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Fuzzy Nonlinear Programming with Applications in Decision Making

Journal of Applied Mathematics

Guest Editors: Tin-Chih Toly Chen, Deng-Feng Li, T. Warren Liao, and Yi-Chi Wang

Fuzzy Nonlinear Programming with Applicationsin Decision Making

Journal of Applied Mathematics

Fuzzy Nonlinear Programming with Applicationsin Decision Making

Guest Editors: Tin-Chih Toly Chen, Deng-Feng Li,T. Warren Liao, and Yi-Chi Wang

Copyright © 2014 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in “Journal of Applied Mathematics.” All articles are open access articles distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the originalwork is properly cited.

Editorial Board

Saeid Abbasbandy, IranMina B. Abd-El-Malek, EgyptMohamed A. Abdou, EgyptSubhas Abel, IndiaMostafa Adimy, FranceCarlos J. S. Alves, PortugalMohamad Alwash, USAIgor Andrianov, GermanySabri Arik, TurkeyFrancis T.K. Au, Hong KongOlivier Bahn, CanadaRoberto Barrio, SpainAlfredo Bellen, ItalyJafar Biazar, IranHester Bijl, The NetherlandsAnjan Biswas, Saudi ArabiaS. P.A. Bordas, USAJames Robert Buchanan, USAAlberto Cabada, SpainXiao Chuan Cai, USAJinde Cao, ChinaAlexandre Carvalho, BrazilSong Cen, ChinaQ. S. Chang, ChinaTai-Ping Chang, TaiwanShih-sen Chang, ChinaRushan Chen, ChinaXinfu Chen, USAKe Chen, UKEric Cheng, Hong KongFrancisco Chiclana, UKJ.-T. Chien, TaiwanC. S. Chien, TaiwanHan H. Choi, Republic of KoreaTin-Tai Chow, ChinaM. S. H. Chowdhury, MalaysiaCarlos Conca, ChileVitor Costa, PortugalLivija Cveticanin, SerbiaEric de Sturler, USAOrazio Descalzi, ChileKai Diethelm, GermanyVit Dolejsi, Czech RepublicBo-Qing Dong, ChinaMagdy A. Ezzat, Egypt

Meng Fan, ChinaYa Ping Fang, ChinaAntonio J. M. Ferreira, PortugalMichel Fliess, FranceM. A. Fontelos, SpainHuijun Gao, ChinaB. J. Geurts, The NetherlandsJamshid Ghaboussi, USAPablo González-Vera, SpainLaurent Gosse, ItalyK. S. Govinder, South AfricaJose L. Gracia, SpainYuantong Gu, AustraliaZhihong GUAN, ChinaNicola Guglielmi, ItalyF. G. Guimarães, BrazilVijay Gupta, IndiaBo Han, ChinaMaoan Han, ChinaPierre Hansen, CanadaFerenc Hartung, HungaryXiaoqiao He, Hong KongLuis Javier Herrera, SpainJ. Hoenderkamp, The NetherlandsYing Hu, FranceNing Hu, JapanZhilong L. Huang, ChinaKazufumi Ito, USATakeshi Iwamoto, JapanGeorge Jaiani, GeorgiaZhongxiao Jia, ChinaTarun Kant, IndiaIdo Kanter, IsraelAbdul Hamid Kara, South AfricaHamid Reza Karimi, NorwayJae-Wook Kim, UKJong H. Kim, Republic of KoreaKazutake Komori, JapanFanrong Kong, USAVadim . Krysko, RussiaJin L. Kuang, SingaporeMiroslaw Lachowicz, PolandHak-Keung Lam, UKTak-Wah Lam, Hong KongPGL Leach, Cyprus

Yongkun Li, ChinaWan-Tong Li, ChinaJ. Liang, ChinaChing-Jong Liao, TaiwanChong Lin, ChinaMingzhu Liu, ChinaChein-Shan Liu, TaiwanKang Liu, USAYansheng Liu, ChinaFawang Liu, AustraliaShutian Liu, ChinaZhijun Liu, ChinaJ. López-Gómez, SpainShiping Lu, ChinaGert Lube, GermanyNazim I. Mahmudov, TurkeyOluwole D. Makinde, South AfricaF. J. Marcellán, SpainG. Mart́ın-Herrán, SpainNicola Mastronardi, ItalyM. McAleer, The NetherlandsStephane Metens, FranceMichael Meylan, AustraliaAlain Miranville, FranceRam N. Mohapatra, USAJaime E. M. Rivera, BrazilJavier Murillo, SpainRoberto Natalini, ItalySrinivasan Natesan, IndiaJiri Nedoma, Czech RepublicJianlei Niu, Hong KongRoger Ohayon, FranceJavier Oliver, SpainDonal O’Regan, IrelandMartin Ostoja-Starzewski, USATurgut Öziş, TurkeyClaudio Padra, ArgentinaReinaldo M. Palhares, BrazilFrancesco Pellicano, ItalyJuan Manuel Peña, SpainRicardo Perera, SpainMalgorzata Peszynska, USAJames F. Peters, CanadaMark A. Petersen, South AfricaMiodrag Petkovic, Serbia

Vu Ngoc Phat, VietnamAndrew Pickering, SpainHector Pomares, SpainMaurizio Porfiri, USAMario Primicerio, ItalyM. Rafei, The NetherlandsRoberto Renò, ItalyJacek Rokicki, PolandDirk Roose, BelgiumCarla Roque, PortugalDebasish Roy, IndiaSamir H. Saker, EgyptMarcelo A. Savi, BrazilWolfgang Schmidt, GermanyEckart Schnack, GermanyMehmet Sezer, TurkeyN. Shahzad, Saudi ArabiaFatemeh Shakeri, IranJian Hua Shen, ChinaHui-Shen Shen, ChinaF. Simões, PortugalTheodore E. Simos, Greece

A.-M. A. Soliman, EgyptXinyu Song, ChinaQiankun Song, ChinaYuri N. Sotskov, BelarusPeter J. C. Spreij, The NetherlandsNiclas Strömberg, SwedenRay KL Su, Hong KongJitao Sun, ChinaWenyu Sun, ChinaXianHua Tang, ChinaAlexander Timokha, NorwayMariano Torrisi, ItalyJung-Fa Tsai, TaiwanCh Tsitouras, GreeceK. Vajravelu, USAAlvaro Valencia, ChileErik Van Vleck, USAEzio Venturino, ItalyJesus Vigo-Aguiar, SpainM. N. Vrahatis, GreeceBaolin Wang, ChinaMingxin Wang, China

Qing-WenWang, ChinaGuangchen Wang, ChinaJunjie Wei, ChinaLi Weili, ChinaMartin Weiser, GermanyFrank Werner, GermanyShanhe Wu, ChinaDongmei Xiao, ChinaGongnan Xie, ChinaYuesheng Xu, USASuh-Yuh Yang, TaiwanBo Yu, ChinaJinyun Yuan, BrazilAlejandro Zarzo, SpainGuisheng Zhai, JapanJianming Zhan, ChinaZhihua Zhang, ChinaJingxin Zhang, AustraliaShan Zhao, USAChongbin Zhao, AustraliaRenat Zhdanov, USAHongping Zhu, China

Contents

Fuzzy Nonlinear Programming with Applications in Decision Making, Tin-Chih Toly Chen,Deng-Feng Li, T. Warren Liao, and Yi-Chi WangVolume 2014, Article ID 509157, 2 pages

Multiple Attribute Decision Making Based on Hesitant Fuzzy Einstein Geometric AggregationOperators, Xiaoqiang Zhou and Qingguo LiVolume 2014, Article ID 745617, 14 pages

Multiproject Resources Allocation Model under Fuzzy Random Environment and Its Application toIndustrial Equipment Installation Engineering, Jun Gang, Jiuping Xu, and Yinfeng XuVolume 2013, Article ID 818731, 19 pages

Optimality Condition andWolfe Duality for Invex Interval-Valued Nonlinear Programming Problems,Jianke ZhangVolume 2013, Article ID 641345, 11 pages

Genetic Algorithm Optimization for Determining Fuzzy Measures from Fuzzy Data, Chen Li,Gong Zeng-tai, and Duan GangVolume 2013, Article ID 542153, 11 pages

A Biobjective Fuzzy Integer-Nonlinear Programming Approach for Creating an IntelligentLocation-Aware Service, Yu-Cheng Lin and Toly ChenVolume 2013, Article ID 423415, 11 pages

Correlation Measures of Dual Hesitant Fuzzy Sets, Lei Wang, Mingfang Ni, and Lei ZhuVolume 2013, Article ID 593739, 12 pages

Stationary Points–I: One-Dimensional p-Fuzzy Dynamical Systems, Joao de Deus M. Silva,Jefferson Leite, Rodney C. Bassanezi, and Moiseis S. CecconelloVolume 2013, Article ID 495864, 11 pages

Generalized Fuzzy Bonferroni Harmonic Mean Operators andTheir Applications in Group DecisionMaking, Jin Han Park and Eun Jin ParkVolume 2013, Article ID 604029, 14 pages

Parametric Extension of the Most Preferred OWAOperator and Its Application in Search Engine’sRank, Xiuzhi Sang and Xinwang LiuVolume 2013, Article ID 273758, 10 pages

A Fuzzy Nonlinear Programming Approach for Optimizing the Performance of a Four-ObjectiveFluctuation Smoothing Rule in a Wafer Fabrication Factory, Horng-Ren Tsai and Toly ChenVolume 2013, Article ID 720607, 15 pages

EditorialFuzzy Nonlinear Programming with Applications inDecision Making

Tin-Chih Toly Chen,1 Deng-Feng Li,2 T. Warren Liao,3 and Yi-Chi Wang1

1 Department of Industrial Engineering and Systems Management, Feng Chia University, Taichung 407, Taiwan2 School of Management, Fuzhou University, Fujian 350108, China3Department of Mechanical and Industrial Engineering, Louisiana State University, Baton Rouge, LA 70803, USA

Correspondence should be addressed to Tin-Chih Toly Chen; [email protected]

Received 14 November 2013; Accepted 14 November 2013; Published 23 January 2014

Copyright © 2014 Tin-Chih Toly Chen et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Fuzzy set theory provides an intuitive and computationallysimple way to deal with uncertain and ambiguous properties.Decisions represented in terms of fuzzy sets also offer flexi-bility in their implementations. On the other hand, nonlinearprogramming relaxes the strict assumptions and constraintsin linear programming and hence is more practicable tohandle many decision making problems, which are betterrepresented in the form of nonlinear programming models.Combining fuzzy set theory with nonlinear programmingenables it to address the issue of uncertain parameters; theresulting fuzzy nonlinear programming problem involvingfuzzy parameters can be viewed as an even more practicableapproach than the conventional nonfuzzy one. In light ofadvanced computing systems, fuzzy nonlinear programmingbecomes one of the most promising approaches to solvepractical application problems. The purpose of this specialissue is to provide recent advances in developing fuzzy non-linear programming methods and their applications to prac-ticable and flexible decision making. The target audiencesare researchers in fuzzy mathematics, operations research,informationmanagement, and system engineering, as well aspracticing managers/engineers. After a strict review process,ten articles from researchers around the world were finallyaccepted. A brief summary of each is described below.

J. d. D. M. Silva et al. discussed the properties of p-fuzzy dynamical systems that are variational systems of whichdynamic behaviors are regulated by a Mamdani-type fuzzysystem. They used a case study to illustrate a 1-dimensionalp-fuzzy dynamical system and presented some theorems on

the conditions of existence and uniqueness of stationarypoints.

H.-R. Tsai and T. Chen proposed a fuzzy nonlinear pro-gramming (FNLP) approach for optimizing the schedulingperformance of a four-factor fluctuation smoothing rulein a wafer fabrication factory. The proposed methodologyconsidered the uncertainty in the remaining cycle time of ajob and optimized a fuzzy four-factor fluctuation-smoothingrule to sequence the jobs in front of each machine. The fuzzyfour-factor fluctuation-smoothing rule had five adjustableparameters, the optimization of which resulted in a FNLPproblem.

Most preferred ordered weighted average (MP-OWA)operators are a new kind of neat OWAoperators in the aggre-gation operator families. It considers the preferences of allalternatives across the criteria and provides unique aggrega-tion characteristics in decision making. X. Sang and X. Liuestablished the parametric form of the MP-OWA operatorto deal with uncertain preference information, including themost commonly used maximum, minimum, and averageaggregation operators. A special form of the parametric MP-OWA operator with the power function was also proposed intheir study.

The Bonferroni mean (BM) operator is an importantaggregation technique which reflects the correlation of aggre-gated arguments. J. H. Park andE. J. Park studied the desirableproperties of the fuzzy Bonferroni harmonic mean (FBHM)operator and the fuzzy ordered Bonferroni harmonic mean(FOBHM) operator. To consider the correlation of any three

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014, Article ID 509157, 2 pageshttp://dx.doi.org/10.1155/2014/509157

http://dx.doi.org/10.1155/2014/509157

2 Journal of Applied Mathematics

aggregated arguments, J. H. Park and E. J. Park developedthe generalized fuzzy weighted Bonferroni harmonic mean(GFWBHM) operator and the generalized fuzzy orderedweighted Bonferroni harmonic mean (GFOWBHM) oper-ator. Based on these new operators, they proposed a newapproach to multiple-attribute group decision making andillustrated the new approach with a practical example.

J. Gang et al. studied a multiproject resource allocationproblem in a bilevel organization. To solve the problem, abilevel multiproject resource allocation model with fuzzy orrandom variables was established, in which the opinions ofdecision makers from two levels were considered. On theupper level, the company manager aims to allocate the com-pany’s resources to multiple projects to minimize the totalcosts. On the lower level, each project manager attempts toschedule their resource-constrained projects, withminimiza-tion of the project duration as the main objective. To searchfor the optimal solution, a hybrid approach of adaptive par-ticle swarm optimization, adaptive hybrid genetic algorithm,and fuzzy random simulation was proposed.

Y.-C. Lin and T. Chen created an intelligent location-aware service, in which a timely service was recommended tothe user without changing the user’s pace. To the user, therewere two goals to achieve: one is to reach the service locationjust in time and the other is to get to the destination as soonas possible. To optimize the two objectives at the same timeand to consider the uncertainty in the dynamic environment,a biobjective fuzzy integer-nonlinear programming problemwas formulated and solved.

L. Wang et al. defined the correlation measures of dualhesitant fuzzy sets (DHFSs) and discussed their properties.A direct transfer algorithm for complex matrix synthesis wasalso proposed that helps to construct an equivalent correla-tion matrix for clustering DHFSs. L. Wang et al. also provedthat the direct transfer algorithm is equivalent to the transferclosure algorithm, but its asymptotic time complexity andspace complexity are superior to the latter.

J. Zhang extended the concepts of preinvex and invexto the interval-valued functions. Under the assumption ofinvexity, the Karush-Kuhn-Tucker optimality of sufficientand necessary conditions was also proved for interval-valuednonlinear programming problems. J. Zhang also proved theWolfe duality theorem for invex interval-valued nonlinearprogramming problems, based on the concept of having noduality gaps in weak and strong senses.

Fuzzy measures and fuzzy integrals have been success-fully used in many real applications. However, the determi-nation of fuzzy measures is still a challenging task. L. Chen,Z.-T. Gong, and G. Duan proposed a more generalized typeof fuzzy measures by means of genetic algorithm and theChoquet integral. In this paper, they firstly defined the 𝜎-𝜆rule then defined and characterized the Choquet integrals ofinterval-valued functions and fuzzy functions based on the𝜎-𝜆 rule.

X. Zhou and Q. Li defined an accuracy function of hesi-tant fuzzy elements (HFEs) and developed a new method tocompare two HFEs. Then, based on Einstein operators, theygave some new operational laws on HFEs and some desirableproperties of these operations. Several new hesitant fuzzy

aggregation operators, including the hesitant fuzzy Einsteinweighted geometric (HFEWG

𝜀) operator and the hesitant

fuzzy Einstein ordered weighted geometric (HFEOWG𝜀)

operator, were also developed as the extensions of theweighted geometric operator and the ordered weighted geo-metric (OWG) operator with hesitant fuzzy information,respectively.

Acknowledgment

We sincerely thank all the authors and reviewers for theirvaluable contributions to this special issue.

Tin-Chih Toly ChenDeng-Feng Li

T. Warren LiaoYi-Chi Wang

Research ArticleMultiple Attribute Decision Making Based onHesitant Fuzzy Einstein Geometric Aggregation Operators

Xiaoqiang Zhou1,2 and Qingguo Li1

1 College of Mathematics and Econometrics, Hunan University, Changsha 410082, China2 College of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, China

Correspondence should be addressed to Qingguo Li; [email protected]

Received 25 June 2013; Accepted 12 November 2013; Published 16 January 2014

Academic Editor: Yi-Chi Wang

Copyright © 2014 X. Zhou and Q. Li. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We first define an accuracy function of hesitant fuzzy elements (HFEs) and develop a new method to compare two HFEs. Then,based on Einstein operators, we give some new operational laws on HFEs and some desirable properties of these operations. Wealso develop several new hesitant fuzzy aggregation operators, including the hesitant fuzzy Einstein weighted geometric (HFEWG

𝜀)

operator and the hesitant fuzzy Einstein ordered weighted geometric (HFEWG𝜀) operator, which are the extensions of the

weighted geometric operator and the ordered weighted geometric (OWG) operator with hesitant fuzzy information, respectively.Furthermore, we establish the connections between the proposed and the existing hesitant fuzzy aggregation operators and discussvarious properties of the proposed operators. Finally, we apply the HFEWG

𝜀operator to solve the hesitant fuzzy decision making

problems.

1. Introduction

Atanassov [1, 2] introduced the concept of intuitionistic fuzzyset (IFS) characterized by a membership function and a non-membership function. It is more suitable to deal with fuzzi-ness and uncertainty than the ordinary fuzzy set proposedby Zadeh [3] characterized by one membership function.Information aggregation is an important research topic inmany applications such as fuzzy logic systems and multiat-tribute decisionmaking as discussed byChen andHwang [4].Research on aggregation operators with intuitionistic fuzzyinformation has received increasing attention as shown inthe literature. Xu [5] developed some basic arithmetic aggre-gation operators based on intuitionistic fuzzy values (IFVs),such as the intuitionistic fuzzy weighted averaging operatorand intuitionistic fuzzy ordered weighted averaging operator,while Xu and Yager [6] presented some basic geometricaggregation operators for aggregating IFVs, including theintuitionistic fuzzy weighted geometric operator and intu-itionistic fuzzy ordered weighted geometric operator. Basedon these basic aggregation operators proposed in [6] and [5],

many generalized intuitionistic fuzzy aggregation operatorshave been investigated [5–30]. Recently, Torra and Narukawa[31] and Torra [32] proposed the hesitant fuzzy set (HFS),which is another generalization form of fuzzy set. The char-acteristic of HFS is that it allows membership degree to havea set of possible values.Therefore, HFS is a very useful tool inthe situationswhere there are somedifficulties in determiningthe membership of an element to a set. Lately, research onaggregation methods and multiple attribute decision makingtheories under hesitant fuzzy environment is very active,and a lot of results have been obtained for hesitant fuzzyinformation [33–43]. For example, Xia et al. [38] developedsome confidence induced aggregation operators for hesitantfuzzy information. Xia et al. [37] gave several series of hesitantfuzzy aggregation operators with the help of quasiarithmeticmeans. Wei [35] explored several hesitant fuzzy prioritizedaggregation operators and applied them to hesitant fuzzydecision making problems. Zhu et al. [43] investigated thegeometric Bonferroni mean combining the Bonferroni meanand the geometric mean under hesitant fuzzy environment.Xia and Xu [36] presented some hesitant fuzzy operational

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014, Article ID 745617, 14 pageshttp://dx.doi.org/10.1155/2014/745617

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laws based on the relationship between the HFEs and theIFVs. They also proposed a series of aggregation operators,such as hesitant fuzzy weighted geometric (HFWG) operatorand hesitant fuzzy ordered weighted geometric (HFOWG)operator. Furthermore, they applied the proposed aggrega-tion operators to solve the multiple attribute decisionmakingproblems.

Note that all aggregation operators introduced previouslyare based on the algebraic product and algebraic sum of IFVs(orHFEs) to carry out the combination process. However, thealgebraic operations include algebraic product and algebraicsum, which are not the unique operations that can be used toperform the intersection and union.There aremany instancesof various t-norms and t-conorms families which can bechosen tomodel the corresponding intersections and unions,among which Einstein product and Einstein sum are goodalternatives for they typically give the same smooth approxi-mation as algebraic product and algebraic sum, respectively.For intuitionistic fuzzy information,Wang and Liu [10, 11, 44]and Wei and Zhao [30] developed some new intuitionisticfuzzy aggregation operators with the help of Einstein oper-ations. For hesitant fuzzy information, however, it seems thatin the literature there is little investigation on aggregationtechniques using the Einstein operations to aggregate hesitantfuzzy information. Therefore, it is necessary to develop somehesitant fuzzy information aggregation operators based onEinstein operations.

The remainder of this paper is structured as follows.In Section 2, we briefly review some basic concepts andoperations related to IFS andHFS. we also define an accuracyfunction of HFEs to distinguish the two HFEs having thesame score values, based on which we give the new com-parison laws on HFEs. In Section 3, we present some newoperations for HFEs and discuss some basic properties of theproposed operations. In Section 4, we develop some novelhesitant fuzzy geometric aggregation operators with the helpof Einstein operations, such as theHFEWG

𝜀operator and the

HFEOWG𝜀operator, and we further study various properties

of these operators. Section 5 gives an approach to solve themultiple attribute hesitant fuzzy decision making problemsbased on the HFEOWG

𝜀operator. Finally, Section 6 con-

cludes the paper.

2. Preliminaries

In this section, we briefly introduce Einstein operations andsome notions of IFS and HFS. Meantime, we define anaccuracy function of HFEs and redefine the comparison lawsbetween two HFEs.

2.1. Einstein Operations. Since the appearance of fuzzy settheory, the set theoretical operators have played an importantrole and received more and more attention. It is well knownthat the t-norms and t-conorms are the general conceptsincluding all types of the specific operators, and they satisfythe requirements of the conjunction and disjunction opera-tors, respectively. There are various t-norms and t-conormsfamilies that can be used to perform the corresponding inter-sections and unions. Einstein sum ⊕

𝜀and Einstein product

⊗𝜀are examples of t-conorms and t-norms, respectively.They

are called Einstein operations and defined as [45]

𝑥⊗𝜀𝑦 =

𝑥 ⋅ 𝑦

1 + (1 − 𝑥) ⋅ (1 − 𝑦)

, 𝑥⊗𝜀𝑦 =

𝑥 + 𝑦

1 + 𝑥 ⋅ 𝑦

,

∀𝑥, 𝑦 ∈ [0, 1] .

(1)

2.2. Intuitionistic Fuzzy Set. Atanassov [1, 2] generalizedthe concept of fuzzy set [3] and defined the concept ofintuitionistic fuzzy set (IFS) as follows.

Definition 1. Let 𝑈 be fixed an IFS𝐴 on 𝑈 is given by;

𝐴 = {⟨𝑥, 𝜇𝐴(𝑥) , ]

𝐴(𝑥)⟩ | 𝑥 ∈ 𝑈} , (2)

where 𝜇𝐴

: 𝑈 → [0, 1] and ]𝐴

: 𝑈 → [0, 1], with thecondition 0 ≤ 𝜇

𝐴(𝑥) + ]

𝐴(𝑥) ≤ 1 for all 𝑥 ∈ 𝑈. Xu [5] called

𝑎 = (𝜇𝑎, ]𝑎) an IFV.

For IFVs, Wang and Liu [11] introduced some operationsas follows.

Let 𝜆 > 0, 𝑎1= (𝜇𝑎1

, ]𝑎1

) and 𝑎2= (𝜇𝑎2

, ]𝑎2

) be two IFVs;then

(1) 𝑎1⊗𝜀𝑎2= (

𝜇𝑎1

+ 𝜇𝑎2

1 + 𝜇𝑎1

𝜇𝑎2

,

]𝑎1

]𝑎2

1 + (1 − ]𝑎1

) (1 − ]𝑎2

)

)

(2) 𝑎1⊗𝜀𝑎2= (

𝜇𝑎1

𝜇𝑎2

1 + (1 − 𝜇𝑎1

) (1 − 𝜇𝑎2

)

,

]𝑎1

+ ]𝑎2

1 + ]𝑎1

]𝑎2

)

(3) 𝑎∧𝜀𝜆

1= (

2]𝜆𝑎1

(2 − ]𝑎1

)

𝜆

+ ]𝜆𝑎1

,

(1 + 𝜇𝑎1

)

𝜆

− (1 − 𝜇𝑎1

)

𝜆

(1 + 𝜇𝑎1

)

𝜆

+ (1 − 𝜇𝑎1

)

𝜆

) .

(3)

2.3. Hesitant Fuzzy Set. As another generalization of fuzzyset, HFS was first introduced by Torra and Narukawa [31, 32].

Definition 2. Let𝑋 be a reference set; anHFS on𝑋 is in termsof a function that when applied to𝑋 returns a subset of [0, 1].

To be easily understood, Xia and Xu use the followingmathematical symbol to express the HFS:

𝐻 = {

ℎ𝐻(𝑥)

𝑥

| 𝑥 ∈ 𝑋} , (4)

where ℎ𝐻(𝑥) is a set of some values in [0, 1], denoting the

possible membership degrees of the element 𝑥 ∈ 𝑋 to the set𝐻. For convenience, Xu and Xia [40] called ℎ

𝐻(𝑥) a hesitant

fuzzy element (HFE).Let ℎ be an HFE, ℎ− = min{𝛾 | 𝛾 ∈ ℎ}, and ℎ+ = max{𝛾 |

𝛾 ∈ ℎ}. Torra and Narukawa [31, 32] define the IFV 𝐴env(ℎ)as the envelope of ℎ, where 𝐴env(ℎ) = (ℎ

−

, 1 − ℎ+

).

Journal of Applied Mathematics 3

Let𝛼 > 0, ℎ1and ℎ2be twoHFEs. Xia andXu [36] defined

some operations as follows:

(4) ℎ1⨁ℎ2= ⋃

𝛾1∈ℎ1,𝛾2∈ℎ2

{𝛾1+ 𝛾2− 𝛾1𝛾2}

(5) ℎ1⨂ℎ2= ⋃

𝛾1∈ℎ1,𝛾2∈ℎ2

{𝛾1𝛾2}

(6) 𝛼ℎ = ⋃

𝛾∈ℎ

{𝛾𝛼

}

(7) ℎ𝛼

= ⋃

𝛾∈ℎ

{1 − (1 − 𝛾)𝛼

} .

(5)

In [36], Xia and Xu defined the score function of anHFE ℎ to compare the HFEs and gave the comparison laws.

Definition 3. Let ℎ be anHFE; 𝑠(ℎ) = (1/𝑛(ℎ))∑𝛾∈ℎ

𝛾 is calledthe score function of ℎ, where 𝑛(ℎ) is the number of values ofℎ. For two HFEs ℎ

1and ℎ

2, if 𝑠(ℎ

1) > 𝑠(ℎ

2), then ℎ

1> ℎ2; if

𝑠(ℎ1) = 𝑠(ℎ

2), then ℎ

1= ℎ2.

From Definition 3, it can be seen that all HFEs areregarded as the same if their score values are equal. In hesitantfuzzy decision making process, however, we usually need tocompare two HFEs for reordering or ranking. In the casewhere two HFEs have the same score values, they can notbe distinguished by Definition 3. Therefore, it is necessary todevelop a new method to overcome the difficulty.

For an IFV, Hong and Choi [46] showed that the relationbetween the score function and the accuracy function is sim-ilar to the relation between mean and variance in statistics.From Definition 3, we know that the score value of HFE ℎ isjust themean of the values in ℎ.Motivated by the idea ofHongand Choi [46], we can define the accuracy function of HFE ℎby using the variance of the values in ℎ.

Definition 4. Let ℎ be an HFE; 𝑘(ℎ) = 1 −√(1/𝑛(ℎ))∑

𝛾∈ℎ(𝛾 − 𝑠(ℎ))

2 is called the accuracy function ofℎ, where 𝑛(ℎ) is the number of values in ℎ and 𝑠(ℎ) is thescore function of ℎ.

It is well known that an efficient estimator is a measureof the variance of an estimate’s sampling distribution instatistics: the smaller the variance, the better the performanceof the estimator. Motivated by this idea, it is meaningful andappropriate to stipulate that the higher the accuracy degreeof HFE, the better the HFE. Therefore, in the following, wedevelop a new method to compare two HFEs, which is basedon the score function and the accuracy function, defined asfollows.

Definition 5. Let ℎ1and ℎ

2be two HFEs and let 𝑠(⋅) and

𝑘(⋅) be the score function and accuracy function of HFEs,respectively. Then

(1) if 𝑠(ℎ1) < 𝑠(ℎ

2), then ℎ

1is smaller than ℎ

2, denoted by

ℎ1≺ ℎ2;

(2) if 𝑠(ℎ1) = 𝑠(ℎ

2), then

(i) if 𝑘(ℎ1) < 𝑘(ℎ

2), then ℎ

1is smaller than ℎ

2,

denoted by ℎ1≺ ℎ2;

(ii) if 𝑘(ℎ1) = 𝑘(ℎ

2), then ℎ

1and ℎ

2represent

the same information, denoted by ℎ1≐ ℎ2. In

particular, if 𝛾1= 𝛾2for any 𝛾

1∈ ℎ1and 𝛾2∈ ℎ2,

then ℎ1is equal to ℎ

2, denoted by ℎ

1= ℎ2.

Example 6. Let ℎ1= {0.5}, ℎ

2= {0.1, 0.9}, ℎ

3= {0.3, 0.7},

ℎ4

= {0.1, 0.3, 0.7, 0.9}, ℎ5

= {0.2, 0.4, 0.6, 0.8}, and ℎ6

=

{0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9}; then 𝑠(ℎ1) = 𝑠(ℎ

2) =

𝑠(ℎ3) = 𝑠(ℎ

4) = 𝑠(ℎ

5) = 𝑠(ℎ

6) = 0.5, 𝑘(ℎ

1) = 1, 𝑘(ℎ

2) = 0.6,

𝑘(ℎ3) = 0.8, 𝑘(ℎ

4) = 0.6838, 𝑘(ℎ

5) = 0.7764, and 𝑘(ℎ

6) =

0.7418. By Definition 5, we have ℎ1≻ ℎ3≻ ℎ5≻ ℎ6≻ ℎ4≻ ℎ2.

3. Einstein Operations of Hesitant Fuzzy Sets

In this section, we will introduce the Einstein operationson HFEs and analyze some desirable properties of theseoperations.Motivated by the operational laws (1)–(3) on IFVsand based on the interconnection between HFEs and IFVs,we give some new operations of HFEs as follows.

Let 𝛼 > 0, ℎ, ℎ1, and ℎ

2be three HFEs; then

(8) ℎ1⊗𝜀ℎ2= ⋃

𝛾1∈ℎ1,𝛾2∈ℎ2

{

𝛾1+ 𝛾2

1 + 𝛾1𝛾2

} ,

(9) ℎ1⊗𝜀ℎ2= ⋃

𝛾1∈ℎ1,𝛾2∈ℎ2

{

𝛾1𝛾2

1 + (1 − 𝛾1) (1 − 𝛾

2)

} ,

(10) ℎ∧𝜀𝛼

= ⋃

𝛾∈ℎ

{

2𝛾𝛼

(2 − 𝛾)𝛼

+ 𝛾𝛼

} .

(6)

Proposition 7. Let 𝛼 > 0, 𝛼1> 0, 𝛼

2> 0, ℎ, ℎ

1and ℎ2be three

HFEs; then

(1) ℎ1⊗𝜀ℎ2= ℎ2⊗𝜀ℎ1,

(2) (ℎ1⊗𝜀ℎ2)⊗𝜀ℎ3= ℎ1⊗𝜀(ℎ2⊗𝜀ℎ3),

(3) (ℎ1⊗𝜀ℎ2)∧𝜀𝛼

= ℎ∧𝜀𝛼

1⊗𝜀ℎ∧𝜀𝛼

2,

(4) (ℎ∧𝜀𝛼1)∧𝜀𝛼2 = ℎ∧𝜀(𝛼1𝛼2);

(5) 𝐴 env(ℎ∧𝜀𝛼

) = (𝐴 env(ℎ))∧𝜀𝛼,

(6) 𝐴 env(ℎ1⊗𝜀ℎ2) = 𝐴 env(ℎ1)⊗𝜀𝐴 env(ℎ2).

Proof. (1) It is trivial.(2) By the operational law (9), we have

(ℎ1⊗𝜀ℎ2) ⊗𝜀ℎ3

= ⋃

𝛾1∈ℎ1,𝛾2∈ℎ2,𝛾3∈ℎ3

{ ((𝛾1𝛾2/ (1 + (1 − 𝛾

1) (1 − 𝛾

2))) 𝛾3)

× (1 + (1 − (𝛾1𝛾2/ (1 + (1 − 𝛾

1) (1 − 𝛾

2))))

× (1 − 𝛾3))−1

}


= ⋃

𝛾1∈ℎ1,𝛾2∈ℎ2,𝛾3∈ℎ3

{ (𝛾1𝛾2𝛾3) × (1 + (1 − 𝛾

1) (1 − 𝛾

2)

+ (1 − 𝛾1) (1 − 𝛾

3) + (1 − 𝛾

2)

× (1 − 𝛾3))−1

}

= ⋃

𝛾1∈ℎ1,𝛾2∈ℎ2,𝛾3∈ℎ3

{ (𝛾1(𝛾2𝛾3/ (1 + (1 − 𝛾

2) (1 − 𝛾

3))))

× (1 + (1 − 𝛾1)

× (1 − (𝛾2𝛾3/ (1 + (1 − 𝛾

2) (1 − 𝛾

3)))))−1

}

= ℎ1⊗𝜀(ℎ2⊗𝜀ℎ3) .

(7)

(3) Let ℎ = ℎ1⊗𝜀ℎ2; then ℎ = ℎ

1⊗𝜀ℎ2

=

⋃𝛾1∈ℎ1,𝛾2∈ℎ2

{𝛾1𝛾2/(1 + (1 − 𝛾

1)(1 − 𝛾

2))}

(ℎ1⊗𝜀ℎ2)∧𝜀𝛼

= ℎ∧𝜀𝛼

= ⋃

𝛾∈ℎ

{

2𝛾𝛼

(2 − 𝛾)𝛼

+ 𝛾𝛼

}

= ⋃

𝛾1∈ℎ1,𝛾2∈ℎ2

{ (2(𝛾1𝛾2/ (1 + (1 − 𝛾

1) (1 − 𝛾

2)))𝛼

)

× ((2 − (𝛾1𝛾2/ (1 + (1 − 𝛾

1) (1 − 𝛾

2))))𝛼

+ (𝛾1𝛾2/ (1 + (1 − 𝛾

1) (1 − 𝛾

2)))𝛼

)

−1

}

= ⋃

𝛾1∈ℎ1,𝛾2∈ℎ2

{

2(𝛾1𝛾2)𝛼

(4 − 2𝛾1− 2𝛾2+ 𝛾1𝛾2)𝛼

+ (𝛾1𝛾2)𝛼} ,

= ⋃

𝛾1∈ℎ1,𝛾2∈ℎ2

{

2(𝛾1𝛾2)𝛼

(2 − 𝛾1)𝛼

(2 − 𝛾2)𝛼

+ (𝛾1𝛾2)𝛼} .

(8)

Since ℎ∧𝜀𝛼1

= ⋃𝛾1∈ℎ{2𝛾𝛼

1/((2 − 𝛾

1)𝛼

+ 𝛾𝛼

1)} and ℎ∧𝜀𝛼

2=

⋃𝛾2∈ℎ{2𝛾𝛼

2/((2 − 𝛾

2)𝛼

+ 𝛾𝛼

2)}, then

ℎ∧𝜀𝛼


2

= ⋃

𝛾1∈ℎ1,𝛾2∈ℎ2

{ ((2𝛾𝛼

1/ ((2 − 𝛾

1)𝛼

+ 𝛾𝛼

1))

⋅ (2𝛾𝛼

2/ ((2 − 𝛾

2)𝛼

+ 𝛾𝛼

2)))

× (1 + (1 − (2𝛾𝛼

1/ ((2 − 𝛾

1)𝛼

+ 𝛾𝛼

1)))

× (1 − (2𝛾𝛼

2/ ((2 − 𝛾

2)𝛼

+ 𝛾𝛼

2))))

−1

}

= ⋃

𝛾1∈ℎ1,𝛾2∈ℎ2

{

2(𝛾1𝛾2)𝛼

(2 − 𝛾1)𝛼

(2 − 𝛾2)𝛼

+ (𝛾1𝛾2)𝛼} .

(9)

Thus (ℎ1⊗𝜀ℎ2)∧𝜀𝛼

= ℎ∧𝜀𝛼


2.

(4) Since ℎ∧𝜀𝛼1 = ⋃𝛾∈ℎ

{2𝛾𝛼1/((2 − 𝛾)

𝛼1+ 𝛾𝛼1)}, then

(ℎ∧𝜀𝛼1)∧𝜀𝛼2

= ⋃

𝛾∈ℎ

{ (2(2𝛾𝛼1/ ((2 − 𝛾)

𝛼1

+ 𝛾𝛼1))

𝛼2

)

× ((2 − (2𝛾𝛼1/ ((2 − 𝛾)

𝛼1

+ 𝛾𝛼1)))

𝛼2

+ (2𝛾𝛼1/ ((2 − 𝛾)

𝛼1

+ 𝛾𝛼1))

𝛼2

)

−1

}

= ⋃

𝛾∈ℎ

{

2𝛾(𝛼1𝛼2)

(2 − 𝛾)(𝛼1𝛼2)

+ 𝛾(𝛼1𝛼2)

}

= ℎ∧𝜀(𝛼1𝛼2)

.

(10)

(5) By the definition of the envelope of an HFE and theoperation laws (3) and (10), we have

(𝐴env (ℎ))∧𝜀𝛼

= (ℎ−

, 1 − ℎ+

)

∧𝜀𝛼

= (

2(ℎ−

)𝛼

(2 − ℎ−)𝛼

+ (ℎ−)𝛼,

[1 + (1 − ℎ+

)]𝛼

− [1 − (1 − ℎ+

)]

𝛼

[1 + (1 − ℎ+)]𝛼

+ [1 − (1 − ℎ+)]𝛼)

= (

2(ℎ−

)𝛼

(2 − ℎ−)𝛼

+ (ℎ−)𝛼,

(2 − ℎ+

)𝛼

− (ℎ+

)𝛼

(2 − ℎ+)𝛼

+ (ℎ+)𝛼) .

𝐴env (ℎ∧𝜀𝛼

)

= 𝐴env (⋃𝛾∈ℎ

{

2𝛾𝛼

(2 − 𝛾)𝛼

+ 𝛾𝛼

})

= (

2(ℎ−

)𝛼

(2 − ℎ−)𝛼

+ (ℎ−)𝛼, 1 −

2(ℎ+

)𝛼

(2 − ℎ+)𝛼

+ (ℎ+)𝛼)

= (

2(ℎ−

)𝛼

(2 − ℎ−)𝛼

+ (ℎ−)𝛼,

(2 − ℎ+

)𝛼

− (ℎ+

)𝛼

(2 − ℎ+)𝛼

+ (ℎ+)𝛼) .

(11)

Thus, 𝐴env(ℎ∧𝜀𝛼

) = (𝐴env(ℎ))∧𝜀𝛼.

(6) By the definition of the envelope of an HFE and theoperation laws (2) and (9), we have

𝐴env (ℎ1) ⊗𝜀𝐴env (ℎ2)

= (ℎ−

1, 1 − ℎ

+

1) ⊗𝜀(ℎ−

2, 1 − ℎ

+

2)

= (

ℎ−

1ℎ−

2

1 + (1 − ℎ−

1) (1 − ℎ

−

2)

,

(1 − ℎ+

1) + (1 − ℎ

+

2)

1 + (1 − ℎ+

1) (1 − ℎ

+

2)

)


𝐴env (ℎ1⊗𝜀ℎ2)

= 𝐴env ( ⋃𝛾1∈ℎ1,𝛾2∈ℎ2

{

𝛾1𝛾2

1 + (1 − 𝛾1) (1 − 𝛾

2)

})

= (

ℎ−

1ℎ−

2

1 + (1 − ℎ−

1) (1 − ℎ

−

2)

, 1 −

ℎ+

1ℎ+

2

1 + (1 − ℎ+

1) (1 − ℎ

+

2)

)

= (

ℎ−

1ℎ−

2

1 + (1 − ℎ−

1) (1 − ℎ

−

2)

,

(1 − ℎ+

1) + (1 − ℎ

+

2)

1 + (1 − ℎ+

1) (1 − ℎ

+

2)

) .

(12)

Thus, 𝐴env(ℎ1⊗𝜀ℎ2) = 𝐴env(ℎ1)⊗𝜀𝐴env(ℎ2).

Remark 8. Let 𝛼1> 0, 𝛼

2> 0, and ℎ be an HFE. It is worth

noting that ℎ∧𝜀𝛼1⊗𝜀ℎ∧𝜀𝛼2≐ ℎ∧𝜀(𝛼1+𝛼2) does not hold necessarily

in general. To illustrate that, an example is given as follows.

Example 9. Let ℎ = (0.3, 0.5), 𝛼1

= 𝛼2

= 1; thenℎ∧𝜀𝛼1⊗𝜀ℎ∧𝜀𝛼2

= ℎ⊗𝜀ℎ = ⋃

𝛾𝑖∈ℎ,𝛾𝑗∈ℎ,(𝑖,𝑗=1,2)

{𝛾𝑖𝛾𝑗/(1 + (1 −

𝛾𝑖)(1 − 𝛾

𝑗))} = (0.0604, 0.1111, 0.2), and ℎ∧𝜀(𝛼1+𝛼2) =

ℎ∧𝜀2

= ⋃𝛾∈ℎ

{2𝛾2

/((2 − 𝛾)2

+ 𝛾2

)} = (0.0604, 0.2). Clearly,𝑠(ℎ∧𝜀𝛼1⊗𝜀ℎ∧𝜀𝛼2) = 0.1238 < 0.1302 = 𝑠(ℎ

∧𝜀(𝛼1+𝛼2)

). Thusℎ∧𝜀𝛼1⊗𝜀ℎ∧𝜀𝛼1≺ ℎ∧𝜀(𝛼1+𝛼2).

However, if the number of the values in ℎ is only one, thatis, HFE ℎ is reduced to a fuzzy value, then the above resultholds.

Proposition 10. Let 𝛼1> 0, 𝛼

2> 0, and ℎ be an HFE, in

which the number of the values is only one, that is, ℎ = {𝛾};then ℎ∧𝜀𝛼1⊗

𝜀ℎ∧𝜀𝛼2= ℎ∧𝜀(𝛼1+𝛼2).

Proof. Since ℎ∧𝜀𝛼1 = ⋃𝛾∈ℎ

{2𝛾𝛼1/((2 − 𝛾)

𝛼1+ 𝛾𝛼1)} and ℎ∧𝜀𝛼2 =

⋃𝛾∈ℎ

{2𝛾𝛼2/((2 − 𝛾)

𝛼2+ 𝛾𝛼2)}, then

ℎ∧𝜀𝛼1⊗𝜀ℎ∧𝜀𝛼1

= ⋃

𝛾∈ℎ

{ ((2𝛾𝛼1/ ((2 − 𝛾)

𝛼1

+ 𝛾𝛼1))

⋅ (2𝛾𝛼2/ ((2 − 𝛾)

𝛼2

+ 𝛾𝛼2)))

× (1 + (1 − (2𝛾𝛼1/ ((2 − 𝛾)

𝛼1

+ 𝛾𝛼1)))

× (1 − (2𝛾𝛼2/ ((2 − 𝛾)

𝛼2

+ 𝛾𝛼2))))

−1

}

= ⋃

𝛾∈ℎ

{ (2𝛾𝛼1⋅ 2𝛾𝛼2) × ([(2 − 𝛾)

𝛼1

+ 𝛾𝛼1] ⋅ [(2 − 𝛾)

𝛼2

+ 𝛾𝛼2]

+ [(2 − 𝛾)𝛼1

− 𝛾𝛼1]

⋅ [(2 − 𝛾)𝛼2

− 𝛾𝛼2])

−1

}

= ⋃

𝛾∈ℎ

{

2𝛾𝛼1+𝛼2

(2 − 𝛾)𝛼1+𝛼2

+ 𝛾𝛼1+𝛼2

}

= ℎ∧𝜀(𝛼1+𝛼2)

.

(13)

Proposition 10 shows that it is consistent with the result(iii) in Theorem 2 in the literature [11].

4. Hesitant Fuzzy Einstein GeometricAggregation Operators

The weighted geometric operator [47] and the orderedweighted geometric operator [48] are two of the most com-mon and basic aggregation operators. Since their appearance,they have received more and more attention. In this section,we extend them to aggregate hesitant fuzzy information usingEinstein operations.

4.1. Hesitant Fuzzy Einstein Geometric Weighted AggregationOperator. Based on the operational laws (5) and (7) onHFEs,Xia and Xu [36] developed some hesitant fuzzy aggregationoperators as listed below.

Let ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛) be a collection of HFEs; then.

(1) the hesitant fuzzy weighted geometric (HFWG) oper-ator

HFWG (ℎ1, ℎ2, . . . , ℎ

𝑛) =

𝑛

⨂

𝑗=1

ℎ𝑗

𝜔𝑗

= ⋃

𝛾1∈ℎ1,𝛾2∈ℎ2,...,𝛾𝑛∈ℎ𝑛

{

{

{

𝑛

∏

𝑗=1

𝛾𝑗

𝜔𝑗

}

}

}

,

(14)

where 𝜔 = (𝜔1, 𝜔2, . . . , 𝜔

𝑛)𝑇 is the weight vector of ℎ

𝑗(𝑗 =

1, 2, . . . , 𝑛) with 𝜔𝑗∈ [0, 1] and ∑𝑛

𝑗=1𝜔𝑗= 1.

(2) the hesitant fuzzy ordered weighted geometric(HFOWG) operator

HFOWG (ℎ1, ℎ2, . . . , ℎ

𝑛)

=

𝑛

⨂

𝑗=1

𝑤𝑗

ℎ𝜎(𝑗)

= ⋃

𝛾𝜎(1)∈ℎ𝜎(1),𝛾𝜎(2)∈ℎ𝜎(2),...,𝛾𝜎(𝑛)∈ℎ𝜎(𝑛)

{

{

{

𝑛

∏

𝑗=1

𝛾

𝑤𝑗

𝜎(𝑗)

}

}

}

,

(15)

where 𝜎(1), 𝜎(2), . . . , 𝜎(𝑛) is a permutation of 1, 2, . . . , 𝑛,such that ℎ

𝜎(𝑗−1)> ℎ𝜎(𝑗)

for all 𝑗 = 2, . . . , 𝑛 and 𝑤 =(𝑤1, 𝑤2, . . . , 𝑤

𝑛)𝑇 is aggregation-associated vector with 𝑤

𝑗∈

[0, 1] and ∑𝑛𝑗=1

𝑤𝑗= 1.

For convenience, let𝐻 be the set of all HFEs. Based on theproposed Einstein operations onHFEs, we develop some newaggregation operators for HFEs and discuss their desirableproperties.


Definition 11. Let ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛) be a collection of HFEs.

A hesitant fuzzy Einstein weighted geometric (HFEWG𝜀)

operator of dimension 𝑛 is a mapping HFEWG𝜀: 𝐻𝑛

→ 𝐻

defined as follows:

HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ

𝑛)

=

𝑛

⨂𝜀

𝑗=1

ℎ

∧𝜀𝜔𝑗

𝑗

= ⋃

𝛾1∈ℎ1,𝛾2∈ℎ2,...,𝛾𝑛∈ℎ𝑛

{

{

{

2∏𝑛

𝑗=1𝛾

𝜔𝑗

𝑗

∏𝑛

𝑗=1(2 − 𝛾

𝑗)

𝜔𝑗

+∏𝑛

𝑗=1𝛾

𝜔𝑗

𝑗

}

}

}

,

(16)

where 𝑤 = (𝑤1, 𝑤2, . . . , 𝑤

𝑛)𝑇 is the weight vector of ℎ

𝑗(𝑗 =

1, 2, . . . , 𝑛) and 𝑤𝑗> 0,∑𝑛

𝑗=1𝑤𝑗= 1. In particular, when 𝑤

𝑗=

1/𝑛, 𝑗 = 1, 2, . . . , 𝑛, the HFEWG𝜀operator is reduced to the

hesitant fuzzy Einstein geometric (HFEG𝜀) operator:

HFEG𝜀(ℎ1, ℎ2, . . . , ℎ

𝑛)

= ⋃

𝛾1∈ℎ1,𝛾2∈ℎ2,...,𝛾𝑛∈ℎ𝑛

{

{

{

2∏𝑛

𝑗=1𝛾1/𝑛

𝑗

∏𝑛

𝑗=1(2 − 𝛾

𝑗)

1/𝑛

+∏𝑛

𝑗=1𝛾1/𝑛

𝑗

}

}

}

.

(17)

From Proposition 10, we easily get the following result.

Corollary 12. If all ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛) are equal and the

number of values in ℎ𝑗is only one, that is, ℎ

𝑗= ℎ = {𝛾} for

all 𝑗 = 1, 2, . . . , 𝑛, then

HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ

𝑛) = ℎ. (18)

Note that the HFEWG𝜀operator is not idempotent in

general; we give the following example to illustrate this case.

Example 13. Let ℎ1

= ℎ2

= ℎ3

= ℎ = (0.3, 0.7), 𝑤 =(0.4, 0.25, 0.35)

𝑇; then HFEWG𝜀(ℎ1, ℎ2, ℎ3) = {0.3, 0.4137,

0.3782, 0.5126, 0.4323, 0.579, 0.5342, 0.7}. ByDefinition 3, wehave 𝑠(HFEWG

𝜀(ℎ1, ℎ2, ℎ3)) = 0.4812 < 0.5 = 𝑠(ℎ). Hence

HFEWG𝜀(ℎ1, ℎ2, ℎ3) ≺ ℎ.

Lemma 14 (see [18, 49]). Let 𝛾𝑗> 0, 𝑤

𝑗> 0, 𝑗 = 1, 2, . . . , 𝑛,

and ∑𝑛𝑗=1

𝑤𝑗= 1. Then

𝑛

∏

𝑗=1

𝛾

𝑤𝑗

𝑗≤

𝑛

∑

𝑗=1

𝑤𝑗𝛾𝑗 (19)

with equality if and only if 𝛾1= 𝛾2= ⋅ ⋅ ⋅ = 𝛾

𝑛.

Theorem 15. Let ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛) be a collection of HFEs

and 𝑤 = (𝑤1, 𝑤2, . . . , 𝑤

𝑛)𝑇 the weight vector of ℎ

𝑗(𝑗 =

1, 2, . . . , 𝑛) with 𝑤𝑗∈ [0, 1] and ∑𝑛

𝑗=1𝑤𝑗= 1. Then

HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ

𝑛) ⪰ HFWG (ℎ

1, ℎ2, . . . , ℎ

𝑛) , (20)

where the equality holds if only if all ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛) are

equal and the number of values in ℎ𝑗is only one.

Proof. For any 𝛾𝑗∈ ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛), by Lemma 14, we have

𝑛

∏

𝑗=1

(2 − 𝛾𝑗)

𝑤𝑗

+

𝑛

∏

𝑗=1

𝛾

𝑤𝑗

𝑗≤

𝑛

∑

𝑗=1

𝑤𝑗(2 − 𝛾

𝑗) +

𝑛

∑

𝑗=1

𝑤𝑗𝛾𝑗= 2. (21)

Then

2∏𝑛

𝑗=1𝛾

𝜔𝑗

𝑗

∏𝑛

𝑗=1(2 − 𝛾

𝑗)

𝜔𝑗

+∏𝑛

𝑗=1𝛾

𝜔𝑗

𝑗

≥

𝑛

∏

𝑗=1

𝛾

𝜔𝑗

𝑗. (22)

It follows that 𝑠(⊗𝜀

𝑛

𝑗=1ℎ

∧𝜀𝜔𝑗

𝑗) ≥ 𝑠(⊗

𝜀

𝑛

𝑗=1ℎ

𝜔𝑗

𝑗), which completes

the proof of Theorem 15.

Theorem 15 tells us the result that the HFEWG𝜀operator

shows the decision maker’s more optimistic attitude than theHFWA operator proposed by Xia and Xu [36] (i.e., (15)) inaggregation process. To illustrate that, we give an exampleadopted from Example 1 in [36] as follows.

Example 16. Let ℎ1= (0.2, 0.3, 0.5), ℎ

2= (0.4, 0.6) be two

HFEs, and let 𝑤 = (0.7, 0.3)𝑇 be the weight vector of ℎ𝑗(𝑗 =

1, 2); then by Definition 11, we have

HFEWG𝜀(ℎ1, ℎ2) = ⋃

𝛾1∈ℎ1,𝛾2∈ℎ2

{

{

{

2∏2

𝑗=1𝛾

𝜔𝑗

𝑗

∏2

𝑗=1(2 − 𝛾

𝑗)

𝜔𝑗

+∏2

𝑗=1𝛾

𝜔𝑗

𝑗

}

}

}

= {0.2482, 0.2856, 0.3276, 0.3744,

0.4683, 0.5288} .

(23)

However, Xia and Xu [36] used the HFWG operator toaggregate the ℎ

𝑗(𝑗 = 1, 2) and got

HFEG (ℎ1, ℎ2)

= ⋃

𝛾1∈ℎ1,𝛾2∈ℎ2

{

{

{

2

∏

𝑗=1

𝛾

𝑤𝑗

𝑗

}

}

}

= {0.2462, 0.2781, 0.3270, 0.3693, 0.4676, 0.5281} .

(24)

It is clear that 𝑠(HFEWG𝜀(ℎ1, ℎ2)) = 0.3722 > 0.3694 =

𝑠(HFEG(ℎ1, ℎ2)). Thus HFEWG

𝜀(ℎ1, ℎ2) ≻ HFEG(ℎ

1, ℎ2).

Based onDefinition 11 and the proposed operational laws,we can obtain the following properties onHFEWG

𝜀operator.

Theorem 17. Let 𝛼 > 0, ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛), be a collection of

HFEs and 𝑤 = (𝑤1, 𝑤2, . . . , 𝑤


𝑗(𝑗 =

1, 2, . . . , 𝑛) with 𝑤𝑗∈ [0, 1] and ∑𝑛

𝑖=1𝑤𝑗= 1. Then

HFEWG𝜀(ℎ∧𝜀𝛼

1, ℎ∧𝜀𝛼

2, . . . , ℎ

∧𝜀𝛼

𝑛)

= (HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ

𝑛))∧𝜀𝛼

.

(25)


Proof. Since ℎ∧𝜀𝛼𝑗

= ⋃𝛾∈ℎ𝑗

{2𝛾𝛼

𝑗/((2 − 𝛾

𝑗)𝛼

+ 𝛾𝛼

𝑗)} for all 𝑗 =

1, 2, . . . , 𝑛, by the definition of HFEWG𝜀, we have


1, ℎ∧𝜀𝛼

2, . . . , ℎ

∧𝜀𝛼

𝑛)

= ⋃

𝛾1∈ℎ1,𝛾2∈ℎ2,...,𝛾𝑛∈ℎ𝑛

{

{

{

(2

𝑛

∏

𝑗=1

(2𝛾𝛼

𝑗/ ((2 − 𝛾

𝑗)

𝛼

+ 𝛾𝛼

𝑗))

𝜔𝑗

)

× (

𝑛

∏

𝑗=1

(2 − (2𝛾𝛼

𝑗/ ((2 − 𝛾

𝑗)

𝛼

+ 𝛾𝛼

𝑗)))

𝜔𝑗

+

𝑛

∏

𝑗=1

(2𝛾𝛼

𝑗/ ((2 − 𝛾

𝑗)

𝛼

+ 𝛾𝛼

𝑗))

𝜔𝑗

)

−1

}

}

}

= ⋃

𝛾1∈ℎ1,𝛾2∈ℎ2,...,𝛾𝑛∈ℎ𝑛

{

{

{

2∏𝑛

𝑗=1𝛾

𝛼𝜔𝑗

𝑗

∏𝑛

𝑗=1(2 − 𝛾

𝑗)

𝛼𝜔𝑗

+∏𝑛

𝑗=1𝛾

𝛼𝜔𝑗

𝑗

}

}

}

.

(26)

Since HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ

𝑛) =

⋃𝛾1∈ℎ1,𝛾2∈ℎ2,...,𝛾𝑛∈ℎ𝑛

{2∏𝑛

𝑗=1𝛾

𝜔𝑗

𝑗/(∏𝑛

𝑗=1(2 − 𝛾

𝑗)𝜔𝑗

+ ∏𝑛

𝑗=1𝛾

𝜔𝑗

𝑗)},

then

(HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ

𝑛))∧𝜀𝛼

= ⋃

𝛾1∈ℎ1,𝛾2∈ℎ2,...,𝛾𝑛∈ℎ𝑛

{

{

{

(2(2

𝑛

∏

𝑗=1

𝛾

𝜔𝑗

𝑗/(

𝑛

∏

𝑗=1

(2 − 𝛾𝑗)

𝜔𝑗

+

𝑛

∏

𝑗=1

𝛾

𝜔𝑗

𝑗))

𝛼

)

× ((2 − (2

𝑛

∏

𝑗=1

𝛾

𝜔𝑗

𝑗/(

𝑛

∏

𝑗=1

(2 − 𝛾𝑗)

𝜔𝑗

+

𝑛

∏

𝑗=1

𝛾

𝜔𝑗

𝑗)))

𝛼

+ (2

𝑛

∏

𝑗=1

𝛾

𝜔𝑗

𝑗/(

𝑛

∏

𝑗=1

(2 − 𝛾𝑗)

𝜔𝑗

+

𝑛

∏

𝑗=1

𝛾

𝜔𝑗

𝑗))

𝛼

)

−1

}

}

}

= ⋃

𝛾1∈ℎ1,𝛾2∈ℎ2,...,𝛾𝑛∈ℎ𝑛

{

{

{

2(∏𝑛

𝑗=1𝛾

𝜔𝑗

𝑗)

𝛼

(∏𝑛

𝑗=1(2 − 𝛾

𝑗)

𝜔𝑗

)

𝛼

+ (∏𝑛

𝑗=1𝛾

𝛼𝜔𝑗

𝑗)

𝛼

}

}

}

= ⋃

𝛾1∈ℎ1,𝛾2∈ℎ2,...,𝛾𝑛∈ℎ𝑛

{

{

{

2∏𝑛

𝑗=1𝛾

𝛼𝜔𝑗

𝑗

∏𝑛

𝑗=1(2 − 𝛾

𝑗)

𝛼𝜔𝑗

+∏𝑛

𝑗=1𝛾

𝛼𝜔𝑗

𝑗

}

}

}

.

(27)

Theorem 18. Let ℎ be an𝐻𝐹𝐸, ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛) a collection

of HFEs, and 𝑤 = (𝑤1, 𝑤2, . . . , 𝑤


𝑗

(𝑗 = 1, 2, . . . , 𝑛) with 𝑤𝑗∈ [0, 1] and ∑𝑛


HFEWG𝜀(ℎ1⊗𝜀ℎ, ℎ2⊗𝜀ℎ, . . . , ℎ

𝑛⊗𝜀ℎ)

= HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ

𝑛) ⊗𝜀ℎ.

(28)

Proof. By the definition of HFEWG𝜀and Einstein product

operator of HFEs, we have

HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ

𝑛) ⊗𝜀ℎ

= ⋃

𝛾∈ℎ,𝛾𝑗∈ℎ𝑗,𝑗=1,...,𝑛

{

{

{

(2∏𝑛

𝑗=1𝛾

𝜔𝑗

𝑗/ (∏𝑛

𝑗=1(2 − 𝛾

𝑗)

𝜔𝑗

+∏𝑛

𝑗=1𝛾

𝜔𝑗

𝑗)) ⋅ 𝛾

1 + (1 − (2∏𝑛

𝑗=1𝛾

𝜔𝑗

𝑗/ (∏𝑛

𝑗=1(2 − 𝛾

𝑗)

𝜔𝑗

+∏𝑛

𝑗=1𝛾

𝜔𝑗

𝑗))) (1 − 𝛾)

}

}

}

= ⋃


{

{

{

2𝛾∏𝑛

𝑗=1𝛾

𝜔𝑗

𝑗

(2 − 𝛾)∏𝑛

𝑗=1(2 − 𝛾

𝑗)

𝜔𝑗

+ 𝛾∏𝑛

𝑗=1𝛾

𝜔𝑗

𝑗

}

}

}

.

(29)

Since ℎ𝑗⊗𝜀ℎ = ⋃

𝛾𝑗∈ℎ𝑗,𝛾∈ℎ

{𝛾𝑗𝛾/(1 + (1 − 𝛾

𝑗)(1 − 𝛾))} for all 𝑗 =

1, 2, . . . , 𝑛, by the definition of HFEWG𝜀, we have

HFEWG𝜀(ℎ1⊗𝜀ℎ, ℎ2⊗𝜀ℎ, . . . , ℎ

𝑛⊕𝜀ℎ)

= ⋃


{

{

{

(2

𝑛

∏

𝑗=1

(𝛾𝑗𝛾/ (1 + (1 − 𝛾

𝑗) (1 − 𝛾)))

𝜔𝑗

)

× (

𝑛

∏

𝑗=1

(2 − (𝛾𝑗𝛾/ (1 + (1 − 𝛾

𝑗)

× (1 − 𝛾))))𝜔𝑗

+

𝑛

∏

𝑗=1

(𝛾𝑗𝛾/ (1 + (1 − 𝛾

𝑗)

× (1 − 𝛾)))𝑤𝑗

)

−1

}

}

}

= ⋃


{

{

{

2∏𝑛

𝑗=1(𝛾𝑗𝛾)

𝜔𝑗

∏𝑛

𝑗=1((2 − 𝛾

𝑗) (2 − 𝛾))

𝜔𝑗

+∏𝑛

𝑗=1(𝛾𝑗𝛾)

𝜔𝑗

}

}

}


= ⋃


{

{

{

(2

𝑛

∏

𝑗=1

𝛾𝜔𝑗⋅

𝑛

∏

𝑗=1

𝛾

𝜔𝑗

𝑗)

× (

𝑛

∏

𝑗=1

(2 − 𝛾)𝜔𝑗

⋅

𝑛

∏

𝑗=1

(2 − 𝛾𝑗)

𝜔𝑗

+

𝑛

∏

𝑗=1

𝛾𝜔𝑗

⋅

𝑛

∏

𝑗=1

𝛾

𝜔𝑗

𝑗)

−1

}

}

}

= ⋃


{

{

{

(2𝛾∑𝑛

𝑗=1𝜔𝑗⋅

𝑛

∏

𝑗=1

𝛾

𝜔𝑗

𝑗)

× ((2 − 𝛾)∑𝑛

𝑗=1𝜔𝑗

⋅

𝑛

∏

𝑗=1

(2 − 𝛾𝑗)

𝜔𝑗

+ 𝛾∑𝑛

𝑗=1𝜔𝑗⋅

𝑛

∏

𝑗=1

𝛾

𝜔𝑗

𝑗)

−1

}

}

}

= ⋃


{

{

{

2𝛾∏𝑛

𝑗=1𝛾

𝜔𝑗

𝑗

(2 − 𝛾)∏𝑛

𝑗=1(2 − 𝛾

𝑗)

𝜔𝑗

+ 𝛾∏𝑛

𝑗=1𝛾

𝜔𝑗

𝑗

}

}

}

.

(30)

Based onTheorems 17 and 18, the following property canbe obtained easily.

Theorem 19. Let 𝛼 > 0, ℎ be an HFE, let ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛)

be a collection of HFEs, and let 𝑤 = (𝑤1, 𝑤2, . . . , 𝑤

𝑛)𝑇 be the

weight vector of ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛) with 𝑤

𝑗∈ [0, 1] and

∑𝑛



1⊗𝜀ℎ, ℎ∧𝜀𝛼

1⊗𝜀ℎ, . . . , ℎ

∧𝜀𝛼

𝑛⊗𝜀ℎ)

= (HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ

𝑛) ⊗𝜀ℎ)∧𝜀𝛼

.

(31)

Theorem 20. Let ℎ𝑗and ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛) be two collections

of HFEs and𝑤 = (𝑤1, 𝑤2, . . . , 𝑤


𝑗(𝑗 =

1, 2, . . . , 𝑛) with 𝑤𝑗∈ [0, 1] and ∑𝑛


HFEWG𝜀(ℎ1⊗𝜀ℎ

1, ℎ2⊗𝜀ℎ

2, . . . , ℎ

𝑛⊗𝜀ℎ

𝑛)

= HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ

𝑛) ⊗𝜀HFEWG

𝜀(ℎ

1, ℎ

2, . . . , ℎ

𝑛) .

(32)

Proof. By the definition of HFEWG𝜀and Einstein product

operator of HFEs, we have

HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ

𝑛) ⊗𝜀HFEWG

𝜀(ℎ

1, ℎ

2, . . . , ℎ

𝑛)

= ⋃

𝛾𝑗∈ℎ𝑗,𝛾

𝑗∈ℎ

𝑗,𝑗=1,...,𝑛

{

{

{

(2∏𝑛

𝑗=1𝛾

𝜔𝑗

𝑗/ (∏𝑛

𝑗=1(2 − 𝛾

𝑗)

𝜔𝑗

+∏𝑛

𝑗=1𝛾

𝜔𝑗

𝑗)) ⋅ (2∏

𝑛

𝑗=1𝛾

𝑗

𝜔𝑗

/ (∏𝑛

𝑗=1(2 − 𝛾

𝑗)

𝜔𝑗

+∏𝑛

𝑗=1𝛾

𝑗

𝜔𝑗

))

1 + (1 − (2∏𝑛

𝑗=1𝛾

𝜔𝑗

𝑗/ (∏𝑛

𝑗=1(2 − 𝛾

𝑗)

𝜔𝑗

+∏𝑛

𝑗=1𝛾

𝜔𝑗

𝑗))) (1 − (2∏

𝑛

𝑗=1𝛾

𝑗

𝜔𝑗/ (∏𝑛

𝑗=1(2 − 𝛾

𝑗)

𝜔𝑗

+∏𝑛

𝑗=1𝛾

𝑗

𝜔𝑗)))

}

}

}

= ⋃


𝑗∈ℎ

𝑗,𝑗=1,...,𝑛

{

{

{

2∏𝑛

𝑗=1𝛾

𝜔𝑗

𝑗⋅ ∏𝑛

𝑗=1𝛾

𝑗

𝜔𝑗

∏𝑛

𝑗=1(2 − 𝛾

𝑗)

𝜔𝑗

⋅ ∏𝑛

𝑗=1(2 − 𝛾

𝑗)

𝜔𝑗

+∏𝑛

𝑗=1𝛾

𝜔𝑗

𝑗⋅ ∏𝑛

𝑗=1𝛾

𝑗

𝜔𝑗

}

}

}

.

(33)

Since ℎ𝑗⊗𝜀ℎ

𝑗= ⋃𝛾𝑗∈ℎ𝑗,𝛾

𝑗∈ℎ

𝑗

{𝛾𝑗𝛾

𝑗/(1 + (1 − 𝛾

𝑗)(1 − 𝛾

𝑗))} for all

𝑗 = 1, 2, . . . , 𝑛, by the definition of HFEWG𝜀, we have

HFEWG𝜀(ℎ1⨂

𝜀

ℎ

1, ℎ2⨂

𝜀

ℎ

2, . . . , ℎ

𝑛⨂

𝜀

ℎ

𝑛)

= ⋃


𝑗∈ℎ

𝑗,𝑗=1,...,𝑛

{

{

{

(2

𝑛

∏

𝑗=1

(𝛾𝑗𝛾

𝑗/ (1 + (1 − 𝛾

𝑗) (1 − 𝛾

𝑗)))

𝜔𝑗

)

× (

𝑛

∏

𝑗=1

(2 − (𝛾𝑗𝛾

𝑗/ (1 + (1 − 𝛾

𝑗)

× (1 − 𝛾

𝑗))))

𝜔𝑗

+

𝑛

∏

𝑗=1

(𝛾𝑗𝛾

𝑗/ (1 + (1 − 𝛾

𝑗)

× (1 − 𝛾

𝑗)))

𝜔𝑗

)

−1

}

}

}

= ⋃


𝑗∈ℎ

𝑗,𝑗=1,...,𝑛

{

{

{

(2

𝑛

∏

𝑗=1

(𝛾𝑗𝛾

𝑗)

𝜔𝑗

)

× (

𝑛

∏

𝑗=1

[(2 − 𝛾𝑗) (2 − 𝛾

𝑗)]

𝜔𝑗

+

𝑛

∏

𝑗=1

(𝛾𝑗𝛾

𝑗)

𝜔𝑗

)

−1

}

}

}

= ⋃


𝑗∈ℎ

𝑗,𝑗=1,...,𝑛

{

{

{

(2

𝑛

∏

𝑗=1

𝛾

𝜔𝑗

𝑗⋅

𝑛

∏

𝑗=1

𝛾

𝑗

𝜔𝑗

)


× (

𝑛

∏

𝑗=1

(2 − 𝛾𝑗)

𝜔𝑗

⋅

𝑛

∏

𝑗=1

(2 − 𝛾

𝑗)

𝜔𝑗

+

𝑛

∏

𝑗=1

𝛾

𝜔𝑗

𝑗⋅

𝑛

∏

𝑗=1

𝛾

𝑗

𝜔𝑗

)

−1

}

}

}

.

(34)

Theorem 21. Let ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛) be a collection of HFEs,

ℎ−

min = min𝑗{ℎ−

𝑗| ℎ−

𝑗= min{𝛾

𝑗∈ ℎ𝑗}}, and ℎ+max = max𝑗{ℎ

+

𝑗|

ℎ+

𝑗= max{𝛾

𝑗∈ ℎ𝑗}}, and let 𝑤 = (𝑤

1, 𝑤2, . . . , 𝑤

𝑛)𝑇 be the

weight vector of ℎ𝑗(𝑗 = 1, 2, . . . , 𝑛) with 𝑤

𝑗∈ [0, 1] and

∑𝑛


ℎ−

min ⪯ HFEWG𝜀 (ℎ1, ℎ2, . . . , ℎ𝑛) ⪯ ℎ+

max, (35)



Proof. Let 𝑓(𝑡) = (2 − 𝑡)/𝑡, 𝑡 ∈ [0, 1]. Then 𝑓(𝑡) = −2/𝑡2 < 0.Hence 𝑓(𝑡) is a decreasing function. Since ℎ−min ≤ ℎ

−

𝑗≤ 𝛾𝑗≤

ℎ+

𝑗≤ ℎ+

max for any 𝛾𝑗 ∈ ℎ𝑗 (𝑗 = 1, 2, . . . , 𝑛), then 𝑓(ℎ+

max) ≤

𝑓(𝛾𝑗) ≤ 𝑓(ℎ

−

min); that is, (2 − ℎ+

max)/ℎ+

max ≤ (2 − 𝛾𝑗)/𝛾𝑗 ≤

(2−ℎ−

min)/ℎ−

min. Then for any 𝛾𝑗 ∈ ℎ𝑗 (𝑗 = 1, 2, . . . , 𝑛), we have

𝑛

∏

𝑗=1

(

2 − ℎ+

maxℎ+

max)

𝑤𝑗

≤

𝑛

∏

𝑗=1

(

2 − 𝛾𝑗

𝛾𝑗

)

𝑤𝑗

≤

𝑛

∏

𝑗=1

(

1 − ℎ−

min1 + ℎ−

min)

𝑤𝑗

⇐⇒ (

2 − ℎ+

maxℎ+

max)

∑𝑛

𝑗=1𝑤𝑗

≤

𝑛

∏

𝑗=1

(

2 − 𝛾𝑗

𝛾𝑗

)

𝑤𝑗

≤ (

1 − ℎ−

min1 + ℎ−

min)

∑𝑛

𝑗=1𝑤𝑗

⇐⇒ (

2 − ℎ+

maxℎ+

max)

≤

𝑛

∏

𝑗=1

(

2 − 𝛾𝑗

𝛾𝑗

)

𝑤𝑗

≤ (

1 − ℎ−

min1 + ℎ−

min) ⇐⇒

2

ℎ+

max

≤

𝑛

∏

𝑗=1

(

2 − 𝛾𝑗

𝛾𝑗

)

𝑤𝑗

+ 1 ≤

2

ℎ−

min⇐⇒

ℎ−

min2

≤

1

∏𝑛

𝑗=1((2 − 𝛾

𝑗) /𝛾𝑗)

𝑤𝑗

+ 1

≤ (

ℎ+

max2

) ⇐⇒ ℎ−

min

≤

2

∏𝑛

𝑗=1((2 − 𝛾

𝑗) /𝛾𝑗)

𝑤𝑗

+ 1

≤ ℎ+

max ⇐⇒ ℎ−

min

≤

2∏𝑛

𝑗=1𝛾

𝜔𝑗

𝑗

∏𝑛

𝑗=1(2 − 𝛾

𝑗)

𝜔𝑗

+∏𝑛

𝑗=1𝛾

𝜔𝑗

𝑗

≤ ℎ+

max.

(36)

It follows that ℎ−min ≤ 𝑠(HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ𝑛)) ≤ ℎ+

max.Thus we have ℎ−min ⪯ HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ𝑛) ⪯ ℎ

+

max.

Remark 22. Let ℎ𝑗and ℎ

𝑗(𝑗 = 1, 2, . . . , 𝑛) be two collections

ofHFEs, and ℎ𝑗≺ ℎ

𝑗for all 𝑗; thenHFEWG

𝜀(ℎ1, ℎ2, . . . , ℎ

𝑛) ≺

HFEWG𝜀(ℎ

1, ℎ

2, . . . , ℎ

𝑛) does not hold necessarily in general.

To illustrate that, an example is given as follows.


= (0.45, 0.6), ℎ2

= (0.6, 0.7), ℎ3

=

(0.5, 0.6), ℎ1= (0.2, 0.9), ℎ

2= (0.45, 0.95), ℎ

3= (0.35, 0.8),

and 𝑤 = (0.5, 0.3, 0.2)𝑇; then HFEWG𝜀(ℎ1, ℎ2, ℎ3) = {0.5024,

0.5215, 0.5286, 0.5483, 0.5791, 0.6, 0.6077, 0.6291} andHFEWG

𝜀(ℎ

1, ℎ

2, ℎ

3) = {0.2778, 0.3372, 0.3835, 0.4595,

0.6088, 0.7099, 0.7833, 0.8947}. By Definition 3, we have𝑠(HFEWG

𝜀(ℎ1, ℎ2, ℎ3)) = 0.5646 and 𝑠(HFEWG

𝜀(ℎ

1, ℎ

2,

ℎ

3)) = 0.5568. It follows that HFEWG

𝜀(ℎ1, ℎ2, ℎ3) ≻

HFEWG𝜀(ℎ

1, ℎ

2, ℎ

3). Clearly, ℎ

𝑗≺ ℎ

𝑗for 𝑗 = 1, 2, 3, but

HFEWG𝜀(ℎ1, ℎ2, ℎ3) ≻ HFEWG

𝜀(ℎ

1, ℎ

2, ℎ

3).

4.2. Hesitant Fuzzy Einstein Ordered Weighted AveragingOperator. Similar to the HFOWG operator introduced byXia and Xu [36] (i.e., (15)), in what follows, we developan (HFEOWG

𝜀) operator, which is an extension of OWA

operator proposed by Yager [50].

Definition 24. For a collection of the HFEs ℎ𝑗(𝑗 =

1, 2, . . . , 𝑛), a hesitant fuzzy Einstein ordered weighted aver-aging (HFEOWG

𝜀) operator is a mapping HFEWG

𝜀: 𝐻𝑛

→

𝐻 such that

HFEOWG𝜀(ℎ1, ℎ2, . . . , ℎ

𝑛)

=

𝑛

⨂𝜀

𝑗=1

ℎ

∧𝜀𝑤𝑗

𝜎(𝑗)

= ⋃

𝛾𝜎(1)∈ℎ𝜎(1),𝛾𝜎(2)∈ℎ𝜎(2),...,𝛾𝜎(𝑛)∈ℎ𝜎(𝑛)

{

{

{

(2

𝑛

∏

𝑗=1

𝛾

𝑤𝑗

𝜎(𝑗))

× (

𝑛

∏

𝑗=1

(2 − 𝛾𝜎(𝑗)

)

𝑤𝑗

+

𝑛

∏

𝑗=1

𝛾

𝑤𝑗

𝜎(𝑗))

−1

}

}

}

,

(37)

where (𝜎(1), 𝜎(2), . . . , 𝜎(𝑛)) is a permutation of (1, 2, . . . , 𝑛),such that ℎ

𝜎(𝑗−1)≻ ℎ

𝜎(𝑗)for all 𝑗 = 2, . . . , 𝑛 and

𝑤 = (𝑤1, 𝑤2, . . . , 𝑤

𝑛)𝑇 is aggregation-associated vector with

𝑤𝑗

∈ [0, 1] and ∑𝑛𝑗=1

𝑤𝑗

= 1. In particular, if 𝑤 =(1/𝑛, 1/𝑛, . . . , 1/𝑛)

𝑇, then the HFEOWG𝜀operator is reduced

to the HFEA𝜀operator of dimension 𝑛 (i.e., (17)).


Note that the HFEOWG𝜀weights can be obtained similar

to the OWA weights. Several methods have been introducedto determine the OWA weights in [20, 21, 50–53].

Similar to the HFEWG𝜀operator, the HFEOWG

𝜀opera-

tor has the following properties.


and 𝑤 = (𝑤1, 𝑤2, . . . , 𝑤


𝑗(𝑗 =

1, 2, . . . , 𝑛) with 𝑤𝑗∈ [0, 1] and ∑𝑛



𝑛) ⪰ HFOWG (ℎ

1, ℎ2, . . . , ℎ

𝑛) ,

(38)



From Theorem 25, we can conclude that the valuesobtained by the HFEOWG

𝜀operator are not less than the

ones obtained by the HFOWA operator proposed by Xiaand Xu [36]. To illustrate that, let us consider the followingexample.


= (0.1, 0.4, 0.7), ℎ2

= (0.3, 0.5), andℎ3

= (0.2, 0.6) be three HFEs and suppose that 𝑤 =(0.2, 0.45, 0.35)

𝑇 is the associated vector of the aggregationoperator.

By Definitions 3 and 4, we calculate the score values andthe accuracy values of ℎ

1, ℎ2, and ℎ

3as follows, respectively:

𝑠(ℎ1) = 𝑠(ℎ

2) = 𝑠(ℎ

3) = 0.5, 𝑘(ℎ

1) = 0.7551, 𝑘(ℎ

2) = 0.9,

𝑘(ℎ3) = 0.8.According to Definition 5, we have ℎ

2≺ ℎ3≺ ℎ1. Then

ℎ𝜎(1)

= ℎ2, ℎ𝜎(2)

= ℎ3, ℎ𝜎(3)

= ℎ1.

By the definition of HFEOWG𝜀, we have

HFEOWG𝜀(ℎ1, ℎ2, ℎ3)

=

3

⨂𝜀

𝑗=1

ℎ

∧𝜀𝑤𝑗

𝜎(𝑗)

= ⋃

𝛾𝜎(1)∈ℎ𝜎(1),𝛾𝜎(2)∈ℎ𝜎(2),𝛾𝜎(3)∈ℎ𝜎(3)

{{

{{

{

2∏3

𝑗=1𝛾

𝑤𝑗

𝜎(𝑗)

∏3

𝑗=1(2 − 𝛾

𝜎(𝑗))

𝑤𝑗

+∏3

𝑗=1𝛾

𝑤𝑗

𝜎(𝑗)

}}

}}

}

= {0.1716, 0.2787, 0.3495, 0.2939, 0.4582, 0.5598, 0.1926,

0.3106, 0.3877, 0.3272, 0.5047, 0.6125} .

(39)

If we use the HFOWA operator, which was given by Xia andXu [36] (i.e., (15)), to aggregate the HFEs ℎ

𝑗(𝑖 = 1, 2, 3), then

we have

HFOWG (ℎ1, ℎ2, . . . , ℎ

𝑛)

=

3

⨂

𝑗=1

ℎ

𝑤𝑗

𝜎(𝑗)= ⋃

𝛾𝜎(1)∈ℎ𝜎(1),𝛾𝜎(2)∈ℎ𝜎(2),𝛾𝜎(3)∈ℎ𝜎(3)

{

{

{

3

∏

𝑗=1

𝛾

𝑤𝑗

𝜎(𝑗)

}

}

}

= {0.1702, 0.2764, 0.3363, 0.2790, 0.4532, 0.5513,

0.1885, 0.3062, 0.3724, 0.3090, 0.5020, 0.6106} .

(40)

Clearly, 𝑠(HFEOWG𝜀(ℎ1, ℎ2, ℎ3)) = 0.3706 > 0.3629 =

𝑠(HFOWG(ℎ1, ℎ2, . . . , ℎ

𝑛)). By Definition 3, we have

HFEOWG𝜀(ℎ1, ℎ2, ℎ3) ≻ HFOWG(ℎ

1, ℎ2, ℎ3).

Theorem 27. Let 𝛼 > 0, ℎ be an HFE, let ℎ𝑗and ℎ

𝑗

(𝑗 = 1, 2, . . . , 𝑛) be two collection of HFEs, and let 𝑤 =(𝑤1, 𝑤2, . . . , 𝑤

𝑛)𝑇 be an aggregation-associated vector with

𝑤𝑗∈ [0, 1] and ∑𝑛


(1) HFEWG𝜀(ℎ∧𝜀𝛼

1, ℎ∧𝜀𝛼

2, . . . , ℎ

∧𝜀𝛼

𝑛) =

(HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ

𝑛))∧𝜀𝛼,

(2) HFEWG𝜀(ℎ1⊗𝜀ℎ, ℎ2⊗𝜀ℎ, . . . , ℎ

𝑛⊗𝜀ℎ) = HFEWG

𝜀(ℎ1,

ℎ2, . . . , ℎ

𝑛)⊗𝜀ℎ,

(3) HFEWG𝜀(ℎ∧𝜀𝛼

1⊗𝜀ℎ, ℎ∧𝜀𝛼

1⊗𝜀ℎ, . . . , ℎ

∧𝜀𝛼

𝑛⊗𝜀ℎ) =

(HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ

𝑛)⊗𝜀ℎ)∧𝜀𝛼,

(4) HFEWG𝜀(ℎ1⊗𝜀ℎ

1, ℎ2⊗𝜀ℎ

2, . . . , ℎ

𝑛⊗𝜀ℎ

𝑛) =

HFEWG𝜀(ℎ1, ℎ2, . . . , ℎ

𝑛)⊗𝜀HFEWG

𝜀(ℎ

1, ℎ

2, . . . , ℎ

𝑛).


and let 𝑤 = (𝑤1, 𝑤2, . . . , 𝑤

𝑛)𝑇 be an aggregation-associated

vector with 𝑤𝑗∈ [0, 1] and ∑𝑛


ℎ−

min ⪯ HFEOWG𝜀 (ℎ1, ℎ2, . . . , ℎ𝑛) ⪯ ℎ+

max, (41)

where ℎ−min = min𝑗{ℎ−

𝑗| ℎ−

𝑗= min{𝛾

𝑗∈ ℎ𝑗}} and ℎ+max =

max𝑗{ℎ+

𝑗| ℎ+

𝑗= max{𝛾

𝑗∈ ℎ𝑗}}.

Besides the above properties, we can get the followingdesirable results on the HFOWG

𝜀operator.


and let 𝑤 = (𝑤1, 𝑤2, . . . , 𝑤





𝑛) = HFEOWG

𝜀(ℎ

1, ℎ

2, . . . , ℎ

𝑛) ,

(42)

where (ℎ1, ℎ

2, . . . , ℎ

𝑛) is any permutation of (ℎ

1, ℎ2, . . . , ℎ

𝑛).

Proof. Let HFEOWG𝜀(ℎ1, ℎ2, . . . , ℎ

𝑛) = ⊗

𝜀

𝑛

𝑗=1ℎ

∧𝜀𝑤𝑗

𝜎(𝑗)

and HFEOWG𝜀(ℎ

1, ℎ

2, . . . , ℎ

𝑛) = ⊗

𝜀

𝑛

𝑗=1ℎ

𝜎(𝑗)

∧𝜀𝑤𝑗 . Since

(ℎ

1, ℎ

2, . . . , ℎ

𝑛) is any permutation of (ℎ

1, ℎ2, . . . , ℎ

𝑛),

then we have ℎ𝜎(𝑗)

= ℎ

𝜎(𝑗)(𝑗 = 1, 2, . . . , 𝑛). Thus


𝑛) = HFEOWG

𝜀(ℎ

1, ℎ

2, . . . , ℎ

𝑛).


and let 𝑤 = (𝑤1, 𝑤2, . . . , 𝑤




(1) if 𝑤 = (0, 0, . . . , 1), then HFOWG𝜀(ℎ1, ℎ2, . . . , ℎ

𝑛)=

min{ℎ1, ℎ2, . . . , ℎ

𝑛};


(2) if 𝑤 = (1, 0, . . . , 0), then HFOWG𝜀(ℎ1, ℎ2, . . .,

ℎ𝑛)=max{ℎ

1, ℎ2, . . . , ℎ

𝑛};

(3) if 𝑤𝑗= 1 and 𝑤

𝑖= 0 (𝑖 ̸= 𝑗), then HFOWG

𝜀(ℎ1, ℎ2, . . .,

ℎ𝑛) = ℎ

𝜎(𝑗), where ℎ

𝜎(𝑗)is the 𝑗th largest of ℎ

𝑖(𝑖 =

1, 2, . . . , 𝑛).

5. An Application in HesitantFuzzy Decision Making

In this section, we apply the HFEWG𝜀and HFEOWG

𝜀

operators to multiple attribute decision making with hesitantfuzzy information.

For hesitant fuzzy multiple attribute decision makingproblems, let 𝑌 = {𝑌

1, 𝑌2, . . . , 𝑌

𝑚} be a discrete set of

alternatives, let 𝐴 = {𝐴1, 𝐴2, . . . , 𝐴

𝑛} be a collection of

attributes, and let 𝜔 = (𝜔1, 𝜔2, . . . , 𝜔

𝑛)𝑇 be the weight vector

of 𝐴𝑗(𝑗 = 1, 2, . . . , 𝑛) with 𝜔

𝑗≥ 0, 𝑗 = 1, 2, . . . , 𝑛, and

∑𝑛

𝑗=1𝜔𝑗= 1. If the decision makers provide several values for

the alternative 𝑌𝑖(𝑖 = 1, 2, . . . , 𝑚) under the attribute 𝐴

𝑗(𝑗 =

1, 2, . . . , 𝑛)with anonymity, these values can be considered asan HFE ℎ

𝑖𝑗. In the case where two decision makers provide

the same value, the value emerges only once in ℎ𝑖𝑗. Suppose

that the decision matrix 𝐻 = (ℎ𝑖𝑗)𝑚×𝑛

is the hesitant fuzzydecision matrix, where ℎ

𝑖𝑗(𝑖 = 1, 2, . . . , 𝑚, 𝑗 = 1, 2, . . . , 𝑛) are

in the form of HFEs.To get the best alternative, we can utilize the HFEWG

𝜀

operator or the HFEOWG𝜀operator; that is,

ℎ𝑖= HFEWG

𝜀(ℎ𝑖1, ℎ𝑖2, . . . , ℎ

𝑖𝑛)

= ⋃

𝛾𝑖1∈ℎ𝑖1,𝛾𝑖2∈ℎ𝑖2,...,𝛾𝑖𝑛∈ℎ𝑖𝑛

{

{

{

2∏𝑛

𝑗=1𝛾

𝜔𝑗

𝑖𝑗

∏𝑛

𝑗=1(2 − 𝛾

𝑖𝑗)

𝜔𝑗

+∏𝑛

𝑗=1𝛾

𝜔𝑗

𝑖𝑗

}

}

}

(43)

or

ℎ𝑖= HFEOWG

𝜀(ℎ𝑖1, ℎ𝑖2, . . . , ℎ

𝑖𝑛)

= ⋃

𝛾𝑖𝜎(𝑗)∈ℎ𝑖𝜎(𝑗),𝑗=1,2,...,𝑛

{{

{{

{

2∏𝑛

𝑗=1𝛾

𝑤𝑗

𝑖𝜎(𝑗)

∏𝑛

𝑗=1(2 − 𝛾

𝑖𝜎(𝑗))

𝑤𝑗

+∏𝑛

𝑗=1𝛾

𝑤𝑗

𝑖𝜎(𝑗)

}}

}}

}

(44)

to derive the overall value ℎ𝑖of the alternatives 𝑌

𝑖(𝑖 =

1, 2, . . . , 𝑚), where𝑤 = (𝑤1, 𝑤2, . . . , 𝑤

𝑛)𝑇 is the weight vector

related to the HFEOWA𝜀operator, such that 𝑤

𝑗≥ 0, 𝑗 =

1, 2, . . . , 𝑛, and ∑𝑛𝑗=1

𝑤𝑗= 1, which can be obtained by the

normal distribution based method [20].Then by Definition 3, we compute the scores 𝑠(ℎ

𝑖) (𝑖 =

1, 2, . . . , 𝑚) of the overall values ℎ𝑖(𝑖 = 1, 2, . . . , 𝑚) and use

the scores 𝑠(ℎ𝑖) (𝑖 = 1, 2, . . . , 𝑚) to rank the alternatives

𝑌 = {𝑌1, 𝑌2, . . . , 𝑌

𝑚} and then select the best one (note that

if there is no difference between the two scores ℎ𝑖and ℎ

𝑗,

thenwe need to compute the accuracy degrees 𝑘(ℎ𝑖) and 𝑘(ℎ

𝑗)

of the overall values ℎ𝑖and ℎ

𝑗by Definition 4, respectively,

and then rank the alternatives 𝑌𝑖and 𝑌

𝑗in accordance with

Definition 5).In the following, an example on multiple attribute deci-

sion making problem involving a customer buying a car,

which is adopted from Herrera and Martinez [54], is givento illustrate the proposed method using the HFEOWG

𝜀

operator.

Example 31. Consider that a customer wants to buy a car,which will be chosen from five types 𝑌

𝑖(𝑖 = 1, 2, . . . , 5).

In the process of choosing one of the cars, four factors areconsidered:𝐴

1is the consumption petrol,𝐴

2is the price,𝐴

3

is the degree of comfort, and 𝐴4is the safety factor. Suppose

that the characteristic information of the alternatives 𝑌𝑖(𝑖 =

1, 2, . . . , 5) can be represented by HFEs ℎ𝑖𝑗(𝑖 = 1, 2, . . . , 5; 𝑗 =

1, 2, . . . , 4), and the hesitant fuzzy decision matrix is given inTable 1.

To use HFEOWG𝜀operator, we first reorder the ℎ

𝑖𝑗(𝑗 =

1, 2, . . . , 4) for each alternative 𝑌𝑖(𝑖 = 1, 2, . . . , 5). According

to Definitions 3 and 4, we compute the score values andaccuracy degrees of 𝑠(ℎ

𝑖𝑗) (𝑖 = 1, 2, . . . , 5; 𝑗 = 1, 2, . . . , 4) as

follows:

𝑠 (ℎ11) = 0.45, 𝑠 (ℎ

12) = 0.75, 𝑠 (ℎ

13) = 0.3,

𝑠 (ℎ14) = 0.3, 𝑘 (ℎ

13) = 0.9184, 𝑘 (ℎ

14) = 0.9;

𝑠 (ℎ21) = 0.5, 𝑠 (ℎ

22) = 0.7, 𝑠 (ℎ

23) = 0.7,

𝑠 (ℎ24) = 0.5, 𝑘 (ℎ

21) = 0.7551, 𝑘 (ℎ

24) = 0.8129,

𝑘 (ℎ22) = 0.8367, 𝑘 (ℎ

23) = 0.9;

𝑠 (ℎ31) = 0.85, 𝑠 (ℎ

32) = 0.4, 𝑠 (ℎ

33) = 0.35,

𝑠 (ℎ34) = 0.4, 𝑘 (ℎ

32) = 0.8367, 𝑘 (ℎ

34) = 0.7764;

𝑠 (ℎ41) = 0.6, 𝑠 (ℎ

42) = 0.6, 𝑠 (ℎ

43) = 0.3,

𝑠 (ℎ44) = 0.4, 𝑘 (ℎ

41) = 0.772, 𝑘 (ℎ

42) = 0.8367;

𝑠 (ℎ51) = 0.5, 𝑠 (ℎ

52) = 0.3, 𝑠 (ℎ

53) = 0.5,

𝑠 (ℎ54) = 0.35, 𝑘 (ℎ

51) = 0.8367, 𝑘 (ℎ

53) = 0.8129.

(45)

Then by Definition 5, we have

ℎ1𝜎(1)

= ℎ12, ℎ1𝜎(2)

= ℎ11, ℎ1𝜎(3)

= ℎ13, ℎ1𝜎(4)

= ℎ14;

ℎ2𝜎(1)

= ℎ23, ℎ2𝜎(2)

= ℎ22, ℎ2𝜎(3)

= ℎ24, ℎ2𝜎(4)

= ℎ21;

ℎ3𝜎(1)

= ℎ31, ℎ3𝜎(2)

= ℎ32, ℎ3𝜎(3)

= ℎ34, ℎ3𝜎(4)

= ℎ33;

ℎ4𝜎(1)

= ℎ42, ℎ4𝜎(2)

= ℎ41, ℎ4𝜎(3)

= ℎ44, ℎ4𝜎(4)

= ℎ43;

ℎ5𝜎(1)

= ℎ51, ℎ5𝜎(2)

= ℎ53, ℎ5𝜎(3)

= ℎ54, ℎ5𝜎(4)

= ℎ52.

(46)

Suppose that 𝑤 = (0.1835, 0.3165, 0.3165, 0.1835)𝑇 is theweighted vector related to the HFEOWA

𝜀operator and it

is derived by the normal distribution based method [20].Thenwe utilize theHFEOWA

𝜀operator to obtain the hesitant


Table 1: Hesitant fuzzy decision making matrix.

𝐴1

𝐴2

𝐴3

𝐴4

𝑌1

{0.4, 0.5} {0.7, 0.8} {0.2, 0.3, 0.4} {0.2, 0.4}

𝑌2

{0.2, 0.5, 0.8} {0.5, 0.7, 0.9} {0.6, 0.8} {0.2, 0.5, 0.6, 0.7}

𝑌3

{0.8, 0.9} {0.2, 0.4, 0.6} {0.2, 0.3, 0.4, 0.5} {0.1, 0.3, 0.5, 0.7}

𝑌4

{0.3, 0.4, 0.6, 0.8, 0.9} {0.4, 0.6, 0.8} {0.1, 0.2, 0.4, 0.5} {0.2, 0.3, 0.5, 0.6}

𝑌5

{0.3, 0.5, 0.7} {0.2, 0.3, 0.4} {0.2, 0.5, 0.6, 0.7} {0.1, 0.3, 0.4, 0.6}

fuzzy elements ℎ𝑖(𝑖 = 1, 2, 3, 4, 5) for the alternatives 𝑋

𝑖

(𝑖 = 1, 2, 3, 4, 5). Take alternative𝑋1for an example; we have

ℎ1= HFEOWG

𝜀(ℎ11, ℎ12, . . . , ℎ

14)

= ⋃

𝛾1𝜎(𝑗)∈ℎ1𝜎(𝑗),𝑗=1,2,3,4

{{

{{

{

2∏4

𝑗=1𝛾

𝑤𝑗

1𝜎(𝑗)

∏4

𝑗=1(2 − 𝛾

1𝜎(𝑗))

𝑤𝑗

+∏4

𝑗=1𝛾

𝑤𝑗

1𝜎(𝑗)

}}

}}

}

= {0.3220, 0.3642, 0.3635, 0.4099, 0.3974, 0.4470,

0.3473, 0.3921, 0.3914, 0.4403, 0.4272, 0.4794,

0.3327, 0.3760, 0.3753, 0.4228, 0.4101, 0.4607,

0.3587, 0.4046, 0.4039, 0.4539, 0.4405, 0.4938} .

(47)

The results can be obtained similarly for the other alterna-tives; here we will not list them for vast amounts of data. ByDefinition 3, the score values 𝑠(ℎ

𝑖) of ℎ𝑖(𝑖 = 1, 2, 3, 4, 5) can

be computed as follows:

𝑠 (ℎ1) = 0.4048, 𝑠 (ℎ

2) = 0.5758, 𝑠 (ℎ

3) = 0.4311,

𝑠 (ℎ4) = 0.4479, 𝑠 (ℎ

5) = 0.3620.

(48)

According to the scores 𝑠(ℎ𝑖) of the overall hesitant fuzzy

values ℎ𝑖(𝑖 = 1, 2, 3, 4, 5), we can rank all the alternatives 𝑋

𝑖:

𝑋2≻ 𝑋4≻ 𝑋3≻ 𝑋1≻ 𝑋5. Thus the optimal alternative is

𝑋2.If we use the HFWG operator introduced by Xia and Xu

[36] to aggregate the hesitant fuzzy values, then

𝑠 (ℎ1) = 0.3960, 𝑠 (ℎ

2) = 0.5630, 𝑠 (ℎ

3) = 0.4164,

𝑠 (ℎ4) = 0.4344, 𝑠 (ℎ

5) = 0.3548.

(49)

By Definition 5, we have𝑋2≻ 𝑋4≻ 𝑋3≻ 𝑋1≻ 𝑋5.

Note that the rankings are the same in such two cases, butthe overall values of alternatives by the HFEOWG

𝜀operator

are not smaller than the ones by the HFOWG operator.It shows that the attitude of the decision maker using theproposed HFEOWG

𝜀operator is more optimistic than the

one using the HFOWG operator introduced by Xia andXu [36] in aggregation process. Therefore, according to thedecision makers’ optimistic (or pessimistic) attitudes, thedifferent hesitant fuzzy aggregation operators can be used toaggregate the hesitant fuzzy information in decision makingprocess.

6. Conclusions

The purpose of multicriteria decision making is to select theoptimal alternative from several alternatives or to get theirranking by aggregating the performances of each alternativeunder some attributes, which is the pervasive phenomenonin modern life. Hesitancy is the most common problemin decision making, for which hesitant fuzzy set can beconsidered as a suitable means allowing several possibledegrees for an element to a set. Therefore, the hesitant fuzzymultiple attribute decision making problems have receivedmore and more attention. In this paper, an accuracy functionof HFEs has been defined for distinguishing between thetwo HFEs having the same score values, and a new orderrelation between two HFEs has been provided. Some Ein-stein operations on HFEs and their basic properties havebeen presented. With the help of the proposed operations,several new hesitant fuzzy aggregation operators includingthe HFEWG

𝜀operator and HFEOWG

𝜀operator have been

developed, which are extensions of the weighted geometricoperator and the OWGoperator with hesitant fuzzy informa-tion, respectively. Moreover, some desirable properties of theproposed operators have been discussed and the relationshipsbetween the proposed operators and the existing hesitantfuzzy aggregation operators introduced by Xia and Xu [36]have been established. Finally, based on the HFEOWG

𝜀

operator, an approach of hesitant fuzzy decision making hasbeen given and a practical example has been presented todemonstrate its practicality and effectiveness.

Conflict of Interests

The authors declared that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The authors are very grateful to the editors and the anony-mous reviewers for their insightful and constructive com-ments and suggestions that have led to an improved version ofthis paper. This work was supported by the National NaturalScience Foundation of China (nos. 11071061 and 11101135)and the National Basic Research Program of China (no.2011CB311808).

References

[1] K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets andSystems, vol. 20, no. 1, pp. 87–96, 1986.


[2] K. T. Atanassov, “More on intuitionistic fuzzy sets,” Fuzzy Setsand Systems, vol. 33, no. 1, pp. 37–45, 1989.

[3] L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3,pp. 338–353, 1965.

[4] S. Chen and C. Hwang, Fuzzy Multiple Attribute DecisionMaking, Springer, Berlin, Germany, 1992.

[5] Z. Xu, “Intuitionistic fuzzy aggregation operators,” IEEE Trans-actions on Fuzzy Systems, vol. 15, no. 6, pp. 1179–1187, 2007.

[6] Z. Xu and R. R. Yager, “Some geometric aggregation operatorsbased on intuitionistic fuzzy sets,” International Journal ofGeneral Systems, vol. 35, no. 4, pp. 417–433, 2006.

[7] D.-F. Li, “Multiattribute decision making method based ongeneralized OWA operators with intuitionistic fuzzy sets,”Expert Systems with Applications, vol. 37, no. 12, pp. 8673–8678,2010.

[8] D.-F. Li, “TheGOWAoperator based approach tomultiattributedecision making using intuitionistic fuzzy sets,” Mathematicaland Computer Modelling, vol. 53, no. 5-6, pp. 1182–1196, 2011.

[9] J. M. Merigó and M. Casanovas, “Fuzzy generalized hybridaggregation operators and its application in fuzzy decisionmaking,” International Journal of Fuzzy Systems, vol. 12, no. 1,pp. 15–24, 2010.

[10] W. Wang and X. Liu, “Intuitionistic fuzzy information aggre-gation using einstein operations,” IEEE Transactions on FuzzySystems, vol. 20, no. 5, pp. 923–938, 2012.

[11] W.Wang andX. Liu, “Intuitionistic fuzzy geometric aggregationoperators based on einstein operations,” International Journal ofIntelligent Systems, vol. 26, no. 11, pp. 1049–1075, 2011.

[12] G. Wei, “Some induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecisionmaking,”Applied Soft Computing Journal, vol. 10, no. 2,pp. 423–431, 2010.

[13] G. W. Wei, “Some geometric aggregation functions and theirapplication to dynamic multiple attribute decision makingin the intuitionistic fuzzy setting,” International Journal ofUncertainty, Fuzziness and Knowlege-Based Systems, vol. 17, no.2, pp. 179–196, 2009.

[14] G. Wei, “Some induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecisionmaking,”Applied Soft Computing Journal, vol. 10, no. 2,pp. 423–431, 2010.

[15] G. Wei and X. Zhao, “Some induced correlated aggregatingoperators with intuitionistic fuzzy information and their appli-cation to multiple attribute group decision making,” ExpertSystems with Applications, vol. 39, no. 2, pp. 2026–2034, 2012.

[16] M. Xia and Z. Xu, “Entropy/cross entropy-based group decisionmaking under intuitionistic fuzzy environment,” InformationFusion, vol. 13, no. 1, pp. 31–47, 2012.

[17] Z. Xu, “Some similarity measures of intuitionistic fuzzy sets andtheir applications to multiple attribute decision making,” FuzzyOptimization and Decision Making, vol. 6, no. 2, pp. 109–121,2007.

[18] Z. Xu, “On consistency of the weighted geometric meancomplex judgement matrix in AHP,” European Journal of Oper-ational Research, vol. 126, no. 3, pp. 683–687, 2000.

[19] Z. Xu, “Approaches to multiple attribute group decisionmakingbased on intuitionistic fuzzy power aggregation operators,”Knowledge-Based Systems, vol. 24, no. 6, pp. 749–760, 2011.

[20] Z. Xu, “An overview ofmethods for determiningOWAweights,”International Journal of Intelligent Systems, vol. 20, no. 8, pp.843–865, 2005.

[21] Z. S. Xu and Q. L. Da, “The uncertain OWA operator,” Interna-tional Journal of Intelligent Systems, vol. 17, no. 6, pp. 569–575,2002.

[22] Z. S. Xu, “Models for multiple attribute decision makingwith intuitionistic fuzzy information,” International Journal ofUncertainty, Fuzziness and Knowlege-Based Systems, vol. 15, no.3, pp. 285–297, 2007.

[23] Z. Xu, “Multi-person multi-attribute decision making modelsunder intuitionistic fuzzy environment,” Fuzzy Optimizationand Decision Making, vol. 6, no. 3, pp. 221–236, 2007.

[24] Z. Xu and R. R. Yager, “Intuitionistic fuzzy bonferroni means,”IEEE Transactions on Systems, Man, and Cybernetics B, vol. 41,no. 2, pp. 568–578, 2011.

[25] Z. Xu and X. Cai, “Recent advances in intuitionistic fuzzyinformation aggregation,” Fuzzy Optimization and DecisionMaking, vol. 9, no. 4, pp. 359–381, 2010.

[26] R. R. Yager, “Some aspects of intuitionistic fuzzy sets,” FuzzyOptimization andDecisionMaking, vol. 8, no. 1, pp. 67–90, 2009.

[27] J. Ye, “Fuzzy decision-making method based on the weightedcorrelation coefficient under intuitionistic fuzzy environment,”European Journal of Operational Research, vol. 205, no. 1, pp.202–204, 2010.

[28] J. Ye, “Multicriteria fuzzy decision-making method usingentropy weights-based correlation coefficients of interval-valued intuitionistic fuzzy sets,” Applied Mathematical Mod-elling, vol. 34, no. 12, pp. 3864–3870, 2010.

[29] J. Ye, “Cosine similarity measures for intuitionistic fuzzy setsand their applications,”Mathematical and Computer Modelling,vol. 53, no. 1-2, pp. 91–97, 2011.

[30] G. Wei and X. Zhao, �

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