01364 046891198603 04081 01756 05244 037791198604 04507 01409 01064 013221198605 00697 02203 03753 01161

Table 10 Order-inducing vectors obtained by our TOPSIS-based method for problem in [39]

1198661 1198662 1198663 11986641198601 04548 04969 04923 047321198602 05092 047 04768 046451198603 07192 04888 04953 049151198604 06722 06163 05706 049291198605 06104 04942 06463 04946

Table 11 Ranking results obtained by applying different order-inducing vectors to [38 39]

Problems Approaches Ranking results

Problem in [38] IHFLOWA [38] + order-inducing vectors in [38] 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198605 ≻ 1198604

IHFLOWA [38] + order-inducing vectors in Table 9 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198604 ≻ 1198605

Problem in [39] I-IVDHFLOWA [39] + order-inducing vectors in [39] 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198602 ≻ 1198601

I-IVDHFLOWA [39] + order-inducing vectors in Table 10 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602

permutation of 1199092 ≻ 1199091 ≻ 1199093 which verifies that ourproposed TOPSIS-based order-inducing approach is prac-tical and useful in helping work out decision results morereasonably and robustly

63 Application Studies of Proposed TOPSIS-Based Order-Inducing Method When confronting complex decision-making situations that need additional attitudinal charactersof decision-makers Yager and Filev [88] and Yager [102]introduced and encouraged to employ order-inducing vectorto represent those attitudinal characters Since then theidea of order-inducing vector has been extended to variousdecision-making environments [98 103] such as those basedon intuitionistic fuzzy information [104 105] linguistic infor-mation [106] hesitant fuzzy linguistic information [38] andinterval-valued dual hesitant fuzzy uncertain linguistic infor-mation [39] Generally in existing literature order-inducingvectors rely on experts to directly give [38 39 104ndash106]

In order to examine the adaptability of our TOPSIS-based order-inducing method to linguistic decision-makingenvironments we respectively apply the method to resolvethe same problems in references [38 39] Due to fact thatthere is no descriptions about how the order-inducing vectorswere derived in both references [38 39] we here applyextremums into our TOPSIS-based method to obtain order-inducing vectors that is (1199047 1 0) and (1199041 0 1) for[38] ([1199046 1199046] 1 0) and ([1199040 1199040] 0 1) for [39] Thenwe obtain our order-inducing vectors listed in Table 9 forthe problem adopted in [38] and our order-inducing vectorslisted in Table 10 for [39]

Now we apply the order-inducing vectors as shownin Tables 9 and 10 to resolve the problems in [38] and[39] respectively Regarding the problem in [38] the scorevalues of all alternatives are calculated as 119878(1198601)=02602119878(1198602)=02417 119878(1198603)=03577 119878(1198604)=0221 and 119878(1198605)=0185so the corresponding ranking result is 1198603 ≻ 1198601 ≻ 1198602 ≻1198604 ≻ 1198605 As for the problem in [39] the score values ofall alternatives are computed as 119878(1198601)=39954 119878(1198602)=3783119878(1198603)=70827 119878(1198604)=71792 and 119878(1198605)=5813 we thus obtainthe ranking result 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602 Formore clarity brief descriptions of comparative approachesand obtained ranking results also have been put together inTable 11

According to Table 11 the approaches that employ ourorder-inducing vectors derive the same ranking order ofthe better three alternatives for both comparative problemsfrom [38 39] As seen our proposed TOPSIS-based order-inducing method exhibits effective and more understand-able Furthermore when decision-makers are incapable ofdirectly put forward all desired order-inducing vectors due tocomplexity in ill-structured problems our method providesa feasible directions to obtain order-inducing vectors byfiguring out ideal solutions Comparatively our proposedTOPSIS-based order-inducing method behaves effective andmore operational in practical use

64 Comparison with Decision-Making Approach Based onDegraded form of IVDHFULS Note that our approachis intrinsically adaptive to decision situations based ondegraded forms of IVDHF UUBLS such as the dual hesitant

24 Complexity

fuzzy linguistic sets introduced by Yang and Ju [54] Tofurther inspect feasibility of our approach we here conductcomparative study on the same case adopted by Yang and Ju[54] We firstly convert their original decision matrix intothe form of IVDHF UUBLS crisp membership and non-membership degrees are equivalently changed to intervalswith same upper and lower limits (eg 02 turns out to be[02 02]) and linguistic variables are rewritten as uncertainlinguistic variables (eg 1199042 turns out to be [11990421199042])

Since the case only got three attributes among whichinterdependency could be not significant under generalassumption weighting vector 120596119860119868 is not calculated Andfor straight comparison order inducing vector is notincluded either Then the argument-dependent attributeweighting vector can be obtained from Eq(29) as 120596119860119863 =(03192 03132 03676)119879 By use of Algorithm Iwith operatorIVDHFUUBLWA (ie GIVDHFUUBLWA in case of 120582 = 1see Appendix B) we have the scores of alternatives 119904(1198601) =03412 119904(1198602) = 01354 119904(1198603) = 03932 119904(1198604) = 07352Therefore the ordering of alternatives is1198604 ≻ 1198603 ≻ 1198601 ≻ 1198602which is mainly in alignment with the ordering of1198604 ≻ 1198601 ≻1198603 ≻ 1198602 in Yang and Ju [54] It is worth noticing that 120596119860119863used in our approach is derived objectively from assessmentswhile attribute weights 120596lowast = (035 025 040)119879used by Yangand Ju [54] were assigned empirically Differences between120596119860119863 and 120596lowast lead to different ordering of1198601 and1198603 Compar-atively speaking our approach holds more adaptability andobjectivity

In general our proposed MADM approach in this paperexhibits an effective way to tackle complex green contractorselection problems in which subjective attributes weightingvector (such as120596119860119868) and objective attributes weighting vector(such as 120596119860119863) exist To comprehensively utilize these twofundamental facets of attributes weighting information ourcase study and computational experiments suggest that linearcombination method (ie 120596 = 120572120596119860119868 + 120573120596119860119863) is effectiveto reflect effects of 120596119860119868 and 120596119860119863 by configuring parameter 120572or 120573 When there is no specific expert opinions to specify orhard to obtain we suggest that parameter 120572 chooses the valueset of 0 02 05 08 1 Furthermore generalized aggrega-tion operators are suggested to inspect score trajectoriesof alternative contractors along increasing of the operatorsrsquoamplification parameter 120582 If some alternative contractorsrsquoscore values get very close or crossed so that they cannot bedifferentiated from each other significantly we suggest to askthe panel of experts to derive opinions of ideal solutions andutilize our TOPSIS-based order-inducing method to includethose expert opinions as attitudinal decision informationthereby obtaining more rational and robust decision results

7 Conclusions

Aiming at tackling complex decision-making problems ofgreen contractor selection that often hold simultaneouslythe characteristics of decision hesitancy attributes inter-dependency and group complex attitudinal characters wehave developed an effective IVDHF UUBLS-based approach

We utilize IVDHF UUBLS to elicit complicate decision-makersrsquo assessments more objectively and comprehensivelywe employ DEMATEL for addressing attribute interdepen-dency more precisely and we determine group attitudinalcharacters as order-inducing vectors based on ideal attitudesof experts rather than by empirical assignment The mainadvantages of our approach are (i) IVDHF UUBLS offers anadequate way to depict decision hesitancy (ii) the designeddistance measure for IVDHF UUBLS avoids informationdistortion of traditional ones which artificially add mis-matching membership or nonmembership degrees (iii) thedeveloped generalized operators include traditional ones asspecial cases so as to provide flexibility in MADM basedon IVDHF UUBLS (iv) the devised synthesized attributeweighting scheme provides an adaptive way to consider twonecessary aspects attributes interdependency-based weightsand argument-dependent weights (v) the TOPSIS-basedmethod can deduce reasonable order-inducing vectors ratherthan empirical ones and holds generality in other linguisticdecision environments Results of case studies and experi-ments have verified the validity of the proposed approach

When multiple decision organizations are involvedgroup decision-making based on IVDHF UUBLS is animportant research direction Besides inducing preferencesfor decision-making problems of high uncertainty interactivemechanisms should also be studied to extract preferences Inaddition empirical studies of the proposed approach can becarried out to various areas such as investment evaluationproject management and vehicle selection in fleet operations

Appendix

A Basic Notions for IVDHFS and UnbalancedLinguistic Term Set

Definition A1 (see [47]) Letting 119883 be a fixed set then anIVDHFS on 119883 is defined as119863 = ⟨119909 ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (A1)

where ℎ(119909) = ⋃[120583119871120583119880]isinℎ(119909)120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and119892(119909) = ⋃[]119871]119880]isin119892(119909)] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two setsof interval values in [0 1] denoting possible membershipand non-membership degrees of element 119909 isin 119883 to theset 119863 respectively with conditions 120583 ] isin [0 1] and0 le (120583119880)+ + (]119880)+ le 1 and for all 119909 isin 119883 (120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880

For convenience normally 119889 = ℎ 119892 is called an interval-valued dual hesitant fuzzy element (IVDHFE) and 119863 is theset of all IVDHFEs

Definition A2 (see [107]) Suppose 119878 = 119904119895 | 119895 = 0 1 119892 minus1 is a finite and totally ordered discrete linguistic term setwhere 119904119895 represent possible values for a linguistic variableand 119892 is an odd cardinality 119878 is uniformly and symmetricallydistributed if the following two conditions are satisfied (1)

Complexity 25

there exists a unique constant 120582 gt 0 such that 119873119878(119904119894) minus119873119878(119904119895) = 119897(119894 minus 119895) for all 119894 119895 = 0 1 119892 minus 1 (2) Let 119878119877 =119904 | 119904 isin 119878 119904 gt 119878119862 and 119878119871 = 119904 | 119904 isin 119878 119904 lt 119878119862 Let C(119878119877) andC(119878119871) be the cardinality of 119878119877 and 119878119871 then C(119878119877) = C(119878119871)If 119878 is uniformly and symmetrically distributed then 119878 iscalled a balanced linguistic term set Otherwise 119878 is calledan unbalanced linguistic term set

The following 2-tuple fuzzy linguistic representationmodel extends traditional linguistic term set to a continuouscase so as to facilitate computing with linguistic variables

Definition A3 (see [59]) Let 119878 = 1199040 1199041 119904119892minus1 be alinguistic term set and 120573 isin [0 119892] Then a 2-tuple fuzzylinguistic variable expresses the equivalent information to 120573is defined as

Δ [0 119892] 997888rarr 119878 times [minus05 05) (A2)

Δ (120573) = (119904119894 120572) with

119904119894 119894 = 119903119900119906119899119889 (120573)120572 = 120573 minus 119894 120572 isin [minus05 05)

(A3)

Δminus1 (119904119894 120572) = 119894 + 120572 = 120573 (A4)

where 119903119900119906119899119889() is the usual rounding operation and120572 is calledsymbolic translation

To obtain 2-tuple fuzzy linguistic representations ofunbalanced linguistic terms the concept of linguistic hierar-chies ie 119871119867 = ⋃119905 119897(119905 119899(119905)) is used 119897(119905 119899(119905)) is a linguistichierarchy with 119905 indicating the level of hierarchy and 119899(119905)denotes the granularity of the linguistic term set of 119905 Herreraet al [59] defined the following transformation functionsbetween labels from different levels in multigranular linguis-tic information contexts without loss of information

Definition A4 (see [59]) In linguistic hierarchies 119871119867 =⋃119905 119897(119905 119899(119905)) whose linguistic term sets are represented by119878119899(119905) = 119904119899(119905)0 119904119899(119905)119899(119905)minus1 the transformation function froma linguistic label in level 119905 to a label in consecutive level 1199051015840 isdefined as 1198791198651199051199051015840 119897(119905 119899(119905)) 997888rarr 119897(1199051015840 119899(1199051015840)) such that

1198791198651199051199051015840 (119904119899(119905)119894 120572119899(119905))= Δ 1199051015840 (Δminus1119905 (119904119899(119905)119894 120572119899(119905)) (119899 (1199051015840) minus 1)119899 (119905) minus 1 ) (A5)

By use of the above transformation function any 2-tuplelinguistic representation can be transformed into a term in119871119867 Detailed transformation procedures are listed as follows

(1) Representation in linguistic hierarchy To transformthe unbalanced terms of 119878 into the corresponding terms inthe 119871119867 transformation function 120595 is employed to associateeach unbalanced linguistic 2-tuple (119904119894 120572) with its linguistic 2-tuple in 119871119867(119878) ie

120595 119878 997888rarr 119871119867(119878) (A6)

so that 120595(119904119894 120572) = (119904119866(119894)119868(119894) 120582) for forall(119904119894 120572) isin 119878(2) Computational phase Firstly transform (119904119866(119894)119868(119894) 120582) intolinguistic 2-tuples denoted as (119904119899(1199051015840)

1198681015840(119894) 1205821015840) in 119878119899(1199051015840) where

(119904119899(1199051015840)1198681015840(119894)

1205821015840) = 119879119865 (119904119866(119894)119868(119894) 120582)= Δ(Δminus1 (119904119866(119894)119868(119894) 120582) sdot (119899 (1199051015840) minus 1)119866 (119894) minus 1 ) (A7)

Then a computational model is used over 119878119899(1199051015840) with aresult denoted as (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)(3) Retranslation process The result (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)is transformed into the unbalanced term in 119878 by using thetransformation function 120595minus1 ie

120595minus1 119871119867 (119878) 997888rarr 119878 (A8)

so that 120595minus1(119904119899(1199051015840)119903 120582119903) = (119904119903119890119904119906119897119905 120582119903119890119904119906119897119905) isin 119878B Details of Theorem 14

Proof Given a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector for119904119889119895 and 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 120582 gt 0 then(1) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLWA and GIVDHFUUBLWG operators respectively(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLOWA and GIVDHFUUBLOWG operators respectively

(2) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to theIVDHF UUBL weighted average (IVDHFUUBLWA)and IVDHF UUBL weighted geometric (IVDHFUUBLWG)operators respectively where

26 Complexity

119868119881119863119867119865119880119880119861119871119882119860120596 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

120596119895119904119889119895= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus

119899prod119895=1

(1 minus 120583119871119895 )120596119895 1 minus 119899prod119895=1

(1 minus 120583119880119895 )120596119895]] [[

119899prod119895=1

(]119871119895)120596119895 119899prod119895=1

(]119880119895 )120596119895]])

(B1)

119868119881119863119867119865119880119880119861119871119882119866120596 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889119895120596119895

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[

119899prod119895=1

(120583119871119895 )120596119895 119899prod119895=1

(120583119880119895 )120596119895]] [[1 minus

119899prod119895=1

(1 minus ]119871119895)120596119895 1 minus 119899prod119895=1

(1 minus ]119880119895 )120596119895]])

(B2)

(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHA andGIVDHFUUBLHG operators reduce to the IVDHF UUBL

orderedweighted average (IVDHFUUBLOWA) and IVDHFUUBL ordered weighted geometric (IVDHFUUBLOWG)operators respectively where

119868119881119863119867119865119880119880119861119871119874119882119860119908 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

119908119895119904119889120590(119895)= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) )))

119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) )))

]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus

119899prod119895=1

(1 minus 120583119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus 120583119880120590(119895))119908119895]] [[

119899prod119895=1

(]119871120590(119895))119908119895 119899prod119895=1

(]119880120590(119895))119908119895]])

(B3)

119868119881119863119867119865119880119880119861119871119874119882119866119908 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889120590(119895)119908119895

= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[

119899prod119895=1

(120583119871120590(119895))119908119895 119899prod119895=1

(120583119880120590(119895))119908119895]] [[1 minus

119899prod119895=1

(1 minus ]119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus ]119880120590(119895))119908119895]])

(B4)

Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

References

[1] V W Y Tam and C M Tam ldquoWaste reduction throughincentivesA case studyrdquoBuilding Research and Information vol36 no 1 pp 37ndash43 2008

[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

[11] X Qi C Liang and J Zhang ldquoGeneralized cross-entropy basedgroup decision making with unknown expert and attributeweights under interval-valued intuitionistic fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 79 pp 52ndash642015

[12] H Zimmermann Fuzzy Set Theorymdashand Its ApplicationsKluwer Academic Boston Mass USA 2001

[13] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 pp 199ndash249 1975

[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

[16] B T Nuong K-W Kim L Prathumratana et al ldquoSustainabledevelopment in the mining sector and its evaluation usingfuzzy AHP (Analytic Hierarchy Process) approachrdquo GeosystemEngineering vol 14 no 1 pp 43ndash50 2011

[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

[18] E K Zavadskas Z Turskis and J Antucheviciene ldquoSelectinga contractor by using a novel method for multiple attributeanalysis Weighted aggregated sum product assessment withgrey values (WASPAS-G)rdquo Studies in Informatics and Controlvol 24 no 2 pp 141ndash150 2015

[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

[21] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

[25] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009

[26] Z Hatush and M Skitmore ldquoContractor Selection UsingMulticriteria Utility Theory An Additive Modelrdquo Building andEnvironment vol 33 no 2-3 pp 105ndash115 1998

[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

[30] AM Anvuur andMM Kumaraswamy ldquoConceptualmodel ofpartnering and alliancingrdquo Journal of Construction Engineeringand Management vol 133 no 3 pp 225ndash234 2007

[31] A H I Lee H Kang C F Hsu and H Hung ldquoA green supplierselection model for high-tech industryrdquo Expert Systems withApplications vol 36 no 4 pp 7917ndash7927 2009

[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

[33] P Humphreys R McIvor and F Chan ldquoUsing case-basedreasoning to evaluate supplier environmental managementperformancerdquo Expert Systems with Applications vol 25 no 2pp 141ndash153 2003

[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

[45] Bin Zhu Zeshui Xu and Meimei Xia ldquoDual Hesitant FuzzySetsrdquo Journal of Applied Mathematics vol 2012 Article ID879629 13 pages 2012

[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

[51] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[52] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

[55] Z Tao X Liu H Chen and L Zhou ldquoUsing new versionof extended t-norms and s-norms for aggregating intervallinguistic labelsrdquo IEEETransactions on SystemsManCyberneticsSystems vol vol p 15 2016

[56] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[57] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[58] L Martınez M Espinilla J Liu L G Perez and P J SanchezldquoAn evaluation model with unbalanced linguistic informationapplied to olive oil sensory evaluationrdquo Journal of Multiple-Valued Logic and Soft Computing vol 15 no 2-3 pp 229ndash2512009

[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

[62] L Marin A Valls D Isern A Moreno and J M MerigoldquoInduced unbalanced linguistic ordered weighted average andits application in multiperson decision makingrdquo The ScientificWorld Journal vol 2014 2014

[63] A Baykasoglu V Kaplanoglu Z D U Durmusoglu and CSahin ldquoIntegrating fuzzy DEMATEL and fuzzy hierarchicalTOPSIS methods for truck selectionrdquo Expert Systems withApplications vol 40 no 3 pp 899ndash907 2013

[64] S K Patil and R Kant ldquoA hybrid approach based on fuzzyDEMATEL and FMCDM to predict success of knowledge man-agement adoption in supply chainrdquoApplied Soft Computing vol18 pp 126ndash135 2014

[65] S Liyin and Y Hong ldquoImproving environmental performanceby means of empowerment of contractorsrdquo Management ofEnvironmental Quality An International Journal vol 17 no 3pp 242ndash257 2006

[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

[69] Y Kazancoglu I Kazancoglu and M Sagnak ldquoA new holisticconceptual framework for green supply chain management

performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

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12 Complexity

When confronted with ill-structured situations wheredecision-makerrsquos complex attitudinal characters need to beincluded order-inducing vectors provide an effective way[88ndash90]Thus we further define following induced operatorsfor IVDHF UUBLNs

Definition 13 For a collection of IVDHF UUBLNs 119904119889119895 (119895 =1 119899) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector

120596119895 isin [0 1] sum119899119895=1 120596119895 = 1 119899 is a balancing coefficient119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weight-ing vector 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1 120576119895 denote a set oforder inducing vectors

(1) Induced Generalized IVDHFUUBL Hybrid Average (I-GIVDHFUUBLHA) Operator

119868-119866119868119881119863119867119865119880119880119861119871119867119860120596119908120582 (lt 1205761 1199041198891 gt lt 1205762 1199041198892 gt lt 120576119899 119904119889119899 gt) = ( 119899⨁119895=1

(119908119895119904119889120587(119895)120582))1120582

= ⋃(119904120599120587(119895) ℎ120587(119895) 119892120587(119895))isin119904119889120587(119895)

([119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120587(119895) ))))

120582)1120582 119904(sum119899119895=1 119908119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587(119895)

))))120582)1120582]

⋃[120583119871120587(119895) 120583

119880120587(119895)]isinℎ120587(119895) []

119871120587(119895) ]119880120587(119895)]isin119892120587(119895)

[[(1 minus

119899prod119895=1

(1 minus (120583119871120587(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (120583119880120587(119895))120582)119908119895)1120582]]

[[1 minus (1 minus119899prod119895=1

(1 minus (1 minus ]119871120587(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus ]119880120587(119895))120582)119908119895)1120582]])

(15)

where (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) is decreasing order of (11990411988911199041198892 119904119889119899) according to the order inducing vector 120576119895 and119904119889119895 = 119899120596119895119904119889119895 = ⋃

(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904119899120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904119899120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

] ⋃

[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[1 minus (1 minus 120583119871119895 )119899120596119895 1 minus (1 minus 120583119880119895 )119899120596119895] [(]119871119895)119899120596119895 (]119880119895 )119899120596119895]) (16)

(2) Induced Generalized IVDHFUUBL Hybrid Geometric(I-GIVDHFUUBLHG) Operator

119868-119866119868119881119863119867119865119880119880119861119871119867119866119908120596120582 (lt 1205761 1199041198891 gt lt 1205762 1199041198892 gt lt 120576119899 119904119889119899 gt) = 1120582 ( 119899⨂119895=1

(120582119904119889120587(119895))119908119895)= ⋃(119904120599120587(119895)

ℎ120587(119895) 119892120587(119895))isin119904119889120587(119895)

([119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989511199050(120595(119904120587(119895) ))))

119908119895 119904(1120582)prod119899119895=1(120582Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587(119895)

))))119908119895]

⋃[120583119871120587(119895)120583119880120587(119895)]isinℎ120587(119895) []119871120587(119895) ]

119880120587(119895)]isin119892120587(119895)

[[1 minus (1 minus

119899prod119895=1

(1 minus (1 minus 120583119871120587(119895))120582)119908119895)1120582 1 minus (1 minus 119899prod119895=1

(1 minus (1 minus 120583119880120587(119895))120582)119908119895)1120582]]

[[(1 minus119899prod119895=1

(1 minus (]119871120587(119895))120582)119908119895)1120582 (1 minus 119899prod119895=1

(1 minus (]119880120587(119895))120582)119908119895)1120582]])

(17)

Complexity 13

Weighted Aggregation

Geometric Arithmetic

GIVDHFUUBLHG GIVDHFUUBLHA

GIVDHFUUBLWG GIVDHFUUBLOWG

IVDHFUUBLWG IVDHFUUBLOWG

GIVDHFUUBLWA GIVDHFUUBLOWA

IVDHFUUBLWA IVDHFUUBLOWA

= 1 w =1

n = 1 =

1

n = 1 =1

n = 1 =

1

n

= 1 = 1 = 1 = 1

Figure 2

where (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) is decreasing order of (11990411988911199041198892 119904119889119899) according to the order inducing variables 120576119895 and

119904119889119895 = 119904119889119895119899120596119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(Δminus11199050 (119879119865

11990511989511199050(120595(119904120572119895 ))))

119899120596119895 119904(Δminus11199050 (119879119865

11990511989521199050(120595(119904120573119895 ))))

119899120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[(120583119871119895 )119899120596119895 (120583119880119895 )119899120596119895] [1 minus (1 minus ]119871119895 )119899120596119895 1 minus (1 minus ]119880119895 )119899120596119895]) (18)

42 Properties of the Proposed GeneralizedAggregation Operators

eorem 14 With special values of 120582 119908 and 120596 the operatorsGIVDHFUUBLHA and GIVDHFUUBLHG can include aseries of traditional aggregation operators as special cases andtheir relationship can be depicted in Figure 2

Proof See Appendix B

As for the induced hybrid aggregation operators we alsohave following theorem

eorem 15 If (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) = (119904119889120590(1) 119904119889120590(2) 119904119889120590(119899)) I-GIVDHFUUBLHA reduces toGIVDHFUUBLHAIf (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) = (119904119889120590(1) 119904119889120590(2) 119904119889120590(119899)) thenI-GIVDHFUUBLHG reduces to the GIVDHFUUBLHG oper-ator

eorem 16 All the proposed generalized operatorsGIVDHFUUBLWAGIVDHFUUBLWGGIVDHFUUBLOWAGIVDHFUUBLOWGGIVDHFUUBLHAGIVDHFUUBLHGI-GIVDHFUUBLHA and I-GIVDHFUUBLHG hold thefollowing properties (1) Commutativity (2) Idempotency (3)Boundedness

Based on above theorems following properties can alsobe derived

eorem 17 For a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weight vector of 119904119889119895with 120596119895 isin [0 1] and sum119899119895=1 120596119895 = 1 120582 gt 0 then we have

(1) 119868119881119863119867119865119880119880119861119871119882119866120596(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119882119860120596(1199041198891 1199041198892 119904119889119899)(2) 119868119881119863119867119865119880119880119861119871119882119866120596(1199041198891 1199041198892 119904119889119899) le119866119868119881119863119867119865119880119880119861119871119882119860120596120582(1199041198891 1199041198892 119904119889119899)(3) 119866119868119881119863119867119865119880119880119861119871119882119866120596120582(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119882119860120596(1199041198891 1199041198892 119904119889119899)

Proof Omitted for concision

eorem 18 For a collection of IVDHF UUBLNs 119904119889119895 = (119904120599119895 ℎ119895 119892119895) 119908 = (1199081 1199082 119908119899)119879 is the weighting vector of 119904119889119895with 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1 120582 gt 0 Then

(1) 119868119881119863119867119865119880119880119861119871119874119882119866119908(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119874119882119860119908(1199041198891 1199041198892 119904119889119899)(2) 119868119881119863119867119865119880119880119861119871119874119882119866119908(1199041198891 1199041198892 119904119889119899) le119866119868119881119863119867119865119880119880119861119871119874119882119860119908120582(1199041198891 1199041198892 119904119889119899)(3) 119866119868119881119863119867119865119880119880119861119871119874119882119866119908120582(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119874119882119860119908(1199041198891 1199041198892 119904119889119899)

Proof Omitted for concision

14 Complexity

5 An Integrated MADM Approach forTackling Complex Green ContractorSelection Problems

As described in Section 2 we take the green contractorselection as a special type of complicate MADM problemsthat synthesizes three characteristics of decision hesitancy[21 22] attributes interdependency [84] and group atti-tudinal characters [61] Therefore in this section we con-struct an integrated MADM approach to tackle the com-plex green contractor selection problems Suppose 119883 =1199091 1199092 119909119898 is the set of alternative green contractors and119860 = 1198601 1198602 119860119899 is the set of evaluative attributes 120596 =(1205961 1205962 120596119899)119879 is weighting vector for the attributes 120596119894 ge 0sum119899119894=1 120596119894 = 1 Let = (119903119894119895)119899times119898 denote the decision matrix inwhich 119903119894119895 = ([119904120572119894119895 119904120573119894119895] ℎ119894119895 119892119894119895) is an IVDHF UUBLN givenby decision-makers for alternative contractor 119909119895 with respectto attribute 119860 119894 According to the mechanism of pair-wisecomparisons among attributes in the DEMATELmethod [6364] the interdependency among attributes can be obtained asamatrix119885 = (119911119894119896)119899times119899 where 119911119894119896 indicates the degree towhich119860 119894 affects119860119896 Subsequently based on the IVDHFUUBLS andits operations we now present detailed steps of our MADMapproach as shown in following Algorithm I

Algorithm I Hesitant fuzzy linguistic MADMwith attributesinterdependency and decision-makersrsquo group attitudinalcharacters

Step 1 Determine argument-dependent weighting vector120596119860119863 according to attribute values by programming modeldeveloped in the following Section 51

Step 2 Obtain the attribute-interdependences based weight-ing vector 120596119860119868 by use of DEMATEL method described in thefollowing Section 52

Step 3 Calculate synthesized attribute weighting vectoraccording to

120596 = 120572120596119860119868 + 120573120596119860119863 (19)

where 120572 and 120573 are parameters to reflect decision character-istics of decision organizations 0 le 120572 120573 le 1 120572 + 120573 =1Step 4 Check requirements for order inducing If no addi-tional order inducing required then go to Step 5 otherwisego to Step 6

Step 5 Utilize generalized aggregation operators to get theoverall IVDHF UUBLNs 119903119895(119895 = 1 119898) for each alternative119909119895 Here we take GIVDHFUUBLHA operator for examplebecause it can include other traditional operators as its specialcases Therefore we have

119866119868119881119863119867119865119880119880119861119871119867119860(1199031119895 1199032119895 119903119899119895) = ( 119899⨁119894=1

(119908119894 (119903120590119894119895)120582))1120582= ⋃(119904120590119894119895ℎ120590119894119895 119892120590119894119895

)isin119903120590119894119895

([119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590119894119895

))))120582)1120582 119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590119894119895

))))120582)1120582]

⋃[120583119871120590119894119895120583

119880120590119894119895]isinℎ120590119894119895[]

119871120590119894119895]119880120590119894119895]isin119892120590119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120590119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120590119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120590119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120590119894119895)120582)119908119894)1120582]])

(20)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of 119903119894119895119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weightvector 119903120590119894119895 is the 119894th largest of 119903119894119895 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))

119904119899120596119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119894119895))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (21)

Complexity 15

Then go to Step 8

Step 6 By use of the TOPSIS-based method developed in thefollowing Section 53 we determine order-inducing vectors 120576to reflect decision-makersrsquo group attitudinal characters

Step 7 Utilizing induced generalized aggregation operatorsto get the overall IVDHF UUBLNs 119903119895(119895 = 1 2 119898) foreach alternative 119909119895 we have

119868-119866119868119881119863119867119865119880119871119867119860(1199031119895 1199032119895 119903119899119898) = ( 119899⨁119894=1

(119908119894 (119903120587119894119895)120582))1120582

= ⋃(119904120587119894119895ℎ120587119894119895119892120587119894119895)isin119903120587119894119895

([119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120587119894119895

))))120582)1120582 119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587119894119895

))))120582)1120582]

⋃[120583119871120587119894119895 120583

119880120587119894119895]isinℎ120587119894119895 []

119871120587119894119895]119880120587119894119895]isin119892120587119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120587119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120587119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120587119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120587119894119895)120582)119908119894)1120582]])

(22)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of119903119894119895 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 119903120587119894119895 is the 119894th largest of 119903119894119895 reordered by theorder-inducing vectors 120576 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))119899120596119894120572119894119895

119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120573119894119895 ))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (23)

Step 8 Calculate scores 119878(119903119895) and accuracy degrees 119875(119903119895) for119903119895(119895 = 1 2 119898)Step 9 According to Definition 5 output ranking order ofalternative contractors

In Algorithm I to objectively derive unknown weight-ing vector for evaluative attributes a synthesized attributeweighting vector is devised by fusing two parts of concern(i) argument-dependent weighting vector and (ii) attribute-interdependences based weighting vector The former is toreflect effects of attribute values on attribute weights whilethe latter is to exploit effects of interdependences amongattributes on their weights For clarity processing flowchartof Algorithm I is depicted in Figure 3

51 Determining Argument-Dependent Attribute WeightingVector 120596119860119863 In order to derive attribute weighting vec-tor appropriately from assessments maximizing deviationmethod [91] is adopted here for its distinguishing ability

and objectivity [92] According to Wang [91] if performancevalues of each alternative have little differences under anattribute it implicates such an attribute plays a less importantrole otherwise it lives a more important role in decision-making Thus if one attribute has similar values across alter-natives it should be assigned a smaller weight contrariwiseit should be a bigger weight

In light of the idea in maximizing deviation methodwe develop a programming model to determine argument-dependent weighting vector 120596119860119863 For attribute 119860 119894 the stan-dard deviation of alternative 119909119895 to all the other alternatives isdenoted as

119865119894119895 (120596119860119863) = 119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (24)

Let119865119894(120596119860119863) represent deviation value of all alternatives toother alternatives according to attribute119860 119894 and be formulatedas follows

16 Complexity

Attribute weighting

Hesitant fuzzy linguistic MADM with attributes interdependency and decision makersrsquo group attitudinal characters

Step 9 Output ranking order of alternatives

interdependency among attributes by DEMATEL in Section 52

attributes values by standard deviation method in Section 51

Step 6 Ensure TOPSIS order inducing variables proposed in Section 53

Step 7 Utilizing proposed induced generalized hybrid aggregation operators developed in Section

41 to get overall value of each alternative

Step 8 Calculate score degrees and accuracy degrees

Step 4 Order inducingYes

Step 5 Choose proposed generalized operators in Section 41 to get overall value of each alternative

No

Collect decision matrix in the form of IVDHF_UUBLS

Complex problems of green contractor selection

Step 1 Weighting vector AD derived from Step 2 Weighting vector AI derived from

Step 3 Obtain synthesized weighting vector of attributes =AI + AD

Figure 3 Processing flowchart of Algorithm I

119865119894 (120596119860119863) = 119898sum119895=1

119865119894119895 (120596119860119863) = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (25)

Based on the analysis above we can get followingprogramming model (M-1) for obtaining optimal attributeweighting vector 120596119860119863

(119872 minus 1)max 119865 (120596119860119863) = 119899sum

119894=1119865119894 (120596119860119863) = 119899sum

119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894st 119899sum

119894=1(120596119860119863119894 )2 = 1 120596119860119863119894 ge 0 119894 = 1 2 119899 (26)

where 119889(119903119894119895 119903119894119897) can be calculated by Definition 6To solve model (M-1) we can construct Lagrange func-

tion

119871 (120596119860119863119894 120577) = 119899sum119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894+ 1205772 ( 119899sum

119894=1

(120596119860119863119894 )2 minus 1) (27)

where 120577 is the Lagrange multiplierDifferentiate with respect to 120596119860119863119894 (119894 = 1 2 119898) and 120577

and set these partial derivatives equal to zero the followingset of equations can be obtained

120597119871120597120596119860119863119894 = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) + 120577120596119860119863119894 = 0120597119871120597120577 = 119899sum

119894=1

(120596119860119863119894 )2 minus 1 = 0 (28)

By normalizing the solutions of above equations we canget a simple and exact formula for determining the attributeweights as follows

120596119860119863119894 = sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897)sum119899119894=1sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897) (29)

52 Deriving Attributes Interdependency-Based AttributeWeighting Vector 120596119860119868 Some efforts [53 60 84 93ndash95]

Complexity 17

have been made to address interdependencies betweenattributes in MADM However they unanimously cannotreveal the interdependences in a more complete way WhileDEMATAEL technique [63 96] gathers collective knowledgeto capture influences between attributes more precisely andcompletely so that weighting vector for attributes can berationally determined according to strength measure ofthose influences [64 97] Therefore in tackling the complexMADMwith interdependent attributes under IVDHF UUBLenvironments we present a DEMATEL-based method as thefollowing Algorithm II to derive attribute weighting vector120596119860119868 from the viewpoint of attributes interdependency

Algorithm II DEMATEL-basedmethod for ensuringweight-ing vector 120596119860119868Step 1 Construct initial direct-influence matrix 119885 = (119911119894119896)119899times119899

For attributes 119860 = 1198601 1198602 119860119899 decision-makersare asked to give direct effect that attribute 119860 119894 has onattribute 119860119896 using a five-point scale with scores representedby natural language (1) 0 mdash lsquoabsolutely no influencersquo (2)1 mdash lsquolow influencersquo (3) 2 mdash lsquomedium influencersquo (4) 3 mdashlsquohigh influencersquo (5) 4mdash lsquovery high influencersquoThen the initialdirect-influence matrix 119885 = (119911119894119896)119899times119899 can be established witheach element being the mean of same factors in differentdirect matrices

Step 2 Calculate normalized direct-influence matrix 119883 =(119909119894119896)119899times119899 Using matrix 119885 the normalized direct-influencematrix 119883 = (119909119894119896)119899times119899 is calculated according to

119909119894119896 = 119911119894119896119904 (30)

where 119904 = maxmax1le119894le119899sum119899119896=1 119911119894119896max1le119896le119899sum119899119895=1 119911119894119896 Allelements in matrix 119883 are complying with 0 le 119909119894119896 le 10 le sum119899119896=1 119909119894119896 le 1 and 0 le sum119899119894=1 119909119894119896 le 1 and at least onerow or column of the summation equals 1

Step 3 Derive total-influence matrix 119879 = (119905119894119896)119899times119899 Basedon matrix 119883 the total-influence matrix 119879 = (119905119894119896)119899times119899 can bederived by summing the direct effects and all of the indirecteffects

119879 = 119883 + 1198832 + 1198833 + sdot sdot sdot + 119883ℎ = 119883 (119868 minus 119883)minus1 (31)

where 119868 represents identity matrix When ℎ 997888rarr infin 119883ℎ =(0)119899times119899Step 4 Compute influencing and influenced degrees of eachattribute The sum of rows and the sum of columns withinthe total-influence matrix 119879 are respectively expressed as thevectors 119903 and 119888

119903 = (119903119894)119899times1 = ( 119899sum119896=1

119905119894119896)119899times1

119888 = (119888119896)119899times1 = ( 119899sum

119895=1

119905119894119896)119879

1times119899

(32)

where 119903119894 denotes the sumof the 119894th row inmatrix119879 and showsthe sum of direct and indirect effects that attribute 119860 119894 hason other attributes Similarly 119888119896 shows the sum of direct andindirect effects that attribute 119860119896 has receivedStep 5 Estimate weights of attributes based on influencedegrees according to

120596119868lowast119894 = [(119903119894 + 119888119894)2 + (119903119894 minus 119888119894)2]12 (33)

Then the weighting vector 120596119860119868 can be obtained by normaliz-ing (33)

120596119860119868119894 = 120596119868lowast119894sum119899119894=1 120596119868lowast119894 (34)

53 Obtaining Order Inducing Variables 120576 by TOPSIS-BasedMethod For ill-defined MADM problems induced aggre-gation operators [89 90 98] provide an effective channelfor experts to express complex attitudinal characters Thekey step in induced aggregation operators is to obtainorder-inducing vectors 120576 which however are presumptivelyassigned in most papers [62] Therefore in this sectionaiming at deducing the order-inducing vectors 120576 rationallywe here devise a TOPSIS-based method in Algorithm IIIObviously the Algorithm III is of generality for MADMwithdifferent forms of linguistic expression

Algorithm III TOPSIS-based method for obtaining orderinducing vectors 120576Step 1 Let the decision maker define IVDHF UUBL idealsolution(s) 119903+119894 and IVDHF UUBL negative ideal solution(s)119903minus119894 corresponding to each attribute in which

119903+119894 = (119903+1198941 119903+1198942 119903+119894119901) 119903minus119894 = (119903minus1198941 119903minus1198942 119903minus119894119902) (35)

where 119901 and 119902 denotes the number of 119903+119894 and 119903minus119894 respectively

Step 2 Calculate distances from each element in decisionmatrix to ideal solution(s) and negative solution(s) respec-tively

119889+119894119895 = max119897=12119901

119889 (119903119894119895 119903+119894119897 ) 119889minus119894119895 = min

119897=12119902119889 (119903119894119895 119903minus119894119897 ) (36)

where 119889(119903119894119895 119903+119894119897 ) and 119889(119903119894119895 119903minus119894119897 ) are calculated by the distancemeasure in Definition 6 If 119901 gt 1 we choose the onewith maximum distance if 119902 gt 1 we choose the one withminimum distance

Step 3 Calculate coefficients according to

119888119894119895 = 119889minus119894119895119889+119894119895 + 119889minus119894119895 119894 = 1 2 119899 119895 = 1 2 119898 (37)

18 Complexity

Table 2 Decision matrix in the form of IVDHF UUBLS1199091 1199092 11990931198601([AHVH] [02 03][02 04] [03 04]) ([HT] [04 05][05 06] [02 03] [02 04]) ([ALAH] [01 02][01 03] [06 07])1198602([MH] [05 06][02 03]) ([AHT] [06 07][01 02] ) ([MVH] [02 03][05 06] [06 07])1198603([VLAL] [03 04][04 05] [05 06]) ([MVH] [06 07] [07 08][01 02]) ([VLL] [04 05][02 03])1198604([MAH] [02 04][05 06]) ([LAL] [03 05][02 03]) ([ALM] [06 07][01 02] [02 03])1198605([ALM] [04 05][01 02] [04 05]) ([AHH] [06 07][01 02]) ([HVH] [04 05][06 07] [01 03])1198606([LM] [01 02][03 05] [03 05]) ([VLL] [02 04] [05 06][02 03]) ([AHH] [03 04][04 05])1198607([VLL] [04 05][05 06] [02 03] [02 04]) ([MAH] [05 06] [07 08][01 02]) ([ALM] [04 05][05 06] [03 04])1198608([ALAH] [01 03][04 06]) ([AHH] [05 07][01 02] [02 03]) ([AHH] [04 06][03 04])

Table 3 Direct influence matrix Z1198601 1198602 1198603 1198604 1198605 1198606 1198607 11986081198601 0 3 4 1 3 1 2 41198602 1 0 2 4 4 2 3 11198603 4 2 0 1 3 2 4 11198604 2 3 1 0 3 4 1 21198605 4 2 1 3 0 1 4 31198606 2 4 3 4 4 0 4 21198607 4 1 2 3 3 1 0 41198608 1 2 4 3 3 2 4 0

Step 4 Derive order inducing variable 120576119894119895 by descendingorder of 119888119894119895 The greater the value 119888119894119895 the bigger the 1205761198941198956 Illustrative Example and Experiments

61 Illustrative Case Study Suppose a mega infrastructureproject is selecting green contractor for one of its subprojectsA panel of decision-makers including managers professorsand competitive intelligence experts has been organized tocomprehensively evaluate three alternative contractors (ie1199091 1199092 and 1199093) under the eight attributes (ie 1198601ndash1198608) listedin Section 2

To elicit true decision preferences we apply theIVDHF UUBLS to assess the three alternatives (1199091ndash1199093)Firstly decision-makers are asked to vote on 119909119895 under eachattribute119860 119894 using different unbalanced linguistic term sets 1198781 1198782 and 1198783 where 1198781 = 119873 119871119860119871119872119860119867119867119876119867119881119867119860119879 1198791198782 = 119873119860119873119881119871 119876119871 119871 119860119871119872119860119867119881119867 119879 1198783 = 119873119881119871119871119860119872119872119876119872119860119867119867 119879 In Figure 4 we show the adoptedlinguistic hierarchies and relationship between 1198781 1198782 and1198783 Attributes 1198601 1198602 1198605 and 1198608 are evaluated by use of 1198781attributes11986031198604 and1198607 are evaluated by 1198782 and attribute1198606is evaluated by 1198783 The highest-voted uncertain unbalancedlinguistic term [119904119898 119904119899] are identified Secondly the panel of

decision-makers is asked to further express their opinionson membership degrees ℎ(119909) and nonmembership degrees119892(119909) to [119904119898 119904119899] Then we can derive compound assessments(ie ⟨119909 119904120599(119909) ℎ(119909) 119892(119909)⟩) in the form of IVDHF UUBLSand obtain the decision matrix in Table 2 Additionallyin order to examine the interdependency among attributesdecision-makers are also required to assess the influentialrelations among the eight attributes using DEMATEL-basedAlgorithm II The direct influence matrix is collected andshown in Table 3

Next we apply the proposed Algorithm I to solve thisgreen contractor selection problem To derive the synthe-sized attribute weighting vector in Algorithm I we assign120572 = 120573 = 05 For scenarios with no group attitudinalcharacters operator GIVDHFUUBLHA in Definition 12 ischosen In contrast for scenarios with group attitudinalcharacters operator I-GIVDHFUUBLHA in Definition 13will be chosen Further the panel of decision-makers is askedto determine their positive and negative ideal alternativesthat are collected in Table 4 where corresponding order-inducing vectors that are derived from Algorithm III arealso listed (see the rightmost column) Thereafter operatorI-GIVDHFUUBLHA in Definition 13 is utilized for aggrega-tion For GIVDHFUUBLHA and I-GIVDHFUUBLHA we

Complexity 19

Table 4 Ideal alternatives (119903+ and 119903minus) and derived order-inducing vectors 120576119903+ 119903minus 1205761199091 1199092 11990931198601 ([VHT] 1 0) ([NL] 1 0) 5 6 61198602 ([MAH] [07 08] [01 02]) ([NL] 1 0) 1 1 41198603 (T 1 0) (L 1 0) 8 7 81198604 ([ALAH] 09 01) (T 1 0) 2 2 11198605 ([HT] [09 1] 0) (N 1 0) 7 5 51198606 (M 1 0) (N 1 0) 6 3 31198607 (M 1 0) ([NL] 1 0) 3 8 71198608 ([MAH] 1 0) (N 1 0) 4 4 2

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 1 2 3 4

0 1 2

N L AH H QH VH AT TM

Level 1 l(13)

Level 2 l(25)

Level 3 l(39)

Level 4 l(417)

Unbalanced linguistic term set S1

N AH VH TM Unbalanced linguistic term set S2VL L ALAN QL

N H TM Unbalanced linguistic term set S3

L AM AHQM

AL

VL

Figure 4 Unbalanced linguistic term sets (1198781 1198782 and 1198783) and their mapping in linguistic hierarchies

assume 120582=2 and 119908 = (0 0 015 025 025 025 01 0)119879Note that 119908 is a position weighting vector which is derivedby the fuzzy semantic quantitative operator [99] Aftercompleting all steps in Algorithm I (see Appendix C) weobtain the ranking results and present them in Table 5 Forthe scores derived from Step 5 we find the ranking order is1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the best solution Step 7gives the ranking order of 1199092 ≻ 1199091 ≻ 1199093 indicating 1199092 is stillthe most appropriate one while 1199091 is better than 1199093 under theinfluence of 12057662 Experiments on Ranking Results along with VariousParameter Configurations In order to inspect the impactsof various parameter configurations on ranking results inthis section more numerical examples are carried out Alleight operators that defined in Section 41 are applied to theabove illustrative case where the parameter 120582 is configured asintegers ranging from 1 to 20 and parameter 120572 is configuredto a value set of 0 03 05 08 09 1

Firstly the four basic aggregation operators of GIVDH-FUUBLWA GIVDHFUUBLWG GIVDHFUUBLOWA and

Table 5 Ranking results of operators GIVDHFUUBLHA and I-GIVDHFUUBLHA

Aggregation operator Ranking orderGIVDHFUUBLHA 1199092 ≻ 1199093 ≻ 1199091I-GIVDHFUUBLHA 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG are applied to the former case Thesame configuration 120572 = 120573 = 05 is chosen while 120582 is testedas integers ranging from 1 to 20 The ranking results derivedfrom different parameter combinations have been collectedin Table 6 As can be seen four operators unanimouslyidentify the contractor 1199092 is the best one no matter how120582 changes Regarding priority relation between 1199091 and 1199093GIVDHFUUBLWA and GIVDHFUUBLOWA generate thesame result of1199093 ≻ 1199091 while results fromGIVDHFUUBLWGandGIVDHFUUBLOWGgot a change to 1199091 ≻ 1199093 at a certainvalue of parameter 120582

Actually GIVDHFUUBLWA and GIVDHFUUBLOWAare generalized version of weighted arithmetic aggregationoperators that producemore favorable results by emphasizing

20 Complexity

Table 6 Ranking orders obtained by GIVDHFUUBLWA GIVDHFUUBLOWA GIVDHFUUBLWG and GIVDHFUUBLOWG along with120582Operators 120582 (Integer) Ranking orderGIVDHFUUBLWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLOWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLWG [1 2] 1199092 ≻ 1199093 ≻ 1199091[3 20] 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG [1 8] 1199092 ≻ 1199093 ≻ 1199091[9 20] 1199092 ≻ 1199091 ≻ 1199093

Table 7 Ranking results obtained by GIVDHFUUBLHA and I-GIVDHFUUBLHA along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHA I-GIVDHFUUBLHA120572=0 120573=1 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=03 120573=07 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=05 120573=05 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=08 120573=02 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=09 120573=01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=1 120573=0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093

Table 8 Ranking results obtained by GIVDHFUUBLHG and I-GIVDHFUUBLHG along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHG I-GIVDHFUUBLHG

120572 = 0 120573 = 1 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 03 120573 = 07 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 05 120573 = 05 [1 8] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[9 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 08 120573 = 02 [1 17] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[18 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 09 120573 = 01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572 = 1 120573 = 0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093membership degrees to a uncertain linguistic term thus canbe considered as optimistic operators while GIVDHFU-UBLWG and GIVDHFUUBLOWG are generalized geomet-ric aggregation operators that producemore favorable resultsby emphasizing nonmembership degrees thus can be consid-ered as pessimistic operators [100 101]Therefore the rankingresults in Table 6 could be perceived as for optimism-baseddecision-making alternative 1199093 always stands out better than1199091 when 120582 gradually amplifies degree of optimism but forpessimism-based decision-making 1199091 turns out better than1199093 when 120582 gradually amplifies degree of pessimism 120582 canbe considered as the parameter for amplifying degree ofoptimism or pessimism [101]

Next to inspect the impacts of various combinationsof 120596119860119868 and 120596119860119863 on ranking results another four aggrega-tion operators of GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG also have beenapplied to the above illustrative case with varying 120572 120573 and 120582

All results have been collected in Tables 7 and 8 for compari-son As can be seen fromTable 7 ranking results generated byoperator GIVDHFUUBLHA keep the same as 1199092 ≻ 1199093 ≻ 1199091for all test combinations of parameters However as shownin Table 8 although GIVDHFUUBLHG is also capable ofidentifying 1199092 as the best contractor the priority relationsbetween 1199091 and 1199093 hold similar changing patterns in rankingresults in Table 6 that were output byGIVDHFUUBLWGandGIVDHFUUBLOWG Regarding decision situations whereobjective attributeweighting vector120596119860119863 is only considered (120572= 0) or its dominant importance (such as 120572 = 03) is accepted1199091 ≻ 1199093 is a clear observation when parameter 120582 is set toan integer in interval [8 20] Contrariwise under decisionscenarios where subjective attribute weighting vector 120596119860119868 isonly needed (120572 = 1) or its dominant importance (such as 120572 =09) is acknowledged 1199093 ≻ 1199091 holds all the way 120582 varies

In order to more clearly show the score trajectories ofalternative contractors along variations of parameter 120582 we

Complexity 21

HAInducedHA

InducedHA-2HA-2

HA-3

InducedHA-1InducedHA-3

= =

HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(a) 120572 = 0

HAInducedHA

= 03 = 07

InducedHA-2

HA-2HA-3

InducedHA-1 InducedHA-3HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(b) 120572 = 03

HAInducedHA

= 5 = 05

InducedHA-2HA-2

HA-3

InducedHA-1 InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(c) 120572 = 05

HAInducedHA

= 08 = 02

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572 = 08

HAInducedHA

= 9 = 01

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(e) 120572 = 09

HAInducedHA

246 2462 2464 2466 2468 247

= 1 = 0InducedHA-2HA-2

HA-3

HA- 3

HA-3

InducedHA-1

InducedHA-3

HA-1

HA-1

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

1085

1086

1087

(f) 120572 = 1

Figure 5 Score trajectories obtained by aggregation operators HA and InducedHA with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

further configure parameter 120582 to the value set of [0 01100]and conduct computational experiments to collect scorevalues and ranking results (as shown in Figures 5 and 6) byfour aggregation operators of GIVDHFUUBLHA GIVDH-FUUBLHG I-GIVDHFUUBLHAand I-GIVDHFUUBLHGFor more clarity in Figures 5 and 6 please note that weuse HA HG Induced-HA and Induced HG to respec-tively denote GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG In Figure 5it can be observed that when dominant importance of 120596119860119863

is emphasized (120572 = 0 120572 = 03 120572 = 05) GIVDHFUUBLHAis capable of differentiating the three alternative contractorsas same ranking order of 1199092 ≻ 1199093 ≻ 1199091 for all 120582 but withincreases in 120572 the margins between scores of 1199091 and 1199093 nar-rows and the rank order changes to 1199091 ≻ 1199093when 120572 gets 09and 1 Interestingly in Figure 6 although GIVDHFUUBLHGis able to significantly differentiate alternative contractors 1199091and 1199093 when dominant importance of 120596119860119863 is emphasized (120572= 0 120572 = 03 120572 = 05) rank order of 1199091 and 1199093 obviouslychanges once from 1199093 ≻ 1199091 to 1199091 ≻ 1199093 when 120582 is bigger

22 Complexity

HGInducedHG

= = 1

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(a) 120572= 0

HGInducedHG

= 3 = 07

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(b) 120572= 03

HGInducedHG

= 05 = 05

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(c) 120572= 05

HGInducedHG

= 8 = 02

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572= 08

HGInducedHG

292 293 294 295 296 297minus08

minus0795

minus079

= 09 = 01

HA-1

HA-3

InducedHG-2

HG-1

HG-1

HG-2

HG-3

HG-3 InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4

minus3

minus2

minus1

0

1

2

scor

e val

ues

(e) 120572= 09

HGInducedHG

50 60 70 80 90 100minus095minus09

minus085

= 1 = 0

HG-x3

HG-x1

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(f) 120572= 1

Figure 6 Score trajectories obtained by aggregation operators HG and InducedHG with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

than 8 When 120572 increases to 1 GIVDHFUUBLHGmaintainsthe rank order of 1199093 ≻ 1199091 but their score trajectorieswalk very close Comparatively I-GIVDHFUUBLHA andI-GIVDHFUUBLHG generate consistent ranking results of1199092 ≻ 1199091 ≻ 1199093 no matter 120582 varies in all the above experiments

In sum when parameters are configured with typical val-ues (ie 120572 = 05 and 120582=1) and no additional attitudinal infor-mation is taken into consideration the proposed aggregationoperators unanimously derive the same rank order 1199092 ≻1199093 ≻ 1199091 of for the three alternative contractors However

with the amplification effect of parameter 120582 analysis resultsfrom linear combination of 120596119860119868 and 120596119860119863 reveal alternative1199092 which consistently is the best choice but rank order of1199091 and 1199093 changes because their score trajectories get closeand crossed which probably indicates that only decision-making informationderived from the decisionmatrix is inad-equate to significantly differentiate alternatives 1199091 and 1199093While by integrating additional decision information aboutattitudinal characters of group experts I-GIVDHFUUBLHAand I-GIVDHFUUBLHG manage to generate the identical

Complexity 23

Table 9 Order-inducing vectors obtained by our TOPSIS-based method for problem in [38]

1198661 1198662 1198663 11986641198601 02522 01629 02282 027141198602 01 05243 01364 046891198603 04081 01756 05244 037791198604 04507 01409 01064 013221198605 00697 02203 03753 01161

Table 10 Order-inducing vectors obtained by our TOPSIS-based method for problem in [39]

1198661 1198662 1198663 11986641198601 04548 04969 04923 047321198602 05092 047 04768 046451198603 07192 04888 04953 049151198604 06722 06163 05706 049291198605 06104 04942 06463 04946

Table 11 Ranking results obtained by applying different order-inducing vectors to [38 39]

Problems Approaches Ranking results

Problem in [38] IHFLOWA [38] + order-inducing vectors in [38] 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198605 ≻ 1198604

IHFLOWA [38] + order-inducing vectors in Table 9 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198604 ≻ 1198605

Problem in [39] I-IVDHFLOWA [39] + order-inducing vectors in [39] 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198602 ≻ 1198601

I-IVDHFLOWA [39] + order-inducing vectors in Table 10 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602

permutation of 1199092 ≻ 1199091 ≻ 1199093 which verifies that ourproposed TOPSIS-based order-inducing approach is prac-tical and useful in helping work out decision results morereasonably and robustly

63 Application Studies of Proposed TOPSIS-Based Order-Inducing Method When confronting complex decision-making situations that need additional attitudinal charactersof decision-makers Yager and Filev [88] and Yager [102]introduced and encouraged to employ order-inducing vectorto represent those attitudinal characters Since then theidea of order-inducing vector has been extended to variousdecision-making environments [98 103] such as those basedon intuitionistic fuzzy information [104 105] linguistic infor-mation [106] hesitant fuzzy linguistic information [38] andinterval-valued dual hesitant fuzzy uncertain linguistic infor-mation [39] Generally in existing literature order-inducingvectors rely on experts to directly give [38 39 104ndash106]

In order to examine the adaptability of our TOPSIS-based order-inducing method to linguistic decision-makingenvironments we respectively apply the method to resolvethe same problems in references [38 39] Due to fact thatthere is no descriptions about how the order-inducing vectorswere derived in both references [38 39] we here applyextremums into our TOPSIS-based method to obtain order-inducing vectors that is (1199047 1 0) and (1199041 0 1) for[38] ([1199046 1199046] 1 0) and ([1199040 1199040] 0 1) for [39] Thenwe obtain our order-inducing vectors listed in Table 9 forthe problem adopted in [38] and our order-inducing vectorslisted in Table 10 for [39]

Now we apply the order-inducing vectors as shownin Tables 9 and 10 to resolve the problems in [38] and[39] respectively Regarding the problem in [38] the scorevalues of all alternatives are calculated as 119878(1198601)=02602119878(1198602)=02417 119878(1198603)=03577 119878(1198604)=0221 and 119878(1198605)=0185so the corresponding ranking result is 1198603 ≻ 1198601 ≻ 1198602 ≻1198604 ≻ 1198605 As for the problem in [39] the score values ofall alternatives are computed as 119878(1198601)=39954 119878(1198602)=3783119878(1198603)=70827 119878(1198604)=71792 and 119878(1198605)=5813 we thus obtainthe ranking result 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602 Formore clarity brief descriptions of comparative approachesand obtained ranking results also have been put together inTable 11

According to Table 11 the approaches that employ ourorder-inducing vectors derive the same ranking order ofthe better three alternatives for both comparative problemsfrom [38 39] As seen our proposed TOPSIS-based order-inducing method exhibits effective and more understand-able Furthermore when decision-makers are incapable ofdirectly put forward all desired order-inducing vectors due tocomplexity in ill-structured problems our method providesa feasible directions to obtain order-inducing vectors byfiguring out ideal solutions Comparatively our proposedTOPSIS-based order-inducing method behaves effective andmore operational in practical use

64 Comparison with Decision-Making Approach Based onDegraded form of IVDHFULS Note that our approachis intrinsically adaptive to decision situations based ondegraded forms of IVDHF UUBLS such as the dual hesitant

24 Complexity

fuzzy linguistic sets introduced by Yang and Ju [54] Tofurther inspect feasibility of our approach we here conductcomparative study on the same case adopted by Yang and Ju[54] We firstly convert their original decision matrix intothe form of IVDHF UUBLS crisp membership and non-membership degrees are equivalently changed to intervalswith same upper and lower limits (eg 02 turns out to be[02 02]) and linguistic variables are rewritten as uncertainlinguistic variables (eg 1199042 turns out to be [11990421199042])

Since the case only got three attributes among whichinterdependency could be not significant under generalassumption weighting vector 120596119860119868 is not calculated Andfor straight comparison order inducing vector is notincluded either Then the argument-dependent attributeweighting vector can be obtained from Eq(29) as 120596119860119863 =(03192 03132 03676)119879 By use of Algorithm Iwith operatorIVDHFUUBLWA (ie GIVDHFUUBLWA in case of 120582 = 1see Appendix B) we have the scores of alternatives 119904(1198601) =03412 119904(1198602) = 01354 119904(1198603) = 03932 119904(1198604) = 07352Therefore the ordering of alternatives is1198604 ≻ 1198603 ≻ 1198601 ≻ 1198602which is mainly in alignment with the ordering of1198604 ≻ 1198601 ≻1198603 ≻ 1198602 in Yang and Ju [54] It is worth noticing that 120596119860119863used in our approach is derived objectively from assessmentswhile attribute weights 120596lowast = (035 025 040)119879used by Yangand Ju [54] were assigned empirically Differences between120596119860119863 and 120596lowast lead to different ordering of1198601 and1198603 Compar-atively speaking our approach holds more adaptability andobjectivity

In general our proposed MADM approach in this paperexhibits an effective way to tackle complex green contractorselection problems in which subjective attributes weightingvector (such as120596119860119868) and objective attributes weighting vector(such as 120596119860119863) exist To comprehensively utilize these twofundamental facets of attributes weighting information ourcase study and computational experiments suggest that linearcombination method (ie 120596 = 120572120596119860119868 + 120573120596119860119863) is effectiveto reflect effects of 120596119860119868 and 120596119860119863 by configuring parameter 120572or 120573 When there is no specific expert opinions to specify orhard to obtain we suggest that parameter 120572 chooses the valueset of 0 02 05 08 1 Furthermore generalized aggrega-tion operators are suggested to inspect score trajectoriesof alternative contractors along increasing of the operatorsrsquoamplification parameter 120582 If some alternative contractorsrsquoscore values get very close or crossed so that they cannot bedifferentiated from each other significantly we suggest to askthe panel of experts to derive opinions of ideal solutions andutilize our TOPSIS-based order-inducing method to includethose expert opinions as attitudinal decision informationthereby obtaining more rational and robust decision results

7 Conclusions

Aiming at tackling complex decision-making problems ofgreen contractor selection that often hold simultaneouslythe characteristics of decision hesitancy attributes inter-dependency and group complex attitudinal characters wehave developed an effective IVDHF UUBLS-based approach

We utilize IVDHF UUBLS to elicit complicate decision-makersrsquo assessments more objectively and comprehensivelywe employ DEMATEL for addressing attribute interdepen-dency more precisely and we determine group attitudinalcharacters as order-inducing vectors based on ideal attitudesof experts rather than by empirical assignment The mainadvantages of our approach are (i) IVDHF UUBLS offers anadequate way to depict decision hesitancy (ii) the designeddistance measure for IVDHF UUBLS avoids informationdistortion of traditional ones which artificially add mis-matching membership or nonmembership degrees (iii) thedeveloped generalized operators include traditional ones asspecial cases so as to provide flexibility in MADM basedon IVDHF UUBLS (iv) the devised synthesized attributeweighting scheme provides an adaptive way to consider twonecessary aspects attributes interdependency-based weightsand argument-dependent weights (v) the TOPSIS-basedmethod can deduce reasonable order-inducing vectors ratherthan empirical ones and holds generality in other linguisticdecision environments Results of case studies and experi-ments have verified the validity of the proposed approach

When multiple decision organizations are involvedgroup decision-making based on IVDHF UUBLS is animportant research direction Besides inducing preferencesfor decision-making problems of high uncertainty interactivemechanisms should also be studied to extract preferences Inaddition empirical studies of the proposed approach can becarried out to various areas such as investment evaluationproject management and vehicle selection in fleet operations

Appendix

A Basic Notions for IVDHFS and UnbalancedLinguistic Term Set

Definition A1 (see [47]) Letting 119883 be a fixed set then anIVDHFS on 119883 is defined as119863 = ⟨119909 ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (A1)

where ℎ(119909) = ⋃[120583119871120583119880]isinℎ(119909)120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and119892(119909) = ⋃[]119871]119880]isin119892(119909)] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two setsof interval values in [0 1] denoting possible membershipand non-membership degrees of element 119909 isin 119883 to theset 119863 respectively with conditions 120583 ] isin [0 1] and0 le (120583119880)+ + (]119880)+ le 1 and for all 119909 isin 119883 (120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880

For convenience normally 119889 = ℎ 119892 is called an interval-valued dual hesitant fuzzy element (IVDHFE) and 119863 is theset of all IVDHFEs

Definition A2 (see [107]) Suppose 119878 = 119904119895 | 119895 = 0 1 119892 minus1 is a finite and totally ordered discrete linguistic term setwhere 119904119895 represent possible values for a linguistic variableand 119892 is an odd cardinality 119878 is uniformly and symmetricallydistributed if the following two conditions are satisfied (1)

Complexity 25

there exists a unique constant 120582 gt 0 such that 119873119878(119904119894) minus119873119878(119904119895) = 119897(119894 minus 119895) for all 119894 119895 = 0 1 119892 minus 1 (2) Let 119878119877 =119904 | 119904 isin 119878 119904 gt 119878119862 and 119878119871 = 119904 | 119904 isin 119878 119904 lt 119878119862 Let C(119878119877) andC(119878119871) be the cardinality of 119878119877 and 119878119871 then C(119878119877) = C(119878119871)If 119878 is uniformly and symmetrically distributed then 119878 iscalled a balanced linguistic term set Otherwise 119878 is calledan unbalanced linguistic term set

The following 2-tuple fuzzy linguistic representationmodel extends traditional linguistic term set to a continuouscase so as to facilitate computing with linguistic variables

Definition A3 (see [59]) Let 119878 = 1199040 1199041 119904119892minus1 be alinguistic term set and 120573 isin [0 119892] Then a 2-tuple fuzzylinguistic variable expresses the equivalent information to 120573is defined as

Δ [0 119892] 997888rarr 119878 times [minus05 05) (A2)

Δ (120573) = (119904119894 120572) with

119904119894 119894 = 119903119900119906119899119889 (120573)120572 = 120573 minus 119894 120572 isin [minus05 05)

(A3)

Δminus1 (119904119894 120572) = 119894 + 120572 = 120573 (A4)

where 119903119900119906119899119889() is the usual rounding operation and120572 is calledsymbolic translation

To obtain 2-tuple fuzzy linguistic representations ofunbalanced linguistic terms the concept of linguistic hierar-chies ie 119871119867 = ⋃119905 119897(119905 119899(119905)) is used 119897(119905 119899(119905)) is a linguistichierarchy with 119905 indicating the level of hierarchy and 119899(119905)denotes the granularity of the linguistic term set of 119905 Herreraet al [59] defined the following transformation functionsbetween labels from different levels in multigranular linguis-tic information contexts without loss of information

Definition A4 (see [59]) In linguistic hierarchies 119871119867 =⋃119905 119897(119905 119899(119905)) whose linguistic term sets are represented by119878119899(119905) = 119904119899(119905)0 119904119899(119905)119899(119905)minus1 the transformation function froma linguistic label in level 119905 to a label in consecutive level 1199051015840 isdefined as 1198791198651199051199051015840 119897(119905 119899(119905)) 997888rarr 119897(1199051015840 119899(1199051015840)) such that

1198791198651199051199051015840 (119904119899(119905)119894 120572119899(119905))= Δ 1199051015840 (Δminus1119905 (119904119899(119905)119894 120572119899(119905)) (119899 (1199051015840) minus 1)119899 (119905) minus 1 ) (A5)

By use of the above transformation function any 2-tuplelinguistic representation can be transformed into a term in119871119867 Detailed transformation procedures are listed as follows

(1) Representation in linguistic hierarchy To transformthe unbalanced terms of 119878 into the corresponding terms inthe 119871119867 transformation function 120595 is employed to associateeach unbalanced linguistic 2-tuple (119904119894 120572) with its linguistic 2-tuple in 119871119867(119878) ie

120595 119878 997888rarr 119871119867(119878) (A6)

so that 120595(119904119894 120572) = (119904119866(119894)119868(119894) 120582) for forall(119904119894 120572) isin 119878(2) Computational phase Firstly transform (119904119866(119894)119868(119894) 120582) intolinguistic 2-tuples denoted as (119904119899(1199051015840)

1198681015840(119894) 1205821015840) in 119878119899(1199051015840) where

(119904119899(1199051015840)1198681015840(119894)

1205821015840) = 119879119865 (119904119866(119894)119868(119894) 120582)= Δ(Δminus1 (119904119866(119894)119868(119894) 120582) sdot (119899 (1199051015840) minus 1)119866 (119894) minus 1 ) (A7)

Then a computational model is used over 119878119899(1199051015840) with aresult denoted as (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)(3) Retranslation process The result (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)is transformed into the unbalanced term in 119878 by using thetransformation function 120595minus1 ie

120595minus1 119871119867 (119878) 997888rarr 119878 (A8)

so that 120595minus1(119904119899(1199051015840)119903 120582119903) = (119904119903119890119904119906119897119905 120582119903119890119904119906119897119905) isin 119878B Details of Theorem 14

Proof Given a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector for119904119889119895 and 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 120582 gt 0 then(1) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLWA and GIVDHFUUBLWG operators respectively(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLOWA and GIVDHFUUBLOWG operators respectively

(2) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to theIVDHF UUBL weighted average (IVDHFUUBLWA)and IVDHF UUBL weighted geometric (IVDHFUUBLWG)operators respectively where

26 Complexity

119868119881119863119867119865119880119880119861119871119882119860120596 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

120596119895119904119889119895= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus

119899prod119895=1

(1 minus 120583119871119895 )120596119895 1 minus 119899prod119895=1

(1 minus 120583119880119895 )120596119895]] [[

119899prod119895=1

(]119871119895)120596119895 119899prod119895=1

(]119880119895 )120596119895]])

(B1)

119868119881119863119867119865119880119880119861119871119882119866120596 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889119895120596119895

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[

119899prod119895=1

(120583119871119895 )120596119895 119899prod119895=1

(120583119880119895 )120596119895]] [[1 minus

119899prod119895=1

(1 minus ]119871119895)120596119895 1 minus 119899prod119895=1

(1 minus ]119880119895 )120596119895]])

(B2)

(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHA andGIVDHFUUBLHG operators reduce to the IVDHF UUBL

orderedweighted average (IVDHFUUBLOWA) and IVDHFUUBL ordered weighted geometric (IVDHFUUBLOWG)operators respectively where

119868119881119863119867119865119880119880119861119871119874119882119860119908 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

119908119895119904119889120590(119895)= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) )))

119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) )))

]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus

119899prod119895=1

(1 minus 120583119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus 120583119880120590(119895))119908119895]] [[

119899prod119895=1

(]119871120590(119895))119908119895 119899prod119895=1

(]119880120590(119895))119908119895]])

(B3)

119868119881119863119867119865119880119880119861119871119874119882119866119908 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889120590(119895)119908119895

= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[

119899prod119895=1

(120583119871120590(119895))119908119895 119899prod119895=1

(120583119880120590(119895))119908119895]] [[1 minus

119899prod119895=1

(1 minus ]119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus ]119880120590(119895))119908119895]])

(B4)

Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

References

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[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

[11] X Qi C Liang and J Zhang ldquoGeneralized cross-entropy basedgroup decision making with unknown expert and attributeweights under interval-valued intuitionistic fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 79 pp 52ndash642015

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[13] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 pp 199ndash249 1975

[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

[16] B T Nuong K-W Kim L Prathumratana et al ldquoSustainabledevelopment in the mining sector and its evaluation usingfuzzy AHP (Analytic Hierarchy Process) approachrdquo GeosystemEngineering vol 14 no 1 pp 43ndash50 2011

[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

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[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

[21] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

[25] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009

[26] Z Hatush and M Skitmore ldquoContractor Selection UsingMulticriteria Utility Theory An Additive Modelrdquo Building andEnvironment vol 33 no 2-3 pp 105ndash115 1998

[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

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[31] A H I Lee H Kang C F Hsu and H Hung ldquoA green supplierselection model for high-tech industryrdquo Expert Systems withApplications vol 36 no 4 pp 7917ndash7927 2009

[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

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[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

[45] Bin Zhu Zeshui Xu and Meimei Xia ldquoDual Hesitant FuzzySetsrdquo Journal of Applied Mathematics vol 2012 Article ID879629 13 pages 2012

[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

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[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

[55] Z Tao X Liu H Chen and L Zhou ldquoUsing new versionof extended t-norms and s-norms for aggregating intervallinguistic labelsrdquo IEEETransactions on SystemsManCyberneticsSystems vol vol p 15 2016

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[57] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

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[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

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[65] S Liyin and Y Hong ldquoImproving environmental performanceby means of empowerment of contractorsrdquo Management ofEnvironmental Quality An International Journal vol 17 no 3pp 242ndash257 2006

[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

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performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

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Complexity 13

Weighted Aggregation

Geometric Arithmetic

GIVDHFUUBLHG GIVDHFUUBLHA

GIVDHFUUBLWG GIVDHFUUBLOWG

IVDHFUUBLWG IVDHFUUBLOWG

GIVDHFUUBLWA GIVDHFUUBLOWA

IVDHFUUBLWA IVDHFUUBLOWA

= 1 w =1

n = 1 =

1

n = 1 =1

n = 1 =

1

n

= 1 = 1 = 1 = 1

Figure 2

where (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) is decreasing order of (11990411988911199041198892 119904119889119899) according to the order inducing variables 120576119895 and

119904119889119895 = 119904119889119895119899120596119895 = ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904(Δminus11199050 (119879119865

11990511989511199050(120595(119904120572119895 ))))

119899120596119895 119904(Δminus11199050 (119879119865

11990511989521199050(120595(119904120573119895 ))))

119899120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[(120583119871119895 )119899120596119895 (120583119880119895 )119899120596119895] [1 minus (1 minus ]119871119895 )119899120596119895 1 minus (1 minus ]119880119895 )119899120596119895]) (18)

42 Properties of the Proposed GeneralizedAggregation Operators

eorem 14 With special values of 120582 119908 and 120596 the operatorsGIVDHFUUBLHA and GIVDHFUUBLHG can include aseries of traditional aggregation operators as special cases andtheir relationship can be depicted in Figure 2

Proof See Appendix B

As for the induced hybrid aggregation operators we alsohave following theorem

eorem 15 If (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) = (119904119889120590(1) 119904119889120590(2) 119904119889120590(119899)) I-GIVDHFUUBLHA reduces toGIVDHFUUBLHAIf (119904119889120587(1) 119904119889120587(2) 119904119889120587(119899)) = (119904119889120590(1) 119904119889120590(2) 119904119889120590(119899)) thenI-GIVDHFUUBLHG reduces to the GIVDHFUUBLHG oper-ator

eorem 16 All the proposed generalized operatorsGIVDHFUUBLWAGIVDHFUUBLWGGIVDHFUUBLOWAGIVDHFUUBLOWGGIVDHFUUBLHAGIVDHFUUBLHGI-GIVDHFUUBLHA and I-GIVDHFUUBLHG hold thefollowing properties (1) Commutativity (2) Idempotency (3)Boundedness

Based on above theorems following properties can alsobe derived

eorem 17 For a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weight vector of 119904119889119895with 120596119895 isin [0 1] and sum119899119895=1 120596119895 = 1 120582 gt 0 then we have

(1) 119868119881119863119867119865119880119880119861119871119882119866120596(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119882119860120596(1199041198891 1199041198892 119904119889119899)(2) 119868119881119863119867119865119880119880119861119871119882119866120596(1199041198891 1199041198892 119904119889119899) le119866119868119881119863119867119865119880119880119861119871119882119860120596120582(1199041198891 1199041198892 119904119889119899)(3) 119866119868119881119863119867119865119880119880119861119871119882119866120596120582(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119882119860120596(1199041198891 1199041198892 119904119889119899)

Proof Omitted for concision

eorem 18 For a collection of IVDHF UUBLNs 119904119889119895 = (119904120599119895 ℎ119895 119892119895) 119908 = (1199081 1199082 119908119899)119879 is the weighting vector of 119904119889119895with 119908119895 isin [0 1] and sum119899119895=1119908119895 = 1 120582 gt 0 Then

(1) 119868119881119863119867119865119880119880119861119871119874119882119866119908(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119874119882119860119908(1199041198891 1199041198892 119904119889119899)(2) 119868119881119863119867119865119880119880119861119871119874119882119866119908(1199041198891 1199041198892 119904119889119899) le119866119868119881119863119867119865119880119880119861119871119874119882119860119908120582(1199041198891 1199041198892 119904119889119899)(3) 119866119868119881119863119867119865119880119880119861119871119874119882119866119908120582(1199041198891 1199041198892 119904119889119899) le119868119881119863119867119865119880119880119861119871119874119882119860119908(1199041198891 1199041198892 119904119889119899)

Proof Omitted for concision

14 Complexity

5 An Integrated MADM Approach forTackling Complex Green ContractorSelection Problems

As described in Section 2 we take the green contractorselection as a special type of complicate MADM problemsthat synthesizes three characteristics of decision hesitancy[21 22] attributes interdependency [84] and group atti-tudinal characters [61] Therefore in this section we con-struct an integrated MADM approach to tackle the com-plex green contractor selection problems Suppose 119883 =1199091 1199092 119909119898 is the set of alternative green contractors and119860 = 1198601 1198602 119860119899 is the set of evaluative attributes 120596 =(1205961 1205962 120596119899)119879 is weighting vector for the attributes 120596119894 ge 0sum119899119894=1 120596119894 = 1 Let = (119903119894119895)119899times119898 denote the decision matrix inwhich 119903119894119895 = ([119904120572119894119895 119904120573119894119895] ℎ119894119895 119892119894119895) is an IVDHF UUBLN givenby decision-makers for alternative contractor 119909119895 with respectto attribute 119860 119894 According to the mechanism of pair-wisecomparisons among attributes in the DEMATELmethod [6364] the interdependency among attributes can be obtained asamatrix119885 = (119911119894119896)119899times119899 where 119911119894119896 indicates the degree towhich119860 119894 affects119860119896 Subsequently based on the IVDHFUUBLS andits operations we now present detailed steps of our MADMapproach as shown in following Algorithm I

Algorithm I Hesitant fuzzy linguistic MADMwith attributesinterdependency and decision-makersrsquo group attitudinalcharacters

Step 1 Determine argument-dependent weighting vector120596119860119863 according to attribute values by programming modeldeveloped in the following Section 51

Step 2 Obtain the attribute-interdependences based weight-ing vector 120596119860119868 by use of DEMATEL method described in thefollowing Section 52

Step 3 Calculate synthesized attribute weighting vectoraccording to

120596 = 120572120596119860119868 + 120573120596119860119863 (19)

where 120572 and 120573 are parameters to reflect decision character-istics of decision organizations 0 le 120572 120573 le 1 120572 + 120573 =1Step 4 Check requirements for order inducing If no addi-tional order inducing required then go to Step 5 otherwisego to Step 6

Step 5 Utilize generalized aggregation operators to get theoverall IVDHF UUBLNs 119903119895(119895 = 1 119898) for each alternative119909119895 Here we take GIVDHFUUBLHA operator for examplebecause it can include other traditional operators as its specialcases Therefore we have

119866119868119881119863119867119865119880119880119861119871119867119860(1199031119895 1199032119895 119903119899119895) = ( 119899⨁119894=1

(119908119894 (119903120590119894119895)120582))1120582= ⋃(119904120590119894119895ℎ120590119894119895 119892120590119894119895

)isin119903120590119894119895

([119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590119894119895

))))120582)1120582 119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590119894119895

))))120582)1120582]

⋃[120583119871120590119894119895120583

119880120590119894119895]isinℎ120590119894119895[]

119871120590119894119895]119880120590119894119895]isin119892120590119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120590119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120590119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120590119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120590119894119895)120582)119908119894)1120582]])

(20)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of 119903119894119895119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weightvector 119903120590119894119895 is the 119894th largest of 119903119894119895 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))

119904119899120596119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119894119895))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (21)

Complexity 15

Then go to Step 8

Step 6 By use of the TOPSIS-based method developed in thefollowing Section 53 we determine order-inducing vectors 120576to reflect decision-makersrsquo group attitudinal characters

Step 7 Utilizing induced generalized aggregation operatorsto get the overall IVDHF UUBLNs 119903119895(119895 = 1 2 119898) foreach alternative 119909119895 we have

119868-119866119868119881119863119867119865119880119871119867119860(1199031119895 1199032119895 119903119899119898) = ( 119899⨁119894=1

(119908119894 (119903120587119894119895)120582))1120582

= ⋃(119904120587119894119895ℎ120587119894119895119892120587119894119895)isin119903120587119894119895

([119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120587119894119895

))))120582)1120582 119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587119894119895

))))120582)1120582]

⋃[120583119871120587119894119895 120583

119880120587119894119895]isinℎ120587119894119895 []

119871120587119894119895]119880120587119894119895]isin119892120587119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120587119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120587119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120587119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120587119894119895)120582)119908119894)1120582]])

(22)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of119903119894119895 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 119903120587119894119895 is the 119894th largest of 119903119894119895 reordered by theorder-inducing vectors 120576 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))119899120596119894120572119894119895

119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120573119894119895 ))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (23)

Step 8 Calculate scores 119878(119903119895) and accuracy degrees 119875(119903119895) for119903119895(119895 = 1 2 119898)Step 9 According to Definition 5 output ranking order ofalternative contractors

In Algorithm I to objectively derive unknown weight-ing vector for evaluative attributes a synthesized attributeweighting vector is devised by fusing two parts of concern(i) argument-dependent weighting vector and (ii) attribute-interdependences based weighting vector The former is toreflect effects of attribute values on attribute weights whilethe latter is to exploit effects of interdependences amongattributes on their weights For clarity processing flowchartof Algorithm I is depicted in Figure 3

51 Determining Argument-Dependent Attribute WeightingVector 120596119860119863 In order to derive attribute weighting vec-tor appropriately from assessments maximizing deviationmethod [91] is adopted here for its distinguishing ability

and objectivity [92] According to Wang [91] if performancevalues of each alternative have little differences under anattribute it implicates such an attribute plays a less importantrole otherwise it lives a more important role in decision-making Thus if one attribute has similar values across alter-natives it should be assigned a smaller weight contrariwiseit should be a bigger weight

In light of the idea in maximizing deviation methodwe develop a programming model to determine argument-dependent weighting vector 120596119860119863 For attribute 119860 119894 the stan-dard deviation of alternative 119909119895 to all the other alternatives isdenoted as

119865119894119895 (120596119860119863) = 119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (24)

Let119865119894(120596119860119863) represent deviation value of all alternatives toother alternatives according to attribute119860 119894 and be formulatedas follows

16 Complexity

Attribute weighting

Hesitant fuzzy linguistic MADM with attributes interdependency and decision makersrsquo group attitudinal characters

Step 9 Output ranking order of alternatives

interdependency among attributes by DEMATEL in Section 52

attributes values by standard deviation method in Section 51

Step 6 Ensure TOPSIS order inducing variables proposed in Section 53

Step 7 Utilizing proposed induced generalized hybrid aggregation operators developed in Section

41 to get overall value of each alternative

Step 8 Calculate score degrees and accuracy degrees

Step 4 Order inducingYes

Step 5 Choose proposed generalized operators in Section 41 to get overall value of each alternative

No

Collect decision matrix in the form of IVDHF_UUBLS

Complex problems of green contractor selection

Step 1 Weighting vector AD derived from Step 2 Weighting vector AI derived from

Step 3 Obtain synthesized weighting vector of attributes =AI + AD

Figure 3 Processing flowchart of Algorithm I

119865119894 (120596119860119863) = 119898sum119895=1

119865119894119895 (120596119860119863) = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (25)

Based on the analysis above we can get followingprogramming model (M-1) for obtaining optimal attributeweighting vector 120596119860119863

(119872 minus 1)max 119865 (120596119860119863) = 119899sum

119894=1119865119894 (120596119860119863) = 119899sum

119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894st 119899sum

119894=1(120596119860119863119894 )2 = 1 120596119860119863119894 ge 0 119894 = 1 2 119899 (26)

where 119889(119903119894119895 119903119894119897) can be calculated by Definition 6To solve model (M-1) we can construct Lagrange func-

tion

119871 (120596119860119863119894 120577) = 119899sum119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894+ 1205772 ( 119899sum

119894=1

(120596119860119863119894 )2 minus 1) (27)

where 120577 is the Lagrange multiplierDifferentiate with respect to 120596119860119863119894 (119894 = 1 2 119898) and 120577

and set these partial derivatives equal to zero the followingset of equations can be obtained

120597119871120597120596119860119863119894 = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) + 120577120596119860119863119894 = 0120597119871120597120577 = 119899sum

119894=1

(120596119860119863119894 )2 minus 1 = 0 (28)

By normalizing the solutions of above equations we canget a simple and exact formula for determining the attributeweights as follows

120596119860119863119894 = sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897)sum119899119894=1sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897) (29)

52 Deriving Attributes Interdependency-Based AttributeWeighting Vector 120596119860119868 Some efforts [53 60 84 93ndash95]

Complexity 17

have been made to address interdependencies betweenattributes in MADM However they unanimously cannotreveal the interdependences in a more complete way WhileDEMATAEL technique [63 96] gathers collective knowledgeto capture influences between attributes more precisely andcompletely so that weighting vector for attributes can berationally determined according to strength measure ofthose influences [64 97] Therefore in tackling the complexMADMwith interdependent attributes under IVDHF UUBLenvironments we present a DEMATEL-based method as thefollowing Algorithm II to derive attribute weighting vector120596119860119868 from the viewpoint of attributes interdependency

Algorithm II DEMATEL-basedmethod for ensuringweight-ing vector 120596119860119868Step 1 Construct initial direct-influence matrix 119885 = (119911119894119896)119899times119899

For attributes 119860 = 1198601 1198602 119860119899 decision-makersare asked to give direct effect that attribute 119860 119894 has onattribute 119860119896 using a five-point scale with scores representedby natural language (1) 0 mdash lsquoabsolutely no influencersquo (2)1 mdash lsquolow influencersquo (3) 2 mdash lsquomedium influencersquo (4) 3 mdashlsquohigh influencersquo (5) 4mdash lsquovery high influencersquoThen the initialdirect-influence matrix 119885 = (119911119894119896)119899times119899 can be established witheach element being the mean of same factors in differentdirect matrices

Step 2 Calculate normalized direct-influence matrix 119883 =(119909119894119896)119899times119899 Using matrix 119885 the normalized direct-influencematrix 119883 = (119909119894119896)119899times119899 is calculated according to

119909119894119896 = 119911119894119896119904 (30)

where 119904 = maxmax1le119894le119899sum119899119896=1 119911119894119896max1le119896le119899sum119899119895=1 119911119894119896 Allelements in matrix 119883 are complying with 0 le 119909119894119896 le 10 le sum119899119896=1 119909119894119896 le 1 and 0 le sum119899119894=1 119909119894119896 le 1 and at least onerow or column of the summation equals 1

Step 3 Derive total-influence matrix 119879 = (119905119894119896)119899times119899 Basedon matrix 119883 the total-influence matrix 119879 = (119905119894119896)119899times119899 can bederived by summing the direct effects and all of the indirecteffects

119879 = 119883 + 1198832 + 1198833 + sdot sdot sdot + 119883ℎ = 119883 (119868 minus 119883)minus1 (31)

where 119868 represents identity matrix When ℎ 997888rarr infin 119883ℎ =(0)119899times119899Step 4 Compute influencing and influenced degrees of eachattribute The sum of rows and the sum of columns withinthe total-influence matrix 119879 are respectively expressed as thevectors 119903 and 119888

119903 = (119903119894)119899times1 = ( 119899sum119896=1

119905119894119896)119899times1

119888 = (119888119896)119899times1 = ( 119899sum

119895=1

119905119894119896)119879

1times119899

(32)

where 119903119894 denotes the sumof the 119894th row inmatrix119879 and showsthe sum of direct and indirect effects that attribute 119860 119894 hason other attributes Similarly 119888119896 shows the sum of direct andindirect effects that attribute 119860119896 has receivedStep 5 Estimate weights of attributes based on influencedegrees according to

120596119868lowast119894 = [(119903119894 + 119888119894)2 + (119903119894 minus 119888119894)2]12 (33)

Then the weighting vector 120596119860119868 can be obtained by normaliz-ing (33)

120596119860119868119894 = 120596119868lowast119894sum119899119894=1 120596119868lowast119894 (34)

53 Obtaining Order Inducing Variables 120576 by TOPSIS-BasedMethod For ill-defined MADM problems induced aggre-gation operators [89 90 98] provide an effective channelfor experts to express complex attitudinal characters Thekey step in induced aggregation operators is to obtainorder-inducing vectors 120576 which however are presumptivelyassigned in most papers [62] Therefore in this sectionaiming at deducing the order-inducing vectors 120576 rationallywe here devise a TOPSIS-based method in Algorithm IIIObviously the Algorithm III is of generality for MADMwithdifferent forms of linguistic expression

Algorithm III TOPSIS-based method for obtaining orderinducing vectors 120576Step 1 Let the decision maker define IVDHF UUBL idealsolution(s) 119903+119894 and IVDHF UUBL negative ideal solution(s)119903minus119894 corresponding to each attribute in which

119903+119894 = (119903+1198941 119903+1198942 119903+119894119901) 119903minus119894 = (119903minus1198941 119903minus1198942 119903minus119894119902) (35)

where 119901 and 119902 denotes the number of 119903+119894 and 119903minus119894 respectively

Step 2 Calculate distances from each element in decisionmatrix to ideal solution(s) and negative solution(s) respec-tively

119889+119894119895 = max119897=12119901

119889 (119903119894119895 119903+119894119897 ) 119889minus119894119895 = min

119897=12119902119889 (119903119894119895 119903minus119894119897 ) (36)

where 119889(119903119894119895 119903+119894119897 ) and 119889(119903119894119895 119903minus119894119897 ) are calculated by the distancemeasure in Definition 6 If 119901 gt 1 we choose the onewith maximum distance if 119902 gt 1 we choose the one withminimum distance

Step 3 Calculate coefficients according to

119888119894119895 = 119889minus119894119895119889+119894119895 + 119889minus119894119895 119894 = 1 2 119899 119895 = 1 2 119898 (37)

18 Complexity

Table 2 Decision matrix in the form of IVDHF UUBLS1199091 1199092 11990931198601([AHVH] [02 03][02 04] [03 04]) ([HT] [04 05][05 06] [02 03] [02 04]) ([ALAH] [01 02][01 03] [06 07])1198602([MH] [05 06][02 03]) ([AHT] [06 07][01 02] ) ([MVH] [02 03][05 06] [06 07])1198603([VLAL] [03 04][04 05] [05 06]) ([MVH] [06 07] [07 08][01 02]) ([VLL] [04 05][02 03])1198604([MAH] [02 04][05 06]) ([LAL] [03 05][02 03]) ([ALM] [06 07][01 02] [02 03])1198605([ALM] [04 05][01 02] [04 05]) ([AHH] [06 07][01 02]) ([HVH] [04 05][06 07] [01 03])1198606([LM] [01 02][03 05] [03 05]) ([VLL] [02 04] [05 06][02 03]) ([AHH] [03 04][04 05])1198607([VLL] [04 05][05 06] [02 03] [02 04]) ([MAH] [05 06] [07 08][01 02]) ([ALM] [04 05][05 06] [03 04])1198608([ALAH] [01 03][04 06]) ([AHH] [05 07][01 02] [02 03]) ([AHH] [04 06][03 04])

Table 3 Direct influence matrix Z1198601 1198602 1198603 1198604 1198605 1198606 1198607 11986081198601 0 3 4 1 3 1 2 41198602 1 0 2 4 4 2 3 11198603 4 2 0 1 3 2 4 11198604 2 3 1 0 3 4 1 21198605 4 2 1 3 0 1 4 31198606 2 4 3 4 4 0 4 21198607 4 1 2 3 3 1 0 41198608 1 2 4 3 3 2 4 0

Step 4 Derive order inducing variable 120576119894119895 by descendingorder of 119888119894119895 The greater the value 119888119894119895 the bigger the 1205761198941198956 Illustrative Example and Experiments

61 Illustrative Case Study Suppose a mega infrastructureproject is selecting green contractor for one of its subprojectsA panel of decision-makers including managers professorsand competitive intelligence experts has been organized tocomprehensively evaluate three alternative contractors (ie1199091 1199092 and 1199093) under the eight attributes (ie 1198601ndash1198608) listedin Section 2

To elicit true decision preferences we apply theIVDHF UUBLS to assess the three alternatives (1199091ndash1199093)Firstly decision-makers are asked to vote on 119909119895 under eachattribute119860 119894 using different unbalanced linguistic term sets 1198781 1198782 and 1198783 where 1198781 = 119873 119871119860119871119872119860119867119867119876119867119881119867119860119879 1198791198782 = 119873119860119873119881119871 119876119871 119871 119860119871119872119860119867119881119867 119879 1198783 = 119873119881119871119871119860119872119872119876119872119860119867119867 119879 In Figure 4 we show the adoptedlinguistic hierarchies and relationship between 1198781 1198782 and1198783 Attributes 1198601 1198602 1198605 and 1198608 are evaluated by use of 1198781attributes11986031198604 and1198607 are evaluated by 1198782 and attribute1198606is evaluated by 1198783 The highest-voted uncertain unbalancedlinguistic term [119904119898 119904119899] are identified Secondly the panel of

decision-makers is asked to further express their opinionson membership degrees ℎ(119909) and nonmembership degrees119892(119909) to [119904119898 119904119899] Then we can derive compound assessments(ie ⟨119909 119904120599(119909) ℎ(119909) 119892(119909)⟩) in the form of IVDHF UUBLSand obtain the decision matrix in Table 2 Additionallyin order to examine the interdependency among attributesdecision-makers are also required to assess the influentialrelations among the eight attributes using DEMATEL-basedAlgorithm II The direct influence matrix is collected andshown in Table 3

Next we apply the proposed Algorithm I to solve thisgreen contractor selection problem To derive the synthe-sized attribute weighting vector in Algorithm I we assign120572 = 120573 = 05 For scenarios with no group attitudinalcharacters operator GIVDHFUUBLHA in Definition 12 ischosen In contrast for scenarios with group attitudinalcharacters operator I-GIVDHFUUBLHA in Definition 13will be chosen Further the panel of decision-makers is askedto determine their positive and negative ideal alternativesthat are collected in Table 4 where corresponding order-inducing vectors that are derived from Algorithm III arealso listed (see the rightmost column) Thereafter operatorI-GIVDHFUUBLHA in Definition 13 is utilized for aggrega-tion For GIVDHFUUBLHA and I-GIVDHFUUBLHA we

Complexity 19

Table 4 Ideal alternatives (119903+ and 119903minus) and derived order-inducing vectors 120576119903+ 119903minus 1205761199091 1199092 11990931198601 ([VHT] 1 0) ([NL] 1 0) 5 6 61198602 ([MAH] [07 08] [01 02]) ([NL] 1 0) 1 1 41198603 (T 1 0) (L 1 0) 8 7 81198604 ([ALAH] 09 01) (T 1 0) 2 2 11198605 ([HT] [09 1] 0) (N 1 0) 7 5 51198606 (M 1 0) (N 1 0) 6 3 31198607 (M 1 0) ([NL] 1 0) 3 8 71198608 ([MAH] 1 0) (N 1 0) 4 4 2

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 1 2 3 4

0 1 2

N L AH H QH VH AT TM

Level 1 l(13)

Level 2 l(25)

Level 3 l(39)

Level 4 l(417)

Unbalanced linguistic term set S1

N AH VH TM Unbalanced linguistic term set S2VL L ALAN QL

N H TM Unbalanced linguistic term set S3

L AM AHQM

AL

VL

Figure 4 Unbalanced linguistic term sets (1198781 1198782 and 1198783) and their mapping in linguistic hierarchies

assume 120582=2 and 119908 = (0 0 015 025 025 025 01 0)119879Note that 119908 is a position weighting vector which is derivedby the fuzzy semantic quantitative operator [99] Aftercompleting all steps in Algorithm I (see Appendix C) weobtain the ranking results and present them in Table 5 Forthe scores derived from Step 5 we find the ranking order is1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the best solution Step 7gives the ranking order of 1199092 ≻ 1199091 ≻ 1199093 indicating 1199092 is stillthe most appropriate one while 1199091 is better than 1199093 under theinfluence of 12057662 Experiments on Ranking Results along with VariousParameter Configurations In order to inspect the impactsof various parameter configurations on ranking results inthis section more numerical examples are carried out Alleight operators that defined in Section 41 are applied to theabove illustrative case where the parameter 120582 is configured asintegers ranging from 1 to 20 and parameter 120572 is configuredto a value set of 0 03 05 08 09 1

Firstly the four basic aggregation operators of GIVDH-FUUBLWA GIVDHFUUBLWG GIVDHFUUBLOWA and

Table 5 Ranking results of operators GIVDHFUUBLHA and I-GIVDHFUUBLHA

Aggregation operator Ranking orderGIVDHFUUBLHA 1199092 ≻ 1199093 ≻ 1199091I-GIVDHFUUBLHA 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG are applied to the former case Thesame configuration 120572 = 120573 = 05 is chosen while 120582 is testedas integers ranging from 1 to 20 The ranking results derivedfrom different parameter combinations have been collectedin Table 6 As can be seen four operators unanimouslyidentify the contractor 1199092 is the best one no matter how120582 changes Regarding priority relation between 1199091 and 1199093GIVDHFUUBLWA and GIVDHFUUBLOWA generate thesame result of1199093 ≻ 1199091 while results fromGIVDHFUUBLWGandGIVDHFUUBLOWGgot a change to 1199091 ≻ 1199093 at a certainvalue of parameter 120582

Actually GIVDHFUUBLWA and GIVDHFUUBLOWAare generalized version of weighted arithmetic aggregationoperators that producemore favorable results by emphasizing

20 Complexity

Table 6 Ranking orders obtained by GIVDHFUUBLWA GIVDHFUUBLOWA GIVDHFUUBLWG and GIVDHFUUBLOWG along with120582Operators 120582 (Integer) Ranking orderGIVDHFUUBLWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLOWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLWG [1 2] 1199092 ≻ 1199093 ≻ 1199091[3 20] 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG [1 8] 1199092 ≻ 1199093 ≻ 1199091[9 20] 1199092 ≻ 1199091 ≻ 1199093

Table 7 Ranking results obtained by GIVDHFUUBLHA and I-GIVDHFUUBLHA along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHA I-GIVDHFUUBLHA120572=0 120573=1 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=03 120573=07 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=05 120573=05 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=08 120573=02 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=09 120573=01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=1 120573=0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093

Table 8 Ranking results obtained by GIVDHFUUBLHG and I-GIVDHFUUBLHG along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHG I-GIVDHFUUBLHG

120572 = 0 120573 = 1 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 03 120573 = 07 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 05 120573 = 05 [1 8] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[9 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 08 120573 = 02 [1 17] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[18 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 09 120573 = 01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572 = 1 120573 = 0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093membership degrees to a uncertain linguistic term thus canbe considered as optimistic operators while GIVDHFU-UBLWG and GIVDHFUUBLOWG are generalized geomet-ric aggregation operators that producemore favorable resultsby emphasizing nonmembership degrees thus can be consid-ered as pessimistic operators [100 101]Therefore the rankingresults in Table 6 could be perceived as for optimism-baseddecision-making alternative 1199093 always stands out better than1199091 when 120582 gradually amplifies degree of optimism but forpessimism-based decision-making 1199091 turns out better than1199093 when 120582 gradually amplifies degree of pessimism 120582 canbe considered as the parameter for amplifying degree ofoptimism or pessimism [101]

Next to inspect the impacts of various combinationsof 120596119860119868 and 120596119860119863 on ranking results another four aggrega-tion operators of GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG also have beenapplied to the above illustrative case with varying 120572 120573 and 120582

All results have been collected in Tables 7 and 8 for compari-son As can be seen fromTable 7 ranking results generated byoperator GIVDHFUUBLHA keep the same as 1199092 ≻ 1199093 ≻ 1199091for all test combinations of parameters However as shownin Table 8 although GIVDHFUUBLHG is also capable ofidentifying 1199092 as the best contractor the priority relationsbetween 1199091 and 1199093 hold similar changing patterns in rankingresults in Table 6 that were output byGIVDHFUUBLWGandGIVDHFUUBLOWG Regarding decision situations whereobjective attributeweighting vector120596119860119863 is only considered (120572= 0) or its dominant importance (such as 120572 = 03) is accepted1199091 ≻ 1199093 is a clear observation when parameter 120582 is set toan integer in interval [8 20] Contrariwise under decisionscenarios where subjective attribute weighting vector 120596119860119868 isonly needed (120572 = 1) or its dominant importance (such as 120572 =09) is acknowledged 1199093 ≻ 1199091 holds all the way 120582 varies

In order to more clearly show the score trajectories ofalternative contractors along variations of parameter 120582 we

Complexity 21

HAInducedHA

InducedHA-2HA-2

HA-3

InducedHA-1InducedHA-3

= =

HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(a) 120572 = 0

HAInducedHA

= 03 = 07

InducedHA-2

HA-2HA-3

InducedHA-1 InducedHA-3HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(b) 120572 = 03

HAInducedHA

= 5 = 05

InducedHA-2HA-2

HA-3

InducedHA-1 InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(c) 120572 = 05

HAInducedHA

= 08 = 02

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572 = 08

HAInducedHA

= 9 = 01

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(e) 120572 = 09

HAInducedHA

246 2462 2464 2466 2468 247

= 1 = 0InducedHA-2HA-2

HA-3

HA- 3

HA-3

InducedHA-1

InducedHA-3

HA-1

HA-1

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

1085

1086

1087

(f) 120572 = 1

Figure 5 Score trajectories obtained by aggregation operators HA and InducedHA with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

further configure parameter 120582 to the value set of [0 01100]and conduct computational experiments to collect scorevalues and ranking results (as shown in Figures 5 and 6) byfour aggregation operators of GIVDHFUUBLHA GIVDH-FUUBLHG I-GIVDHFUUBLHAand I-GIVDHFUUBLHGFor more clarity in Figures 5 and 6 please note that weuse HA HG Induced-HA and Induced HG to respec-tively denote GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG In Figure 5it can be observed that when dominant importance of 120596119860119863

is emphasized (120572 = 0 120572 = 03 120572 = 05) GIVDHFUUBLHAis capable of differentiating the three alternative contractorsas same ranking order of 1199092 ≻ 1199093 ≻ 1199091 for all 120582 but withincreases in 120572 the margins between scores of 1199091 and 1199093 nar-rows and the rank order changes to 1199091 ≻ 1199093when 120572 gets 09and 1 Interestingly in Figure 6 although GIVDHFUUBLHGis able to significantly differentiate alternative contractors 1199091and 1199093 when dominant importance of 120596119860119863 is emphasized (120572= 0 120572 = 03 120572 = 05) rank order of 1199091 and 1199093 obviouslychanges once from 1199093 ≻ 1199091 to 1199091 ≻ 1199093 when 120582 is bigger

22 Complexity

HGInducedHG

= = 1

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(a) 120572= 0

HGInducedHG

= 3 = 07

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(b) 120572= 03

HGInducedHG

= 05 = 05

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(c) 120572= 05

HGInducedHG

= 8 = 02

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572= 08

HGInducedHG

292 293 294 295 296 297minus08

minus0795

minus079

= 09 = 01

HA-1

HA-3

InducedHG-2

HG-1

HG-1

HG-2

HG-3

HG-3 InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4

minus3

minus2

minus1

0

1

2

scor

e val

ues

(e) 120572= 09

HGInducedHG

50 60 70 80 90 100minus095minus09

minus085

= 1 = 0

HG-x3

HG-x1

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(f) 120572= 1

Figure 6 Score trajectories obtained by aggregation operators HG and InducedHG with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

than 8 When 120572 increases to 1 GIVDHFUUBLHGmaintainsthe rank order of 1199093 ≻ 1199091 but their score trajectorieswalk very close Comparatively I-GIVDHFUUBLHA andI-GIVDHFUUBLHG generate consistent ranking results of1199092 ≻ 1199091 ≻ 1199093 no matter 120582 varies in all the above experiments

In sum when parameters are configured with typical val-ues (ie 120572 = 05 and 120582=1) and no additional attitudinal infor-mation is taken into consideration the proposed aggregationoperators unanimously derive the same rank order 1199092 ≻1199093 ≻ 1199091 of for the three alternative contractors However

with the amplification effect of parameter 120582 analysis resultsfrom linear combination of 120596119860119868 and 120596119860119863 reveal alternative1199092 which consistently is the best choice but rank order of1199091 and 1199093 changes because their score trajectories get closeand crossed which probably indicates that only decision-making informationderived from the decisionmatrix is inad-equate to significantly differentiate alternatives 1199091 and 1199093While by integrating additional decision information aboutattitudinal characters of group experts I-GIVDHFUUBLHAand I-GIVDHFUUBLHG manage to generate the identical

Complexity 23

Table 9 Order-inducing vectors obtained by our TOPSIS-based method for problem in [38]

1198661 1198662 1198663 11986641198601 02522 01629 02282 027141198602 01 05243 01364 046891198603 04081 01756 05244 037791198604 04507 01409 01064 013221198605 00697 02203 03753 01161

Table 10 Order-inducing vectors obtained by our TOPSIS-based method for problem in [39]

1198661 1198662 1198663 11986641198601 04548 04969 04923 047321198602 05092 047 04768 046451198603 07192 04888 04953 049151198604 06722 06163 05706 049291198605 06104 04942 06463 04946

Table 11 Ranking results obtained by applying different order-inducing vectors to [38 39]

Problems Approaches Ranking results

Problem in [38] IHFLOWA [38] + order-inducing vectors in [38] 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198605 ≻ 1198604

IHFLOWA [38] + order-inducing vectors in Table 9 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198604 ≻ 1198605

Problem in [39] I-IVDHFLOWA [39] + order-inducing vectors in [39] 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198602 ≻ 1198601

I-IVDHFLOWA [39] + order-inducing vectors in Table 10 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602

permutation of 1199092 ≻ 1199091 ≻ 1199093 which verifies that ourproposed TOPSIS-based order-inducing approach is prac-tical and useful in helping work out decision results morereasonably and robustly

63 Application Studies of Proposed TOPSIS-Based Order-Inducing Method When confronting complex decision-making situations that need additional attitudinal charactersof decision-makers Yager and Filev [88] and Yager [102]introduced and encouraged to employ order-inducing vectorto represent those attitudinal characters Since then theidea of order-inducing vector has been extended to variousdecision-making environments [98 103] such as those basedon intuitionistic fuzzy information [104 105] linguistic infor-mation [106] hesitant fuzzy linguistic information [38] andinterval-valued dual hesitant fuzzy uncertain linguistic infor-mation [39] Generally in existing literature order-inducingvectors rely on experts to directly give [38 39 104ndash106]

In order to examine the adaptability of our TOPSIS-based order-inducing method to linguistic decision-makingenvironments we respectively apply the method to resolvethe same problems in references [38 39] Due to fact thatthere is no descriptions about how the order-inducing vectorswere derived in both references [38 39] we here applyextremums into our TOPSIS-based method to obtain order-inducing vectors that is (1199047 1 0) and (1199041 0 1) for[38] ([1199046 1199046] 1 0) and ([1199040 1199040] 0 1) for [39] Thenwe obtain our order-inducing vectors listed in Table 9 forthe problem adopted in [38] and our order-inducing vectorslisted in Table 10 for [39]

Now we apply the order-inducing vectors as shownin Tables 9 and 10 to resolve the problems in [38] and[39] respectively Regarding the problem in [38] the scorevalues of all alternatives are calculated as 119878(1198601)=02602119878(1198602)=02417 119878(1198603)=03577 119878(1198604)=0221 and 119878(1198605)=0185so the corresponding ranking result is 1198603 ≻ 1198601 ≻ 1198602 ≻1198604 ≻ 1198605 As for the problem in [39] the score values ofall alternatives are computed as 119878(1198601)=39954 119878(1198602)=3783119878(1198603)=70827 119878(1198604)=71792 and 119878(1198605)=5813 we thus obtainthe ranking result 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602 Formore clarity brief descriptions of comparative approachesand obtained ranking results also have been put together inTable 11

According to Table 11 the approaches that employ ourorder-inducing vectors derive the same ranking order ofthe better three alternatives for both comparative problemsfrom [38 39] As seen our proposed TOPSIS-based order-inducing method exhibits effective and more understand-able Furthermore when decision-makers are incapable ofdirectly put forward all desired order-inducing vectors due tocomplexity in ill-structured problems our method providesa feasible directions to obtain order-inducing vectors byfiguring out ideal solutions Comparatively our proposedTOPSIS-based order-inducing method behaves effective andmore operational in practical use

64 Comparison with Decision-Making Approach Based onDegraded form of IVDHFULS Note that our approachis intrinsically adaptive to decision situations based ondegraded forms of IVDHF UUBLS such as the dual hesitant

24 Complexity

fuzzy linguistic sets introduced by Yang and Ju [54] Tofurther inspect feasibility of our approach we here conductcomparative study on the same case adopted by Yang and Ju[54] We firstly convert their original decision matrix intothe form of IVDHF UUBLS crisp membership and non-membership degrees are equivalently changed to intervalswith same upper and lower limits (eg 02 turns out to be[02 02]) and linguistic variables are rewritten as uncertainlinguistic variables (eg 1199042 turns out to be [11990421199042])

Since the case only got three attributes among whichinterdependency could be not significant under generalassumption weighting vector 120596119860119868 is not calculated Andfor straight comparison order inducing vector is notincluded either Then the argument-dependent attributeweighting vector can be obtained from Eq(29) as 120596119860119863 =(03192 03132 03676)119879 By use of Algorithm Iwith operatorIVDHFUUBLWA (ie GIVDHFUUBLWA in case of 120582 = 1see Appendix B) we have the scores of alternatives 119904(1198601) =03412 119904(1198602) = 01354 119904(1198603) = 03932 119904(1198604) = 07352Therefore the ordering of alternatives is1198604 ≻ 1198603 ≻ 1198601 ≻ 1198602which is mainly in alignment with the ordering of1198604 ≻ 1198601 ≻1198603 ≻ 1198602 in Yang and Ju [54] It is worth noticing that 120596119860119863used in our approach is derived objectively from assessmentswhile attribute weights 120596lowast = (035 025 040)119879used by Yangand Ju [54] were assigned empirically Differences between120596119860119863 and 120596lowast lead to different ordering of1198601 and1198603 Compar-atively speaking our approach holds more adaptability andobjectivity

In general our proposed MADM approach in this paperexhibits an effective way to tackle complex green contractorselection problems in which subjective attributes weightingvector (such as120596119860119868) and objective attributes weighting vector(such as 120596119860119863) exist To comprehensively utilize these twofundamental facets of attributes weighting information ourcase study and computational experiments suggest that linearcombination method (ie 120596 = 120572120596119860119868 + 120573120596119860119863) is effectiveto reflect effects of 120596119860119868 and 120596119860119863 by configuring parameter 120572or 120573 When there is no specific expert opinions to specify orhard to obtain we suggest that parameter 120572 chooses the valueset of 0 02 05 08 1 Furthermore generalized aggrega-tion operators are suggested to inspect score trajectoriesof alternative contractors along increasing of the operatorsrsquoamplification parameter 120582 If some alternative contractorsrsquoscore values get very close or crossed so that they cannot bedifferentiated from each other significantly we suggest to askthe panel of experts to derive opinions of ideal solutions andutilize our TOPSIS-based order-inducing method to includethose expert opinions as attitudinal decision informationthereby obtaining more rational and robust decision results

7 Conclusions

Aiming at tackling complex decision-making problems ofgreen contractor selection that often hold simultaneouslythe characteristics of decision hesitancy attributes inter-dependency and group complex attitudinal characters wehave developed an effective IVDHF UUBLS-based approach

We utilize IVDHF UUBLS to elicit complicate decision-makersrsquo assessments more objectively and comprehensivelywe employ DEMATEL for addressing attribute interdepen-dency more precisely and we determine group attitudinalcharacters as order-inducing vectors based on ideal attitudesof experts rather than by empirical assignment The mainadvantages of our approach are (i) IVDHF UUBLS offers anadequate way to depict decision hesitancy (ii) the designeddistance measure for IVDHF UUBLS avoids informationdistortion of traditional ones which artificially add mis-matching membership or nonmembership degrees (iii) thedeveloped generalized operators include traditional ones asspecial cases so as to provide flexibility in MADM basedon IVDHF UUBLS (iv) the devised synthesized attributeweighting scheme provides an adaptive way to consider twonecessary aspects attributes interdependency-based weightsand argument-dependent weights (v) the TOPSIS-basedmethod can deduce reasonable order-inducing vectors ratherthan empirical ones and holds generality in other linguisticdecision environments Results of case studies and experi-ments have verified the validity of the proposed approach

When multiple decision organizations are involvedgroup decision-making based on IVDHF UUBLS is animportant research direction Besides inducing preferencesfor decision-making problems of high uncertainty interactivemechanisms should also be studied to extract preferences Inaddition empirical studies of the proposed approach can becarried out to various areas such as investment evaluationproject management and vehicle selection in fleet operations

Appendix

A Basic Notions for IVDHFS and UnbalancedLinguistic Term Set

Definition A1 (see [47]) Letting 119883 be a fixed set then anIVDHFS on 119883 is defined as119863 = ⟨119909 ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (A1)

where ℎ(119909) = ⋃[120583119871120583119880]isinℎ(119909)120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and119892(119909) = ⋃[]119871]119880]isin119892(119909)] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two setsof interval values in [0 1] denoting possible membershipand non-membership degrees of element 119909 isin 119883 to theset 119863 respectively with conditions 120583 ] isin [0 1] and0 le (120583119880)+ + (]119880)+ le 1 and for all 119909 isin 119883 (120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880

For convenience normally 119889 = ℎ 119892 is called an interval-valued dual hesitant fuzzy element (IVDHFE) and 119863 is theset of all IVDHFEs

Definition A2 (see [107]) Suppose 119878 = 119904119895 | 119895 = 0 1 119892 minus1 is a finite and totally ordered discrete linguistic term setwhere 119904119895 represent possible values for a linguistic variableand 119892 is an odd cardinality 119878 is uniformly and symmetricallydistributed if the following two conditions are satisfied (1)

Complexity 25

there exists a unique constant 120582 gt 0 such that 119873119878(119904119894) minus119873119878(119904119895) = 119897(119894 minus 119895) for all 119894 119895 = 0 1 119892 minus 1 (2) Let 119878119877 =119904 | 119904 isin 119878 119904 gt 119878119862 and 119878119871 = 119904 | 119904 isin 119878 119904 lt 119878119862 Let C(119878119877) andC(119878119871) be the cardinality of 119878119877 and 119878119871 then C(119878119877) = C(119878119871)If 119878 is uniformly and symmetrically distributed then 119878 iscalled a balanced linguistic term set Otherwise 119878 is calledan unbalanced linguistic term set

The following 2-tuple fuzzy linguistic representationmodel extends traditional linguistic term set to a continuouscase so as to facilitate computing with linguistic variables

Definition A3 (see [59]) Let 119878 = 1199040 1199041 119904119892minus1 be alinguistic term set and 120573 isin [0 119892] Then a 2-tuple fuzzylinguistic variable expresses the equivalent information to 120573is defined as

Δ [0 119892] 997888rarr 119878 times [minus05 05) (A2)

Δ (120573) = (119904119894 120572) with

119904119894 119894 = 119903119900119906119899119889 (120573)120572 = 120573 minus 119894 120572 isin [minus05 05)

(A3)

Δminus1 (119904119894 120572) = 119894 + 120572 = 120573 (A4)

where 119903119900119906119899119889() is the usual rounding operation and120572 is calledsymbolic translation

To obtain 2-tuple fuzzy linguistic representations ofunbalanced linguistic terms the concept of linguistic hierar-chies ie 119871119867 = ⋃119905 119897(119905 119899(119905)) is used 119897(119905 119899(119905)) is a linguistichierarchy with 119905 indicating the level of hierarchy and 119899(119905)denotes the granularity of the linguistic term set of 119905 Herreraet al [59] defined the following transformation functionsbetween labels from different levels in multigranular linguis-tic information contexts without loss of information

Definition A4 (see [59]) In linguistic hierarchies 119871119867 =⋃119905 119897(119905 119899(119905)) whose linguistic term sets are represented by119878119899(119905) = 119904119899(119905)0 119904119899(119905)119899(119905)minus1 the transformation function froma linguistic label in level 119905 to a label in consecutive level 1199051015840 isdefined as 1198791198651199051199051015840 119897(119905 119899(119905)) 997888rarr 119897(1199051015840 119899(1199051015840)) such that

1198791198651199051199051015840 (119904119899(119905)119894 120572119899(119905))= Δ 1199051015840 (Δminus1119905 (119904119899(119905)119894 120572119899(119905)) (119899 (1199051015840) minus 1)119899 (119905) minus 1 ) (A5)

By use of the above transformation function any 2-tuplelinguistic representation can be transformed into a term in119871119867 Detailed transformation procedures are listed as follows

(1) Representation in linguistic hierarchy To transformthe unbalanced terms of 119878 into the corresponding terms inthe 119871119867 transformation function 120595 is employed to associateeach unbalanced linguistic 2-tuple (119904119894 120572) with its linguistic 2-tuple in 119871119867(119878) ie

120595 119878 997888rarr 119871119867(119878) (A6)

so that 120595(119904119894 120572) = (119904119866(119894)119868(119894) 120582) for forall(119904119894 120572) isin 119878(2) Computational phase Firstly transform (119904119866(119894)119868(119894) 120582) intolinguistic 2-tuples denoted as (119904119899(1199051015840)

1198681015840(119894) 1205821015840) in 119878119899(1199051015840) where

(119904119899(1199051015840)1198681015840(119894)

1205821015840) = 119879119865 (119904119866(119894)119868(119894) 120582)= Δ(Δminus1 (119904119866(119894)119868(119894) 120582) sdot (119899 (1199051015840) minus 1)119866 (119894) minus 1 ) (A7)

Then a computational model is used over 119878119899(1199051015840) with aresult denoted as (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)(3) Retranslation process The result (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)is transformed into the unbalanced term in 119878 by using thetransformation function 120595minus1 ie

120595minus1 119871119867 (119878) 997888rarr 119878 (A8)

so that 120595minus1(119904119899(1199051015840)119903 120582119903) = (119904119903119890119904119906119897119905 120582119903119890119904119906119897119905) isin 119878B Details of Theorem 14

Proof Given a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector for119904119889119895 and 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 120582 gt 0 then(1) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLWA and GIVDHFUUBLWG operators respectively(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLOWA and GIVDHFUUBLOWG operators respectively

(2) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to theIVDHF UUBL weighted average (IVDHFUUBLWA)and IVDHF UUBL weighted geometric (IVDHFUUBLWG)operators respectively where

26 Complexity

119868119881119863119867119865119880119880119861119871119882119860120596 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

120596119895119904119889119895= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus

119899prod119895=1

(1 minus 120583119871119895 )120596119895 1 minus 119899prod119895=1

(1 minus 120583119880119895 )120596119895]] [[

119899prod119895=1

(]119871119895)120596119895 119899prod119895=1

(]119880119895 )120596119895]])

(B1)

119868119881119863119867119865119880119880119861119871119882119866120596 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889119895120596119895

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[

119899prod119895=1

(120583119871119895 )120596119895 119899prod119895=1

(120583119880119895 )120596119895]] [[1 minus

119899prod119895=1

(1 minus ]119871119895)120596119895 1 minus 119899prod119895=1

(1 minus ]119880119895 )120596119895]])

(B2)

(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHA andGIVDHFUUBLHG operators reduce to the IVDHF UUBL

orderedweighted average (IVDHFUUBLOWA) and IVDHFUUBL ordered weighted geometric (IVDHFUUBLOWG)operators respectively where

119868119881119863119867119865119880119880119861119871119874119882119860119908 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

119908119895119904119889120590(119895)= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) )))

119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) )))

]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus

119899prod119895=1

(1 minus 120583119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus 120583119880120590(119895))119908119895]] [[

119899prod119895=1

(]119871120590(119895))119908119895 119899prod119895=1

(]119880120590(119895))119908119895]])

(B3)

119868119881119863119867119865119880119880119861119871119874119882119866119908 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889120590(119895)119908119895

= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[

119899prod119895=1

(120583119871120590(119895))119908119895 119899prod119895=1

(120583119880120590(119895))119908119895]] [[1 minus

119899prod119895=1

(1 minus ]119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus ]119880120590(119895))119908119895]])

(B4)

Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

References

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[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

[11] X Qi C Liang and J Zhang ldquoGeneralized cross-entropy basedgroup decision making with unknown expert and attributeweights under interval-valued intuitionistic fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 79 pp 52ndash642015

[12] H Zimmermann Fuzzy Set Theorymdashand Its ApplicationsKluwer Academic Boston Mass USA 2001

[13] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 pp 199ndash249 1975

[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

[16] B T Nuong K-W Kim L Prathumratana et al ldquoSustainabledevelopment in the mining sector and its evaluation usingfuzzy AHP (Analytic Hierarchy Process) approachrdquo GeosystemEngineering vol 14 no 1 pp 43ndash50 2011

[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

[18] E K Zavadskas Z Turskis and J Antucheviciene ldquoSelectinga contractor by using a novel method for multiple attributeanalysis Weighted aggregated sum product assessment withgrey values (WASPAS-G)rdquo Studies in Informatics and Controlvol 24 no 2 pp 141ndash150 2015

[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

[21] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

[25] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009

[26] Z Hatush and M Skitmore ldquoContractor Selection UsingMulticriteria Utility Theory An Additive Modelrdquo Building andEnvironment vol 33 no 2-3 pp 105ndash115 1998

[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

[30] AM Anvuur andMM Kumaraswamy ldquoConceptualmodel ofpartnering and alliancingrdquo Journal of Construction Engineeringand Management vol 133 no 3 pp 225ndash234 2007

[31] A H I Lee H Kang C F Hsu and H Hung ldquoA green supplierselection model for high-tech industryrdquo Expert Systems withApplications vol 36 no 4 pp 7917ndash7927 2009

[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

[33] P Humphreys R McIvor and F Chan ldquoUsing case-basedreasoning to evaluate supplier environmental managementperformancerdquo Expert Systems with Applications vol 25 no 2pp 141ndash153 2003

[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

[45] Bin Zhu Zeshui Xu and Meimei Xia ldquoDual Hesitant FuzzySetsrdquo Journal of Applied Mathematics vol 2012 Article ID879629 13 pages 2012

[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

[51] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[52] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

[55] Z Tao X Liu H Chen and L Zhou ldquoUsing new versionof extended t-norms and s-norms for aggregating intervallinguistic labelsrdquo IEEETransactions on SystemsManCyberneticsSystems vol vol p 15 2016

[56] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[57] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[58] L Martınez M Espinilla J Liu L G Perez and P J SanchezldquoAn evaluation model with unbalanced linguistic informationapplied to olive oil sensory evaluationrdquo Journal of Multiple-Valued Logic and Soft Computing vol 15 no 2-3 pp 229ndash2512009

[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

[62] L Marin A Valls D Isern A Moreno and J M MerigoldquoInduced unbalanced linguistic ordered weighted average andits application in multiperson decision makingrdquo The ScientificWorld Journal vol 2014 2014

[63] A Baykasoglu V Kaplanoglu Z D U Durmusoglu and CSahin ldquoIntegrating fuzzy DEMATEL and fuzzy hierarchicalTOPSIS methods for truck selectionrdquo Expert Systems withApplications vol 40 no 3 pp 899ndash907 2013

[64] S K Patil and R Kant ldquoA hybrid approach based on fuzzyDEMATEL and FMCDM to predict success of knowledge man-agement adoption in supply chainrdquoApplied Soft Computing vol18 pp 126ndash135 2014

[65] S Liyin and Y Hong ldquoImproving environmental performanceby means of empowerment of contractorsrdquo Management ofEnvironmental Quality An International Journal vol 17 no 3pp 242ndash257 2006

[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

[69] Y Kazancoglu I Kazancoglu and M Sagnak ldquoA new holisticconceptual framework for green supply chain management

performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

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14 Complexity

5 An Integrated MADM Approach forTackling Complex Green ContractorSelection Problems

As described in Section 2 we take the green contractorselection as a special type of complicate MADM problemsthat synthesizes three characteristics of decision hesitancy[21 22] attributes interdependency [84] and group atti-tudinal characters [61] Therefore in this section we con-struct an integrated MADM approach to tackle the com-plex green contractor selection problems Suppose 119883 =1199091 1199092 119909119898 is the set of alternative green contractors and119860 = 1198601 1198602 119860119899 is the set of evaluative attributes 120596 =(1205961 1205962 120596119899)119879 is weighting vector for the attributes 120596119894 ge 0sum119899119894=1 120596119894 = 1 Let = (119903119894119895)119899times119898 denote the decision matrix inwhich 119903119894119895 = ([119904120572119894119895 119904120573119894119895] ℎ119894119895 119892119894119895) is an IVDHF UUBLN givenby decision-makers for alternative contractor 119909119895 with respectto attribute 119860 119894 According to the mechanism of pair-wisecomparisons among attributes in the DEMATELmethod [6364] the interdependency among attributes can be obtained asamatrix119885 = (119911119894119896)119899times119899 where 119911119894119896 indicates the degree towhich119860 119894 affects119860119896 Subsequently based on the IVDHFUUBLS andits operations we now present detailed steps of our MADMapproach as shown in following Algorithm I

Algorithm I Hesitant fuzzy linguistic MADMwith attributesinterdependency and decision-makersrsquo group attitudinalcharacters

Step 1 Determine argument-dependent weighting vector120596119860119863 according to attribute values by programming modeldeveloped in the following Section 51

Step 2 Obtain the attribute-interdependences based weight-ing vector 120596119860119868 by use of DEMATEL method described in thefollowing Section 52

Step 3 Calculate synthesized attribute weighting vectoraccording to

120596 = 120572120596119860119868 + 120573120596119860119863 (19)

where 120572 and 120573 are parameters to reflect decision character-istics of decision organizations 0 le 120572 120573 le 1 120572 + 120573 =1Step 4 Check requirements for order inducing If no addi-tional order inducing required then go to Step 5 otherwisego to Step 6

Step 5 Utilize generalized aggregation operators to get theoverall IVDHF UUBLNs 119903119895(119895 = 1 119898) for each alternative119909119895 Here we take GIVDHFUUBLHA operator for examplebecause it can include other traditional operators as its specialcases Therefore we have

119866119868119881119863119867119865119880119880119861119871119867119860(1199031119895 1199032119895 119903119899119895) = ( 119899⨁119894=1

(119908119894 (119903120590119894119895)120582))1120582= ⋃(119904120590119894119895ℎ120590119894119895 119892120590119894119895

)isin119903120590119894119895

([119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590119894119895

))))120582)1120582 119904(sum119899119894=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590119894119895

))))120582)1120582]

⋃[120583119871120590119894119895120583

119880120590119894119895]isinℎ120590119894119895[]

119871120590119894119895]119880120590119894119895]isin119892120590119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120590119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120590119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120590119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120590119894119895)120582)119908119894)1120582]])

(20)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of 119903119894119895119908 = (1199081 1199082 119908119899)119879 is the aggregation-associated weightvector 119903120590119894119895 is the 119894th largest of 119903119894119895 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))

119904119899120596119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119894119895))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (21)

Complexity 15

Then go to Step 8

Step 6 By use of the TOPSIS-based method developed in thefollowing Section 53 we determine order-inducing vectors 120576to reflect decision-makersrsquo group attitudinal characters

Step 7 Utilizing induced generalized aggregation operatorsto get the overall IVDHF UUBLNs 119903119895(119895 = 1 2 119898) foreach alternative 119909119895 we have

119868-119866119868119881119863119867119865119880119871119867119860(1199031119895 1199032119895 119903119899119898) = ( 119899⨁119894=1

(119908119894 (119903120587119894119895)120582))1120582

= ⋃(119904120587119894119895ℎ120587119894119895119892120587119894119895)isin119903120587119894119895

([119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120587119894119895

))))120582)1120582 119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587119894119895

))))120582)1120582]

⋃[120583119871120587119894119895 120583

119880120587119894119895]isinℎ120587119894119895 []

119871120587119894119895]119880120587119894119895]isin119892120587119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120587119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120587119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120587119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120587119894119895)120582)119908119894)1120582]])

(22)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of119903119894119895 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 119903120587119894119895 is the 119894th largest of 119903119894119895 reordered by theorder-inducing vectors 120576 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))119899120596119894120572119894119895

119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120573119894119895 ))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (23)

Step 8 Calculate scores 119878(119903119895) and accuracy degrees 119875(119903119895) for119903119895(119895 = 1 2 119898)Step 9 According to Definition 5 output ranking order ofalternative contractors

In Algorithm I to objectively derive unknown weight-ing vector for evaluative attributes a synthesized attributeweighting vector is devised by fusing two parts of concern(i) argument-dependent weighting vector and (ii) attribute-interdependences based weighting vector The former is toreflect effects of attribute values on attribute weights whilethe latter is to exploit effects of interdependences amongattributes on their weights For clarity processing flowchartof Algorithm I is depicted in Figure 3

51 Determining Argument-Dependent Attribute WeightingVector 120596119860119863 In order to derive attribute weighting vec-tor appropriately from assessments maximizing deviationmethod [91] is adopted here for its distinguishing ability

and objectivity [92] According to Wang [91] if performancevalues of each alternative have little differences under anattribute it implicates such an attribute plays a less importantrole otherwise it lives a more important role in decision-making Thus if one attribute has similar values across alter-natives it should be assigned a smaller weight contrariwiseit should be a bigger weight

In light of the idea in maximizing deviation methodwe develop a programming model to determine argument-dependent weighting vector 120596119860119863 For attribute 119860 119894 the stan-dard deviation of alternative 119909119895 to all the other alternatives isdenoted as

119865119894119895 (120596119860119863) = 119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (24)

Let119865119894(120596119860119863) represent deviation value of all alternatives toother alternatives according to attribute119860 119894 and be formulatedas follows

16 Complexity

Attribute weighting

Hesitant fuzzy linguistic MADM with attributes interdependency and decision makersrsquo group attitudinal characters

Step 9 Output ranking order of alternatives

interdependency among attributes by DEMATEL in Section 52

attributes values by standard deviation method in Section 51

Step 6 Ensure TOPSIS order inducing variables proposed in Section 53

Step 7 Utilizing proposed induced generalized hybrid aggregation operators developed in Section

41 to get overall value of each alternative

Step 8 Calculate score degrees and accuracy degrees

Step 4 Order inducingYes

Step 5 Choose proposed generalized operators in Section 41 to get overall value of each alternative

No

Collect decision matrix in the form of IVDHF_UUBLS

Complex problems of green contractor selection

Step 1 Weighting vector AD derived from Step 2 Weighting vector AI derived from

Step 3 Obtain synthesized weighting vector of attributes =AI + AD

Figure 3 Processing flowchart of Algorithm I

119865119894 (120596119860119863) = 119898sum119895=1

119865119894119895 (120596119860119863) = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (25)

Based on the analysis above we can get followingprogramming model (M-1) for obtaining optimal attributeweighting vector 120596119860119863

(119872 minus 1)max 119865 (120596119860119863) = 119899sum

119894=1119865119894 (120596119860119863) = 119899sum

119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894st 119899sum

119894=1(120596119860119863119894 )2 = 1 120596119860119863119894 ge 0 119894 = 1 2 119899 (26)

where 119889(119903119894119895 119903119894119897) can be calculated by Definition 6To solve model (M-1) we can construct Lagrange func-

tion

119871 (120596119860119863119894 120577) = 119899sum119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894+ 1205772 ( 119899sum

119894=1

(120596119860119863119894 )2 minus 1) (27)

where 120577 is the Lagrange multiplierDifferentiate with respect to 120596119860119863119894 (119894 = 1 2 119898) and 120577

and set these partial derivatives equal to zero the followingset of equations can be obtained

120597119871120597120596119860119863119894 = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) + 120577120596119860119863119894 = 0120597119871120597120577 = 119899sum

119894=1

(120596119860119863119894 )2 minus 1 = 0 (28)

By normalizing the solutions of above equations we canget a simple and exact formula for determining the attributeweights as follows

120596119860119863119894 = sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897)sum119899119894=1sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897) (29)

52 Deriving Attributes Interdependency-Based AttributeWeighting Vector 120596119860119868 Some efforts [53 60 84 93ndash95]

Complexity 17

have been made to address interdependencies betweenattributes in MADM However they unanimously cannotreveal the interdependences in a more complete way WhileDEMATAEL technique [63 96] gathers collective knowledgeto capture influences between attributes more precisely andcompletely so that weighting vector for attributes can berationally determined according to strength measure ofthose influences [64 97] Therefore in tackling the complexMADMwith interdependent attributes under IVDHF UUBLenvironments we present a DEMATEL-based method as thefollowing Algorithm II to derive attribute weighting vector120596119860119868 from the viewpoint of attributes interdependency

Algorithm II DEMATEL-basedmethod for ensuringweight-ing vector 120596119860119868Step 1 Construct initial direct-influence matrix 119885 = (119911119894119896)119899times119899

For attributes 119860 = 1198601 1198602 119860119899 decision-makersare asked to give direct effect that attribute 119860 119894 has onattribute 119860119896 using a five-point scale with scores representedby natural language (1) 0 mdash lsquoabsolutely no influencersquo (2)1 mdash lsquolow influencersquo (3) 2 mdash lsquomedium influencersquo (4) 3 mdashlsquohigh influencersquo (5) 4mdash lsquovery high influencersquoThen the initialdirect-influence matrix 119885 = (119911119894119896)119899times119899 can be established witheach element being the mean of same factors in differentdirect matrices

Step 2 Calculate normalized direct-influence matrix 119883 =(119909119894119896)119899times119899 Using matrix 119885 the normalized direct-influencematrix 119883 = (119909119894119896)119899times119899 is calculated according to

119909119894119896 = 119911119894119896119904 (30)

where 119904 = maxmax1le119894le119899sum119899119896=1 119911119894119896max1le119896le119899sum119899119895=1 119911119894119896 Allelements in matrix 119883 are complying with 0 le 119909119894119896 le 10 le sum119899119896=1 119909119894119896 le 1 and 0 le sum119899119894=1 119909119894119896 le 1 and at least onerow or column of the summation equals 1

Step 3 Derive total-influence matrix 119879 = (119905119894119896)119899times119899 Basedon matrix 119883 the total-influence matrix 119879 = (119905119894119896)119899times119899 can bederived by summing the direct effects and all of the indirecteffects

119879 = 119883 + 1198832 + 1198833 + sdot sdot sdot + 119883ℎ = 119883 (119868 minus 119883)minus1 (31)

where 119868 represents identity matrix When ℎ 997888rarr infin 119883ℎ =(0)119899times119899Step 4 Compute influencing and influenced degrees of eachattribute The sum of rows and the sum of columns withinthe total-influence matrix 119879 are respectively expressed as thevectors 119903 and 119888

119903 = (119903119894)119899times1 = ( 119899sum119896=1

119905119894119896)119899times1

119888 = (119888119896)119899times1 = ( 119899sum

119895=1

119905119894119896)119879

1times119899

(32)

where 119903119894 denotes the sumof the 119894th row inmatrix119879 and showsthe sum of direct and indirect effects that attribute 119860 119894 hason other attributes Similarly 119888119896 shows the sum of direct andindirect effects that attribute 119860119896 has receivedStep 5 Estimate weights of attributes based on influencedegrees according to

120596119868lowast119894 = [(119903119894 + 119888119894)2 + (119903119894 minus 119888119894)2]12 (33)

Then the weighting vector 120596119860119868 can be obtained by normaliz-ing (33)

120596119860119868119894 = 120596119868lowast119894sum119899119894=1 120596119868lowast119894 (34)

53 Obtaining Order Inducing Variables 120576 by TOPSIS-BasedMethod For ill-defined MADM problems induced aggre-gation operators [89 90 98] provide an effective channelfor experts to express complex attitudinal characters Thekey step in induced aggregation operators is to obtainorder-inducing vectors 120576 which however are presumptivelyassigned in most papers [62] Therefore in this sectionaiming at deducing the order-inducing vectors 120576 rationallywe here devise a TOPSIS-based method in Algorithm IIIObviously the Algorithm III is of generality for MADMwithdifferent forms of linguistic expression

Algorithm III TOPSIS-based method for obtaining orderinducing vectors 120576Step 1 Let the decision maker define IVDHF UUBL idealsolution(s) 119903+119894 and IVDHF UUBL negative ideal solution(s)119903minus119894 corresponding to each attribute in which

119903+119894 = (119903+1198941 119903+1198942 119903+119894119901) 119903minus119894 = (119903minus1198941 119903minus1198942 119903minus119894119902) (35)

where 119901 and 119902 denotes the number of 119903+119894 and 119903minus119894 respectively

Step 2 Calculate distances from each element in decisionmatrix to ideal solution(s) and negative solution(s) respec-tively

119889+119894119895 = max119897=12119901

119889 (119903119894119895 119903+119894119897 ) 119889minus119894119895 = min

119897=12119902119889 (119903119894119895 119903minus119894119897 ) (36)

where 119889(119903119894119895 119903+119894119897 ) and 119889(119903119894119895 119903minus119894119897 ) are calculated by the distancemeasure in Definition 6 If 119901 gt 1 we choose the onewith maximum distance if 119902 gt 1 we choose the one withminimum distance

Step 3 Calculate coefficients according to

119888119894119895 = 119889minus119894119895119889+119894119895 + 119889minus119894119895 119894 = 1 2 119899 119895 = 1 2 119898 (37)

18 Complexity

Table 2 Decision matrix in the form of IVDHF UUBLS1199091 1199092 11990931198601([AHVH] [02 03][02 04] [03 04]) ([HT] [04 05][05 06] [02 03] [02 04]) ([ALAH] [01 02][01 03] [06 07])1198602([MH] [05 06][02 03]) ([AHT] [06 07][01 02] ) ([MVH] [02 03][05 06] [06 07])1198603([VLAL] [03 04][04 05] [05 06]) ([MVH] [06 07] [07 08][01 02]) ([VLL] [04 05][02 03])1198604([MAH] [02 04][05 06]) ([LAL] [03 05][02 03]) ([ALM] [06 07][01 02] [02 03])1198605([ALM] [04 05][01 02] [04 05]) ([AHH] [06 07][01 02]) ([HVH] [04 05][06 07] [01 03])1198606([LM] [01 02][03 05] [03 05]) ([VLL] [02 04] [05 06][02 03]) ([AHH] [03 04][04 05])1198607([VLL] [04 05][05 06] [02 03] [02 04]) ([MAH] [05 06] [07 08][01 02]) ([ALM] [04 05][05 06] [03 04])1198608([ALAH] [01 03][04 06]) ([AHH] [05 07][01 02] [02 03]) ([AHH] [04 06][03 04])

Table 3 Direct influence matrix Z1198601 1198602 1198603 1198604 1198605 1198606 1198607 11986081198601 0 3 4 1 3 1 2 41198602 1 0 2 4 4 2 3 11198603 4 2 0 1 3 2 4 11198604 2 3 1 0 3 4 1 21198605 4 2 1 3 0 1 4 31198606 2 4 3 4 4 0 4 21198607 4 1 2 3 3 1 0 41198608 1 2 4 3 3 2 4 0

Step 4 Derive order inducing variable 120576119894119895 by descendingorder of 119888119894119895 The greater the value 119888119894119895 the bigger the 1205761198941198956 Illustrative Example and Experiments

61 Illustrative Case Study Suppose a mega infrastructureproject is selecting green contractor for one of its subprojectsA panel of decision-makers including managers professorsand competitive intelligence experts has been organized tocomprehensively evaluate three alternative contractors (ie1199091 1199092 and 1199093) under the eight attributes (ie 1198601ndash1198608) listedin Section 2

To elicit true decision preferences we apply theIVDHF UUBLS to assess the three alternatives (1199091ndash1199093)Firstly decision-makers are asked to vote on 119909119895 under eachattribute119860 119894 using different unbalanced linguistic term sets 1198781 1198782 and 1198783 where 1198781 = 119873 119871119860119871119872119860119867119867119876119867119881119867119860119879 1198791198782 = 119873119860119873119881119871 119876119871 119871 119860119871119872119860119867119881119867 119879 1198783 = 119873119881119871119871119860119872119872119876119872119860119867119867 119879 In Figure 4 we show the adoptedlinguistic hierarchies and relationship between 1198781 1198782 and1198783 Attributes 1198601 1198602 1198605 and 1198608 are evaluated by use of 1198781attributes11986031198604 and1198607 are evaluated by 1198782 and attribute1198606is evaluated by 1198783 The highest-voted uncertain unbalancedlinguistic term [119904119898 119904119899] are identified Secondly the panel of

decision-makers is asked to further express their opinionson membership degrees ℎ(119909) and nonmembership degrees119892(119909) to [119904119898 119904119899] Then we can derive compound assessments(ie ⟨119909 119904120599(119909) ℎ(119909) 119892(119909)⟩) in the form of IVDHF UUBLSand obtain the decision matrix in Table 2 Additionallyin order to examine the interdependency among attributesdecision-makers are also required to assess the influentialrelations among the eight attributes using DEMATEL-basedAlgorithm II The direct influence matrix is collected andshown in Table 3

Next we apply the proposed Algorithm I to solve thisgreen contractor selection problem To derive the synthe-sized attribute weighting vector in Algorithm I we assign120572 = 120573 = 05 For scenarios with no group attitudinalcharacters operator GIVDHFUUBLHA in Definition 12 ischosen In contrast for scenarios with group attitudinalcharacters operator I-GIVDHFUUBLHA in Definition 13will be chosen Further the panel of decision-makers is askedto determine their positive and negative ideal alternativesthat are collected in Table 4 where corresponding order-inducing vectors that are derived from Algorithm III arealso listed (see the rightmost column) Thereafter operatorI-GIVDHFUUBLHA in Definition 13 is utilized for aggrega-tion For GIVDHFUUBLHA and I-GIVDHFUUBLHA we

Complexity 19

Table 4 Ideal alternatives (119903+ and 119903minus) and derived order-inducing vectors 120576119903+ 119903minus 1205761199091 1199092 11990931198601 ([VHT] 1 0) ([NL] 1 0) 5 6 61198602 ([MAH] [07 08] [01 02]) ([NL] 1 0) 1 1 41198603 (T 1 0) (L 1 0) 8 7 81198604 ([ALAH] 09 01) (T 1 0) 2 2 11198605 ([HT] [09 1] 0) (N 1 0) 7 5 51198606 (M 1 0) (N 1 0) 6 3 31198607 (M 1 0) ([NL] 1 0) 3 8 71198608 ([MAH] 1 0) (N 1 0) 4 4 2

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 1 2 3 4

0 1 2

N L AH H QH VH AT TM

Level 1 l(13)

Level 2 l(25)

Level 3 l(39)

Level 4 l(417)

Unbalanced linguistic term set S1

N AH VH TM Unbalanced linguistic term set S2VL L ALAN QL

N H TM Unbalanced linguistic term set S3

L AM AHQM

AL

VL

Figure 4 Unbalanced linguistic term sets (1198781 1198782 and 1198783) and their mapping in linguistic hierarchies

assume 120582=2 and 119908 = (0 0 015 025 025 025 01 0)119879Note that 119908 is a position weighting vector which is derivedby the fuzzy semantic quantitative operator [99] Aftercompleting all steps in Algorithm I (see Appendix C) weobtain the ranking results and present them in Table 5 Forthe scores derived from Step 5 we find the ranking order is1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the best solution Step 7gives the ranking order of 1199092 ≻ 1199091 ≻ 1199093 indicating 1199092 is stillthe most appropriate one while 1199091 is better than 1199093 under theinfluence of 12057662 Experiments on Ranking Results along with VariousParameter Configurations In order to inspect the impactsof various parameter configurations on ranking results inthis section more numerical examples are carried out Alleight operators that defined in Section 41 are applied to theabove illustrative case where the parameter 120582 is configured asintegers ranging from 1 to 20 and parameter 120572 is configuredto a value set of 0 03 05 08 09 1

Firstly the four basic aggregation operators of GIVDH-FUUBLWA GIVDHFUUBLWG GIVDHFUUBLOWA and

Table 5 Ranking results of operators GIVDHFUUBLHA and I-GIVDHFUUBLHA

Aggregation operator Ranking orderGIVDHFUUBLHA 1199092 ≻ 1199093 ≻ 1199091I-GIVDHFUUBLHA 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG are applied to the former case Thesame configuration 120572 = 120573 = 05 is chosen while 120582 is testedas integers ranging from 1 to 20 The ranking results derivedfrom different parameter combinations have been collectedin Table 6 As can be seen four operators unanimouslyidentify the contractor 1199092 is the best one no matter how120582 changes Regarding priority relation between 1199091 and 1199093GIVDHFUUBLWA and GIVDHFUUBLOWA generate thesame result of1199093 ≻ 1199091 while results fromGIVDHFUUBLWGandGIVDHFUUBLOWGgot a change to 1199091 ≻ 1199093 at a certainvalue of parameter 120582

Actually GIVDHFUUBLWA and GIVDHFUUBLOWAare generalized version of weighted arithmetic aggregationoperators that producemore favorable results by emphasizing

20 Complexity

Table 6 Ranking orders obtained by GIVDHFUUBLWA GIVDHFUUBLOWA GIVDHFUUBLWG and GIVDHFUUBLOWG along with120582Operators 120582 (Integer) Ranking orderGIVDHFUUBLWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLOWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLWG [1 2] 1199092 ≻ 1199093 ≻ 1199091[3 20] 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG [1 8] 1199092 ≻ 1199093 ≻ 1199091[9 20] 1199092 ≻ 1199091 ≻ 1199093

Table 7 Ranking results obtained by GIVDHFUUBLHA and I-GIVDHFUUBLHA along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHA I-GIVDHFUUBLHA120572=0 120573=1 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=03 120573=07 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=05 120573=05 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=08 120573=02 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=09 120573=01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=1 120573=0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093

Table 8 Ranking results obtained by GIVDHFUUBLHG and I-GIVDHFUUBLHG along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHG I-GIVDHFUUBLHG

120572 = 0 120573 = 1 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 03 120573 = 07 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 05 120573 = 05 [1 8] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[9 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 08 120573 = 02 [1 17] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[18 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 09 120573 = 01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572 = 1 120573 = 0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093membership degrees to a uncertain linguistic term thus canbe considered as optimistic operators while GIVDHFU-UBLWG and GIVDHFUUBLOWG are generalized geomet-ric aggregation operators that producemore favorable resultsby emphasizing nonmembership degrees thus can be consid-ered as pessimistic operators [100 101]Therefore the rankingresults in Table 6 could be perceived as for optimism-baseddecision-making alternative 1199093 always stands out better than1199091 when 120582 gradually amplifies degree of optimism but forpessimism-based decision-making 1199091 turns out better than1199093 when 120582 gradually amplifies degree of pessimism 120582 canbe considered as the parameter for amplifying degree ofoptimism or pessimism [101]

Next to inspect the impacts of various combinationsof 120596119860119868 and 120596119860119863 on ranking results another four aggrega-tion operators of GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG also have beenapplied to the above illustrative case with varying 120572 120573 and 120582

All results have been collected in Tables 7 and 8 for compari-son As can be seen fromTable 7 ranking results generated byoperator GIVDHFUUBLHA keep the same as 1199092 ≻ 1199093 ≻ 1199091for all test combinations of parameters However as shownin Table 8 although GIVDHFUUBLHG is also capable ofidentifying 1199092 as the best contractor the priority relationsbetween 1199091 and 1199093 hold similar changing patterns in rankingresults in Table 6 that were output byGIVDHFUUBLWGandGIVDHFUUBLOWG Regarding decision situations whereobjective attributeweighting vector120596119860119863 is only considered (120572= 0) or its dominant importance (such as 120572 = 03) is accepted1199091 ≻ 1199093 is a clear observation when parameter 120582 is set toan integer in interval [8 20] Contrariwise under decisionscenarios where subjective attribute weighting vector 120596119860119868 isonly needed (120572 = 1) or its dominant importance (such as 120572 =09) is acknowledged 1199093 ≻ 1199091 holds all the way 120582 varies

In order to more clearly show the score trajectories ofalternative contractors along variations of parameter 120582 we

Complexity 21

HAInducedHA

InducedHA-2HA-2

HA-3

InducedHA-1InducedHA-3

= =

HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(a) 120572 = 0

HAInducedHA

= 03 = 07

InducedHA-2

HA-2HA-3

InducedHA-1 InducedHA-3HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(b) 120572 = 03

HAInducedHA

= 5 = 05

InducedHA-2HA-2

HA-3

InducedHA-1 InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(c) 120572 = 05

HAInducedHA

= 08 = 02

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572 = 08

HAInducedHA

= 9 = 01

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(e) 120572 = 09

HAInducedHA

246 2462 2464 2466 2468 247

= 1 = 0InducedHA-2HA-2

HA-3

HA- 3

HA-3

InducedHA-1

InducedHA-3

HA-1

HA-1

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

1085

1086

1087

(f) 120572 = 1

Figure 5 Score trajectories obtained by aggregation operators HA and InducedHA with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

further configure parameter 120582 to the value set of [0 01100]and conduct computational experiments to collect scorevalues and ranking results (as shown in Figures 5 and 6) byfour aggregation operators of GIVDHFUUBLHA GIVDH-FUUBLHG I-GIVDHFUUBLHAand I-GIVDHFUUBLHGFor more clarity in Figures 5 and 6 please note that weuse HA HG Induced-HA and Induced HG to respec-tively denote GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG In Figure 5it can be observed that when dominant importance of 120596119860119863

is emphasized (120572 = 0 120572 = 03 120572 = 05) GIVDHFUUBLHAis capable of differentiating the three alternative contractorsas same ranking order of 1199092 ≻ 1199093 ≻ 1199091 for all 120582 but withincreases in 120572 the margins between scores of 1199091 and 1199093 nar-rows and the rank order changes to 1199091 ≻ 1199093when 120572 gets 09and 1 Interestingly in Figure 6 although GIVDHFUUBLHGis able to significantly differentiate alternative contractors 1199091and 1199093 when dominant importance of 120596119860119863 is emphasized (120572= 0 120572 = 03 120572 = 05) rank order of 1199091 and 1199093 obviouslychanges once from 1199093 ≻ 1199091 to 1199091 ≻ 1199093 when 120582 is bigger

22 Complexity

HGInducedHG

= = 1

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(a) 120572= 0

HGInducedHG

= 3 = 07

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(b) 120572= 03

HGInducedHG

= 05 = 05

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(c) 120572= 05

HGInducedHG

= 8 = 02

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572= 08

HGInducedHG

292 293 294 295 296 297minus08

minus0795

minus079

= 09 = 01

HA-1

HA-3

InducedHG-2

HG-1

HG-1

HG-2

HG-3

HG-3 InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4

minus3

minus2

minus1

0

1

2

scor

e val

ues

(e) 120572= 09

HGInducedHG

50 60 70 80 90 100minus095minus09

minus085

= 1 = 0

HG-x3

HG-x1

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(f) 120572= 1

Figure 6 Score trajectories obtained by aggregation operators HG and InducedHG with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

than 8 When 120572 increases to 1 GIVDHFUUBLHGmaintainsthe rank order of 1199093 ≻ 1199091 but their score trajectorieswalk very close Comparatively I-GIVDHFUUBLHA andI-GIVDHFUUBLHG generate consistent ranking results of1199092 ≻ 1199091 ≻ 1199093 no matter 120582 varies in all the above experiments

In sum when parameters are configured with typical val-ues (ie 120572 = 05 and 120582=1) and no additional attitudinal infor-mation is taken into consideration the proposed aggregationoperators unanimously derive the same rank order 1199092 ≻1199093 ≻ 1199091 of for the three alternative contractors However

with the amplification effect of parameter 120582 analysis resultsfrom linear combination of 120596119860119868 and 120596119860119863 reveal alternative1199092 which consistently is the best choice but rank order of1199091 and 1199093 changes because their score trajectories get closeand crossed which probably indicates that only decision-making informationderived from the decisionmatrix is inad-equate to significantly differentiate alternatives 1199091 and 1199093While by integrating additional decision information aboutattitudinal characters of group experts I-GIVDHFUUBLHAand I-GIVDHFUUBLHG manage to generate the identical

Complexity 23

Table 9 Order-inducing vectors obtained by our TOPSIS-based method for problem in [38]

1198661 1198662 1198663 11986641198601 02522 01629 02282 027141198602 01 05243 01364 046891198603 04081 01756 05244 037791198604 04507 01409 01064 013221198605 00697 02203 03753 01161

Table 10 Order-inducing vectors obtained by our TOPSIS-based method for problem in [39]

1198661 1198662 1198663 11986641198601 04548 04969 04923 047321198602 05092 047 04768 046451198603 07192 04888 04953 049151198604 06722 06163 05706 049291198605 06104 04942 06463 04946

Table 11 Ranking results obtained by applying different order-inducing vectors to [38 39]

Problems Approaches Ranking results

Problem in [38] IHFLOWA [38] + order-inducing vectors in [38] 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198605 ≻ 1198604

IHFLOWA [38] + order-inducing vectors in Table 9 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198604 ≻ 1198605

Problem in [39] I-IVDHFLOWA [39] + order-inducing vectors in [39] 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198602 ≻ 1198601

I-IVDHFLOWA [39] + order-inducing vectors in Table 10 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602

permutation of 1199092 ≻ 1199091 ≻ 1199093 which verifies that ourproposed TOPSIS-based order-inducing approach is prac-tical and useful in helping work out decision results morereasonably and robustly

63 Application Studies of Proposed TOPSIS-Based Order-Inducing Method When confronting complex decision-making situations that need additional attitudinal charactersof decision-makers Yager and Filev [88] and Yager [102]introduced and encouraged to employ order-inducing vectorto represent those attitudinal characters Since then theidea of order-inducing vector has been extended to variousdecision-making environments [98 103] such as those basedon intuitionistic fuzzy information [104 105] linguistic infor-mation [106] hesitant fuzzy linguistic information [38] andinterval-valued dual hesitant fuzzy uncertain linguistic infor-mation [39] Generally in existing literature order-inducingvectors rely on experts to directly give [38 39 104ndash106]

In order to examine the adaptability of our TOPSIS-based order-inducing method to linguistic decision-makingenvironments we respectively apply the method to resolvethe same problems in references [38 39] Due to fact thatthere is no descriptions about how the order-inducing vectorswere derived in both references [38 39] we here applyextremums into our TOPSIS-based method to obtain order-inducing vectors that is (1199047 1 0) and (1199041 0 1) for[38] ([1199046 1199046] 1 0) and ([1199040 1199040] 0 1) for [39] Thenwe obtain our order-inducing vectors listed in Table 9 forthe problem adopted in [38] and our order-inducing vectorslisted in Table 10 for [39]

Now we apply the order-inducing vectors as shownin Tables 9 and 10 to resolve the problems in [38] and[39] respectively Regarding the problem in [38] the scorevalues of all alternatives are calculated as 119878(1198601)=02602119878(1198602)=02417 119878(1198603)=03577 119878(1198604)=0221 and 119878(1198605)=0185so the corresponding ranking result is 1198603 ≻ 1198601 ≻ 1198602 ≻1198604 ≻ 1198605 As for the problem in [39] the score values ofall alternatives are computed as 119878(1198601)=39954 119878(1198602)=3783119878(1198603)=70827 119878(1198604)=71792 and 119878(1198605)=5813 we thus obtainthe ranking result 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602 Formore clarity brief descriptions of comparative approachesand obtained ranking results also have been put together inTable 11

According to Table 11 the approaches that employ ourorder-inducing vectors derive the same ranking order ofthe better three alternatives for both comparative problemsfrom [38 39] As seen our proposed TOPSIS-based order-inducing method exhibits effective and more understand-able Furthermore when decision-makers are incapable ofdirectly put forward all desired order-inducing vectors due tocomplexity in ill-structured problems our method providesa feasible directions to obtain order-inducing vectors byfiguring out ideal solutions Comparatively our proposedTOPSIS-based order-inducing method behaves effective andmore operational in practical use

64 Comparison with Decision-Making Approach Based onDegraded form of IVDHFULS Note that our approachis intrinsically adaptive to decision situations based ondegraded forms of IVDHF UUBLS such as the dual hesitant

24 Complexity

fuzzy linguistic sets introduced by Yang and Ju [54] Tofurther inspect feasibility of our approach we here conductcomparative study on the same case adopted by Yang and Ju[54] We firstly convert their original decision matrix intothe form of IVDHF UUBLS crisp membership and non-membership degrees are equivalently changed to intervalswith same upper and lower limits (eg 02 turns out to be[02 02]) and linguistic variables are rewritten as uncertainlinguistic variables (eg 1199042 turns out to be [11990421199042])

Since the case only got three attributes among whichinterdependency could be not significant under generalassumption weighting vector 120596119860119868 is not calculated Andfor straight comparison order inducing vector is notincluded either Then the argument-dependent attributeweighting vector can be obtained from Eq(29) as 120596119860119863 =(03192 03132 03676)119879 By use of Algorithm Iwith operatorIVDHFUUBLWA (ie GIVDHFUUBLWA in case of 120582 = 1see Appendix B) we have the scores of alternatives 119904(1198601) =03412 119904(1198602) = 01354 119904(1198603) = 03932 119904(1198604) = 07352Therefore the ordering of alternatives is1198604 ≻ 1198603 ≻ 1198601 ≻ 1198602which is mainly in alignment with the ordering of1198604 ≻ 1198601 ≻1198603 ≻ 1198602 in Yang and Ju [54] It is worth noticing that 120596119860119863used in our approach is derived objectively from assessmentswhile attribute weights 120596lowast = (035 025 040)119879used by Yangand Ju [54] were assigned empirically Differences between120596119860119863 and 120596lowast lead to different ordering of1198601 and1198603 Compar-atively speaking our approach holds more adaptability andobjectivity

In general our proposed MADM approach in this paperexhibits an effective way to tackle complex green contractorselection problems in which subjective attributes weightingvector (such as120596119860119868) and objective attributes weighting vector(such as 120596119860119863) exist To comprehensively utilize these twofundamental facets of attributes weighting information ourcase study and computational experiments suggest that linearcombination method (ie 120596 = 120572120596119860119868 + 120573120596119860119863) is effectiveto reflect effects of 120596119860119868 and 120596119860119863 by configuring parameter 120572or 120573 When there is no specific expert opinions to specify orhard to obtain we suggest that parameter 120572 chooses the valueset of 0 02 05 08 1 Furthermore generalized aggrega-tion operators are suggested to inspect score trajectoriesof alternative contractors along increasing of the operatorsrsquoamplification parameter 120582 If some alternative contractorsrsquoscore values get very close or crossed so that they cannot bedifferentiated from each other significantly we suggest to askthe panel of experts to derive opinions of ideal solutions andutilize our TOPSIS-based order-inducing method to includethose expert opinions as attitudinal decision informationthereby obtaining more rational and robust decision results

7 Conclusions

Aiming at tackling complex decision-making problems ofgreen contractor selection that often hold simultaneouslythe characteristics of decision hesitancy attributes inter-dependency and group complex attitudinal characters wehave developed an effective IVDHF UUBLS-based approach

We utilize IVDHF UUBLS to elicit complicate decision-makersrsquo assessments more objectively and comprehensivelywe employ DEMATEL for addressing attribute interdepen-dency more precisely and we determine group attitudinalcharacters as order-inducing vectors based on ideal attitudesof experts rather than by empirical assignment The mainadvantages of our approach are (i) IVDHF UUBLS offers anadequate way to depict decision hesitancy (ii) the designeddistance measure for IVDHF UUBLS avoids informationdistortion of traditional ones which artificially add mis-matching membership or nonmembership degrees (iii) thedeveloped generalized operators include traditional ones asspecial cases so as to provide flexibility in MADM basedon IVDHF UUBLS (iv) the devised synthesized attributeweighting scheme provides an adaptive way to consider twonecessary aspects attributes interdependency-based weightsand argument-dependent weights (v) the TOPSIS-basedmethod can deduce reasonable order-inducing vectors ratherthan empirical ones and holds generality in other linguisticdecision environments Results of case studies and experi-ments have verified the validity of the proposed approach

When multiple decision organizations are involvedgroup decision-making based on IVDHF UUBLS is animportant research direction Besides inducing preferencesfor decision-making problems of high uncertainty interactivemechanisms should also be studied to extract preferences Inaddition empirical studies of the proposed approach can becarried out to various areas such as investment evaluationproject management and vehicle selection in fleet operations

Appendix

A Basic Notions for IVDHFS and UnbalancedLinguistic Term Set

Definition A1 (see [47]) Letting 119883 be a fixed set then anIVDHFS on 119883 is defined as119863 = ⟨119909 ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (A1)

where ℎ(119909) = ⋃[120583119871120583119880]isinℎ(119909)120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and119892(119909) = ⋃[]119871]119880]isin119892(119909)] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two setsof interval values in [0 1] denoting possible membershipand non-membership degrees of element 119909 isin 119883 to theset 119863 respectively with conditions 120583 ] isin [0 1] and0 le (120583119880)+ + (]119880)+ le 1 and for all 119909 isin 119883 (120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880

For convenience normally 119889 = ℎ 119892 is called an interval-valued dual hesitant fuzzy element (IVDHFE) and 119863 is theset of all IVDHFEs

Definition A2 (see [107]) Suppose 119878 = 119904119895 | 119895 = 0 1 119892 minus1 is a finite and totally ordered discrete linguistic term setwhere 119904119895 represent possible values for a linguistic variableand 119892 is an odd cardinality 119878 is uniformly and symmetricallydistributed if the following two conditions are satisfied (1)

Complexity 25

there exists a unique constant 120582 gt 0 such that 119873119878(119904119894) minus119873119878(119904119895) = 119897(119894 minus 119895) for all 119894 119895 = 0 1 119892 minus 1 (2) Let 119878119877 =119904 | 119904 isin 119878 119904 gt 119878119862 and 119878119871 = 119904 | 119904 isin 119878 119904 lt 119878119862 Let C(119878119877) andC(119878119871) be the cardinality of 119878119877 and 119878119871 then C(119878119877) = C(119878119871)If 119878 is uniformly and symmetrically distributed then 119878 iscalled a balanced linguistic term set Otherwise 119878 is calledan unbalanced linguistic term set

The following 2-tuple fuzzy linguistic representationmodel extends traditional linguistic term set to a continuouscase so as to facilitate computing with linguistic variables

Definition A3 (see [59]) Let 119878 = 1199040 1199041 119904119892minus1 be alinguistic term set and 120573 isin [0 119892] Then a 2-tuple fuzzylinguistic variable expresses the equivalent information to 120573is defined as

Δ [0 119892] 997888rarr 119878 times [minus05 05) (A2)

Δ (120573) = (119904119894 120572) with

119904119894 119894 = 119903119900119906119899119889 (120573)120572 = 120573 minus 119894 120572 isin [minus05 05)

(A3)

Δminus1 (119904119894 120572) = 119894 + 120572 = 120573 (A4)

where 119903119900119906119899119889() is the usual rounding operation and120572 is calledsymbolic translation

To obtain 2-tuple fuzzy linguistic representations ofunbalanced linguistic terms the concept of linguistic hierar-chies ie 119871119867 = ⋃119905 119897(119905 119899(119905)) is used 119897(119905 119899(119905)) is a linguistichierarchy with 119905 indicating the level of hierarchy and 119899(119905)denotes the granularity of the linguistic term set of 119905 Herreraet al [59] defined the following transformation functionsbetween labels from different levels in multigranular linguis-tic information contexts without loss of information

Definition A4 (see [59]) In linguistic hierarchies 119871119867 =⋃119905 119897(119905 119899(119905)) whose linguistic term sets are represented by119878119899(119905) = 119904119899(119905)0 119904119899(119905)119899(119905)minus1 the transformation function froma linguistic label in level 119905 to a label in consecutive level 1199051015840 isdefined as 1198791198651199051199051015840 119897(119905 119899(119905)) 997888rarr 119897(1199051015840 119899(1199051015840)) such that

1198791198651199051199051015840 (119904119899(119905)119894 120572119899(119905))= Δ 1199051015840 (Δminus1119905 (119904119899(119905)119894 120572119899(119905)) (119899 (1199051015840) minus 1)119899 (119905) minus 1 ) (A5)

By use of the above transformation function any 2-tuplelinguistic representation can be transformed into a term in119871119867 Detailed transformation procedures are listed as follows

(1) Representation in linguistic hierarchy To transformthe unbalanced terms of 119878 into the corresponding terms inthe 119871119867 transformation function 120595 is employed to associateeach unbalanced linguistic 2-tuple (119904119894 120572) with its linguistic 2-tuple in 119871119867(119878) ie

120595 119878 997888rarr 119871119867(119878) (A6)

so that 120595(119904119894 120572) = (119904119866(119894)119868(119894) 120582) for forall(119904119894 120572) isin 119878(2) Computational phase Firstly transform (119904119866(119894)119868(119894) 120582) intolinguistic 2-tuples denoted as (119904119899(1199051015840)

1198681015840(119894) 1205821015840) in 119878119899(1199051015840) where

(119904119899(1199051015840)1198681015840(119894)

1205821015840) = 119879119865 (119904119866(119894)119868(119894) 120582)= Δ(Δminus1 (119904119866(119894)119868(119894) 120582) sdot (119899 (1199051015840) minus 1)119866 (119894) minus 1 ) (A7)

Then a computational model is used over 119878119899(1199051015840) with aresult denoted as (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)(3) Retranslation process The result (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)is transformed into the unbalanced term in 119878 by using thetransformation function 120595minus1 ie

120595minus1 119871119867 (119878) 997888rarr 119878 (A8)

so that 120595minus1(119904119899(1199051015840)119903 120582119903) = (119904119903119890119904119906119897119905 120582119903119890119904119906119897119905) isin 119878B Details of Theorem 14

Proof Given a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector for119904119889119895 and 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 120582 gt 0 then(1) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLWA and GIVDHFUUBLWG operators respectively(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLOWA and GIVDHFUUBLOWG operators respectively

(2) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to theIVDHF UUBL weighted average (IVDHFUUBLWA)and IVDHF UUBL weighted geometric (IVDHFUUBLWG)operators respectively where

26 Complexity

119868119881119863119867119865119880119880119861119871119882119860120596 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

120596119895119904119889119895= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus

119899prod119895=1

(1 minus 120583119871119895 )120596119895 1 minus 119899prod119895=1

(1 minus 120583119880119895 )120596119895]] [[

119899prod119895=1

(]119871119895)120596119895 119899prod119895=1

(]119880119895 )120596119895]])

(B1)

119868119881119863119867119865119880119880119861119871119882119866120596 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889119895120596119895

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[

119899prod119895=1

(120583119871119895 )120596119895 119899prod119895=1

(120583119880119895 )120596119895]] [[1 minus

119899prod119895=1

(1 minus ]119871119895)120596119895 1 minus 119899prod119895=1

(1 minus ]119880119895 )120596119895]])

(B2)

(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHA andGIVDHFUUBLHG operators reduce to the IVDHF UUBL

orderedweighted average (IVDHFUUBLOWA) and IVDHFUUBL ordered weighted geometric (IVDHFUUBLOWG)operators respectively where

119868119881119863119867119865119880119880119861119871119874119882119860119908 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

119908119895119904119889120590(119895)= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) )))

119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) )))

]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus

119899prod119895=1

(1 minus 120583119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus 120583119880120590(119895))119908119895]] [[

119899prod119895=1

(]119871120590(119895))119908119895 119899prod119895=1

(]119880120590(119895))119908119895]])

(B3)

119868119881119863119867119865119880119880119861119871119874119882119866119908 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889120590(119895)119908119895

= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[

119899prod119895=1

(120583119871120590(119895))119908119895 119899prod119895=1

(120583119880120590(119895))119908119895]] [[1 minus

119899prod119895=1

(1 minus ]119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus ]119880120590(119895))119908119895]])

(B4)

Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

References

[1] V W Y Tam and C M Tam ldquoWaste reduction throughincentivesA case studyrdquoBuilding Research and Information vol36 no 1 pp 37ndash43 2008

[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

[11] X Qi C Liang and J Zhang ldquoGeneralized cross-entropy basedgroup decision making with unknown expert and attributeweights under interval-valued intuitionistic fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 79 pp 52ndash642015

[12] H Zimmermann Fuzzy Set Theorymdashand Its ApplicationsKluwer Academic Boston Mass USA 2001

[13] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 pp 199ndash249 1975

[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

[16] B T Nuong K-W Kim L Prathumratana et al ldquoSustainabledevelopment in the mining sector and its evaluation usingfuzzy AHP (Analytic Hierarchy Process) approachrdquo GeosystemEngineering vol 14 no 1 pp 43ndash50 2011

[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

[18] E K Zavadskas Z Turskis and J Antucheviciene ldquoSelectinga contractor by using a novel method for multiple attributeanalysis Weighted aggregated sum product assessment withgrey values (WASPAS-G)rdquo Studies in Informatics and Controlvol 24 no 2 pp 141ndash150 2015

[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

[21] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

[25] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009

[26] Z Hatush and M Skitmore ldquoContractor Selection UsingMulticriteria Utility Theory An Additive Modelrdquo Building andEnvironment vol 33 no 2-3 pp 105ndash115 1998

[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

[30] AM Anvuur andMM Kumaraswamy ldquoConceptualmodel ofpartnering and alliancingrdquo Journal of Construction Engineeringand Management vol 133 no 3 pp 225ndash234 2007

[31] A H I Lee H Kang C F Hsu and H Hung ldquoA green supplierselection model for high-tech industryrdquo Expert Systems withApplications vol 36 no 4 pp 7917ndash7927 2009

[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

[33] P Humphreys R McIvor and F Chan ldquoUsing case-basedreasoning to evaluate supplier environmental managementperformancerdquo Expert Systems with Applications vol 25 no 2pp 141ndash153 2003

[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

[45] Bin Zhu Zeshui Xu and Meimei Xia ldquoDual Hesitant FuzzySetsrdquo Journal of Applied Mathematics vol 2012 Article ID879629 13 pages 2012

[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

[51] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[52] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

[55] Z Tao X Liu H Chen and L Zhou ldquoUsing new versionof extended t-norms and s-norms for aggregating intervallinguistic labelsrdquo IEEETransactions on SystemsManCyberneticsSystems vol vol p 15 2016

[56] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[57] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[58] L Martınez M Espinilla J Liu L G Perez and P J SanchezldquoAn evaluation model with unbalanced linguistic informationapplied to olive oil sensory evaluationrdquo Journal of Multiple-Valued Logic and Soft Computing vol 15 no 2-3 pp 229ndash2512009

[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

[62] L Marin A Valls D Isern A Moreno and J M MerigoldquoInduced unbalanced linguistic ordered weighted average andits application in multiperson decision makingrdquo The ScientificWorld Journal vol 2014 2014

[63] A Baykasoglu V Kaplanoglu Z D U Durmusoglu and CSahin ldquoIntegrating fuzzy DEMATEL and fuzzy hierarchicalTOPSIS methods for truck selectionrdquo Expert Systems withApplications vol 40 no 3 pp 899ndash907 2013

[64] S K Patil and R Kant ldquoA hybrid approach based on fuzzyDEMATEL and FMCDM to predict success of knowledge man-agement adoption in supply chainrdquoApplied Soft Computing vol18 pp 126ndash135 2014

[65] S Liyin and Y Hong ldquoImproving environmental performanceby means of empowerment of contractorsrdquo Management ofEnvironmental Quality An International Journal vol 17 no 3pp 242ndash257 2006

[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

[69] Y Kazancoglu I Kazancoglu and M Sagnak ldquoA new holisticconceptual framework for green supply chain management

performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

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Complexity 15

Then go to Step 8

Step 6 By use of the TOPSIS-based method developed in thefollowing Section 53 we determine order-inducing vectors 120576to reflect decision-makersrsquo group attitudinal characters

Step 7 Utilizing induced generalized aggregation operatorsto get the overall IVDHF UUBLNs 119903119895(119895 = 1 2 119898) foreach alternative 119909119895 we have

119868-119866119868119881119863119867119865119880119871119867119860(1199031119895 1199032119895 119903119899119898) = ( 119899⨁119894=1

(119908119894 (119903120587119894119895)120582))1120582

= ⋃(119904120587119894119895ℎ120587119894119895119892120587119894119895)isin119903120587119894119895

([119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120587119894119895

))))120582)1120582 119904(sum119899119895=1 119908119894(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120587119894119895

))))120582)1120582]

⋃[120583119871120587119894119895 120583

119880120587119894119895]isinℎ120587119894119895 []

119871120587119894119895]119880120587119894119895]isin119892120587119894119895

[[(1 minus

119899prod119894=1

(1 minus (120583119871120587119894119895)120582)119908119894)1120582 (1 minus 119899prod119894=1

(1 minus (120583119880120587119894119895)120582)119908119894)1120582]]

[[1 minus (1 minus119899prod119894=1

(1 minus (1 minus ]119871120587119894119895)120582)119908119894)1120582 1 minus (1 minus 119899prod119894=1

(1 minus (1 minus ]119880120587119894119895)120582)119908119894)1120582]])

(22)

where 120596 = (1205961 1205962 120596119899)119879 is the weighting vector of119903119894119895 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 119903120587119894119895 is the 119894th largest of 119903119894119895 reordered by theorder-inducing vectors 120576 and

119903119894119895 = 119899120596119894119903119894119895 = ⋃(119904119894119895ℎ119894119895119892119894119895)isin119903119894119895

([119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119894119895 ))))119899120596119894120572119894119895

119904119899120596119894(Δ

minus11199050(11987911986511990511989511199050(120595(119904120573119894119895 ))))

] ⋃

[120583119871119894119895120583119880119894119895 ]isinℎ119894119895[]

119871119894119895]119880119894119895 ]isin119892119894119895

[1 minus (1 minus 120583119871119894119895)119899120596119894 1 minus (1 minus 120583119880119894119895 )119899120596119894] [(]119871119894119895)119899120596119894 (]119880119894119895)119899120596119894]) (23)

Step 8 Calculate scores 119878(119903119895) and accuracy degrees 119875(119903119895) for119903119895(119895 = 1 2 119898)Step 9 According to Definition 5 output ranking order ofalternative contractors

In Algorithm I to objectively derive unknown weight-ing vector for evaluative attributes a synthesized attributeweighting vector is devised by fusing two parts of concern(i) argument-dependent weighting vector and (ii) attribute-interdependences based weighting vector The former is toreflect effects of attribute values on attribute weights whilethe latter is to exploit effects of interdependences amongattributes on their weights For clarity processing flowchartof Algorithm I is depicted in Figure 3

51 Determining Argument-Dependent Attribute WeightingVector 120596119860119863 In order to derive attribute weighting vec-tor appropriately from assessments maximizing deviationmethod [91] is adopted here for its distinguishing ability

and objectivity [92] According to Wang [91] if performancevalues of each alternative have little differences under anattribute it implicates such an attribute plays a less importantrole otherwise it lives a more important role in decision-making Thus if one attribute has similar values across alter-natives it should be assigned a smaller weight contrariwiseit should be a bigger weight

In light of the idea in maximizing deviation methodwe develop a programming model to determine argument-dependent weighting vector 120596119860119863 For attribute 119860 119894 the stan-dard deviation of alternative 119909119895 to all the other alternatives isdenoted as

119865119894119895 (120596119860119863) = 119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (24)

Let119865119894(120596119860119863) represent deviation value of all alternatives toother alternatives according to attribute119860 119894 and be formulatedas follows

16 Complexity

Attribute weighting

Hesitant fuzzy linguistic MADM with attributes interdependency and decision makersrsquo group attitudinal characters

Step 9 Output ranking order of alternatives

interdependency among attributes by DEMATEL in Section 52

attributes values by standard deviation method in Section 51

Step 6 Ensure TOPSIS order inducing variables proposed in Section 53

Step 7 Utilizing proposed induced generalized hybrid aggregation operators developed in Section

41 to get overall value of each alternative

Step 8 Calculate score degrees and accuracy degrees

Step 4 Order inducingYes

Step 5 Choose proposed generalized operators in Section 41 to get overall value of each alternative

No

Collect decision matrix in the form of IVDHF_UUBLS

Complex problems of green contractor selection

Step 1 Weighting vector AD derived from Step 2 Weighting vector AI derived from

Step 3 Obtain synthesized weighting vector of attributes =AI + AD

Figure 3 Processing flowchart of Algorithm I

119865119894 (120596119860119863) = 119898sum119895=1

119865119894119895 (120596119860119863) = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (25)

Based on the analysis above we can get followingprogramming model (M-1) for obtaining optimal attributeweighting vector 120596119860119863

(119872 minus 1)max 119865 (120596119860119863) = 119899sum

119894=1119865119894 (120596119860119863) = 119899sum

119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894st 119899sum

119894=1(120596119860119863119894 )2 = 1 120596119860119863119894 ge 0 119894 = 1 2 119899 (26)

where 119889(119903119894119895 119903119894119897) can be calculated by Definition 6To solve model (M-1) we can construct Lagrange func-

tion

119871 (120596119860119863119894 120577) = 119899sum119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894+ 1205772 ( 119899sum

119894=1

(120596119860119863119894 )2 minus 1) (27)

where 120577 is the Lagrange multiplierDifferentiate with respect to 120596119860119863119894 (119894 = 1 2 119898) and 120577

and set these partial derivatives equal to zero the followingset of equations can be obtained

120597119871120597120596119860119863119894 = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) + 120577120596119860119863119894 = 0120597119871120597120577 = 119899sum

119894=1

(120596119860119863119894 )2 minus 1 = 0 (28)

By normalizing the solutions of above equations we canget a simple and exact formula for determining the attributeweights as follows

120596119860119863119894 = sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897)sum119899119894=1sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897) (29)

52 Deriving Attributes Interdependency-Based AttributeWeighting Vector 120596119860119868 Some efforts [53 60 84 93ndash95]

Complexity 17

have been made to address interdependencies betweenattributes in MADM However they unanimously cannotreveal the interdependences in a more complete way WhileDEMATAEL technique [63 96] gathers collective knowledgeto capture influences between attributes more precisely andcompletely so that weighting vector for attributes can berationally determined according to strength measure ofthose influences [64 97] Therefore in tackling the complexMADMwith interdependent attributes under IVDHF UUBLenvironments we present a DEMATEL-based method as thefollowing Algorithm II to derive attribute weighting vector120596119860119868 from the viewpoint of attributes interdependency

Algorithm II DEMATEL-basedmethod for ensuringweight-ing vector 120596119860119868Step 1 Construct initial direct-influence matrix 119885 = (119911119894119896)119899times119899

For attributes 119860 = 1198601 1198602 119860119899 decision-makersare asked to give direct effect that attribute 119860 119894 has onattribute 119860119896 using a five-point scale with scores representedby natural language (1) 0 mdash lsquoabsolutely no influencersquo (2)1 mdash lsquolow influencersquo (3) 2 mdash lsquomedium influencersquo (4) 3 mdashlsquohigh influencersquo (5) 4mdash lsquovery high influencersquoThen the initialdirect-influence matrix 119885 = (119911119894119896)119899times119899 can be established witheach element being the mean of same factors in differentdirect matrices

Step 2 Calculate normalized direct-influence matrix 119883 =(119909119894119896)119899times119899 Using matrix 119885 the normalized direct-influencematrix 119883 = (119909119894119896)119899times119899 is calculated according to

119909119894119896 = 119911119894119896119904 (30)

where 119904 = maxmax1le119894le119899sum119899119896=1 119911119894119896max1le119896le119899sum119899119895=1 119911119894119896 Allelements in matrix 119883 are complying with 0 le 119909119894119896 le 10 le sum119899119896=1 119909119894119896 le 1 and 0 le sum119899119894=1 119909119894119896 le 1 and at least onerow or column of the summation equals 1

Step 3 Derive total-influence matrix 119879 = (119905119894119896)119899times119899 Basedon matrix 119883 the total-influence matrix 119879 = (119905119894119896)119899times119899 can bederived by summing the direct effects and all of the indirecteffects

119879 = 119883 + 1198832 + 1198833 + sdot sdot sdot + 119883ℎ = 119883 (119868 minus 119883)minus1 (31)

where 119868 represents identity matrix When ℎ 997888rarr infin 119883ℎ =(0)119899times119899Step 4 Compute influencing and influenced degrees of eachattribute The sum of rows and the sum of columns withinthe total-influence matrix 119879 are respectively expressed as thevectors 119903 and 119888

119903 = (119903119894)119899times1 = ( 119899sum119896=1

119905119894119896)119899times1

119888 = (119888119896)119899times1 = ( 119899sum

119895=1

119905119894119896)119879

1times119899

(32)

where 119903119894 denotes the sumof the 119894th row inmatrix119879 and showsthe sum of direct and indirect effects that attribute 119860 119894 hason other attributes Similarly 119888119896 shows the sum of direct andindirect effects that attribute 119860119896 has receivedStep 5 Estimate weights of attributes based on influencedegrees according to

120596119868lowast119894 = [(119903119894 + 119888119894)2 + (119903119894 minus 119888119894)2]12 (33)

Then the weighting vector 120596119860119868 can be obtained by normaliz-ing (33)

120596119860119868119894 = 120596119868lowast119894sum119899119894=1 120596119868lowast119894 (34)

53 Obtaining Order Inducing Variables 120576 by TOPSIS-BasedMethod For ill-defined MADM problems induced aggre-gation operators [89 90 98] provide an effective channelfor experts to express complex attitudinal characters Thekey step in induced aggregation operators is to obtainorder-inducing vectors 120576 which however are presumptivelyassigned in most papers [62] Therefore in this sectionaiming at deducing the order-inducing vectors 120576 rationallywe here devise a TOPSIS-based method in Algorithm IIIObviously the Algorithm III is of generality for MADMwithdifferent forms of linguistic expression

Algorithm III TOPSIS-based method for obtaining orderinducing vectors 120576Step 1 Let the decision maker define IVDHF UUBL idealsolution(s) 119903+119894 and IVDHF UUBL negative ideal solution(s)119903minus119894 corresponding to each attribute in which

119903+119894 = (119903+1198941 119903+1198942 119903+119894119901) 119903minus119894 = (119903minus1198941 119903minus1198942 119903minus119894119902) (35)

where 119901 and 119902 denotes the number of 119903+119894 and 119903minus119894 respectively

Step 2 Calculate distances from each element in decisionmatrix to ideal solution(s) and negative solution(s) respec-tively

119889+119894119895 = max119897=12119901

119889 (119903119894119895 119903+119894119897 ) 119889minus119894119895 = min

119897=12119902119889 (119903119894119895 119903minus119894119897 ) (36)

where 119889(119903119894119895 119903+119894119897 ) and 119889(119903119894119895 119903minus119894119897 ) are calculated by the distancemeasure in Definition 6 If 119901 gt 1 we choose the onewith maximum distance if 119902 gt 1 we choose the one withminimum distance

Step 3 Calculate coefficients according to

119888119894119895 = 119889minus119894119895119889+119894119895 + 119889minus119894119895 119894 = 1 2 119899 119895 = 1 2 119898 (37)

18 Complexity

Table 2 Decision matrix in the form of IVDHF UUBLS1199091 1199092 11990931198601([AHVH] [02 03][02 04] [03 04]) ([HT] [04 05][05 06] [02 03] [02 04]) ([ALAH] [01 02][01 03] [06 07])1198602([MH] [05 06][02 03]) ([AHT] [06 07][01 02] ) ([MVH] [02 03][05 06] [06 07])1198603([VLAL] [03 04][04 05] [05 06]) ([MVH] [06 07] [07 08][01 02]) ([VLL] [04 05][02 03])1198604([MAH] [02 04][05 06]) ([LAL] [03 05][02 03]) ([ALM] [06 07][01 02] [02 03])1198605([ALM] [04 05][01 02] [04 05]) ([AHH] [06 07][01 02]) ([HVH] [04 05][06 07] [01 03])1198606([LM] [01 02][03 05] [03 05]) ([VLL] [02 04] [05 06][02 03]) ([AHH] [03 04][04 05])1198607([VLL] [04 05][05 06] [02 03] [02 04]) ([MAH] [05 06] [07 08][01 02]) ([ALM] [04 05][05 06] [03 04])1198608([ALAH] [01 03][04 06]) ([AHH] [05 07][01 02] [02 03]) ([AHH] [04 06][03 04])

Table 3 Direct influence matrix Z1198601 1198602 1198603 1198604 1198605 1198606 1198607 11986081198601 0 3 4 1 3 1 2 41198602 1 0 2 4 4 2 3 11198603 4 2 0 1 3 2 4 11198604 2 3 1 0 3 4 1 21198605 4 2 1 3 0 1 4 31198606 2 4 3 4 4 0 4 21198607 4 1 2 3 3 1 0 41198608 1 2 4 3 3 2 4 0

Step 4 Derive order inducing variable 120576119894119895 by descendingorder of 119888119894119895 The greater the value 119888119894119895 the bigger the 1205761198941198956 Illustrative Example and Experiments

61 Illustrative Case Study Suppose a mega infrastructureproject is selecting green contractor for one of its subprojectsA panel of decision-makers including managers professorsand competitive intelligence experts has been organized tocomprehensively evaluate three alternative contractors (ie1199091 1199092 and 1199093) under the eight attributes (ie 1198601ndash1198608) listedin Section 2

To elicit true decision preferences we apply theIVDHF UUBLS to assess the three alternatives (1199091ndash1199093)Firstly decision-makers are asked to vote on 119909119895 under eachattribute119860 119894 using different unbalanced linguistic term sets 1198781 1198782 and 1198783 where 1198781 = 119873 119871119860119871119872119860119867119867119876119867119881119867119860119879 1198791198782 = 119873119860119873119881119871 119876119871 119871 119860119871119872119860119867119881119867 119879 1198783 = 119873119881119871119871119860119872119872119876119872119860119867119867 119879 In Figure 4 we show the adoptedlinguistic hierarchies and relationship between 1198781 1198782 and1198783 Attributes 1198601 1198602 1198605 and 1198608 are evaluated by use of 1198781attributes11986031198604 and1198607 are evaluated by 1198782 and attribute1198606is evaluated by 1198783 The highest-voted uncertain unbalancedlinguistic term [119904119898 119904119899] are identified Secondly the panel of

decision-makers is asked to further express their opinionson membership degrees ℎ(119909) and nonmembership degrees119892(119909) to [119904119898 119904119899] Then we can derive compound assessments(ie ⟨119909 119904120599(119909) ℎ(119909) 119892(119909)⟩) in the form of IVDHF UUBLSand obtain the decision matrix in Table 2 Additionallyin order to examine the interdependency among attributesdecision-makers are also required to assess the influentialrelations among the eight attributes using DEMATEL-basedAlgorithm II The direct influence matrix is collected andshown in Table 3

Next we apply the proposed Algorithm I to solve thisgreen contractor selection problem To derive the synthe-sized attribute weighting vector in Algorithm I we assign120572 = 120573 = 05 For scenarios with no group attitudinalcharacters operator GIVDHFUUBLHA in Definition 12 ischosen In contrast for scenarios with group attitudinalcharacters operator I-GIVDHFUUBLHA in Definition 13will be chosen Further the panel of decision-makers is askedto determine their positive and negative ideal alternativesthat are collected in Table 4 where corresponding order-inducing vectors that are derived from Algorithm III arealso listed (see the rightmost column) Thereafter operatorI-GIVDHFUUBLHA in Definition 13 is utilized for aggrega-tion For GIVDHFUUBLHA and I-GIVDHFUUBLHA we

Complexity 19

Table 4 Ideal alternatives (119903+ and 119903minus) and derived order-inducing vectors 120576119903+ 119903minus 1205761199091 1199092 11990931198601 ([VHT] 1 0) ([NL] 1 0) 5 6 61198602 ([MAH] [07 08] [01 02]) ([NL] 1 0) 1 1 41198603 (T 1 0) (L 1 0) 8 7 81198604 ([ALAH] 09 01) (T 1 0) 2 2 11198605 ([HT] [09 1] 0) (N 1 0) 7 5 51198606 (M 1 0) (N 1 0) 6 3 31198607 (M 1 0) ([NL] 1 0) 3 8 71198608 ([MAH] 1 0) (N 1 0) 4 4 2

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 1 2 3 4

0 1 2

N L AH H QH VH AT TM

Level 1 l(13)

Level 2 l(25)

Level 3 l(39)

Level 4 l(417)

Unbalanced linguistic term set S1

N AH VH TM Unbalanced linguistic term set S2VL L ALAN QL

N H TM Unbalanced linguistic term set S3

L AM AHQM

AL

VL

Figure 4 Unbalanced linguistic term sets (1198781 1198782 and 1198783) and their mapping in linguistic hierarchies

assume 120582=2 and 119908 = (0 0 015 025 025 025 01 0)119879Note that 119908 is a position weighting vector which is derivedby the fuzzy semantic quantitative operator [99] Aftercompleting all steps in Algorithm I (see Appendix C) weobtain the ranking results and present them in Table 5 Forthe scores derived from Step 5 we find the ranking order is1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the best solution Step 7gives the ranking order of 1199092 ≻ 1199091 ≻ 1199093 indicating 1199092 is stillthe most appropriate one while 1199091 is better than 1199093 under theinfluence of 12057662 Experiments on Ranking Results along with VariousParameter Configurations In order to inspect the impactsof various parameter configurations on ranking results inthis section more numerical examples are carried out Alleight operators that defined in Section 41 are applied to theabove illustrative case where the parameter 120582 is configured asintegers ranging from 1 to 20 and parameter 120572 is configuredto a value set of 0 03 05 08 09 1

Firstly the four basic aggregation operators of GIVDH-FUUBLWA GIVDHFUUBLWG GIVDHFUUBLOWA and

Table 5 Ranking results of operators GIVDHFUUBLHA and I-GIVDHFUUBLHA

Aggregation operator Ranking orderGIVDHFUUBLHA 1199092 ≻ 1199093 ≻ 1199091I-GIVDHFUUBLHA 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG are applied to the former case Thesame configuration 120572 = 120573 = 05 is chosen while 120582 is testedas integers ranging from 1 to 20 The ranking results derivedfrom different parameter combinations have been collectedin Table 6 As can be seen four operators unanimouslyidentify the contractor 1199092 is the best one no matter how120582 changes Regarding priority relation between 1199091 and 1199093GIVDHFUUBLWA and GIVDHFUUBLOWA generate thesame result of1199093 ≻ 1199091 while results fromGIVDHFUUBLWGandGIVDHFUUBLOWGgot a change to 1199091 ≻ 1199093 at a certainvalue of parameter 120582

Actually GIVDHFUUBLWA and GIVDHFUUBLOWAare generalized version of weighted arithmetic aggregationoperators that producemore favorable results by emphasizing

20 Complexity

Table 6 Ranking orders obtained by GIVDHFUUBLWA GIVDHFUUBLOWA GIVDHFUUBLWG and GIVDHFUUBLOWG along with120582Operators 120582 (Integer) Ranking orderGIVDHFUUBLWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLOWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLWG [1 2] 1199092 ≻ 1199093 ≻ 1199091[3 20] 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG [1 8] 1199092 ≻ 1199093 ≻ 1199091[9 20] 1199092 ≻ 1199091 ≻ 1199093

Table 7 Ranking results obtained by GIVDHFUUBLHA and I-GIVDHFUUBLHA along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHA I-GIVDHFUUBLHA120572=0 120573=1 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=03 120573=07 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=05 120573=05 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=08 120573=02 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=09 120573=01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=1 120573=0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093

Table 8 Ranking results obtained by GIVDHFUUBLHG and I-GIVDHFUUBLHG along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHG I-GIVDHFUUBLHG

120572 = 0 120573 = 1 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 03 120573 = 07 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 05 120573 = 05 [1 8] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[9 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 08 120573 = 02 [1 17] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[18 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 09 120573 = 01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572 = 1 120573 = 0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093membership degrees to a uncertain linguistic term thus canbe considered as optimistic operators while GIVDHFU-UBLWG and GIVDHFUUBLOWG are generalized geomet-ric aggregation operators that producemore favorable resultsby emphasizing nonmembership degrees thus can be consid-ered as pessimistic operators [100 101]Therefore the rankingresults in Table 6 could be perceived as for optimism-baseddecision-making alternative 1199093 always stands out better than1199091 when 120582 gradually amplifies degree of optimism but forpessimism-based decision-making 1199091 turns out better than1199093 when 120582 gradually amplifies degree of pessimism 120582 canbe considered as the parameter for amplifying degree ofoptimism or pessimism [101]

Next to inspect the impacts of various combinationsof 120596119860119868 and 120596119860119863 on ranking results another four aggrega-tion operators of GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG also have beenapplied to the above illustrative case with varying 120572 120573 and 120582

All results have been collected in Tables 7 and 8 for compari-son As can be seen fromTable 7 ranking results generated byoperator GIVDHFUUBLHA keep the same as 1199092 ≻ 1199093 ≻ 1199091for all test combinations of parameters However as shownin Table 8 although GIVDHFUUBLHG is also capable ofidentifying 1199092 as the best contractor the priority relationsbetween 1199091 and 1199093 hold similar changing patterns in rankingresults in Table 6 that were output byGIVDHFUUBLWGandGIVDHFUUBLOWG Regarding decision situations whereobjective attributeweighting vector120596119860119863 is only considered (120572= 0) or its dominant importance (such as 120572 = 03) is accepted1199091 ≻ 1199093 is a clear observation when parameter 120582 is set toan integer in interval [8 20] Contrariwise under decisionscenarios where subjective attribute weighting vector 120596119860119868 isonly needed (120572 = 1) or its dominant importance (such as 120572 =09) is acknowledged 1199093 ≻ 1199091 holds all the way 120582 varies

In order to more clearly show the score trajectories ofalternative contractors along variations of parameter 120582 we

Complexity 21

HAInducedHA

InducedHA-2HA-2

HA-3

InducedHA-1InducedHA-3

= =

HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(a) 120572 = 0

HAInducedHA

= 03 = 07

InducedHA-2

HA-2HA-3

InducedHA-1 InducedHA-3HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(b) 120572 = 03

HAInducedHA

= 5 = 05

InducedHA-2HA-2

HA-3

InducedHA-1 InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(c) 120572 = 05

HAInducedHA

= 08 = 02

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572 = 08

HAInducedHA

= 9 = 01

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(e) 120572 = 09

HAInducedHA

246 2462 2464 2466 2468 247

= 1 = 0InducedHA-2HA-2

HA-3

HA- 3

HA-3

InducedHA-1

InducedHA-3

HA-1

HA-1

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

1085

1086

1087

(f) 120572 = 1

Figure 5 Score trajectories obtained by aggregation operators HA and InducedHA with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

further configure parameter 120582 to the value set of [0 01100]and conduct computational experiments to collect scorevalues and ranking results (as shown in Figures 5 and 6) byfour aggregation operators of GIVDHFUUBLHA GIVDH-FUUBLHG I-GIVDHFUUBLHAand I-GIVDHFUUBLHGFor more clarity in Figures 5 and 6 please note that weuse HA HG Induced-HA and Induced HG to respec-tively denote GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG In Figure 5it can be observed that when dominant importance of 120596119860119863

is emphasized (120572 = 0 120572 = 03 120572 = 05) GIVDHFUUBLHAis capable of differentiating the three alternative contractorsas same ranking order of 1199092 ≻ 1199093 ≻ 1199091 for all 120582 but withincreases in 120572 the margins between scores of 1199091 and 1199093 nar-rows and the rank order changes to 1199091 ≻ 1199093when 120572 gets 09and 1 Interestingly in Figure 6 although GIVDHFUUBLHGis able to significantly differentiate alternative contractors 1199091and 1199093 when dominant importance of 120596119860119863 is emphasized (120572= 0 120572 = 03 120572 = 05) rank order of 1199091 and 1199093 obviouslychanges once from 1199093 ≻ 1199091 to 1199091 ≻ 1199093 when 120582 is bigger

22 Complexity

HGInducedHG

= = 1

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(a) 120572= 0

HGInducedHG

= 3 = 07

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(b) 120572= 03

HGInducedHG

= 05 = 05

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(c) 120572= 05

HGInducedHG

= 8 = 02

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572= 08

HGInducedHG

292 293 294 295 296 297minus08

minus0795

minus079

= 09 = 01

HA-1

HA-3

InducedHG-2

HG-1

HG-1

HG-2

HG-3

HG-3 InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4

minus3

minus2

minus1

0

1

2

scor

e val

ues

(e) 120572= 09

HGInducedHG

50 60 70 80 90 100minus095minus09

minus085

= 1 = 0

HG-x3

HG-x1

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(f) 120572= 1

Figure 6 Score trajectories obtained by aggregation operators HG and InducedHG with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

than 8 When 120572 increases to 1 GIVDHFUUBLHGmaintainsthe rank order of 1199093 ≻ 1199091 but their score trajectorieswalk very close Comparatively I-GIVDHFUUBLHA andI-GIVDHFUUBLHG generate consistent ranking results of1199092 ≻ 1199091 ≻ 1199093 no matter 120582 varies in all the above experiments

In sum when parameters are configured with typical val-ues (ie 120572 = 05 and 120582=1) and no additional attitudinal infor-mation is taken into consideration the proposed aggregationoperators unanimously derive the same rank order 1199092 ≻1199093 ≻ 1199091 of for the three alternative contractors However

with the amplification effect of parameter 120582 analysis resultsfrom linear combination of 120596119860119868 and 120596119860119863 reveal alternative1199092 which consistently is the best choice but rank order of1199091 and 1199093 changes because their score trajectories get closeand crossed which probably indicates that only decision-making informationderived from the decisionmatrix is inad-equate to significantly differentiate alternatives 1199091 and 1199093While by integrating additional decision information aboutattitudinal characters of group experts I-GIVDHFUUBLHAand I-GIVDHFUUBLHG manage to generate the identical

Complexity 23

Table 9 Order-inducing vectors obtained by our TOPSIS-based method for problem in [38]

1198661 1198662 1198663 11986641198601 02522 01629 02282 027141198602 01 05243 01364 046891198603 04081 01756 05244 037791198604 04507 01409 01064 013221198605 00697 02203 03753 01161

Table 10 Order-inducing vectors obtained by our TOPSIS-based method for problem in [39]

1198661 1198662 1198663 11986641198601 04548 04969 04923 047321198602 05092 047 04768 046451198603 07192 04888 04953 049151198604 06722 06163 05706 049291198605 06104 04942 06463 04946

Table 11 Ranking results obtained by applying different order-inducing vectors to [38 39]

Problems Approaches Ranking results

Problem in [38] IHFLOWA [38] + order-inducing vectors in [38] 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198605 ≻ 1198604

IHFLOWA [38] + order-inducing vectors in Table 9 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198604 ≻ 1198605

Problem in [39] I-IVDHFLOWA [39] + order-inducing vectors in [39] 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198602 ≻ 1198601

I-IVDHFLOWA [39] + order-inducing vectors in Table 10 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602

permutation of 1199092 ≻ 1199091 ≻ 1199093 which verifies that ourproposed TOPSIS-based order-inducing approach is prac-tical and useful in helping work out decision results morereasonably and robustly

63 Application Studies of Proposed TOPSIS-Based Order-Inducing Method When confronting complex decision-making situations that need additional attitudinal charactersof decision-makers Yager and Filev [88] and Yager [102]introduced and encouraged to employ order-inducing vectorto represent those attitudinal characters Since then theidea of order-inducing vector has been extended to variousdecision-making environments [98 103] such as those basedon intuitionistic fuzzy information [104 105] linguistic infor-mation [106] hesitant fuzzy linguistic information [38] andinterval-valued dual hesitant fuzzy uncertain linguistic infor-mation [39] Generally in existing literature order-inducingvectors rely on experts to directly give [38 39 104ndash106]

In order to examine the adaptability of our TOPSIS-based order-inducing method to linguistic decision-makingenvironments we respectively apply the method to resolvethe same problems in references [38 39] Due to fact thatthere is no descriptions about how the order-inducing vectorswere derived in both references [38 39] we here applyextremums into our TOPSIS-based method to obtain order-inducing vectors that is (1199047 1 0) and (1199041 0 1) for[38] ([1199046 1199046] 1 0) and ([1199040 1199040] 0 1) for [39] Thenwe obtain our order-inducing vectors listed in Table 9 forthe problem adopted in [38] and our order-inducing vectorslisted in Table 10 for [39]

Now we apply the order-inducing vectors as shownin Tables 9 and 10 to resolve the problems in [38] and[39] respectively Regarding the problem in [38] the scorevalues of all alternatives are calculated as 119878(1198601)=02602119878(1198602)=02417 119878(1198603)=03577 119878(1198604)=0221 and 119878(1198605)=0185so the corresponding ranking result is 1198603 ≻ 1198601 ≻ 1198602 ≻1198604 ≻ 1198605 As for the problem in [39] the score values ofall alternatives are computed as 119878(1198601)=39954 119878(1198602)=3783119878(1198603)=70827 119878(1198604)=71792 and 119878(1198605)=5813 we thus obtainthe ranking result 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602 Formore clarity brief descriptions of comparative approachesand obtained ranking results also have been put together inTable 11

According to Table 11 the approaches that employ ourorder-inducing vectors derive the same ranking order ofthe better three alternatives for both comparative problemsfrom [38 39] As seen our proposed TOPSIS-based order-inducing method exhibits effective and more understand-able Furthermore when decision-makers are incapable ofdirectly put forward all desired order-inducing vectors due tocomplexity in ill-structured problems our method providesa feasible directions to obtain order-inducing vectors byfiguring out ideal solutions Comparatively our proposedTOPSIS-based order-inducing method behaves effective andmore operational in practical use

64 Comparison with Decision-Making Approach Based onDegraded form of IVDHFULS Note that our approachis intrinsically adaptive to decision situations based ondegraded forms of IVDHF UUBLS such as the dual hesitant

24 Complexity

fuzzy linguistic sets introduced by Yang and Ju [54] Tofurther inspect feasibility of our approach we here conductcomparative study on the same case adopted by Yang and Ju[54] We firstly convert their original decision matrix intothe form of IVDHF UUBLS crisp membership and non-membership degrees are equivalently changed to intervalswith same upper and lower limits (eg 02 turns out to be[02 02]) and linguistic variables are rewritten as uncertainlinguistic variables (eg 1199042 turns out to be [11990421199042])

Since the case only got three attributes among whichinterdependency could be not significant under generalassumption weighting vector 120596119860119868 is not calculated Andfor straight comparison order inducing vector is notincluded either Then the argument-dependent attributeweighting vector can be obtained from Eq(29) as 120596119860119863 =(03192 03132 03676)119879 By use of Algorithm Iwith operatorIVDHFUUBLWA (ie GIVDHFUUBLWA in case of 120582 = 1see Appendix B) we have the scores of alternatives 119904(1198601) =03412 119904(1198602) = 01354 119904(1198603) = 03932 119904(1198604) = 07352Therefore the ordering of alternatives is1198604 ≻ 1198603 ≻ 1198601 ≻ 1198602which is mainly in alignment with the ordering of1198604 ≻ 1198601 ≻1198603 ≻ 1198602 in Yang and Ju [54] It is worth noticing that 120596119860119863used in our approach is derived objectively from assessmentswhile attribute weights 120596lowast = (035 025 040)119879used by Yangand Ju [54] were assigned empirically Differences between120596119860119863 and 120596lowast lead to different ordering of1198601 and1198603 Compar-atively speaking our approach holds more adaptability andobjectivity

In general our proposed MADM approach in this paperexhibits an effective way to tackle complex green contractorselection problems in which subjective attributes weightingvector (such as120596119860119868) and objective attributes weighting vector(such as 120596119860119863) exist To comprehensively utilize these twofundamental facets of attributes weighting information ourcase study and computational experiments suggest that linearcombination method (ie 120596 = 120572120596119860119868 + 120573120596119860119863) is effectiveto reflect effects of 120596119860119868 and 120596119860119863 by configuring parameter 120572or 120573 When there is no specific expert opinions to specify orhard to obtain we suggest that parameter 120572 chooses the valueset of 0 02 05 08 1 Furthermore generalized aggrega-tion operators are suggested to inspect score trajectoriesof alternative contractors along increasing of the operatorsrsquoamplification parameter 120582 If some alternative contractorsrsquoscore values get very close or crossed so that they cannot bedifferentiated from each other significantly we suggest to askthe panel of experts to derive opinions of ideal solutions andutilize our TOPSIS-based order-inducing method to includethose expert opinions as attitudinal decision informationthereby obtaining more rational and robust decision results

7 Conclusions

Aiming at tackling complex decision-making problems ofgreen contractor selection that often hold simultaneouslythe characteristics of decision hesitancy attributes inter-dependency and group complex attitudinal characters wehave developed an effective IVDHF UUBLS-based approach

We utilize IVDHF UUBLS to elicit complicate decision-makersrsquo assessments more objectively and comprehensivelywe employ DEMATEL for addressing attribute interdepen-dency more precisely and we determine group attitudinalcharacters as order-inducing vectors based on ideal attitudesof experts rather than by empirical assignment The mainadvantages of our approach are (i) IVDHF UUBLS offers anadequate way to depict decision hesitancy (ii) the designeddistance measure for IVDHF UUBLS avoids informationdistortion of traditional ones which artificially add mis-matching membership or nonmembership degrees (iii) thedeveloped generalized operators include traditional ones asspecial cases so as to provide flexibility in MADM basedon IVDHF UUBLS (iv) the devised synthesized attributeweighting scheme provides an adaptive way to consider twonecessary aspects attributes interdependency-based weightsand argument-dependent weights (v) the TOPSIS-basedmethod can deduce reasonable order-inducing vectors ratherthan empirical ones and holds generality in other linguisticdecision environments Results of case studies and experi-ments have verified the validity of the proposed approach

When multiple decision organizations are involvedgroup decision-making based on IVDHF UUBLS is animportant research direction Besides inducing preferencesfor decision-making problems of high uncertainty interactivemechanisms should also be studied to extract preferences Inaddition empirical studies of the proposed approach can becarried out to various areas such as investment evaluationproject management and vehicle selection in fleet operations

Appendix

A Basic Notions for IVDHFS and UnbalancedLinguistic Term Set

Definition A1 (see [47]) Letting 119883 be a fixed set then anIVDHFS on 119883 is defined as119863 = ⟨119909 ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (A1)

where ℎ(119909) = ⋃[120583119871120583119880]isinℎ(119909)120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and119892(119909) = ⋃[]119871]119880]isin119892(119909)] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two setsof interval values in [0 1] denoting possible membershipand non-membership degrees of element 119909 isin 119883 to theset 119863 respectively with conditions 120583 ] isin [0 1] and0 le (120583119880)+ + (]119880)+ le 1 and for all 119909 isin 119883 (120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880

For convenience normally 119889 = ℎ 119892 is called an interval-valued dual hesitant fuzzy element (IVDHFE) and 119863 is theset of all IVDHFEs

Definition A2 (see [107]) Suppose 119878 = 119904119895 | 119895 = 0 1 119892 minus1 is a finite and totally ordered discrete linguistic term setwhere 119904119895 represent possible values for a linguistic variableand 119892 is an odd cardinality 119878 is uniformly and symmetricallydistributed if the following two conditions are satisfied (1)

Complexity 25

there exists a unique constant 120582 gt 0 such that 119873119878(119904119894) minus119873119878(119904119895) = 119897(119894 minus 119895) for all 119894 119895 = 0 1 119892 minus 1 (2) Let 119878119877 =119904 | 119904 isin 119878 119904 gt 119878119862 and 119878119871 = 119904 | 119904 isin 119878 119904 lt 119878119862 Let C(119878119877) andC(119878119871) be the cardinality of 119878119877 and 119878119871 then C(119878119877) = C(119878119871)If 119878 is uniformly and symmetrically distributed then 119878 iscalled a balanced linguistic term set Otherwise 119878 is calledan unbalanced linguistic term set

The following 2-tuple fuzzy linguistic representationmodel extends traditional linguistic term set to a continuouscase so as to facilitate computing with linguistic variables

Definition A3 (see [59]) Let 119878 = 1199040 1199041 119904119892minus1 be alinguistic term set and 120573 isin [0 119892] Then a 2-tuple fuzzylinguistic variable expresses the equivalent information to 120573is defined as

Δ [0 119892] 997888rarr 119878 times [minus05 05) (A2)

Δ (120573) = (119904119894 120572) with

119904119894 119894 = 119903119900119906119899119889 (120573)120572 = 120573 minus 119894 120572 isin [minus05 05)

(A3)

Δminus1 (119904119894 120572) = 119894 + 120572 = 120573 (A4)

where 119903119900119906119899119889() is the usual rounding operation and120572 is calledsymbolic translation

To obtain 2-tuple fuzzy linguistic representations ofunbalanced linguistic terms the concept of linguistic hierar-chies ie 119871119867 = ⋃119905 119897(119905 119899(119905)) is used 119897(119905 119899(119905)) is a linguistichierarchy with 119905 indicating the level of hierarchy and 119899(119905)denotes the granularity of the linguistic term set of 119905 Herreraet al [59] defined the following transformation functionsbetween labels from different levels in multigranular linguis-tic information contexts without loss of information

Definition A4 (see [59]) In linguistic hierarchies 119871119867 =⋃119905 119897(119905 119899(119905)) whose linguistic term sets are represented by119878119899(119905) = 119904119899(119905)0 119904119899(119905)119899(119905)minus1 the transformation function froma linguistic label in level 119905 to a label in consecutive level 1199051015840 isdefined as 1198791198651199051199051015840 119897(119905 119899(119905)) 997888rarr 119897(1199051015840 119899(1199051015840)) such that

1198791198651199051199051015840 (119904119899(119905)119894 120572119899(119905))= Δ 1199051015840 (Δminus1119905 (119904119899(119905)119894 120572119899(119905)) (119899 (1199051015840) minus 1)119899 (119905) minus 1 ) (A5)

By use of the above transformation function any 2-tuplelinguistic representation can be transformed into a term in119871119867 Detailed transformation procedures are listed as follows

(1) Representation in linguistic hierarchy To transformthe unbalanced terms of 119878 into the corresponding terms inthe 119871119867 transformation function 120595 is employed to associateeach unbalanced linguistic 2-tuple (119904119894 120572) with its linguistic 2-tuple in 119871119867(119878) ie

120595 119878 997888rarr 119871119867(119878) (A6)

so that 120595(119904119894 120572) = (119904119866(119894)119868(119894) 120582) for forall(119904119894 120572) isin 119878(2) Computational phase Firstly transform (119904119866(119894)119868(119894) 120582) intolinguistic 2-tuples denoted as (119904119899(1199051015840)

1198681015840(119894) 1205821015840) in 119878119899(1199051015840) where

(119904119899(1199051015840)1198681015840(119894)

1205821015840) = 119879119865 (119904119866(119894)119868(119894) 120582)= Δ(Δminus1 (119904119866(119894)119868(119894) 120582) sdot (119899 (1199051015840) minus 1)119866 (119894) minus 1 ) (A7)

Then a computational model is used over 119878119899(1199051015840) with aresult denoted as (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)(3) Retranslation process The result (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)is transformed into the unbalanced term in 119878 by using thetransformation function 120595minus1 ie

120595minus1 119871119867 (119878) 997888rarr 119878 (A8)

so that 120595minus1(119904119899(1199051015840)119903 120582119903) = (119904119903119890119904119906119897119905 120582119903119890119904119906119897119905) isin 119878B Details of Theorem 14

Proof Given a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector for119904119889119895 and 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 120582 gt 0 then(1) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLWA and GIVDHFUUBLWG operators respectively(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLOWA and GIVDHFUUBLOWG operators respectively

(2) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to theIVDHF UUBL weighted average (IVDHFUUBLWA)and IVDHF UUBL weighted geometric (IVDHFUUBLWG)operators respectively where

26 Complexity

119868119881119863119867119865119880119880119861119871119882119860120596 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

120596119895119904119889119895= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus

119899prod119895=1

(1 minus 120583119871119895 )120596119895 1 minus 119899prod119895=1

(1 minus 120583119880119895 )120596119895]] [[

119899prod119895=1

(]119871119895)120596119895 119899prod119895=1

(]119880119895 )120596119895]])

(B1)

119868119881119863119867119865119880119880119861119871119882119866120596 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889119895120596119895

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[

119899prod119895=1

(120583119871119895 )120596119895 119899prod119895=1

(120583119880119895 )120596119895]] [[1 minus

119899prod119895=1

(1 minus ]119871119895)120596119895 1 minus 119899prod119895=1

(1 minus ]119880119895 )120596119895]])

(B2)

(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHA andGIVDHFUUBLHG operators reduce to the IVDHF UUBL

orderedweighted average (IVDHFUUBLOWA) and IVDHFUUBL ordered weighted geometric (IVDHFUUBLOWG)operators respectively where

119868119881119863119867119865119880119880119861119871119874119882119860119908 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

119908119895119904119889120590(119895)= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) )))

119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) )))

]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus

119899prod119895=1

(1 minus 120583119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus 120583119880120590(119895))119908119895]] [[

119899prod119895=1

(]119871120590(119895))119908119895 119899prod119895=1

(]119880120590(119895))119908119895]])

(B3)

119868119881119863119867119865119880119880119861119871119874119882119866119908 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889120590(119895)119908119895

= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[

119899prod119895=1

(120583119871120590(119895))119908119895 119899prod119895=1

(120583119880120590(119895))119908119895]] [[1 minus

119899prod119895=1

(1 minus ]119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus ]119880120590(119895))119908119895]])

(B4)

Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

References

[1] V W Y Tam and C M Tam ldquoWaste reduction throughincentivesA case studyrdquoBuilding Research and Information vol36 no 1 pp 37ndash43 2008

[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

[11] X Qi C Liang and J Zhang ldquoGeneralized cross-entropy basedgroup decision making with unknown expert and attributeweights under interval-valued intuitionistic fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 79 pp 52ndash642015

[12] H Zimmermann Fuzzy Set Theorymdashand Its ApplicationsKluwer Academic Boston Mass USA 2001

[13] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 pp 199ndash249 1975

[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

[16] B T Nuong K-W Kim L Prathumratana et al ldquoSustainabledevelopment in the mining sector and its evaluation usingfuzzy AHP (Analytic Hierarchy Process) approachrdquo GeosystemEngineering vol 14 no 1 pp 43ndash50 2011

[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

[18] E K Zavadskas Z Turskis and J Antucheviciene ldquoSelectinga contractor by using a novel method for multiple attributeanalysis Weighted aggregated sum product assessment withgrey values (WASPAS-G)rdquo Studies in Informatics and Controlvol 24 no 2 pp 141ndash150 2015

[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

[21] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

[25] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009

[26] Z Hatush and M Skitmore ldquoContractor Selection UsingMulticriteria Utility Theory An Additive Modelrdquo Building andEnvironment vol 33 no 2-3 pp 105ndash115 1998

[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

[30] AM Anvuur andMM Kumaraswamy ldquoConceptualmodel ofpartnering and alliancingrdquo Journal of Construction Engineeringand Management vol 133 no 3 pp 225ndash234 2007

[31] A H I Lee H Kang C F Hsu and H Hung ldquoA green supplierselection model for high-tech industryrdquo Expert Systems withApplications vol 36 no 4 pp 7917ndash7927 2009

[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

[33] P Humphreys R McIvor and F Chan ldquoUsing case-basedreasoning to evaluate supplier environmental managementperformancerdquo Expert Systems with Applications vol 25 no 2pp 141ndash153 2003

[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

[45] Bin Zhu Zeshui Xu and Meimei Xia ldquoDual Hesitant FuzzySetsrdquo Journal of Applied Mathematics vol 2012 Article ID879629 13 pages 2012

[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

[51] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[52] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

[55] Z Tao X Liu H Chen and L Zhou ldquoUsing new versionof extended t-norms and s-norms for aggregating intervallinguistic labelsrdquo IEEETransactions on SystemsManCyberneticsSystems vol vol p 15 2016

[56] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[57] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[58] L Martınez M Espinilla J Liu L G Perez and P J SanchezldquoAn evaluation model with unbalanced linguistic informationapplied to olive oil sensory evaluationrdquo Journal of Multiple-Valued Logic and Soft Computing vol 15 no 2-3 pp 229ndash2512009

[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

[62] L Marin A Valls D Isern A Moreno and J M MerigoldquoInduced unbalanced linguistic ordered weighted average andits application in multiperson decision makingrdquo The ScientificWorld Journal vol 2014 2014

[63] A Baykasoglu V Kaplanoglu Z D U Durmusoglu and CSahin ldquoIntegrating fuzzy DEMATEL and fuzzy hierarchicalTOPSIS methods for truck selectionrdquo Expert Systems withApplications vol 40 no 3 pp 899ndash907 2013

[64] S K Patil and R Kant ldquoA hybrid approach based on fuzzyDEMATEL and FMCDM to predict success of knowledge man-agement adoption in supply chainrdquoApplied Soft Computing vol18 pp 126ndash135 2014

[65] S Liyin and Y Hong ldquoImproving environmental performanceby means of empowerment of contractorsrdquo Management ofEnvironmental Quality An International Journal vol 17 no 3pp 242ndash257 2006

[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

[69] Y Kazancoglu I Kazancoglu and M Sagnak ldquoA new holisticconceptual framework for green supply chain management

performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

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16 Complexity

Attribute weighting

Hesitant fuzzy linguistic MADM with attributes interdependency and decision makersrsquo group attitudinal characters

Step 9 Output ranking order of alternatives

interdependency among attributes by DEMATEL in Section 52

attributes values by standard deviation method in Section 51

Step 6 Ensure TOPSIS order inducing variables proposed in Section 53

Step 7 Utilizing proposed induced generalized hybrid aggregation operators developed in Section

41 to get overall value of each alternative

Step 8 Calculate score degrees and accuracy degrees

Step 4 Order inducingYes

Step 5 Choose proposed generalized operators in Section 41 to get overall value of each alternative

No

Collect decision matrix in the form of IVDHF_UUBLS

Complex problems of green contractor selection

Step 1 Weighting vector AD derived from Step 2 Weighting vector AI derived from

Step 3 Obtain synthesized weighting vector of attributes =AI + AD

Figure 3 Processing flowchart of Algorithm I

119865119894 (120596119860119863) = 119898sum119895=1

119865119894119895 (120596119860119863) = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894 119894 = 1 2 119899 119895 = 1 2 119898 (25)

Based on the analysis above we can get followingprogramming model (M-1) for obtaining optimal attributeweighting vector 120596119860119863

(119872 minus 1)max 119865 (120596119860119863) = 119899sum

119894=1119865119894 (120596119860119863) = 119899sum

119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897)120596119860119863119894st 119899sum

119894=1(120596119860119863119894 )2 = 1 120596119860119863119894 ge 0 119894 = 1 2 119899 (26)

where 119889(119903119894119895 119903119894119897) can be calculated by Definition 6To solve model (M-1) we can construct Lagrange func-

tion

119871 (120596119860119863119894 120577) = 119899sum119894=1

119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) 120596119860119863119894+ 1205772 ( 119899sum

119894=1

(120596119860119863119894 )2 minus 1) (27)

where 120577 is the Lagrange multiplierDifferentiate with respect to 120596119860119863119894 (119894 = 1 2 119898) and 120577

and set these partial derivatives equal to zero the followingset of equations can be obtained

120597119871120597120596119860119863119894 = 119898sum119895=1

119898sum119897=1

119889 (119903119894119895 119903119894119897) + 120577120596119860119863119894 = 0120597119871120597120577 = 119899sum

119894=1

(120596119860119863119894 )2 minus 1 = 0 (28)

By normalizing the solutions of above equations we canget a simple and exact formula for determining the attributeweights as follows

120596119860119863119894 = sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897)sum119899119894=1sum119898119895=1sum119898119897=1 119889 (119903119894119895 119903119894119897) (29)

52 Deriving Attributes Interdependency-Based AttributeWeighting Vector 120596119860119868 Some efforts [53 60 84 93ndash95]

Complexity 17

have been made to address interdependencies betweenattributes in MADM However they unanimously cannotreveal the interdependences in a more complete way WhileDEMATAEL technique [63 96] gathers collective knowledgeto capture influences between attributes more precisely andcompletely so that weighting vector for attributes can berationally determined according to strength measure ofthose influences [64 97] Therefore in tackling the complexMADMwith interdependent attributes under IVDHF UUBLenvironments we present a DEMATEL-based method as thefollowing Algorithm II to derive attribute weighting vector120596119860119868 from the viewpoint of attributes interdependency

Algorithm II DEMATEL-basedmethod for ensuringweight-ing vector 120596119860119868Step 1 Construct initial direct-influence matrix 119885 = (119911119894119896)119899times119899

For attributes 119860 = 1198601 1198602 119860119899 decision-makersare asked to give direct effect that attribute 119860 119894 has onattribute 119860119896 using a five-point scale with scores representedby natural language (1) 0 mdash lsquoabsolutely no influencersquo (2)1 mdash lsquolow influencersquo (3) 2 mdash lsquomedium influencersquo (4) 3 mdashlsquohigh influencersquo (5) 4mdash lsquovery high influencersquoThen the initialdirect-influence matrix 119885 = (119911119894119896)119899times119899 can be established witheach element being the mean of same factors in differentdirect matrices

Step 2 Calculate normalized direct-influence matrix 119883 =(119909119894119896)119899times119899 Using matrix 119885 the normalized direct-influencematrix 119883 = (119909119894119896)119899times119899 is calculated according to

119909119894119896 = 119911119894119896119904 (30)

where 119904 = maxmax1le119894le119899sum119899119896=1 119911119894119896max1le119896le119899sum119899119895=1 119911119894119896 Allelements in matrix 119883 are complying with 0 le 119909119894119896 le 10 le sum119899119896=1 119909119894119896 le 1 and 0 le sum119899119894=1 119909119894119896 le 1 and at least onerow or column of the summation equals 1

Step 3 Derive total-influence matrix 119879 = (119905119894119896)119899times119899 Basedon matrix 119883 the total-influence matrix 119879 = (119905119894119896)119899times119899 can bederived by summing the direct effects and all of the indirecteffects

119879 = 119883 + 1198832 + 1198833 + sdot sdot sdot + 119883ℎ = 119883 (119868 minus 119883)minus1 (31)

where 119868 represents identity matrix When ℎ 997888rarr infin 119883ℎ =(0)119899times119899Step 4 Compute influencing and influenced degrees of eachattribute The sum of rows and the sum of columns withinthe total-influence matrix 119879 are respectively expressed as thevectors 119903 and 119888

119903 = (119903119894)119899times1 = ( 119899sum119896=1

119905119894119896)119899times1

119888 = (119888119896)119899times1 = ( 119899sum

119895=1

119905119894119896)119879

1times119899

(32)

where 119903119894 denotes the sumof the 119894th row inmatrix119879 and showsthe sum of direct and indirect effects that attribute 119860 119894 hason other attributes Similarly 119888119896 shows the sum of direct andindirect effects that attribute 119860119896 has receivedStep 5 Estimate weights of attributes based on influencedegrees according to

120596119868lowast119894 = [(119903119894 + 119888119894)2 + (119903119894 minus 119888119894)2]12 (33)

Then the weighting vector 120596119860119868 can be obtained by normaliz-ing (33)

120596119860119868119894 = 120596119868lowast119894sum119899119894=1 120596119868lowast119894 (34)

53 Obtaining Order Inducing Variables 120576 by TOPSIS-BasedMethod For ill-defined MADM problems induced aggre-gation operators [89 90 98] provide an effective channelfor experts to express complex attitudinal characters Thekey step in induced aggregation operators is to obtainorder-inducing vectors 120576 which however are presumptivelyassigned in most papers [62] Therefore in this sectionaiming at deducing the order-inducing vectors 120576 rationallywe here devise a TOPSIS-based method in Algorithm IIIObviously the Algorithm III is of generality for MADMwithdifferent forms of linguistic expression

Algorithm III TOPSIS-based method for obtaining orderinducing vectors 120576Step 1 Let the decision maker define IVDHF UUBL idealsolution(s) 119903+119894 and IVDHF UUBL negative ideal solution(s)119903minus119894 corresponding to each attribute in which

119903+119894 = (119903+1198941 119903+1198942 119903+119894119901) 119903minus119894 = (119903minus1198941 119903minus1198942 119903minus119894119902) (35)

where 119901 and 119902 denotes the number of 119903+119894 and 119903minus119894 respectively

Step 2 Calculate distances from each element in decisionmatrix to ideal solution(s) and negative solution(s) respec-tively

119889+119894119895 = max119897=12119901

119889 (119903119894119895 119903+119894119897 ) 119889minus119894119895 = min

119897=12119902119889 (119903119894119895 119903minus119894119897 ) (36)

where 119889(119903119894119895 119903+119894119897 ) and 119889(119903119894119895 119903minus119894119897 ) are calculated by the distancemeasure in Definition 6 If 119901 gt 1 we choose the onewith maximum distance if 119902 gt 1 we choose the one withminimum distance

Step 3 Calculate coefficients according to

119888119894119895 = 119889minus119894119895119889+119894119895 + 119889minus119894119895 119894 = 1 2 119899 119895 = 1 2 119898 (37)

18 Complexity

Table 2 Decision matrix in the form of IVDHF UUBLS1199091 1199092 11990931198601([AHVH] [02 03][02 04] [03 04]) ([HT] [04 05][05 06] [02 03] [02 04]) ([ALAH] [01 02][01 03] [06 07])1198602([MH] [05 06][02 03]) ([AHT] [06 07][01 02] ) ([MVH] [02 03][05 06] [06 07])1198603([VLAL] [03 04][04 05] [05 06]) ([MVH] [06 07] [07 08][01 02]) ([VLL] [04 05][02 03])1198604([MAH] [02 04][05 06]) ([LAL] [03 05][02 03]) ([ALM] [06 07][01 02] [02 03])1198605([ALM] [04 05][01 02] [04 05]) ([AHH] [06 07][01 02]) ([HVH] [04 05][06 07] [01 03])1198606([LM] [01 02][03 05] [03 05]) ([VLL] [02 04] [05 06][02 03]) ([AHH] [03 04][04 05])1198607([VLL] [04 05][05 06] [02 03] [02 04]) ([MAH] [05 06] [07 08][01 02]) ([ALM] [04 05][05 06] [03 04])1198608([ALAH] [01 03][04 06]) ([AHH] [05 07][01 02] [02 03]) ([AHH] [04 06][03 04])

Table 3 Direct influence matrix Z1198601 1198602 1198603 1198604 1198605 1198606 1198607 11986081198601 0 3 4 1 3 1 2 41198602 1 0 2 4 4 2 3 11198603 4 2 0 1 3 2 4 11198604 2 3 1 0 3 4 1 21198605 4 2 1 3 0 1 4 31198606 2 4 3 4 4 0 4 21198607 4 1 2 3 3 1 0 41198608 1 2 4 3 3 2 4 0

Step 4 Derive order inducing variable 120576119894119895 by descendingorder of 119888119894119895 The greater the value 119888119894119895 the bigger the 1205761198941198956 Illustrative Example and Experiments

61 Illustrative Case Study Suppose a mega infrastructureproject is selecting green contractor for one of its subprojectsA panel of decision-makers including managers professorsand competitive intelligence experts has been organized tocomprehensively evaluate three alternative contractors (ie1199091 1199092 and 1199093) under the eight attributes (ie 1198601ndash1198608) listedin Section 2

To elicit true decision preferences we apply theIVDHF UUBLS to assess the three alternatives (1199091ndash1199093)Firstly decision-makers are asked to vote on 119909119895 under eachattribute119860 119894 using different unbalanced linguistic term sets 1198781 1198782 and 1198783 where 1198781 = 119873 119871119860119871119872119860119867119867119876119867119881119867119860119879 1198791198782 = 119873119860119873119881119871 119876119871 119871 119860119871119872119860119867119881119867 119879 1198783 = 119873119881119871119871119860119872119872119876119872119860119867119867 119879 In Figure 4 we show the adoptedlinguistic hierarchies and relationship between 1198781 1198782 and1198783 Attributes 1198601 1198602 1198605 and 1198608 are evaluated by use of 1198781attributes11986031198604 and1198607 are evaluated by 1198782 and attribute1198606is evaluated by 1198783 The highest-voted uncertain unbalancedlinguistic term [119904119898 119904119899] are identified Secondly the panel of

decision-makers is asked to further express their opinionson membership degrees ℎ(119909) and nonmembership degrees119892(119909) to [119904119898 119904119899] Then we can derive compound assessments(ie ⟨119909 119904120599(119909) ℎ(119909) 119892(119909)⟩) in the form of IVDHF UUBLSand obtain the decision matrix in Table 2 Additionallyin order to examine the interdependency among attributesdecision-makers are also required to assess the influentialrelations among the eight attributes using DEMATEL-basedAlgorithm II The direct influence matrix is collected andshown in Table 3

Next we apply the proposed Algorithm I to solve thisgreen contractor selection problem To derive the synthe-sized attribute weighting vector in Algorithm I we assign120572 = 120573 = 05 For scenarios with no group attitudinalcharacters operator GIVDHFUUBLHA in Definition 12 ischosen In contrast for scenarios with group attitudinalcharacters operator I-GIVDHFUUBLHA in Definition 13will be chosen Further the panel of decision-makers is askedto determine their positive and negative ideal alternativesthat are collected in Table 4 where corresponding order-inducing vectors that are derived from Algorithm III arealso listed (see the rightmost column) Thereafter operatorI-GIVDHFUUBLHA in Definition 13 is utilized for aggrega-tion For GIVDHFUUBLHA and I-GIVDHFUUBLHA we

Complexity 19

Table 4 Ideal alternatives (119903+ and 119903minus) and derived order-inducing vectors 120576119903+ 119903minus 1205761199091 1199092 11990931198601 ([VHT] 1 0) ([NL] 1 0) 5 6 61198602 ([MAH] [07 08] [01 02]) ([NL] 1 0) 1 1 41198603 (T 1 0) (L 1 0) 8 7 81198604 ([ALAH] 09 01) (T 1 0) 2 2 11198605 ([HT] [09 1] 0) (N 1 0) 7 5 51198606 (M 1 0) (N 1 0) 6 3 31198607 (M 1 0) ([NL] 1 0) 3 8 71198608 ([MAH] 1 0) (N 1 0) 4 4 2

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 1 2 3 4

0 1 2

N L AH H QH VH AT TM

Level 1 l(13)

Level 2 l(25)

Level 3 l(39)

Level 4 l(417)

Unbalanced linguistic term set S1

N AH VH TM Unbalanced linguistic term set S2VL L ALAN QL

N H TM Unbalanced linguistic term set S3

L AM AHQM

AL

VL

Figure 4 Unbalanced linguistic term sets (1198781 1198782 and 1198783) and their mapping in linguistic hierarchies

assume 120582=2 and 119908 = (0 0 015 025 025 025 01 0)119879Note that 119908 is a position weighting vector which is derivedby the fuzzy semantic quantitative operator [99] Aftercompleting all steps in Algorithm I (see Appendix C) weobtain the ranking results and present them in Table 5 Forthe scores derived from Step 5 we find the ranking order is1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the best solution Step 7gives the ranking order of 1199092 ≻ 1199091 ≻ 1199093 indicating 1199092 is stillthe most appropriate one while 1199091 is better than 1199093 under theinfluence of 12057662 Experiments on Ranking Results along with VariousParameter Configurations In order to inspect the impactsof various parameter configurations on ranking results inthis section more numerical examples are carried out Alleight operators that defined in Section 41 are applied to theabove illustrative case where the parameter 120582 is configured asintegers ranging from 1 to 20 and parameter 120572 is configuredto a value set of 0 03 05 08 09 1

Firstly the four basic aggregation operators of GIVDH-FUUBLWA GIVDHFUUBLWG GIVDHFUUBLOWA and

Table 5 Ranking results of operators GIVDHFUUBLHA and I-GIVDHFUUBLHA

Aggregation operator Ranking orderGIVDHFUUBLHA 1199092 ≻ 1199093 ≻ 1199091I-GIVDHFUUBLHA 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG are applied to the former case Thesame configuration 120572 = 120573 = 05 is chosen while 120582 is testedas integers ranging from 1 to 20 The ranking results derivedfrom different parameter combinations have been collectedin Table 6 As can be seen four operators unanimouslyidentify the contractor 1199092 is the best one no matter how120582 changes Regarding priority relation between 1199091 and 1199093GIVDHFUUBLWA and GIVDHFUUBLOWA generate thesame result of1199093 ≻ 1199091 while results fromGIVDHFUUBLWGandGIVDHFUUBLOWGgot a change to 1199091 ≻ 1199093 at a certainvalue of parameter 120582

Actually GIVDHFUUBLWA and GIVDHFUUBLOWAare generalized version of weighted arithmetic aggregationoperators that producemore favorable results by emphasizing

20 Complexity

Table 6 Ranking orders obtained by GIVDHFUUBLWA GIVDHFUUBLOWA GIVDHFUUBLWG and GIVDHFUUBLOWG along with120582Operators 120582 (Integer) Ranking orderGIVDHFUUBLWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLOWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLWG [1 2] 1199092 ≻ 1199093 ≻ 1199091[3 20] 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG [1 8] 1199092 ≻ 1199093 ≻ 1199091[9 20] 1199092 ≻ 1199091 ≻ 1199093

Table 7 Ranking results obtained by GIVDHFUUBLHA and I-GIVDHFUUBLHA along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHA I-GIVDHFUUBLHA120572=0 120573=1 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=03 120573=07 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=05 120573=05 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=08 120573=02 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=09 120573=01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=1 120573=0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093

Table 8 Ranking results obtained by GIVDHFUUBLHG and I-GIVDHFUUBLHG along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHG I-GIVDHFUUBLHG

120572 = 0 120573 = 1 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 03 120573 = 07 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 05 120573 = 05 [1 8] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[9 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 08 120573 = 02 [1 17] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[18 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 09 120573 = 01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572 = 1 120573 = 0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093membership degrees to a uncertain linguistic term thus canbe considered as optimistic operators while GIVDHFU-UBLWG and GIVDHFUUBLOWG are generalized geomet-ric aggregation operators that producemore favorable resultsby emphasizing nonmembership degrees thus can be consid-ered as pessimistic operators [100 101]Therefore the rankingresults in Table 6 could be perceived as for optimism-baseddecision-making alternative 1199093 always stands out better than1199091 when 120582 gradually amplifies degree of optimism but forpessimism-based decision-making 1199091 turns out better than1199093 when 120582 gradually amplifies degree of pessimism 120582 canbe considered as the parameter for amplifying degree ofoptimism or pessimism [101]

Next to inspect the impacts of various combinationsof 120596119860119868 and 120596119860119863 on ranking results another four aggrega-tion operators of GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG also have beenapplied to the above illustrative case with varying 120572 120573 and 120582

All results have been collected in Tables 7 and 8 for compari-son As can be seen fromTable 7 ranking results generated byoperator GIVDHFUUBLHA keep the same as 1199092 ≻ 1199093 ≻ 1199091for all test combinations of parameters However as shownin Table 8 although GIVDHFUUBLHG is also capable ofidentifying 1199092 as the best contractor the priority relationsbetween 1199091 and 1199093 hold similar changing patterns in rankingresults in Table 6 that were output byGIVDHFUUBLWGandGIVDHFUUBLOWG Regarding decision situations whereobjective attributeweighting vector120596119860119863 is only considered (120572= 0) or its dominant importance (such as 120572 = 03) is accepted1199091 ≻ 1199093 is a clear observation when parameter 120582 is set toan integer in interval [8 20] Contrariwise under decisionscenarios where subjective attribute weighting vector 120596119860119868 isonly needed (120572 = 1) or its dominant importance (such as 120572 =09) is acknowledged 1199093 ≻ 1199091 holds all the way 120582 varies

In order to more clearly show the score trajectories ofalternative contractors along variations of parameter 120582 we

Complexity 21

HAInducedHA

InducedHA-2HA-2

HA-3

InducedHA-1InducedHA-3

= =

HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(a) 120572 = 0

HAInducedHA

= 03 = 07

InducedHA-2

HA-2HA-3

InducedHA-1 InducedHA-3HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(b) 120572 = 03

HAInducedHA

= 5 = 05

InducedHA-2HA-2

HA-3

InducedHA-1 InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(c) 120572 = 05

HAInducedHA

= 08 = 02

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572 = 08

HAInducedHA

= 9 = 01

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(e) 120572 = 09

HAInducedHA

246 2462 2464 2466 2468 247

= 1 = 0InducedHA-2HA-2

HA-3

HA- 3

HA-3

InducedHA-1

InducedHA-3

HA-1

HA-1

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

1085

1086

1087

(f) 120572 = 1

Figure 5 Score trajectories obtained by aggregation operators HA and InducedHA with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

further configure parameter 120582 to the value set of [0 01100]and conduct computational experiments to collect scorevalues and ranking results (as shown in Figures 5 and 6) byfour aggregation operators of GIVDHFUUBLHA GIVDH-FUUBLHG I-GIVDHFUUBLHAand I-GIVDHFUUBLHGFor more clarity in Figures 5 and 6 please note that weuse HA HG Induced-HA and Induced HG to respec-tively denote GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG In Figure 5it can be observed that when dominant importance of 120596119860119863

is emphasized (120572 = 0 120572 = 03 120572 = 05) GIVDHFUUBLHAis capable of differentiating the three alternative contractorsas same ranking order of 1199092 ≻ 1199093 ≻ 1199091 for all 120582 but withincreases in 120572 the margins between scores of 1199091 and 1199093 nar-rows and the rank order changes to 1199091 ≻ 1199093when 120572 gets 09and 1 Interestingly in Figure 6 although GIVDHFUUBLHGis able to significantly differentiate alternative contractors 1199091and 1199093 when dominant importance of 120596119860119863 is emphasized (120572= 0 120572 = 03 120572 = 05) rank order of 1199091 and 1199093 obviouslychanges once from 1199093 ≻ 1199091 to 1199091 ≻ 1199093 when 120582 is bigger

22 Complexity

HGInducedHG

= = 1

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(a) 120572= 0

HGInducedHG

= 3 = 07

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(b) 120572= 03

HGInducedHG

= 05 = 05

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(c) 120572= 05

HGInducedHG

= 8 = 02

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572= 08

HGInducedHG

292 293 294 295 296 297minus08

minus0795

minus079

= 09 = 01

HA-1

HA-3

InducedHG-2

HG-1

HG-1

HG-2

HG-3

HG-3 InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4

minus3

minus2

minus1

0

1

2

scor

e val

ues

(e) 120572= 09

HGInducedHG

50 60 70 80 90 100minus095minus09

minus085

= 1 = 0

HG-x3

HG-x1

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(f) 120572= 1

Figure 6 Score trajectories obtained by aggregation operators HG and InducedHG with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

than 8 When 120572 increases to 1 GIVDHFUUBLHGmaintainsthe rank order of 1199093 ≻ 1199091 but their score trajectorieswalk very close Comparatively I-GIVDHFUUBLHA andI-GIVDHFUUBLHG generate consistent ranking results of1199092 ≻ 1199091 ≻ 1199093 no matter 120582 varies in all the above experiments

In sum when parameters are configured with typical val-ues (ie 120572 = 05 and 120582=1) and no additional attitudinal infor-mation is taken into consideration the proposed aggregationoperators unanimously derive the same rank order 1199092 ≻1199093 ≻ 1199091 of for the three alternative contractors However

with the amplification effect of parameter 120582 analysis resultsfrom linear combination of 120596119860119868 and 120596119860119863 reveal alternative1199092 which consistently is the best choice but rank order of1199091 and 1199093 changes because their score trajectories get closeand crossed which probably indicates that only decision-making informationderived from the decisionmatrix is inad-equate to significantly differentiate alternatives 1199091 and 1199093While by integrating additional decision information aboutattitudinal characters of group experts I-GIVDHFUUBLHAand I-GIVDHFUUBLHG manage to generate the identical

Complexity 23

Table 9 Order-inducing vectors obtained by our TOPSIS-based method for problem in [38]

1198661 1198662 1198663 11986641198601 02522 01629 02282 027141198602 01 05243 01364 046891198603 04081 01756 05244 037791198604 04507 01409 01064 013221198605 00697 02203 03753 01161

Table 10 Order-inducing vectors obtained by our TOPSIS-based method for problem in [39]

1198661 1198662 1198663 11986641198601 04548 04969 04923 047321198602 05092 047 04768 046451198603 07192 04888 04953 049151198604 06722 06163 05706 049291198605 06104 04942 06463 04946

Table 11 Ranking results obtained by applying different order-inducing vectors to [38 39]

Problems Approaches Ranking results

Problem in [38] IHFLOWA [38] + order-inducing vectors in [38] 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198605 ≻ 1198604

IHFLOWA [38] + order-inducing vectors in Table 9 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198604 ≻ 1198605

Problem in [39] I-IVDHFLOWA [39] + order-inducing vectors in [39] 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198602 ≻ 1198601

I-IVDHFLOWA [39] + order-inducing vectors in Table 10 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602

permutation of 1199092 ≻ 1199091 ≻ 1199093 which verifies that ourproposed TOPSIS-based order-inducing approach is prac-tical and useful in helping work out decision results morereasonably and robustly

63 Application Studies of Proposed TOPSIS-Based Order-Inducing Method When confronting complex decision-making situations that need additional attitudinal charactersof decision-makers Yager and Filev [88] and Yager [102]introduced and encouraged to employ order-inducing vectorto represent those attitudinal characters Since then theidea of order-inducing vector has been extended to variousdecision-making environments [98 103] such as those basedon intuitionistic fuzzy information [104 105] linguistic infor-mation [106] hesitant fuzzy linguistic information [38] andinterval-valued dual hesitant fuzzy uncertain linguistic infor-mation [39] Generally in existing literature order-inducingvectors rely on experts to directly give [38 39 104ndash106]

In order to examine the adaptability of our TOPSIS-based order-inducing method to linguistic decision-makingenvironments we respectively apply the method to resolvethe same problems in references [38 39] Due to fact thatthere is no descriptions about how the order-inducing vectorswere derived in both references [38 39] we here applyextremums into our TOPSIS-based method to obtain order-inducing vectors that is (1199047 1 0) and (1199041 0 1) for[38] ([1199046 1199046] 1 0) and ([1199040 1199040] 0 1) for [39] Thenwe obtain our order-inducing vectors listed in Table 9 forthe problem adopted in [38] and our order-inducing vectorslisted in Table 10 for [39]

Now we apply the order-inducing vectors as shownin Tables 9 and 10 to resolve the problems in [38] and[39] respectively Regarding the problem in [38] the scorevalues of all alternatives are calculated as 119878(1198601)=02602119878(1198602)=02417 119878(1198603)=03577 119878(1198604)=0221 and 119878(1198605)=0185so the corresponding ranking result is 1198603 ≻ 1198601 ≻ 1198602 ≻1198604 ≻ 1198605 As for the problem in [39] the score values ofall alternatives are computed as 119878(1198601)=39954 119878(1198602)=3783119878(1198603)=70827 119878(1198604)=71792 and 119878(1198605)=5813 we thus obtainthe ranking result 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602 Formore clarity brief descriptions of comparative approachesand obtained ranking results also have been put together inTable 11

According to Table 11 the approaches that employ ourorder-inducing vectors derive the same ranking order ofthe better three alternatives for both comparative problemsfrom [38 39] As seen our proposed TOPSIS-based order-inducing method exhibits effective and more understand-able Furthermore when decision-makers are incapable ofdirectly put forward all desired order-inducing vectors due tocomplexity in ill-structured problems our method providesa feasible directions to obtain order-inducing vectors byfiguring out ideal solutions Comparatively our proposedTOPSIS-based order-inducing method behaves effective andmore operational in practical use

64 Comparison with Decision-Making Approach Based onDegraded form of IVDHFULS Note that our approachis intrinsically adaptive to decision situations based ondegraded forms of IVDHF UUBLS such as the dual hesitant

24 Complexity

fuzzy linguistic sets introduced by Yang and Ju [54] Tofurther inspect feasibility of our approach we here conductcomparative study on the same case adopted by Yang and Ju[54] We firstly convert their original decision matrix intothe form of IVDHF UUBLS crisp membership and non-membership degrees are equivalently changed to intervalswith same upper and lower limits (eg 02 turns out to be[02 02]) and linguistic variables are rewritten as uncertainlinguistic variables (eg 1199042 turns out to be [11990421199042])

Since the case only got three attributes among whichinterdependency could be not significant under generalassumption weighting vector 120596119860119868 is not calculated Andfor straight comparison order inducing vector is notincluded either Then the argument-dependent attributeweighting vector can be obtained from Eq(29) as 120596119860119863 =(03192 03132 03676)119879 By use of Algorithm Iwith operatorIVDHFUUBLWA (ie GIVDHFUUBLWA in case of 120582 = 1see Appendix B) we have the scores of alternatives 119904(1198601) =03412 119904(1198602) = 01354 119904(1198603) = 03932 119904(1198604) = 07352Therefore the ordering of alternatives is1198604 ≻ 1198603 ≻ 1198601 ≻ 1198602which is mainly in alignment with the ordering of1198604 ≻ 1198601 ≻1198603 ≻ 1198602 in Yang and Ju [54] It is worth noticing that 120596119860119863used in our approach is derived objectively from assessmentswhile attribute weights 120596lowast = (035 025 040)119879used by Yangand Ju [54] were assigned empirically Differences between120596119860119863 and 120596lowast lead to different ordering of1198601 and1198603 Compar-atively speaking our approach holds more adaptability andobjectivity

In general our proposed MADM approach in this paperexhibits an effective way to tackle complex green contractorselection problems in which subjective attributes weightingvector (such as120596119860119868) and objective attributes weighting vector(such as 120596119860119863) exist To comprehensively utilize these twofundamental facets of attributes weighting information ourcase study and computational experiments suggest that linearcombination method (ie 120596 = 120572120596119860119868 + 120573120596119860119863) is effectiveto reflect effects of 120596119860119868 and 120596119860119863 by configuring parameter 120572or 120573 When there is no specific expert opinions to specify orhard to obtain we suggest that parameter 120572 chooses the valueset of 0 02 05 08 1 Furthermore generalized aggrega-tion operators are suggested to inspect score trajectoriesof alternative contractors along increasing of the operatorsrsquoamplification parameter 120582 If some alternative contractorsrsquoscore values get very close or crossed so that they cannot bedifferentiated from each other significantly we suggest to askthe panel of experts to derive opinions of ideal solutions andutilize our TOPSIS-based order-inducing method to includethose expert opinions as attitudinal decision informationthereby obtaining more rational and robust decision results

7 Conclusions

Aiming at tackling complex decision-making problems ofgreen contractor selection that often hold simultaneouslythe characteristics of decision hesitancy attributes inter-dependency and group complex attitudinal characters wehave developed an effective IVDHF UUBLS-based approach

We utilize IVDHF UUBLS to elicit complicate decision-makersrsquo assessments more objectively and comprehensivelywe employ DEMATEL for addressing attribute interdepen-dency more precisely and we determine group attitudinalcharacters as order-inducing vectors based on ideal attitudesof experts rather than by empirical assignment The mainadvantages of our approach are (i) IVDHF UUBLS offers anadequate way to depict decision hesitancy (ii) the designeddistance measure for IVDHF UUBLS avoids informationdistortion of traditional ones which artificially add mis-matching membership or nonmembership degrees (iii) thedeveloped generalized operators include traditional ones asspecial cases so as to provide flexibility in MADM basedon IVDHF UUBLS (iv) the devised synthesized attributeweighting scheme provides an adaptive way to consider twonecessary aspects attributes interdependency-based weightsand argument-dependent weights (v) the TOPSIS-basedmethod can deduce reasonable order-inducing vectors ratherthan empirical ones and holds generality in other linguisticdecision environments Results of case studies and experi-ments have verified the validity of the proposed approach

When multiple decision organizations are involvedgroup decision-making based on IVDHF UUBLS is animportant research direction Besides inducing preferencesfor decision-making problems of high uncertainty interactivemechanisms should also be studied to extract preferences Inaddition empirical studies of the proposed approach can becarried out to various areas such as investment evaluationproject management and vehicle selection in fleet operations

Appendix

A Basic Notions for IVDHFS and UnbalancedLinguistic Term Set

Definition A1 (see [47]) Letting 119883 be a fixed set then anIVDHFS on 119883 is defined as119863 = ⟨119909 ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (A1)

where ℎ(119909) = ⋃[120583119871120583119880]isinℎ(119909)120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and119892(119909) = ⋃[]119871]119880]isin119892(119909)] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two setsof interval values in [0 1] denoting possible membershipand non-membership degrees of element 119909 isin 119883 to theset 119863 respectively with conditions 120583 ] isin [0 1] and0 le (120583119880)+ + (]119880)+ le 1 and for all 119909 isin 119883 (120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880

For convenience normally 119889 = ℎ 119892 is called an interval-valued dual hesitant fuzzy element (IVDHFE) and 119863 is theset of all IVDHFEs

Definition A2 (see [107]) Suppose 119878 = 119904119895 | 119895 = 0 1 119892 minus1 is a finite and totally ordered discrete linguistic term setwhere 119904119895 represent possible values for a linguistic variableand 119892 is an odd cardinality 119878 is uniformly and symmetricallydistributed if the following two conditions are satisfied (1)

Complexity 25

there exists a unique constant 120582 gt 0 such that 119873119878(119904119894) minus119873119878(119904119895) = 119897(119894 minus 119895) for all 119894 119895 = 0 1 119892 minus 1 (2) Let 119878119877 =119904 | 119904 isin 119878 119904 gt 119878119862 and 119878119871 = 119904 | 119904 isin 119878 119904 lt 119878119862 Let C(119878119877) andC(119878119871) be the cardinality of 119878119877 and 119878119871 then C(119878119877) = C(119878119871)If 119878 is uniformly and symmetrically distributed then 119878 iscalled a balanced linguistic term set Otherwise 119878 is calledan unbalanced linguistic term set

The following 2-tuple fuzzy linguistic representationmodel extends traditional linguistic term set to a continuouscase so as to facilitate computing with linguistic variables

Definition A3 (see [59]) Let 119878 = 1199040 1199041 119904119892minus1 be alinguistic term set and 120573 isin [0 119892] Then a 2-tuple fuzzylinguistic variable expresses the equivalent information to 120573is defined as

Δ [0 119892] 997888rarr 119878 times [minus05 05) (A2)

Δ (120573) = (119904119894 120572) with

119904119894 119894 = 119903119900119906119899119889 (120573)120572 = 120573 minus 119894 120572 isin [minus05 05)

(A3)

Δminus1 (119904119894 120572) = 119894 + 120572 = 120573 (A4)

where 119903119900119906119899119889() is the usual rounding operation and120572 is calledsymbolic translation

To obtain 2-tuple fuzzy linguistic representations ofunbalanced linguistic terms the concept of linguistic hierar-chies ie 119871119867 = ⋃119905 119897(119905 119899(119905)) is used 119897(119905 119899(119905)) is a linguistichierarchy with 119905 indicating the level of hierarchy and 119899(119905)denotes the granularity of the linguistic term set of 119905 Herreraet al [59] defined the following transformation functionsbetween labels from different levels in multigranular linguis-tic information contexts without loss of information

Definition A4 (see [59]) In linguistic hierarchies 119871119867 =⋃119905 119897(119905 119899(119905)) whose linguistic term sets are represented by119878119899(119905) = 119904119899(119905)0 119904119899(119905)119899(119905)minus1 the transformation function froma linguistic label in level 119905 to a label in consecutive level 1199051015840 isdefined as 1198791198651199051199051015840 119897(119905 119899(119905)) 997888rarr 119897(1199051015840 119899(1199051015840)) such that

1198791198651199051199051015840 (119904119899(119905)119894 120572119899(119905))= Δ 1199051015840 (Δminus1119905 (119904119899(119905)119894 120572119899(119905)) (119899 (1199051015840) minus 1)119899 (119905) minus 1 ) (A5)

By use of the above transformation function any 2-tuplelinguistic representation can be transformed into a term in119871119867 Detailed transformation procedures are listed as follows

(1) Representation in linguistic hierarchy To transformthe unbalanced terms of 119878 into the corresponding terms inthe 119871119867 transformation function 120595 is employed to associateeach unbalanced linguistic 2-tuple (119904119894 120572) with its linguistic 2-tuple in 119871119867(119878) ie

120595 119878 997888rarr 119871119867(119878) (A6)

so that 120595(119904119894 120572) = (119904119866(119894)119868(119894) 120582) for forall(119904119894 120572) isin 119878(2) Computational phase Firstly transform (119904119866(119894)119868(119894) 120582) intolinguistic 2-tuples denoted as (119904119899(1199051015840)

1198681015840(119894) 1205821015840) in 119878119899(1199051015840) where

(119904119899(1199051015840)1198681015840(119894)

1205821015840) = 119879119865 (119904119866(119894)119868(119894) 120582)= Δ(Δminus1 (119904119866(119894)119868(119894) 120582) sdot (119899 (1199051015840) minus 1)119866 (119894) minus 1 ) (A7)

Then a computational model is used over 119878119899(1199051015840) with aresult denoted as (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)(3) Retranslation process The result (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)is transformed into the unbalanced term in 119878 by using thetransformation function 120595minus1 ie

120595minus1 119871119867 (119878) 997888rarr 119878 (A8)

so that 120595minus1(119904119899(1199051015840)119903 120582119903) = (119904119903119890119904119906119897119905 120582119903119890119904119906119897119905) isin 119878B Details of Theorem 14

Proof Given a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector for119904119889119895 and 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 120582 gt 0 then(1) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLWA and GIVDHFUUBLWG operators respectively(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLOWA and GIVDHFUUBLOWG operators respectively

(2) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to theIVDHF UUBL weighted average (IVDHFUUBLWA)and IVDHF UUBL weighted geometric (IVDHFUUBLWG)operators respectively where

26 Complexity

119868119881119863119867119865119880119880119861119871119882119860120596 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

120596119895119904119889119895= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus

119899prod119895=1

(1 minus 120583119871119895 )120596119895 1 minus 119899prod119895=1

(1 minus 120583119880119895 )120596119895]] [[

119899prod119895=1

(]119871119895)120596119895 119899prod119895=1

(]119880119895 )120596119895]])

(B1)

119868119881119863119867119865119880119880119861119871119882119866120596 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889119895120596119895

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[

119899prod119895=1

(120583119871119895 )120596119895 119899prod119895=1

(120583119880119895 )120596119895]] [[1 minus

119899prod119895=1

(1 minus ]119871119895)120596119895 1 minus 119899prod119895=1

(1 minus ]119880119895 )120596119895]])

(B2)

(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHA andGIVDHFUUBLHG operators reduce to the IVDHF UUBL

orderedweighted average (IVDHFUUBLOWA) and IVDHFUUBL ordered weighted geometric (IVDHFUUBLOWG)operators respectively where

119868119881119863119867119865119880119880119861119871119874119882119860119908 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

119908119895119904119889120590(119895)= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) )))

119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) )))

]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus

119899prod119895=1

(1 minus 120583119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus 120583119880120590(119895))119908119895]] [[

119899prod119895=1

(]119871120590(119895))119908119895 119899prod119895=1

(]119880120590(119895))119908119895]])

(B3)

119868119881119863119867119865119880119880119861119871119874119882119866119908 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889120590(119895)119908119895

= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[

119899prod119895=1

(120583119871120590(119895))119908119895 119899prod119895=1

(120583119880120590(119895))119908119895]] [[1 minus

119899prod119895=1

(1 minus ]119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus ]119880120590(119895))119908119895]])

(B4)

Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

References

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[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

[11] X Qi C Liang and J Zhang ldquoGeneralized cross-entropy basedgroup decision making with unknown expert and attributeweights under interval-valued intuitionistic fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 79 pp 52ndash642015

[12] H Zimmermann Fuzzy Set Theorymdashand Its ApplicationsKluwer Academic Boston Mass USA 2001

[13] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 pp 199ndash249 1975

[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

[16] B T Nuong K-W Kim L Prathumratana et al ldquoSustainabledevelopment in the mining sector and its evaluation usingfuzzy AHP (Analytic Hierarchy Process) approachrdquo GeosystemEngineering vol 14 no 1 pp 43ndash50 2011

[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

[18] E K Zavadskas Z Turskis and J Antucheviciene ldquoSelectinga contractor by using a novel method for multiple attributeanalysis Weighted aggregated sum product assessment withgrey values (WASPAS-G)rdquo Studies in Informatics and Controlvol 24 no 2 pp 141ndash150 2015

[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

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[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

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[26] Z Hatush and M Skitmore ldquoContractor Selection UsingMulticriteria Utility Theory An Additive Modelrdquo Building andEnvironment vol 33 no 2-3 pp 105ndash115 1998

[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

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[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

[33] P Humphreys R McIvor and F Chan ldquoUsing case-basedreasoning to evaluate supplier environmental managementperformancerdquo Expert Systems with Applications vol 25 no 2pp 141ndash153 2003

[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

[45] Bin Zhu Zeshui Xu and Meimei Xia ldquoDual Hesitant FuzzySetsrdquo Journal of Applied Mathematics vol 2012 Article ID879629 13 pages 2012

[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

[51] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[52] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

[55] Z Tao X Liu H Chen and L Zhou ldquoUsing new versionof extended t-norms and s-norms for aggregating intervallinguistic labelsrdquo IEEETransactions on SystemsManCyberneticsSystems vol vol p 15 2016

[56] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[57] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[58] L Martınez M Espinilla J Liu L G Perez and P J SanchezldquoAn evaluation model with unbalanced linguistic informationapplied to olive oil sensory evaluationrdquo Journal of Multiple-Valued Logic and Soft Computing vol 15 no 2-3 pp 229ndash2512009

[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

[62] L Marin A Valls D Isern A Moreno and J M MerigoldquoInduced unbalanced linguistic ordered weighted average andits application in multiperson decision makingrdquo The ScientificWorld Journal vol 2014 2014

[63] A Baykasoglu V Kaplanoglu Z D U Durmusoglu and CSahin ldquoIntegrating fuzzy DEMATEL and fuzzy hierarchicalTOPSIS methods for truck selectionrdquo Expert Systems withApplications vol 40 no 3 pp 899ndash907 2013

[64] S K Patil and R Kant ldquoA hybrid approach based on fuzzyDEMATEL and FMCDM to predict success of knowledge man-agement adoption in supply chainrdquoApplied Soft Computing vol18 pp 126ndash135 2014

[65] S Liyin and Y Hong ldquoImproving environmental performanceby means of empowerment of contractorsrdquo Management ofEnvironmental Quality An International Journal vol 17 no 3pp 242ndash257 2006

[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

[69] Y Kazancoglu I Kazancoglu and M Sagnak ldquoA new holisticconceptual framework for green supply chain management

performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

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Complexity 17

have been made to address interdependencies betweenattributes in MADM However they unanimously cannotreveal the interdependences in a more complete way WhileDEMATAEL technique [63 96] gathers collective knowledgeto capture influences between attributes more precisely andcompletely so that weighting vector for attributes can berationally determined according to strength measure ofthose influences [64 97] Therefore in tackling the complexMADMwith interdependent attributes under IVDHF UUBLenvironments we present a DEMATEL-based method as thefollowing Algorithm II to derive attribute weighting vector120596119860119868 from the viewpoint of attributes interdependency

Algorithm II DEMATEL-basedmethod for ensuringweight-ing vector 120596119860119868Step 1 Construct initial direct-influence matrix 119885 = (119911119894119896)119899times119899

For attributes 119860 = 1198601 1198602 119860119899 decision-makersare asked to give direct effect that attribute 119860 119894 has onattribute 119860119896 using a five-point scale with scores representedby natural language (1) 0 mdash lsquoabsolutely no influencersquo (2)1 mdash lsquolow influencersquo (3) 2 mdash lsquomedium influencersquo (4) 3 mdashlsquohigh influencersquo (5) 4mdash lsquovery high influencersquoThen the initialdirect-influence matrix 119885 = (119911119894119896)119899times119899 can be established witheach element being the mean of same factors in differentdirect matrices

Step 2 Calculate normalized direct-influence matrix 119883 =(119909119894119896)119899times119899 Using matrix 119885 the normalized direct-influencematrix 119883 = (119909119894119896)119899times119899 is calculated according to

119909119894119896 = 119911119894119896119904 (30)

where 119904 = maxmax1le119894le119899sum119899119896=1 119911119894119896max1le119896le119899sum119899119895=1 119911119894119896 Allelements in matrix 119883 are complying with 0 le 119909119894119896 le 10 le sum119899119896=1 119909119894119896 le 1 and 0 le sum119899119894=1 119909119894119896 le 1 and at least onerow or column of the summation equals 1

Step 3 Derive total-influence matrix 119879 = (119905119894119896)119899times119899 Basedon matrix 119883 the total-influence matrix 119879 = (119905119894119896)119899times119899 can bederived by summing the direct effects and all of the indirecteffects

119879 = 119883 + 1198832 + 1198833 + sdot sdot sdot + 119883ℎ = 119883 (119868 minus 119883)minus1 (31)

where 119868 represents identity matrix When ℎ 997888rarr infin 119883ℎ =(0)119899times119899Step 4 Compute influencing and influenced degrees of eachattribute The sum of rows and the sum of columns withinthe total-influence matrix 119879 are respectively expressed as thevectors 119903 and 119888

119903 = (119903119894)119899times1 = ( 119899sum119896=1

119905119894119896)119899times1

119888 = (119888119896)119899times1 = ( 119899sum

119895=1

119905119894119896)119879

1times119899

(32)

where 119903119894 denotes the sumof the 119894th row inmatrix119879 and showsthe sum of direct and indirect effects that attribute 119860 119894 hason other attributes Similarly 119888119896 shows the sum of direct andindirect effects that attribute 119860119896 has receivedStep 5 Estimate weights of attributes based on influencedegrees according to

120596119868lowast119894 = [(119903119894 + 119888119894)2 + (119903119894 minus 119888119894)2]12 (33)

Then the weighting vector 120596119860119868 can be obtained by normaliz-ing (33)

120596119860119868119894 = 120596119868lowast119894sum119899119894=1 120596119868lowast119894 (34)

53 Obtaining Order Inducing Variables 120576 by TOPSIS-BasedMethod For ill-defined MADM problems induced aggre-gation operators [89 90 98] provide an effective channelfor experts to express complex attitudinal characters Thekey step in induced aggregation operators is to obtainorder-inducing vectors 120576 which however are presumptivelyassigned in most papers [62] Therefore in this sectionaiming at deducing the order-inducing vectors 120576 rationallywe here devise a TOPSIS-based method in Algorithm IIIObviously the Algorithm III is of generality for MADMwithdifferent forms of linguistic expression

Algorithm III TOPSIS-based method for obtaining orderinducing vectors 120576Step 1 Let the decision maker define IVDHF UUBL idealsolution(s) 119903+119894 and IVDHF UUBL negative ideal solution(s)119903minus119894 corresponding to each attribute in which

119903+119894 = (119903+1198941 119903+1198942 119903+119894119901) 119903minus119894 = (119903minus1198941 119903minus1198942 119903minus119894119902) (35)

where 119901 and 119902 denotes the number of 119903+119894 and 119903minus119894 respectively

Step 2 Calculate distances from each element in decisionmatrix to ideal solution(s) and negative solution(s) respec-tively

119889+119894119895 = max119897=12119901

119889 (119903119894119895 119903+119894119897 ) 119889minus119894119895 = min

119897=12119902119889 (119903119894119895 119903minus119894119897 ) (36)

where 119889(119903119894119895 119903+119894119897 ) and 119889(119903119894119895 119903minus119894119897 ) are calculated by the distancemeasure in Definition 6 If 119901 gt 1 we choose the onewith maximum distance if 119902 gt 1 we choose the one withminimum distance

Step 3 Calculate coefficients according to

119888119894119895 = 119889minus119894119895119889+119894119895 + 119889minus119894119895 119894 = 1 2 119899 119895 = 1 2 119898 (37)

18 Complexity

Table 2 Decision matrix in the form of IVDHF UUBLS1199091 1199092 11990931198601([AHVH] [02 03][02 04] [03 04]) ([HT] [04 05][05 06] [02 03] [02 04]) ([ALAH] [01 02][01 03] [06 07])1198602([MH] [05 06][02 03]) ([AHT] [06 07][01 02] ) ([MVH] [02 03][05 06] [06 07])1198603([VLAL] [03 04][04 05] [05 06]) ([MVH] [06 07] [07 08][01 02]) ([VLL] [04 05][02 03])1198604([MAH] [02 04][05 06]) ([LAL] [03 05][02 03]) ([ALM] [06 07][01 02] [02 03])1198605([ALM] [04 05][01 02] [04 05]) ([AHH] [06 07][01 02]) ([HVH] [04 05][06 07] [01 03])1198606([LM] [01 02][03 05] [03 05]) ([VLL] [02 04] [05 06][02 03]) ([AHH] [03 04][04 05])1198607([VLL] [04 05][05 06] [02 03] [02 04]) ([MAH] [05 06] [07 08][01 02]) ([ALM] [04 05][05 06] [03 04])1198608([ALAH] [01 03][04 06]) ([AHH] [05 07][01 02] [02 03]) ([AHH] [04 06][03 04])

Table 3 Direct influence matrix Z1198601 1198602 1198603 1198604 1198605 1198606 1198607 11986081198601 0 3 4 1 3 1 2 41198602 1 0 2 4 4 2 3 11198603 4 2 0 1 3 2 4 11198604 2 3 1 0 3 4 1 21198605 4 2 1 3 0 1 4 31198606 2 4 3 4 4 0 4 21198607 4 1 2 3 3 1 0 41198608 1 2 4 3 3 2 4 0

Step 4 Derive order inducing variable 120576119894119895 by descendingorder of 119888119894119895 The greater the value 119888119894119895 the bigger the 1205761198941198956 Illustrative Example and Experiments

61 Illustrative Case Study Suppose a mega infrastructureproject is selecting green contractor for one of its subprojectsA panel of decision-makers including managers professorsand competitive intelligence experts has been organized tocomprehensively evaluate three alternative contractors (ie1199091 1199092 and 1199093) under the eight attributes (ie 1198601ndash1198608) listedin Section 2

To elicit true decision preferences we apply theIVDHF UUBLS to assess the three alternatives (1199091ndash1199093)Firstly decision-makers are asked to vote on 119909119895 under eachattribute119860 119894 using different unbalanced linguistic term sets 1198781 1198782 and 1198783 where 1198781 = 119873 119871119860119871119872119860119867119867119876119867119881119867119860119879 1198791198782 = 119873119860119873119881119871 119876119871 119871 119860119871119872119860119867119881119867 119879 1198783 = 119873119881119871119871119860119872119872119876119872119860119867119867 119879 In Figure 4 we show the adoptedlinguistic hierarchies and relationship between 1198781 1198782 and1198783 Attributes 1198601 1198602 1198605 and 1198608 are evaluated by use of 1198781attributes11986031198604 and1198607 are evaluated by 1198782 and attribute1198606is evaluated by 1198783 The highest-voted uncertain unbalancedlinguistic term [119904119898 119904119899] are identified Secondly the panel of

decision-makers is asked to further express their opinionson membership degrees ℎ(119909) and nonmembership degrees119892(119909) to [119904119898 119904119899] Then we can derive compound assessments(ie ⟨119909 119904120599(119909) ℎ(119909) 119892(119909)⟩) in the form of IVDHF UUBLSand obtain the decision matrix in Table 2 Additionallyin order to examine the interdependency among attributesdecision-makers are also required to assess the influentialrelations among the eight attributes using DEMATEL-basedAlgorithm II The direct influence matrix is collected andshown in Table 3

Next we apply the proposed Algorithm I to solve thisgreen contractor selection problem To derive the synthe-sized attribute weighting vector in Algorithm I we assign120572 = 120573 = 05 For scenarios with no group attitudinalcharacters operator GIVDHFUUBLHA in Definition 12 ischosen In contrast for scenarios with group attitudinalcharacters operator I-GIVDHFUUBLHA in Definition 13will be chosen Further the panel of decision-makers is askedto determine their positive and negative ideal alternativesthat are collected in Table 4 where corresponding order-inducing vectors that are derived from Algorithm III arealso listed (see the rightmost column) Thereafter operatorI-GIVDHFUUBLHA in Definition 13 is utilized for aggrega-tion For GIVDHFUUBLHA and I-GIVDHFUUBLHA we

Complexity 19

Table 4 Ideal alternatives (119903+ and 119903minus) and derived order-inducing vectors 120576119903+ 119903minus 1205761199091 1199092 11990931198601 ([VHT] 1 0) ([NL] 1 0) 5 6 61198602 ([MAH] [07 08] [01 02]) ([NL] 1 0) 1 1 41198603 (T 1 0) (L 1 0) 8 7 81198604 ([ALAH] 09 01) (T 1 0) 2 2 11198605 ([HT] [09 1] 0) (N 1 0) 7 5 51198606 (M 1 0) (N 1 0) 6 3 31198607 (M 1 0) ([NL] 1 0) 3 8 71198608 ([MAH] 1 0) (N 1 0) 4 4 2

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 1 2 3 4

0 1 2

N L AH H QH VH AT TM

Level 1 l(13)

Level 2 l(25)

Level 3 l(39)

Level 4 l(417)

Unbalanced linguistic term set S1

N AH VH TM Unbalanced linguistic term set S2VL L ALAN QL

N H TM Unbalanced linguistic term set S3

L AM AHQM

AL

VL

Figure 4 Unbalanced linguistic term sets (1198781 1198782 and 1198783) and their mapping in linguistic hierarchies

assume 120582=2 and 119908 = (0 0 015 025 025 025 01 0)119879Note that 119908 is a position weighting vector which is derivedby the fuzzy semantic quantitative operator [99] Aftercompleting all steps in Algorithm I (see Appendix C) weobtain the ranking results and present them in Table 5 Forthe scores derived from Step 5 we find the ranking order is1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the best solution Step 7gives the ranking order of 1199092 ≻ 1199091 ≻ 1199093 indicating 1199092 is stillthe most appropriate one while 1199091 is better than 1199093 under theinfluence of 12057662 Experiments on Ranking Results along with VariousParameter Configurations In order to inspect the impactsof various parameter configurations on ranking results inthis section more numerical examples are carried out Alleight operators that defined in Section 41 are applied to theabove illustrative case where the parameter 120582 is configured asintegers ranging from 1 to 20 and parameter 120572 is configuredto a value set of 0 03 05 08 09 1

Firstly the four basic aggregation operators of GIVDH-FUUBLWA GIVDHFUUBLWG GIVDHFUUBLOWA and

Table 5 Ranking results of operators GIVDHFUUBLHA and I-GIVDHFUUBLHA

Aggregation operator Ranking orderGIVDHFUUBLHA 1199092 ≻ 1199093 ≻ 1199091I-GIVDHFUUBLHA 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG are applied to the former case Thesame configuration 120572 = 120573 = 05 is chosen while 120582 is testedas integers ranging from 1 to 20 The ranking results derivedfrom different parameter combinations have been collectedin Table 6 As can be seen four operators unanimouslyidentify the contractor 1199092 is the best one no matter how120582 changes Regarding priority relation between 1199091 and 1199093GIVDHFUUBLWA and GIVDHFUUBLOWA generate thesame result of1199093 ≻ 1199091 while results fromGIVDHFUUBLWGandGIVDHFUUBLOWGgot a change to 1199091 ≻ 1199093 at a certainvalue of parameter 120582

Actually GIVDHFUUBLWA and GIVDHFUUBLOWAare generalized version of weighted arithmetic aggregationoperators that producemore favorable results by emphasizing

20 Complexity

Table 6 Ranking orders obtained by GIVDHFUUBLWA GIVDHFUUBLOWA GIVDHFUUBLWG and GIVDHFUUBLOWG along with120582Operators 120582 (Integer) Ranking orderGIVDHFUUBLWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLOWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLWG [1 2] 1199092 ≻ 1199093 ≻ 1199091[3 20] 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG [1 8] 1199092 ≻ 1199093 ≻ 1199091[9 20] 1199092 ≻ 1199091 ≻ 1199093

Table 7 Ranking results obtained by GIVDHFUUBLHA and I-GIVDHFUUBLHA along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHA I-GIVDHFUUBLHA120572=0 120573=1 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=03 120573=07 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=05 120573=05 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=08 120573=02 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=09 120573=01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=1 120573=0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093

Table 8 Ranking results obtained by GIVDHFUUBLHG and I-GIVDHFUUBLHG along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHG I-GIVDHFUUBLHG

120572 = 0 120573 = 1 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 03 120573 = 07 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 05 120573 = 05 [1 8] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[9 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 08 120573 = 02 [1 17] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[18 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 09 120573 = 01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572 = 1 120573 = 0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093membership degrees to a uncertain linguistic term thus canbe considered as optimistic operators while GIVDHFU-UBLWG and GIVDHFUUBLOWG are generalized geomet-ric aggregation operators that producemore favorable resultsby emphasizing nonmembership degrees thus can be consid-ered as pessimistic operators [100 101]Therefore the rankingresults in Table 6 could be perceived as for optimism-baseddecision-making alternative 1199093 always stands out better than1199091 when 120582 gradually amplifies degree of optimism but forpessimism-based decision-making 1199091 turns out better than1199093 when 120582 gradually amplifies degree of pessimism 120582 canbe considered as the parameter for amplifying degree ofoptimism or pessimism [101]

Next to inspect the impacts of various combinationsof 120596119860119868 and 120596119860119863 on ranking results another four aggrega-tion operators of GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG also have beenapplied to the above illustrative case with varying 120572 120573 and 120582

All results have been collected in Tables 7 and 8 for compari-son As can be seen fromTable 7 ranking results generated byoperator GIVDHFUUBLHA keep the same as 1199092 ≻ 1199093 ≻ 1199091for all test combinations of parameters However as shownin Table 8 although GIVDHFUUBLHG is also capable ofidentifying 1199092 as the best contractor the priority relationsbetween 1199091 and 1199093 hold similar changing patterns in rankingresults in Table 6 that were output byGIVDHFUUBLWGandGIVDHFUUBLOWG Regarding decision situations whereobjective attributeweighting vector120596119860119863 is only considered (120572= 0) or its dominant importance (such as 120572 = 03) is accepted1199091 ≻ 1199093 is a clear observation when parameter 120582 is set toan integer in interval [8 20] Contrariwise under decisionscenarios where subjective attribute weighting vector 120596119860119868 isonly needed (120572 = 1) or its dominant importance (such as 120572 =09) is acknowledged 1199093 ≻ 1199091 holds all the way 120582 varies

In order to more clearly show the score trajectories ofalternative contractors along variations of parameter 120582 we

Complexity 21

HAInducedHA

InducedHA-2HA-2

HA-3

InducedHA-1InducedHA-3

= =

HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(a) 120572 = 0

HAInducedHA

= 03 = 07

InducedHA-2

HA-2HA-3

InducedHA-1 InducedHA-3HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(b) 120572 = 03

HAInducedHA

= 5 = 05

InducedHA-2HA-2

HA-3

InducedHA-1 InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(c) 120572 = 05

HAInducedHA

= 08 = 02

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572 = 08

HAInducedHA

= 9 = 01

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(e) 120572 = 09

HAInducedHA

246 2462 2464 2466 2468 247

= 1 = 0InducedHA-2HA-2

HA-3

HA- 3

HA-3

InducedHA-1

InducedHA-3

HA-1

HA-1

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

1085

1086

1087

(f) 120572 = 1

Figure 5 Score trajectories obtained by aggregation operators HA and InducedHA with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

further configure parameter 120582 to the value set of [0 01100]and conduct computational experiments to collect scorevalues and ranking results (as shown in Figures 5 and 6) byfour aggregation operators of GIVDHFUUBLHA GIVDH-FUUBLHG I-GIVDHFUUBLHAand I-GIVDHFUUBLHGFor more clarity in Figures 5 and 6 please note that weuse HA HG Induced-HA and Induced HG to respec-tively denote GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG In Figure 5it can be observed that when dominant importance of 120596119860119863

is emphasized (120572 = 0 120572 = 03 120572 = 05) GIVDHFUUBLHAis capable of differentiating the three alternative contractorsas same ranking order of 1199092 ≻ 1199093 ≻ 1199091 for all 120582 but withincreases in 120572 the margins between scores of 1199091 and 1199093 nar-rows and the rank order changes to 1199091 ≻ 1199093when 120572 gets 09and 1 Interestingly in Figure 6 although GIVDHFUUBLHGis able to significantly differentiate alternative contractors 1199091and 1199093 when dominant importance of 120596119860119863 is emphasized (120572= 0 120572 = 03 120572 = 05) rank order of 1199091 and 1199093 obviouslychanges once from 1199093 ≻ 1199091 to 1199091 ≻ 1199093 when 120582 is bigger

22 Complexity

HGInducedHG

= = 1

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(a) 120572= 0

HGInducedHG

= 3 = 07

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(b) 120572= 03

HGInducedHG

= 05 = 05

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(c) 120572= 05

HGInducedHG

= 8 = 02

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572= 08

HGInducedHG

292 293 294 295 296 297minus08

minus0795

minus079

= 09 = 01

HA-1

HA-3

InducedHG-2

HG-1

HG-1

HG-2

HG-3

HG-3 InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4

minus3

minus2

minus1

0

1

2

scor

e val

ues

(e) 120572= 09

HGInducedHG

50 60 70 80 90 100minus095minus09

minus085

= 1 = 0

HG-x3

HG-x1

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(f) 120572= 1

Figure 6 Score trajectories obtained by aggregation operators HG and InducedHG with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

than 8 When 120572 increases to 1 GIVDHFUUBLHGmaintainsthe rank order of 1199093 ≻ 1199091 but their score trajectorieswalk very close Comparatively I-GIVDHFUUBLHA andI-GIVDHFUUBLHG generate consistent ranking results of1199092 ≻ 1199091 ≻ 1199093 no matter 120582 varies in all the above experiments

In sum when parameters are configured with typical val-ues (ie 120572 = 05 and 120582=1) and no additional attitudinal infor-mation is taken into consideration the proposed aggregationoperators unanimously derive the same rank order 1199092 ≻1199093 ≻ 1199091 of for the three alternative contractors However

with the amplification effect of parameter 120582 analysis resultsfrom linear combination of 120596119860119868 and 120596119860119863 reveal alternative1199092 which consistently is the best choice but rank order of1199091 and 1199093 changes because their score trajectories get closeand crossed which probably indicates that only decision-making informationderived from the decisionmatrix is inad-equate to significantly differentiate alternatives 1199091 and 1199093While by integrating additional decision information aboutattitudinal characters of group experts I-GIVDHFUUBLHAand I-GIVDHFUUBLHG manage to generate the identical

Complexity 23

Table 9 Order-inducing vectors obtained by our TOPSIS-based method for problem in [38]

1198661 1198662 1198663 11986641198601 02522 01629 02282 027141198602 01 05243 01364 046891198603 04081 01756 05244 037791198604 04507 01409 01064 013221198605 00697 02203 03753 01161

Table 10 Order-inducing vectors obtained by our TOPSIS-based method for problem in [39]

1198661 1198662 1198663 11986641198601 04548 04969 04923 047321198602 05092 047 04768 046451198603 07192 04888 04953 049151198604 06722 06163 05706 049291198605 06104 04942 06463 04946

Table 11 Ranking results obtained by applying different order-inducing vectors to [38 39]

Problems Approaches Ranking results

Problem in [38] IHFLOWA [38] + order-inducing vectors in [38] 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198605 ≻ 1198604

IHFLOWA [38] + order-inducing vectors in Table 9 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198604 ≻ 1198605

Problem in [39] I-IVDHFLOWA [39] + order-inducing vectors in [39] 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198602 ≻ 1198601

I-IVDHFLOWA [39] + order-inducing vectors in Table 10 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602

permutation of 1199092 ≻ 1199091 ≻ 1199093 which verifies that ourproposed TOPSIS-based order-inducing approach is prac-tical and useful in helping work out decision results morereasonably and robustly

63 Application Studies of Proposed TOPSIS-Based Order-Inducing Method When confronting complex decision-making situations that need additional attitudinal charactersof decision-makers Yager and Filev [88] and Yager [102]introduced and encouraged to employ order-inducing vectorto represent those attitudinal characters Since then theidea of order-inducing vector has been extended to variousdecision-making environments [98 103] such as those basedon intuitionistic fuzzy information [104 105] linguistic infor-mation [106] hesitant fuzzy linguistic information [38] andinterval-valued dual hesitant fuzzy uncertain linguistic infor-mation [39] Generally in existing literature order-inducingvectors rely on experts to directly give [38 39 104ndash106]

In order to examine the adaptability of our TOPSIS-based order-inducing method to linguistic decision-makingenvironments we respectively apply the method to resolvethe same problems in references [38 39] Due to fact thatthere is no descriptions about how the order-inducing vectorswere derived in both references [38 39] we here applyextremums into our TOPSIS-based method to obtain order-inducing vectors that is (1199047 1 0) and (1199041 0 1) for[38] ([1199046 1199046] 1 0) and ([1199040 1199040] 0 1) for [39] Thenwe obtain our order-inducing vectors listed in Table 9 forthe problem adopted in [38] and our order-inducing vectorslisted in Table 10 for [39]

Now we apply the order-inducing vectors as shownin Tables 9 and 10 to resolve the problems in [38] and[39] respectively Regarding the problem in [38] the scorevalues of all alternatives are calculated as 119878(1198601)=02602119878(1198602)=02417 119878(1198603)=03577 119878(1198604)=0221 and 119878(1198605)=0185so the corresponding ranking result is 1198603 ≻ 1198601 ≻ 1198602 ≻1198604 ≻ 1198605 As for the problem in [39] the score values ofall alternatives are computed as 119878(1198601)=39954 119878(1198602)=3783119878(1198603)=70827 119878(1198604)=71792 and 119878(1198605)=5813 we thus obtainthe ranking result 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602 Formore clarity brief descriptions of comparative approachesand obtained ranking results also have been put together inTable 11

According to Table 11 the approaches that employ ourorder-inducing vectors derive the same ranking order ofthe better three alternatives for both comparative problemsfrom [38 39] As seen our proposed TOPSIS-based order-inducing method exhibits effective and more understand-able Furthermore when decision-makers are incapable ofdirectly put forward all desired order-inducing vectors due tocomplexity in ill-structured problems our method providesa feasible directions to obtain order-inducing vectors byfiguring out ideal solutions Comparatively our proposedTOPSIS-based order-inducing method behaves effective andmore operational in practical use

64 Comparison with Decision-Making Approach Based onDegraded form of IVDHFULS Note that our approachis intrinsically adaptive to decision situations based ondegraded forms of IVDHF UUBLS such as the dual hesitant

24 Complexity

fuzzy linguistic sets introduced by Yang and Ju [54] Tofurther inspect feasibility of our approach we here conductcomparative study on the same case adopted by Yang and Ju[54] We firstly convert their original decision matrix intothe form of IVDHF UUBLS crisp membership and non-membership degrees are equivalently changed to intervalswith same upper and lower limits (eg 02 turns out to be[02 02]) and linguistic variables are rewritten as uncertainlinguistic variables (eg 1199042 turns out to be [11990421199042])

Since the case only got three attributes among whichinterdependency could be not significant under generalassumption weighting vector 120596119860119868 is not calculated Andfor straight comparison order inducing vector is notincluded either Then the argument-dependent attributeweighting vector can be obtained from Eq(29) as 120596119860119863 =(03192 03132 03676)119879 By use of Algorithm Iwith operatorIVDHFUUBLWA (ie GIVDHFUUBLWA in case of 120582 = 1see Appendix B) we have the scores of alternatives 119904(1198601) =03412 119904(1198602) = 01354 119904(1198603) = 03932 119904(1198604) = 07352Therefore the ordering of alternatives is1198604 ≻ 1198603 ≻ 1198601 ≻ 1198602which is mainly in alignment with the ordering of1198604 ≻ 1198601 ≻1198603 ≻ 1198602 in Yang and Ju [54] It is worth noticing that 120596119860119863used in our approach is derived objectively from assessmentswhile attribute weights 120596lowast = (035 025 040)119879used by Yangand Ju [54] were assigned empirically Differences between120596119860119863 and 120596lowast lead to different ordering of1198601 and1198603 Compar-atively speaking our approach holds more adaptability andobjectivity

In general our proposed MADM approach in this paperexhibits an effective way to tackle complex green contractorselection problems in which subjective attributes weightingvector (such as120596119860119868) and objective attributes weighting vector(such as 120596119860119863) exist To comprehensively utilize these twofundamental facets of attributes weighting information ourcase study and computational experiments suggest that linearcombination method (ie 120596 = 120572120596119860119868 + 120573120596119860119863) is effectiveto reflect effects of 120596119860119868 and 120596119860119863 by configuring parameter 120572or 120573 When there is no specific expert opinions to specify orhard to obtain we suggest that parameter 120572 chooses the valueset of 0 02 05 08 1 Furthermore generalized aggrega-tion operators are suggested to inspect score trajectoriesof alternative contractors along increasing of the operatorsrsquoamplification parameter 120582 If some alternative contractorsrsquoscore values get very close or crossed so that they cannot bedifferentiated from each other significantly we suggest to askthe panel of experts to derive opinions of ideal solutions andutilize our TOPSIS-based order-inducing method to includethose expert opinions as attitudinal decision informationthereby obtaining more rational and robust decision results

7 Conclusions

Aiming at tackling complex decision-making problems ofgreen contractor selection that often hold simultaneouslythe characteristics of decision hesitancy attributes inter-dependency and group complex attitudinal characters wehave developed an effective IVDHF UUBLS-based approach

We utilize IVDHF UUBLS to elicit complicate decision-makersrsquo assessments more objectively and comprehensivelywe employ DEMATEL for addressing attribute interdepen-dency more precisely and we determine group attitudinalcharacters as order-inducing vectors based on ideal attitudesof experts rather than by empirical assignment The mainadvantages of our approach are (i) IVDHF UUBLS offers anadequate way to depict decision hesitancy (ii) the designeddistance measure for IVDHF UUBLS avoids informationdistortion of traditional ones which artificially add mis-matching membership or nonmembership degrees (iii) thedeveloped generalized operators include traditional ones asspecial cases so as to provide flexibility in MADM basedon IVDHF UUBLS (iv) the devised synthesized attributeweighting scheme provides an adaptive way to consider twonecessary aspects attributes interdependency-based weightsand argument-dependent weights (v) the TOPSIS-basedmethod can deduce reasonable order-inducing vectors ratherthan empirical ones and holds generality in other linguisticdecision environments Results of case studies and experi-ments have verified the validity of the proposed approach

When multiple decision organizations are involvedgroup decision-making based on IVDHF UUBLS is animportant research direction Besides inducing preferencesfor decision-making problems of high uncertainty interactivemechanisms should also be studied to extract preferences Inaddition empirical studies of the proposed approach can becarried out to various areas such as investment evaluationproject management and vehicle selection in fleet operations

Appendix

A Basic Notions for IVDHFS and UnbalancedLinguistic Term Set

Definition A1 (see [47]) Letting 119883 be a fixed set then anIVDHFS on 119883 is defined as119863 = ⟨119909 ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (A1)

where ℎ(119909) = ⋃[120583119871120583119880]isinℎ(119909)120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and119892(119909) = ⋃[]119871]119880]isin119892(119909)] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two setsof interval values in [0 1] denoting possible membershipand non-membership degrees of element 119909 isin 119883 to theset 119863 respectively with conditions 120583 ] isin [0 1] and0 le (120583119880)+ + (]119880)+ le 1 and for all 119909 isin 119883 (120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880

For convenience normally 119889 = ℎ 119892 is called an interval-valued dual hesitant fuzzy element (IVDHFE) and 119863 is theset of all IVDHFEs

Definition A2 (see [107]) Suppose 119878 = 119904119895 | 119895 = 0 1 119892 minus1 is a finite and totally ordered discrete linguistic term setwhere 119904119895 represent possible values for a linguistic variableand 119892 is an odd cardinality 119878 is uniformly and symmetricallydistributed if the following two conditions are satisfied (1)

Complexity 25

there exists a unique constant 120582 gt 0 such that 119873119878(119904119894) minus119873119878(119904119895) = 119897(119894 minus 119895) for all 119894 119895 = 0 1 119892 minus 1 (2) Let 119878119877 =119904 | 119904 isin 119878 119904 gt 119878119862 and 119878119871 = 119904 | 119904 isin 119878 119904 lt 119878119862 Let C(119878119877) andC(119878119871) be the cardinality of 119878119877 and 119878119871 then C(119878119877) = C(119878119871)If 119878 is uniformly and symmetrically distributed then 119878 iscalled a balanced linguistic term set Otherwise 119878 is calledan unbalanced linguistic term set

The following 2-tuple fuzzy linguistic representationmodel extends traditional linguistic term set to a continuouscase so as to facilitate computing with linguistic variables

Definition A3 (see [59]) Let 119878 = 1199040 1199041 119904119892minus1 be alinguistic term set and 120573 isin [0 119892] Then a 2-tuple fuzzylinguistic variable expresses the equivalent information to 120573is defined as

Δ [0 119892] 997888rarr 119878 times [minus05 05) (A2)

Δ (120573) = (119904119894 120572) with

119904119894 119894 = 119903119900119906119899119889 (120573)120572 = 120573 minus 119894 120572 isin [minus05 05)

(A3)

Δminus1 (119904119894 120572) = 119894 + 120572 = 120573 (A4)

where 119903119900119906119899119889() is the usual rounding operation and120572 is calledsymbolic translation

To obtain 2-tuple fuzzy linguistic representations ofunbalanced linguistic terms the concept of linguistic hierar-chies ie 119871119867 = ⋃119905 119897(119905 119899(119905)) is used 119897(119905 119899(119905)) is a linguistichierarchy with 119905 indicating the level of hierarchy and 119899(119905)denotes the granularity of the linguistic term set of 119905 Herreraet al [59] defined the following transformation functionsbetween labels from different levels in multigranular linguis-tic information contexts without loss of information

Definition A4 (see [59]) In linguistic hierarchies 119871119867 =⋃119905 119897(119905 119899(119905)) whose linguistic term sets are represented by119878119899(119905) = 119904119899(119905)0 119904119899(119905)119899(119905)minus1 the transformation function froma linguistic label in level 119905 to a label in consecutive level 1199051015840 isdefined as 1198791198651199051199051015840 119897(119905 119899(119905)) 997888rarr 119897(1199051015840 119899(1199051015840)) such that

1198791198651199051199051015840 (119904119899(119905)119894 120572119899(119905))= Δ 1199051015840 (Δminus1119905 (119904119899(119905)119894 120572119899(119905)) (119899 (1199051015840) minus 1)119899 (119905) minus 1 ) (A5)

By use of the above transformation function any 2-tuplelinguistic representation can be transformed into a term in119871119867 Detailed transformation procedures are listed as follows

(1) Representation in linguistic hierarchy To transformthe unbalanced terms of 119878 into the corresponding terms inthe 119871119867 transformation function 120595 is employed to associateeach unbalanced linguistic 2-tuple (119904119894 120572) with its linguistic 2-tuple in 119871119867(119878) ie

120595 119878 997888rarr 119871119867(119878) (A6)

so that 120595(119904119894 120572) = (119904119866(119894)119868(119894) 120582) for forall(119904119894 120572) isin 119878(2) Computational phase Firstly transform (119904119866(119894)119868(119894) 120582) intolinguistic 2-tuples denoted as (119904119899(1199051015840)

1198681015840(119894) 1205821015840) in 119878119899(1199051015840) where

(119904119899(1199051015840)1198681015840(119894)

1205821015840) = 119879119865 (119904119866(119894)119868(119894) 120582)= Δ(Δminus1 (119904119866(119894)119868(119894) 120582) sdot (119899 (1199051015840) minus 1)119866 (119894) minus 1 ) (A7)

Then a computational model is used over 119878119899(1199051015840) with aresult denoted as (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)(3) Retranslation process The result (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)is transformed into the unbalanced term in 119878 by using thetransformation function 120595minus1 ie

120595minus1 119871119867 (119878) 997888rarr 119878 (A8)

so that 120595minus1(119904119899(1199051015840)119903 120582119903) = (119904119903119890119904119906119897119905 120582119903119890119904119906119897119905) isin 119878B Details of Theorem 14

Proof Given a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector for119904119889119895 and 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 120582 gt 0 then(1) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLWA and GIVDHFUUBLWG operators respectively(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLOWA and GIVDHFUUBLOWG operators respectively

(2) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to theIVDHF UUBL weighted average (IVDHFUUBLWA)and IVDHF UUBL weighted geometric (IVDHFUUBLWG)operators respectively where

26 Complexity

119868119881119863119867119865119880119880119861119871119882119860120596 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

120596119895119904119889119895= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus

119899prod119895=1

(1 minus 120583119871119895 )120596119895 1 minus 119899prod119895=1

(1 minus 120583119880119895 )120596119895]] [[

119899prod119895=1

(]119871119895)120596119895 119899prod119895=1

(]119880119895 )120596119895]])

(B1)

119868119881119863119867119865119880119880119861119871119882119866120596 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889119895120596119895

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[

119899prod119895=1

(120583119871119895 )120596119895 119899prod119895=1

(120583119880119895 )120596119895]] [[1 minus

119899prod119895=1

(1 minus ]119871119895)120596119895 1 minus 119899prod119895=1

(1 minus ]119880119895 )120596119895]])

(B2)

(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHA andGIVDHFUUBLHG operators reduce to the IVDHF UUBL

orderedweighted average (IVDHFUUBLOWA) and IVDHFUUBL ordered weighted geometric (IVDHFUUBLOWG)operators respectively where

119868119881119863119867119865119880119880119861119871119874119882119860119908 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

119908119895119904119889120590(119895)= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) )))

119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) )))

]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus

119899prod119895=1

(1 minus 120583119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus 120583119880120590(119895))119908119895]] [[

119899prod119895=1

(]119871120590(119895))119908119895 119899prod119895=1

(]119880120590(119895))119908119895]])

(B3)

119868119881119863119867119865119880119880119861119871119874119882119866119908 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889120590(119895)119908119895

= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[

119899prod119895=1

(120583119871120590(119895))119908119895 119899prod119895=1

(120583119880120590(119895))119908119895]] [[1 minus

119899prod119895=1

(1 minus ]119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus ]119880120590(119895))119908119895]])

(B4)

Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

References

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[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

[11] X Qi C Liang and J Zhang ldquoGeneralized cross-entropy basedgroup decision making with unknown expert and attributeweights under interval-valued intuitionistic fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 79 pp 52ndash642015

[12] H Zimmermann Fuzzy Set Theorymdashand Its ApplicationsKluwer Academic Boston Mass USA 2001

[13] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 pp 199ndash249 1975

[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

[16] B T Nuong K-W Kim L Prathumratana et al ldquoSustainabledevelopment in the mining sector and its evaluation usingfuzzy AHP (Analytic Hierarchy Process) approachrdquo GeosystemEngineering vol 14 no 1 pp 43ndash50 2011

[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

[18] E K Zavadskas Z Turskis and J Antucheviciene ldquoSelectinga contractor by using a novel method for multiple attributeanalysis Weighted aggregated sum product assessment withgrey values (WASPAS-G)rdquo Studies in Informatics and Controlvol 24 no 2 pp 141ndash150 2015

[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

[21] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

[25] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009

[26] Z Hatush and M Skitmore ldquoContractor Selection UsingMulticriteria Utility Theory An Additive Modelrdquo Building andEnvironment vol 33 no 2-3 pp 105ndash115 1998

[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

[30] AM Anvuur andMM Kumaraswamy ldquoConceptualmodel ofpartnering and alliancingrdquo Journal of Construction Engineeringand Management vol 133 no 3 pp 225ndash234 2007

[31] A H I Lee H Kang C F Hsu and H Hung ldquoA green supplierselection model for high-tech industryrdquo Expert Systems withApplications vol 36 no 4 pp 7917ndash7927 2009

[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

[33] P Humphreys R McIvor and F Chan ldquoUsing case-basedreasoning to evaluate supplier environmental managementperformancerdquo Expert Systems with Applications vol 25 no 2pp 141ndash153 2003

[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

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[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

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[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

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[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

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[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

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performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

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[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

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[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

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[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

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[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

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18 Complexity

Table 2 Decision matrix in the form of IVDHF UUBLS1199091 1199092 11990931198601([AHVH] [02 03][02 04] [03 04]) ([HT] [04 05][05 06] [02 03] [02 04]) ([ALAH] [01 02][01 03] [06 07])1198602([MH] [05 06][02 03]) ([AHT] [06 07][01 02] ) ([MVH] [02 03][05 06] [06 07])1198603([VLAL] [03 04][04 05] [05 06]) ([MVH] [06 07] [07 08][01 02]) ([VLL] [04 05][02 03])1198604([MAH] [02 04][05 06]) ([LAL] [03 05][02 03]) ([ALM] [06 07][01 02] [02 03])1198605([ALM] [04 05][01 02] [04 05]) ([AHH] [06 07][01 02]) ([HVH] [04 05][06 07] [01 03])1198606([LM] [01 02][03 05] [03 05]) ([VLL] [02 04] [05 06][02 03]) ([AHH] [03 04][04 05])1198607([VLL] [04 05][05 06] [02 03] [02 04]) ([MAH] [05 06] [07 08][01 02]) ([ALM] [04 05][05 06] [03 04])1198608([ALAH] [01 03][04 06]) ([AHH] [05 07][01 02] [02 03]) ([AHH] [04 06][03 04])

Table 3 Direct influence matrix Z1198601 1198602 1198603 1198604 1198605 1198606 1198607 11986081198601 0 3 4 1 3 1 2 41198602 1 0 2 4 4 2 3 11198603 4 2 0 1 3 2 4 11198604 2 3 1 0 3 4 1 21198605 4 2 1 3 0 1 4 31198606 2 4 3 4 4 0 4 21198607 4 1 2 3 3 1 0 41198608 1 2 4 3 3 2 4 0

Step 4 Derive order inducing variable 120576119894119895 by descendingorder of 119888119894119895 The greater the value 119888119894119895 the bigger the 1205761198941198956 Illustrative Example and Experiments

61 Illustrative Case Study Suppose a mega infrastructureproject is selecting green contractor for one of its subprojectsA panel of decision-makers including managers professorsand competitive intelligence experts has been organized tocomprehensively evaluate three alternative contractors (ie1199091 1199092 and 1199093) under the eight attributes (ie 1198601ndash1198608) listedin Section 2

To elicit true decision preferences we apply theIVDHF UUBLS to assess the three alternatives (1199091ndash1199093)Firstly decision-makers are asked to vote on 119909119895 under eachattribute119860 119894 using different unbalanced linguistic term sets 1198781 1198782 and 1198783 where 1198781 = 119873 119871119860119871119872119860119867119867119876119867119881119867119860119879 1198791198782 = 119873119860119873119881119871 119876119871 119871 119860119871119872119860119867119881119867 119879 1198783 = 119873119881119871119871119860119872119872119876119872119860119867119867 119879 In Figure 4 we show the adoptedlinguistic hierarchies and relationship between 1198781 1198782 and1198783 Attributes 1198601 1198602 1198605 and 1198608 are evaluated by use of 1198781attributes11986031198604 and1198607 are evaluated by 1198782 and attribute1198606is evaluated by 1198783 The highest-voted uncertain unbalancedlinguistic term [119904119898 119904119899] are identified Secondly the panel of

decision-makers is asked to further express their opinionson membership degrees ℎ(119909) and nonmembership degrees119892(119909) to [119904119898 119904119899] Then we can derive compound assessments(ie ⟨119909 119904120599(119909) ℎ(119909) 119892(119909)⟩) in the form of IVDHF UUBLSand obtain the decision matrix in Table 2 Additionallyin order to examine the interdependency among attributesdecision-makers are also required to assess the influentialrelations among the eight attributes using DEMATEL-basedAlgorithm II The direct influence matrix is collected andshown in Table 3

Next we apply the proposed Algorithm I to solve thisgreen contractor selection problem To derive the synthe-sized attribute weighting vector in Algorithm I we assign120572 = 120573 = 05 For scenarios with no group attitudinalcharacters operator GIVDHFUUBLHA in Definition 12 ischosen In contrast for scenarios with group attitudinalcharacters operator I-GIVDHFUUBLHA in Definition 13will be chosen Further the panel of decision-makers is askedto determine their positive and negative ideal alternativesthat are collected in Table 4 where corresponding order-inducing vectors that are derived from Algorithm III arealso listed (see the rightmost column) Thereafter operatorI-GIVDHFUUBLHA in Definition 13 is utilized for aggrega-tion For GIVDHFUUBLHA and I-GIVDHFUUBLHA we

Complexity 19

Table 4 Ideal alternatives (119903+ and 119903minus) and derived order-inducing vectors 120576119903+ 119903minus 1205761199091 1199092 11990931198601 ([VHT] 1 0) ([NL] 1 0) 5 6 61198602 ([MAH] [07 08] [01 02]) ([NL] 1 0) 1 1 41198603 (T 1 0) (L 1 0) 8 7 81198604 ([ALAH] 09 01) (T 1 0) 2 2 11198605 ([HT] [09 1] 0) (N 1 0) 7 5 51198606 (M 1 0) (N 1 0) 6 3 31198607 (M 1 0) ([NL] 1 0) 3 8 71198608 ([MAH] 1 0) (N 1 0) 4 4 2

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 1 2 3 4

0 1 2

N L AH H QH VH AT TM

Level 1 l(13)

Level 2 l(25)

Level 3 l(39)

Level 4 l(417)

Unbalanced linguistic term set S1

N AH VH TM Unbalanced linguistic term set S2VL L ALAN QL

N H TM Unbalanced linguistic term set S3

L AM AHQM

AL

VL

Figure 4 Unbalanced linguistic term sets (1198781 1198782 and 1198783) and their mapping in linguistic hierarchies

assume 120582=2 and 119908 = (0 0 015 025 025 025 01 0)119879Note that 119908 is a position weighting vector which is derivedby the fuzzy semantic quantitative operator [99] Aftercompleting all steps in Algorithm I (see Appendix C) weobtain the ranking results and present them in Table 5 Forthe scores derived from Step 5 we find the ranking order is1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the best solution Step 7gives the ranking order of 1199092 ≻ 1199091 ≻ 1199093 indicating 1199092 is stillthe most appropriate one while 1199091 is better than 1199093 under theinfluence of 12057662 Experiments on Ranking Results along with VariousParameter Configurations In order to inspect the impactsof various parameter configurations on ranking results inthis section more numerical examples are carried out Alleight operators that defined in Section 41 are applied to theabove illustrative case where the parameter 120582 is configured asintegers ranging from 1 to 20 and parameter 120572 is configuredto a value set of 0 03 05 08 09 1

Firstly the four basic aggregation operators of GIVDH-FUUBLWA GIVDHFUUBLWG GIVDHFUUBLOWA and

Table 5 Ranking results of operators GIVDHFUUBLHA and I-GIVDHFUUBLHA

Aggregation operator Ranking orderGIVDHFUUBLHA 1199092 ≻ 1199093 ≻ 1199091I-GIVDHFUUBLHA 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG are applied to the former case Thesame configuration 120572 = 120573 = 05 is chosen while 120582 is testedas integers ranging from 1 to 20 The ranking results derivedfrom different parameter combinations have been collectedin Table 6 As can be seen four operators unanimouslyidentify the contractor 1199092 is the best one no matter how120582 changes Regarding priority relation between 1199091 and 1199093GIVDHFUUBLWA and GIVDHFUUBLOWA generate thesame result of1199093 ≻ 1199091 while results fromGIVDHFUUBLWGandGIVDHFUUBLOWGgot a change to 1199091 ≻ 1199093 at a certainvalue of parameter 120582

Actually GIVDHFUUBLWA and GIVDHFUUBLOWAare generalized version of weighted arithmetic aggregationoperators that producemore favorable results by emphasizing

20 Complexity

Table 6 Ranking orders obtained by GIVDHFUUBLWA GIVDHFUUBLOWA GIVDHFUUBLWG and GIVDHFUUBLOWG along with120582Operators 120582 (Integer) Ranking orderGIVDHFUUBLWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLOWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLWG [1 2] 1199092 ≻ 1199093 ≻ 1199091[3 20] 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG [1 8] 1199092 ≻ 1199093 ≻ 1199091[9 20] 1199092 ≻ 1199091 ≻ 1199093

Table 7 Ranking results obtained by GIVDHFUUBLHA and I-GIVDHFUUBLHA along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHA I-GIVDHFUUBLHA120572=0 120573=1 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=03 120573=07 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=05 120573=05 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=08 120573=02 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=09 120573=01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=1 120573=0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093

Table 8 Ranking results obtained by GIVDHFUUBLHG and I-GIVDHFUUBLHG along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHG I-GIVDHFUUBLHG

120572 = 0 120573 = 1 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 03 120573 = 07 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 05 120573 = 05 [1 8] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[9 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 08 120573 = 02 [1 17] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[18 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 09 120573 = 01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572 = 1 120573 = 0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093membership degrees to a uncertain linguistic term thus canbe considered as optimistic operators while GIVDHFU-UBLWG and GIVDHFUUBLOWG are generalized geomet-ric aggregation operators that producemore favorable resultsby emphasizing nonmembership degrees thus can be consid-ered as pessimistic operators [100 101]Therefore the rankingresults in Table 6 could be perceived as for optimism-baseddecision-making alternative 1199093 always stands out better than1199091 when 120582 gradually amplifies degree of optimism but forpessimism-based decision-making 1199091 turns out better than1199093 when 120582 gradually amplifies degree of pessimism 120582 canbe considered as the parameter for amplifying degree ofoptimism or pessimism [101]

Next to inspect the impacts of various combinationsof 120596119860119868 and 120596119860119863 on ranking results another four aggrega-tion operators of GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG also have beenapplied to the above illustrative case with varying 120572 120573 and 120582

All results have been collected in Tables 7 and 8 for compari-son As can be seen fromTable 7 ranking results generated byoperator GIVDHFUUBLHA keep the same as 1199092 ≻ 1199093 ≻ 1199091for all test combinations of parameters However as shownin Table 8 although GIVDHFUUBLHG is also capable ofidentifying 1199092 as the best contractor the priority relationsbetween 1199091 and 1199093 hold similar changing patterns in rankingresults in Table 6 that were output byGIVDHFUUBLWGandGIVDHFUUBLOWG Regarding decision situations whereobjective attributeweighting vector120596119860119863 is only considered (120572= 0) or its dominant importance (such as 120572 = 03) is accepted1199091 ≻ 1199093 is a clear observation when parameter 120582 is set toan integer in interval [8 20] Contrariwise under decisionscenarios where subjective attribute weighting vector 120596119860119868 isonly needed (120572 = 1) or its dominant importance (such as 120572 =09) is acknowledged 1199093 ≻ 1199091 holds all the way 120582 varies

In order to more clearly show the score trajectories ofalternative contractors along variations of parameter 120582 we

Complexity 21

HAInducedHA

InducedHA-2HA-2

HA-3

InducedHA-1InducedHA-3

= =

HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(a) 120572 = 0

HAInducedHA

= 03 = 07

InducedHA-2

HA-2HA-3

InducedHA-1 InducedHA-3HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(b) 120572 = 03

HAInducedHA

= 5 = 05

InducedHA-2HA-2

HA-3

InducedHA-1 InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(c) 120572 = 05

HAInducedHA

= 08 = 02

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572 = 08

HAInducedHA

= 9 = 01

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(e) 120572 = 09

HAInducedHA

246 2462 2464 2466 2468 247

= 1 = 0InducedHA-2HA-2

HA-3

HA- 3

HA-3

InducedHA-1

InducedHA-3

HA-1

HA-1

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

1085

1086

1087

(f) 120572 = 1

Figure 5 Score trajectories obtained by aggregation operators HA and InducedHA with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

further configure parameter 120582 to the value set of [0 01100]and conduct computational experiments to collect scorevalues and ranking results (as shown in Figures 5 and 6) byfour aggregation operators of GIVDHFUUBLHA GIVDH-FUUBLHG I-GIVDHFUUBLHAand I-GIVDHFUUBLHGFor more clarity in Figures 5 and 6 please note that weuse HA HG Induced-HA and Induced HG to respec-tively denote GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG In Figure 5it can be observed that when dominant importance of 120596119860119863

is emphasized (120572 = 0 120572 = 03 120572 = 05) GIVDHFUUBLHAis capable of differentiating the three alternative contractorsas same ranking order of 1199092 ≻ 1199093 ≻ 1199091 for all 120582 but withincreases in 120572 the margins between scores of 1199091 and 1199093 nar-rows and the rank order changes to 1199091 ≻ 1199093when 120572 gets 09and 1 Interestingly in Figure 6 although GIVDHFUUBLHGis able to significantly differentiate alternative contractors 1199091and 1199093 when dominant importance of 120596119860119863 is emphasized (120572= 0 120572 = 03 120572 = 05) rank order of 1199091 and 1199093 obviouslychanges once from 1199093 ≻ 1199091 to 1199091 ≻ 1199093 when 120582 is bigger

22 Complexity

HGInducedHG

= = 1

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(a) 120572= 0

HGInducedHG

= 3 = 07

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(b) 120572= 03

HGInducedHG

= 05 = 05

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(c) 120572= 05

HGInducedHG

= 8 = 02

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572= 08

HGInducedHG

292 293 294 295 296 297minus08

minus0795

minus079

= 09 = 01

HA-1

HA-3

InducedHG-2

HG-1

HG-1

HG-2

HG-3

HG-3 InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4

minus3

minus2

minus1

0

1

2

scor

e val

ues

(e) 120572= 09

HGInducedHG

50 60 70 80 90 100minus095minus09

minus085

= 1 = 0

HG-x3

HG-x1

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(f) 120572= 1

Figure 6 Score trajectories obtained by aggregation operators HG and InducedHG with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

than 8 When 120572 increases to 1 GIVDHFUUBLHGmaintainsthe rank order of 1199093 ≻ 1199091 but their score trajectorieswalk very close Comparatively I-GIVDHFUUBLHA andI-GIVDHFUUBLHG generate consistent ranking results of1199092 ≻ 1199091 ≻ 1199093 no matter 120582 varies in all the above experiments

In sum when parameters are configured with typical val-ues (ie 120572 = 05 and 120582=1) and no additional attitudinal infor-mation is taken into consideration the proposed aggregationoperators unanimously derive the same rank order 1199092 ≻1199093 ≻ 1199091 of for the three alternative contractors However

with the amplification effect of parameter 120582 analysis resultsfrom linear combination of 120596119860119868 and 120596119860119863 reveal alternative1199092 which consistently is the best choice but rank order of1199091 and 1199093 changes because their score trajectories get closeand crossed which probably indicates that only decision-making informationderived from the decisionmatrix is inad-equate to significantly differentiate alternatives 1199091 and 1199093While by integrating additional decision information aboutattitudinal characters of group experts I-GIVDHFUUBLHAand I-GIVDHFUUBLHG manage to generate the identical

Complexity 23

Table 9 Order-inducing vectors obtained by our TOPSIS-based method for problem in [38]

1198661 1198662 1198663 11986641198601 02522 01629 02282 027141198602 01 05243 01364 046891198603 04081 01756 05244 037791198604 04507 01409 01064 013221198605 00697 02203 03753 01161

Table 10 Order-inducing vectors obtained by our TOPSIS-based method for problem in [39]

1198661 1198662 1198663 11986641198601 04548 04969 04923 047321198602 05092 047 04768 046451198603 07192 04888 04953 049151198604 06722 06163 05706 049291198605 06104 04942 06463 04946

Table 11 Ranking results obtained by applying different order-inducing vectors to [38 39]

Problems Approaches Ranking results

Problem in [38] IHFLOWA [38] + order-inducing vectors in [38] 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198605 ≻ 1198604

IHFLOWA [38] + order-inducing vectors in Table 9 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198604 ≻ 1198605

Problem in [39] I-IVDHFLOWA [39] + order-inducing vectors in [39] 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198602 ≻ 1198601

I-IVDHFLOWA [39] + order-inducing vectors in Table 10 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602

permutation of 1199092 ≻ 1199091 ≻ 1199093 which verifies that ourproposed TOPSIS-based order-inducing approach is prac-tical and useful in helping work out decision results morereasonably and robustly

63 Application Studies of Proposed TOPSIS-Based Order-Inducing Method When confronting complex decision-making situations that need additional attitudinal charactersof decision-makers Yager and Filev [88] and Yager [102]introduced and encouraged to employ order-inducing vectorto represent those attitudinal characters Since then theidea of order-inducing vector has been extended to variousdecision-making environments [98 103] such as those basedon intuitionistic fuzzy information [104 105] linguistic infor-mation [106] hesitant fuzzy linguistic information [38] andinterval-valued dual hesitant fuzzy uncertain linguistic infor-mation [39] Generally in existing literature order-inducingvectors rely on experts to directly give [38 39 104ndash106]

In order to examine the adaptability of our TOPSIS-based order-inducing method to linguistic decision-makingenvironments we respectively apply the method to resolvethe same problems in references [38 39] Due to fact thatthere is no descriptions about how the order-inducing vectorswere derived in both references [38 39] we here applyextremums into our TOPSIS-based method to obtain order-inducing vectors that is (1199047 1 0) and (1199041 0 1) for[38] ([1199046 1199046] 1 0) and ([1199040 1199040] 0 1) for [39] Thenwe obtain our order-inducing vectors listed in Table 9 forthe problem adopted in [38] and our order-inducing vectorslisted in Table 10 for [39]

Now we apply the order-inducing vectors as shownin Tables 9 and 10 to resolve the problems in [38] and[39] respectively Regarding the problem in [38] the scorevalues of all alternatives are calculated as 119878(1198601)=02602119878(1198602)=02417 119878(1198603)=03577 119878(1198604)=0221 and 119878(1198605)=0185so the corresponding ranking result is 1198603 ≻ 1198601 ≻ 1198602 ≻1198604 ≻ 1198605 As for the problem in [39] the score values ofall alternatives are computed as 119878(1198601)=39954 119878(1198602)=3783119878(1198603)=70827 119878(1198604)=71792 and 119878(1198605)=5813 we thus obtainthe ranking result 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602 Formore clarity brief descriptions of comparative approachesand obtained ranking results also have been put together inTable 11

According to Table 11 the approaches that employ ourorder-inducing vectors derive the same ranking order ofthe better three alternatives for both comparative problemsfrom [38 39] As seen our proposed TOPSIS-based order-inducing method exhibits effective and more understand-able Furthermore when decision-makers are incapable ofdirectly put forward all desired order-inducing vectors due tocomplexity in ill-structured problems our method providesa feasible directions to obtain order-inducing vectors byfiguring out ideal solutions Comparatively our proposedTOPSIS-based order-inducing method behaves effective andmore operational in practical use

64 Comparison with Decision-Making Approach Based onDegraded form of IVDHFULS Note that our approachis intrinsically adaptive to decision situations based ondegraded forms of IVDHF UUBLS such as the dual hesitant

24 Complexity

fuzzy linguistic sets introduced by Yang and Ju [54] Tofurther inspect feasibility of our approach we here conductcomparative study on the same case adopted by Yang and Ju[54] We firstly convert their original decision matrix intothe form of IVDHF UUBLS crisp membership and non-membership degrees are equivalently changed to intervalswith same upper and lower limits (eg 02 turns out to be[02 02]) and linguistic variables are rewritten as uncertainlinguistic variables (eg 1199042 turns out to be [11990421199042])

Since the case only got three attributes among whichinterdependency could be not significant under generalassumption weighting vector 120596119860119868 is not calculated Andfor straight comparison order inducing vector is notincluded either Then the argument-dependent attributeweighting vector can be obtained from Eq(29) as 120596119860119863 =(03192 03132 03676)119879 By use of Algorithm Iwith operatorIVDHFUUBLWA (ie GIVDHFUUBLWA in case of 120582 = 1see Appendix B) we have the scores of alternatives 119904(1198601) =03412 119904(1198602) = 01354 119904(1198603) = 03932 119904(1198604) = 07352Therefore the ordering of alternatives is1198604 ≻ 1198603 ≻ 1198601 ≻ 1198602which is mainly in alignment with the ordering of1198604 ≻ 1198601 ≻1198603 ≻ 1198602 in Yang and Ju [54] It is worth noticing that 120596119860119863used in our approach is derived objectively from assessmentswhile attribute weights 120596lowast = (035 025 040)119879used by Yangand Ju [54] were assigned empirically Differences between120596119860119863 and 120596lowast lead to different ordering of1198601 and1198603 Compar-atively speaking our approach holds more adaptability andobjectivity

In general our proposed MADM approach in this paperexhibits an effective way to tackle complex green contractorselection problems in which subjective attributes weightingvector (such as120596119860119868) and objective attributes weighting vector(such as 120596119860119863) exist To comprehensively utilize these twofundamental facets of attributes weighting information ourcase study and computational experiments suggest that linearcombination method (ie 120596 = 120572120596119860119868 + 120573120596119860119863) is effectiveto reflect effects of 120596119860119868 and 120596119860119863 by configuring parameter 120572or 120573 When there is no specific expert opinions to specify orhard to obtain we suggest that parameter 120572 chooses the valueset of 0 02 05 08 1 Furthermore generalized aggrega-tion operators are suggested to inspect score trajectoriesof alternative contractors along increasing of the operatorsrsquoamplification parameter 120582 If some alternative contractorsrsquoscore values get very close or crossed so that they cannot bedifferentiated from each other significantly we suggest to askthe panel of experts to derive opinions of ideal solutions andutilize our TOPSIS-based order-inducing method to includethose expert opinions as attitudinal decision informationthereby obtaining more rational and robust decision results

7 Conclusions

Aiming at tackling complex decision-making problems ofgreen contractor selection that often hold simultaneouslythe characteristics of decision hesitancy attributes inter-dependency and group complex attitudinal characters wehave developed an effective IVDHF UUBLS-based approach

We utilize IVDHF UUBLS to elicit complicate decision-makersrsquo assessments more objectively and comprehensivelywe employ DEMATEL for addressing attribute interdepen-dency more precisely and we determine group attitudinalcharacters as order-inducing vectors based on ideal attitudesof experts rather than by empirical assignment The mainadvantages of our approach are (i) IVDHF UUBLS offers anadequate way to depict decision hesitancy (ii) the designeddistance measure for IVDHF UUBLS avoids informationdistortion of traditional ones which artificially add mis-matching membership or nonmembership degrees (iii) thedeveloped generalized operators include traditional ones asspecial cases so as to provide flexibility in MADM basedon IVDHF UUBLS (iv) the devised synthesized attributeweighting scheme provides an adaptive way to consider twonecessary aspects attributes interdependency-based weightsand argument-dependent weights (v) the TOPSIS-basedmethod can deduce reasonable order-inducing vectors ratherthan empirical ones and holds generality in other linguisticdecision environments Results of case studies and experi-ments have verified the validity of the proposed approach

When multiple decision organizations are involvedgroup decision-making based on IVDHF UUBLS is animportant research direction Besides inducing preferencesfor decision-making problems of high uncertainty interactivemechanisms should also be studied to extract preferences Inaddition empirical studies of the proposed approach can becarried out to various areas such as investment evaluationproject management and vehicle selection in fleet operations

Appendix

A Basic Notions for IVDHFS and UnbalancedLinguistic Term Set

Definition A1 (see [47]) Letting 119883 be a fixed set then anIVDHFS on 119883 is defined as119863 = ⟨119909 ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (A1)

where ℎ(119909) = ⋃[120583119871120583119880]isinℎ(119909)120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and119892(119909) = ⋃[]119871]119880]isin119892(119909)] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two setsof interval values in [0 1] denoting possible membershipand non-membership degrees of element 119909 isin 119883 to theset 119863 respectively with conditions 120583 ] isin [0 1] and0 le (120583119880)+ + (]119880)+ le 1 and for all 119909 isin 119883 (120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880

For convenience normally 119889 = ℎ 119892 is called an interval-valued dual hesitant fuzzy element (IVDHFE) and 119863 is theset of all IVDHFEs

Definition A2 (see [107]) Suppose 119878 = 119904119895 | 119895 = 0 1 119892 minus1 is a finite and totally ordered discrete linguistic term setwhere 119904119895 represent possible values for a linguistic variableand 119892 is an odd cardinality 119878 is uniformly and symmetricallydistributed if the following two conditions are satisfied (1)

Complexity 25

there exists a unique constant 120582 gt 0 such that 119873119878(119904119894) minus119873119878(119904119895) = 119897(119894 minus 119895) for all 119894 119895 = 0 1 119892 minus 1 (2) Let 119878119877 =119904 | 119904 isin 119878 119904 gt 119878119862 and 119878119871 = 119904 | 119904 isin 119878 119904 lt 119878119862 Let C(119878119877) andC(119878119871) be the cardinality of 119878119877 and 119878119871 then C(119878119877) = C(119878119871)If 119878 is uniformly and symmetrically distributed then 119878 iscalled a balanced linguistic term set Otherwise 119878 is calledan unbalanced linguistic term set

The following 2-tuple fuzzy linguistic representationmodel extends traditional linguistic term set to a continuouscase so as to facilitate computing with linguistic variables

Definition A3 (see [59]) Let 119878 = 1199040 1199041 119904119892minus1 be alinguistic term set and 120573 isin [0 119892] Then a 2-tuple fuzzylinguistic variable expresses the equivalent information to 120573is defined as

Δ [0 119892] 997888rarr 119878 times [minus05 05) (A2)

Δ (120573) = (119904119894 120572) with

119904119894 119894 = 119903119900119906119899119889 (120573)120572 = 120573 minus 119894 120572 isin [minus05 05)

(A3)

Δminus1 (119904119894 120572) = 119894 + 120572 = 120573 (A4)

where 119903119900119906119899119889() is the usual rounding operation and120572 is calledsymbolic translation

To obtain 2-tuple fuzzy linguistic representations ofunbalanced linguistic terms the concept of linguistic hierar-chies ie 119871119867 = ⋃119905 119897(119905 119899(119905)) is used 119897(119905 119899(119905)) is a linguistichierarchy with 119905 indicating the level of hierarchy and 119899(119905)denotes the granularity of the linguistic term set of 119905 Herreraet al [59] defined the following transformation functionsbetween labels from different levels in multigranular linguis-tic information contexts without loss of information

Definition A4 (see [59]) In linguistic hierarchies 119871119867 =⋃119905 119897(119905 119899(119905)) whose linguistic term sets are represented by119878119899(119905) = 119904119899(119905)0 119904119899(119905)119899(119905)minus1 the transformation function froma linguistic label in level 119905 to a label in consecutive level 1199051015840 isdefined as 1198791198651199051199051015840 119897(119905 119899(119905)) 997888rarr 119897(1199051015840 119899(1199051015840)) such that

1198791198651199051199051015840 (119904119899(119905)119894 120572119899(119905))= Δ 1199051015840 (Δminus1119905 (119904119899(119905)119894 120572119899(119905)) (119899 (1199051015840) minus 1)119899 (119905) minus 1 ) (A5)

By use of the above transformation function any 2-tuplelinguistic representation can be transformed into a term in119871119867 Detailed transformation procedures are listed as follows

(1) Representation in linguistic hierarchy To transformthe unbalanced terms of 119878 into the corresponding terms inthe 119871119867 transformation function 120595 is employed to associateeach unbalanced linguistic 2-tuple (119904119894 120572) with its linguistic 2-tuple in 119871119867(119878) ie

120595 119878 997888rarr 119871119867(119878) (A6)

so that 120595(119904119894 120572) = (119904119866(119894)119868(119894) 120582) for forall(119904119894 120572) isin 119878(2) Computational phase Firstly transform (119904119866(119894)119868(119894) 120582) intolinguistic 2-tuples denoted as (119904119899(1199051015840)

1198681015840(119894) 1205821015840) in 119878119899(1199051015840) where

(119904119899(1199051015840)1198681015840(119894)

1205821015840) = 119879119865 (119904119866(119894)119868(119894) 120582)= Δ(Δminus1 (119904119866(119894)119868(119894) 120582) sdot (119899 (1199051015840) minus 1)119866 (119894) minus 1 ) (A7)

Then a computational model is used over 119878119899(1199051015840) with aresult denoted as (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)(3) Retranslation process The result (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)is transformed into the unbalanced term in 119878 by using thetransformation function 120595minus1 ie

120595minus1 119871119867 (119878) 997888rarr 119878 (A8)

so that 120595minus1(119904119899(1199051015840)119903 120582119903) = (119904119903119890119904119906119897119905 120582119903119890119904119906119897119905) isin 119878B Details of Theorem 14

Proof Given a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector for119904119889119895 and 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 120582 gt 0 then(1) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLWA and GIVDHFUUBLWG operators respectively(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLOWA and GIVDHFUUBLOWG operators respectively

(2) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to theIVDHF UUBL weighted average (IVDHFUUBLWA)and IVDHF UUBL weighted geometric (IVDHFUUBLWG)operators respectively where

26 Complexity

119868119881119863119867119865119880119880119861119871119882119860120596 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

120596119895119904119889119895= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus

119899prod119895=1

(1 minus 120583119871119895 )120596119895 1 minus 119899prod119895=1

(1 minus 120583119880119895 )120596119895]] [[

119899prod119895=1

(]119871119895)120596119895 119899prod119895=1

(]119880119895 )120596119895]])

(B1)

119868119881119863119867119865119880119880119861119871119882119866120596 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889119895120596119895

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[

119899prod119895=1

(120583119871119895 )120596119895 119899prod119895=1

(120583119880119895 )120596119895]] [[1 minus

119899prod119895=1

(1 minus ]119871119895)120596119895 1 minus 119899prod119895=1

(1 minus ]119880119895 )120596119895]])

(B2)

(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHA andGIVDHFUUBLHG operators reduce to the IVDHF UUBL

orderedweighted average (IVDHFUUBLOWA) and IVDHFUUBL ordered weighted geometric (IVDHFUUBLOWG)operators respectively where

119868119881119863119867119865119880119880119861119871119874119882119860119908 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

119908119895119904119889120590(119895)= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) )))

119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) )))

]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus

119899prod119895=1

(1 minus 120583119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus 120583119880120590(119895))119908119895]] [[

119899prod119895=1

(]119871120590(119895))119908119895 119899prod119895=1

(]119880120590(119895))119908119895]])

(B3)

119868119881119863119867119865119880119880119861119871119874119882119866119908 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889120590(119895)119908119895

= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[

119899prod119895=1

(120583119871120590(119895))119908119895 119899prod119895=1

(120583119880120590(119895))119908119895]] [[1 minus

119899prod119895=1

(1 minus ]119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus ]119880120590(119895))119908119895]])

(B4)

Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

References

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[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

[11] X Qi C Liang and J Zhang ldquoGeneralized cross-entropy basedgroup decision making with unknown expert and attributeweights under interval-valued intuitionistic fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 79 pp 52ndash642015

[12] H Zimmermann Fuzzy Set Theorymdashand Its ApplicationsKluwer Academic Boston Mass USA 2001

[13] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 pp 199ndash249 1975

[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

[16] B T Nuong K-W Kim L Prathumratana et al ldquoSustainabledevelopment in the mining sector and its evaluation usingfuzzy AHP (Analytic Hierarchy Process) approachrdquo GeosystemEngineering vol 14 no 1 pp 43ndash50 2011

[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

[18] E K Zavadskas Z Turskis and J Antucheviciene ldquoSelectinga contractor by using a novel method for multiple attributeanalysis Weighted aggregated sum product assessment withgrey values (WASPAS-G)rdquo Studies in Informatics and Controlvol 24 no 2 pp 141ndash150 2015

[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

[21] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

[25] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009

[26] Z Hatush and M Skitmore ldquoContractor Selection UsingMulticriteria Utility Theory An Additive Modelrdquo Building andEnvironment vol 33 no 2-3 pp 105ndash115 1998

[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

[30] AM Anvuur andMM Kumaraswamy ldquoConceptualmodel ofpartnering and alliancingrdquo Journal of Construction Engineeringand Management vol 133 no 3 pp 225ndash234 2007

[31] A H I Lee H Kang C F Hsu and H Hung ldquoA green supplierselection model for high-tech industryrdquo Expert Systems withApplications vol 36 no 4 pp 7917ndash7927 2009

[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

[33] P Humphreys R McIvor and F Chan ldquoUsing case-basedreasoning to evaluate supplier environmental managementperformancerdquo Expert Systems with Applications vol 25 no 2pp 141ndash153 2003

[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

[45] Bin Zhu Zeshui Xu and Meimei Xia ldquoDual Hesitant FuzzySetsrdquo Journal of Applied Mathematics vol 2012 Article ID879629 13 pages 2012

[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

[51] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[52] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

[55] Z Tao X Liu H Chen and L Zhou ldquoUsing new versionof extended t-norms and s-norms for aggregating intervallinguistic labelsrdquo IEEETransactions on SystemsManCyberneticsSystems vol vol p 15 2016

[56] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[57] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[58] L Martınez M Espinilla J Liu L G Perez and P J SanchezldquoAn evaluation model with unbalanced linguistic informationapplied to olive oil sensory evaluationrdquo Journal of Multiple-Valued Logic and Soft Computing vol 15 no 2-3 pp 229ndash2512009

[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

[62] L Marin A Valls D Isern A Moreno and J M MerigoldquoInduced unbalanced linguistic ordered weighted average andits application in multiperson decision makingrdquo The ScientificWorld Journal vol 2014 2014

[63] A Baykasoglu V Kaplanoglu Z D U Durmusoglu and CSahin ldquoIntegrating fuzzy DEMATEL and fuzzy hierarchicalTOPSIS methods for truck selectionrdquo Expert Systems withApplications vol 40 no 3 pp 899ndash907 2013

[64] S K Patil and R Kant ldquoA hybrid approach based on fuzzyDEMATEL and FMCDM to predict success of knowledge man-agement adoption in supply chainrdquoApplied Soft Computing vol18 pp 126ndash135 2014

[65] S Liyin and Y Hong ldquoImproving environmental performanceby means of empowerment of contractorsrdquo Management ofEnvironmental Quality An International Journal vol 17 no 3pp 242ndash257 2006

[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

[69] Y Kazancoglu I Kazancoglu and M Sagnak ldquoA new holisticconceptual framework for green supply chain management

performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

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Complexity 19

Table 4 Ideal alternatives (119903+ and 119903minus) and derived order-inducing vectors 120576119903+ 119903minus 1205761199091 1199092 11990931198601 ([VHT] 1 0) ([NL] 1 0) 5 6 61198602 ([MAH] [07 08] [01 02]) ([NL] 1 0) 1 1 41198603 (T 1 0) (L 1 0) 8 7 81198604 ([ALAH] 09 01) (T 1 0) 2 2 11198605 ([HT] [09 1] 0) (N 1 0) 7 5 51198606 (M 1 0) (N 1 0) 6 3 31198607 (M 1 0) ([NL] 1 0) 3 8 71198608 ([MAH] 1 0) (N 1 0) 4 4 2

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 1 2 3 4

0 1 2

N L AH H QH VH AT TM

Level 1 l(13)

Level 2 l(25)

Level 3 l(39)

Level 4 l(417)

Unbalanced linguistic term set S1

N AH VH TM Unbalanced linguistic term set S2VL L ALAN QL

N H TM Unbalanced linguistic term set S3

L AM AHQM

AL

VL

Figure 4 Unbalanced linguistic term sets (1198781 1198782 and 1198783) and their mapping in linguistic hierarchies

assume 120582=2 and 119908 = (0 0 015 025 025 025 01 0)119879Note that 119908 is a position weighting vector which is derivedby the fuzzy semantic quantitative operator [99] Aftercompleting all steps in Algorithm I (see Appendix C) weobtain the ranking results and present them in Table 5 Forthe scores derived from Step 5 we find the ranking order is1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the best solution Step 7gives the ranking order of 1199092 ≻ 1199091 ≻ 1199093 indicating 1199092 is stillthe most appropriate one while 1199091 is better than 1199093 under theinfluence of 12057662 Experiments on Ranking Results along with VariousParameter Configurations In order to inspect the impactsof various parameter configurations on ranking results inthis section more numerical examples are carried out Alleight operators that defined in Section 41 are applied to theabove illustrative case where the parameter 120582 is configured asintegers ranging from 1 to 20 and parameter 120572 is configuredto a value set of 0 03 05 08 09 1

Firstly the four basic aggregation operators of GIVDH-FUUBLWA GIVDHFUUBLWG GIVDHFUUBLOWA and

Table 5 Ranking results of operators GIVDHFUUBLHA and I-GIVDHFUUBLHA

Aggregation operator Ranking orderGIVDHFUUBLHA 1199092 ≻ 1199093 ≻ 1199091I-GIVDHFUUBLHA 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG are applied to the former case Thesame configuration 120572 = 120573 = 05 is chosen while 120582 is testedas integers ranging from 1 to 20 The ranking results derivedfrom different parameter combinations have been collectedin Table 6 As can be seen four operators unanimouslyidentify the contractor 1199092 is the best one no matter how120582 changes Regarding priority relation between 1199091 and 1199093GIVDHFUUBLWA and GIVDHFUUBLOWA generate thesame result of1199093 ≻ 1199091 while results fromGIVDHFUUBLWGandGIVDHFUUBLOWGgot a change to 1199091 ≻ 1199093 at a certainvalue of parameter 120582

Actually GIVDHFUUBLWA and GIVDHFUUBLOWAare generalized version of weighted arithmetic aggregationoperators that producemore favorable results by emphasizing

20 Complexity

Table 6 Ranking orders obtained by GIVDHFUUBLWA GIVDHFUUBLOWA GIVDHFUUBLWG and GIVDHFUUBLOWG along with120582Operators 120582 (Integer) Ranking orderGIVDHFUUBLWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLOWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLWG [1 2] 1199092 ≻ 1199093 ≻ 1199091[3 20] 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG [1 8] 1199092 ≻ 1199093 ≻ 1199091[9 20] 1199092 ≻ 1199091 ≻ 1199093

Table 7 Ranking results obtained by GIVDHFUUBLHA and I-GIVDHFUUBLHA along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHA I-GIVDHFUUBLHA120572=0 120573=1 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=03 120573=07 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=05 120573=05 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=08 120573=02 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=09 120573=01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=1 120573=0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093

Table 8 Ranking results obtained by GIVDHFUUBLHG and I-GIVDHFUUBLHG along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHG I-GIVDHFUUBLHG

120572 = 0 120573 = 1 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 03 120573 = 07 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 05 120573 = 05 [1 8] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[9 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 08 120573 = 02 [1 17] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[18 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 09 120573 = 01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572 = 1 120573 = 0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093membership degrees to a uncertain linguistic term thus canbe considered as optimistic operators while GIVDHFU-UBLWG and GIVDHFUUBLOWG are generalized geomet-ric aggregation operators that producemore favorable resultsby emphasizing nonmembership degrees thus can be consid-ered as pessimistic operators [100 101]Therefore the rankingresults in Table 6 could be perceived as for optimism-baseddecision-making alternative 1199093 always stands out better than1199091 when 120582 gradually amplifies degree of optimism but forpessimism-based decision-making 1199091 turns out better than1199093 when 120582 gradually amplifies degree of pessimism 120582 canbe considered as the parameter for amplifying degree ofoptimism or pessimism [101]

Next to inspect the impacts of various combinationsof 120596119860119868 and 120596119860119863 on ranking results another four aggrega-tion operators of GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG also have beenapplied to the above illustrative case with varying 120572 120573 and 120582

All results have been collected in Tables 7 and 8 for compari-son As can be seen fromTable 7 ranking results generated byoperator GIVDHFUUBLHA keep the same as 1199092 ≻ 1199093 ≻ 1199091for all test combinations of parameters However as shownin Table 8 although GIVDHFUUBLHG is also capable ofidentifying 1199092 as the best contractor the priority relationsbetween 1199091 and 1199093 hold similar changing patterns in rankingresults in Table 6 that were output byGIVDHFUUBLWGandGIVDHFUUBLOWG Regarding decision situations whereobjective attributeweighting vector120596119860119863 is only considered (120572= 0) or its dominant importance (such as 120572 = 03) is accepted1199091 ≻ 1199093 is a clear observation when parameter 120582 is set toan integer in interval [8 20] Contrariwise under decisionscenarios where subjective attribute weighting vector 120596119860119868 isonly needed (120572 = 1) or its dominant importance (such as 120572 =09) is acknowledged 1199093 ≻ 1199091 holds all the way 120582 varies

In order to more clearly show the score trajectories ofalternative contractors along variations of parameter 120582 we

Complexity 21

HAInducedHA

InducedHA-2HA-2

HA-3

InducedHA-1InducedHA-3

= =

HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(a) 120572 = 0

HAInducedHA

= 03 = 07

InducedHA-2

HA-2HA-3

InducedHA-1 InducedHA-3HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(b) 120572 = 03

HAInducedHA

= 5 = 05

InducedHA-2HA-2

HA-3

InducedHA-1 InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(c) 120572 = 05

HAInducedHA

= 08 = 02

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572 = 08

HAInducedHA

= 9 = 01

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(e) 120572 = 09

HAInducedHA

246 2462 2464 2466 2468 247

= 1 = 0InducedHA-2HA-2

HA-3

HA- 3

HA-3

InducedHA-1

InducedHA-3

HA-1

HA-1

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

1085

1086

1087

(f) 120572 = 1

Figure 5 Score trajectories obtained by aggregation operators HA and InducedHA with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

further configure parameter 120582 to the value set of [0 01100]and conduct computational experiments to collect scorevalues and ranking results (as shown in Figures 5 and 6) byfour aggregation operators of GIVDHFUUBLHA GIVDH-FUUBLHG I-GIVDHFUUBLHAand I-GIVDHFUUBLHGFor more clarity in Figures 5 and 6 please note that weuse HA HG Induced-HA and Induced HG to respec-tively denote GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG In Figure 5it can be observed that when dominant importance of 120596119860119863

is emphasized (120572 = 0 120572 = 03 120572 = 05) GIVDHFUUBLHAis capable of differentiating the three alternative contractorsas same ranking order of 1199092 ≻ 1199093 ≻ 1199091 for all 120582 but withincreases in 120572 the margins between scores of 1199091 and 1199093 nar-rows and the rank order changes to 1199091 ≻ 1199093when 120572 gets 09and 1 Interestingly in Figure 6 although GIVDHFUUBLHGis able to significantly differentiate alternative contractors 1199091and 1199093 when dominant importance of 120596119860119863 is emphasized (120572= 0 120572 = 03 120572 = 05) rank order of 1199091 and 1199093 obviouslychanges once from 1199093 ≻ 1199091 to 1199091 ≻ 1199093 when 120582 is bigger

22 Complexity

HGInducedHG

= = 1

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(a) 120572= 0

HGInducedHG

= 3 = 07

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(b) 120572= 03

HGInducedHG

= 05 = 05

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(c) 120572= 05

HGInducedHG

= 8 = 02

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572= 08

HGInducedHG

292 293 294 295 296 297minus08

minus0795

minus079

= 09 = 01

HA-1

HA-3

InducedHG-2

HG-1

HG-1

HG-2

HG-3

HG-3 InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4

minus3

minus2

minus1

0

1

2

scor

e val

ues

(e) 120572= 09

HGInducedHG

50 60 70 80 90 100minus095minus09

minus085

= 1 = 0

HG-x3

HG-x1

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(f) 120572= 1

Figure 6 Score trajectories obtained by aggregation operators HG and InducedHG with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

than 8 When 120572 increases to 1 GIVDHFUUBLHGmaintainsthe rank order of 1199093 ≻ 1199091 but their score trajectorieswalk very close Comparatively I-GIVDHFUUBLHA andI-GIVDHFUUBLHG generate consistent ranking results of1199092 ≻ 1199091 ≻ 1199093 no matter 120582 varies in all the above experiments

In sum when parameters are configured with typical val-ues (ie 120572 = 05 and 120582=1) and no additional attitudinal infor-mation is taken into consideration the proposed aggregationoperators unanimously derive the same rank order 1199092 ≻1199093 ≻ 1199091 of for the three alternative contractors However

with the amplification effect of parameter 120582 analysis resultsfrom linear combination of 120596119860119868 and 120596119860119863 reveal alternative1199092 which consistently is the best choice but rank order of1199091 and 1199093 changes because their score trajectories get closeand crossed which probably indicates that only decision-making informationderived from the decisionmatrix is inad-equate to significantly differentiate alternatives 1199091 and 1199093While by integrating additional decision information aboutattitudinal characters of group experts I-GIVDHFUUBLHAand I-GIVDHFUUBLHG manage to generate the identical

Complexity 23

Table 9 Order-inducing vectors obtained by our TOPSIS-based method for problem in [38]

1198661 1198662 1198663 11986641198601 02522 01629 02282 027141198602 01 05243 01364 046891198603 04081 01756 05244 037791198604 04507 01409 01064 013221198605 00697 02203 03753 01161

Table 10 Order-inducing vectors obtained by our TOPSIS-based method for problem in [39]

1198661 1198662 1198663 11986641198601 04548 04969 04923 047321198602 05092 047 04768 046451198603 07192 04888 04953 049151198604 06722 06163 05706 049291198605 06104 04942 06463 04946

Table 11 Ranking results obtained by applying different order-inducing vectors to [38 39]

Problems Approaches Ranking results

Problem in [38] IHFLOWA [38] + order-inducing vectors in [38] 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198605 ≻ 1198604

IHFLOWA [38] + order-inducing vectors in Table 9 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198604 ≻ 1198605

Problem in [39] I-IVDHFLOWA [39] + order-inducing vectors in [39] 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198602 ≻ 1198601

I-IVDHFLOWA [39] + order-inducing vectors in Table 10 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602

permutation of 1199092 ≻ 1199091 ≻ 1199093 which verifies that ourproposed TOPSIS-based order-inducing approach is prac-tical and useful in helping work out decision results morereasonably and robustly

63 Application Studies of Proposed TOPSIS-Based Order-Inducing Method When confronting complex decision-making situations that need additional attitudinal charactersof decision-makers Yager and Filev [88] and Yager [102]introduced and encouraged to employ order-inducing vectorto represent those attitudinal characters Since then theidea of order-inducing vector has been extended to variousdecision-making environments [98 103] such as those basedon intuitionistic fuzzy information [104 105] linguistic infor-mation [106] hesitant fuzzy linguistic information [38] andinterval-valued dual hesitant fuzzy uncertain linguistic infor-mation [39] Generally in existing literature order-inducingvectors rely on experts to directly give [38 39 104ndash106]

In order to examine the adaptability of our TOPSIS-based order-inducing method to linguistic decision-makingenvironments we respectively apply the method to resolvethe same problems in references [38 39] Due to fact thatthere is no descriptions about how the order-inducing vectorswere derived in both references [38 39] we here applyextremums into our TOPSIS-based method to obtain order-inducing vectors that is (1199047 1 0) and (1199041 0 1) for[38] ([1199046 1199046] 1 0) and ([1199040 1199040] 0 1) for [39] Thenwe obtain our order-inducing vectors listed in Table 9 forthe problem adopted in [38] and our order-inducing vectorslisted in Table 10 for [39]

Now we apply the order-inducing vectors as shownin Tables 9 and 10 to resolve the problems in [38] and[39] respectively Regarding the problem in [38] the scorevalues of all alternatives are calculated as 119878(1198601)=02602119878(1198602)=02417 119878(1198603)=03577 119878(1198604)=0221 and 119878(1198605)=0185so the corresponding ranking result is 1198603 ≻ 1198601 ≻ 1198602 ≻1198604 ≻ 1198605 As for the problem in [39] the score values ofall alternatives are computed as 119878(1198601)=39954 119878(1198602)=3783119878(1198603)=70827 119878(1198604)=71792 and 119878(1198605)=5813 we thus obtainthe ranking result 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602 Formore clarity brief descriptions of comparative approachesand obtained ranking results also have been put together inTable 11

According to Table 11 the approaches that employ ourorder-inducing vectors derive the same ranking order ofthe better three alternatives for both comparative problemsfrom [38 39] As seen our proposed TOPSIS-based order-inducing method exhibits effective and more understand-able Furthermore when decision-makers are incapable ofdirectly put forward all desired order-inducing vectors due tocomplexity in ill-structured problems our method providesa feasible directions to obtain order-inducing vectors byfiguring out ideal solutions Comparatively our proposedTOPSIS-based order-inducing method behaves effective andmore operational in practical use

64 Comparison with Decision-Making Approach Based onDegraded form of IVDHFULS Note that our approachis intrinsically adaptive to decision situations based ondegraded forms of IVDHF UUBLS such as the dual hesitant

24 Complexity

fuzzy linguistic sets introduced by Yang and Ju [54] Tofurther inspect feasibility of our approach we here conductcomparative study on the same case adopted by Yang and Ju[54] We firstly convert their original decision matrix intothe form of IVDHF UUBLS crisp membership and non-membership degrees are equivalently changed to intervalswith same upper and lower limits (eg 02 turns out to be[02 02]) and linguistic variables are rewritten as uncertainlinguistic variables (eg 1199042 turns out to be [11990421199042])

Since the case only got three attributes among whichinterdependency could be not significant under generalassumption weighting vector 120596119860119868 is not calculated Andfor straight comparison order inducing vector is notincluded either Then the argument-dependent attributeweighting vector can be obtained from Eq(29) as 120596119860119863 =(03192 03132 03676)119879 By use of Algorithm Iwith operatorIVDHFUUBLWA (ie GIVDHFUUBLWA in case of 120582 = 1see Appendix B) we have the scores of alternatives 119904(1198601) =03412 119904(1198602) = 01354 119904(1198603) = 03932 119904(1198604) = 07352Therefore the ordering of alternatives is1198604 ≻ 1198603 ≻ 1198601 ≻ 1198602which is mainly in alignment with the ordering of1198604 ≻ 1198601 ≻1198603 ≻ 1198602 in Yang and Ju [54] It is worth noticing that 120596119860119863used in our approach is derived objectively from assessmentswhile attribute weights 120596lowast = (035 025 040)119879used by Yangand Ju [54] were assigned empirically Differences between120596119860119863 and 120596lowast lead to different ordering of1198601 and1198603 Compar-atively speaking our approach holds more adaptability andobjectivity

In general our proposed MADM approach in this paperexhibits an effective way to tackle complex green contractorselection problems in which subjective attributes weightingvector (such as120596119860119868) and objective attributes weighting vector(such as 120596119860119863) exist To comprehensively utilize these twofundamental facets of attributes weighting information ourcase study and computational experiments suggest that linearcombination method (ie 120596 = 120572120596119860119868 + 120573120596119860119863) is effectiveto reflect effects of 120596119860119868 and 120596119860119863 by configuring parameter 120572or 120573 When there is no specific expert opinions to specify orhard to obtain we suggest that parameter 120572 chooses the valueset of 0 02 05 08 1 Furthermore generalized aggrega-tion operators are suggested to inspect score trajectoriesof alternative contractors along increasing of the operatorsrsquoamplification parameter 120582 If some alternative contractorsrsquoscore values get very close or crossed so that they cannot bedifferentiated from each other significantly we suggest to askthe panel of experts to derive opinions of ideal solutions andutilize our TOPSIS-based order-inducing method to includethose expert opinions as attitudinal decision informationthereby obtaining more rational and robust decision results

7 Conclusions

Aiming at tackling complex decision-making problems ofgreen contractor selection that often hold simultaneouslythe characteristics of decision hesitancy attributes inter-dependency and group complex attitudinal characters wehave developed an effective IVDHF UUBLS-based approach

We utilize IVDHF UUBLS to elicit complicate decision-makersrsquo assessments more objectively and comprehensivelywe employ DEMATEL for addressing attribute interdepen-dency more precisely and we determine group attitudinalcharacters as order-inducing vectors based on ideal attitudesof experts rather than by empirical assignment The mainadvantages of our approach are (i) IVDHF UUBLS offers anadequate way to depict decision hesitancy (ii) the designeddistance measure for IVDHF UUBLS avoids informationdistortion of traditional ones which artificially add mis-matching membership or nonmembership degrees (iii) thedeveloped generalized operators include traditional ones asspecial cases so as to provide flexibility in MADM basedon IVDHF UUBLS (iv) the devised synthesized attributeweighting scheme provides an adaptive way to consider twonecessary aspects attributes interdependency-based weightsand argument-dependent weights (v) the TOPSIS-basedmethod can deduce reasonable order-inducing vectors ratherthan empirical ones and holds generality in other linguisticdecision environments Results of case studies and experi-ments have verified the validity of the proposed approach

When multiple decision organizations are involvedgroup decision-making based on IVDHF UUBLS is animportant research direction Besides inducing preferencesfor decision-making problems of high uncertainty interactivemechanisms should also be studied to extract preferences Inaddition empirical studies of the proposed approach can becarried out to various areas such as investment evaluationproject management and vehicle selection in fleet operations

Appendix

A Basic Notions for IVDHFS and UnbalancedLinguistic Term Set

Definition A1 (see [47]) Letting 119883 be a fixed set then anIVDHFS on 119883 is defined as119863 = ⟨119909 ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (A1)

where ℎ(119909) = ⋃[120583119871120583119880]isinℎ(119909)120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and119892(119909) = ⋃[]119871]119880]isin119892(119909)] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two setsof interval values in [0 1] denoting possible membershipand non-membership degrees of element 119909 isin 119883 to theset 119863 respectively with conditions 120583 ] isin [0 1] and0 le (120583119880)+ + (]119880)+ le 1 and for all 119909 isin 119883 (120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880

For convenience normally 119889 = ℎ 119892 is called an interval-valued dual hesitant fuzzy element (IVDHFE) and 119863 is theset of all IVDHFEs

Definition A2 (see [107]) Suppose 119878 = 119904119895 | 119895 = 0 1 119892 minus1 is a finite and totally ordered discrete linguistic term setwhere 119904119895 represent possible values for a linguistic variableand 119892 is an odd cardinality 119878 is uniformly and symmetricallydistributed if the following two conditions are satisfied (1)

Complexity 25

there exists a unique constant 120582 gt 0 such that 119873119878(119904119894) minus119873119878(119904119895) = 119897(119894 minus 119895) for all 119894 119895 = 0 1 119892 minus 1 (2) Let 119878119877 =119904 | 119904 isin 119878 119904 gt 119878119862 and 119878119871 = 119904 | 119904 isin 119878 119904 lt 119878119862 Let C(119878119877) andC(119878119871) be the cardinality of 119878119877 and 119878119871 then C(119878119877) = C(119878119871)If 119878 is uniformly and symmetrically distributed then 119878 iscalled a balanced linguistic term set Otherwise 119878 is calledan unbalanced linguistic term set

The following 2-tuple fuzzy linguistic representationmodel extends traditional linguistic term set to a continuouscase so as to facilitate computing with linguistic variables

Definition A3 (see [59]) Let 119878 = 1199040 1199041 119904119892minus1 be alinguistic term set and 120573 isin [0 119892] Then a 2-tuple fuzzylinguistic variable expresses the equivalent information to 120573is defined as

Δ [0 119892] 997888rarr 119878 times [minus05 05) (A2)

Δ (120573) = (119904119894 120572) with

119904119894 119894 = 119903119900119906119899119889 (120573)120572 = 120573 minus 119894 120572 isin [minus05 05)

(A3)

Δminus1 (119904119894 120572) = 119894 + 120572 = 120573 (A4)

where 119903119900119906119899119889() is the usual rounding operation and120572 is calledsymbolic translation

To obtain 2-tuple fuzzy linguistic representations ofunbalanced linguistic terms the concept of linguistic hierar-chies ie 119871119867 = ⋃119905 119897(119905 119899(119905)) is used 119897(119905 119899(119905)) is a linguistichierarchy with 119905 indicating the level of hierarchy and 119899(119905)denotes the granularity of the linguistic term set of 119905 Herreraet al [59] defined the following transformation functionsbetween labels from different levels in multigranular linguis-tic information contexts without loss of information

Definition A4 (see [59]) In linguistic hierarchies 119871119867 =⋃119905 119897(119905 119899(119905)) whose linguistic term sets are represented by119878119899(119905) = 119904119899(119905)0 119904119899(119905)119899(119905)minus1 the transformation function froma linguistic label in level 119905 to a label in consecutive level 1199051015840 isdefined as 1198791198651199051199051015840 119897(119905 119899(119905)) 997888rarr 119897(1199051015840 119899(1199051015840)) such that

1198791198651199051199051015840 (119904119899(119905)119894 120572119899(119905))= Δ 1199051015840 (Δminus1119905 (119904119899(119905)119894 120572119899(119905)) (119899 (1199051015840) minus 1)119899 (119905) minus 1 ) (A5)

By use of the above transformation function any 2-tuplelinguistic representation can be transformed into a term in119871119867 Detailed transformation procedures are listed as follows

(1) Representation in linguistic hierarchy To transformthe unbalanced terms of 119878 into the corresponding terms inthe 119871119867 transformation function 120595 is employed to associateeach unbalanced linguistic 2-tuple (119904119894 120572) with its linguistic 2-tuple in 119871119867(119878) ie

120595 119878 997888rarr 119871119867(119878) (A6)

so that 120595(119904119894 120572) = (119904119866(119894)119868(119894) 120582) for forall(119904119894 120572) isin 119878(2) Computational phase Firstly transform (119904119866(119894)119868(119894) 120582) intolinguistic 2-tuples denoted as (119904119899(1199051015840)

1198681015840(119894) 1205821015840) in 119878119899(1199051015840) where

(119904119899(1199051015840)1198681015840(119894)

1205821015840) = 119879119865 (119904119866(119894)119868(119894) 120582)= Δ(Δminus1 (119904119866(119894)119868(119894) 120582) sdot (119899 (1199051015840) minus 1)119866 (119894) minus 1 ) (A7)

Then a computational model is used over 119878119899(1199051015840) with aresult denoted as (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)(3) Retranslation process The result (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)is transformed into the unbalanced term in 119878 by using thetransformation function 120595minus1 ie

120595minus1 119871119867 (119878) 997888rarr 119878 (A8)

so that 120595minus1(119904119899(1199051015840)119903 120582119903) = (119904119903119890119904119906119897119905 120582119903119890119904119906119897119905) isin 119878B Details of Theorem 14

Proof Given a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector for119904119889119895 and 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 120582 gt 0 then(1) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLWA and GIVDHFUUBLWG operators respectively(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLOWA and GIVDHFUUBLOWG operators respectively

(2) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to theIVDHF UUBL weighted average (IVDHFUUBLWA)and IVDHF UUBL weighted geometric (IVDHFUUBLWG)operators respectively where

26 Complexity

119868119881119863119867119865119880119880119861119871119882119860120596 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

120596119895119904119889119895= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus

119899prod119895=1

(1 minus 120583119871119895 )120596119895 1 minus 119899prod119895=1

(1 minus 120583119880119895 )120596119895]] [[

119899prod119895=1

(]119871119895)120596119895 119899prod119895=1

(]119880119895 )120596119895]])

(B1)

119868119881119863119867119865119880119880119861119871119882119866120596 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889119895120596119895

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[

119899prod119895=1

(120583119871119895 )120596119895 119899prod119895=1

(120583119880119895 )120596119895]] [[1 minus

119899prod119895=1

(1 minus ]119871119895)120596119895 1 minus 119899prod119895=1

(1 minus ]119880119895 )120596119895]])

(B2)

(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHA andGIVDHFUUBLHG operators reduce to the IVDHF UUBL

orderedweighted average (IVDHFUUBLOWA) and IVDHFUUBL ordered weighted geometric (IVDHFUUBLOWG)operators respectively where

119868119881119863119867119865119880119880119861119871119874119882119860119908 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

119908119895119904119889120590(119895)= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) )))

119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) )))

]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus

119899prod119895=1

(1 minus 120583119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus 120583119880120590(119895))119908119895]] [[

119899prod119895=1

(]119871120590(119895))119908119895 119899prod119895=1

(]119880120590(119895))119908119895]])

(B3)

119868119881119863119867119865119880119880119861119871119874119882119866119908 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889120590(119895)119908119895

= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[

119899prod119895=1

(120583119871120590(119895))119908119895 119899prod119895=1

(120583119880120590(119895))119908119895]] [[1 minus

119899prod119895=1

(1 minus ]119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus ]119880120590(119895))119908119895]])

(B4)

Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

References

[1] V W Y Tam and C M Tam ldquoWaste reduction throughincentivesA case studyrdquoBuilding Research and Information vol36 no 1 pp 37ndash43 2008

[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

[11] X Qi C Liang and J Zhang ldquoGeneralized cross-entropy basedgroup decision making with unknown expert and attributeweights under interval-valued intuitionistic fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 79 pp 52ndash642015

[12] H Zimmermann Fuzzy Set Theorymdashand Its ApplicationsKluwer Academic Boston Mass USA 2001

[13] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 pp 199ndash249 1975

[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

[16] B T Nuong K-W Kim L Prathumratana et al ldquoSustainabledevelopment in the mining sector and its evaluation usingfuzzy AHP (Analytic Hierarchy Process) approachrdquo GeosystemEngineering vol 14 no 1 pp 43ndash50 2011

[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

[18] E K Zavadskas Z Turskis and J Antucheviciene ldquoSelectinga contractor by using a novel method for multiple attributeanalysis Weighted aggregated sum product assessment withgrey values (WASPAS-G)rdquo Studies in Informatics and Controlvol 24 no 2 pp 141ndash150 2015

[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

[21] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

[25] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009

[26] Z Hatush and M Skitmore ldquoContractor Selection UsingMulticriteria Utility Theory An Additive Modelrdquo Building andEnvironment vol 33 no 2-3 pp 105ndash115 1998

[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

[30] AM Anvuur andMM Kumaraswamy ldquoConceptualmodel ofpartnering and alliancingrdquo Journal of Construction Engineeringand Management vol 133 no 3 pp 225ndash234 2007

[31] A H I Lee H Kang C F Hsu and H Hung ldquoA green supplierselection model for high-tech industryrdquo Expert Systems withApplications vol 36 no 4 pp 7917ndash7927 2009

[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

[33] P Humphreys R McIvor and F Chan ldquoUsing case-basedreasoning to evaluate supplier environmental managementperformancerdquo Expert Systems with Applications vol 25 no 2pp 141ndash153 2003

[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

[45] Bin Zhu Zeshui Xu and Meimei Xia ldquoDual Hesitant FuzzySetsrdquo Journal of Applied Mathematics vol 2012 Article ID879629 13 pages 2012

[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

[51] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[52] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

[55] Z Tao X Liu H Chen and L Zhou ldquoUsing new versionof extended t-norms and s-norms for aggregating intervallinguistic labelsrdquo IEEETransactions on SystemsManCyberneticsSystems vol vol p 15 2016

[56] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[57] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[58] L Martınez M Espinilla J Liu L G Perez and P J SanchezldquoAn evaluation model with unbalanced linguistic informationapplied to olive oil sensory evaluationrdquo Journal of Multiple-Valued Logic and Soft Computing vol 15 no 2-3 pp 229ndash2512009

[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

[62] L Marin A Valls D Isern A Moreno and J M MerigoldquoInduced unbalanced linguistic ordered weighted average andits application in multiperson decision makingrdquo The ScientificWorld Journal vol 2014 2014

[63] A Baykasoglu V Kaplanoglu Z D U Durmusoglu and CSahin ldquoIntegrating fuzzy DEMATEL and fuzzy hierarchicalTOPSIS methods for truck selectionrdquo Expert Systems withApplications vol 40 no 3 pp 899ndash907 2013

[64] S K Patil and R Kant ldquoA hybrid approach based on fuzzyDEMATEL and FMCDM to predict success of knowledge man-agement adoption in supply chainrdquoApplied Soft Computing vol18 pp 126ndash135 2014

[65] S Liyin and Y Hong ldquoImproving environmental performanceby means of empowerment of contractorsrdquo Management ofEnvironmental Quality An International Journal vol 17 no 3pp 242ndash257 2006

[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

[69] Y Kazancoglu I Kazancoglu and M Sagnak ldquoA new holisticconceptual framework for green supply chain management

performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

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20 Complexity

Table 6 Ranking orders obtained by GIVDHFUUBLWA GIVDHFUUBLOWA GIVDHFUUBLWG and GIVDHFUUBLOWG along with120582Operators 120582 (Integer) Ranking orderGIVDHFUUBLWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLOWA [1 20] 1199092 ≻ 1199093 ≻ 1199091GIVDHFUUBLWG [1 2] 1199092 ≻ 1199093 ≻ 1199091[3 20] 1199092 ≻ 1199091 ≻ 1199093GIVDHFUUBLOWG [1 8] 1199092 ≻ 1199093 ≻ 1199091[9 20] 1199092 ≻ 1199091 ≻ 1199093

Table 7 Ranking results obtained by GIVDHFUUBLHA and I-GIVDHFUUBLHA along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHA I-GIVDHFUUBLHA120572=0 120573=1 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=03 120573=07 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=05 120573=05 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=08 120573=02 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=09 120573=01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572=1 120573=0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093

Table 8 Ranking results obtained by GIVDHFUUBLHG and I-GIVDHFUUBLHG along with 120572 120573 and 120582120572 120573 120582 (Integer) GIVDHFUUBLHG I-GIVDHFUUBLHG

120572 = 0 120573 = 1 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 03 120573 = 07 [1 7] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[8 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 05 120573 = 05 [1 8] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[9 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 08 120573 = 02 [1 17] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093[18 20] 1199092 ≻ 1199091 ≻ 1199093 1199092 ≻ 1199091 ≻ 1199093120572 = 09 120573 = 01 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093120572 = 1 120573 = 0 [1 20] 1199092 ≻ 1199093 ≻ 1199091 1199092 ≻ 1199091 ≻ 1199093membership degrees to a uncertain linguistic term thus canbe considered as optimistic operators while GIVDHFU-UBLWG and GIVDHFUUBLOWG are generalized geomet-ric aggregation operators that producemore favorable resultsby emphasizing nonmembership degrees thus can be consid-ered as pessimistic operators [100 101]Therefore the rankingresults in Table 6 could be perceived as for optimism-baseddecision-making alternative 1199093 always stands out better than1199091 when 120582 gradually amplifies degree of optimism but forpessimism-based decision-making 1199091 turns out better than1199093 when 120582 gradually amplifies degree of pessimism 120582 canbe considered as the parameter for amplifying degree ofoptimism or pessimism [101]

Next to inspect the impacts of various combinationsof 120596119860119868 and 120596119860119863 on ranking results another four aggrega-tion operators of GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG also have beenapplied to the above illustrative case with varying 120572 120573 and 120582

All results have been collected in Tables 7 and 8 for compari-son As can be seen fromTable 7 ranking results generated byoperator GIVDHFUUBLHA keep the same as 1199092 ≻ 1199093 ≻ 1199091for all test combinations of parameters However as shownin Table 8 although GIVDHFUUBLHG is also capable ofidentifying 1199092 as the best contractor the priority relationsbetween 1199091 and 1199093 hold similar changing patterns in rankingresults in Table 6 that were output byGIVDHFUUBLWGandGIVDHFUUBLOWG Regarding decision situations whereobjective attributeweighting vector120596119860119863 is only considered (120572= 0) or its dominant importance (such as 120572 = 03) is accepted1199091 ≻ 1199093 is a clear observation when parameter 120582 is set toan integer in interval [8 20] Contrariwise under decisionscenarios where subjective attribute weighting vector 120596119860119868 isonly needed (120572 = 1) or its dominant importance (such as 120572 =09) is acknowledged 1199093 ≻ 1199091 holds all the way 120582 varies

In order to more clearly show the score trajectories ofalternative contractors along variations of parameter 120582 we

Complexity 21

HAInducedHA

InducedHA-2HA-2

HA-3

InducedHA-1InducedHA-3

= =

HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(a) 120572 = 0

HAInducedHA

= 03 = 07

InducedHA-2

HA-2HA-3

InducedHA-1 InducedHA-3HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(b) 120572 = 03

HAInducedHA

= 5 = 05

InducedHA-2HA-2

HA-3

InducedHA-1 InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(c) 120572 = 05

HAInducedHA

= 08 = 02

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572 = 08

HAInducedHA

= 9 = 01

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(e) 120572 = 09

HAInducedHA

246 2462 2464 2466 2468 247

= 1 = 0InducedHA-2HA-2

HA-3

HA- 3

HA-3

InducedHA-1

InducedHA-3

HA-1

HA-1

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

1085

1086

1087

(f) 120572 = 1

Figure 5 Score trajectories obtained by aggregation operators HA and InducedHA with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

further configure parameter 120582 to the value set of [0 01100]and conduct computational experiments to collect scorevalues and ranking results (as shown in Figures 5 and 6) byfour aggregation operators of GIVDHFUUBLHA GIVDH-FUUBLHG I-GIVDHFUUBLHAand I-GIVDHFUUBLHGFor more clarity in Figures 5 and 6 please note that weuse HA HG Induced-HA and Induced HG to respec-tively denote GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG In Figure 5it can be observed that when dominant importance of 120596119860119863

is emphasized (120572 = 0 120572 = 03 120572 = 05) GIVDHFUUBLHAis capable of differentiating the three alternative contractorsas same ranking order of 1199092 ≻ 1199093 ≻ 1199091 for all 120582 but withincreases in 120572 the margins between scores of 1199091 and 1199093 nar-rows and the rank order changes to 1199091 ≻ 1199093when 120572 gets 09and 1 Interestingly in Figure 6 although GIVDHFUUBLHGis able to significantly differentiate alternative contractors 1199091and 1199093 when dominant importance of 120596119860119863 is emphasized (120572= 0 120572 = 03 120572 = 05) rank order of 1199091 and 1199093 obviouslychanges once from 1199093 ≻ 1199091 to 1199091 ≻ 1199093 when 120582 is bigger

22 Complexity

HGInducedHG

= = 1

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(a) 120572= 0

HGInducedHG

= 3 = 07

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(b) 120572= 03

HGInducedHG

= 05 = 05

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(c) 120572= 05

HGInducedHG

= 8 = 02

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572= 08

HGInducedHG

292 293 294 295 296 297minus08

minus0795

minus079

= 09 = 01

HA-1

HA-3

InducedHG-2

HG-1

HG-1

HG-2

HG-3

HG-3 InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4

minus3

minus2

minus1

0

1

2

scor

e val

ues

(e) 120572= 09

HGInducedHG

50 60 70 80 90 100minus095minus09

minus085

= 1 = 0

HG-x3

HG-x1

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(f) 120572= 1

Figure 6 Score trajectories obtained by aggregation operators HG and InducedHG with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

than 8 When 120572 increases to 1 GIVDHFUUBLHGmaintainsthe rank order of 1199093 ≻ 1199091 but their score trajectorieswalk very close Comparatively I-GIVDHFUUBLHA andI-GIVDHFUUBLHG generate consistent ranking results of1199092 ≻ 1199091 ≻ 1199093 no matter 120582 varies in all the above experiments

In sum when parameters are configured with typical val-ues (ie 120572 = 05 and 120582=1) and no additional attitudinal infor-mation is taken into consideration the proposed aggregationoperators unanimously derive the same rank order 1199092 ≻1199093 ≻ 1199091 of for the three alternative contractors However

with the amplification effect of parameter 120582 analysis resultsfrom linear combination of 120596119860119868 and 120596119860119863 reveal alternative1199092 which consistently is the best choice but rank order of1199091 and 1199093 changes because their score trajectories get closeand crossed which probably indicates that only decision-making informationderived from the decisionmatrix is inad-equate to significantly differentiate alternatives 1199091 and 1199093While by integrating additional decision information aboutattitudinal characters of group experts I-GIVDHFUUBLHAand I-GIVDHFUUBLHG manage to generate the identical

Complexity 23

Table 9 Order-inducing vectors obtained by our TOPSIS-based method for problem in [38]

1198661 1198662 1198663 11986641198601 02522 01629 02282 027141198602 01 05243 01364 046891198603 04081 01756 05244 037791198604 04507 01409 01064 013221198605 00697 02203 03753 01161

Table 10 Order-inducing vectors obtained by our TOPSIS-based method for problem in [39]

1198661 1198662 1198663 11986641198601 04548 04969 04923 047321198602 05092 047 04768 046451198603 07192 04888 04953 049151198604 06722 06163 05706 049291198605 06104 04942 06463 04946

Table 11 Ranking results obtained by applying different order-inducing vectors to [38 39]

Problems Approaches Ranking results

Problem in [38] IHFLOWA [38] + order-inducing vectors in [38] 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198605 ≻ 1198604

IHFLOWA [38] + order-inducing vectors in Table 9 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198604 ≻ 1198605

Problem in [39] I-IVDHFLOWA [39] + order-inducing vectors in [39] 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198602 ≻ 1198601

I-IVDHFLOWA [39] + order-inducing vectors in Table 10 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602

permutation of 1199092 ≻ 1199091 ≻ 1199093 which verifies that ourproposed TOPSIS-based order-inducing approach is prac-tical and useful in helping work out decision results morereasonably and robustly

63 Application Studies of Proposed TOPSIS-Based Order-Inducing Method When confronting complex decision-making situations that need additional attitudinal charactersof decision-makers Yager and Filev [88] and Yager [102]introduced and encouraged to employ order-inducing vectorto represent those attitudinal characters Since then theidea of order-inducing vector has been extended to variousdecision-making environments [98 103] such as those basedon intuitionistic fuzzy information [104 105] linguistic infor-mation [106] hesitant fuzzy linguistic information [38] andinterval-valued dual hesitant fuzzy uncertain linguistic infor-mation [39] Generally in existing literature order-inducingvectors rely on experts to directly give [38 39 104ndash106]

In order to examine the adaptability of our TOPSIS-based order-inducing method to linguistic decision-makingenvironments we respectively apply the method to resolvethe same problems in references [38 39] Due to fact thatthere is no descriptions about how the order-inducing vectorswere derived in both references [38 39] we here applyextremums into our TOPSIS-based method to obtain order-inducing vectors that is (1199047 1 0) and (1199041 0 1) for[38] ([1199046 1199046] 1 0) and ([1199040 1199040] 0 1) for [39] Thenwe obtain our order-inducing vectors listed in Table 9 forthe problem adopted in [38] and our order-inducing vectorslisted in Table 10 for [39]

Now we apply the order-inducing vectors as shownin Tables 9 and 10 to resolve the problems in [38] and[39] respectively Regarding the problem in [38] the scorevalues of all alternatives are calculated as 119878(1198601)=02602119878(1198602)=02417 119878(1198603)=03577 119878(1198604)=0221 and 119878(1198605)=0185so the corresponding ranking result is 1198603 ≻ 1198601 ≻ 1198602 ≻1198604 ≻ 1198605 As for the problem in [39] the score values ofall alternatives are computed as 119878(1198601)=39954 119878(1198602)=3783119878(1198603)=70827 119878(1198604)=71792 and 119878(1198605)=5813 we thus obtainthe ranking result 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602 Formore clarity brief descriptions of comparative approachesand obtained ranking results also have been put together inTable 11

According to Table 11 the approaches that employ ourorder-inducing vectors derive the same ranking order ofthe better three alternatives for both comparative problemsfrom [38 39] As seen our proposed TOPSIS-based order-inducing method exhibits effective and more understand-able Furthermore when decision-makers are incapable ofdirectly put forward all desired order-inducing vectors due tocomplexity in ill-structured problems our method providesa feasible directions to obtain order-inducing vectors byfiguring out ideal solutions Comparatively our proposedTOPSIS-based order-inducing method behaves effective andmore operational in practical use

64 Comparison with Decision-Making Approach Based onDegraded form of IVDHFULS Note that our approachis intrinsically adaptive to decision situations based ondegraded forms of IVDHF UUBLS such as the dual hesitant

24 Complexity

fuzzy linguistic sets introduced by Yang and Ju [54] Tofurther inspect feasibility of our approach we here conductcomparative study on the same case adopted by Yang and Ju[54] We firstly convert their original decision matrix intothe form of IVDHF UUBLS crisp membership and non-membership degrees are equivalently changed to intervalswith same upper and lower limits (eg 02 turns out to be[02 02]) and linguistic variables are rewritten as uncertainlinguistic variables (eg 1199042 turns out to be [11990421199042])

Since the case only got three attributes among whichinterdependency could be not significant under generalassumption weighting vector 120596119860119868 is not calculated Andfor straight comparison order inducing vector is notincluded either Then the argument-dependent attributeweighting vector can be obtained from Eq(29) as 120596119860119863 =(03192 03132 03676)119879 By use of Algorithm Iwith operatorIVDHFUUBLWA (ie GIVDHFUUBLWA in case of 120582 = 1see Appendix B) we have the scores of alternatives 119904(1198601) =03412 119904(1198602) = 01354 119904(1198603) = 03932 119904(1198604) = 07352Therefore the ordering of alternatives is1198604 ≻ 1198603 ≻ 1198601 ≻ 1198602which is mainly in alignment with the ordering of1198604 ≻ 1198601 ≻1198603 ≻ 1198602 in Yang and Ju [54] It is worth noticing that 120596119860119863used in our approach is derived objectively from assessmentswhile attribute weights 120596lowast = (035 025 040)119879used by Yangand Ju [54] were assigned empirically Differences between120596119860119863 and 120596lowast lead to different ordering of1198601 and1198603 Compar-atively speaking our approach holds more adaptability andobjectivity

In general our proposed MADM approach in this paperexhibits an effective way to tackle complex green contractorselection problems in which subjective attributes weightingvector (such as120596119860119868) and objective attributes weighting vector(such as 120596119860119863) exist To comprehensively utilize these twofundamental facets of attributes weighting information ourcase study and computational experiments suggest that linearcombination method (ie 120596 = 120572120596119860119868 + 120573120596119860119863) is effectiveto reflect effects of 120596119860119868 and 120596119860119863 by configuring parameter 120572or 120573 When there is no specific expert opinions to specify orhard to obtain we suggest that parameter 120572 chooses the valueset of 0 02 05 08 1 Furthermore generalized aggrega-tion operators are suggested to inspect score trajectoriesof alternative contractors along increasing of the operatorsrsquoamplification parameter 120582 If some alternative contractorsrsquoscore values get very close or crossed so that they cannot bedifferentiated from each other significantly we suggest to askthe panel of experts to derive opinions of ideal solutions andutilize our TOPSIS-based order-inducing method to includethose expert opinions as attitudinal decision informationthereby obtaining more rational and robust decision results

7 Conclusions

Aiming at tackling complex decision-making problems ofgreen contractor selection that often hold simultaneouslythe characteristics of decision hesitancy attributes inter-dependency and group complex attitudinal characters wehave developed an effective IVDHF UUBLS-based approach

We utilize IVDHF UUBLS to elicit complicate decision-makersrsquo assessments more objectively and comprehensivelywe employ DEMATEL for addressing attribute interdepen-dency more precisely and we determine group attitudinalcharacters as order-inducing vectors based on ideal attitudesof experts rather than by empirical assignment The mainadvantages of our approach are (i) IVDHF UUBLS offers anadequate way to depict decision hesitancy (ii) the designeddistance measure for IVDHF UUBLS avoids informationdistortion of traditional ones which artificially add mis-matching membership or nonmembership degrees (iii) thedeveloped generalized operators include traditional ones asspecial cases so as to provide flexibility in MADM basedon IVDHF UUBLS (iv) the devised synthesized attributeweighting scheme provides an adaptive way to consider twonecessary aspects attributes interdependency-based weightsand argument-dependent weights (v) the TOPSIS-basedmethod can deduce reasonable order-inducing vectors ratherthan empirical ones and holds generality in other linguisticdecision environments Results of case studies and experi-ments have verified the validity of the proposed approach

When multiple decision organizations are involvedgroup decision-making based on IVDHF UUBLS is animportant research direction Besides inducing preferencesfor decision-making problems of high uncertainty interactivemechanisms should also be studied to extract preferences Inaddition empirical studies of the proposed approach can becarried out to various areas such as investment evaluationproject management and vehicle selection in fleet operations

Appendix

A Basic Notions for IVDHFS and UnbalancedLinguistic Term Set

Definition A1 (see [47]) Letting 119883 be a fixed set then anIVDHFS on 119883 is defined as119863 = ⟨119909 ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (A1)

where ℎ(119909) = ⋃[120583119871120583119880]isinℎ(119909)120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and119892(119909) = ⋃[]119871]119880]isin119892(119909)] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two setsof interval values in [0 1] denoting possible membershipand non-membership degrees of element 119909 isin 119883 to theset 119863 respectively with conditions 120583 ] isin [0 1] and0 le (120583119880)+ + (]119880)+ le 1 and for all 119909 isin 119883 (120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880

For convenience normally 119889 = ℎ 119892 is called an interval-valued dual hesitant fuzzy element (IVDHFE) and 119863 is theset of all IVDHFEs

Definition A2 (see [107]) Suppose 119878 = 119904119895 | 119895 = 0 1 119892 minus1 is a finite and totally ordered discrete linguistic term setwhere 119904119895 represent possible values for a linguistic variableand 119892 is an odd cardinality 119878 is uniformly and symmetricallydistributed if the following two conditions are satisfied (1)

Complexity 25

there exists a unique constant 120582 gt 0 such that 119873119878(119904119894) minus119873119878(119904119895) = 119897(119894 minus 119895) for all 119894 119895 = 0 1 119892 minus 1 (2) Let 119878119877 =119904 | 119904 isin 119878 119904 gt 119878119862 and 119878119871 = 119904 | 119904 isin 119878 119904 lt 119878119862 Let C(119878119877) andC(119878119871) be the cardinality of 119878119877 and 119878119871 then C(119878119877) = C(119878119871)If 119878 is uniformly and symmetrically distributed then 119878 iscalled a balanced linguistic term set Otherwise 119878 is calledan unbalanced linguistic term set

The following 2-tuple fuzzy linguistic representationmodel extends traditional linguistic term set to a continuouscase so as to facilitate computing with linguistic variables

Definition A3 (see [59]) Let 119878 = 1199040 1199041 119904119892minus1 be alinguistic term set and 120573 isin [0 119892] Then a 2-tuple fuzzylinguistic variable expresses the equivalent information to 120573is defined as

Δ [0 119892] 997888rarr 119878 times [minus05 05) (A2)

Δ (120573) = (119904119894 120572) with

119904119894 119894 = 119903119900119906119899119889 (120573)120572 = 120573 minus 119894 120572 isin [minus05 05)

(A3)

Δminus1 (119904119894 120572) = 119894 + 120572 = 120573 (A4)

where 119903119900119906119899119889() is the usual rounding operation and120572 is calledsymbolic translation

To obtain 2-tuple fuzzy linguistic representations ofunbalanced linguistic terms the concept of linguistic hierar-chies ie 119871119867 = ⋃119905 119897(119905 119899(119905)) is used 119897(119905 119899(119905)) is a linguistichierarchy with 119905 indicating the level of hierarchy and 119899(119905)denotes the granularity of the linguistic term set of 119905 Herreraet al [59] defined the following transformation functionsbetween labels from different levels in multigranular linguis-tic information contexts without loss of information

Definition A4 (see [59]) In linguistic hierarchies 119871119867 =⋃119905 119897(119905 119899(119905)) whose linguistic term sets are represented by119878119899(119905) = 119904119899(119905)0 119904119899(119905)119899(119905)minus1 the transformation function froma linguistic label in level 119905 to a label in consecutive level 1199051015840 isdefined as 1198791198651199051199051015840 119897(119905 119899(119905)) 997888rarr 119897(1199051015840 119899(1199051015840)) such that

1198791198651199051199051015840 (119904119899(119905)119894 120572119899(119905))= Δ 1199051015840 (Δminus1119905 (119904119899(119905)119894 120572119899(119905)) (119899 (1199051015840) minus 1)119899 (119905) minus 1 ) (A5)

By use of the above transformation function any 2-tuplelinguistic representation can be transformed into a term in119871119867 Detailed transformation procedures are listed as follows

(1) Representation in linguistic hierarchy To transformthe unbalanced terms of 119878 into the corresponding terms inthe 119871119867 transformation function 120595 is employed to associateeach unbalanced linguistic 2-tuple (119904119894 120572) with its linguistic 2-tuple in 119871119867(119878) ie

120595 119878 997888rarr 119871119867(119878) (A6)

so that 120595(119904119894 120572) = (119904119866(119894)119868(119894) 120582) for forall(119904119894 120572) isin 119878(2) Computational phase Firstly transform (119904119866(119894)119868(119894) 120582) intolinguistic 2-tuples denoted as (119904119899(1199051015840)

1198681015840(119894) 1205821015840) in 119878119899(1199051015840) where

(119904119899(1199051015840)1198681015840(119894)

1205821015840) = 119879119865 (119904119866(119894)119868(119894) 120582)= Δ(Δminus1 (119904119866(119894)119868(119894) 120582) sdot (119899 (1199051015840) minus 1)119866 (119894) minus 1 ) (A7)

Then a computational model is used over 119878119899(1199051015840) with aresult denoted as (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)(3) Retranslation process The result (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)is transformed into the unbalanced term in 119878 by using thetransformation function 120595minus1 ie

120595minus1 119871119867 (119878) 997888rarr 119878 (A8)

so that 120595minus1(119904119899(1199051015840)119903 120582119903) = (119904119903119890119904119906119897119905 120582119903119890119904119906119897119905) isin 119878B Details of Theorem 14

Proof Given a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector for119904119889119895 and 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 120582 gt 0 then(1) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLWA and GIVDHFUUBLWG operators respectively(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLOWA and GIVDHFUUBLOWG operators respectively

(2) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to theIVDHF UUBL weighted average (IVDHFUUBLWA)and IVDHF UUBL weighted geometric (IVDHFUUBLWG)operators respectively where

26 Complexity

119868119881119863119867119865119880119880119861119871119882119860120596 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

120596119895119904119889119895= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus

119899prod119895=1

(1 minus 120583119871119895 )120596119895 1 minus 119899prod119895=1

(1 minus 120583119880119895 )120596119895]] [[

119899prod119895=1

(]119871119895)120596119895 119899prod119895=1

(]119880119895 )120596119895]])

(B1)

119868119881119863119867119865119880119880119861119871119882119866120596 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889119895120596119895

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[

119899prod119895=1

(120583119871119895 )120596119895 119899prod119895=1

(120583119880119895 )120596119895]] [[1 minus

119899prod119895=1

(1 minus ]119871119895)120596119895 1 minus 119899prod119895=1

(1 minus ]119880119895 )120596119895]])

(B2)

(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHA andGIVDHFUUBLHG operators reduce to the IVDHF UUBL

orderedweighted average (IVDHFUUBLOWA) and IVDHFUUBL ordered weighted geometric (IVDHFUUBLOWG)operators respectively where

119868119881119863119867119865119880119880119861119871119874119882119860119908 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

119908119895119904119889120590(119895)= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) )))

119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) )))

]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus

119899prod119895=1

(1 minus 120583119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus 120583119880120590(119895))119908119895]] [[

119899prod119895=1

(]119871120590(119895))119908119895 119899prod119895=1

(]119880120590(119895))119908119895]])

(B3)

119868119881119863119867119865119880119880119861119871119874119882119866119908 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889120590(119895)119908119895

= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[

119899prod119895=1

(120583119871120590(119895))119908119895 119899prod119895=1

(120583119880120590(119895))119908119895]] [[1 minus

119899prod119895=1

(1 minus ]119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus ]119880120590(119895))119908119895]])

(B4)

Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

References

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[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

[11] X Qi C Liang and J Zhang ldquoGeneralized cross-entropy basedgroup decision making with unknown expert and attributeweights under interval-valued intuitionistic fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 79 pp 52ndash642015

[12] H Zimmermann Fuzzy Set Theorymdashand Its ApplicationsKluwer Academic Boston Mass USA 2001

[13] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 pp 199ndash249 1975

[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

[16] B T Nuong K-W Kim L Prathumratana et al ldquoSustainabledevelopment in the mining sector and its evaluation usingfuzzy AHP (Analytic Hierarchy Process) approachrdquo GeosystemEngineering vol 14 no 1 pp 43ndash50 2011

[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

[18] E K Zavadskas Z Turskis and J Antucheviciene ldquoSelectinga contractor by using a novel method for multiple attributeanalysis Weighted aggregated sum product assessment withgrey values (WASPAS-G)rdquo Studies in Informatics and Controlvol 24 no 2 pp 141ndash150 2015

[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

[21] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

[25] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009

[26] Z Hatush and M Skitmore ldquoContractor Selection UsingMulticriteria Utility Theory An Additive Modelrdquo Building andEnvironment vol 33 no 2-3 pp 105ndash115 1998

[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

[30] AM Anvuur andMM Kumaraswamy ldquoConceptualmodel ofpartnering and alliancingrdquo Journal of Construction Engineeringand Management vol 133 no 3 pp 225ndash234 2007

[31] A H I Lee H Kang C F Hsu and H Hung ldquoA green supplierselection model for high-tech industryrdquo Expert Systems withApplications vol 36 no 4 pp 7917ndash7927 2009

[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

[33] P Humphreys R McIvor and F Chan ldquoUsing case-basedreasoning to evaluate supplier environmental managementperformancerdquo Expert Systems with Applications vol 25 no 2pp 141ndash153 2003

[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

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[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

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[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

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[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

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[64] S K Patil and R Kant ldquoA hybrid approach based on fuzzyDEMATEL and FMCDM to predict success of knowledge man-agement adoption in supply chainrdquoApplied Soft Computing vol18 pp 126ndash135 2014

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[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

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performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

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[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

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Complexity 21

HAInducedHA

InducedHA-2HA-2

HA-3

InducedHA-1InducedHA-3

= =

HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(a) 120572 = 0

HAInducedHA

= 03 = 07

InducedHA-2

HA-2HA-3

InducedHA-1 InducedHA-3HA-1

10 20 30 40 50 60 70 80 90 1000lambda

minus3minus2minus1

012345

scor

e val

ues

(b) 120572 = 03

HAInducedHA

= 5 = 05

InducedHA-2HA-2

HA-3

InducedHA-1 InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(c) 120572 = 05

HAInducedHA

= 08 = 02

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572 = 08

HAInducedHA

= 9 = 01

InducedHA-2

HA-2

HA-3

InducedHA-1

InducedHA-3

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(e) 120572 = 09

HAInducedHA

246 2462 2464 2466 2468 247

= 1 = 0InducedHA-2HA-2

HA-3

HA- 3

HA-3

InducedHA-1

InducedHA-3

HA-1

HA-1

HA-1

minus3minus2minus1

012345

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

1085

1086

1087

(f) 120572 = 1

Figure 5 Score trajectories obtained by aggregation operators HA and InducedHA with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

further configure parameter 120582 to the value set of [0 01100]and conduct computational experiments to collect scorevalues and ranking results (as shown in Figures 5 and 6) byfour aggregation operators of GIVDHFUUBLHA GIVDH-FUUBLHG I-GIVDHFUUBLHAand I-GIVDHFUUBLHGFor more clarity in Figures 5 and 6 please note that weuse HA HG Induced-HA and Induced HG to respec-tively denote GIVDHFUUBLHA GIVDHFUUBLHG I-GIVDHFUUBLHA and I-GIVDHFUUBLHG In Figure 5it can be observed that when dominant importance of 120596119860119863

is emphasized (120572 = 0 120572 = 03 120572 = 05) GIVDHFUUBLHAis capable of differentiating the three alternative contractorsas same ranking order of 1199092 ≻ 1199093 ≻ 1199091 for all 120582 but withincreases in 120572 the margins between scores of 1199091 and 1199093 nar-rows and the rank order changes to 1199091 ≻ 1199093when 120572 gets 09and 1 Interestingly in Figure 6 although GIVDHFUUBLHGis able to significantly differentiate alternative contractors 1199091and 1199093 when dominant importance of 120596119860119863 is emphasized (120572= 0 120572 = 03 120572 = 05) rank order of 1199091 and 1199093 obviouslychanges once from 1199093 ≻ 1199091 to 1199091 ≻ 1199093 when 120582 is bigger

22 Complexity

HGInducedHG

= = 1

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(a) 120572= 0

HGInducedHG

= 3 = 07

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(b) 120572= 03

HGInducedHG

= 05 = 05

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(c) 120572= 05

HGInducedHG

= 8 = 02

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572= 08

HGInducedHG

292 293 294 295 296 297minus08

minus0795

minus079

= 09 = 01

HA-1

HA-3

InducedHG-2

HG-1

HG-1

HG-2

HG-3

HG-3 InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4

minus3

minus2

minus1

0

1

2

scor

e val

ues

(e) 120572= 09

HGInducedHG

50 60 70 80 90 100minus095minus09

minus085

= 1 = 0

HG-x3

HG-x1

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(f) 120572= 1

Figure 6 Score trajectories obtained by aggregation operators HG and InducedHG with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

than 8 When 120572 increases to 1 GIVDHFUUBLHGmaintainsthe rank order of 1199093 ≻ 1199091 but their score trajectorieswalk very close Comparatively I-GIVDHFUUBLHA andI-GIVDHFUUBLHG generate consistent ranking results of1199092 ≻ 1199091 ≻ 1199093 no matter 120582 varies in all the above experiments

In sum when parameters are configured with typical val-ues (ie 120572 = 05 and 120582=1) and no additional attitudinal infor-mation is taken into consideration the proposed aggregationoperators unanimously derive the same rank order 1199092 ≻1199093 ≻ 1199091 of for the three alternative contractors However

with the amplification effect of parameter 120582 analysis resultsfrom linear combination of 120596119860119868 and 120596119860119863 reveal alternative1199092 which consistently is the best choice but rank order of1199091 and 1199093 changes because their score trajectories get closeand crossed which probably indicates that only decision-making informationderived from the decisionmatrix is inad-equate to significantly differentiate alternatives 1199091 and 1199093While by integrating additional decision information aboutattitudinal characters of group experts I-GIVDHFUUBLHAand I-GIVDHFUUBLHG manage to generate the identical

Complexity 23

Table 9 Order-inducing vectors obtained by our TOPSIS-based method for problem in [38]

1198661 1198662 1198663 11986641198601 02522 01629 02282 027141198602 01 05243 01364 046891198603 04081 01756 05244 037791198604 04507 01409 01064 013221198605 00697 02203 03753 01161

Table 10 Order-inducing vectors obtained by our TOPSIS-based method for problem in [39]

1198661 1198662 1198663 11986641198601 04548 04969 04923 047321198602 05092 047 04768 046451198603 07192 04888 04953 049151198604 06722 06163 05706 049291198605 06104 04942 06463 04946

Table 11 Ranking results obtained by applying different order-inducing vectors to [38 39]

Problems Approaches Ranking results

Problem in [38] IHFLOWA [38] + order-inducing vectors in [38] 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198605 ≻ 1198604

IHFLOWA [38] + order-inducing vectors in Table 9 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198604 ≻ 1198605

Problem in [39] I-IVDHFLOWA [39] + order-inducing vectors in [39] 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198602 ≻ 1198601

I-IVDHFLOWA [39] + order-inducing vectors in Table 10 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602

permutation of 1199092 ≻ 1199091 ≻ 1199093 which verifies that ourproposed TOPSIS-based order-inducing approach is prac-tical and useful in helping work out decision results morereasonably and robustly

63 Application Studies of Proposed TOPSIS-Based Order-Inducing Method When confronting complex decision-making situations that need additional attitudinal charactersof decision-makers Yager and Filev [88] and Yager [102]introduced and encouraged to employ order-inducing vectorto represent those attitudinal characters Since then theidea of order-inducing vector has been extended to variousdecision-making environments [98 103] such as those basedon intuitionistic fuzzy information [104 105] linguistic infor-mation [106] hesitant fuzzy linguistic information [38] andinterval-valued dual hesitant fuzzy uncertain linguistic infor-mation [39] Generally in existing literature order-inducingvectors rely on experts to directly give [38 39 104ndash106]

In order to examine the adaptability of our TOPSIS-based order-inducing method to linguistic decision-makingenvironments we respectively apply the method to resolvethe same problems in references [38 39] Due to fact thatthere is no descriptions about how the order-inducing vectorswere derived in both references [38 39] we here applyextremums into our TOPSIS-based method to obtain order-inducing vectors that is (1199047 1 0) and (1199041 0 1) for[38] ([1199046 1199046] 1 0) and ([1199040 1199040] 0 1) for [39] Thenwe obtain our order-inducing vectors listed in Table 9 forthe problem adopted in [38] and our order-inducing vectorslisted in Table 10 for [39]

Now we apply the order-inducing vectors as shownin Tables 9 and 10 to resolve the problems in [38] and[39] respectively Regarding the problem in [38] the scorevalues of all alternatives are calculated as 119878(1198601)=02602119878(1198602)=02417 119878(1198603)=03577 119878(1198604)=0221 and 119878(1198605)=0185so the corresponding ranking result is 1198603 ≻ 1198601 ≻ 1198602 ≻1198604 ≻ 1198605 As for the problem in [39] the score values ofall alternatives are computed as 119878(1198601)=39954 119878(1198602)=3783119878(1198603)=70827 119878(1198604)=71792 and 119878(1198605)=5813 we thus obtainthe ranking result 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602 Formore clarity brief descriptions of comparative approachesand obtained ranking results also have been put together inTable 11

According to Table 11 the approaches that employ ourorder-inducing vectors derive the same ranking order ofthe better three alternatives for both comparative problemsfrom [38 39] As seen our proposed TOPSIS-based order-inducing method exhibits effective and more understand-able Furthermore when decision-makers are incapable ofdirectly put forward all desired order-inducing vectors due tocomplexity in ill-structured problems our method providesa feasible directions to obtain order-inducing vectors byfiguring out ideal solutions Comparatively our proposedTOPSIS-based order-inducing method behaves effective andmore operational in practical use

64 Comparison with Decision-Making Approach Based onDegraded form of IVDHFULS Note that our approachis intrinsically adaptive to decision situations based ondegraded forms of IVDHF UUBLS such as the dual hesitant

24 Complexity

fuzzy linguistic sets introduced by Yang and Ju [54] Tofurther inspect feasibility of our approach we here conductcomparative study on the same case adopted by Yang and Ju[54] We firstly convert their original decision matrix intothe form of IVDHF UUBLS crisp membership and non-membership degrees are equivalently changed to intervalswith same upper and lower limits (eg 02 turns out to be[02 02]) and linguistic variables are rewritten as uncertainlinguistic variables (eg 1199042 turns out to be [11990421199042])

Since the case only got three attributes among whichinterdependency could be not significant under generalassumption weighting vector 120596119860119868 is not calculated Andfor straight comparison order inducing vector is notincluded either Then the argument-dependent attributeweighting vector can be obtained from Eq(29) as 120596119860119863 =(03192 03132 03676)119879 By use of Algorithm Iwith operatorIVDHFUUBLWA (ie GIVDHFUUBLWA in case of 120582 = 1see Appendix B) we have the scores of alternatives 119904(1198601) =03412 119904(1198602) = 01354 119904(1198603) = 03932 119904(1198604) = 07352Therefore the ordering of alternatives is1198604 ≻ 1198603 ≻ 1198601 ≻ 1198602which is mainly in alignment with the ordering of1198604 ≻ 1198601 ≻1198603 ≻ 1198602 in Yang and Ju [54] It is worth noticing that 120596119860119863used in our approach is derived objectively from assessmentswhile attribute weights 120596lowast = (035 025 040)119879used by Yangand Ju [54] were assigned empirically Differences between120596119860119863 and 120596lowast lead to different ordering of1198601 and1198603 Compar-atively speaking our approach holds more adaptability andobjectivity

In general our proposed MADM approach in this paperexhibits an effective way to tackle complex green contractorselection problems in which subjective attributes weightingvector (such as120596119860119868) and objective attributes weighting vector(such as 120596119860119863) exist To comprehensively utilize these twofundamental facets of attributes weighting information ourcase study and computational experiments suggest that linearcombination method (ie 120596 = 120572120596119860119868 + 120573120596119860119863) is effectiveto reflect effects of 120596119860119868 and 120596119860119863 by configuring parameter 120572or 120573 When there is no specific expert opinions to specify orhard to obtain we suggest that parameter 120572 chooses the valueset of 0 02 05 08 1 Furthermore generalized aggrega-tion operators are suggested to inspect score trajectoriesof alternative contractors along increasing of the operatorsrsquoamplification parameter 120582 If some alternative contractorsrsquoscore values get very close or crossed so that they cannot bedifferentiated from each other significantly we suggest to askthe panel of experts to derive opinions of ideal solutions andutilize our TOPSIS-based order-inducing method to includethose expert opinions as attitudinal decision informationthereby obtaining more rational and robust decision results

7 Conclusions

Aiming at tackling complex decision-making problems ofgreen contractor selection that often hold simultaneouslythe characteristics of decision hesitancy attributes inter-dependency and group complex attitudinal characters wehave developed an effective IVDHF UUBLS-based approach

We utilize IVDHF UUBLS to elicit complicate decision-makersrsquo assessments more objectively and comprehensivelywe employ DEMATEL for addressing attribute interdepen-dency more precisely and we determine group attitudinalcharacters as order-inducing vectors based on ideal attitudesof experts rather than by empirical assignment The mainadvantages of our approach are (i) IVDHF UUBLS offers anadequate way to depict decision hesitancy (ii) the designeddistance measure for IVDHF UUBLS avoids informationdistortion of traditional ones which artificially add mis-matching membership or nonmembership degrees (iii) thedeveloped generalized operators include traditional ones asspecial cases so as to provide flexibility in MADM basedon IVDHF UUBLS (iv) the devised synthesized attributeweighting scheme provides an adaptive way to consider twonecessary aspects attributes interdependency-based weightsand argument-dependent weights (v) the TOPSIS-basedmethod can deduce reasonable order-inducing vectors ratherthan empirical ones and holds generality in other linguisticdecision environments Results of case studies and experi-ments have verified the validity of the proposed approach

When multiple decision organizations are involvedgroup decision-making based on IVDHF UUBLS is animportant research direction Besides inducing preferencesfor decision-making problems of high uncertainty interactivemechanisms should also be studied to extract preferences Inaddition empirical studies of the proposed approach can becarried out to various areas such as investment evaluationproject management and vehicle selection in fleet operations

Appendix

A Basic Notions for IVDHFS and UnbalancedLinguistic Term Set

Definition A1 (see [47]) Letting 119883 be a fixed set then anIVDHFS on 119883 is defined as119863 = ⟨119909 ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (A1)

where ℎ(119909) = ⋃[120583119871120583119880]isinℎ(119909)120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and119892(119909) = ⋃[]119871]119880]isin119892(119909)] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two setsof interval values in [0 1] denoting possible membershipand non-membership degrees of element 119909 isin 119883 to theset 119863 respectively with conditions 120583 ] isin [0 1] and0 le (120583119880)+ + (]119880)+ le 1 and for all 119909 isin 119883 (120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880

For convenience normally 119889 = ℎ 119892 is called an interval-valued dual hesitant fuzzy element (IVDHFE) and 119863 is theset of all IVDHFEs

Definition A2 (see [107]) Suppose 119878 = 119904119895 | 119895 = 0 1 119892 minus1 is a finite and totally ordered discrete linguistic term setwhere 119904119895 represent possible values for a linguistic variableand 119892 is an odd cardinality 119878 is uniformly and symmetricallydistributed if the following two conditions are satisfied (1)

Complexity 25

there exists a unique constant 120582 gt 0 such that 119873119878(119904119894) minus119873119878(119904119895) = 119897(119894 minus 119895) for all 119894 119895 = 0 1 119892 minus 1 (2) Let 119878119877 =119904 | 119904 isin 119878 119904 gt 119878119862 and 119878119871 = 119904 | 119904 isin 119878 119904 lt 119878119862 Let C(119878119877) andC(119878119871) be the cardinality of 119878119877 and 119878119871 then C(119878119877) = C(119878119871)If 119878 is uniformly and symmetrically distributed then 119878 iscalled a balanced linguistic term set Otherwise 119878 is calledan unbalanced linguistic term set

The following 2-tuple fuzzy linguistic representationmodel extends traditional linguistic term set to a continuouscase so as to facilitate computing with linguistic variables

Definition A3 (see [59]) Let 119878 = 1199040 1199041 119904119892minus1 be alinguistic term set and 120573 isin [0 119892] Then a 2-tuple fuzzylinguistic variable expresses the equivalent information to 120573is defined as

Δ [0 119892] 997888rarr 119878 times [minus05 05) (A2)

Δ (120573) = (119904119894 120572) with

119904119894 119894 = 119903119900119906119899119889 (120573)120572 = 120573 minus 119894 120572 isin [minus05 05)

(A3)

Δminus1 (119904119894 120572) = 119894 + 120572 = 120573 (A4)

where 119903119900119906119899119889() is the usual rounding operation and120572 is calledsymbolic translation

To obtain 2-tuple fuzzy linguistic representations ofunbalanced linguistic terms the concept of linguistic hierar-chies ie 119871119867 = ⋃119905 119897(119905 119899(119905)) is used 119897(119905 119899(119905)) is a linguistichierarchy with 119905 indicating the level of hierarchy and 119899(119905)denotes the granularity of the linguistic term set of 119905 Herreraet al [59] defined the following transformation functionsbetween labels from different levels in multigranular linguis-tic information contexts without loss of information

Definition A4 (see [59]) In linguistic hierarchies 119871119867 =⋃119905 119897(119905 119899(119905)) whose linguistic term sets are represented by119878119899(119905) = 119904119899(119905)0 119904119899(119905)119899(119905)minus1 the transformation function froma linguistic label in level 119905 to a label in consecutive level 1199051015840 isdefined as 1198791198651199051199051015840 119897(119905 119899(119905)) 997888rarr 119897(1199051015840 119899(1199051015840)) such that

1198791198651199051199051015840 (119904119899(119905)119894 120572119899(119905))= Δ 1199051015840 (Δminus1119905 (119904119899(119905)119894 120572119899(119905)) (119899 (1199051015840) minus 1)119899 (119905) minus 1 ) (A5)

By use of the above transformation function any 2-tuplelinguistic representation can be transformed into a term in119871119867 Detailed transformation procedures are listed as follows

(1) Representation in linguistic hierarchy To transformthe unbalanced terms of 119878 into the corresponding terms inthe 119871119867 transformation function 120595 is employed to associateeach unbalanced linguistic 2-tuple (119904119894 120572) with its linguistic 2-tuple in 119871119867(119878) ie

120595 119878 997888rarr 119871119867(119878) (A6)

so that 120595(119904119894 120572) = (119904119866(119894)119868(119894) 120582) for forall(119904119894 120572) isin 119878(2) Computational phase Firstly transform (119904119866(119894)119868(119894) 120582) intolinguistic 2-tuples denoted as (119904119899(1199051015840)

1198681015840(119894) 1205821015840) in 119878119899(1199051015840) where

(119904119899(1199051015840)1198681015840(119894)

1205821015840) = 119879119865 (119904119866(119894)119868(119894) 120582)= Δ(Δminus1 (119904119866(119894)119868(119894) 120582) sdot (119899 (1199051015840) minus 1)119866 (119894) minus 1 ) (A7)

Then a computational model is used over 119878119899(1199051015840) with aresult denoted as (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)(3) Retranslation process The result (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)is transformed into the unbalanced term in 119878 by using thetransformation function 120595minus1 ie

120595minus1 119871119867 (119878) 997888rarr 119878 (A8)

so that 120595minus1(119904119899(1199051015840)119903 120582119903) = (119904119903119890119904119906119897119905 120582119903119890119904119906119897119905) isin 119878B Details of Theorem 14

Proof Given a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector for119904119889119895 and 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 120582 gt 0 then(1) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLWA and GIVDHFUUBLWG operators respectively(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLOWA and GIVDHFUUBLOWG operators respectively

(2) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to theIVDHF UUBL weighted average (IVDHFUUBLWA)and IVDHF UUBL weighted geometric (IVDHFUUBLWG)operators respectively where

26 Complexity

119868119881119863119867119865119880119880119861119871119882119860120596 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

120596119895119904119889119895= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus

119899prod119895=1

(1 minus 120583119871119895 )120596119895 1 minus 119899prod119895=1

(1 minus 120583119880119895 )120596119895]] [[

119899prod119895=1

(]119871119895)120596119895 119899prod119895=1

(]119880119895 )120596119895]])

(B1)

119868119881119863119867119865119880119880119861119871119882119866120596 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889119895120596119895

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[

119899prod119895=1

(120583119871119895 )120596119895 119899prod119895=1

(120583119880119895 )120596119895]] [[1 minus

119899prod119895=1

(1 minus ]119871119895)120596119895 1 minus 119899prod119895=1

(1 minus ]119880119895 )120596119895]])

(B2)

(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHA andGIVDHFUUBLHG operators reduce to the IVDHF UUBL

orderedweighted average (IVDHFUUBLOWA) and IVDHFUUBL ordered weighted geometric (IVDHFUUBLOWG)operators respectively where

119868119881119863119867119865119880119880119861119871119874119882119860119908 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

119908119895119904119889120590(119895)= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) )))

119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) )))

]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus

119899prod119895=1

(1 minus 120583119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus 120583119880120590(119895))119908119895]] [[

119899prod119895=1

(]119871120590(119895))119908119895 119899prod119895=1

(]119880120590(119895))119908119895]])

(B3)

119868119881119863119867119865119880119880119861119871119874119882119866119908 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889120590(119895)119908119895

= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[

119899prod119895=1

(120583119871120590(119895))119908119895 119899prod119895=1

(120583119880120590(119895))119908119895]] [[1 minus

119899prod119895=1

(1 minus ]119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus ]119880120590(119895))119908119895]])

(B4)

Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

References

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[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

[11] X Qi C Liang and J Zhang ldquoGeneralized cross-entropy basedgroup decision making with unknown expert and attributeweights under interval-valued intuitionistic fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 79 pp 52ndash642015

[12] H Zimmermann Fuzzy Set Theorymdashand Its ApplicationsKluwer Academic Boston Mass USA 2001

[13] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 pp 199ndash249 1975

[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

[16] B T Nuong K-W Kim L Prathumratana et al ldquoSustainabledevelopment in the mining sector and its evaluation usingfuzzy AHP (Analytic Hierarchy Process) approachrdquo GeosystemEngineering vol 14 no 1 pp 43ndash50 2011

[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

[18] E K Zavadskas Z Turskis and J Antucheviciene ldquoSelectinga contractor by using a novel method for multiple attributeanalysis Weighted aggregated sum product assessment withgrey values (WASPAS-G)rdquo Studies in Informatics and Controlvol 24 no 2 pp 141ndash150 2015

[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

[21] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

[25] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009

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[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

[30] AM Anvuur andMM Kumaraswamy ldquoConceptualmodel ofpartnering and alliancingrdquo Journal of Construction Engineeringand Management vol 133 no 3 pp 225ndash234 2007

[31] A H I Lee H Kang C F Hsu and H Hung ldquoA green supplierselection model for high-tech industryrdquo Expert Systems withApplications vol 36 no 4 pp 7917ndash7927 2009

[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

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[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

[45] Bin Zhu Zeshui Xu and Meimei Xia ldquoDual Hesitant FuzzySetsrdquo Journal of Applied Mathematics vol 2012 Article ID879629 13 pages 2012

[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

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[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

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[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

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[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

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performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

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[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

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[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

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[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

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[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

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22 Complexity

HGInducedHG

= = 1

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(a) 120572= 0

HGInducedHG

= 3 = 07

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(b) 120572= 03

HGInducedHG

= 05 = 05

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4minus3minus2minus1

0123

scor

e val

ues

(c) 120572= 05

HGInducedHG

= 8 = 02

InducedHG-2HG-1

HG-2

HG-3

InducedHG-1

InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(d) 120572= 08

HGInducedHG

292 293 294 295 296 297minus08

minus0795

minus079

= 09 = 01

HA-1

HA-3

InducedHG-2

HG-1

HG-1

HG-2

HG-3

HG-3 InducedHG-1

InducedHG-3

10 20 30 40 50 60 70 80 90 1000lambda

minus4

minus3

minus2

minus1

0

1

2

scor

e val

ues

(e) 120572= 09

HGInducedHG

50 60 70 80 90 100minus095minus09

minus085

= 1 = 0

HG-x3

HG-x1

InducedHG-2

HG-1

HG-2

HG-3

InducedHG-1InducedHG-3

minus4minus3minus2minus1

0123

scor

e val

ues

10 20 30 40 50 60 70 80 90 1000lambda

(f) 120572= 1

Figure 6 Score trajectories obtained by aggregation operators HG and InducedHG with different configuration of 120572 (a) 120572 = 0 (b) 120572 = 03(c) 120572 = 05 (d) 120572 = 07 (e) 120572 = 09 and (f) 120572 = 1

than 8 When 120572 increases to 1 GIVDHFUUBLHGmaintainsthe rank order of 1199093 ≻ 1199091 but their score trajectorieswalk very close Comparatively I-GIVDHFUUBLHA andI-GIVDHFUUBLHG generate consistent ranking results of1199092 ≻ 1199091 ≻ 1199093 no matter 120582 varies in all the above experiments

In sum when parameters are configured with typical val-ues (ie 120572 = 05 and 120582=1) and no additional attitudinal infor-mation is taken into consideration the proposed aggregationoperators unanimously derive the same rank order 1199092 ≻1199093 ≻ 1199091 of for the three alternative contractors However

with the amplification effect of parameter 120582 analysis resultsfrom linear combination of 120596119860119868 and 120596119860119863 reveal alternative1199092 which consistently is the best choice but rank order of1199091 and 1199093 changes because their score trajectories get closeand crossed which probably indicates that only decision-making informationderived from the decisionmatrix is inad-equate to significantly differentiate alternatives 1199091 and 1199093While by integrating additional decision information aboutattitudinal characters of group experts I-GIVDHFUUBLHAand I-GIVDHFUUBLHG manage to generate the identical

Complexity 23

Table 9 Order-inducing vectors obtained by our TOPSIS-based method for problem in [38]

1198661 1198662 1198663 11986641198601 02522 01629 02282 027141198602 01 05243 01364 046891198603 04081 01756 05244 037791198604 04507 01409 01064 013221198605 00697 02203 03753 01161

Table 10 Order-inducing vectors obtained by our TOPSIS-based method for problem in [39]

1198661 1198662 1198663 11986641198601 04548 04969 04923 047321198602 05092 047 04768 046451198603 07192 04888 04953 049151198604 06722 06163 05706 049291198605 06104 04942 06463 04946

Table 11 Ranking results obtained by applying different order-inducing vectors to [38 39]

Problems Approaches Ranking results

Problem in [38] IHFLOWA [38] + order-inducing vectors in [38] 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198605 ≻ 1198604

IHFLOWA [38] + order-inducing vectors in Table 9 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198604 ≻ 1198605

Problem in [39] I-IVDHFLOWA [39] + order-inducing vectors in [39] 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198602 ≻ 1198601

I-IVDHFLOWA [39] + order-inducing vectors in Table 10 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602

permutation of 1199092 ≻ 1199091 ≻ 1199093 which verifies that ourproposed TOPSIS-based order-inducing approach is prac-tical and useful in helping work out decision results morereasonably and robustly

63 Application Studies of Proposed TOPSIS-Based Order-Inducing Method When confronting complex decision-making situations that need additional attitudinal charactersof decision-makers Yager and Filev [88] and Yager [102]introduced and encouraged to employ order-inducing vectorto represent those attitudinal characters Since then theidea of order-inducing vector has been extended to variousdecision-making environments [98 103] such as those basedon intuitionistic fuzzy information [104 105] linguistic infor-mation [106] hesitant fuzzy linguistic information [38] andinterval-valued dual hesitant fuzzy uncertain linguistic infor-mation [39] Generally in existing literature order-inducingvectors rely on experts to directly give [38 39 104ndash106]

In order to examine the adaptability of our TOPSIS-based order-inducing method to linguistic decision-makingenvironments we respectively apply the method to resolvethe same problems in references [38 39] Due to fact thatthere is no descriptions about how the order-inducing vectorswere derived in both references [38 39] we here applyextremums into our TOPSIS-based method to obtain order-inducing vectors that is (1199047 1 0) and (1199041 0 1) for[38] ([1199046 1199046] 1 0) and ([1199040 1199040] 0 1) for [39] Thenwe obtain our order-inducing vectors listed in Table 9 forthe problem adopted in [38] and our order-inducing vectorslisted in Table 10 for [39]

Now we apply the order-inducing vectors as shownin Tables 9 and 10 to resolve the problems in [38] and[39] respectively Regarding the problem in [38] the scorevalues of all alternatives are calculated as 119878(1198601)=02602119878(1198602)=02417 119878(1198603)=03577 119878(1198604)=0221 and 119878(1198605)=0185so the corresponding ranking result is 1198603 ≻ 1198601 ≻ 1198602 ≻1198604 ≻ 1198605 As for the problem in [39] the score values ofall alternatives are computed as 119878(1198601)=39954 119878(1198602)=3783119878(1198603)=70827 119878(1198604)=71792 and 119878(1198605)=5813 we thus obtainthe ranking result 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602 Formore clarity brief descriptions of comparative approachesand obtained ranking results also have been put together inTable 11

According to Table 11 the approaches that employ ourorder-inducing vectors derive the same ranking order ofthe better three alternatives for both comparative problemsfrom [38 39] As seen our proposed TOPSIS-based order-inducing method exhibits effective and more understand-able Furthermore when decision-makers are incapable ofdirectly put forward all desired order-inducing vectors due tocomplexity in ill-structured problems our method providesa feasible directions to obtain order-inducing vectors byfiguring out ideal solutions Comparatively our proposedTOPSIS-based order-inducing method behaves effective andmore operational in practical use

64 Comparison with Decision-Making Approach Based onDegraded form of IVDHFULS Note that our approachis intrinsically adaptive to decision situations based ondegraded forms of IVDHF UUBLS such as the dual hesitant

24 Complexity

fuzzy linguistic sets introduced by Yang and Ju [54] Tofurther inspect feasibility of our approach we here conductcomparative study on the same case adopted by Yang and Ju[54] We firstly convert their original decision matrix intothe form of IVDHF UUBLS crisp membership and non-membership degrees are equivalently changed to intervalswith same upper and lower limits (eg 02 turns out to be[02 02]) and linguistic variables are rewritten as uncertainlinguistic variables (eg 1199042 turns out to be [11990421199042])

Since the case only got three attributes among whichinterdependency could be not significant under generalassumption weighting vector 120596119860119868 is not calculated Andfor straight comparison order inducing vector is notincluded either Then the argument-dependent attributeweighting vector can be obtained from Eq(29) as 120596119860119863 =(03192 03132 03676)119879 By use of Algorithm Iwith operatorIVDHFUUBLWA (ie GIVDHFUUBLWA in case of 120582 = 1see Appendix B) we have the scores of alternatives 119904(1198601) =03412 119904(1198602) = 01354 119904(1198603) = 03932 119904(1198604) = 07352Therefore the ordering of alternatives is1198604 ≻ 1198603 ≻ 1198601 ≻ 1198602which is mainly in alignment with the ordering of1198604 ≻ 1198601 ≻1198603 ≻ 1198602 in Yang and Ju [54] It is worth noticing that 120596119860119863used in our approach is derived objectively from assessmentswhile attribute weights 120596lowast = (035 025 040)119879used by Yangand Ju [54] were assigned empirically Differences between120596119860119863 and 120596lowast lead to different ordering of1198601 and1198603 Compar-atively speaking our approach holds more adaptability andobjectivity

In general our proposed MADM approach in this paperexhibits an effective way to tackle complex green contractorselection problems in which subjective attributes weightingvector (such as120596119860119868) and objective attributes weighting vector(such as 120596119860119863) exist To comprehensively utilize these twofundamental facets of attributes weighting information ourcase study and computational experiments suggest that linearcombination method (ie 120596 = 120572120596119860119868 + 120573120596119860119863) is effectiveto reflect effects of 120596119860119868 and 120596119860119863 by configuring parameter 120572or 120573 When there is no specific expert opinions to specify orhard to obtain we suggest that parameter 120572 chooses the valueset of 0 02 05 08 1 Furthermore generalized aggrega-tion operators are suggested to inspect score trajectoriesof alternative contractors along increasing of the operatorsrsquoamplification parameter 120582 If some alternative contractorsrsquoscore values get very close or crossed so that they cannot bedifferentiated from each other significantly we suggest to askthe panel of experts to derive opinions of ideal solutions andutilize our TOPSIS-based order-inducing method to includethose expert opinions as attitudinal decision informationthereby obtaining more rational and robust decision results

7 Conclusions

Aiming at tackling complex decision-making problems ofgreen contractor selection that often hold simultaneouslythe characteristics of decision hesitancy attributes inter-dependency and group complex attitudinal characters wehave developed an effective IVDHF UUBLS-based approach

We utilize IVDHF UUBLS to elicit complicate decision-makersrsquo assessments more objectively and comprehensivelywe employ DEMATEL for addressing attribute interdepen-dency more precisely and we determine group attitudinalcharacters as order-inducing vectors based on ideal attitudesof experts rather than by empirical assignment The mainadvantages of our approach are (i) IVDHF UUBLS offers anadequate way to depict decision hesitancy (ii) the designeddistance measure for IVDHF UUBLS avoids informationdistortion of traditional ones which artificially add mis-matching membership or nonmembership degrees (iii) thedeveloped generalized operators include traditional ones asspecial cases so as to provide flexibility in MADM basedon IVDHF UUBLS (iv) the devised synthesized attributeweighting scheme provides an adaptive way to consider twonecessary aspects attributes interdependency-based weightsand argument-dependent weights (v) the TOPSIS-basedmethod can deduce reasonable order-inducing vectors ratherthan empirical ones and holds generality in other linguisticdecision environments Results of case studies and experi-ments have verified the validity of the proposed approach

When multiple decision organizations are involvedgroup decision-making based on IVDHF UUBLS is animportant research direction Besides inducing preferencesfor decision-making problems of high uncertainty interactivemechanisms should also be studied to extract preferences Inaddition empirical studies of the proposed approach can becarried out to various areas such as investment evaluationproject management and vehicle selection in fleet operations

Appendix

A Basic Notions for IVDHFS and UnbalancedLinguistic Term Set

Definition A1 (see [47]) Letting 119883 be a fixed set then anIVDHFS on 119883 is defined as119863 = ⟨119909 ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (A1)

where ℎ(119909) = ⋃[120583119871120583119880]isinℎ(119909)120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and119892(119909) = ⋃[]119871]119880]isin119892(119909)] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two setsof interval values in [0 1] denoting possible membershipand non-membership degrees of element 119909 isin 119883 to theset 119863 respectively with conditions 120583 ] isin [0 1] and0 le (120583119880)+ + (]119880)+ le 1 and for all 119909 isin 119883 (120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880

For convenience normally 119889 = ℎ 119892 is called an interval-valued dual hesitant fuzzy element (IVDHFE) and 119863 is theset of all IVDHFEs

Definition A2 (see [107]) Suppose 119878 = 119904119895 | 119895 = 0 1 119892 minus1 is a finite and totally ordered discrete linguistic term setwhere 119904119895 represent possible values for a linguistic variableand 119892 is an odd cardinality 119878 is uniformly and symmetricallydistributed if the following two conditions are satisfied (1)

Complexity 25

there exists a unique constant 120582 gt 0 such that 119873119878(119904119894) minus119873119878(119904119895) = 119897(119894 minus 119895) for all 119894 119895 = 0 1 119892 minus 1 (2) Let 119878119877 =119904 | 119904 isin 119878 119904 gt 119878119862 and 119878119871 = 119904 | 119904 isin 119878 119904 lt 119878119862 Let C(119878119877) andC(119878119871) be the cardinality of 119878119877 and 119878119871 then C(119878119877) = C(119878119871)If 119878 is uniformly and symmetrically distributed then 119878 iscalled a balanced linguistic term set Otherwise 119878 is calledan unbalanced linguistic term set

The following 2-tuple fuzzy linguistic representationmodel extends traditional linguistic term set to a continuouscase so as to facilitate computing with linguistic variables

Definition A3 (see [59]) Let 119878 = 1199040 1199041 119904119892minus1 be alinguistic term set and 120573 isin [0 119892] Then a 2-tuple fuzzylinguistic variable expresses the equivalent information to 120573is defined as

Δ [0 119892] 997888rarr 119878 times [minus05 05) (A2)

Δ (120573) = (119904119894 120572) with

119904119894 119894 = 119903119900119906119899119889 (120573)120572 = 120573 minus 119894 120572 isin [minus05 05)

(A3)

Δminus1 (119904119894 120572) = 119894 + 120572 = 120573 (A4)

where 119903119900119906119899119889() is the usual rounding operation and120572 is calledsymbolic translation

To obtain 2-tuple fuzzy linguistic representations ofunbalanced linguistic terms the concept of linguistic hierar-chies ie 119871119867 = ⋃119905 119897(119905 119899(119905)) is used 119897(119905 119899(119905)) is a linguistichierarchy with 119905 indicating the level of hierarchy and 119899(119905)denotes the granularity of the linguistic term set of 119905 Herreraet al [59] defined the following transformation functionsbetween labels from different levels in multigranular linguis-tic information contexts without loss of information

Definition A4 (see [59]) In linguistic hierarchies 119871119867 =⋃119905 119897(119905 119899(119905)) whose linguistic term sets are represented by119878119899(119905) = 119904119899(119905)0 119904119899(119905)119899(119905)minus1 the transformation function froma linguistic label in level 119905 to a label in consecutive level 1199051015840 isdefined as 1198791198651199051199051015840 119897(119905 119899(119905)) 997888rarr 119897(1199051015840 119899(1199051015840)) such that

1198791198651199051199051015840 (119904119899(119905)119894 120572119899(119905))= Δ 1199051015840 (Δminus1119905 (119904119899(119905)119894 120572119899(119905)) (119899 (1199051015840) minus 1)119899 (119905) minus 1 ) (A5)

By use of the above transformation function any 2-tuplelinguistic representation can be transformed into a term in119871119867 Detailed transformation procedures are listed as follows

(1) Representation in linguistic hierarchy To transformthe unbalanced terms of 119878 into the corresponding terms inthe 119871119867 transformation function 120595 is employed to associateeach unbalanced linguistic 2-tuple (119904119894 120572) with its linguistic 2-tuple in 119871119867(119878) ie

120595 119878 997888rarr 119871119867(119878) (A6)

so that 120595(119904119894 120572) = (119904119866(119894)119868(119894) 120582) for forall(119904119894 120572) isin 119878(2) Computational phase Firstly transform (119904119866(119894)119868(119894) 120582) intolinguistic 2-tuples denoted as (119904119899(1199051015840)

1198681015840(119894) 1205821015840) in 119878119899(1199051015840) where

(119904119899(1199051015840)1198681015840(119894)

1205821015840) = 119879119865 (119904119866(119894)119868(119894) 120582)= Δ(Δminus1 (119904119866(119894)119868(119894) 120582) sdot (119899 (1199051015840) minus 1)119866 (119894) minus 1 ) (A7)

Then a computational model is used over 119878119899(1199051015840) with aresult denoted as (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)(3) Retranslation process The result (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)is transformed into the unbalanced term in 119878 by using thetransformation function 120595minus1 ie

120595minus1 119871119867 (119878) 997888rarr 119878 (A8)

so that 120595minus1(119904119899(1199051015840)119903 120582119903) = (119904119903119890119904119906119897119905 120582119903119890119904119906119897119905) isin 119878B Details of Theorem 14

Proof Given a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector for119904119889119895 and 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 120582 gt 0 then(1) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLWA and GIVDHFUUBLWG operators respectively(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLOWA and GIVDHFUUBLOWG operators respectively

(2) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to theIVDHF UUBL weighted average (IVDHFUUBLWA)and IVDHF UUBL weighted geometric (IVDHFUUBLWG)operators respectively where

26 Complexity

119868119881119863119867119865119880119880119861119871119882119860120596 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

120596119895119904119889119895= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus

119899prod119895=1

(1 minus 120583119871119895 )120596119895 1 minus 119899prod119895=1

(1 minus 120583119880119895 )120596119895]] [[

119899prod119895=1

(]119871119895)120596119895 119899prod119895=1

(]119880119895 )120596119895]])

(B1)

119868119881119863119867119865119880119880119861119871119882119866120596 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889119895120596119895

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[

119899prod119895=1

(120583119871119895 )120596119895 119899prod119895=1

(120583119880119895 )120596119895]] [[1 minus

119899prod119895=1

(1 minus ]119871119895)120596119895 1 minus 119899prod119895=1

(1 minus ]119880119895 )120596119895]])

(B2)

(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHA andGIVDHFUUBLHG operators reduce to the IVDHF UUBL

orderedweighted average (IVDHFUUBLOWA) and IVDHFUUBL ordered weighted geometric (IVDHFUUBLOWG)operators respectively where

119868119881119863119867119865119880119880119861119871119874119882119860119908 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

119908119895119904119889120590(119895)= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) )))

119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) )))

]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus

119899prod119895=1

(1 minus 120583119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus 120583119880120590(119895))119908119895]] [[

119899prod119895=1

(]119871120590(119895))119908119895 119899prod119895=1

(]119880120590(119895))119908119895]])

(B3)

119868119881119863119867119865119880119880119861119871119874119882119866119908 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889120590(119895)119908119895

= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[

119899prod119895=1

(120583119871120590(119895))119908119895 119899prod119895=1

(120583119880120590(119895))119908119895]] [[1 minus

119899prod119895=1

(1 minus ]119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus ]119880120590(119895))119908119895]])

(B4)

Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

References

[1] V W Y Tam and C M Tam ldquoWaste reduction throughincentivesA case studyrdquoBuilding Research and Information vol36 no 1 pp 37ndash43 2008

[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

[11] X Qi C Liang and J Zhang ldquoGeneralized cross-entropy basedgroup decision making with unknown expert and attributeweights under interval-valued intuitionistic fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 79 pp 52ndash642015

[12] H Zimmermann Fuzzy Set Theorymdashand Its ApplicationsKluwer Academic Boston Mass USA 2001

[13] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 pp 199ndash249 1975

[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

[16] B T Nuong K-W Kim L Prathumratana et al ldquoSustainabledevelopment in the mining sector and its evaluation usingfuzzy AHP (Analytic Hierarchy Process) approachrdquo GeosystemEngineering vol 14 no 1 pp 43ndash50 2011

[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

[18] E K Zavadskas Z Turskis and J Antucheviciene ldquoSelectinga contractor by using a novel method for multiple attributeanalysis Weighted aggregated sum product assessment withgrey values (WASPAS-G)rdquo Studies in Informatics and Controlvol 24 no 2 pp 141ndash150 2015

[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

[21] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

[25] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009

[26] Z Hatush and M Skitmore ldquoContractor Selection UsingMulticriteria Utility Theory An Additive Modelrdquo Building andEnvironment vol 33 no 2-3 pp 105ndash115 1998

[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

[30] AM Anvuur andMM Kumaraswamy ldquoConceptualmodel ofpartnering and alliancingrdquo Journal of Construction Engineeringand Management vol 133 no 3 pp 225ndash234 2007

[31] A H I Lee H Kang C F Hsu and H Hung ldquoA green supplierselection model for high-tech industryrdquo Expert Systems withApplications vol 36 no 4 pp 7917ndash7927 2009

[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

[33] P Humphreys R McIvor and F Chan ldquoUsing case-basedreasoning to evaluate supplier environmental managementperformancerdquo Expert Systems with Applications vol 25 no 2pp 141ndash153 2003

[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

[45] Bin Zhu Zeshui Xu and Meimei Xia ldquoDual Hesitant FuzzySetsrdquo Journal of Applied Mathematics vol 2012 Article ID879629 13 pages 2012

[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

[51] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[52] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

[55] Z Tao X Liu H Chen and L Zhou ldquoUsing new versionof extended t-norms and s-norms for aggregating intervallinguistic labelsrdquo IEEETransactions on SystemsManCyberneticsSystems vol vol p 15 2016

[56] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[57] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[58] L Martınez M Espinilla J Liu L G Perez and P J SanchezldquoAn evaluation model with unbalanced linguistic informationapplied to olive oil sensory evaluationrdquo Journal of Multiple-Valued Logic and Soft Computing vol 15 no 2-3 pp 229ndash2512009

[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

[62] L Marin A Valls D Isern A Moreno and J M MerigoldquoInduced unbalanced linguistic ordered weighted average andits application in multiperson decision makingrdquo The ScientificWorld Journal vol 2014 2014

[63] A Baykasoglu V Kaplanoglu Z D U Durmusoglu and CSahin ldquoIntegrating fuzzy DEMATEL and fuzzy hierarchicalTOPSIS methods for truck selectionrdquo Expert Systems withApplications vol 40 no 3 pp 899ndash907 2013

[64] S K Patil and R Kant ldquoA hybrid approach based on fuzzyDEMATEL and FMCDM to predict success of knowledge man-agement adoption in supply chainrdquoApplied Soft Computing vol18 pp 126ndash135 2014

[65] S Liyin and Y Hong ldquoImproving environmental performanceby means of empowerment of contractorsrdquo Management ofEnvironmental Quality An International Journal vol 17 no 3pp 242ndash257 2006

[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

[69] Y Kazancoglu I Kazancoglu and M Sagnak ldquoA new holisticconceptual framework for green supply chain management

performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

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Complexity 23

Table 9 Order-inducing vectors obtained by our TOPSIS-based method for problem in [38]

1198661 1198662 1198663 11986641198601 02522 01629 02282 027141198602 01 05243 01364 046891198603 04081 01756 05244 037791198604 04507 01409 01064 013221198605 00697 02203 03753 01161

Table 10 Order-inducing vectors obtained by our TOPSIS-based method for problem in [39]

1198661 1198662 1198663 11986641198601 04548 04969 04923 047321198602 05092 047 04768 046451198603 07192 04888 04953 049151198604 06722 06163 05706 049291198605 06104 04942 06463 04946

Table 11 Ranking results obtained by applying different order-inducing vectors to [38 39]

Problems Approaches Ranking results

Problem in [38] IHFLOWA [38] + order-inducing vectors in [38] 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198605 ≻ 1198604

IHFLOWA [38] + order-inducing vectors in Table 9 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198604 ≻ 1198605

Problem in [39] I-IVDHFLOWA [39] + order-inducing vectors in [39] 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198602 ≻ 1198601

I-IVDHFLOWA [39] + order-inducing vectors in Table 10 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602

permutation of 1199092 ≻ 1199091 ≻ 1199093 which verifies that ourproposed TOPSIS-based order-inducing approach is prac-tical and useful in helping work out decision results morereasonably and robustly

63 Application Studies of Proposed TOPSIS-Based Order-Inducing Method When confronting complex decision-making situations that need additional attitudinal charactersof decision-makers Yager and Filev [88] and Yager [102]introduced and encouraged to employ order-inducing vectorto represent those attitudinal characters Since then theidea of order-inducing vector has been extended to variousdecision-making environments [98 103] such as those basedon intuitionistic fuzzy information [104 105] linguistic infor-mation [106] hesitant fuzzy linguistic information [38] andinterval-valued dual hesitant fuzzy uncertain linguistic infor-mation [39] Generally in existing literature order-inducingvectors rely on experts to directly give [38 39 104ndash106]

In order to examine the adaptability of our TOPSIS-based order-inducing method to linguistic decision-makingenvironments we respectively apply the method to resolvethe same problems in references [38 39] Due to fact thatthere is no descriptions about how the order-inducing vectorswere derived in both references [38 39] we here applyextremums into our TOPSIS-based method to obtain order-inducing vectors that is (1199047 1 0) and (1199041 0 1) for[38] ([1199046 1199046] 1 0) and ([1199040 1199040] 0 1) for [39] Thenwe obtain our order-inducing vectors listed in Table 9 forthe problem adopted in [38] and our order-inducing vectorslisted in Table 10 for [39]

Now we apply the order-inducing vectors as shownin Tables 9 and 10 to resolve the problems in [38] and[39] respectively Regarding the problem in [38] the scorevalues of all alternatives are calculated as 119878(1198601)=02602119878(1198602)=02417 119878(1198603)=03577 119878(1198604)=0221 and 119878(1198605)=0185so the corresponding ranking result is 1198603 ≻ 1198601 ≻ 1198602 ≻1198604 ≻ 1198605 As for the problem in [39] the score values ofall alternatives are computed as 119878(1198601)=39954 119878(1198602)=3783119878(1198603)=70827 119878(1198604)=71792 and 119878(1198605)=5813 we thus obtainthe ranking result 1198604 ≻ 1198603 ≻ 1198605 ≻ 1198601 ≻ 1198602 Formore clarity brief descriptions of comparative approachesand obtained ranking results also have been put together inTable 11

According to Table 11 the approaches that employ ourorder-inducing vectors derive the same ranking order ofthe better three alternatives for both comparative problemsfrom [38 39] As seen our proposed TOPSIS-based order-inducing method exhibits effective and more understand-able Furthermore when decision-makers are incapable ofdirectly put forward all desired order-inducing vectors due tocomplexity in ill-structured problems our method providesa feasible directions to obtain order-inducing vectors byfiguring out ideal solutions Comparatively our proposedTOPSIS-based order-inducing method behaves effective andmore operational in practical use

64 Comparison with Decision-Making Approach Based onDegraded form of IVDHFULS Note that our approachis intrinsically adaptive to decision situations based ondegraded forms of IVDHF UUBLS such as the dual hesitant

24 Complexity

fuzzy linguistic sets introduced by Yang and Ju [54] Tofurther inspect feasibility of our approach we here conductcomparative study on the same case adopted by Yang and Ju[54] We firstly convert their original decision matrix intothe form of IVDHF UUBLS crisp membership and non-membership degrees are equivalently changed to intervalswith same upper and lower limits (eg 02 turns out to be[02 02]) and linguistic variables are rewritten as uncertainlinguistic variables (eg 1199042 turns out to be [11990421199042])

Since the case only got three attributes among whichinterdependency could be not significant under generalassumption weighting vector 120596119860119868 is not calculated Andfor straight comparison order inducing vector is notincluded either Then the argument-dependent attributeweighting vector can be obtained from Eq(29) as 120596119860119863 =(03192 03132 03676)119879 By use of Algorithm Iwith operatorIVDHFUUBLWA (ie GIVDHFUUBLWA in case of 120582 = 1see Appendix B) we have the scores of alternatives 119904(1198601) =03412 119904(1198602) = 01354 119904(1198603) = 03932 119904(1198604) = 07352Therefore the ordering of alternatives is1198604 ≻ 1198603 ≻ 1198601 ≻ 1198602which is mainly in alignment with the ordering of1198604 ≻ 1198601 ≻1198603 ≻ 1198602 in Yang and Ju [54] It is worth noticing that 120596119860119863used in our approach is derived objectively from assessmentswhile attribute weights 120596lowast = (035 025 040)119879used by Yangand Ju [54] were assigned empirically Differences between120596119860119863 and 120596lowast lead to different ordering of1198601 and1198603 Compar-atively speaking our approach holds more adaptability andobjectivity

In general our proposed MADM approach in this paperexhibits an effective way to tackle complex green contractorselection problems in which subjective attributes weightingvector (such as120596119860119868) and objective attributes weighting vector(such as 120596119860119863) exist To comprehensively utilize these twofundamental facets of attributes weighting information ourcase study and computational experiments suggest that linearcombination method (ie 120596 = 120572120596119860119868 + 120573120596119860119863) is effectiveto reflect effects of 120596119860119868 and 120596119860119863 by configuring parameter 120572or 120573 When there is no specific expert opinions to specify orhard to obtain we suggest that parameter 120572 chooses the valueset of 0 02 05 08 1 Furthermore generalized aggrega-tion operators are suggested to inspect score trajectoriesof alternative contractors along increasing of the operatorsrsquoamplification parameter 120582 If some alternative contractorsrsquoscore values get very close or crossed so that they cannot bedifferentiated from each other significantly we suggest to askthe panel of experts to derive opinions of ideal solutions andutilize our TOPSIS-based order-inducing method to includethose expert opinions as attitudinal decision informationthereby obtaining more rational and robust decision results

7 Conclusions

Aiming at tackling complex decision-making problems ofgreen contractor selection that often hold simultaneouslythe characteristics of decision hesitancy attributes inter-dependency and group complex attitudinal characters wehave developed an effective IVDHF UUBLS-based approach

We utilize IVDHF UUBLS to elicit complicate decision-makersrsquo assessments more objectively and comprehensivelywe employ DEMATEL for addressing attribute interdepen-dency more precisely and we determine group attitudinalcharacters as order-inducing vectors based on ideal attitudesof experts rather than by empirical assignment The mainadvantages of our approach are (i) IVDHF UUBLS offers anadequate way to depict decision hesitancy (ii) the designeddistance measure for IVDHF UUBLS avoids informationdistortion of traditional ones which artificially add mis-matching membership or nonmembership degrees (iii) thedeveloped generalized operators include traditional ones asspecial cases so as to provide flexibility in MADM basedon IVDHF UUBLS (iv) the devised synthesized attributeweighting scheme provides an adaptive way to consider twonecessary aspects attributes interdependency-based weightsand argument-dependent weights (v) the TOPSIS-basedmethod can deduce reasonable order-inducing vectors ratherthan empirical ones and holds generality in other linguisticdecision environments Results of case studies and experi-ments have verified the validity of the proposed approach

When multiple decision organizations are involvedgroup decision-making based on IVDHF UUBLS is animportant research direction Besides inducing preferencesfor decision-making problems of high uncertainty interactivemechanisms should also be studied to extract preferences Inaddition empirical studies of the proposed approach can becarried out to various areas such as investment evaluationproject management and vehicle selection in fleet operations

Appendix

A Basic Notions for IVDHFS and UnbalancedLinguistic Term Set

Definition A1 (see [47]) Letting 119883 be a fixed set then anIVDHFS on 119883 is defined as119863 = ⟨119909 ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (A1)

where ℎ(119909) = ⋃[120583119871120583119880]isinℎ(119909)120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and119892(119909) = ⋃[]119871]119880]isin119892(119909)] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two setsof interval values in [0 1] denoting possible membershipand non-membership degrees of element 119909 isin 119883 to theset 119863 respectively with conditions 120583 ] isin [0 1] and0 le (120583119880)+ + (]119880)+ le 1 and for all 119909 isin 119883 (120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880

For convenience normally 119889 = ℎ 119892 is called an interval-valued dual hesitant fuzzy element (IVDHFE) and 119863 is theset of all IVDHFEs

Definition A2 (see [107]) Suppose 119878 = 119904119895 | 119895 = 0 1 119892 minus1 is a finite and totally ordered discrete linguistic term setwhere 119904119895 represent possible values for a linguistic variableand 119892 is an odd cardinality 119878 is uniformly and symmetricallydistributed if the following two conditions are satisfied (1)

Complexity 25

there exists a unique constant 120582 gt 0 such that 119873119878(119904119894) minus119873119878(119904119895) = 119897(119894 minus 119895) for all 119894 119895 = 0 1 119892 minus 1 (2) Let 119878119877 =119904 | 119904 isin 119878 119904 gt 119878119862 and 119878119871 = 119904 | 119904 isin 119878 119904 lt 119878119862 Let C(119878119877) andC(119878119871) be the cardinality of 119878119877 and 119878119871 then C(119878119877) = C(119878119871)If 119878 is uniformly and symmetrically distributed then 119878 iscalled a balanced linguistic term set Otherwise 119878 is calledan unbalanced linguistic term set

The following 2-tuple fuzzy linguistic representationmodel extends traditional linguistic term set to a continuouscase so as to facilitate computing with linguistic variables

Definition A3 (see [59]) Let 119878 = 1199040 1199041 119904119892minus1 be alinguistic term set and 120573 isin [0 119892] Then a 2-tuple fuzzylinguistic variable expresses the equivalent information to 120573is defined as

Δ [0 119892] 997888rarr 119878 times [minus05 05) (A2)

Δ (120573) = (119904119894 120572) with

119904119894 119894 = 119903119900119906119899119889 (120573)120572 = 120573 minus 119894 120572 isin [minus05 05)

(A3)

Δminus1 (119904119894 120572) = 119894 + 120572 = 120573 (A4)

where 119903119900119906119899119889() is the usual rounding operation and120572 is calledsymbolic translation

To obtain 2-tuple fuzzy linguistic representations ofunbalanced linguistic terms the concept of linguistic hierar-chies ie 119871119867 = ⋃119905 119897(119905 119899(119905)) is used 119897(119905 119899(119905)) is a linguistichierarchy with 119905 indicating the level of hierarchy and 119899(119905)denotes the granularity of the linguistic term set of 119905 Herreraet al [59] defined the following transformation functionsbetween labels from different levels in multigranular linguis-tic information contexts without loss of information

Definition A4 (see [59]) In linguistic hierarchies 119871119867 =⋃119905 119897(119905 119899(119905)) whose linguistic term sets are represented by119878119899(119905) = 119904119899(119905)0 119904119899(119905)119899(119905)minus1 the transformation function froma linguistic label in level 119905 to a label in consecutive level 1199051015840 isdefined as 1198791198651199051199051015840 119897(119905 119899(119905)) 997888rarr 119897(1199051015840 119899(1199051015840)) such that

1198791198651199051199051015840 (119904119899(119905)119894 120572119899(119905))= Δ 1199051015840 (Δminus1119905 (119904119899(119905)119894 120572119899(119905)) (119899 (1199051015840) minus 1)119899 (119905) minus 1 ) (A5)

By use of the above transformation function any 2-tuplelinguistic representation can be transformed into a term in119871119867 Detailed transformation procedures are listed as follows

(1) Representation in linguistic hierarchy To transformthe unbalanced terms of 119878 into the corresponding terms inthe 119871119867 transformation function 120595 is employed to associateeach unbalanced linguistic 2-tuple (119904119894 120572) with its linguistic 2-tuple in 119871119867(119878) ie

120595 119878 997888rarr 119871119867(119878) (A6)

so that 120595(119904119894 120572) = (119904119866(119894)119868(119894) 120582) for forall(119904119894 120572) isin 119878(2) Computational phase Firstly transform (119904119866(119894)119868(119894) 120582) intolinguistic 2-tuples denoted as (119904119899(1199051015840)

1198681015840(119894) 1205821015840) in 119878119899(1199051015840) where

(119904119899(1199051015840)1198681015840(119894)

1205821015840) = 119879119865 (119904119866(119894)119868(119894) 120582)= Δ(Δminus1 (119904119866(119894)119868(119894) 120582) sdot (119899 (1199051015840) minus 1)119866 (119894) minus 1 ) (A7)

Then a computational model is used over 119878119899(1199051015840) with aresult denoted as (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)(3) Retranslation process The result (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)is transformed into the unbalanced term in 119878 by using thetransformation function 120595minus1 ie

120595minus1 119871119867 (119878) 997888rarr 119878 (A8)

so that 120595minus1(119904119899(1199051015840)119903 120582119903) = (119904119903119890119904119906119897119905 120582119903119890119904119906119897119905) isin 119878B Details of Theorem 14

Proof Given a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector for119904119889119895 and 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 120582 gt 0 then(1) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLWA and GIVDHFUUBLWG operators respectively(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLOWA and GIVDHFUUBLOWG operators respectively

(2) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to theIVDHF UUBL weighted average (IVDHFUUBLWA)and IVDHF UUBL weighted geometric (IVDHFUUBLWG)operators respectively where

26 Complexity

119868119881119863119867119865119880119880119861119871119882119860120596 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

120596119895119904119889119895= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus

119899prod119895=1

(1 minus 120583119871119895 )120596119895 1 minus 119899prod119895=1

(1 minus 120583119880119895 )120596119895]] [[

119899prod119895=1

(]119871119895)120596119895 119899prod119895=1

(]119880119895 )120596119895]])

(B1)

119868119881119863119867119865119880119880119861119871119882119866120596 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889119895120596119895

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[

119899prod119895=1

(120583119871119895 )120596119895 119899prod119895=1

(120583119880119895 )120596119895]] [[1 minus

119899prod119895=1

(1 minus ]119871119895)120596119895 1 minus 119899prod119895=1

(1 minus ]119880119895 )120596119895]])

(B2)

(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHA andGIVDHFUUBLHG operators reduce to the IVDHF UUBL

orderedweighted average (IVDHFUUBLOWA) and IVDHFUUBL ordered weighted geometric (IVDHFUUBLOWG)operators respectively where

119868119881119863119867119865119880119880119861119871119874119882119860119908 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

119908119895119904119889120590(119895)= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) )))

119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) )))

]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus

119899prod119895=1

(1 minus 120583119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus 120583119880120590(119895))119908119895]] [[

119899prod119895=1

(]119871120590(119895))119908119895 119899prod119895=1

(]119880120590(119895))119908119895]])

(B3)

119868119881119863119867119865119880119880119861119871119874119882119866119908 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889120590(119895)119908119895

= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[

119899prod119895=1

(120583119871120590(119895))119908119895 119899prod119895=1

(120583119880120590(119895))119908119895]] [[1 minus

119899prod119895=1

(1 minus ]119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus ]119880120590(119895))119908119895]])

(B4)

Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

References

[1] V W Y Tam and C M Tam ldquoWaste reduction throughincentivesA case studyrdquoBuilding Research and Information vol36 no 1 pp 37ndash43 2008

[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

[11] X Qi C Liang and J Zhang ldquoGeneralized cross-entropy basedgroup decision making with unknown expert and attributeweights under interval-valued intuitionistic fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 79 pp 52ndash642015

[12] H Zimmermann Fuzzy Set Theorymdashand Its ApplicationsKluwer Academic Boston Mass USA 2001

[13] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 pp 199ndash249 1975

[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

[16] B T Nuong K-W Kim L Prathumratana et al ldquoSustainabledevelopment in the mining sector and its evaluation usingfuzzy AHP (Analytic Hierarchy Process) approachrdquo GeosystemEngineering vol 14 no 1 pp 43ndash50 2011

[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

[18] E K Zavadskas Z Turskis and J Antucheviciene ldquoSelectinga contractor by using a novel method for multiple attributeanalysis Weighted aggregated sum product assessment withgrey values (WASPAS-G)rdquo Studies in Informatics and Controlvol 24 no 2 pp 141ndash150 2015

[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

[21] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

[25] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009

[26] Z Hatush and M Skitmore ldquoContractor Selection UsingMulticriteria Utility Theory An Additive Modelrdquo Building andEnvironment vol 33 no 2-3 pp 105ndash115 1998

[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

[30] AM Anvuur andMM Kumaraswamy ldquoConceptualmodel ofpartnering and alliancingrdquo Journal of Construction Engineeringand Management vol 133 no 3 pp 225ndash234 2007

[31] A H I Lee H Kang C F Hsu and H Hung ldquoA green supplierselection model for high-tech industryrdquo Expert Systems withApplications vol 36 no 4 pp 7917ndash7927 2009

[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

[33] P Humphreys R McIvor and F Chan ldquoUsing case-basedreasoning to evaluate supplier environmental managementperformancerdquo Expert Systems with Applications vol 25 no 2pp 141ndash153 2003

[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

[45] Bin Zhu Zeshui Xu and Meimei Xia ldquoDual Hesitant FuzzySetsrdquo Journal of Applied Mathematics vol 2012 Article ID879629 13 pages 2012

[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

[51] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[52] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

[55] Z Tao X Liu H Chen and L Zhou ldquoUsing new versionof extended t-norms and s-norms for aggregating intervallinguistic labelsrdquo IEEETransactions on SystemsManCyberneticsSystems vol vol p 15 2016

[56] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[57] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[58] L Martınez M Espinilla J Liu L G Perez and P J SanchezldquoAn evaluation model with unbalanced linguistic informationapplied to olive oil sensory evaluationrdquo Journal of Multiple-Valued Logic and Soft Computing vol 15 no 2-3 pp 229ndash2512009

[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

[62] L Marin A Valls D Isern A Moreno and J M MerigoldquoInduced unbalanced linguistic ordered weighted average andits application in multiperson decision makingrdquo The ScientificWorld Journal vol 2014 2014

[63] A Baykasoglu V Kaplanoglu Z D U Durmusoglu and CSahin ldquoIntegrating fuzzy DEMATEL and fuzzy hierarchicalTOPSIS methods for truck selectionrdquo Expert Systems withApplications vol 40 no 3 pp 899ndash907 2013

[64] S K Patil and R Kant ldquoA hybrid approach based on fuzzyDEMATEL and FMCDM to predict success of knowledge man-agement adoption in supply chainrdquoApplied Soft Computing vol18 pp 126ndash135 2014

[65] S Liyin and Y Hong ldquoImproving environmental performanceby means of empowerment of contractorsrdquo Management ofEnvironmental Quality An International Journal vol 17 no 3pp 242ndash257 2006

[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

[69] Y Kazancoglu I Kazancoglu and M Sagnak ldquoA new holisticconceptual framework for green supply chain management

performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

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24 Complexity

fuzzy linguistic sets introduced by Yang and Ju [54] Tofurther inspect feasibility of our approach we here conductcomparative study on the same case adopted by Yang and Ju[54] We firstly convert their original decision matrix intothe form of IVDHF UUBLS crisp membership and non-membership degrees are equivalently changed to intervalswith same upper and lower limits (eg 02 turns out to be[02 02]) and linguistic variables are rewritten as uncertainlinguistic variables (eg 1199042 turns out to be [11990421199042])

Since the case only got three attributes among whichinterdependency could be not significant under generalassumption weighting vector 120596119860119868 is not calculated Andfor straight comparison order inducing vector is notincluded either Then the argument-dependent attributeweighting vector can be obtained from Eq(29) as 120596119860119863 =(03192 03132 03676)119879 By use of Algorithm Iwith operatorIVDHFUUBLWA (ie GIVDHFUUBLWA in case of 120582 = 1see Appendix B) we have the scores of alternatives 119904(1198601) =03412 119904(1198602) = 01354 119904(1198603) = 03932 119904(1198604) = 07352Therefore the ordering of alternatives is1198604 ≻ 1198603 ≻ 1198601 ≻ 1198602which is mainly in alignment with the ordering of1198604 ≻ 1198601 ≻1198603 ≻ 1198602 in Yang and Ju [54] It is worth noticing that 120596119860119863used in our approach is derived objectively from assessmentswhile attribute weights 120596lowast = (035 025 040)119879used by Yangand Ju [54] were assigned empirically Differences between120596119860119863 and 120596lowast lead to different ordering of1198601 and1198603 Compar-atively speaking our approach holds more adaptability andobjectivity

In general our proposed MADM approach in this paperexhibits an effective way to tackle complex green contractorselection problems in which subjective attributes weightingvector (such as120596119860119868) and objective attributes weighting vector(such as 120596119860119863) exist To comprehensively utilize these twofundamental facets of attributes weighting information ourcase study and computational experiments suggest that linearcombination method (ie 120596 = 120572120596119860119868 + 120573120596119860119863) is effectiveto reflect effects of 120596119860119868 and 120596119860119863 by configuring parameter 120572or 120573 When there is no specific expert opinions to specify orhard to obtain we suggest that parameter 120572 chooses the valueset of 0 02 05 08 1 Furthermore generalized aggrega-tion operators are suggested to inspect score trajectoriesof alternative contractors along increasing of the operatorsrsquoamplification parameter 120582 If some alternative contractorsrsquoscore values get very close or crossed so that they cannot bedifferentiated from each other significantly we suggest to askthe panel of experts to derive opinions of ideal solutions andutilize our TOPSIS-based order-inducing method to includethose expert opinions as attitudinal decision informationthereby obtaining more rational and robust decision results

7 Conclusions

Aiming at tackling complex decision-making problems ofgreen contractor selection that often hold simultaneouslythe characteristics of decision hesitancy attributes inter-dependency and group complex attitudinal characters wehave developed an effective IVDHF UUBLS-based approach

We utilize IVDHF UUBLS to elicit complicate decision-makersrsquo assessments more objectively and comprehensivelywe employ DEMATEL for addressing attribute interdepen-dency more precisely and we determine group attitudinalcharacters as order-inducing vectors based on ideal attitudesof experts rather than by empirical assignment The mainadvantages of our approach are (i) IVDHF UUBLS offers anadequate way to depict decision hesitancy (ii) the designeddistance measure for IVDHF UUBLS avoids informationdistortion of traditional ones which artificially add mis-matching membership or nonmembership degrees (iii) thedeveloped generalized operators include traditional ones asspecial cases so as to provide flexibility in MADM basedon IVDHF UUBLS (iv) the devised synthesized attributeweighting scheme provides an adaptive way to consider twonecessary aspects attributes interdependency-based weightsand argument-dependent weights (v) the TOPSIS-basedmethod can deduce reasonable order-inducing vectors ratherthan empirical ones and holds generality in other linguisticdecision environments Results of case studies and experi-ments have verified the validity of the proposed approach

When multiple decision organizations are involvedgroup decision-making based on IVDHF UUBLS is animportant research direction Besides inducing preferencesfor decision-making problems of high uncertainty interactivemechanisms should also be studied to extract preferences Inaddition empirical studies of the proposed approach can becarried out to various areas such as investment evaluationproject management and vehicle selection in fleet operations

Appendix

A Basic Notions for IVDHFS and UnbalancedLinguistic Term Set

Definition A1 (see [47]) Letting 119883 be a fixed set then anIVDHFS on 119883 is defined as119863 = ⟨119909 ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (A1)

where ℎ(119909) = ⋃[120583119871120583119880]isinℎ(119909)120583 = ⋃[120583119871120583119880]isinℎ(119909)[120583119871 120583119880] and119892(119909) = ⋃[]119871]119880]isin119892(119909)] = ⋃[]119871]119880]isin119892(119909)[]119871 ]119880] are two setsof interval values in [0 1] denoting possible membershipand non-membership degrees of element 119909 isin 119883 to theset 119863 respectively with conditions 120583 ] isin [0 1] and0 le (120583119880)+ + (]119880)+ le 1 and for all 119909 isin 119883 (120583119880)+ isinℎ+(119909) = ⋃[120583119871120583119880]isinℎ(119909)max120583119880 and (]119880)+ isin 119892+(119909) =⋃[]119871]119880]isin119892(119909)max]119880

For convenience normally 119889 = ℎ 119892 is called an interval-valued dual hesitant fuzzy element (IVDHFE) and 119863 is theset of all IVDHFEs

Definition A2 (see [107]) Suppose 119878 = 119904119895 | 119895 = 0 1 119892 minus1 is a finite and totally ordered discrete linguistic term setwhere 119904119895 represent possible values for a linguistic variableand 119892 is an odd cardinality 119878 is uniformly and symmetricallydistributed if the following two conditions are satisfied (1)

Complexity 25

there exists a unique constant 120582 gt 0 such that 119873119878(119904119894) minus119873119878(119904119895) = 119897(119894 minus 119895) for all 119894 119895 = 0 1 119892 minus 1 (2) Let 119878119877 =119904 | 119904 isin 119878 119904 gt 119878119862 and 119878119871 = 119904 | 119904 isin 119878 119904 lt 119878119862 Let C(119878119877) andC(119878119871) be the cardinality of 119878119877 and 119878119871 then C(119878119877) = C(119878119871)If 119878 is uniformly and symmetrically distributed then 119878 iscalled a balanced linguistic term set Otherwise 119878 is calledan unbalanced linguistic term set

The following 2-tuple fuzzy linguistic representationmodel extends traditional linguistic term set to a continuouscase so as to facilitate computing with linguistic variables

Definition A3 (see [59]) Let 119878 = 1199040 1199041 119904119892minus1 be alinguistic term set and 120573 isin [0 119892] Then a 2-tuple fuzzylinguistic variable expresses the equivalent information to 120573is defined as

Δ [0 119892] 997888rarr 119878 times [minus05 05) (A2)

Δ (120573) = (119904119894 120572) with

119904119894 119894 = 119903119900119906119899119889 (120573)120572 = 120573 minus 119894 120572 isin [minus05 05)

(A3)

Δminus1 (119904119894 120572) = 119894 + 120572 = 120573 (A4)

where 119903119900119906119899119889() is the usual rounding operation and120572 is calledsymbolic translation

To obtain 2-tuple fuzzy linguistic representations ofunbalanced linguistic terms the concept of linguistic hierar-chies ie 119871119867 = ⋃119905 119897(119905 119899(119905)) is used 119897(119905 119899(119905)) is a linguistichierarchy with 119905 indicating the level of hierarchy and 119899(119905)denotes the granularity of the linguistic term set of 119905 Herreraet al [59] defined the following transformation functionsbetween labels from different levels in multigranular linguis-tic information contexts without loss of information

Definition A4 (see [59]) In linguistic hierarchies 119871119867 =⋃119905 119897(119905 119899(119905)) whose linguistic term sets are represented by119878119899(119905) = 119904119899(119905)0 119904119899(119905)119899(119905)minus1 the transformation function froma linguistic label in level 119905 to a label in consecutive level 1199051015840 isdefined as 1198791198651199051199051015840 119897(119905 119899(119905)) 997888rarr 119897(1199051015840 119899(1199051015840)) such that

1198791198651199051199051015840 (119904119899(119905)119894 120572119899(119905))= Δ 1199051015840 (Δminus1119905 (119904119899(119905)119894 120572119899(119905)) (119899 (1199051015840) minus 1)119899 (119905) minus 1 ) (A5)

By use of the above transformation function any 2-tuplelinguistic representation can be transformed into a term in119871119867 Detailed transformation procedures are listed as follows

(1) Representation in linguistic hierarchy To transformthe unbalanced terms of 119878 into the corresponding terms inthe 119871119867 transformation function 120595 is employed to associateeach unbalanced linguistic 2-tuple (119904119894 120572) with its linguistic 2-tuple in 119871119867(119878) ie

120595 119878 997888rarr 119871119867(119878) (A6)

so that 120595(119904119894 120572) = (119904119866(119894)119868(119894) 120582) for forall(119904119894 120572) isin 119878(2) Computational phase Firstly transform (119904119866(119894)119868(119894) 120582) intolinguistic 2-tuples denoted as (119904119899(1199051015840)

1198681015840(119894) 1205821015840) in 119878119899(1199051015840) where

(119904119899(1199051015840)1198681015840(119894)

1205821015840) = 119879119865 (119904119866(119894)119868(119894) 120582)= Δ(Δminus1 (119904119866(119894)119868(119894) 120582) sdot (119899 (1199051015840) minus 1)119866 (119894) minus 1 ) (A7)

Then a computational model is used over 119878119899(1199051015840) with aresult denoted as (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)(3) Retranslation process The result (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)is transformed into the unbalanced term in 119878 by using thetransformation function 120595minus1 ie

120595minus1 119871119867 (119878) 997888rarr 119878 (A8)

so that 120595minus1(119904119899(1199051015840)119903 120582119903) = (119904119903119890119904119906119897119905 120582119903119890119904119906119897119905) isin 119878B Details of Theorem 14

Proof Given a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector for119904119889119895 and 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 120582 gt 0 then(1) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLWA and GIVDHFUUBLWG operators respectively(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLOWA and GIVDHFUUBLOWG operators respectively

(2) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to theIVDHF UUBL weighted average (IVDHFUUBLWA)and IVDHF UUBL weighted geometric (IVDHFUUBLWG)operators respectively where

26 Complexity

119868119881119863119867119865119880119880119861119871119882119860120596 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

120596119895119904119889119895= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus

119899prod119895=1

(1 minus 120583119871119895 )120596119895 1 minus 119899prod119895=1

(1 minus 120583119880119895 )120596119895]] [[

119899prod119895=1

(]119871119895)120596119895 119899prod119895=1

(]119880119895 )120596119895]])

(B1)

119868119881119863119867119865119880119880119861119871119882119866120596 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889119895120596119895

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[

119899prod119895=1

(120583119871119895 )120596119895 119899prod119895=1

(120583119880119895 )120596119895]] [[1 minus

119899prod119895=1

(1 minus ]119871119895)120596119895 1 minus 119899prod119895=1

(1 minus ]119880119895 )120596119895]])

(B2)

(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHA andGIVDHFUUBLHG operators reduce to the IVDHF UUBL

orderedweighted average (IVDHFUUBLOWA) and IVDHFUUBL ordered weighted geometric (IVDHFUUBLOWG)operators respectively where

119868119881119863119867119865119880119880119861119871119874119882119860119908 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

119908119895119904119889120590(119895)= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) )))

119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) )))

]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus

119899prod119895=1

(1 minus 120583119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus 120583119880120590(119895))119908119895]] [[

119899prod119895=1

(]119871120590(119895))119908119895 119899prod119895=1

(]119880120590(119895))119908119895]])

(B3)

119868119881119863119867119865119880119880119861119871119874119882119866119908 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889120590(119895)119908119895

= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[

119899prod119895=1

(120583119871120590(119895))119908119895 119899prod119895=1

(120583119880120590(119895))119908119895]] [[1 minus

119899prod119895=1

(1 minus ]119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus ]119880120590(119895))119908119895]])

(B4)

Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

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[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

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[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

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[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

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[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

[21] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

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[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

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[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

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[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

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[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

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[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

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[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

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[64] S K Patil and R Kant ldquoA hybrid approach based on fuzzyDEMATEL and FMCDM to predict success of knowledge man-agement adoption in supply chainrdquoApplied Soft Computing vol18 pp 126ndash135 2014

[65] S Liyin and Y Hong ldquoImproving environmental performanceby means of empowerment of contractorsrdquo Management ofEnvironmental Quality An International Journal vol 17 no 3pp 242ndash257 2006

[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

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performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

Hindawiwwwhindawicom Volume 2018

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Complexity 25

there exists a unique constant 120582 gt 0 such that 119873119878(119904119894) minus119873119878(119904119895) = 119897(119894 minus 119895) for all 119894 119895 = 0 1 119892 minus 1 (2) Let 119878119877 =119904 | 119904 isin 119878 119904 gt 119878119862 and 119878119871 = 119904 | 119904 isin 119878 119904 lt 119878119862 Let C(119878119877) andC(119878119871) be the cardinality of 119878119877 and 119878119871 then C(119878119877) = C(119878119871)If 119878 is uniformly and symmetrically distributed then 119878 iscalled a balanced linguistic term set Otherwise 119878 is calledan unbalanced linguistic term set

The following 2-tuple fuzzy linguistic representationmodel extends traditional linguistic term set to a continuouscase so as to facilitate computing with linguistic variables

Definition A3 (see [59]) Let 119878 = 1199040 1199041 119904119892minus1 be alinguistic term set and 120573 isin [0 119892] Then a 2-tuple fuzzylinguistic variable expresses the equivalent information to 120573is defined as

Δ [0 119892] 997888rarr 119878 times [minus05 05) (A2)

Δ (120573) = (119904119894 120572) with

119904119894 119894 = 119903119900119906119899119889 (120573)120572 = 120573 minus 119894 120572 isin [minus05 05)

(A3)

Δminus1 (119904119894 120572) = 119894 + 120572 = 120573 (A4)

where 119903119900119906119899119889() is the usual rounding operation and120572 is calledsymbolic translation

To obtain 2-tuple fuzzy linguistic representations ofunbalanced linguistic terms the concept of linguistic hierar-chies ie 119871119867 = ⋃119905 119897(119905 119899(119905)) is used 119897(119905 119899(119905)) is a linguistichierarchy with 119905 indicating the level of hierarchy and 119899(119905)denotes the granularity of the linguistic term set of 119905 Herreraet al [59] defined the following transformation functionsbetween labels from different levels in multigranular linguis-tic information contexts without loss of information

Definition A4 (see [59]) In linguistic hierarchies 119871119867 =⋃119905 119897(119905 119899(119905)) whose linguistic term sets are represented by119878119899(119905) = 119904119899(119905)0 119904119899(119905)119899(119905)minus1 the transformation function froma linguistic label in level 119905 to a label in consecutive level 1199051015840 isdefined as 1198791198651199051199051015840 119897(119905 119899(119905)) 997888rarr 119897(1199051015840 119899(1199051015840)) such that

1198791198651199051199051015840 (119904119899(119905)119894 120572119899(119905))= Δ 1199051015840 (Δminus1119905 (119904119899(119905)119894 120572119899(119905)) (119899 (1199051015840) minus 1)119899 (119905) minus 1 ) (A5)

By use of the above transformation function any 2-tuplelinguistic representation can be transformed into a term in119871119867 Detailed transformation procedures are listed as follows

(1) Representation in linguistic hierarchy To transformthe unbalanced terms of 119878 into the corresponding terms inthe 119871119867 transformation function 120595 is employed to associateeach unbalanced linguistic 2-tuple (119904119894 120572) with its linguistic 2-tuple in 119871119867(119878) ie

120595 119878 997888rarr 119871119867(119878) (A6)

so that 120595(119904119894 120572) = (119904119866(119894)119868(119894) 120582) for forall(119904119894 120572) isin 119878(2) Computational phase Firstly transform (119904119866(119894)119868(119894) 120582) intolinguistic 2-tuples denoted as (119904119899(1199051015840)

1198681015840(119894) 1205821015840) in 119878119899(1199051015840) where

(119904119899(1199051015840)1198681015840(119894)

1205821015840) = 119879119865 (119904119866(119894)119868(119894) 120582)= Δ(Δminus1 (119904119866(119894)119868(119894) 120582) sdot (119899 (1199051015840) minus 1)119866 (119894) minus 1 ) (A7)

Then a computational model is used over 119878119899(1199051015840) with aresult denoted as (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)(3) Retranslation process The result (119904119899(1199051015840)119903 120582119903) isin 119878119899(1199051015840)is transformed into the unbalanced term in 119878 by using thetransformation function 120595minus1 ie

120595minus1 119871119867 (119878) 997888rarr 119878 (A8)

so that 120595minus1(119904119899(1199051015840)119903 120582119903) = (119904119903119890119904119906119897119905 120582119903119890119904119906119897119905) isin 119878B Details of Theorem 14

Proof Given a collection of IVDHF UUBLNs 119904119889119895 =(119904120599119895 ℎ119895 119892119895) 120596 = (1205961 1205962 120596119899)119879 is the weighting vector for119904119889119895 and 119908 = (1199081 1199082 119908119899)119879 is the aggregation-associatedweighting vector 120582 gt 0 then(1) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLWA and GIVDHFUUBLWG operators respectively(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to the GIVDHFU-UBLOWA and GIVDHFUUBLOWG operators respectively

(2) If 120582 = 1(1) If 119908 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHAand GIVDHFUUBLHG operators reduce to theIVDHF UUBL weighted average (IVDHFUUBLWA)and IVDHF UUBL weighted geometric (IVDHFUUBLWG)operators respectively where

26 Complexity

119868119881119863119867119865119880119880119861119871119882119860120596 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

120596119895119904119889119895= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus

119899prod119895=1

(1 minus 120583119871119895 )120596119895 1 minus 119899prod119895=1

(1 minus 120583119880119895 )120596119895]] [[

119899prod119895=1

(]119871119895)120596119895 119899prod119895=1

(]119880119895 )120596119895]])

(B1)

119868119881119863119867119865119880119880119861119871119882119866120596 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889119895120596119895

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[

119899prod119895=1

(120583119871119895 )120596119895 119899prod119895=1

(120583119880119895 )120596119895]] [[1 minus

119899prod119895=1

(1 minus ]119871119895)120596119895 1 minus 119899prod119895=1

(1 minus ]119880119895 )120596119895]])

(B2)

(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHA andGIVDHFUUBLHG operators reduce to the IVDHF UUBL

orderedweighted average (IVDHFUUBLOWA) and IVDHFUUBL ordered weighted geometric (IVDHFUUBLOWG)operators respectively where

119868119881119863119867119865119880119880119861119871119874119882119860119908 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

119908119895119904119889120590(119895)= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) )))

119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) )))

]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus

119899prod119895=1

(1 minus 120583119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus 120583119880120590(119895))119908119895]] [[

119899prod119895=1

(]119871120590(119895))119908119895 119899prod119895=1

(]119880120590(119895))119908119895]])

(B3)

119868119881119863119867119865119880119880119861119871119874119882119866119908 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889120590(119895)119908119895

= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[

119899prod119895=1

(120583119871120590(119895))119908119895 119899prod119895=1

(120583119880120590(119895))119908119895]] [[1 minus

119899prod119895=1

(1 minus ]119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus ]119880120590(119895))119908119895]])

(B4)

Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

References

[1] V W Y Tam and C M Tam ldquoWaste reduction throughincentivesA case studyrdquoBuilding Research and Information vol36 no 1 pp 37ndash43 2008

[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

[11] X Qi C Liang and J Zhang ldquoGeneralized cross-entropy basedgroup decision making with unknown expert and attributeweights under interval-valued intuitionistic fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 79 pp 52ndash642015

[12] H Zimmermann Fuzzy Set Theorymdashand Its ApplicationsKluwer Academic Boston Mass USA 2001

[13] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 pp 199ndash249 1975

[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

[16] B T Nuong K-W Kim L Prathumratana et al ldquoSustainabledevelopment in the mining sector and its evaluation usingfuzzy AHP (Analytic Hierarchy Process) approachrdquo GeosystemEngineering vol 14 no 1 pp 43ndash50 2011

[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

[18] E K Zavadskas Z Turskis and J Antucheviciene ldquoSelectinga contractor by using a novel method for multiple attributeanalysis Weighted aggregated sum product assessment withgrey values (WASPAS-G)rdquo Studies in Informatics and Controlvol 24 no 2 pp 141ndash150 2015

[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

[21] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

[25] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009

[26] Z Hatush and M Skitmore ldquoContractor Selection UsingMulticriteria Utility Theory An Additive Modelrdquo Building andEnvironment vol 33 no 2-3 pp 105ndash115 1998

[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

[30] AM Anvuur andMM Kumaraswamy ldquoConceptualmodel ofpartnering and alliancingrdquo Journal of Construction Engineeringand Management vol 133 no 3 pp 225ndash234 2007

[31] A H I Lee H Kang C F Hsu and H Hung ldquoA green supplierselection model for high-tech industryrdquo Expert Systems withApplications vol 36 no 4 pp 7917ndash7927 2009

[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

[33] P Humphreys R McIvor and F Chan ldquoUsing case-basedreasoning to evaluate supplier environmental managementperformancerdquo Expert Systems with Applications vol 25 no 2pp 141ndash153 2003

[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

[45] Bin Zhu Zeshui Xu and Meimei Xia ldquoDual Hesitant FuzzySetsrdquo Journal of Applied Mathematics vol 2012 Article ID879629 13 pages 2012

[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

[51] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[52] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

[55] Z Tao X Liu H Chen and L Zhou ldquoUsing new versionof extended t-norms and s-norms for aggregating intervallinguistic labelsrdquo IEEETransactions on SystemsManCyberneticsSystems vol vol p 15 2016

[56] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[57] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[58] L Martınez M Espinilla J Liu L G Perez and P J SanchezldquoAn evaluation model with unbalanced linguistic informationapplied to olive oil sensory evaluationrdquo Journal of Multiple-Valued Logic and Soft Computing vol 15 no 2-3 pp 229ndash2512009

[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

[62] L Marin A Valls D Isern A Moreno and J M MerigoldquoInduced unbalanced linguistic ordered weighted average andits application in multiperson decision makingrdquo The ScientificWorld Journal vol 2014 2014

[63] A Baykasoglu V Kaplanoglu Z D U Durmusoglu and CSahin ldquoIntegrating fuzzy DEMATEL and fuzzy hierarchicalTOPSIS methods for truck selectionrdquo Expert Systems withApplications vol 40 no 3 pp 899ndash907 2013

[64] S K Patil and R Kant ldquoA hybrid approach based on fuzzyDEMATEL and FMCDM to predict success of knowledge man-agement adoption in supply chainrdquoApplied Soft Computing vol18 pp 126ndash135 2014

[65] S Liyin and Y Hong ldquoImproving environmental performanceby means of empowerment of contractorsrdquo Management ofEnvironmental Quality An International Journal vol 17 no 3pp 242ndash257 2006

[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

[69] Y Kazancoglu I Kazancoglu and M Sagnak ldquoA new holisticconceptual framework for green supply chain management

performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

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26 Complexity

119868119881119863119867119865119880119880119861119871119882119860120596 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

120596119895119904119889119895= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

119904sum119899119895=1 120596119895(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[1 minus

119899prod119895=1

(1 minus 120583119871119895 )120596119895 1 minus 119899prod119895=1

(1 minus 120583119880119895 )120596119895]] [[

119899prod119895=1

(]119871119895)120596119895 119899prod119895=1

(]119880119895 )120596119895]])

(B1)

119868119881119863119867119865119880119880119861119871119882119866120596 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889119895120596119895

= ⋃(119904120599119895 ℎ119895119892119895)isin119904119889119895

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572119895 ))))

120596119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573119895 ))))

120596119895]

⋃[120583119871119895 120583119880119895 ]isinℎ119895[]

119871119895 ]119880119895 ]isin119892119895

[[

119899prod119895=1

(120583119871119895 )120596119895 119899prod119895=1

(120583119880119895 )120596119895]] [[1 minus

119899prod119895=1

(1 minus ]119871119895)120596119895 1 minus 119899prod119895=1

(1 minus ]119880119895 )120596119895]])

(B2)

(2) If 120596 = (1119899 1119899 1119899)119879 then GIVDHFUUBLHA andGIVDHFUUBLHG operators reduce to the IVDHF UUBL

orderedweighted average (IVDHFUUBLOWA) and IVDHFUUBL ordered weighted geometric (IVDHFUUBLOWG)operators respectively where

119868119881119863119867119865119880119880119861119871119874119882119860119908 (1199041198891 1199041198892 119904119889119899) = 119899⨁119895=1

119908119895119904119889120590(119895)= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) )))

119904sum119899119895=1 119908119895Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) )))

]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[1 minus

119899prod119895=1

(1 minus 120583119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus 120583119880120590(119895))119908119895]] [[

119899prod119895=1

(]119871120590(119895))119908119895 119899prod119895=1

(]119880120590(119895))119908119895]])

(B3)

119868119881119863119867119865119880119880119861119871119874119882119866119908 (1199041198891 1199041198892 119904119889119899) = 119899⨂119895=1

119904119889120590(119895)119908119895

= ⋃(119904120599120590(119895) ℎ120590(119895)119892120590(119895))isin119904119889120590(119895)

([119904prod119899119895=1(Δ

minus11199050(11987911986511990511989511199050(120595(119904120572120590(119895) ))))

119908119895 119904prod119899119895=1(Δ

minus11199050(11987911986511990511989521199050(120595(119904120573120590(119895) ))))

119908119895]

⋃[120583119871120590(119895)120583119880120590(119895)]isinℎ120590(119895)[]119871120590(119895) ]

119880120590(119895)]isin119892120590(119895)

[[

119899prod119895=1

(120583119871120590(119895))119908119895 119899prod119895=1

(120583119880120590(119895))119908119895]] [[1 minus

119899prod119895=1

(1 minus ]119871120590(119895))119908119895 1 minus 119899prod119895=1

(1 minus ]119880120590(119895))119908119895]])

(B4)

Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

References

[1] V W Y Tam and C M Tam ldquoWaste reduction throughincentivesA case studyrdquoBuilding Research and Information vol36 no 1 pp 37ndash43 2008

[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

[11] X Qi C Liang and J Zhang ldquoGeneralized cross-entropy basedgroup decision making with unknown expert and attributeweights under interval-valued intuitionistic fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 79 pp 52ndash642015

[12] H Zimmermann Fuzzy Set Theorymdashand Its ApplicationsKluwer Academic Boston Mass USA 2001

[13] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 pp 199ndash249 1975

[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

[16] B T Nuong K-W Kim L Prathumratana et al ldquoSustainabledevelopment in the mining sector and its evaluation usingfuzzy AHP (Analytic Hierarchy Process) approachrdquo GeosystemEngineering vol 14 no 1 pp 43ndash50 2011

[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

[18] E K Zavadskas Z Turskis and J Antucheviciene ldquoSelectinga contractor by using a novel method for multiple attributeanalysis Weighted aggregated sum product assessment withgrey values (WASPAS-G)rdquo Studies in Informatics and Controlvol 24 no 2 pp 141ndash150 2015

[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

[21] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

[25] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009

[26] Z Hatush and M Skitmore ldquoContractor Selection UsingMulticriteria Utility Theory An Additive Modelrdquo Building andEnvironment vol 33 no 2-3 pp 105ndash115 1998

[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

[30] AM Anvuur andMM Kumaraswamy ldquoConceptualmodel ofpartnering and alliancingrdquo Journal of Construction Engineeringand Management vol 133 no 3 pp 225ndash234 2007

[31] A H I Lee H Kang C F Hsu and H Hung ldquoA green supplierselection model for high-tech industryrdquo Expert Systems withApplications vol 36 no 4 pp 7917ndash7927 2009

[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

[33] P Humphreys R McIvor and F Chan ldquoUsing case-basedreasoning to evaluate supplier environmental managementperformancerdquo Expert Systems with Applications vol 25 no 2pp 141ndash153 2003

[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

[45] Bin Zhu Zeshui Xu and Meimei Xia ldquoDual Hesitant FuzzySetsrdquo Journal of Applied Mathematics vol 2012 Article ID879629 13 pages 2012

[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

[51] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[52] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

[55] Z Tao X Liu H Chen and L Zhou ldquoUsing new versionof extended t-norms and s-norms for aggregating intervallinguistic labelsrdquo IEEETransactions on SystemsManCyberneticsSystems vol vol p 15 2016

[56] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[57] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[58] L Martınez M Espinilla J Liu L G Perez and P J SanchezldquoAn evaluation model with unbalanced linguistic informationapplied to olive oil sensory evaluationrdquo Journal of Multiple-Valued Logic and Soft Computing vol 15 no 2-3 pp 229ndash2512009

[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

[62] L Marin A Valls D Isern A Moreno and J M MerigoldquoInduced unbalanced linguistic ordered weighted average andits application in multiperson decision makingrdquo The ScientificWorld Journal vol 2014 2014

[63] A Baykasoglu V Kaplanoglu Z D U Durmusoglu and CSahin ldquoIntegrating fuzzy DEMATEL and fuzzy hierarchicalTOPSIS methods for truck selectionrdquo Expert Systems withApplications vol 40 no 3 pp 899ndash907 2013

[64] S K Patil and R Kant ldquoA hybrid approach based on fuzzyDEMATEL and FMCDM to predict success of knowledge man-agement adoption in supply chainrdquoApplied Soft Computing vol18 pp 126ndash135 2014

[65] S Liyin and Y Hong ldquoImproving environmental performanceby means of empowerment of contractorsrdquo Management ofEnvironmental Quality An International Journal vol 17 no 3pp 242ndash257 2006

[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

[69] Y Kazancoglu I Kazancoglu and M Sagnak ldquoA new holisticconceptual framework for green supply chain management

performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

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Complexity 27

Overall we can conclude the relationship between all theseproposed aggregation operators as shown inTheorem 14

C Computation Steps of Algorithm I

Step 1 Determine argument-dependent weighting vector120596119860119863

Utilize the programming model (M-1) in Section 41 bycalculating (29) we get

120596119860119863 = (01383 0171 01597 01059 0102 00902 01202 01127)119879 (C1)

Step 2 Determine attributes interdependency-based weight-ing vector 120596119860119868

After processing Step 4 in Algorithm II the influencingand influenced degrees of each attribute can be obtained as

shown in Table 2 Then according to (33) and (34) weightingvector based on attributes interdependency is determinedas

120596119860119868 = (01248 01149 01158 0121 01379 01255 0135 01251)119879 (C2)

Step 3 Obtain synthesized attributes weighting vectorTo comprehensively consider attribute weighting infor-

mation synthesized weighting vector is computed by (19) inwhich 120572 and 120573 are set as 05

120596= (01316 0143 01377 01135 012 01078 01276 01188)119879 (C3)

Step 4 Check order inducing requirementsAfter consulting the expert team if there is no group

attitudinal characters (ie order inducing vectors) then goto Step 5 otherwise go to Step 6

Step 5 Derive overall IVDHF UUBLNs 119903119895 of each alternative119909119895 (119895 = 1 2 3)Here we choose GIVDHFUUBLHA operator in Defini-

tion 12 and assign 120582 = 2 The associated position weightingvector is assigned as 119908 = (0 0 015 025 025 025 01 0)119879which is acquired by commonly used fuzzy semantic quan-titative operator with parameters (a b) set as (03 08) [99]Thenwe get the values of 1199031 1199032 and 1199033 Taking 1199033 for examplewe have

1199033 = ([11990435754 11990448345] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [036594 048336] [036594 048336] [039856 051409] [039856 051409] [029622 039832] [030123 04036] [029622 039832] [030123 04036])

(C4)

Later on 1199031 1199032 and 1199033 will be fed into Step 8 to calculate scoresand accuracy degrees of each contractor

Step 6 Derive order-inducing vectors 120576According to the TOPSIS-based method described in

Algorithm III decision-makers are required to put forwardboth IVDHF UUBL ideal alternatives and IVDHF UUBLnegative ideal alternatives As listed in Table 3 the idealsolution and negative ideal solution are collected based onwhich the order-inducing vectors 120576 are derived by sorting thecoefficients generated out of (37)

Step 7 Utilize induced generalized aggregation operators toget the overall IVDHFUBBLNs 119903119895(119895 = 1 2 119898) of eachalternative 119909119895

After order-inducing vector 120576 is obtained I-GIVDHFUUBLHA operator defined in Definition 13 ischosen for information aggregation 120582 and position weightsare set same as in Step 5 and then we can obtain 1199031 1199032 and1199033 Taking 1199033 as an example we have

1199033 = ([11990440762 1199045877] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [031061 043001] [031061 043548] [034999 046638] [034999 047123] [039075 049203] [039075 049203] [040918 051064] [040918 051064])

(C5)

Step 8 Calculate scores and accuracy degrees of each solu-tion

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

References

[1] V W Y Tam and C M Tam ldquoWaste reduction throughincentivesA case studyrdquoBuilding Research and Information vol36 no 1 pp 37ndash43 2008

[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

[11] X Qi C Liang and J Zhang ldquoGeneralized cross-entropy basedgroup decision making with unknown expert and attributeweights under interval-valued intuitionistic fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 79 pp 52ndash642015

[12] H Zimmermann Fuzzy Set Theorymdashand Its ApplicationsKluwer Academic Boston Mass USA 2001

[13] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 pp 199ndash249 1975

[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

[16] B T Nuong K-W Kim L Prathumratana et al ldquoSustainabledevelopment in the mining sector and its evaluation usingfuzzy AHP (Analytic Hierarchy Process) approachrdquo GeosystemEngineering vol 14 no 1 pp 43ndash50 2011

[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

[18] E K Zavadskas Z Turskis and J Antucheviciene ldquoSelectinga contractor by using a novel method for multiple attributeanalysis Weighted aggregated sum product assessment withgrey values (WASPAS-G)rdquo Studies in Informatics and Controlvol 24 no 2 pp 141ndash150 2015

[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

[21] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

[25] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009

[26] Z Hatush and M Skitmore ldquoContractor Selection UsingMulticriteria Utility Theory An Additive Modelrdquo Building andEnvironment vol 33 no 2-3 pp 105ndash115 1998

[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

[30] AM Anvuur andMM Kumaraswamy ldquoConceptualmodel ofpartnering and alliancingrdquo Journal of Construction Engineeringand Management vol 133 no 3 pp 225ndash234 2007

[31] A H I Lee H Kang C F Hsu and H Hung ldquoA green supplierselection model for high-tech industryrdquo Expert Systems withApplications vol 36 no 4 pp 7917ndash7927 2009

[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

[33] P Humphreys R McIvor and F Chan ldquoUsing case-basedreasoning to evaluate supplier environmental managementperformancerdquo Expert Systems with Applications vol 25 no 2pp 141ndash153 2003

[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

[45] Bin Zhu Zeshui Xu and Meimei Xia ldquoDual Hesitant FuzzySetsrdquo Journal of Applied Mathematics vol 2012 Article ID879629 13 pages 2012

[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

[51] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[52] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

[55] Z Tao X Liu H Chen and L Zhou ldquoUsing new versionof extended t-norms and s-norms for aggregating intervallinguistic labelsrdquo IEEETransactions on SystemsManCyberneticsSystems vol vol p 15 2016

[56] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[57] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[58] L Martınez M Espinilla J Liu L G Perez and P J SanchezldquoAn evaluation model with unbalanced linguistic informationapplied to olive oil sensory evaluationrdquo Journal of Multiple-Valued Logic and Soft Computing vol 15 no 2-3 pp 229ndash2512009

[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

[62] L Marin A Valls D Isern A Moreno and J M MerigoldquoInduced unbalanced linguistic ordered weighted average andits application in multiperson decision makingrdquo The ScientificWorld Journal vol 2014 2014

[63] A Baykasoglu V Kaplanoglu Z D U Durmusoglu and CSahin ldquoIntegrating fuzzy DEMATEL and fuzzy hierarchicalTOPSIS methods for truck selectionrdquo Expert Systems withApplications vol 40 no 3 pp 899ndash907 2013

[64] S K Patil and R Kant ldquoA hybrid approach based on fuzzyDEMATEL and FMCDM to predict success of knowledge man-agement adoption in supply chainrdquoApplied Soft Computing vol18 pp 126ndash135 2014

[65] S Liyin and Y Hong ldquoImproving environmental performanceby means of empowerment of contractorsrdquo Management ofEnvironmental Quality An International Journal vol 17 no 3pp 242ndash257 2006

[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

[69] Y Kazancoglu I Kazancoglu and M Sagnak ldquoA new holisticconceptual framework for green supply chain management

performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

Hindawiwwwhindawicom Volume 2018

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Mathematical Problems in Engineering

Applied MathematicsJournal of

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

28 Complexity

Now we can utilize the score function and accuracyfunction defined in Definition 4 to calculate scores 119878(119903119895)and accuracy degrees 119875(119903119895) of each 119903119895 For each 119903119895 gener-ated from Step 5 we have 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = minus0146119878119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 2157 and 119878119866119868119881119863119867119865119880119880119861119871119867119860(1199033) =0381 And for each 119903119895 generated from Step 7 we have119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199031) = 0136 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199032) = 1962and 119878119868minus119866119868119881119863119867119865119880119880119861119871119867119860(1199033) = minus0299Step 9 Generate final ranking results

According to the scores obtained from Step 5 and thecomparative rules defined in Definition 5 the ranking orderis 1199092 ≻ 1199093 ≻ 1199091 indicating that 1199092 is the most appropriatesolution Regarding the scores obtained from Step 7 theranking order is 1199092 ≻ 1199091 ≻ 1199093 indicating that 1199092 is stillthe most appropriate solution and solution 1199091 is better thansolution 1199093 Ranking results are then collected in Table 4 forclarity

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to greatly thank joint-support by theNational Natural Science Foundation of China (nos 7170118171771075 and 71331002) the Social Science Foundation ofMinistry of Education of China (no 16YJC630094) and theNatural Science Foundation of Zhejiang Province of China(LQ17G010002 and LY18G010010)

References

[1] V W Y Tam and C M Tam ldquoWaste reduction throughincentivesA case studyrdquoBuilding Research and Information vol36 no 1 pp 37ndash43 2008

[2] Z Majdalani M Ajam and T Mezher ldquoSustainability in theconstruction industry A Lebanese case studyrdquo ConstructionInnovation vol 6 no 1 pp 33ndash46 2006

[3] V W Y Tam L Y Shen I W H Fung and J Y WangldquoControlling construction waste by implementing governmen-tal ordinances in Hong Kongrdquo Construction Innovation vol 7no 2 pp 149ndash166 2007

[4] P O Akadiri P O Olomolaiye and E A Chinyio ldquoMulti-criteria evaluation model for the selection of sustainable mate-rials for building projectsrdquo Automation in Construction vol 30pp 113ndash125 2013

[5] M Safa A Shahi C T Haas et al ldquoCompetitive intelligence(CI) for evaluation of construction contractorsrdquoAutomation inConstruction vol 59 article no 1862 pp 149ndash157 2015

[6] G Y Qi L Y Shen S X Zeng and O J Jorge ldquoThe drivers forcontractorsrsquo green innovation An industry perspectiverdquo Journalof Cleaner Production vol 18 no 14 pp 1358ndash1365 2010

[7] A Rostami and C F Oduoza ldquoKey risks in constructionprojects in Italy Contractorsrsquo perspectiverdquo Engineering Con-struction and Architectural Management vol 24 no 3 pp 451ndash462 2017

[8] R Liang Z Sheng and X Wang ldquoMethods Dealing withComplexity in Selecting Joint Venture Contractors for Large-Scale Infrastructure Projectsrdquo Complexity vol 2018 Article ID8705134 14 pages 2018

[9] M Safa C T Haas K W Hipel and J Gray ldquoFront EndPlanning Tool (FEPT) Based on an Electronic ProcessManage-mentrdquo Journal of Construction Engineering and Project Manage-ment vol 3 no 2 pp 1ndash12 2013

[10] A Nieto-Morote and F Ruz-Vila ldquoA fuzzy multi-criteriadecision-making model for construction contractor prequali-ficationrdquo Automation in Construction vol 25 pp 8ndash19 2012

[11] X Qi C Liang and J Zhang ldquoGeneralized cross-entropy basedgroup decision making with unknown expert and attributeweights under interval-valued intuitionistic fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 79 pp 52ndash642015

[12] H Zimmermann Fuzzy Set Theorymdashand Its ApplicationsKluwer Academic Boston Mass USA 2001

[13] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 pp 199ndash249 1975

[14] Z Xu ldquoA method for multiple attribute decision makingwith incomplete weight information in linguistic settingrdquoKnowledge-Based Systems vol 20 no 8 pp 719ndash725 2007

[15] Y Ju A Wang and X Liu ldquoEvaluating emergency responsecapacity by fuzzy AHP and 2-tuple fuzzy linguistic approachrdquoExpert Systems with Applications vol 39 no 8 pp 6972ndash69812012

[16] B T Nuong K-W Kim L Prathumratana et al ldquoSustainabledevelopment in the mining sector and its evaluation usingfuzzy AHP (Analytic Hierarchy Process) approachrdquo GeosystemEngineering vol 14 no 1 pp 43ndash50 2011

[17] S Kusi-Sarpong C Bai J Sarkis and X Wang ldquoGreen supplychain practices evaluation in the mining industry using a jointrough sets and fuzzy TOPSIS methodologyrdquo Resources Policyvol 46 pp 86ndash100 2015

[18] E K Zavadskas Z Turskis and J Antucheviciene ldquoSelectinga contractor by using a novel method for multiple attributeanalysis Weighted aggregated sum product assessment withgrey values (WASPAS-G)rdquo Studies in Informatics and Controlvol 24 no 2 pp 141ndash150 2015

[19] B Vahdani SMMousavi H HashemiMMousakhani andRTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[20] H M Alhumaidi ldquoConstruction contractors ranking methodusing multiple decision-makers and multiattribute fuzzyweighted averagerdquo Journal of Construction Engineering andManagement vol 141 no 4 2015

[21] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[22] Z XuHesitant Fuzzy SetsTheory vol 314 of Studies in Fuzzinessand Soft Computing Springer Berlin Germany 2014

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

[25] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009

[26] Z Hatush and M Skitmore ldquoContractor Selection UsingMulticriteria Utility Theory An Additive Modelrdquo Building andEnvironment vol 33 no 2-3 pp 105ndash115 1998

[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

[30] AM Anvuur andMM Kumaraswamy ldquoConceptualmodel ofpartnering and alliancingrdquo Journal of Construction Engineeringand Management vol 133 no 3 pp 225ndash234 2007

[31] A H I Lee H Kang C F Hsu and H Hung ldquoA green supplierselection model for high-tech industryrdquo Expert Systems withApplications vol 36 no 4 pp 7917ndash7927 2009

[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

[33] P Humphreys R McIvor and F Chan ldquoUsing case-basedreasoning to evaluate supplier environmental managementperformancerdquo Expert Systems with Applications vol 25 no 2pp 141ndash153 2003

[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

[45] Bin Zhu Zeshui Xu and Meimei Xia ldquoDual Hesitant FuzzySetsrdquo Journal of Applied Mathematics vol 2012 Article ID879629 13 pages 2012

[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

[51] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[52] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

[55] Z Tao X Liu H Chen and L Zhou ldquoUsing new versionof extended t-norms and s-norms for aggregating intervallinguistic labelsrdquo IEEETransactions on SystemsManCyberneticsSystems vol vol p 15 2016

[56] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[57] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[58] L Martınez M Espinilla J Liu L G Perez and P J SanchezldquoAn evaluation model with unbalanced linguistic informationapplied to olive oil sensory evaluationrdquo Journal of Multiple-Valued Logic and Soft Computing vol 15 no 2-3 pp 229ndash2512009

[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

[62] L Marin A Valls D Isern A Moreno and J M MerigoldquoInduced unbalanced linguistic ordered weighted average andits application in multiperson decision makingrdquo The ScientificWorld Journal vol 2014 2014

[63] A Baykasoglu V Kaplanoglu Z D U Durmusoglu and CSahin ldquoIntegrating fuzzy DEMATEL and fuzzy hierarchicalTOPSIS methods for truck selectionrdquo Expert Systems withApplications vol 40 no 3 pp 899ndash907 2013

[64] S K Patil and R Kant ldquoA hybrid approach based on fuzzyDEMATEL and FMCDM to predict success of knowledge man-agement adoption in supply chainrdquoApplied Soft Computing vol18 pp 126ndash135 2014

[65] S Liyin and Y Hong ldquoImproving environmental performanceby means of empowerment of contractorsrdquo Management ofEnvironmental Quality An International Journal vol 17 no 3pp 242ndash257 2006

[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

[69] Y Kazancoglu I Kazancoglu and M Sagnak ldquoA new holisticconceptual framework for green supply chain management

performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 29

[23] M P Borujeni and H Gitinavard ldquoEvaluating the sustain-able mining contractor selection problems An imprecise lastaggregation preference selection index methodrdquo Journal ofSustainable Mining vol 16 no 4 pp 207ndash218 2017

[24] P S-W Fong and S K-Y Choi ldquoFinal contractor selection usingthe analytical hierarchy processrdquoConstructionManagement andEconomics vol 18 no 5 pp 547ndash557 2000

[25] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009

[26] Z Hatush and M Skitmore ldquoContractor Selection UsingMulticriteria Utility Theory An Additive Modelrdquo Building andEnvironment vol 33 no 2-3 pp 105ndash115 1998

[27] B-G Hwang X Zhao and G S Yu ldquoRisk identification andallocation in underground rail construction joint venturescontractorsrsquo perspectiverdquo Journal of Civil Engineering andMan-agement vol 22 no 6 pp 758ndash767 2016

[28] L Y Shen W Lu Q Shen and H Li ldquoA computer-aideddecision support system for assessing a contractorrsquos competi-tivenessrdquoAutomation in Construction vol 12 no 5 pp 577ndash5872003

[29] B G Hwang and H B Ng ldquoProject network managementrisks and contributors from the viewpoint of contractors andsub-contractorsrdquo Technological and Economic Development ofEconomy vol 22 no 4 pp 631ndash648 2016

[30] AM Anvuur andMM Kumaraswamy ldquoConceptualmodel ofpartnering and alliancingrdquo Journal of Construction Engineeringand Management vol 133 no 3 pp 225ndash234 2007

[31] A H I Lee H Kang C F Hsu and H Hung ldquoA green supplierselection model for high-tech industryrdquo Expert Systems withApplications vol 36 no 4 pp 7917ndash7927 2009

[32] C Y Chiou C W Hsu and W Y Hwang ldquoComparativeinvestigation on green supplier selection of the AmericanJapanese and Taiwanese Electronics Industry in Chinardquo inProceedings of the 2008 IEEE International Conference on Indus-trial Engineering and Engineering Management IEEM 2008 pp1909ndash1914 Singapore December 2008

[33] P Humphreys R McIvor and F Chan ldquoUsing case-basedreasoning to evaluate supplier environmental managementperformancerdquo Expert Systems with Applications vol 25 no 2pp 141ndash153 2003

[34] P Humphreys A McCloskey R McIvor L Maguire and CGlackin ldquoEmploying dynamic fuzzy membership functions toassess environmental performance in the supplier selectionprocessrdquo International Journal of Production Research vol 44no 12 pp 2379ndash2419 2006

[35] R J Lin ldquoUsing fuzzy DEMATEL to evaluate the green supplychain management practicesrdquo Journal of Cleaner Productionvol 40 pp 32ndash39 2013

[36] YYang andLWu ldquoExtensionMethod forGreen Supplier Selec-tionrdquo in Proceedings of the 2008 4th International Conference onWireless Communications Networking and Mobile Computing(WiCOM) pp 1ndash4 Dalian China October 2008

[37] B Liu and H-J Liu ldquoA research on supplier assessment indicessystem of green purchasingrdquo in Proceedings of the InternationalConference onMeasuring Technology andMechatronics Automa-tion ICMTMA 2010 pp 314ndash317 China March 2010

[38] R Lin X Zhao HWang andGWei ldquoHesitant fuzzy linguisticaggregation operators and their application tomultiple attribute

decision makingrdquo Journal of Intelligent amp Fuzzy Systems Appli-cations in Engineering and Technology vol 27 no 1 pp 49ndash632014

[39] GWei ldquoInterval-valueddual hesitant fuzzy uncertain linguisticaggregation operators in multiple attribute decision makingrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 33 no 3 pp 1881ndash1893 2017

[40] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy multi-ple criteria decision-making techniques and applicationsmdashtwodecades review from 1994 to 2014rdquo Expert Systems with Appli-cations vol 42 no 8 pp 4126ndash4148 2015

[41] F Meng X Chen and Q Zhang ldquoGeneralized hesitant fuzzygeneralized Shapley-Choquet integral operators and their appli-cation in decision makingrdquo International Journal of FuzzySystems vol 16 no 3 pp 400ndash410 2014

[42] S CevikOnar BOztaysi andCKahraman ldquoStrategicDecisionSelection Using Hesitant fuzzy TOPSIS and Interval Type-2Fuzzy AHP A case studyrdquo International Journal of Computa-tional Intelligence Systems vol 7 no 5 pp 1002ndash1021 2014

[43] C TanW Yi and X Chen ldquoHesitant fuzzy Hamacher aggrega-tion operators for multicriteria decision makingrdquo Applied SoftComputing vol 26 pp 325ndash349 2015

[44] X Zhang and Z Xu ldquoHesitant fuzzy QUALIFLEX approachwith a signed distance-based comparison method for multiplecriteria decision analysisrdquo Expert Systems with Applications vol42 no 2 pp 873ndash884 2015

[45] Bin Zhu Zeshui Xu and Meimei Xia ldquoDual Hesitant FuzzySetsrdquo Journal of Applied Mathematics vol 2012 Article ID879629 13 pages 2012

[46] B Zhu and Z Xu ldquoSome results for dual hesitant fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 26 no 4 pp 1657ndash1668 2014

[47] Y Ju X Liu and S Yang ldquoInterval-valued dual hesitantfuzzy aggregation operators and their applications to multipleattribute decision makingrdquo Journal of Intelligent amp Fuzzy Sys-tems Applications in Engineering and Technology vol 27 no 3pp 1203ndash1218 2014

[48] W Zhang X Li and Y Ju ldquoSome Aggregation Operators Basedon Einstein Operations under Interval-Valued Dual HesitantFuzzy Setting andTheir ApplicationrdquoMathematical Problems inEngineering vol 2014 Article ID 958927 21 pages 2014

[49] B Farhadinia ldquoStudy on division and subtraction operationsfor hesitant fuzzy sets interval-valued hesitant fuzzy sets andtypical dual hesitant fuzzy setsrdquo Journal of Intelligent amp FuzzySystems Applications in Engineering and Technology vol 28 no3 pp 1393ndash1402 2015

[50] S Yang and Y Ju ldquoA GRA method for investment alternativeselection under dual hesitant fuzzy environment with incom-plete weight informationrdquo Journal of IntelligentampFuzzy SystemsApplications in Engineering and Technology vol 28 no 4 pp1533ndash1543 2015

[51] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[52] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[53] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

[55] Z Tao X Liu H Chen and L Zhou ldquoUsing new versionof extended t-norms and s-norms for aggregating intervallinguistic labelsrdquo IEEETransactions on SystemsManCyberneticsSystems vol vol p 15 2016

[56] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[57] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[58] L Martınez M Espinilla J Liu L G Perez and P J SanchezldquoAn evaluation model with unbalanced linguistic informationapplied to olive oil sensory evaluationrdquo Journal of Multiple-Valued Logic and Soft Computing vol 15 no 2-3 pp 229ndash2512009

[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

[62] L Marin A Valls D Isern A Moreno and J M MerigoldquoInduced unbalanced linguistic ordered weighted average andits application in multiperson decision makingrdquo The ScientificWorld Journal vol 2014 2014

[63] A Baykasoglu V Kaplanoglu Z D U Durmusoglu and CSahin ldquoIntegrating fuzzy DEMATEL and fuzzy hierarchicalTOPSIS methods for truck selectionrdquo Expert Systems withApplications vol 40 no 3 pp 899ndash907 2013

[64] S K Patil and R Kant ldquoA hybrid approach based on fuzzyDEMATEL and FMCDM to predict success of knowledge man-agement adoption in supply chainrdquoApplied Soft Computing vol18 pp 126ndash135 2014

[65] S Liyin and Y Hong ldquoImproving environmental performanceby means of empowerment of contractorsrdquo Management ofEnvironmental Quality An International Journal vol 17 no 3pp 242ndash257 2006

[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

[69] Y Kazancoglu I Kazancoglu and M Sagnak ldquoA new holisticconceptual framework for green supply chain management

performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

30 Complexity

[54] S Yang and Y Ju ldquoDual hesitant fuzzy linguistic aggregationoperators and their applications to multi-attribute decisionmakingrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 27 no 4 pp 1935ndash1947 2014

[55] Z Tao X Liu H Chen and L Zhou ldquoUsing new versionof extended t-norms and s-norms for aggregating intervallinguistic labelsrdquo IEEETransactions on SystemsManCyberneticsSystems vol vol p 15 2016

[56] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[57] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[58] L Martınez M Espinilla J Liu L G Perez and P J SanchezldquoAn evaluation model with unbalanced linguistic informationapplied to olive oil sensory evaluationrdquo Journal of Multiple-Valued Logic and Soft Computing vol 15 no 2-3 pp 229ndash2512009

[59] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzylinguistic methodology to deal with unbalanced linguistic termsetsrdquo IEEETransactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[60] D Joshi and S Kumar ldquoInterval-valued intuitionistic hesitantfuzzy Choquet integral based TOPSIS method formulti-criteriagroup decision makingrdquo European Journal of OperationalResearch vol 248 no 1 pp 183ndash191 2016

[61] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquoComputersamp Industrial Engineering vol 60 no 1 pp 66ndash76 2011

[62] L Marin A Valls D Isern A Moreno and J M MerigoldquoInduced unbalanced linguistic ordered weighted average andits application in multiperson decision makingrdquo The ScientificWorld Journal vol 2014 2014

[63] A Baykasoglu V Kaplanoglu Z D U Durmusoglu and CSahin ldquoIntegrating fuzzy DEMATEL and fuzzy hierarchicalTOPSIS methods for truck selectionrdquo Expert Systems withApplications vol 40 no 3 pp 899ndash907 2013

[64] S K Patil and R Kant ldquoA hybrid approach based on fuzzyDEMATEL and FMCDM to predict success of knowledge man-agement adoption in supply chainrdquoApplied Soft Computing vol18 pp 126ndash135 2014

[65] S Liyin and Y Hong ldquoImproving environmental performanceby means of empowerment of contractorsrdquo Management ofEnvironmental Quality An International Journal vol 17 no 3pp 242ndash257 2006

[66] ZWuA TW Yu and L Shen ldquoInvestigating the determinantsof contractorrsquos construction and demolition wastemanagementbehavior in Mainland Chinardquo Waste Management vol 60 pp290ndash300 2017

[67] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[68] Y Tan L Shen and H Yao ldquoSustainable construction practiceand contractorsrsquo competitiveness A preliminary studyrdquoHabitatInternational vol 35 no 2 pp 225ndash230 2011

[69] Y Kazancoglu I Kazancoglu and M Sagnak ldquoA new holisticconceptual framework for green supply chain management

performance assessment based on circular economyrdquo Journal ofCleaner Production vol 195 pp 1282ndash1299 2018

[70] A Nureize and J Watada ldquoMulti-attribute decision makingin contractor selection under hybrid uncertaintyrdquo Journal ofAdvancedComputational Intelligence and Intelligent Informaticsvol 15 no 4 pp 465ndash472 2011

[71] J Sarkis L M Meade and A R Presley ldquoIncorporatingsustainability into contractor evaluation and team formation inthe built environmentrdquo Journal of Cleaner Production vol 31pp 40ndash53 2012

[72] N Romeli F M Halil F Ismail and A S Shukor ldquoEconomicChallenges in Joint Venture Infrastructure Projects TowardsContractorrsquos Quality of Liferdquo Procedia - Social and BehavioralSciences vol 234 pp 19ndash27 2016

[73] R K Mavi M Goh and N Zarbakhshnia ldquoSustainable third-party reverse logistic provider selection with fuzzy SWARA andfuzzy MOORA in plastic industryrdquoThe International Journal ofAdvanced Manufacturing Technology vol 91 no 5-8 pp 2401ndash2418 2017

[74] Z-Y Zhao C Tang X Zhang and M Skitmore ldquoAgglomer-ation and Competitive Position of Contractors in the Interna-tional Construction Sectorrdquo Journal of Construction Engineeringand Management vol 143 no 6 2017

[75] E U Olugu K Y Wong and A M Shaharoun ldquoDevelopmentof key performance measures for the automobile green supplychainrdquo Resources Conservation amp Recycling vol 55 no 6 pp567ndash579 2011

[76] Q Zhu and J Sarkis ldquoAn inter-sectoral comparison of greensupply chain management in China drivers and practicesrdquoJournal of Cleaner Production vol 14 no 5 pp 472ndash486 2006

[77] K Govindan S Rajendran J Sarkis and P Murugesan ldquoMulticriteria decision making approaches for green supplier eval-uation and selection a literature reviewrdquo Journal of CleanerProduction vol 98 pp 66ndash83 2015

[78] O Feyzioglu and G Buyukozkan ldquoEvaluation of Green Suppli-ers Considering Decision Criteria Dependenciesrdquo in MultipleCriteria Decision Making for Sustainable Energy and Trans-portation Systems vol 634 of Lecture Notes in Economics andMathematical Systems pp 145ndash154 Springer Berlin HeidelbergBerlin Heidelberg 2010

[79] P D Rwelamila A A Talukhaba and A B Ngowi ldquoProjectprocurement systems in the attainment of sustainable construc-tionrdquo Sustainable Development vol 8 no 1 pp 39ndash50 2000

[80] H Fergusson and D A Langford ldquoStrategies for managingenvironmental issues in construction organizationsrdquo Engineer-ing Construction and Architectural Management vol 13 no 2pp 171ndash185 2006

[81] S Sharma and H Vredenburg ldquoProactive corporate environ-mental strategy and the development of competitively valuableorganizational capabilitiesrdquo Strategic Management Journal vol19 no 8 pp 729ndash753 1998

[82] C M Tam V W Y Tam L Y Tam and D Drew ldquoMappingapproach for examining waste management on constructionsitesrdquo Journal of Construction Engineering andManagement vol130 no 4 pp 472ndash481 2004

[83] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[84] CQ Tan andXHChen ldquoIntuitionistic fuzzyChoquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 31

[85] P P Wakker Additive representations of preferences Theoryand Decision Library Series C Game Theory MathematicalProgramming and Operations Research Kluwer AcademicPublishers Group Dordrecht 1989

[86] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[87] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[88] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEETransactions on SystemsMan andCyberneticsPart B Cybernetics vol 29 no 2 pp 141ndash150 1999

[89] JMMerigo andMCasanovas ldquoInduced aggregation operatorsin the Euclidean distance and its application in financialdecision makingrdquo Expert Systems with Applications vol 38 no6 pp 7603ndash7608 2011

[90] Z Xu and M Xia ldquoInduced generalized intuitionistic fuzzyoperatorsrdquoKnowledge-Based Systems vol 24 no 2 pp 197ndash2092011

[91] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 8 pp 21ndash26 1997

[92] C T Chen P F Pai and W Z Hung ldquoAn integratedmethodology using linguistic PROMETHEE and maximumdeviation method for third-party logistics supplier selectionrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 438ndash451 2010

[93] Z S Xu ldquoApproaches tomultiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[94] Z S Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[95] Y Ju S Yang and X Liu ldquoSome new dual hesitant fuzzyaggregation operators based on Choquet integral and theirapplications to multiple attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems Applications in Engineering andTechnology vol 27 no 6 pp 2857ndash2868 2014

[96] D Dalalah M Hayajneh and F Batieha ldquoA fuzzy multi-criteriadecision making model for supplier selectionrdquo Expert Systemswith Applications vol 38 no 7 pp 8384ndash8391 2011

[97] H Liu J X You Ch Lu andY Z Chen ldquoEvaluatinghealth-carewaste treatment technologies using a hybrid multi-criteria deci-sion making modelrdquo Renewable amp Sustainable Energy Reviewsvol 41 pp 932ndash942 2015

[98] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[99] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computers amp Mathematics with Applica-tions vol 9 no 1 pp 149ndash184 1983

[100] Z Zhang ldquoGeneralized Atanassovrsquos intuitionistic fuzzy powergeometric operators and their application to multiple attributegroup decision makingrdquo Information Fusion vol 14 no 4 pp460ndash486 2013

[101] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[102] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003

[103] J M Merigo and M Casanovas ldquoInduced and heavy aggre-gation operators with distance measuresrdquo Journal of SystemsEngineering and Electronics vol 21 no 3 pp 431ndash439 2010

[104] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[105] Jun-Ling Zhang and Xiao-Wen Qi ldquoInduced Interval-ValuedIntuitionistic Fuzzy Hybrid Aggregation Operators with TOP-SIS Order-Inducing Variablesrdquo Journal of Applied Mathematicsvol 2012 Article ID 245732 24 pages 2012

[106] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[107] Y Dong C-C Li and F Herrera ldquoConnecting the numericalscale model to the unbalanced linguistic term setsrdquo in Proceed-ings of the 2014 IEEE International Conference on Fuzzy SystemsFUZZ-IEEE 2014 pp 455ndash462 China July 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom