riskassessmentofawindturbineusinganahp-mabac

Research ArticleRisk Assessment of a Wind Turbine Using an AHP-MABACApproach with Grey System Theory A Case Study of Morocco

Rim Bakhat 1 and Mohammed Rajaa2

1Management Logistics and Applied Management Department School of Economics Sciences University of Abdel Malek EssaidiTangier City 90060 Morocco2Management Logistics and Applied Management Department School of Economics Sciences University of Abdel Malek EssaidiTetouan City 93000 Morocco

Correspondence should be addressed to Rim Bakhat rbakhatuaeacma

Received 11 February 2020 Revised 17 June 2020 Accepted 8 July 2020 Published 13 August 2020

Academic Editor Sajad Azizi

Copyright copy 2020 Rim Bakhat and Mohammed Rajaa is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

Clean energy has become a growing concern and many organizations pay attention to environmental protection and energyproduction as well In the last few decades the wind turbine has become the core of clean energy production and has advanced ingenerating electricity from 40 kW to 5mW However the new design of the wind turbine causes several potential failures whichfrequently lead to the inability to accomplish the operational requirements intended to meet the customersrsquo expectations Asa solution to this problem the present paper proposes a novel systematic approach that combines Multicriteria Decision-Making(MCDM) techniques and Failure Mode Effects and Criticality Analysis (FMECA) tool to reveal the fatal failures and optimize themaintenance actions To further develop the preceding framework this work will not only rely on the three risk factors that areinvolved in the traditional Risk Priority Numbers (RPN) approach but also will consider the economic aspect of the system In theproposed approach the grey Analytic Hierarchy Process (AHP) method is applied in the first place to calculate the weights of thefour risk factors criteria Second the grey Multiattribute Border Approximation area Comparison (MABAC) technique is appliedto rank the failure modes and their criticality on the whole system e proposed model is verified within an organization ofrenewable energy production in Morocco Furthermore the results of the comparative and the sensitivity analysis affirm that theproposed research framework is adequate for enhancing other complex systems design especially in a developing world wherefunds and resources are scarce

1 Introduction

In the recent decades maintenance has become a widelyused concept and an indispensable research issue for bothmanagement and engineering disciplines e exploitationof reliability and maintainability concepts has significantlyincreased to ameliorate the preventive measures and thecapability of the system in responding to emergenciesTechnically most systems embody as an amalgamation ofmany other interconnected components which must allfunction together to accomplish the operational re-quirements [1]

As a whole several organizations seek to determine themajor causes of the failures measure their risks and adopt

adequate practical measures to alleviate and control themAccording to IEC 31010 2019 numerous risk assessmentprocedures could be used including Root Cause Analysis(RCA) Fault Tree Analysis (FTA) Event Tree Analysis(ETA) Hazard and Operability Analysis (HAZOP) andFailure Mode and Effect Analysis (FMEA) [2]

Failure Mode and Effect Analysis (FMEA) is a reliabilityanalysis tool and a design ldquobottom-uprdquo approach devoted toexamine the failure modes and determine their effects on theoverall system functioning [3] However it has been appliedin various industrial fields such as the renewable energy[4ndash6] nuclear [7] automotive [8] and manufacturing [9]e latter also comprises a Criticality Analysis (CA) which isa qualitative analysis technique employed to measure the

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 2496914 22 pageshttpsdoiorg10115520202496914

probability of the failure modes of a system [10] eCriticality Analysis (CA) relies on the development of theRisk Priority Number (RPN) approach attained through themultiplication of the three main risk factors ldquoseverity (S)occurrence (O) and detection (D)rdquo [11]

Clean energy notably wind power presents a currentdebate and that goes back to the increasing awareness ofcountries toward environmental protection According tothe US Department of Commerce the sustainablemanufacturing is ldquothe creation of manufactured productsthat use processes that minimize negative environmentalimpacts conserve energy and natural resources are safe foremployees communities and consumers and are eco-nomically soundrdquo [12] Until now the worldwide mountednumerous wind turbines with a capacity of 21 per year Atthe beginning of 2020 the wind power generation will havebecome the pillar of clean energy and will have exceeded15 of the global electricity production [13] However thetraditional risk factors of the RPN approach have beencriticized due to the following motives [14ndash17]

(i) traditional RPN approach takes into account onlythree risk factors abandoning other essentialldquoeconomic aspectsrdquo

(ii) emultiplicity of the risk factors for measuring thefailures risk evaluation is questionable and variantto the critical factors deviations

(iii) Multiplication operation of the risk factors is unableto detect the inherent risk in each failure mode andthat implies the lack in themathematical foundation

(iv) Traditional attributes of the RPN approach lead toimprecise results due more to the fuzzy data col-lected from experts and decision-makers throughthe process of the risk evaluation

(v) traditional RPN approach cannot correctly measurethe weights of the failure modes

(vi) e risk factors are evaluated by cross-functionaldecision-makers using linguistic variables which aretranslated into crisp numbers ranging from 1 to 10neglecting the ingrained ambiguity

A need remains for developing a simplified methodicalrisk assessment approach that serves to identify the potentialfailure modes and mitigate their effects on the whole systemAs an answer to this challenge this paper proposesa methodology that combines multicriteria decision-making(MCDM) methods namely AHP-MABAC and FMECA tocontrol fatal failure modes e proposed framework issupported by the use of the Grey Systems eory (GST) tohandle the fuzziness and the ambiguity of the decision-makersrsquo preference judgements [18] e main contributionof this paper is to take into consideration the economicaspect (expected costs) of the system in the RPN approachand reduce the risk factors integers to a (1ndash10) scale in orderto decrease the risk in the assessment process [19] Fur-thermore the wind turbine has been designated as a com-plex system with hierarchical levels to verify the proposedgrey model Finally a comparative study is performed to

verify the feasibility of the proposed model and a sensitivityanalysis is carried out to check the criteria weights variationimpact on the ranking results

e remainder of the present paper is structured asfollows a literature review of relevant existing studies ispresented in Section 2 In Section 3 the flowchart and re-search framework are provided In Section 4 the grey AHPand grey MABAC models are developed e application ofthe proposed framework in a real-world case study is fur-nished in Section 5 e comparative analysis and theconsequences of the sensitivity analysis are outlined inSection 6 Finally Section 7 highlights the closing remarksstudy contributions conclusions and recommendation forfuture researches

2 Literature Review

21 Risk Factors in FMECA Nowadays the FMECA is usedin several ldquorisk-basedrdquo industries including ldquoengineeringautomotive medical and chemicalrdquo [20]is approach wasinitially introduced by the US Military and it implies aninductive ldquobottom-uprdquo risk analysis [10] e FMECA isdefined as ldquoan analysis tool that identifies all the waysa particular component can fail what its effects would be atthe subsystem level and ultimately on the system and whatthe criticality isrdquo [21] On the other hand Lee defined theFMEA as ldquoa procedure for analysis of potential failure modeswithin a system using the classification by the severity orevaluation of the failurersquos effect upon the system where thefailure modes refer categories of detailed failures accordingto the mechanism under a certain circumstance of the wholesystemrdquo [22]e Criticality Analysis (CA) is as an extensionof the FMEAwhich allows analysts to quantitatively rank thepotential failure modes regarding their criticality on thewhole system [23] e latter could be performed quanti-tatively or qualitatively and also achieved with or withoutavailable data [24] Braglia [25] acknowledged that theFMECA is anMCDM issue and highlighted four risk factorsnamely severity detection occurrence and estimated costserefore the MCDM methodology is relevant to prioritizethe failure modes and eliminate the inherent risks in thetraditional FMECA [26]

22 Recent Applications of the AHP and MABAC MethodIn general the Multicriteria Decision-making (MCDM)methodology is regarded as one of the most practical ap-proaches in decision science and comprises a wide set oftechniques that qualitatively and quantitatively answer theresearch questions [27] Due to its practicability severalstudies have used MCDM approaches to solve complexproblems in the wind turbines [28ndash31] Besides the AHPmethod has been also employed to overcome difficulties indifferent areas such as ldquoinformation technologyrdquo [32]ldquomanagementrdquo [33] ldquomaintenancerdquo [34] and ldquosustainableconstructionrdquo [35] On the other hand the multiattributeborder approximation area comparison (MABAC) is one ofthe most precise MCDM techniques that make the in-formation as exact as possible by calculating the closeness

2 Mathematical Problems in Engineering

coefficient values However the MABAC technique hascaptivated numerous researchers since its commencementand it has been successfully applied in different researchdomains such as ldquotransport and logisticsrdquo [36] ldquomaterialevaluation and selectionrdquo [37] ldquosystem engineer evaluationrdquo[38] and ldquoposition selection of wind farmsrdquo [39] Based onthe literature review any research which has used a greymodel namely AHP-MABAC is yet to assess the failuremodes in the wind turbine

23 Grey Systems -eory (GST) Artificial IntelligenceModels (AIMs) embrace numerous practical models whichcan be used with the help of computer systems such as GreySystems eory (GST) Fuzzy Set eory (FST) Rough Seteory (RST) Decision Tree (DT) Support Vector Machine(SVM) Bayesian Networks (BN) Association Rule (AR)DempsterndashShafer eory (DST) and Case-Based Reasoning(CBR) [40] e AIMs can solve different problems based onexpertsrsquo knowledge and experiences erefore the GreySystems eory is adopted to handle the uncertainty in thedecision-making process through discrete information andpartial data [41] e latter is determined by grey formulasgrey numbers and grey matrices and it can be merged withthe FMECA [42] In this regard grey system theory has beenselected as the convenient artificial intelligence model due toits capacity to handle the fuzziness of expertsrsquo judgment andassess the failure modes [43 44] As such the advantages ofthe Grey Systems eory over other models are brieflypresented in Table 1

e major benefit of adopting grey systems theory ismanifested in its capability to formulate rational resultsemploying a slight quantity of data [45 46] However thegrey numbers may contain unfinished or partial in-formation but the interval that comprises their values isdetermined [18] e Grey Systems eory (GST) has beeneffectively employed in various research areas such as ldquotheoutsourcing logistics activitiesrdquo ldquoproject risks managementrdquoldquoautomotive industryrdquo ldquoinformation technologyrdquo ldquobusinessprocessrdquo and ldquoinsurance industryrdquo [47ndash51]

24 FailureModes in aWind Turbine Wind energy is one ofthe available powers in the world and could provide2600TWh by 2020 which represents 123 of the globalelectricity provision increasing to 218 by 2030 [52] Aswind energy exploitation expands organizations requireefficient solutions for expenditures management In generalthe operations and maintenance (OampM) of a 750 kW on-shore wind turbine costs approximately about 75ndash90 of thetotal investment costs [13 53]emain reason behind theseenormous costs is the lack of revealing the most criticalfailure modes in the assembly However two sorts of windturbine systems are globally manufactured horizontal axiswind turbine (HAWT) and vertical axis wind turbine(VAWT) depending on the sector requirements [54] emost used sort of wind turbine in the world is 3-bladeentities containing the essential parts as illustrated inFigure 1

e main role of the rotor and blades is to transfer thewind power into mechanical power through the main shaftby the help of a gearbox and the generatore gearbox helpsthe generator to accelerate the speed close to electricityproduction us the role of the main shaft is to support thewind turbine bearings Placement of the wind turbine withthe wind direction is made by the yaw system and nacelles[56] A wind turbine is a complex system with hierarchicallevels that transforms natural wind energy into electricalenergy e wind turbine system structure is hierarchicaland entails numerous attached parts and components toform a united assembly to generate the electricity Table 2shows the essential components and parts of the windturbine assembly that work altogether to produce electricalpower e rapport is developed based on deep relevantliterature and expertsrsquo knowledge in this area to explain therelations among the indenture levels and the final indenturelevels [54ndash56]

3 Research Framework

31 Projected Flowchart e projected flowchart of failuremode evaluation for the wind turbine based on the greyAHP-MABAC methodology involves two essential phaseswhich are graphically represented in Figure 2 e researchframework can assist decision-makers and managers in theterm of

(i) Determining the risk factors that cause fatal failuremodes in a system

(ii) Calculating the criteria weights of the failure modes(iii) Ranking the most critical failure modes and en-

hance the system design

e details of the proposed research framework areexplained as follows

Phase I define the research problem and structure therisk factors and failure modes the identification of therisk factors and the failure modes comes after the re-search problem definition e risk factors and thefailure modes have been designated through a review ofthe relevant literature and contributions of expertsworking in the clean energy production field As a re-sult four risk factors have been selected namely oc-currence severity detection and expected costsBesides 25 failure modes have been arranged as themost important parts of the wind turbine Further-more a hierarchical framework containing the evalu-ation of the risk factors and the failure modes of thewind turbine is furnished and presented in Figure 2Phase II grey AHP model the grey AHP model isapplied with the support of the grey systems theory toevaluate the risk factor weights by following the cal-culation steps described in Figure 2 is model revealsthe weights of each risk factor and checks the consis-tency rate of the decision-makers judgements as wellPhase III greyMABACmodel the obtained risk factorsweights from the grey AHP model are integrated with

Mathematical Problems in Engineering 3

the grey MABAC model to select the most criticalfailure modes in the wind turbine e initial decisionmatrix is constructed with the support of the experts

32 Proposed Framework e main role of the proposedframework within this study is to support the decision-

making process and guide managers to reveal the fatalfailures in the wind turbine Also this framework will lead tothe application of a rational approach that relies on the riskpriority number (RPN) In theory the RPN considers onlythree types of attributes namely ldquoSeverity (S) Occurrence(O) and Detection (D)rdquo Practically the research frameworkhas taken into account the ldquoexpected costs (ECs)rdquo to pre-cisely measure the failure modes As explained in Figure 3the proposed RPN approach applies linguistic variables toprioritize the probability of the failure modes ldquooccurrencerdquothe severity of the failure modes and their critical effectldquorigorousnessrdquo the opportunity of the failure mode beingrevealed ldquodetectionrdquo and the expected costs of the main-tenance actions ldquoexpected costsrdquo

4 Methodology

41 Preliminaries A grey numberotimesN refers to as an intervalwith defined upper and lower limits and undefined distri-bution information for N [57] In the following equation N

and N denote the lower and upper limits of otimesN corre-spondingly [58]

otimesN N N1113858 1113859 Nprime isin N | N leNprime leN1113858 1113859 (1)

In the following equations four main grey numbermathematical operations are given [43]

Addition otimesN1 + otimesN2 N1 + N2 N1 + N21113960 1113961

Subtraction otimesN1 minus otimesN2 N1 minusN2 N1 minus N21113960 1113961

Division otimesN1 divideotimesN2 N1 N11113960 1113961 times1

N21

N2

⎡⎣ ⎤⎦

Multiplication otimesN1 times otimesN2

min N1 N2 N1 N2 N1 N2 N1N21113872 1113873

max N1 N2 N1 N2 N1 N2 N1N21113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(2)

When it comes to a crisp number the grey aggregationmethod is necessary to be applied In the present worka ldquodegreyingrdquo technique is hereafter applied with the supportof the translating fuzzy data into Crisp Scores (CFCS)

[59 60] us otimesNrij denotes the grey number of a cross-

functional decision-maker who will assess the impact of riski on a risk j where Nr

ij and Nrij represent the lower and

upper grey values of the grey number otimesNrij similarly [58]

Blade

Coupler 1

Coupler 2

Hub

Low speed shaft

High speed shaft

Yaw system

Power cable

Tower

Generator

Nacelle

Controller

TransformerWind vane

Gearbox

Figure 1 Onshore wind turbine hierarchical structure [55]

Table 1 Core difference between the grey set theory and other mathematical models

FeaturesMethods

Grey systems theory Fuzzy mathematics Probability statisticsStudy objects Reduced data uncertainty Rational uncertainty Stochastic uncertaintyModel sets Grey hazy sets Fuzzy sets Cantor setsConditions Any distribution Experience Exact distributionTechniques Data analysis Function of association Probability distributionAim Laws of reality Rational formulas Laws of statisticsStructures Slight samples Experience Vast samples


otimesNrij N

rij N

r

ij]1113960 (3)

e convertion of ambiguous data into crisp scoresentails three main steps represented as follows

Step 1 normalization

1113957Nr

ij Nr

ij minus minjNrij1113872 1113873

Δmaxmin

1113957Nr

ij N

r

ij minus minjNr

ij1113872 1113873

Δminmax

(4)

where

Δmaxmin max

jN

r

ij minus minj

Nrij (5)

Table 2 Rapport among wind turbine parts

Failure category Subsystem Code Failure modes (components) Failure effect and criticality

Aerodynamic system

Blade FM1 ldquoGear teeth sliprdquo ldquoIncapability to regulate the anglerdquoFM2 ldquoBlade crackrdquo ldquoSystem halts functioningrdquo

Hub assembly FM3 ldquoError in positioningrdquo ldquoBlade detach from the hubrdquoFM4 ldquoFracture in the shellrdquo ldquoRotor breaks downrdquo

Tower and nacelle FM5 ldquoFracturerdquo ldquoInsecure structural integrityrdquo

Pitch systemFM6 ldquoElectrical overloadrdquo

ldquoCollapse of pitchrdquoFM7 ldquoLow insulation levelrdquoFM8 ldquoExcessive loadingrdquo

Yaw system FM9 ldquoFatigue or excessive loadingrdquo ldquoCollapse of yawrdquo

Mechanical system

System brakes FM10 ldquoFull speedrdquo ldquoDownfall of other componentsrdquoFM11 ldquoOver heatingrdquo ldquoYaw and rotor breakdownrdquo

Shaft setFM12 ldquoMain shaft vibrationrdquo ldquoParts joining vibrationrdquoFM13 ldquoMain shaft malfunctionrdquo ldquoShaft breakdownrdquoFM14 ldquoFatigue cracksrdquo ldquoRuin of the systemrdquo

GearboxFM15 ldquoGearbox vibrationrdquo ldquoParts connection vibrationrdquoFM16 ldquoGearbox malfunctionrdquo ldquoGearbox breakdownrdquoFM17 ldquoGearbox abnormal noiserdquo ldquoRev of damagerdquo

Electrical system

GeneratorFM18 ldquoExcessive fatiguerdquo ldquoFailure of the generatorrdquoFM19 ldquoPoor lubricationrdquo ldquoRift in shaft rotor bearing and statorrdquoFM20 ldquoOverheatingrdquo ldquoLoss of stator and rotorrdquo

Converter system FM21 ldquoShorting the circuitrdquo ldquoIn capability to transmit the energyto the transformerrdquo

Centralized lubrication system FM22 ldquoOverloadingrdquo ldquoCollapse of the pumprdquoFM23 ldquoShorting circuitrdquo ldquoRisk on electronic componentsrdquo

Power electrical system FM24 ldquoFatiguerdquo ldquoFailure of the power systemrdquoFM25 ldquoDistortingrdquo ldquoCollapse of the systemrdquo

Occurrence (O)

Severity (S)

Detection (D)

Expected costs (EC)

Basic set of criteria

Primary criteria

Adaptive criteria

Required set of criteria

Dimensions

FMECA

Qualitative criticality analysis (CA) method

Failure mode 3

Failure mode n

Failure mode 2

Failure mode 1

Failure mode n ndash 1

Risk priority numberRPN

Figure 2 Hierarchical framework


Determine the scale of the dimension (RPN)

Aggregate the failure modes and collect the data set from experts

The grey analytic hierarchy process G-AHP

The grey multiattribute border approximation area comparison

MABAC-G

Structuring the grey decision hierarchy

No

Convert the linguistic variables into the grey values

Make a pairwise comparison and establish the grey comparison matrix

Calculate the grey weights of the criteria

Check the CR of the grey comparison matrix

No

Normalize the grey row sums define the grey weight of each criterion and transfer the grey

values into MABAC-G

Formulate the decision matrices


Calculate and normalize the grey aggregated matrix

Compute the grey weighted decision matrix

Calculate the closeness coefficient

Yes

Yes

Phas

e I

dim

ensio

n an

dcr

iteria

Phas

e III

the g

rey

syste

ms t

heor

y to

supp

ort M

ABA

C

Phas

e II t

he g

rey

syste

ms t

heor

y to

supp

ort A

HP

Determine critical parts in the system

Report the maintenance department improve the system and design

Rank all failure modes

Phas

e IV

appl

icat

ion

Determine the grey BAA matrix

Compute the alternatives distance

Approve thedecision

hierarchy

0 lt CR lt 01

Figure 3 Projected flowchart


Table 4 Linguistic terms and the evaluation scale for ldquooccurrencerdquo [43 49 64]

Probability of failure Code Grey value Probability of failure occurrenceExtremely less (EL) [1 2] lt10minus5

Remote (R) [1 3] 10minus5

Very slight (VS) [2 4] 10minus5

Slight (S) [3 5] 4 times 10minus4

Occasional (O) [4 6] 2 times 10minus3

Moderate (M) [5 7] 1 times 10minus2

Frequent (F) [6 8] 4 times 10minus2

High (H) [7 9] 020Very high (VH) [8 10] 033Extremely high (EH) [9 10] ge05

Table 6 Linguistic terms and the evaluation scale for ldquodetectionrdquo [43 49 64]

Detection Code Grey value Possibility of detecting potential failuresAlmost certain (AC) [1 2] ldquoCertain detectedrdquoVery high (VH) [1 3] ldquoVery high detectionrdquoHigh (H) [2 4] ldquoHigh detectionrdquoModerately high (MH) [3 5] ldquoModerate detectionrdquoMedium (M) [4 6] ldquoMedium detectionrdquoLow chance (LC) [5 7] ldquoLow detectionrdquoSlight (S) [6 8] ldquoVery low detectionrdquoRemote (R) [7 9] ldquoRemote detectionrdquoVery remote (VR) [8 10] ldquoVery remote detectionrdquoAbsolute uncertainty (AU) [9 10] ldquoNo inspection and no chancerdquo

Table 7 Linguistic terms and the evaluation scale for ldquoexpected costsrdquo

Cost Code Grey value System repair costsClose to original price (O) [9 10] Cost is similar to the originalEnormously important (EI) [8 10] Cost is enormously importantrsquoVery important (VI) [7 9] Cost is very importantrsquoImportant (I) [6 8] Cost is importantrsquoModerately important (MI) [5 7] Cost is moderately importantrsquoModerate (M) [4 6] Cost is moderatersquoModerately low (ML) [3 5] Cost is quite lowrsquoSoft (S) [2 4] Cost is acceptablersquoDistant (D) [1 3] Cost is affordablersquoApproximately no cost (N) [1 2] Almost no cost is requiredrsquo

Table 5 Linguistic terms and the evaluation scale for ldquoseverityrdquo [43 49 64]

Failure effect Code Grey value Severity effectVery minor (VM) [1 3] Effect is not noticedrsquoMinor (MI) [2 4] Very slight effect noticedrsquoLow (L) [3 5] Slight effect causing annoyancersquoModerate (MO) [4 6] Moderate effected and maintenance is requiredrsquoSignificant (S) [5 7] Significant effect and system performance is degradedrsquoMajor (MA) [6 8] Major effect and system performance is affectedrsquoExtreme (E) [7 9] Extreme effect system inoperable and safety issuersquoHazardous (H) [8 10] ldquoCritical effect and system safety riskrdquoVery hazardous (VH) [9 10] ldquoHigher severity ranking of a potential failure moderdquo

Table 3 Grey AHP model linguistic terms and grey weights [61]

Linguistic weights Code Grey weightsAbsolute important (AI) [7 9]More important (MI) [5 7]Important (I) [3 5]Moderately important (DI) [1 3]Equal important (EI) [1 1]


Step 2 computation of the total normalized crispvalues

Crij

1113957Nr

ij 1 minus 1113957Nr

ij1113872 1113873 + 1113957Nr

ij times 1113957Nr

ij1113874 11138751113874 1113875

1 minus 1113957Nr

ij + 1113957Nr

ij1113874 1113875

(6)

Step 3 computation of the crisp values

Vrij min

jN

rij + C

rijΔ

maxmin (7)

42 Grey AHP Model for the Risk Factors Evaluation egrey AHP model is used to calculate the weights of riskfactors It entails three essential paces that could beexplained as follows [61]

Step 1 formulation of the grey comparison matrix thedecision-makers dispense a list of linguistic terms as il-lustrated in Table 3 en these linguistic terms aretranslated into grey weights as indicated in Table 3 usthe grey comparisonmatrix (otimesQ) is constructed as follows

otimesQ otimesQij1113872 1113873ntimesn

(8)

where

otimesQij Qij Qij1113876 1113877

Qminus1ij

1Qij

1

Qij

⎡⎢⎢⎣ ⎤⎥⎥⎦

(9)

e values Qij and Qij presented in equation (9) denotethe minimum and the maximum values of the constructedgrey comparison matrix otimesQij respectively

Step 2 calculation of the consistency rate the obtainedgrey values are transformed into crisp values by uti-lizing equation (10) Afterwards the consistency rateof the constructed grey matrix is verified with the helpof equations (11) and (12) [62] In this pace the de-cision-makers calculate the consistency index (CI) andcompare it to the random index (RI) for the

Select wind turbine failure modes by priority

Occurrence Severity Detection ExpectedCosts

Main objective

Risk factors criteria

FM 01

FM 02

FM n ndash 1

FM n

FM 03

FM 01

FM 02

FM n ndash 1

FM n

FM 03

FM 01

FM 02

FM n ndash 1

FM n

FM 03

FM 01

FM 02

FM n ndash 1

FM n

FM 03 Alternatives

(faults)

Figure 4 Hierarchical tree of the risk assessment

Table 9 Grey AHP model results

CriteriaResults

otimesYi otimeswi

O [2286 24] [0061 0064]S [2222 2286] [0060 0061]D [14 20] [0376 0537]EC [13333 18] [0358 0483]

CR 005638lt 01

Table 8 Grey comparison matrix

CriteriaCriteria

O S D ECO [1 1] [1 1] [0143 02] [0143 02]S [1 1] [1 1] [0111 0143] [0111 0143]D [5 7] [7 9] [1 1] [1 3]EC [5 7] [7 9] [0333 1] [1 1]


Table 11 Average grey decision matrix (1113954T)

AlternativesCriteria

O S D ECAlternative 1 (FM1) [1 3] [9 10] [3 5] [9 10]Alternative 2 (FM2) [1 2] [9 10] [1 3] [5 7]Alternative 3 (FM3) [2 4] [9 10] [4 6] [7 9]Alternative 4 (FM4) [1 2] [9 10] [1 2] [9 10]Alternative 5 (FM5) [1 2] [1 3] [4 6] [3 5]Alternative 6 (FM6) [4 6] [4 6] [6 8] [3 5]Alternative 7 (FM7) [2 4] [5 7] [4 6] [3 5]Alternative 8 (FM8) [2 4] [4 6] [7 9] [4 6]Alternative 9 (FM9) [5 7] [9 10] [1 3] [5 7]Alternative 10 (FM10) [1 3] [8 10] [1 3] [5 7]Alternative 11 (FM11) [7 9] [4 6] [8 10] [3 5]Alternative 12 (FM12) [1 2] [1 3] [8 10] [1 2]Alternative 13 (FM13) [1 2] [1 3] [6 8] [6 8]Alternative 14 (FM14) [1 2] [2 4] [5 7] [6 8]Alternative 15 (FM15) [2 4] [2 4] [7 9] [1 2]Alternative 16 (FM16) [2 4] [9 10] [1 3] [9 10]Alternative 17 (FM17) [3 5] [2 4] [7 9] [2 4]Alternative 18 (FM18) [3 5] [9 10] [2 4] [9 10]Alternative 19 (FM19) [6 8] [9 10] [4 6] [9 10]Alternative 20 (FM20) [5 7] [9 10] [3 5] [9 10]Alternative 21 (FM21) [3 5] [9 10] [7 9] [1 2]Alternative 22 (FM22) [6 8] [6 8] [6 8] [4 6]Alternative 23 (FM23) [7 9] [4 6] [5 7] [2 4]Alternative 24 (FM24) [7 9] [5 7] [7 9] [1 3]Alternative 25 (FM25) [8 10] [8 10] [4 6] [2 4]

Table 10 Performance rating of the decision-makers


O S D ECFM1 R VS VS EL VS VH VH H VH VH MH M LC H M O O EI O OFM2 EL R EL R R VH VH VH H H VH H S VH LC M M I I IFM3 VS S R R R VH VH VH H VH M L S VH VH VI I EI VI EIFM4 EL R R EL EL VH VH H VH VH AC VH VH VH VH O O O EI OFM5 EL R EL R R VM MI MI MI MI M C MH LC LC ML ML L M MFM6 O M M S F MO S S L S S R R R ML M ML ML MLFM7 VS S R O S S MA MO E MO M MH MH S LC ML ML L D IFM8 VS VS VS EL VS MO S S S L R VR R VR M M M MI MIFM9 M O O O O VH H H MA H VH H H VH AC MI M M I MIFM10 R VS VS EL EL H H E MA E VH H H H VH MI MI M S MFM11 H H F H VH MO S S MO S VR R R AU R ML S ML ML IFM12 EL EL EL EL EL VM MI MI L VM VR R VR R VR N D N S DFM13 EL EL EL R VS VM MI MI MI MI S LC M LC LC I VI VI I VIFM14 EL EL R EL R MI L L VM MI LCM M I I VI I IFM15 VS S S VS S MI L VM L MO R S VR S M N D D N DFM16 VS R VS O VS VH VH MI VH H VH VH AC VH AC O O O O EIFM17 S VS O VS O MI L VH MO L R VR S LC S S ML D D SFM18 S VS S VS S VH H H L H H VH VH H AU O O EI EI OFM19 F VS H F H VH H H H VH M H R R R VR O O EI VI EIFM20 M F F M F VH VH VH VH H MH VH S AU R O O O O OFM21 S VS S S VS VH H E H E R H LC VR VR N D ML D MIFM22 F VH S F F MA E S E MO S R MH LC S M MI M M MFM23 H VH VH H H MO S MO S E LC S AU S M S ML D S DFM24 H EH VH VH VH S MA MO L S R VR R VR R D D S ML SFM25 VH EH EH M VH H H E E E R VR R VR R D ML D D D M


Table 12 Normalized grey decision matrix (1113954P)



Table 13 Weighted grey decision matrix (1113954G)




Table 15 Grey BAA matrix (X)


O S D ECAlternative 1 (FM1) [minus00215 minus00133] [00292 00212] [01198 01058] [02181 01908]Alternative 2 (FM2) [minus00215 minus00209] [00292 00212] [02575 02311] [00297 00460]Alternative 3 (FM3) [minus00108 minus00057] [00292 00212] [00510 00431] [01239 01425]Alternative 4 (FM4) [minus00215 minus00209] [00292 00212] [02575 02937] [02181 01908]Alternative 5 (FM5) [minus00215 minus00209] [minus00371 minus00304] [00510 00431] [minus00646 minus00506]Alternative 6 (FM6) [00106 00095] [minus00122 minus00083] [minus00867 minus00822] [minus00646 minus00506]Alternative 7 (FM7) [minus00108 minus00057] [minus00040 minus00009] [00510 00431] [minus00646 minus00506]Alternative 8 (FM8) [minus00108 minus00057] [minus00122 minus00083] [minus01556 minus01449] [minus00174 00023]Alternative 9 (FM9) [00213 00172] [00292 00212] [02575 02311] [00297 00460]Alternative 10 (FM10) [minus00215 minus00133] [00209 00212] [02575 02311] [00297 00460]Alternative 11 (FM11) [00427 00324] [minus00122 minus00083] [02244 minus02075] [minus00646 minus00506]Alternative 12 (FM12) [minus00215 minus00209] [minus00371 minus00304] [02244 minus02075] [minus01588 minus01954]Alternative 13 (FM13) [minus00215 minus00209] [minus00371 minus00304] [minus00867 minus00822] [00768 00942]Alternative 14 (FM14) [minus00215 minus00209] [minus00288 minus00231] [minus00179 minus00196] [00768 00942]Alternative 15 (FM15) [minus00108 minus00057] [minus00288 minus00231] [minus01556 minus01449] [minus01588 minus01954]Alternative 16 (FM16) [minus00108 minus00057] [00292 00212] [02575 02311] [02181 01908]Alternative 17 (FM17) [minus00001 00019] [minus00288 minus00231] [minus01556 minus01449] [minus01117 minus00988]Alternative 18 (FM18) [minus00001 00019] [00292 00212] [01887 01684] [02181 01908]Alternative 19 (FM19) [00320 00248] [00292 00212] [00510 00431] [02181 01908]Alternative 20 (FM20) [00213 00172] [00292 00212] [01198 01058] [02181 01908]Alternative 21 (FM21) [minus00001 00019] [00292 00212] [minus01556 minus01449] [minus01588 minus01954]Alternative 22 (FM22) [00320 00248] [00043 00064] [minus00867 minus00822] [minus00174 minus00023]Alternative 23 (FM23) [00427 00324] [minus00122 minus00083] [minus00179 minus00196] [minus01117 minus00988]Alternative 24 (FM24) [00427 00324] [minus00040 minus00009] [minus01556 minus01449] [minus01588 minus01471]Alternative 25 (FM25) [00534 00400] [00209 00212] [00510 00431] [minus01117 minus00988]

Table 16 Closeness coefficient (CC) and ranking of the failure modes

Alternatives CCi RankAlternative 1 (FM1) 06500 6Alternative 2 (FM2) 05722 8Alternative 3 (FM3) 03944 10Alternative 4 (FM4) 09681 1Alternative 5 (FM5) minus01310 16Alternative 6 (FM6) minus02845 18Alternative 7 (FM7) minus00424 13Alternative 8 (FM8) minus03572 19Alternative 9 (FM9) 06531 5Alternative 10 (FM10) 05715 9Alternative 11 (FM11) minus04926 20Alternative 12 (FM12) minus08961 25Alternative 13 (FM13) minus01079 14Alternative 14 (FM14) 00393 11Alternative 15 (FM15) minus07230 24Alternative 16 (FM16) 09314 2Alternative 17 (FM17) minus05610 22Alternative 18 (FM18) 08182 3Alternative 19 (FM19) 06101 7Alternative 20 (FM20) 07233 4Alternative 21 (FM21) minus06024 23Alternative 22 (FM22) minus01212 15Alternative 23 (FM23) minus01934 17Alternative 24 (FM24) minus05362 21Alternative 25 (FM25) 00190 12

Table 14 Grey BAA matrix (1113954B)


O S D ECGEO [00963 00819] [01033 00821] [07064 07088] [05357 05815]


appropriate value of criteria n To move to the nextpace the consistency rate of the study must be minorthan 01

Qij 12

times Qij +Qij1113874 1113875 (10)

CI nablamax minus n( 1113857

(n minus 1) (11)

CR CIRI

1113874 1113875 (12)

Step 3 normalization of the grey weights the grey rowsums (otimesSi) are normalized by applying equations (13)(15) and (16) given below to determine the final greyweights (otimesxi) of criterion respectively e obtainedgrey values are subsequently integrated with the greyMABAC method to rank the alternatives

otimesSi 1113944n

j1Qij Qij1113876 1113877 (13)

otimesSi Si Si1113960 1113961 (14)

Slowasti

2 times Si

1113936ni1 Si + 1113936

ni1 Si

⎡⎢⎣ ⎤⎥⎦ (15)

Slowasti 2 times Si

1113936ni1 Si + 1113936

ni1 Si

⎡⎢⎣ ⎤⎥⎦ (16)

otimesXi Slowasti Slowasti1113876 1113877 xi xi1113960 1113961 (17)

43 Grey MABAC Model for the Failure Modes PrioritizingAfter obtaining the weights coefficients from the grey AHPmodel the ground is ready to develop the mathematicalequations of the grey MABAC model e process of

ndash10000

ndash05000

00000

05000

10000

15000

0 5 10 15 20 25 30Risk

fact

ors f

unct

ions

AlternativesA1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13A14 A15 A16 A17 A18 A19 A20 A21 A22 A23 A24 A25

G+ upper approximation area

G border approximation area

Gndash lower approximation area

Figure 5 Grey MABAC ranking results

Table 17 Summary of the comparative analysis

Applied modelsRanking order

Most critical Less criticalClassical RPN FM25 gt FM19 gt FM22 mdashClassical RPN with the help of the developed integrated G-AHP and MABAC-G model FM12 lt FM8 lt FM21 FM9 gt FM16 gt FM4Proposed RPN with the help of the developed integrated G-AHP and MABAC-G model FM12 lt FM15 lt FM21 FM4 gt FM16 gt FM18


Table 18 Comparative analysis results

Alternatives

Approaches

Traditional RPN Traditional RPN with thegrey model

Proposed RPN with thegrey model

(OtimesStimesD) Rank CCi Rank CCi RankAlternative 1 (FM1) 72 14 01132 7 06500 6Alternative 2 (FM2) 20 20 01910 5 05722 8Alternative 3 (FM3) 150 10 00386 9 03944 10Alternative 4 (FM4) 9 22 02446 3 09681 1Alternative 5 (FM5) 5 25 minus00887 21 minus01310 16Alternative 6 (FM6) 175 8 minus00336 17 minus02845 18Alternative 7 (FM7) 60 16 minus00078 15 minus00424 13Alternative 8 (FM8) 120 11 minus02098 24 minus03572 19Alternative 9 (FM9) 8 13 03357 1 06531 5Alternative 10 (FM10) 36 19 02157 4 05715 9Alternative 11 (FM11) 288 5 00079 12 minus04926 20Alternative 12 (FM12) 9 22 minus02794 25 minus08961 25Alternative 13 (FM13) 7 24 minus00680 20 minus01079 14Alternative 14 (FM14) 12 21 minus00349 18 00393 11Alternative 15 (FM15) 48 18 00460 8 minus07230 24Alternative 16 (FM16) 60 16 02942 2 09314 2Alternative 17 (FM17) 64 15 minus00606 19 minus05610 22Alternative 18 (FM18) 120 11 minus01159 22 08182 3Alternative 19 (FM19) 350 2 00016 13 06101 7Alternative 20 (FM20) 240 6 00255 11 07233 4Alternative 21 (FM21) 160 9 minus01311 23 minus06024 23Alternative 22 (FM22) 336 3 00006 14 minus01212 15Alternative 23 (FM23) 192 7 01881 6 minus01934 17Alternative 24 (FM24) 320 4 00276 10 minus05362 21Alternative 25 (FM25) 360 1 minus00318 16 00190 12

y = 05269x + 615R2 = 02776

Rank

ing

of th

e tra

ditio

nal R

PN w

ith th

e use

of th

e int

egra

ted

mod

el

0

5

10

15

20

25

30

5 10 15 20 25 300Ranking of the proposed model

Figure 7 Positive correlation

y = ndash0058x + 13747R2 = 00033


0

5

10

15

20

25

30

Rank

ing

of th

e tra

ditio

nal R

PN

Figure 6 Negative correlation


applying the grey MABAC model consists of seven mainpaces as given below

Step 1 generating the initial decision matrix (1113954T) theresearch problem considers m number of alternatives(Ai i 1 2 3 m) and n number of

criteria(Cj j 1 2 3 n) Here 1113954Tk

[otimestkij]mtimesn de-

notes the grey decision matrix generated by the de-cision-maker Rk and with the support of Tables 4ndash6

1113954Tk

otimestkij1113960 1113961

mtimesn

tk11 t

k

11

tk11 t

k

11

⎡⎢⎢⎣ ⎤⎥⎥⎦tk12 t

k

12

tk12 t

k

12

⎡⎢⎢⎣ ⎤⎥⎥⎦tk13 t

k

13

tk13 t

k

13

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ middot middot middottk1n t

k

1n

tk1n t

k

1n

⎡⎢⎢⎣ ⎤⎥⎥⎦

middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot

tkm1 t

k

m11113960 1113961 tkm2 t

k

m21113960 1113961 tkm3 t

k

m31113960 1113961 middot middot middot tkmn t

k

mn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(18)

ndash15000

ndash10000

ndash05000

00000

05000

10000

15000

0 1 2 3 4 5 6

AlternativesA8A17

A9A18

A1A10A19

A2A11A20

A3A12A21

A4A13A22

A5A14A23

A6A15A24

A7A16A25

Upper approximation area

Border approximation area

Lower approximation area

Crite

ria fu

nctio

ns

Figure 8 Developed model sensitivity analysis

Table 19 Diverse scenarios of risk factors weights

RPN criteriaCriteria weights scenarios

Original Scenario 1 Scenario 2 Scenario 3 Scenario 4C1 (occurrence) 00679 00793 00788 1061 00774C1 (severity) 00589 00516 00718 00783 00699C3 (detection) 04916 04896 02416 02696 02154C4 (expected cost) 03815 03795 06078 05461 06373


where otimestkij denotes the performance grade of Ai with respect

to criterion Cj according to Rk(k 1 2 3 K)us thecross-functional decision-makers K are involved in theevaluation procedure Each Rs is given equal importance εk(where 1113936

Kkminus1 αk 1)en the grey systems theory is applied

to handle the fuzziness of the collected data e linguisticvariables and the grey scale for the four risk factors are givenin Tables 4ndash7

Step 2 formation of the grey decision matrix (1113954T) thedecision matrices are gathered from the cross-func-tional decision-makers to aggregate the initial decisionmatrices 1113954T

k(k 1 2 3 K) into a grey decision

matrix set 1113954T [otimestij]mtimesn as follows

1113954Tk

otimestij1113960 1113961mtimesn

t11 t111113858 1113859 t12 t121113858 1113859 t13 t131113858 1113859 middot middot middot t1n t1n1113858 1113859



middot middot middot middot middot middot middot middot middot middot middot middot

tm1 tm11113858 1113859 tm2 tm21113858 1113859 tm3 tm31113858 1113859 middot middot middot tmn tmn1113858 1113859

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(19)

tij 1113944K

k1αk times t

kij tij 1113944

K

k1αk times t

k

ij (20)

where m denotes the total number of the alternatives and n

denotes the total number of the criteriaStep 3 normalization of the elements from the greyaggregated decision matrix (1113954P) the normalization of

Table 20 Ranking results based on diverse scenarios of risk factors weights

Failure modesAlternatives ranking response scenarios

Original Scenario 1 Scenario 2 Scenario 3 Scenario 4CCi (rank) CCi (rank) CCi (rank) CCi (rank) CCi (rank)

A1 (FM1) 06500 (6) 05410 (6) 06217 (5) 08459 (3) 07212 (3)A2 (FM2) 05722 (8) 03467 (9) 04794 (8) 05050 (7) 03772 (8)A3 (FM3) 03944 (10) 01699 (10) 05511 (7) 04236 (8) 05660 (5)A4 (FM4) 09681 (1) 09408 (3) 09203 (2) 08487 (2) 08526 (1)A5 (FM5) minus01310 (16) minus02778 (18) minus03132 (19) minus01106 (15) minus02436 (19)A6 (FM6) minus02845 (18) minus03288 (19) minus02572 (17) minus03148 (19) minus03591 (20)A7 (FM7) minus00424 (13) 01055 (13) minus02655 (18) 4036 (20) 01784 (12)A8 (FM8) minus03572 (19) minus05281 (21) minus02248 (15) minus00571 (13) minus01942 (18)A9 (FM9) 06531 (5) 03987 (8) 02345 (11) 05798 (5) 03513 (9)A10 (FM10) 05715 (9) 04606 (7) 01760 (12) minus00957 (14) 00608 (14)A11 (FM11) minus04926 (20) minus04923 (20) minus02375 (16) 2835 (18) 01942 (11)A12 (FM12) minus08961 (25) minus07710 (25) minus08425 (25) minus05213 (23) minus09461 (25)A13 (FM13) minus01079 (14) 01182 (11) 03861 (9) 1624 (10) 02889 (10)A14 (FM14) 00393 (11) minus01073 (15) 02939 (10) 0646 (12) 01514 (13)A15 (FM15) minus07230 (24) minus05652 (22) minus07375 (24) minus06103 (25) minus06206 (24)A16 (FM16) 09314 (2) 09408 (3) 09718 (1) 10169 (1) 08216 (2)A17 (FM17) minus05610 (22) minus06616 (23) minus04534 (21) minus04502 (21) minus05567 (23)A18 (FM18) 08182 (3) 095319 (2) 08830 (3) 07857 (4) 04921 (7)A19 (FM19) 06101 (7) 08132 (5) 05993 (6) 04205 (9) 05290 (6)A20 (FM20) 07233 (4) 10399 (1) 07262 (4) 05063 (6) 06335 (4)A21 (FM21) minus06024 (23) minus008569 (14) minus01375 (14) minus05565 (24) minus01154 (15)A22 (FM22) minus01212 (15) minus01358 (16) 00970 (13) 01355 (11) minus01543 (16)A23 (FM23) minus01934 (17) minus02310 (17) minus06102 (23) minus02233 (16) minus04785 (22)A24 (FM24) minus05362 (21) minus06884 (24) minus03692 (20) minus02452 (17) minus04684 (21)A25 (FM25) 00190 (12) 01061 (12) minus05139 (22) minus05204 (22) minus01867 (17)


the elements of the grey aggregated matrix (1113954P) areidentified from the initial matrix 1113954T applying the fol-lowing equation

otimespij pij

pij1113876 1113877

tij

t+j

tij

t+j

tminusj

tij

tminusj

tij

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(21)

where t+j max1leilem(tij) denotes the benefit category cri-

teria where maximal value of the criterion is required whiletminusj min1leilem(tij) indicates the cost category criteria wherethe minimum value of the criterion is required ereforethe normalized grey decision matrix is illustrated as follows

1113954P otimestij1113960 1113961mtimesn

p11 p111113960 1113961 p12 p121113960 1113961 p13 p131113960 1113961 middot middot middot t1n t1n1113858 1113859

p21 p211113960 1113961 p22 p221113960 1113961 p23 p231113960 1113961 middot middot middot p2n p2n1113960 1113961



pm1 pm11113960 1113961 p

m2 pm21113960 1113961 pm3 pm31113960 1113961 middot middot middot p

mn pmn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(22)

Step 4 computation of the elements from the grey-weighted decision mrix (1113954G) the elements of theweighted matrix (1113954G) are computed based on equation(23) us Wj denotes the weighted coefficients of the

criterion j By applying equation (24) the weightedmatrix (1113954G) is formulated as follows

otimesgij gij

gij1113876 1113877 Wj times otimespij Wj times p11 Wj times p111113960 1113961 (23)

1113954G otimesgij1113960 1113961mtimesn

g11 g111113960 1113961 g12 g121113960 1113961 g13 g131113960 1113961 middot middot middot g1ng1n1113960 1113961




gm1 gm11113960 1113961 g

m2 gm21113960 1113961 gm3 gm31113960 1113961 middot middot middot g

mngmn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(24)

Step 5 calculation of the grey Border ApproximationArea (BAA) matrix (1113954B) the grey Border Approxima-tion Area (BAA) for each criterion is calculated byusing equation (25)

otimesbij bj bj1113960 1113961 1113945m

i1g

ij⎛⎝ ⎞⎠

1m

1113945m

i1gij

⎛⎝ ⎞⎠

1m⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ (25)

us [gij

gij] denotes the elements of the weightedmatrix (1113954G) and m indicates the total number of al-ternatives Once the grey value otimesbij [bj bj] is cal-culated for each criterion a border approximation areavector is generated However the BAA is an orientationpoint for each alternative Furthermore the grey vector1113954b (otimesg1otimesg2otimesg3 otimesgn)1timesn is used in the greyBorder Approximation Area (BAA)matrix (1113954B) as rowsof the following matrix


1113954B

b1 b11113960 1113961 b2 b21113960 1113961 b3 b31113960 1113961 middot middot middot bn bn1113960 1113961





⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(26)

Step 6 computation of the alternatives distance fromthe BAA matrix for the matrix elements (X)e distance of the alternatives is calculated employingthe ldquoEuclidean distancerdquo between the grey numbers

otimesgij and otimesbjen the elements matrix X is carried outas follows

X 1113954G minus 1113954B xij1113960 1113961mtimesn

otimesg11 minus otimesb1 otimesg12 minus otimesb2 otimesg13 minus otimesb3 middot middot middot otimesg1n minus otimesbn




otimesgm1 minus otimesb1 otimesgm2otimesb2 otimesgm3otimesb3 middot middot middot otimesgmn minus otimesbn

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(27)

where

X

x11 x12 x13 middot middot middot x1n




xm1 xm2 xm3 middot middot middot xmn

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)

e distance of the alternatives from the Border Ap-proximation Area (BAA) of each criterion is defined as thedifference between the elements in the grey-weighted matrix(1113954G) and the value of the border approximation area (1113954B)erefore otimesbj denotes the border approximation area ofcriterion Cj and otimesgij denotes the elements of the grey-weighted matrix (1113954G) e coefficient correlation is carriedout as follows

CC Ai( 1113857 1113944n

j1xij i 1 2 3 m j 1 2 3 n

(29)

Equation (30) denotes the sum of the distance of thealternatives from the Border Approximation Area (bij) CCindicates the closeness coefficient of each alternative fromthe BAA Alternative Ai (i 1 2 3 m) belongs to theBorder Approximation Area (BAA) ere are three mainareas and are pined as follows

Ai isin

G+ if xij gt 0 upper approximation area

G if xij 0 border approximation area

Gminus if xij gt 0 lower approximation area

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)

where G+ denotes the upper approximation area for alter-natives that are equal or close to the ideal solution mean-while Gminus denotes the lower approximation area foralternatives that are equal or close to the anti-ideal solution

Step 7 ranking the Alternatives the ranking of thealternatives is performed with the help of the followingequation

1113954Ri 1113944n

j1xij i 1 2 3 j 1 2 3 (31)

5 Case Study

e proposed grey model is applied and verified withina multinational company of renewable energy productionimplemented in the north region of Morocco (Tangier andTangier Med Port) e type of wind turbine that has beenstudied and selected for the risk assessment is ldquoB63 witha 294 megawatts power rating and 63 meters long and 17tonnes of weightrdquo However it is one of North Africarsquoslargest and most powerful implanted power generationsystems In the present work data have been collected fromfive cross-functional decision-makers (R1 R2 R3 R4 R5)who are working for the called organization with workexperience ranging from 1 to 25 years andmost of them havegot postgraduation qualifications As mentioned earlier thispaper focuses on fourmain risk factors explained in Figure 4

51 Grey AHPModel Application As stated earlier the greyAHP model application for evaluating the risk priorityentails three principal steps e first step considers the


construction of the grey pairwise comparison matrix esecond step is about normalizing the pairwise comparisonmatrix e third step consists of the consistency index (CI)calculation en compare the latter to the random index(RI) for the appropriate value of n

Step 1 application the grey comparison matrix isconstructed with the support and the assessments of thequalified team e grey comparison matrix isexplained in detail in Table 8Step 2 application the row sums of the grey com-parison matrix and the grey weights of criteria of theproposed model are given in Table 9 e consistencyrate (CR) is calculated and the random index for thefour criteria is 090Step 3 application due to the space limits only oneapplication is provided Table 9 presents the results ofthe Grey AHP model However the degree of con-sistency is satisfactory CIRI 005638 e rate in-dicates that the collected data from decision-makers isreliable and feasible

52 GMABACModel Application As mentioned above thealternatives used in the present work range from the failuremode (FM1) to (FM25) e assessment of the alternatives inthe grey MABAC is related to the four risk factors eobtained weights of the four risk factors are integrated withgrey MABAC to rank the critical failure modes in the windturbine However Table 10 presents the performance ratingsof the five cross-functional decision-makers

Step 1 and 2 applications the linguistic terms aretranslated into a grey number and presented in Ta-ble 11 e aggregated grey decision matrix is obtainedby applying equations (18)ndash(20)Step 3 application the aggregated grey decision matrix(1113954T) is converted into the normalized grey decisionmatrix (1113954P) by using equations (21) and (22) Table 12illustrates the normalized grey decision matrix (1113954P)Step 4 application the weighted grey decision matrix(1113954G) is computed by using the weight vector andequations (19) and (20) However the weighted matrixis given as presented in Table 13Step 5 application this step considers the calculation ofthe Boer Appoximation Area (BAA) by applyingequations (25) and (26) erefore the grey BAAmatrix(1113954B) is performed in this work in order to preciselydetermine the criticality of the failure modes in the windturbine e latter is calculated with the help of thegeometric average as per shown in Table 14Step 6 application the preference index matrix (X) iscalculated by utilizing equation (27) as per viewed inTable 15 However the obtained results are divided intotwo categories e negative values denote the weak-ness (critical failures modes) and the positive valuesrefer to the strength (noncritical failure modes) in thewind turbine assembly

Step 7 application the closeness coefficients for eachfailure mode (CC) are calculated by applying equation(29) and given in Table 16 Furthermore the failuremodes are ranked to the descending order (from theupper value to the lowest value) However the obtainedresults through the application of the proposed greymodel reveal the inherent risk in the wind turbine whereFM4 (fracture in the shell) is less critical than (FM16gearbox malfunction) and which in its turn is less criticalthan (FM18 excessive fatigue of the generator) mean-while the failure mode (FM12 main shaft vibration) is themost critical part in the wind turbine assembly

A graphical representation of the closeness coefficients isnecessary to complete the grey model application Figure 5illustrates the final results of the grey AHP-MABAC modelHowever the failure modes which belong to the upper ap-proximation area G+ are noncritical failures (high-perfor-mance componentsparts) while the failure modes whichbelong to the lower approximation area Gminus are the mostcritical in the system (weak componentsparts)

6 Results and Discussions

A comparative analysis of the three different approaches isfurnished in the first place en the consequences of thesensitivity analysis of the proposed grey model are presented

61 Comparative Analysis A comparison of failure modesranking utilizing different approaches is conducted in thispresent paper to demonstrate the practicability of theproposed grey integrated model First the traditional RPNapproach is applied by multiplying the three classical riskfactors namely lsquooccurrence severity and detectionrsquo with thesupport of the cross-functional decision-makers Second thetraditional RPN is applied by using the proposed grey in-tegrated model ird the proposed RPN approach is appliedto the use of the proposed grey integrated model e rankingresults of the three different approaches are presented in Ta-bles 17 and 18 Here the results validate the statistical sig-nificance of the grey model among the ranking acquiredthrough the application of different methods SubsequentlyPearsonrsquos correlation coefficient [63] is used to comparemutual correspondence between the three approaches Herethe obtained results validate the statistical significance of thegrey risk-basedmodel among the ranking acquired through theapplication of different methods e comparative analysisexplanation of the three approaches is as follows

(i) Traditional RPN approach the most critical failuremodes in the wind turbine according to the firstcomparative approach are ranked as followsFM12 gt FM8 gt FM21 e details are presented inTable 18 Figure 6 shows the negative correlationbetween the first comparative approach and thethird comparative approach e negative correla-tion affirms that the ranking order of both ap-proaches moves in the opposite direction


(ii) Grey integrated model with the traditional RPN ap-proach the most critical failure modes in the windturbine according to the second comparative ap-proach are ranked as follows FM12 lt FM8 ltFM21e first worst-ranked failuremode in this comparativeapproach is the main shaft vibration Next the secondworst-ranked failure mode is Pitch system excessiveloading en the third worse-ranked failure mode isa frequency converter set Meanwhile the less-criticalfailure modes are FM9 gtFM16 gtFM4 Figure 7 showsthe positive correlation between the second compar-ative approach and the third comparative approache positive correlation affirms that the ranking orderof both approaches moves in the same direction edetails are presented in Table 18

(iii) Grey integrated model with the proposed RPN Ap-proach the most critical failure modes in the windturbine according to the third comparative approachare ranked as follows FM12 lt FM15 ltFM21 e firstworst-ranked failure mode in the study is the mainshaft set which has a negative critical effect on theremaining parts of the systemus the secondworst-ranked failure mode is the gearbox vibration whichleads to the collapse of the assembly for a long timeHence the third worse-ranked failure mode is thefrequency converter set that restrains the conversionof the energy to the transformer which in turn plugsthe power generation and causes economic damagesto the company Meanwhile the less-critical failuremodes are FM4 gtFM16 gtFM18e FM4 (the fracturein the shell) is less critical than the FM16 (the gearboxmalfunction) which in turn is less critical than theFM18(the excessive fatigue of the generator) edetails are presented in Table 18

62 Sensitivity Analysis e alternatives ranking results arerelated to the criteria weight coefficientsus it is necessaryto conduct a sensitivity analysis by changing the weightcoefficients of the criteria to test the reliability and practi-cability of the proposed methodology However the alter-ation in criteria weight coefficients drives soft changes in theranking order of the alternatives Based on that a sensitivityanalysis is conducted in the present paper to control theunity amongst the risk factor criteria ldquooccurrence severitydetection and expected costsrdquo observing the diversificationin failure modes ranking with the diversification in criteriaweights e results of the sensitivity analysis for assessingthe n criteria weights (Cj j 1 2 4) are presented inTable 19 and their impact on the alternatives rankingorder (Ai i 1 2 25) is given in Table 19

e consequences of the sensitivity analysis are shown inFigure 8 First the results in Table 19 illustrate that thealteration in criteria weights (Cj j 1 2 4) throughoutfour different scenarios S1 minus S41113864 1113865 affect the ranking order ofthe alternatives is changing demonstrates that the pro-posed approach is sensitive to modification in criteriaweights Second the results of the failure modes ranking as

illustrated in Table 20 prove that FM12 is the first mostcritical failure mode and retains its rank in three scenariosS1 S2 S41113864 1113865 while in scenario S31113864 1113865 it is ranked the third mostcritical failure mode in the wind turbine ird the FM16 isthe less-critical failure mode in the wind turbine and itretains its rank in two scenarios S2 S31113864 1113865 meanwhile inscenario S41113864 1113865 it is ranked as the second most critical and inscenario S11113864 1113865 it is ranked as the third most critical

However it is observed that the ranking of the failuremode FM12 belongs to the lower approximation area whichrefers to the anti-ideal situation while the failure mode FM16belongs to the upper approximation area which refers to theideal situation e ranking of the failure modes in most ofthe scenarios remains steady unless some enormous mod-ifications are made in criteria weights alteration e sen-sitivity analysis conducted in the present work wasmeaningful to prioritize the failure modes for the windturbine system

7 Conclusions

e complex systems are designed to deliver the manu-factured products the delineated services and to have longlife cycles to meet the user expectation Renewable energyproduction is an interesting field entailing a large number ofhigh technology complex systems e inherent risks affectthe economic activity of the organization which in turnindirectly affects the productivity of the latter resulting indelays and insufficient energy With these organizationalperceptions the effective solution to this challenge was topropose a grey multicriteria approach to support theFMECA A novel RPN approach has been developed toovercome the results of the traditional RPN approachHowever the wind turbine has been selected as a convenientbackground to verify the proposed grey methodology egrey AHP model has been applied to evaluate the re-lationship between the risk factors of the proposed RPNapproach and calculate their weights en the greyMABAC model has been applied to prioritize the mostcritical failure mode in the wind turbine e weightsgenerated from the grey AHP model has been employed asinputs in the grey MABAC model to assess the twenty-fivefailure modes

In the traditional RPN approach the failure modes evalu-ation is performed using crisp ratings and the cross-functionaldecision-makers react inflexibly and imprecisely by using onlywhite numbers us Pearsonrsquos correlation coefficient value ofthe classical RPN rating is negative which confirms that thismethod is not reliable On the other hand Pearsonrsquos correlationcoefficient value of the proposedRPNapproachwith the supportof the grey AHP-MABAC is positive which indicates that theproposed model is an imperative mechanism to envisage theinterrelations among the severity detection occurrence andexpected costs criteria is research study provides a series ofsignificant contributions to the field of asset management

(i) Grey systems theory (GST) is considered as a sup-portive tool to overcome the fuzziness and theuncertainty of poor data and small samples


Subsequently it was capable in this work to driveand address ambiguous data weighting risk factorsand alternatives in a systematic approach

(ii) e grey AHP model has a foremost benefit due toits flexibility and certainty It involves the consis-tency rate (CR) calculation which was a compulsoryphase during the model application process toevaluate the reliability of the decision-makerrsquosevaluation

(iii) e grey MABAC model has a remarkable advan-tage over other MCDM techniques It shows how toaggregate the decision-makersrsquo judgements ina systematic approach It allowed us to divide thecriteria into beneficial (eg detecting the failuremodes) and nonbeneficial (severity occurrence andexpected costs of the failure modes) is modelconsiders the calculation of the geometric averagewhich was used to determine the performance re-sults of a portfolio andor an investment of theorganization Also it considers the calculation ofthe closeness coefficient values (CCi) and catego-rized them into three different intervals (upperapproximation area for positive values border ap-proximation area equal to zero and lower ap-proximation area negative values)

(iv) Graphical representation of the grey MABACmodel will help the executive officials of the cleanenergy production management to appraise andselect the most and the less-critical failure mode inthe wind turbine Unlike the traditional RPN theproposed grey model highlights the mutual in-fluences between the operational risk factors (se-verity-occurrence-detection) and the economiccriteria (expected costs)

(v) Study findings are grounded on a single case in-stitute (wind turbine type B63) and the judgementsare assessed in rigorouslyis helps managers fromother national and international institutes to usethis framework as a reference to analyze the in-herent risks in the assemblies

(iv) Integration of the engineering and managementdisciplines will allow managers to implement themost appropriate preventive solutions implementthe efficient reliability strategies for maintaining thesystem components in good work conditions andenhance their designs in the future With theseorganizational insights the optimal investmentstrategy will have a positive impact needed for boththe environment and the organization

In future research the proposed framework needs to bemodified and includes the subcriteria considering hiddenrisks in the wind turbine and further be applied to otherrenewable energy production systems such as farm windturbines and offshore wind turbines Furthermore nu-merous methods and MCDM techniques (based ANPDEMATE COPRAS ARAS grey connective maps and Dnumbers) could be pragmatic to analyze such risk evaluation

issues and support the relationship among the main andsubcriteria

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] Y Fu X Liu and Z Yuan ldquoLife-cycle assessment of multi-crystalline photovoltaic (PV) systems in Chinardquo Journal ofCleaner Production vol 86 pp 180ndash190 2015

[2] R B Santos and U R de Oliveira ldquoAnalysis of occupationalrisk management tools for the film and television industryrdquoInternational Journal of Industrial Ergonomics vol 72pp 199ndash211 2019

[3] T Bjerga T Aven and E Zio ldquoUncertainty treatment in riskanalysis of complex systems the cases of STAMP and FRAMrdquoReliability Engineering amp System Safety vol 156 pp 203ndash2092016

[4] F Dinmohammadi and M Shafiee ldquoA fuzzy-FMEA riskassessment approach for offshore wind turbinesrdquo In-ternational Journal of Prognostics and Health Managementvol 4 no 13 pp 59ndash68 2013

[5] A Zhou D Yu andW Zhang ldquoA research on intelligent faultdiagnosis of wind turbines based on ontology and FMECArdquoAdvanced Engineering Informatics vol 29 no 1 pp 115ndash1252015

[6] M G Bharatbhai ldquoFailure mode and effect analysis of re-power 5M wind turbinerdquo International Journal of AdvancedScientific and Technical Research vol 2 no 5 pp 2394ndash24442015

[7] W Song X Ming Z Wu and B Zhu ldquoFailure modes andeffects analysis using integrated weight-based fuzzy TOPSISrdquoInternational Journal of Computer Integrated Manufacturingvol 26 no 12 pp 1172ndash1186 2013

[8] S Carpitella A Certa J Izquierdo and C M La Fata ldquoAcombined multi-criteria approach to support FMECA ana-lyses a real-world caserdquo Reliability Engineering amp SystemSafety vol 169 pp 394ndash402 2018

[9] H-W Lo J J H Liou C-N Huang and Y-C Chuang ldquoAnovel failure mode and effect analysis model for machine toolrisk analysisrdquo Reliability Engineering amp System Safety vol 183pp 173ndash183 2019

[10] L Jun and X Huibin ldquoReliability analysis of aircraft equip-ment based on FMECA methodrdquo Physics Procedia vol 25pp 1816ndash1822 2012

[11] H-C Liu L Liu and N Liu ldquoRisk evaluation approaches infailure mode and effects analysis a literature reviewrdquo ExpertSystems with Applications vol 40 no 2 pp 828ndash838 2013

[12] W Faulkner and F Badurdeen ldquoSustainable value streammapping (Sus-VSM) methodology to visualize and assessmanufacturing sustainability performancerdquo Journal ofCleaner Production vol 85 pp 8ndash18 2014

[13] F P G Marquez A M Tobias J M P Perez andM Papaelias ldquoCondition monitoring of wind turbinestechniques and methodsrdquo Renewable Energy vol 46pp 169ndash178 2012


[14] A Pillay and J Wang ldquoModified failure mode and effectsanalysis using approximate reasoningrdquo Reliability Engineeringamp System Safety vol 79 no 1 pp 69ndash85 2003

[15] S M Seyed-Hosseini N Safaei and M J AsgharpourldquoReprioritization of failures in a system failure mode andeffects analysis by decision making trial and evaluation lab-oratory techniquerdquo Reliability Engineering amp System Safetyvol 91 no 8 pp 872ndash881 2006

[16] H Gargama and S K Chaturvedi ldquoCriticality assessmentmodels for failure mode effects and criticality analysis usingfuzzy logicrdquo IEEE Transactions on Reliability vol 60 no 1pp 102ndash110 2011

[17] E V Kumar and S K Chaturvedi ldquoPrioritization of main-tenance tasks on industrial equipment for reliabilityrdquo In-ternational Journal of Quality amp Reliability Managementvol 28 no 1 pp 109ndash126 2011

[18] G-D Li D Yamaguchi and M Nagai ldquoApplication of grey-based rough decision-making approach to suppliers selec-tionrdquo Journal of Modelling in Management vol 2 no 2pp 131ndash142 2007

[19] H-W Lo and J J H Liou ldquoA novel multiple-criteria de-cision-making-based FMEA model for risk assessmentrdquoApplied Soft Computing vol 73 pp 684ndash696 2018

[20] M Abolghasemi V Khodakarami and H Tehranifard ldquoAnew approach for supply chain risk management mappingSCOR into Bayesian networkrdquo Journal of Industrial Engi-neering and Management (JIEM) vol 8 no 1 pp 280ndash3022015

[21] N J Bahr System Safety Engineering and Risk Assessment APractical Approach CRC Press Boca Raton FL USA 2018

[22] Y-S Lee D-J Kim J-O Kim and H Kim ldquoNew FMECAmethodology using structural importance and fuzzy theoryrdquoIEEE Transactions on Power Systems vol 26 no 4pp 2364ndash2370 2011

[23] V Holley M Jankovic and B Yannou ldquoMultiple-domaindesign scorecards a method for architecture generation andevaluation through interface characterisationrdquo vol 23 no 10-11 pp 746ndash766 2014

[24] H Hwang K Lansey and D R Quintanar ldquoResilience-basedfailure mode effects and criticality analysis for regional watersupply systemrdquo Journal of Hydroinformatics vol 17 no 2pp 193ndash210 2014

[25] M Braglia ldquoMAFMA multi-attribute failure mode analysisrdquoInternational Journal of Quality amp Reliability Managementvol 17 no 9 pp 1017ndash1033 2000

[26] H-C Liu L-E Wang Z Li and Y-P Hu ldquoImproving riskevaluation in FMEA with cloud model and hierarchicalTOPSIS methodrdquo IEEE Transactions on Fuzzy Systemsvol 27 no 1 pp 84ndash95 2018

[27] E K Zavadskas Z Turskis and S Kildiene ldquoState of artsurveys of overviews on MCDMMADM methodsrdquo Tech-nological and Economic Development of Economy vol 20no 1 pp 165ndash179 2014

[28] M Farajzadeh and A Taghilo ldquoe wind energy potentialzoning using GIS and fuzzy MCDM-based approach (studyarea Zanjan province Iran)rdquo -e International Journal ofHumanities vol 20 no 2 pp 45ndash60 2013

[29] M Rezaei-Shouroki ldquoe location optimization of windturbine sites with using the MCDM approach a case studyrdquoEnergy Equipment and Systems vol 5 no 2 pp 165ndash1872017

[30] G Bos and N Chatterjee ldquoFuzzy hybrid MCDM approach forselection of wind turbine service techniciansrdquo ManagementScience Letters vol 6 no 1 pp 1ndash18 2016

[31] A Beskese A Camci G T Temur and E Erturk ldquoWindturbine evaluation using the hesitant fuzzy AHP-TOPSISmethod with a case in Turkeyrdquo Journal of Intelligent amp FuzzySystems vol 38 no 1 pp 997ndash1011 2020

[32] X Deng and Y Deng ldquoD-AHP method with differentcredibility of informationrdquo Soft Computing vol 23 no 2pp 683ndash691 2019

[33] R Bakhat and M Rajaa ldquoDeveloping a novel grey integratedmulti-criteria approach for enhancing the supplier selectionprocedure a real-world case of textile companyrdquo DecisionScience Letters vol 8 no 3 pp 211ndash224 2019

[34] N Hemmati M R Galankashi D M Imani and F M RafieildquoAn integrated fuzzy-AHP and TOPSIS approach for main-tenance policy selectionrdquo International Journal of Quality ampReliability Management vol 1 no 1 pp 1ndash24 2019

[35] M Waris S Panigrahi A Mengal et al ldquoAn application ofanalytic hierarchy process (AHP) for sustainable procurementof construction equipment multicriteria-based decisionframework for Malaysiardquo Mathematical Problems in Engi-neering vol 2019 Article ID 6391431 20 pages 2019

[36] D Pamucar and G Cirovic ldquoe selection of transport andhandling resources in logistics centers using multi-attrib-utive border approximation area comparison (MABAC)rdquoExpert Systems with Applications vol 42 no 6 pp 3016ndash3028 2015

[37] Y-X Xue J-X You X-D Lai and H-C Liu ldquoAn interval-valued intuitionistic fuzzy MABAC approach for materialselection with incomplete weight informationrdquo Applied SoftComputing vol 38 pp 703ndash713 2016

[38] A Debnath J Roy S Kar E Zavadskas andJ Antucheviciene ldquoA hybrid MCDM approach for strategicproject portfolio selection of agro by-productsrdquo Sustain-ability vol 9 no 8 p 1302 2017

[39] L Gigovic D Pamucar D Bozanic and S LjubojevicldquoApplication of the GIS-DANP-MABACmulti-criteria modelfor selecting the location of wind farms a case study ofVojvodina Serbiardquo Renewable Energy vol 103 pp 501ndash5212017

[40] L De Boer E Labro and P Morlacchi ldquoA review of methodssupporting supplier selectionrdquo European Journal of Pur-chasing amp Supply Management vol 7 no 2 pp 75ndash89 2001

[41] D Julong ldquoIntroduction to grey system theoryrdquo -e Journalof Grey System vol 1 no 1 pp 1ndash24 1989

[42] S Liu L Tao N Xie and Y Yang ldquoOn the new model systemand framework of grey system theoryrdquo in Proceedings of the2015 IEEE International Conference on Grey Systems andIntelligent Services (GSIS) pp 1ndash11 IEEE Leicester UKAugust 2015

[43] H-C Liu J-X You X-J Fan and Q-L Lin ldquoFailure modeand effects analysis using D numbers and grey relationalprojection methodrdquo Expert Systems with Applications vol 41no 10 pp 4670ndash4679 2014

[44] N Pancholi and M G Bhatt ldquoMulticriteria FMECA baseddecision-making for aluminium wire process rolling millthrough COPRAS-Grdquo Journal of Quality and ReliabilityEngineering vol 2016 Article ID 8421916 8 pages 2016

[45] M-L Tseng ldquoA causal and effect decision making model ofservice quality expectation using grey-fuzzy DEMATEL ap-proachrdquo Expert Systems with Applications vol 36 no 4pp 7738ndash7748 2009

[46] C Bai and J Sarkis ldquoA grey-based DEMATEL model forevaluating business process management critical successfactorsrdquo International Journal of Production Economicsvol 146 no 1 pp 281ndash292 2013


[47] X Xia K Govindan and Q Zhu ldquoAnalyzing internal barriersfor automotive parts remanufacturers in China using grey-DEMATEL approachrdquo Journal of Cleaner Production vol 87pp 811ndash825 2015

[48] M Mendonccedila Silva T Poleto L Camara e SilvaA P Henriques de Gusmao and A P Cabral Seixas Costa ldquoAgrey theory based approach to big data risk managementusing FMEArdquo Mathematical Problems in Engineeringvol 2016 Article ID 9175418 15 pages 2016

[49] Q Zhou and V V ai ldquoFuzzy and grey theories in failuremode and effect analysis for tanker equipment failure pre-dictionrdquo Safety Science vol 83 pp 74ndash79 2016

[50] M Omidvar and F Nirumand ldquoRisk assessment using FMEAmethod and on the basis of MCDM fuzzy logic and greytheory a case study of overhead cranesrdquo Health and Safety atWork vol 7 no 1 pp 63ndash76 2017

[51] G M Duman E Kongar and S M Gupta ldquoA holistic grey-MCDM approach for green supplier elicitation in responsiblemanufacturingrdquo in Responsible Manufacturing pp 105ndash118CRC Press Boca Raton FL USA 2019

[52] S Soua P Van Lieshout A Perera T-H Gan and B BridgeldquoDetermination of the combined vibrational and acousticemission signature of a wind turbine gearbox and generatorshaft in service as a pre-requisite for effective conditionmonitoringrdquo Renewable Energy vol 51 pp 175ndash181 2013

[53] D Milborrow ldquoOperation and maintenance costs comparedand revealedrdquo Wind Statistics vol 19 no 3 p 3 2006

[54] W T Chong K C Pan S C Poh et al ldquoPerformance in-vestigation of a power augmented vertical axis wind turbinefor urban high-rise applicationrdquo Renewable Energy vol 51pp 388ndash397 2013

[55] Y Sinha and J A Steel ldquoA progressive study into offshorewind farm maintenance optimisation using risk based failureanalysisrdquo Renewable and Sustainable Energy Reviews vol 42pp 735ndash742 2015

[56] W Y Liu B P Tang J G Han X N Lu N N Hu andZ Z He ldquoe structure healthy condition monitoring andfault diagnosis methods in wind turbines a reviewrdquo Re-newable and Sustainable Energy Reviews vol 44 pp 466ndash4722015

[57] D Julong ldquoIntroduction to grey system theoryrdquo -e Journalof grey system vol 1 pp 1ndash24 1989

[58] A VafadarnikjooMMobin C Salmon andN Javadian ldquoAnintegrated gray-fuzzy cause and effect approach to determinethe most significant categories of project risksrdquo in Proceedingsof the IIE Annual Conference p 987 Institute of Industrialand Systems Engineers (IISE) Nashville TN USA May 2015

[59] S Opricovic and G-H Tzeng ldquoDefuzzification withina multicriteria decision modelrdquo International Journal ofUncertainty Fuzziness and Knowledge-Based Systems vol 11no 5 pp 635ndash652 2003

[60] Y Dou Q Zhu and J Sarkis ldquoEvaluating green supplierdevelopment programs with a grey-analytical network pro-cess-based methodologyrdquo European Journal of OperationalResearch vol 233 no 2 pp 420ndash431 2014

[61] A Ulutas N Shukla S Kiridena and P Gibson ldquoA utility-driven approach to supplier evaluation and selection em-pirical validation of an integrated solution frameworkrdquo In-ternational Journal of Production Research vol 54 no 5pp 1554ndash1567 2016

[62] T L Saaty ldquoHow to make a decision the analytic hierarchyprocessrdquo European Journal of Operational Research vol 48no 1 pp 9ndash26 1990

[63] J Benesty J Chen Y Huang and I Cohen ldquoPearson cor-relation coefficientrdquo in Noise Reduction in Speech Processingpp 1ndash4 Springer Berlin Heidelberg 2009

[64] H S Tooranloo and A sadat Ayatollah ldquoA model for failuremode and effects analysis based on intuitionistic fuzzy ap-proachrdquo Applied Soft Computing vol 49 pp 238ndash247 2016


probability of the failure modes of a system [10] eCriticality Analysis (CA) relies on the development of theRisk Priority Number (RPN) approach attained through themultiplication of the three main risk factors ldquoseverity (S)occurrence (O) and detection (D)rdquo [11]

Clean energy notably wind power presents a currentdebate and that goes back to the increasing awareness ofcountries toward environmental protection According tothe US Department of Commerce the sustainablemanufacturing is ldquothe creation of manufactured productsthat use processes that minimize negative environmentalimpacts conserve energy and natural resources are safe foremployees communities and consumers and are eco-nomically soundrdquo [12] Until now the worldwide mountednumerous wind turbines with a capacity of 21 per year Atthe beginning of 2020 the wind power generation will havebecome the pillar of clean energy and will have exceeded15 of the global electricity production [13] However thetraditional risk factors of the RPN approach have beencriticized due to the following motives [14ndash17]

(i) traditional RPN approach takes into account onlythree risk factors abandoning other essentialldquoeconomic aspectsrdquo

(ii) emultiplicity of the risk factors for measuring thefailures risk evaluation is questionable and variantto the critical factors deviations

(iii) Multiplication operation of the risk factors is unableto detect the inherent risk in each failure mode andthat implies the lack in themathematical foundation

(iv) Traditional attributes of the RPN approach lead toimprecise results due more to the fuzzy data col-lected from experts and decision-makers throughthe process of the risk evaluation

(v) traditional RPN approach cannot correctly measurethe weights of the failure modes

(vi) e risk factors are evaluated by cross-functionaldecision-makers using linguistic variables which aretranslated into crisp numbers ranging from 1 to 10neglecting the ingrained ambiguity

A need remains for developing a simplified methodicalrisk assessment approach that serves to identify the potentialfailure modes and mitigate their effects on the whole systemAs an answer to this challenge this paper proposesa methodology that combines multicriteria decision-making(MCDM) methods namely AHP-MABAC and FMECA tocontrol fatal failure modes e proposed framework issupported by the use of the Grey Systems eory (GST) tohandle the fuzziness and the ambiguity of the decision-makersrsquo preference judgements [18] e main contributionof this paper is to take into consideration the economicaspect (expected costs) of the system in the RPN approachand reduce the risk factors integers to a (1ndash10) scale in orderto decrease the risk in the assessment process [19] Fur-thermore the wind turbine has been designated as a com-plex system with hierarchical levels to verify the proposedgrey model Finally a comparative study is performed to

verify the feasibility of the proposed model and a sensitivityanalysis is carried out to check the criteria weights variationimpact on the ranking results

e remainder of the present paper is structured asfollows a literature review of relevant existing studies ispresented in Section 2 In Section 3 the flowchart and re-search framework are provided In Section 4 the grey AHPand grey MABAC models are developed e application ofthe proposed framework in a real-world case study is fur-nished in Section 5 e comparative analysis and theconsequences of the sensitivity analysis are outlined inSection 6 Finally Section 7 highlights the closing remarksstudy contributions conclusions and recommendation forfuture researches

2 Literature Review

21 Risk Factors in FMECA Nowadays the FMECA is usedin several ldquorisk-basedrdquo industries including ldquoengineeringautomotive medical and chemicalrdquo [20]is approach wasinitially introduced by the US Military and it implies aninductive ldquobottom-uprdquo risk analysis [10] e FMECA isdefined as ldquoan analysis tool that identifies all the waysa particular component can fail what its effects would be atthe subsystem level and ultimately on the system and whatthe criticality isrdquo [21] On the other hand Lee defined theFMEA as ldquoa procedure for analysis of potential failure modeswithin a system using the classification by the severity orevaluation of the failurersquos effect upon the system where thefailure modes refer categories of detailed failures accordingto the mechanism under a certain circumstance of the wholesystemrdquo [22]e Criticality Analysis (CA) is as an extensionof the FMEAwhich allows analysts to quantitatively rank thepotential failure modes regarding their criticality on thewhole system [23] e latter could be performed quanti-tatively or qualitatively and also achieved with or withoutavailable data [24] Braglia [25] acknowledged that theFMECA is anMCDM issue and highlighted four risk factorsnamely severity detection occurrence and estimated costserefore the MCDM methodology is relevant to prioritizethe failure modes and eliminate the inherent risks in thetraditional FMECA [26]

22 Recent Applications of the AHP and MABAC MethodIn general the Multicriteria Decision-making (MCDM)methodology is regarded as one of the most practical ap-proaches in decision science and comprises a wide set oftechniques that qualitatively and quantitatively answer theresearch questions [27] Due to its practicability severalstudies have used MCDM approaches to solve complexproblems in the wind turbines [28ndash31] Besides the AHPmethod has been also employed to overcome difficulties indifferent areas such as ldquoinformation technologyrdquo [32]ldquomanagementrdquo [33] ldquomaintenancerdquo [34] and ldquosustainableconstructionrdquo [35] On the other hand the multiattributeborder approximation area comparison (MABAC) is one ofthe most precise MCDM techniques that make the in-formation as exact as possible by calculating the closeness


















4 Methodology








N21

N2

⎡⎣ ⎤⎦


min N1 N2 N1 N2 N1 N2 N1N21113872 1113873

max N1 N2 N1 N2 N1 N2 N1N21113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(2)






Blade

Coupler 1

Coupler 2

Hub

Low speed shaft

High speed shaft

Yaw system

Power cable

Tower

Generator

Nacelle

Controller


Gearbox



FeaturesMethods



otimesNrij N

rij N

r

ij]1113960 (3)



1113957Nr

ij Nr


Δmaxmin

1113957Nr

ij N

r

ij minus minjNr

ij1113872 1113873

Δminmax

(4)

where

Δmaxmin max

jN

r

ij minus minj

Nrij (5)



Aerodynamic system







Mechanical system




Electrical system





Occurrence (O)

Severity (S)

Detection (D)

Expected costs (EC)


Primary criteria

Adaptive criteria


Dimensions

FMECA


Failure mode 3

Failure mode n

Failure mode 2

Failure mode 1









MABAC-G


No





No


values into MABAC-G






Yes

Yes

Phas

e I

dim

ensio

n an

dcr

iteria

Phas

e III

the g

rey

syste

ms t

heor

y to

supp

ort M

ABA

C

Phas

e II t

he g

rey

syste

ms t

heor

y to

supp

ort A

HP




Phas

e IV

appl

icat

ion



Approve thedecision

hierarchy

0 lt CR lt 01






















Crij

1113957Nr


ij1113872 1113873 + 1113957Nr

ij times 1113957Nr

ij1113874 11138751113874 1113875

1 minus 1113957Nr

ij + 1113957Nr

ij1113874 1113875

(6)


Vrij min

jN

rij + C

rijΔ

maxmin (7)




(8)

where


Qminus1ij

1Qij

1

Qij

⎡⎢⎢⎣ ⎤⎥⎥⎦

(9)





Main objective


FM 01

FM 02

FM n ndash 1

FM n

FM 03

FM 01

FM 02

FM n ndash 1

FM n

FM 03

FM 01

FM 02

FM n ndash 1

FM n

FM 03

FM 01

FM 02

FM n ndash 1

FM n

FM 03 Alternatives

(faults)



CriteriaResults

otimesYi otimeswi

O [2286 24] [0061 0064]S [2222 2286] [0060 0061]D [14 20] [0376 0537]EC [13333 18] [0358 0483]

CR 005638lt 01


CriteriaCriteria

O S D ECO [1 1] [1 1] [0143 02] [0143 02]S [1 1] [1 1] [0111 0143] [0111 0143]D [5 7] [7 9] [1 1] [1 3]EC [5 7] [7 9] [0333 1] [1 1]























O S D ECGEO [00963 00819] [01033 00821] [07064 07088] [05357 05815]



Qij 12

times Qij +Qij1113874 1113875 (10)


(n minus 1) (11)

CR CIRI

1113874 1113875 (12)


otimesSi 1113944n

j1Qij Qij1113876 1113877 (13)

otimesSi Si Si1113960 1113961 (14)

Slowasti

2 times Si

1113936ni1 Si + 1113936

ni1 Si

⎡⎢⎣ ⎤⎥⎦ (15)

Slowasti 2 times Si

1113936ni1 Si + 1113936

ni1 Si

⎡⎢⎣ ⎤⎥⎦ (16)



ndash10000

ndash05000

00000

05000

10000

15000

0 5 10 15 20 25 30Risk

fact

ors f

unct

ions











Alternatives

Approaches




y = 05269x + 615R2 = 02776

Rank

ing

of th

e tra

ditio

nal R

PN w

ith th

e use

of th

e int

egra

ted

mod

el

0

5

10

15

20

25

30



y = ndash0058x + 13747R2 = 00033


0

5

10

15

20

25

30

Rank

ing

of th

e tra

ditio

nal R

PN








1113954Tk


mtimesn

tk11 t

k

11

tk11 t

k

11

⎡⎢⎢⎣ ⎤⎥⎥⎦tk12 t

k

12

tk12 t

k

12

⎡⎢⎢⎣ ⎤⎥⎥⎦tk13 t

k

13

tk13 t

k

13


k

1n

tk1n t

k

1n

⎡⎢⎢⎣ ⎤⎥⎥⎦


tkm1 t

k

m11113960 1113961 tkm2 t

k

m21113960 1113961 tkm3 t

k


k

mn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(18)

ndash15000

ndash10000

ndash05000

00000

05000

10000

15000

0 1 2 3 4 5 6

AlternativesA8A17

A9A18

A1A10A19

A2A11A20

A3A12A21

A4A13A22

A5A14A23

A6A15A24

A7A16A25




Crite

ria fu

nctio

ns













1113954Tk







⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(19)

tij 1113944K

k1αk times t

kij tij 1113944

K

k1αk times t

k

ij (20)









otimespij pij

pij1113876 1113877

tij

t+j

tij

t+j

tminusj

tij

tminusj

tij

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(21)








pm1 pm11113960 1113961 p


mn pmn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(22)



otimesgij gij







gm1 gm11113960 1113961 g


mngmn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(24)


otimesbij bj bj1113960 1113961 1113945m

i1g

ij⎛⎝ ⎞⎠

1m

1113945m

i1gij

⎛⎝ ⎞⎠

1m⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ (25)

us [gij



1113954B






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(26)









⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(27)

where

X






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


CC Ai( 1113857 1113944n

j1xij i 1 2 3 m j 1 2 3 n

(29)


Ai isin




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)



1113954Ri 1113944n

j1xij i 1 2 3 j 1 2 3 (31)

5 Case Study





















7 Conclusions













Data Availability




References




















































































4 Methodology








N21

N2

⎡⎣ ⎤⎦


min N1 N2 N1 N2 N1 N2 N1N21113872 1113873

max N1 N2 N1 N2 N1 N2 N1N21113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(2)






Blade

Coupler 1

Coupler 2

Hub

Low speed shaft

High speed shaft

Yaw system

Power cable

Tower

Generator

Nacelle

Controller


Gearbox



FeaturesMethods



otimesNrij N

rij N

r

ij]1113960 (3)



1113957Nr

ij Nr


Δmaxmin

1113957Nr

ij N

r

ij minus minjNr

ij1113872 1113873

Δminmax

(4)

where

Δmaxmin max

jN

r

ij minus minj

Nrij (5)



Aerodynamic system







Mechanical system




Electrical system





Occurrence (O)

Severity (S)

Detection (D)

Expected costs (EC)


Primary criteria

Adaptive criteria


Dimensions

FMECA


Failure mode 3

Failure mode n

Failure mode 2

Failure mode 1









MABAC-G


No





No


values into MABAC-G






Yes

Yes

Phas

e I

dim

ensio

n an

dcr

iteria

Phas

e III

the g

rey

syste

ms t

heor

y to

supp

ort M

ABA

C

Phas

e II t

he g

rey

syste

ms t

heor

y to

supp

ort A

HP




Phas

e IV

appl

icat

ion



Approve thedecision

hierarchy

0 lt CR lt 01






















Crij

1113957Nr


ij1113872 1113873 + 1113957Nr

ij times 1113957Nr

ij1113874 11138751113874 1113875

1 minus 1113957Nr

ij + 1113957Nr

ij1113874 1113875

(6)


Vrij min

jN

rij + C

rijΔ

maxmin (7)




(8)

where


Qminus1ij

1Qij

1

Qij

⎡⎢⎢⎣ ⎤⎥⎥⎦

(9)





Main objective


FM 01

FM 02

FM n ndash 1

FM n

FM 03

FM 01

FM 02

FM n ndash 1

FM n

FM 03

FM 01

FM 02

FM n ndash 1

FM n

FM 03

FM 01

FM 02

FM n ndash 1

FM n

FM 03 Alternatives

(faults)



CriteriaResults

otimesYi otimeswi

O [2286 24] [0061 0064]S [2222 2286] [0060 0061]D [14 20] [0376 0537]EC [13333 18] [0358 0483]

CR 005638lt 01


CriteriaCriteria

O S D ECO [1 1] [1 1] [0143 02] [0143 02]S [1 1] [1 1] [0111 0143] [0111 0143]D [5 7] [7 9] [1 1] [1 3]EC [5 7] [7 9] [0333 1] [1 1]























O S D ECGEO [00963 00819] [01033 00821] [07064 07088] [05357 05815]



Qij 12

times Qij +Qij1113874 1113875 (10)


(n minus 1) (11)

CR CIRI

1113874 1113875 (12)


otimesSi 1113944n

j1Qij Qij1113876 1113877 (13)

otimesSi Si Si1113960 1113961 (14)

Slowasti

2 times Si

1113936ni1 Si + 1113936

ni1 Si

⎡⎢⎣ ⎤⎥⎦ (15)

Slowasti 2 times Si

1113936ni1 Si + 1113936

ni1 Si

⎡⎢⎣ ⎤⎥⎦ (16)



ndash10000

ndash05000

00000

05000

10000

15000

0 5 10 15 20 25 30Risk

fact

ors f

unct

ions











Alternatives

Approaches




y = 05269x + 615R2 = 02776

Rank

ing

of th

e tra

ditio

nal R

PN w

ith th

e use

of th

e int

egra

ted

mod

el

0

5

10

15

20

25

30



y = ndash0058x + 13747R2 = 00033


0

5

10

15

20

25

30

Rank

ing

of th

e tra

ditio

nal R

PN








1113954Tk


mtimesn

tk11 t

k

11

tk11 t

k

11

⎡⎢⎢⎣ ⎤⎥⎥⎦tk12 t

k

12

tk12 t

k

12

⎡⎢⎢⎣ ⎤⎥⎥⎦tk13 t

k

13

tk13 t

k

13


k

1n

tk1n t

k

1n

⎡⎢⎢⎣ ⎤⎥⎥⎦


tkm1 t

k

m11113960 1113961 tkm2 t

k

m21113960 1113961 tkm3 t

k


k

mn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(18)

ndash15000

ndash10000

ndash05000

00000

05000

10000

15000

0 1 2 3 4 5 6

AlternativesA8A17

A9A18

A1A10A19

A2A11A20

A3A12A21

A4A13A22

A5A14A23

A6A15A24

A7A16A25




Crite

ria fu

nctio

ns













1113954Tk







⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(19)

tij 1113944K

k1αk times t

kij tij 1113944

K

k1αk times t

k

ij (20)









otimespij pij

pij1113876 1113877

tij

t+j

tij

t+j

tminusj

tij

tminusj

tij

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(21)








pm1 pm11113960 1113961 p


mn pmn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(22)



otimesgij gij







gm1 gm11113960 1113961 g


mngmn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(24)


otimesbij bj bj1113960 1113961 1113945m

i1g

ij⎛⎝ ⎞⎠

1m

1113945m

i1gij

⎛⎝ ⎞⎠

1m⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ (25)

us [gij



1113954B






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(26)









⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(27)

where

X






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


CC Ai( 1113857 1113944n

j1xij i 1 2 3 m j 1 2 3 n

(29)


Ai isin




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)



1113954Ri 1113944n

j1xij i 1 2 3 j 1 2 3 (31)

5 Case Study





















7 Conclusions













Data Availability




References




































































otimesNrij N

rij N

r

ij]1113960 (3)



1113957Nr

ij Nr


Δmaxmin

1113957Nr

ij N

r

ij minus minjNr

ij1113872 1113873

Δminmax

(4)

where

Δmaxmin max

jN

r

ij minus minj

Nrij (5)



Aerodynamic system







Mechanical system




Electrical system





Occurrence (O)

Severity (S)

Detection (D)

Expected costs (EC)


Primary criteria

Adaptive criteria


Dimensions

FMECA


Failure mode 3

Failure mode n

Failure mode 2

Failure mode 1









MABAC-G


No





No


values into MABAC-G






Yes

Yes

Phas

e I

dim

ensio

n an

dcr

iteria

Phas

e III

the g

rey

syste

ms t

heor

y to

supp

ort M

ABA

C

Phas

e II t

he g

rey

syste

ms t

heor

y to

supp

ort A

HP




Phas

e IV

appl

icat

ion



Approve thedecision

hierarchy

0 lt CR lt 01






















Crij

1113957Nr


ij1113872 1113873 + 1113957Nr

ij times 1113957Nr

ij1113874 11138751113874 1113875

1 minus 1113957Nr

ij + 1113957Nr

ij1113874 1113875

(6)


Vrij min

jN

rij + C

rijΔ

maxmin (7)




(8)

where


Qminus1ij

1Qij

1

Qij

⎡⎢⎢⎣ ⎤⎥⎥⎦

(9)





Main objective


FM 01

FM 02

FM n ndash 1

FM n

FM 03

FM 01

FM 02

FM n ndash 1

FM n

FM 03

FM 01

FM 02

FM n ndash 1

FM n

FM 03

FM 01

FM 02

FM n ndash 1

FM n

FM 03 Alternatives

(faults)



CriteriaResults

otimesYi otimeswi

O [2286 24] [0061 0064]S [2222 2286] [0060 0061]D [14 20] [0376 0537]EC [13333 18] [0358 0483]

CR 005638lt 01


CriteriaCriteria

O S D ECO [1 1] [1 1] [0143 02] [0143 02]S [1 1] [1 1] [0111 0143] [0111 0143]D [5 7] [7 9] [1 1] [1 3]EC [5 7] [7 9] [0333 1] [1 1]























O S D ECGEO [00963 00819] [01033 00821] [07064 07088] [05357 05815]



Qij 12

times Qij +Qij1113874 1113875 (10)


(n minus 1) (11)

CR CIRI

1113874 1113875 (12)


otimesSi 1113944n

j1Qij Qij1113876 1113877 (13)

otimesSi Si Si1113960 1113961 (14)

Slowasti

2 times Si

1113936ni1 Si + 1113936

ni1 Si

⎡⎢⎣ ⎤⎥⎦ (15)

Slowasti 2 times Si

1113936ni1 Si + 1113936

ni1 Si

⎡⎢⎣ ⎤⎥⎦ (16)



ndash10000

ndash05000

00000

05000

10000

15000

0 5 10 15 20 25 30Risk

fact

ors f

unct

ions











Alternatives

Approaches




y = 05269x + 615R2 = 02776

Rank

ing

of th

e tra

ditio

nal R

PN w

ith th

e use

of th

e int

egra

ted

mod

el

0

5

10

15

20

25

30



y = ndash0058x + 13747R2 = 00033


0

5

10

15

20

25

30

Rank

ing

of th

e tra

ditio

nal R

PN








1113954Tk


mtimesn

tk11 t

k

11

tk11 t

k

11

⎡⎢⎢⎣ ⎤⎥⎥⎦tk12 t

k

12

tk12 t

k

12

⎡⎢⎢⎣ ⎤⎥⎥⎦tk13 t

k

13

tk13 t

k

13


k

1n

tk1n t

k

1n

⎡⎢⎢⎣ ⎤⎥⎥⎦


tkm1 t

k

m11113960 1113961 tkm2 t

k

m21113960 1113961 tkm3 t

k


k

mn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(18)

ndash15000

ndash10000

ndash05000

00000

05000

10000

15000

0 1 2 3 4 5 6

AlternativesA8A17

A9A18

A1A10A19

A2A11A20

A3A12A21

A4A13A22

A5A14A23

A6A15A24

A7A16A25




Crite

ria fu

nctio

ns













1113954Tk







⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(19)

tij 1113944K

k1αk times t

kij tij 1113944

K

k1αk times t

k

ij (20)









otimespij pij

pij1113876 1113877

tij

t+j

tij

t+j

tminusj

tij

tminusj

tij

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(21)








pm1 pm11113960 1113961 p


mn pmn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(22)



otimesgij gij







gm1 gm11113960 1113961 g


mngmn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(24)


otimesbij bj bj1113960 1113961 1113945m

i1g

ij⎛⎝ ⎞⎠

1m

1113945m

i1gij

⎛⎝ ⎞⎠

1m⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ (25)

us [gij



1113954B






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(26)









⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(27)

where

X






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


CC Ai( 1113857 1113944n

j1xij i 1 2 3 m j 1 2 3 n

(29)


Ai isin




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)



1113954Ri 1113944n

j1xij i 1 2 3 j 1 2 3 (31)

5 Case Study





















7 Conclusions













Data Availability




References








































































MABAC-G


No





No


values into MABAC-G






Yes

Yes

Phas

e I

dim

ensio

n an

dcr

iteria

Phas

e III

the g

rey

syste

ms t

heor

y to

supp

ort M

ABA

C

Phas

e II t

he g

rey

syste

ms t

heor

y to

supp

ort A

HP




Phas

e IV

appl

icat

ion



Approve thedecision

hierarchy

0 lt CR lt 01






















Crij

1113957Nr


ij1113872 1113873 + 1113957Nr

ij times 1113957Nr

ij1113874 11138751113874 1113875

1 minus 1113957Nr

ij + 1113957Nr

ij1113874 1113875

(6)


Vrij min

jN

rij + C

rijΔ

maxmin (7)




(8)

where


Qminus1ij

1Qij

1

Qij

⎡⎢⎢⎣ ⎤⎥⎥⎦

(9)





Main objective


FM 01

FM 02

FM n ndash 1

FM n

FM 03

FM 01

FM 02

FM n ndash 1

FM n

FM 03

FM 01

FM 02

FM n ndash 1

FM n

FM 03

FM 01

FM 02

FM n ndash 1

FM n

FM 03 Alternatives

(faults)



CriteriaResults

otimesYi otimeswi

O [2286 24] [0061 0064]S [2222 2286] [0060 0061]D [14 20] [0376 0537]EC [13333 18] [0358 0483]

CR 005638lt 01


CriteriaCriteria

O S D ECO [1 1] [1 1] [0143 02] [0143 02]S [1 1] [1 1] [0111 0143] [0111 0143]D [5 7] [7 9] [1 1] [1 3]EC [5 7] [7 9] [0333 1] [1 1]























O S D ECGEO [00963 00819] [01033 00821] [07064 07088] [05357 05815]



Qij 12

times Qij +Qij1113874 1113875 (10)


(n minus 1) (11)

CR CIRI

1113874 1113875 (12)


otimesSi 1113944n

j1Qij Qij1113876 1113877 (13)

otimesSi Si Si1113960 1113961 (14)

Slowasti

2 times Si

1113936ni1 Si + 1113936

ni1 Si

⎡⎢⎣ ⎤⎥⎦ (15)

Slowasti 2 times Si

1113936ni1 Si + 1113936

ni1 Si

⎡⎢⎣ ⎤⎥⎦ (16)



ndash10000

ndash05000

00000

05000

10000

15000

0 5 10 15 20 25 30Risk

fact

ors f

unct

ions











Alternatives

Approaches




y = 05269x + 615R2 = 02776

Rank

ing

of th

e tra

ditio

nal R

PN w

ith th

e use

of th

e int

egra

ted

mod

el

0

5

10

15

20

25

30



y = ndash0058x + 13747R2 = 00033


0

5

10

15

20

25

30

Rank

ing

of th

e tra

ditio

nal R

PN








1113954Tk


mtimesn

tk11 t

k

11

tk11 t

k

11

⎡⎢⎢⎣ ⎤⎥⎥⎦tk12 t

k

12

tk12 t

k

12

⎡⎢⎢⎣ ⎤⎥⎥⎦tk13 t

k

13

tk13 t

k

13


k

1n

tk1n t

k

1n

⎡⎢⎢⎣ ⎤⎥⎥⎦


tkm1 t

k

m11113960 1113961 tkm2 t

k

m21113960 1113961 tkm3 t

k


k

mn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(18)

ndash15000

ndash10000

ndash05000

00000

05000

10000

15000

0 1 2 3 4 5 6

AlternativesA8A17

A9A18

A1A10A19

A2A11A20

A3A12A21

A4A13A22

A5A14A23

A6A15A24

A7A16A25




Crite

ria fu

nctio

ns













1113954Tk







⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(19)

tij 1113944K

k1αk times t

kij tij 1113944

K

k1αk times t

k

ij (20)









otimespij pij

pij1113876 1113877

tij

t+j

tij

t+j

tminusj

tij

tminusj

tij

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(21)








pm1 pm11113960 1113961 p


mn pmn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(22)



otimesgij gij







gm1 gm11113960 1113961 g


mngmn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(24)


otimesbij bj bj1113960 1113961 1113945m

i1g

ij⎛⎝ ⎞⎠

1m

1113945m

i1gij

⎛⎝ ⎞⎠

1m⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ (25)

us [gij



1113954B






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(26)









⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(27)

where

X






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


CC Ai( 1113857 1113944n

j1xij i 1 2 3 m j 1 2 3 n

(29)


Ai isin




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)



1113954Ri 1113944n

j1xij i 1 2 3 j 1 2 3 (31)

5 Case Study





















7 Conclusions













Data Availability




References























































































Crij

1113957Nr


ij1113872 1113873 + 1113957Nr

ij times 1113957Nr

ij1113874 11138751113874 1113875

1 minus 1113957Nr

ij + 1113957Nr

ij1113874 1113875

(6)


Vrij min

jN

rij + C

rijΔ

maxmin (7)




(8)

where


Qminus1ij

1Qij

1

Qij

⎡⎢⎢⎣ ⎤⎥⎥⎦

(9)





Main objective


FM 01

FM 02

FM n ndash 1

FM n

FM 03

FM 01

FM 02

FM n ndash 1

FM n

FM 03

FM 01

FM 02

FM n ndash 1

FM n

FM 03

FM 01

FM 02

FM n ndash 1

FM n

FM 03 Alternatives

(faults)



CriteriaResults

otimesYi otimeswi

O [2286 24] [0061 0064]S [2222 2286] [0060 0061]D [14 20] [0376 0537]EC [13333 18] [0358 0483]

CR 005638lt 01


CriteriaCriteria

O S D ECO [1 1] [1 1] [0143 02] [0143 02]S [1 1] [1 1] [0111 0143] [0111 0143]D [5 7] [7 9] [1 1] [1 3]EC [5 7] [7 9] [0333 1] [1 1]























O S D ECGEO [00963 00819] [01033 00821] [07064 07088] [05357 05815]



Qij 12

times Qij +Qij1113874 1113875 (10)


(n minus 1) (11)

CR CIRI

1113874 1113875 (12)


otimesSi 1113944n

j1Qij Qij1113876 1113877 (13)

otimesSi Si Si1113960 1113961 (14)

Slowasti

2 times Si

1113936ni1 Si + 1113936

ni1 Si

⎡⎢⎣ ⎤⎥⎦ (15)

Slowasti 2 times Si

1113936ni1 Si + 1113936

ni1 Si

⎡⎢⎣ ⎤⎥⎦ (16)



ndash10000

ndash05000

00000

05000

10000

15000

0 5 10 15 20 25 30Risk

fact

ors f

unct

ions











Alternatives

Approaches




y = 05269x + 615R2 = 02776

Rank

ing

of th

e tra

ditio

nal R

PN w

ith th

e use

of th

e int

egra

ted

mod

el

0

5

10

15

20

25

30



y = ndash0058x + 13747R2 = 00033


0

5

10

15

20

25

30

Rank

ing

of th

e tra

ditio

nal R

PN








1113954Tk


mtimesn

tk11 t

k

11

tk11 t

k

11

⎡⎢⎢⎣ ⎤⎥⎥⎦tk12 t

k

12

tk12 t

k

12

⎡⎢⎢⎣ ⎤⎥⎥⎦tk13 t

k

13

tk13 t

k

13


k

1n

tk1n t

k

1n

⎡⎢⎢⎣ ⎤⎥⎥⎦


tkm1 t

k

m11113960 1113961 tkm2 t

k

m21113960 1113961 tkm3 t

k


k

mn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(18)

ndash15000

ndash10000

ndash05000

00000

05000

10000

15000

0 1 2 3 4 5 6

AlternativesA8A17

A9A18

A1A10A19

A2A11A20

A3A12A21

A4A13A22

A5A14A23

A6A15A24

A7A16A25




Crite

ria fu

nctio

ns













1113954Tk







⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(19)

tij 1113944K

k1αk times t

kij tij 1113944

K

k1αk times t

k

ij (20)









otimespij pij

pij1113876 1113877

tij

t+j

tij

t+j

tminusj

tij

tminusj

tij

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(21)








pm1 pm11113960 1113961 p


mn pmn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(22)



otimesgij gij







gm1 gm11113960 1113961 g


mngmn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(24)


otimesbij bj bj1113960 1113961 1113945m

i1g

ij⎛⎝ ⎞⎠

1m

1113945m

i1gij

⎛⎝ ⎞⎠

1m⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ (25)

us [gij



1113954B






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(26)









⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(27)

where

X






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


CC Ai( 1113857 1113944n

j1xij i 1 2 3 m j 1 2 3 n

(29)


Ai isin




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)



1113954Ri 1113944n

j1xij i 1 2 3 j 1 2 3 (31)

5 Case Study





















7 Conclusions













Data Availability




References

























































































O S D ECGEO [00963 00819] [01033 00821] [07064 07088] [05357 05815]



Qij 12

times Qij +Qij1113874 1113875 (10)


(n minus 1) (11)

CR CIRI

1113874 1113875 (12)


otimesSi 1113944n

j1Qij Qij1113876 1113877 (13)

otimesSi Si Si1113960 1113961 (14)

Slowasti

2 times Si

1113936ni1 Si + 1113936

ni1 Si

⎡⎢⎣ ⎤⎥⎦ (15)

Slowasti 2 times Si

1113936ni1 Si + 1113936

ni1 Si

⎡⎢⎣ ⎤⎥⎦ (16)



ndash10000

ndash05000

00000

05000

10000

15000

0 5 10 15 20 25 30Risk

fact

ors f

unct

ions











Alternatives

Approaches




y = 05269x + 615R2 = 02776

Rank

ing

of th

e tra

ditio

nal R

PN w

ith th

e use

of th

e int

egra

ted

mod

el

0

5

10

15

20

25

30



y = ndash0058x + 13747R2 = 00033


0

5

10

15

20

25

30

Rank

ing

of th

e tra

ditio

nal R

PN








1113954Tk


mtimesn

tk11 t

k

11

tk11 t

k

11

⎡⎢⎢⎣ ⎤⎥⎥⎦tk12 t

k

12

tk12 t

k

12

⎡⎢⎢⎣ ⎤⎥⎥⎦tk13 t

k

13

tk13 t

k

13


k

1n

tk1n t

k

1n

⎡⎢⎢⎣ ⎤⎥⎥⎦


tkm1 t

k

m11113960 1113961 tkm2 t

k

m21113960 1113961 tkm3 t

k


k

mn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(18)

ndash15000

ndash10000

ndash05000

00000

05000

10000

15000

0 1 2 3 4 5 6

AlternativesA8A17

A9A18

A1A10A19

A2A11A20

A3A12A21

A4A13A22

A5A14A23

A6A15A24

A7A16A25




Crite

ria fu

nctio

ns













1113954Tk







⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(19)

tij 1113944K

k1αk times t

kij tij 1113944

K

k1αk times t

k

ij (20)









otimespij pij

pij1113876 1113877

tij

t+j

tij

t+j

tminusj

tij

tminusj

tij

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(21)








pm1 pm11113960 1113961 p


mn pmn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(22)



otimesgij gij







gm1 gm11113960 1113961 g


mngmn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(24)


otimesbij bj bj1113960 1113961 1113945m

i1g

ij⎛⎝ ⎞⎠

1m

1113945m

i1gij

⎛⎝ ⎞⎠

1m⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ (25)

us [gij



1113954B






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(26)









⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(27)

where

X






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


CC Ai( 1113857 1113944n

j1xij i 1 2 3 m j 1 2 3 n

(29)


Ai isin




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)



1113954Ri 1113944n

j1xij i 1 2 3 j 1 2 3 (31)

5 Case Study





















7 Conclusions













Data Availability




References





































































Qij 12

times Qij +Qij1113874 1113875 (10)


(n minus 1) (11)

CR CIRI

1113874 1113875 (12)


otimesSi 1113944n

j1Qij Qij1113876 1113877 (13)

otimesSi Si Si1113960 1113961 (14)

Slowasti

2 times Si

1113936ni1 Si + 1113936

ni1 Si

⎡⎢⎣ ⎤⎥⎦ (15)

Slowasti 2 times Si

1113936ni1 Si + 1113936

ni1 Si

⎡⎢⎣ ⎤⎥⎦ (16)



ndash10000

ndash05000

00000

05000

10000

15000

0 5 10 15 20 25 30Risk

fact

ors f

unct

ions











Alternatives

Approaches




y = 05269x + 615R2 = 02776

Rank

ing

of th

e tra

ditio

nal R

PN w

ith th

e use

of th

e int

egra

ted

mod

el

0

5

10

15

20

25

30



y = ndash0058x + 13747R2 = 00033


0

5

10

15

20

25

30

Rank

ing

of th

e tra

ditio

nal R

PN








1113954Tk


mtimesn

tk11 t

k

11

tk11 t

k

11

⎡⎢⎢⎣ ⎤⎥⎥⎦tk12 t

k

12

tk12 t

k

12

⎡⎢⎢⎣ ⎤⎥⎥⎦tk13 t

k

13

tk13 t

k

13


k

1n

tk1n t

k

1n

⎡⎢⎢⎣ ⎤⎥⎥⎦


tkm1 t

k

m11113960 1113961 tkm2 t

k

m21113960 1113961 tkm3 t

k


k

mn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(18)

ndash15000

ndash10000

ndash05000

00000

05000

10000

15000

0 1 2 3 4 5 6

AlternativesA8A17

A9A18

A1A10A19

A2A11A20

A3A12A21

A4A13A22

A5A14A23

A6A15A24

A7A16A25




Crite

ria fu

nctio

ns













1113954Tk







⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(19)

tij 1113944K

k1αk times t

kij tij 1113944

K

k1αk times t

k

ij (20)









otimespij pij

pij1113876 1113877

tij

t+j

tij

t+j

tminusj

tij

tminusj

tij

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(21)








pm1 pm11113960 1113961 p


mn pmn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(22)



otimesgij gij







gm1 gm11113960 1113961 g


mngmn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(24)


otimesbij bj bj1113960 1113961 1113945m

i1g

ij⎛⎝ ⎞⎠

1m

1113945m

i1gij

⎛⎝ ⎞⎠

1m⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ (25)

us [gij



1113954B






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(26)









⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(27)

where

X






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


CC Ai( 1113857 1113944n

j1xij i 1 2 3 m j 1 2 3 n

(29)


Ai isin




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)



1113954Ri 1113944n

j1xij i 1 2 3 j 1 2 3 (31)

5 Case Study





















7 Conclusions













Data Availability




References





































































Alternatives

Approaches




y = 05269x + 615R2 = 02776

Rank

ing

of th

e tra

ditio

nal R

PN w

ith th

e use

of th

e int

egra

ted

mod

el

0

5

10

15

20

25

30



y = ndash0058x + 13747R2 = 00033


0

5

10

15

20

25

30

Rank

ing

of th

e tra

ditio

nal R

PN








1113954Tk


mtimesn

tk11 t

k

11

tk11 t

k

11

⎡⎢⎢⎣ ⎤⎥⎥⎦tk12 t

k

12

tk12 t

k

12

⎡⎢⎢⎣ ⎤⎥⎥⎦tk13 t

k

13

tk13 t

k

13


k

1n

tk1n t

k

1n

⎡⎢⎢⎣ ⎤⎥⎥⎦


tkm1 t

k

m11113960 1113961 tkm2 t

k

m21113960 1113961 tkm3 t

k


k

mn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(18)

ndash15000

ndash10000

ndash05000

00000

05000

10000

15000

0 1 2 3 4 5 6

AlternativesA8A17

A9A18

A1A10A19

A2A11A20

A3A12A21

A4A13A22

A5A14A23

A6A15A24

A7A16A25




Crite

ria fu

nctio

ns













1113954Tk







⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(19)

tij 1113944K

k1αk times t

kij tij 1113944

K

k1αk times t

k

ij (20)









otimespij pij

pij1113876 1113877

tij

t+j

tij

t+j

tminusj

tij

tminusj

tij

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(21)








pm1 pm11113960 1113961 p


mn pmn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(22)



otimesgij gij







gm1 gm11113960 1113961 g


mngmn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(24)


otimesbij bj bj1113960 1113961 1113945m

i1g

ij⎛⎝ ⎞⎠

1m

1113945m

i1gij

⎛⎝ ⎞⎠

1m⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ (25)

us [gij



1113954B






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(26)









⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(27)

where

X






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


CC Ai( 1113857 1113944n

j1xij i 1 2 3 m j 1 2 3 n

(29)


Ai isin




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)



1113954Ri 1113944n

j1xij i 1 2 3 j 1 2 3 (31)

5 Case Study





















7 Conclusions













Data Availability




References









































































1113954Tk


mtimesn

tk11 t

k

11

tk11 t

k

11

⎡⎢⎢⎣ ⎤⎥⎥⎦tk12 t

k

12

tk12 t

k

12

⎡⎢⎢⎣ ⎤⎥⎥⎦tk13 t

k

13

tk13 t

k

13


k

1n

tk1n t

k

1n

⎡⎢⎢⎣ ⎤⎥⎥⎦


tkm1 t

k

m11113960 1113961 tkm2 t

k

m21113960 1113961 tkm3 t

k


k

mn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(18)

ndash15000

ndash10000

ndash05000

00000

05000

10000

15000

0 1 2 3 4 5 6

AlternativesA8A17

A9A18

A1A10A19

A2A11A20

A3A12A21

A4A13A22

A5A14A23

A6A15A24

A7A16A25




Crite

ria fu

nctio

ns













1113954Tk







⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(19)

tij 1113944K

k1αk times t

kij tij 1113944

K

k1αk times t

k

ij (20)









otimespij pij

pij1113876 1113877

tij

t+j

tij

t+j

tminusj

tij

tminusj

tij

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(21)








pm1 pm11113960 1113961 p


mn pmn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(22)



otimesgij gij







gm1 gm11113960 1113961 g


mngmn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(24)


otimesbij bj bj1113960 1113961 1113945m

i1g

ij⎛⎝ ⎞⎠

1m

1113945m

i1gij

⎛⎝ ⎞⎠

1m⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ (25)

us [gij



1113954B






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(26)









⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(27)

where

X






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


CC Ai( 1113857 1113944n

j1xij i 1 2 3 m j 1 2 3 n

(29)


Ai isin




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)



1113954Ri 1113944n

j1xij i 1 2 3 j 1 2 3 (31)

5 Case Study





















7 Conclusions













Data Availability




References











































































1113954Tk







⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(19)

tij 1113944K

k1αk times t

kij tij 1113944

K

k1αk times t

k

ij (20)









otimespij pij

pij1113876 1113877

tij

t+j

tij

t+j

tminusj

tij

tminusj

tij

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(21)








pm1 pm11113960 1113961 p


mn pmn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(22)



otimesgij gij







gm1 gm11113960 1113961 g


mngmn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(24)


otimesbij bj bj1113960 1113961 1113945m

i1g

ij⎛⎝ ⎞⎠

1m

1113945m

i1gij

⎛⎝ ⎞⎠

1m⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ (25)

us [gij



1113954B






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(26)









⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(27)

where

X






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


CC Ai( 1113857 1113944n

j1xij i 1 2 3 m j 1 2 3 n

(29)


Ai isin




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)



1113954Ri 1113944n

j1xij i 1 2 3 j 1 2 3 (31)

5 Case Study





















7 Conclusions













Data Availability




References





































































otimespij pij

pij1113876 1113877

tij

t+j

tij

t+j

tminusj

tij

tminusj

tij

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(21)








pm1 pm11113960 1113961 p


mn pmn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(22)



otimesgij gij







gm1 gm11113960 1113961 g


mngmn1113960 1113961

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(24)


otimesbij bj bj1113960 1113961 1113945m

i1g

ij⎛⎝ ⎞⎠

1m

1113945m

i1gij

⎛⎝ ⎞⎠

1m⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ (25)

us [gij



1113954B






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(26)









⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(27)

where

X






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


CC Ai( 1113857 1113944n

j1xij i 1 2 3 m j 1 2 3 n

(29)


Ai isin




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)



1113954Ri 1113944n

j1xij i 1 2 3 j 1 2 3 (31)

5 Case Study





















7 Conclusions













Data Availability




References




































































1113954B






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(26)









⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(27)

where

X






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


CC Ai( 1113857 1113944n

j1xij i 1 2 3 m j 1 2 3 n

(29)


Ai isin




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)



1113954Ri 1113944n

j1xij i 1 2 3 j 1 2 3 (31)

5 Case Study





















7 Conclusions













Data Availability




References





















































































7 Conclusions













Data Availability




References












































































Data Availability




References




































































riskassessmentofawindturbineusinganahp-mabac

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