Computers,Materials&Continua DOI:10.32604/cmc.2021.013092
Article
MohamedAbdel-Basset1,*,RehabMohamed1,FlorentinSmarandache2 andMohamedElhoseny3
1FacultyofComputersandInformatics,ZagazigUniversity,Zagazig,44519,Egypt
2Math&ScienceDepartment,UniversityofNewMexico,Gallup,NewMexico,87301,USA
3CollegeofComputerInformationTechnology,AmericanUniversityintheEmirates,Dubai,TheUnitedArabEmirates
CorrespondingAuthor:MohamedAbdel-Basset.Email:analyst_mohamed@yahoo.com Received:25July2020;Accepted:21September2020
Abstract: Supplierselectionisacommonandrelevantphasetoinitializethesupplychainprocessesandensureitssustainability.Thechoiceofsupplierisamulticriteriadecisionmaking(MCDM)toobtaintheoptimaldecisionbasedonagroup ofcriteria.Thehealthcaresectorfacesseveraltypesofproblems,andoneofthe mostimportantisselectinganappropriatesupplierthat fi tsthedesiredperformancelevel.Thedevelopmentofservice/productqualityinhealthcarefacilities inacountrywillimprovethequalityofthelifeofitspopulation.Thispaperproposesanintegratedmulti-attribute borderapproximationareacomparison (MABAC)basedonthebest-worstmethod(BWM),plithogenicset,andrough numbers.BWMisappliedtoregulatetheweightvectorofthemeasuresingroup decision-makingproblemswithahighlevelofconsistency.Forthetreatmentof uncertainty,aplithogenicsetandroughnumber(RN)areusedtoimprovethe accuracyofresults.Plithogenicsetoperationsareusedtodealwithinformation inthedesiredmannerthathandlesuncertaintyandvagueness.Then,basedon theplithogenicaggregationandtheresultsofBWMevaluation,weuseMABAC to findtheoptimalalternativeaccordingtodefinedcriteria.Toexaminetheproposedintegratedalgorithm,anempiricalexampleisproducedtoselectanoptimal supplierwithin fiveoptionsinthehealthcareindustry.
Keywords: Supplierselection;roughsettheory;MABAC;MCDM;BWM; plithogenicset
1Introduction
Theprocessofevaluatingasetofcriteriaunderaseriesofconstraintstoobtaintheoptimalalternative becamepopularandsignifi cantinmanydecision-makingissues.InMCDMproblems,thedecision-maker triestodecidetheoptimalalternativethatful fi llsmostofthecriteriaconsideringconditionsand constraints.Inmostcurrentpractices,supplychain(SC)managersfocusonselectingtheproper suppliertoimproveperformanceinthesupplyphase,suchasproductquality,deliveryconsistency,and prices.Appropriatesupp lierselectioncansignifi cantlyincreaseproductivity,meetcustomer expectations,increaseprofi tability,andreducethesupplycosts.But,duetothegrowthinthenumber ofsuppliers,thefullrangeofproductsavailable,andtheincreaseofcustomerexpectations,supplier
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selectionbecamecomplex,andarealchallengeforth esupplychainmanagers,thusitneedstobestudied underuncertainenvironment.
ResearchersuseddifferentMCDMtechniquesforselectingthesupplier,wheretheproblemofsupplier selectionamongalternativesispresentedbasedonasetofcriteriaconsistentwiththenatureofthe field.Itis worthpresentingtheproblemofselectingsuppliersintheformofamulti-criteriadecision-makingproblem astheselectioncriteriadifferfromonedomaintoanother.SeveralMCDMmethodswereusedinmaking proposedmodelstosolvetheproblemofsupplierselectioninseveralareas;forexample,acombined modelofBWMandfuzzygreycognitivemapstoevaluategreensuppliers[1].Inthe fieldofsupplier selectiontosupplychainsustainability,anextensionoftheDataEnvelopmentAnalysis(DEA)model wassuggested[2].Ananalyticalhierarchyprocess(AHP),additiveratioassessment(ARAS),andmultichoicegoalprogramming(MCGP)toselectacateringsupplierisproposedin[3].Asustainablesupplier selectionisevaluatedwiththeintuitionisticfuzzyTechniquesforOrderPreferencesbySimilaritytoIdeal Solution(TOPSIS)method[4].
Thesupplierselectionproblemisbasedonasetofcriteriathataredefinedaccordingtobusinessnature. TherearemanyMCDMmethods,suchasBWM,thatcanbeusedtoevaluatethesecriteriaandobtainthe optimalalternative.TheBest-Worstisasimplepairwisecomparisonmethodthatshowsreliableresultsin manytopics.Gupta(2018)combinedBWMwithFuzzyTOPSIStoevaluategreenhumanresource management(GHRM)[5].TheselectionoftheconceptualdesignoftheproductswasevaluatedusingBWM underfuzzyenvironment[6].SupplychainsustainabilitywasmeasuredbyBWM[7].Inthispaper,BWMis usedtodecideandevaluatetheweightoftheselectionstandardsthataredefinedtomeasurethesuppliers.
Theaimofthispaperistosolveasupplierselectionproblemaccordingtoplithogenicset,whichisto aggregatethegroupdecision-makers.Toachievethisaim,weproposeanintegratedplithogenicapproach basedonBWMto findtheweightvectorofthecriteriaandtheMABACmethodtoobtaintheoptimalalternative.
Thispaperisstructuredasfollows:asummarizedpresentationofstudiesonsupplierselectionand healthcareindustryinSection2.Section3consistsofdefinitionsanddetailsofthetechniquesappliedin theresearch.Section4givesthedetailsoftheproposedapproachtohandlethesupplierselection problem.Ourmethodisappliedtonumerousexamples,andtheresultsarediscussedinSection5.The conclusionoftheresearchisgiveninSection6.
2LiteratureReview
SupplierselectionisasuperiorresponsibilityofSCmanagers.Supplierselectionisconsideredas MCDMprocessthatinvolvescomparisonsamonggroupsofcriteriatochoosethepreferablesupplier, providingthehighestlevelofperformancetotheorganization.Appropriatesupplierselectioncan improvetheprofit,decreasecosts,satisfycustomerexpectations,andstimulatethecompetitivenessofthe organization[8].Thatiswhythesupplierselectionprocessbecameacriticaldecisionthatmayinfluence thewholesupplychainperformance.Inthisstudy,agroupofMCDMmethodswerecombinedtoarrive atthebestdecision,whichistochoosetheoptimalsupplier.Theimportanceofcombiningthesetheories isduetothecreationofamoreaccuratemodelofdecisions,wheretheBWMischaracterizedbyits abilitytodeterminetheweightsofmeasurementcriteriathatrelatetoselectingthebestsupplier.MABAC ischaracterizedbytheabilitytochoosethebestalternativefromtheproposedalternatives.Thisproposed hybridmodelwasdevelopedunderuncertaintyenvironmentbasedonroughnumbersandplithogenicset toavoidproblemsofambiguityofinformationthatcharacterizesmostdecision-makingproblems.Health careproblemsareamongthemostsignificantproblemsthatneedtobestudiedcarefullyindecision-making.
Thesupplierselectioncanberesumedinthreemainsteps[9].Firstly,theselectioncriteriashouldbe chosen,suchasproductquality,cost,productreliability,ordeliveryperformance.Ofcourse,thesetof
criteriaisnotstandardforallsupplierselectionsituations.Secondly,theweightofeachcriterionpreviously definedshouldbedetermined,andthecollectionofsupplieralternativesthatmay fittheproductionprocess shouldbedecided.Finally,anMCDMmodelshouldbeconstructedtoassessthealternativesbasedonthe evaluationmetricsto findtheoptimalsupplier.Theaimofthesupplierevaluationprocessinthesupplychain istorecognizetheoptimalsupplierthatcanprovidetheacceptedqualityofproduct/serviceandtheright capacityattherightstageandrightlocation[10].
Healthcareproductsaccomplishedgrowthinpastyears.Theseproductshavemanystandardsand guidelinesthatrestrictcompaniesinthesupplierselectionprocess.Thedistributionofhealthcare facilitiesbytheuseofGISandMCDMtechniquesisstudiedin[11].Also,toselectthebesttreatment technologyinhealth-carewaste(HCW)management,amodelbasedonMCDMmethodswasproposed [12].WeightedaverageoperatorandTOPSISwereusedtoofferamodelthatassiststhehierarchical medicalsystem[13].Anexpertsystembasedonthefuzzysetforassessmentofhealthcarestructures wasintroducedin[14].Besides,fuzzy(QualityFunctionDevelopment)QFDwasintegratedwithordered weightedaveraging(OWA)operatortosolvethesupplierselectionprobleminhealthcarefacilities[15].
Selectingtheoptimalsupplierprocessimpliesalargeamountofuncertainty,whichrequiresdifferent methodologiestoaccuratetheresultsoftheselection.TherearevarioustypesofMCDMtechniquesthat canbeinvolvedinsupplierselection,butthequestioniswhichapproachormethodtoapply.RNisan appropriatetooltodealwithuncertaintyortohandlesubjectivejudgmentsofcustomers,andalsoto determinetheboundaryintervals[16].IthelpsmanyMCDMmethodstoimproveefficientresultsinan uncertainenvironment.Consequently,thequalitative flexiblemultiplecriteriamethodwascombinedwith aroughnumbertodisbandthetroubleofsheltersiteselection.Theassessmentofthird-partylogistics wasmeasuredbyMABACandBWMbasedontheroughnumberintheresearchof[17].Chenetal. (2019)combinedaroughnumberwiththefuzzy-DEMATELandanalyticalnetworkprocessmethod (ANP)toevaluatetheproduct-servicesystem[18].
Plithogenytheoryreferstocreating,improving,andgrowthofnovelobjectsfromgroupsofconflicting ornon-conflictingoldobjects[19].Itisconsideredasageneralizationofneutrosophicsettheory.Plithogenic sethastwofeaturesthatenhancetheimportanceofitsoperations.The firstfeatureisthecontradictiondegree betweenthesetofelements,whichimprovestheplithogenicoperationaccuracy.Thisfeaturecomparesthe dissimilaritiesbetweenthedominantattributesandthesetofattributes.Thesecondfeatureisthe appurtenancedegreeoftheattributevalue,whichwediscussindetailinthenextsection.
3MethodsandDefinitions
3.1PlithogenicSet
Thethreerepresentationsofc(v,D)canbefuzzyCF,intuitionisticfuzzy(CIF:V×V → [0,1]2),or neutrosophic(CN:V×V → [0,1]3).Thecontradictiondegreefunctionusedinplithogenicsetoperatorsis neededfortheIntersection(AND),Union(OR),Implication(=>),Equivalence(5),andother plithogenicaggregationoperatorsthatcombinetwoormoreattribute-valuedegrees[20].
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Theplithogenicunionis:
ai1 ; ai2 ; ai3 ðÞ; 1 i n ðÞ_ p bi1 ; bi2 ; bi3 ðÞ; 1 i n ðÞ ¼ ai1 _F bi1 ; 1 2 ai2 ^F bi2 ðÞþ 1 2 ai2 _F bi2 ðÞ; ai2 ^F bi3 ; 1 i n: (2) where ai1 ^ p bi1 ¼ 1 cvD ; v1 ½ðÞ tnorm vD ; v1 ðÞþ cvD ; v1 ðÞ tconorm vD ; v1 ðÞ (3) ai1 _ p bi1 ¼ 1 cvD ; v1 ½ðÞ :tconorm vD ; v1 ðÞþ cvD ; v1 ðÞ:tnorm vD ; v1 ðÞ (4) where,tnorm ¼^Fb ¼ ab; tconorm a _ Fb ¼ a þ b ab
Theplithogeniccomplement(negation)is: : ai1 ; ai2 ; ai3 ðÞ; 1 i n ðÞ¼ ai3 ; ai2 ; ai1 ðÞ; 1 i n ðÞ (5)
3.2RoughNumber
RNistheapproximationoftheupperandlowervaluesoftheoriginalcrispvalue.It’sanefficienttheory indecisionmakingbecausethedecisionmustbedeterminedbyagroupofdecision-makersratherthana singleone.Theroughnumberisinspiredbytheroughsettheoryproposedin[21].Also,RNisregulated bylowerandupperboundsvagueinformation.Thereforeitcaneffectivelyobtaintherealdecisionmakers’ expectationsandcombinetheminanaccuratemanner[22].
Definition2.SupposeUistheuniversecontainingobjects,andtherearenclassesexpressedas G ¼ A1 ; A2 ; ; An fg orderedas A1 < A2 < < An ;then,thelowerapproximation AprAnðÞ andthe upperapproximation AprAnðÞ of An willbedefinedas:
AprAnðÞ¼[ Y 2 U =GYðÞ An fg (6)
AprAnðÞ¼[ Y 2 U =GYðÞ An fg (7)
Then, An canbeexpressedbyRNas RNAnðÞ¼ RNAnðÞ; RNAnðÞ ,where RNAnðÞ isthelowerlimit and RNAnðÞ istheupperlimit,as:
RNAnðÞ¼ 1 ML X GYðÞjY 2 AprAnðÞ (8)
RNAnðÞ¼ 1 MU X GYðÞjY 2 AprAnðÞ (9) whereMLandMUarethenumberofobjectscontainedin AprAnðÞ and AprAnðÞ,respectively.Thebasic operationsofroughnumberswereproposed[22].
Definition3.Let a1 ½¼½a1 ; a1 and a2 ½¼½a2 ; a2 betworoughnumbers,where a1 ; a1 ; a2 ; a2 > 0and a > 0;so: a a1 ½¼½a a1 ; a a1 (10) a1 ½þ a2 ½¼ a1 þ a2 ; a1 þ a2 (11) a1 ½ a2 ½¼ a1 a2 ; a1 a2 (12)
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a1 ½ a2 ½ ¼ a1 a2 ; a1 a2 (13)
Definition4.Let si ¼ an þ an ,whichisthesummationoftheupperandlowerlimits,and Di ¼ an an , whichisthesubtractionofthelowerandupperlimits.For a1 ½¼½a1 ; a1 and a2 ½¼½a2 ; a2 tworough numbers,
If sum1 > sum2 ,then a1 > a2 ;
If sum1 ¼ sum2 and D1 ¼ D2 ,then a1 ¼ a2 ;
If sum1 ¼ sum2 and D1 < D2 ,then a1 > a2
3.3Best-WorstMethod(BWM)
BWMisanefficientandstraightforwardpairwisecomparisonofMCDMproblemsthatcomparethe mostpreferred(best)criterionandtheleastdesired(worst)criterionwiththerestoftheproblemcriteria [23].AframeworkbasedontheBWMtoevaluatethe financialperformancetocomparedynamic analysisandcross-sectionalanalysiswassuggestedin[24].Also,sustainablesupplierselectioncriteria wereevaluatedusingBWM[25].BWMconsistsof fivesteps:
Step1:ThesetofcriteriaAisdefinedbyacommitteeofdecision-makers k ¼ k1 ; k2 ; ; km fg based onthescopeoftheproblemas A ¼ c1 ; c2 ; ... ; cn fg
Step2:DeterminetheBestABandWorstAWcriterionfromthesetofcriteriaAaccordingtothe decision-makerpreferences.
Step3:Constructthebest-to-othervector AB ¼faB1; aB2; ... aBn },where aBn isthepreferenceofcriteria ncomparedbytheBestcriterionBusinga(1–9)scale.
Step4:Constructtheothers-to-worstvector Aw ¼faw1; aw2; ... awn },where awn isthepreferenceof criteriancomparedbytheWorstcriterionWusinga(1–9)scale.
Step5:ProposetheBWMmodelthatevaluatestheweightofthecriteria wn : Min e s.t. wB wj aBj e,forallj wj ww ajW e,forallj X j wj ¼ 1 (14) wj 0,forallj
3.4MABACMethod
MABACmethodisarecentMCDMtechniquethatevaluatesasetofcriteriato findthebestalternative. Thismethodcomparestherelevantdistanceidealandanti-idealsolution,anditisvaluablewhentheproblem hasenormousalternativesorcriteria[10].
Step1:Constructthedecision-makingmatrix ~ D basedonthecommitteeofdecision-makers(DMs), whichevaluatesthealternativesmaccordingtothesetofcriterian.
D ¼ dij m n ¼ d11 d12 ... d1n d21 ... dm1
2 6 4 3 7 5m n
d22 d21 dm2 ... dmn
(15)
Step2:Computethenormalizeddecisionmatrix: N ¼ xij m n ¼ x11 x12 ... x1n x21 xm1
2 6 4 3 7 5m n
x22 ... x21 xm2 ... xmn
8 > > > < > > > : (17)
(16) where, xij ¼ xij xmin ij xmax ij xmin ij ifxij isbenefitcriteria xmax ij xij xmax ij xmin ij ifxij iscostcriteria
Step3:Calculateweightedmatrixelement: V ¼ vij m n ¼ v11 v12 ... v1n v21 ... vm1
2 6 4 3 7 5m n ¼ w2 : x11 þ 1 ðÞ w2 : x12 þ 1 ðÞ ... wn : x1n þ 1 ðÞ w2 x21 þ 1 ðÞ w2 xm1 þ 1 ðÞ w2 x22 þ 1 ðÞ wn x2n þ 1 ðÞ w2 xm2 þ 1 ðÞ wn xmn þ 1 ðÞ 2 6 4 3 7 5m n
v22 v21 vm2 ... vmn
(18)
Step4:Determinetheborderapproximationarea(BAA)matrix: Ggn ½ 1 n ¼ g1 g2 ... gn ½ (19) gi ¼ Y m j¼1 vij ! 1=n (20)
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Alternativemmaybelongtothreeregions m 2 G; Gþ ; G fg as Fig.1 shows: 1.BelongtotheBAA(G) 2.Upperapproximationarea(G+) 3.Lowerapproximationarea(G-) Step6:Rankthealternativesand findthebestsolution. si ¼ Pm j¼1 qmj (22)
G+ G A A2 A4
A1 A5 A6
G-
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 A A-
Upper approximation area Border approximation area
Lower approximation area
Figure1: Borderapproximationarea
4ProposedApproach
Thisresearchproposesanintegratedapproachtodisbandasupplierselectionproblemtakinginto considerationambiguousinformation.Thisapproachintegratesthefeaturesofthefourtechniquesand methods.Firstly,theplithogenicsetaggregationfeaturesconcentratesonprovidingmoreaccurate aggregationresultswhileconsideringuncertainty.Secondly,roughnumbersconsiderthehigherandlower limitsoftheinformationtohandlevagueinformation;thirdly,BWMevaluatesasetofcriteriainMCDM problems;and finally,theMABACmethoddeterminestheoptimalalternativecomparedwithagroupof criteriabymeasuringthedistanceofalternativeswiththeBAA.Thevalueofintegratingthesemethods liesinthedevelopmentofamoreeffectivedecisionmodel.Roughnumbersandplithogenicsetssupport theproposedmodeltohaveamoreaccurateandconsistentdecisionandsolvetheproblemofinformation ambiguityinanuncertaintyenvironment.Theprocessoftheproposedmodelissummarizedin Fig.2
Step1:AgroupofDMswithexperienceintheproblemscopemustbechosen, D ¼ d1 ; d2 ; ; dk fg; andthesetofcriteriathatcontroltheproblem C ¼ c1 ; c2 ; ; cn fg must bedetermined,togetherwiththealternativesthatneedtobecomparedto findthebest one A ¼ a1 ; a2 ; ; am fg
Figure2: Proposedmodel’smainsteps
Table1: Linguisticscale
SignificanceLinguisticVariableTriangularNeutrosophicScale
VeryLowsignificant(VLS)((0.10,0.30,0.35),0.1,0.2,0.15)
Lowsignificant(LS)((0.15,0.25,0.10),0.6,0.2,0.3)
Partiallysignificant(PS)((0.40,0.35,0.50),0.6,0.1,0.2)
Equalsignificant(ES)(0.65,0.60,0.70),0.8,0.1,0.1)
Highsignificant(HS)((0.70,0.65,0.80),0.9,0.2,0.1)
VeryHighsignificant(VHS)((0.90,0.85,0.90),0.7,0.2,0.2)
Absolutelysignificant(AS)((0.95,0.90,0.95),0.9,0.10,0.10)
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InordertoevaluatethealternativestothecriteriabyagroupofDMs,identifyanevaluationscale.In thisresearch,TNNisemployedusedinalinguisticscale(Tab.1).
Step2:Expresstheevaluationmatrixvaluesasroughnumberstoconsidertheuncertaininformation anddefinetheupperandlowerlimitsoftheevaluation,asexplainedindetailinSection3.3.
Step3:Aggregatethedecisionmaker ’sevaluationusingplithogenicaggregationoperatoras discussedin Eqs.(1), (3), (4).
Definethecontradictiondegreethatestablishestherelationbetweenthemostpreferred(dominant) criterionandothercriteria.Thisfeatureimprovestheaccuracyoftheaggregationoperation.
Step4:Normalizethedecisionmatrixusing Eqs.(16), (17),consideringwhetherthecriterionis benefitandcostcriteria.
Converttheneutrosophicevaluationvaluestocrispvalues,asin Eq.(23)
S ðaÞ¼ 1 8 a1 þ b1 þ c1 ðÞ 2 þ a h b ðÞ (23)
Step5:FindtheweighteddecisionmatrixbasedonBWM.
Findthebest,andtheworstcriterionaccordingdodecision-makersexperienceorpreference
Constructthebest-to-othervectorandothers-to-worstvectorbasedonthescale[0,1]
UsetheBWMmodel(24,25)to findtheweightvector
Step6:CalculatetheBAAmatrixusing Eqs.(19), (20)
Step7:FindthedistancebetweenthealternativesandBAAmatrixtodefinethealternativesinthe upperapproximationarea,BAA,andlowerapproximationarea,asin Eq.(21).
Step8:Using Eq.(22),rankthealternatives.
5NumericalApplicationandResults
TheproposedapproachevaluatesamajorMCDMproblem,whichisthesupplierselectioninthe healthcareindustry.Asacasestudy,wesimulateasupplierselectionataprivatehealthcare firmin Malaysiathatownsbothawholesalerandachainofmedicalclinics.Wholesalerincludesan administrativecenterandasinglewarehousethatcollectsproductsfrom fivedifferentsuppliers.Theyare seekingto findthebestsuppliertoexecutealargesupplyorder.
Acommitteeoffourexpertswhohavealongexperienceinhealthcaredevicesmettohelpinthis decision.Theydefinedasetofninecriteriathatmustbeconsideredwhileevaluatingthe fivealternatives. Thesecriteriaare:
– productquality(C1), – supplierexpeditesemergencyorders(C2), – supplieradequatelytestnewproducts(C3), – technologyservice:problem-solving(C4), – technologyservice:responsiveness(C5), – thesupplierprovidestechnicalassistance(C6), – thesupplierprovidesnoticeofproductproblems(C7), – consistencyofdeliveredproduct(C8),and – accuracyin fillingorders(C9).
Thehierarchyofthiscaseispresentedin Fig.3.
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Figure3: Healthcareindustryhierarchy
ApplyingtheproposedintegratedapproachinthecaseofahealthcarecompanyinMalaysiawillbeas follows:
Accordingtotheneutrosophiclinguisticscalein Tab.1,the fivesupplierswillbeevaluatedbythegroup ofdecision-makers,asin Tab.2:
Table2: Evaluationmatrix
AlternativesDMC1C2C3C4C5C6C7C8C9
A1
DM1HSESESHSPSESESPSPS
DM2VHSHSVHSESPSESHSPSHS
DM3ASVHSPSESHSPSESESES
DM4HSESHSESESESPSPSES
A2
DM1ESESHSHSHSESESHSPS
DM2HSESVHSHSHSPSPSESHS
DM3HSVHSHSPSPSPSHSESPS
DM4VHSHSPSESHSESESPSES
A3
DM1ASVHSVHSHSVHSHSESHSHS
DM2HSESHSHSHSESHSHSHS
DM3ASVHSESVHSHSESHSESPS
DM4VHSVHSVHSHSHSHSPSESES
A4
DM1ESESHSPSPSESPSPSES
DM2HSESPSPSHSHSESESPS
DM3ESHSHSHSESESHSPSPS
DM4HSPSHSHSHSPSESHSPS
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Table2(continued).
AlternativesDMC1C2C3C4C5C6C7C8C9 A5 DM1HSVHSHSHSHSESESESES DM2HSESVHSPSESPSHSESPS DM3ASESHSPSPSHSESPSPS DM4ESVHSESVHSESESPSESES Table3: Evaluationmatrixbyroughnumbers C1C2C3
Tohandletheuncertaintyofinformation,theevaluationvalueswillbetransformedintoroughnumbers todefinetheupperandlowerlimitsoftheevaluation,asdiscussedinSection3.3.Theevaluationmatrixis representedinroughnumbersin Tab.3.Then,theplithogenicaggregationoperator-assistedincombiningthe assessmentofthefourdecision-makers.Thecontradictiondegreeofthecriteriawasdefinedtoensuremore accurategathering,asshownin Tab.4.Afterconvertingtheevaluationtocrispvaluesasin Eq.(23),the normalizeddecisionmatrixmustbecalculatedbasedonthenatureofthecriteria,asexpressedin Eqs.(16), (17),andtheresultispresentedin Tab.5.
A1 DM1[((0.70,0.65,0.80),0.9,0.2,0.1), ((0.81,0.76,0.86),0.85,0.18,0.13)]
DM2 [((0.77,0.72,0.83),0.83,0.2,0.13), ((0.93,0.88,0.93),0.8,0.15,0.15)]
DM3 [((0.81,0.76,0.86),0.85,0.18,0.13), ((0.95,0.90,0.95),0.9,0.10,0.10)]
DM4[((0.70,0.65,0.80),0.9,0.2,0.1), ((0.81,0.76,0.86),0.85,0.18,0.13)]
[(0.65,0.60,0.70),0.8,0.1,0.1), ((0.73,0.68,0.78),0.8,0.15,0.13)]
[((0.67,0.62,0.73),0.83,0.13,0.1), ((0.8,0.75,0.85),0.8,0.2,0.15)]
[((0.73,0.68,0.78),0.8,0.15,0.13), ((0.90,0.85,0.90),0.7,0.2,0.2)]
[(0.65,0.60,0.70),0.8,0.1,0.1), ((0.73,0.68,0.78),0.8,0.15,0.13)]
[((0.53,0.48,0.6),0.7,0.1,0.5), ((0.75,0.7,0.8),0.8,0.17,0.13)]
[((0.66,0.61,0.73),0.75,0.15,0.15), ((0.90,0.85,0.90),0.7,0.2,0.2)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.66,0.61,0.73),0.75,0.15,0.15)]
[((0.58,0.53,0.67),0.77,0.13,0.13), ((0.8,0.75,0.85),0.8,0.2,0.15)]
A2
DM1[((0.65,0.60,0.70),0.8,0.1,0.1), ((0.74,0.69,0.8),0.83,0.18,0.13)]
DM2 [((0.68,0.63,0.77),0.87,0.17,0.1), ((0.77,0.72,0.83),0.83,0.2,0.13)]
DM3 [((0.68,0.63,0.77),0.87,0.17,0.1), ((0.77,0.72,0.83),0.83,0.2,0.13)]
DM4[((0.74,0.69,0.8),0.83,0.18,0.13), ((0.90,0.85,0.90),0.7,0.2,0.2)]
[(0.65,0.60,0.70),0.8,0.1,0.1), ((0.73,0.68,0.78),0.8,0.15,0.13)]
[((0.67,0.62,0.73),0.83,0.13,0.1), ((0.8,0.75,0.85),0.8,0.2,0.15)]
[((0.73,0.68,0.78),0.8,0.15,0.13), ((0.90,0.85,0.90),0.7,0.2,0.2)]
[(0.65,0.60,0.70),0.8,0.1,0.1), ((0.73,0.68,0.78),0.8,0.15,0.13)]
[((0.6,0.55,0.7),0.8,0.17,0.13), ((0.77,0.72,0.83),0.83,0.2,0.13)]
[((0.68,0.63,0.75),0.78,0.18,0.15), ((0.90,0.85,0.90),0.7,0.2,0.2)]
[((0.6,0.55,0.7),0.8,0.17,0.13), ((0.77,0.72,0.83),0.83,0.2,0.13)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.68,0.63,0.75),0.78,0.18,0.15)]
A3
DM1[((0.88,0.83,0.9),0.85,0.15,0.13), ((0.95,0.90,0.95),0.9,0.10,0.10)]
DM2 [((0.70,0.65,0.80),0.9,0.2,0.1), ((0.88,0.83,0.9),0.85,0.15,0.13)]
DM3 [((0.88,0.83,0.9),0.85,0.15,0.13), ((0.95,0.90,0.95),0.9,0.10,0.10)]
DM4 [((0.8,0.75,0.85),0.8,0.2,0.15), ((0.93,0.88,0.93),0.83,0.13,0.13)]
[((0.84,0.79,0.85),0.73,0.18,0.18), ((0.90,0.85,0.90),0.7,0.2,0.2)]
[((0.65,0.60,0.70),0.8,0.1,0.1), ((0.84,0.79,0.85),0.73,0.18,0.18)]
[((0.84,0.79,0.85),0.73,0.18,0.18), ((0.90,0.85,0.90),0.7,0.2,0.2)]
[((0.84,0.79,0.85),0.73,0.18,0.18), ((0.90,0.85,0.90),0.7,0.2,0.2)]
[((0.79,0.74,0.83),0.78,0.18,0.15), ((0.90,0.85,0.90),0.7,0.2,0.2)]
[((0.68,0.63,0.75),0.85,0.15,0.1), ((0.83,0.78,0.87),0.77,0.2,0.17)]
[((0.65,0.60,0.70),0.8,0.1,0.1), ((0.79,0.74,0.83),0.78,0.18,0.15)]
[((0.79,0.74,0.83),0.78,0.18,0.15), ((0.90,0.85,0.90),0.7,0.2,0.2)]
Table3(continued).
A4
DM1[((0.65,0.60,0.70),0.8,0.1,0.1), ((0.68,0.63,0.75),0.85,0.15,0.1)]
DM2 [((0.68,0.63,0.75),0.85,0.15,0.1), ((0.70,0.65,0.80),0.9,0.2,0.1)]
DM3 [((0.65,0.60,0.70),0.8,0.1,0.1), ((0.68,0.63,0.75),0.85,0.15,0.1)]
DM4[((0.68,0.63,0.75),0.85,0.15,0.1), ((0.70,0.65,0.80),0.9,0.2,0.1)]
A5 DM1[((0.68,0.63,0.77),0.87,0.17,0.1), ((0.78,0.73,0.85),0.9,0.17,0.1)]
DM2 [((0.68,0.63,0.77),0.87,0.17,0.1), ((0.78,0.73,0.85),0.9,0.17,0.1)]
DM3 [((0.75,0.7,0.81),0.88,0.15,0.1), ((0.95,0.90,0.95),0.9,0.10,0.10)]
DM4[((0.65,0.60,0.70),0.8,0.1,0.1), ((0.75,0.7,0.81),0.88,0.15,0.1)]
[((0.6,0.55,0.65),0.78,0.13,0.13), ((0.70,0.65,0.80),0.9,0.2,0.1)]
[((0.57,0.52,0.63),0.73,0.1,0.13), ((0.67,0.62,0.73),0.83,0.13,0.1)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.6,0.55,0.65),0.78,0.13,0.13)]
[((0.57,0.52,0.63),0.73,0.1,0.13), ((0.67,0.62,0.73),0.83,0.13,0.1)]
[((0.78,0.73,0.8),0.75,0.15,0.15), ((0.90,0.85,0.90),0.7,0.2,0.2)]
[((0.65,0.60,0.70),0.8,0.1,0.1), ((0.78,0.73,0.8),0.75,0.15,0.15)]
[((0.65,0.60,0.70),0.8,0.1,0.1), ((0.78,0.73,0.8),0.75,0.15,0.15)]
A1
A2
A3
CMC,2021,vol.66,no.3
[((0.55,0.5,0.65),0.75,0.15,0.15), ((0.70,0.65,0.80),0.9,0.2,0.1)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.55,0.5,0.65),0.75,0.15,0.15)]
[((0.55,0.5,0.65),0.75,0.15,0.15), ((0.70,0.65,0.80),0.9,0.2,0.1)]
[((0.55,0.5,0.65),0.75,0.15,0.15), ((0.70,0.65,0.80),0.9,0.2,0.1)]
[((0.65,0.60,0.70),0.8,0.1,0.1), ((0.74,0.69,0.8),0.83,0.18,0.13)]
[((0.68,0.63,0.77),0.87,0.17,0.1), ((0.77,0.72,0.83),0.83,0.2,0.13)]
[((0.68,0.63,0.77),0.87,0.17,0.1), ((0.77,0.72,0.83),0.83,0.2,0.13)]
[((0.74,0.69,0.8),0.83,0.18,0.13), ((0.90,0.85,0.90),0.7,0.2,0.2)] C4C5C6
DM1[((0.66,0.61,0.73),0.83,0.13,0.1), ((0.70,0.65,0.80),0.9,0.2,0.1)]
DM2 [((0.65,0.60,0.70),0.8,0.1,0.1), ((0.66,0.61,0.73),0.83,0.13,0.1)]
DM3 [((0.65,0.60,0.70),0.8,0.1,0.1), ((0.66,0.61,0.73),0.83,0.13,0.1)]
DM4[((0.65,0.60,0.70),0.8,0.1,0.1), ((0.66,0.61,0.73),0.83,0.13,0.1)]
DM1[((0.61,0.56,0.7),0.8,0.15,0.13), ((0.70,0.65,0.80),0.9,0.2,0.1)]
DM2 [((0.61,0.56,0.7),0.8,0.15,0.13), ((0.70,0.65,0.80),0.9,0.2,0.1)]
DM3 [((0.40,0.35,0.50),0.6,0.1,0.2), ((0.61,0.56,0.7),0.8,0.15,0.13)]
DM4[((0.53,0.48,0.6),0.7,0.1,0.15), ((0.68,0.63,0.77),0.87,0.17,0.1)]
DM1[((0.70,0.65,0.80),0.9,0.2,0.1), ((0.75,0.7,0.83),0.85,0.2,0.13)]
DM2 [((0.70,0.65,0.80),0.9,0.2,0.1), ((0.75,0.7,0.83),0.85,0.2,0.13)]
DM3 [((0.75,0.7,0.83),0.85,0.2,0.13), ((0.90,0.85,0.90),0.7,0.2,0.2)]
DM4 [((0.70,0.65,0.80),0.9,0.2,0.1), ((0.75,0.7,0.83),0.85,0.2,0.13)]
[((0.78,0.73,0.8),0.75,0.15,0.15), ((0.90,0.85,0.90),0.7,0.2,0.2)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.54,0.49,0.63),0.73,0.13,0.15)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.54,0.49,0.63),0.73,0.13,0.15)]
[((0.54,0.49,0.63),0.73,0.13,0.15), ((0.70,0.65,0.80),0.9,0.2,0.1)]
[((0.45,0.43,0.57),0.65,0.1,0.17), ((0.68,0.63,0.75),0.85,0.15,0.1)]
[((0.55,0.5,0.65),0.75,0.15,0.15), ((0.70,0.65,0.80),0.9,0.2,0.1)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.55,0.5,0.65),0.75,0.15,0.15)]
[((0.55,0.5,0.65),0.75,0.15,0.15), ((0.70,0.65,0.80),0.9,0.2,0.1)]
[((0.55,0.5,0.65),0.75,0.15,0.15), ((0.70,0.65,0.80),0.9,0.2,0.1)]
[((0.70,0.65,0.80),0.9,0.2,0.1), ((0.75,0.7,0.83),0.85,0.2,0.13)]
[((0.70,0.65,0.80),0.9,0.2,0.1), ((0.75,0.7,0.83),0.85,0.2,0.13)]
[((0.75,0.7,0.83),0.85,0.2,0.13), ((0.90,0.85,0.90),0.7,0.2,0.2)]
[((0.70,0.65,0.80),0.9,0.2,0.1), ((0.75,0.7,0.83),0.85,0.2,0.13)]
[((0.59,0.54,0.65),0.75,0.1,0.13), ((0.65,0.60,0.70),0.8,0.1,0.1)]
[((0.59,0.54,0.65),0.75,0.1,0.13), ((0.65,0.60,0.70),0.8,0.1,0.1)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.59,0.54,0.65),0.75,0.1,0.13)]
[((0.59,0.54,0.65),0.75,0.1,0.13), ((0.65,0.60,0.70),0.8,0.1,0.1)]
[((0.53,0.48,0.6),0.7,0.1,0.15), (0.65,0.60,0.70),0.8,0.1,0.1)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.53,0.48,0.6),0.7,0.1,0.15)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.53,0.48,0.6),0.7,0.1,0.15)]
[((0.53,0.48,0.6),0.7,0.1,0.15), (0.65,0.60,0.70),0.8,0.1,0.1)]
[((0.66,0.61,0.73),0.83,0.13,0.1), ((0.70,0.65,0.80),0.9,0.2,0.1)]
[((0.65,0.60,0.70),0.8,0.1,0.1), ((0.66,0.61,0.73),0.83,0.13,0.1)]
[((0.65,0.60,0.70),0.8,0.1,0.1), ((0.66,0.61,0.73),0.83,0.13,0.1)]
[((0.65,0.60,0.70),0.8,0.1,0.1), ((0.66,0.61,0.73),0.83,0.13,0.1)]
CMC,2021,vol.66,no.3
Table3(continued).
A4
DM1[((0.53,0.48,0.6),0.7,0.1,0.15), (0.65,0.60,0.70),0.8,0.1,0.1)]
DM2 [((0.40,0.35,0.50),0.6,0.1,0.2), ((0.53,0.48,0.6),0.7,0.1,0.15)]
DM3 [((0.40,0.35,0.50),0.6,0.1,0.2), ((0.53,0.48,0.6),0.7,0.1,0.15)]
DM4[((0.53,0.48,0.6),0.7,0.1,0.15), (0.65,0.60,0.70),0.8,0.1,0.1)]
A5 DM1[((0.5,0.45,0.6),0.7,0.13,0.17), ((0.8,0.75,0.85),0.8,0.2,0.15)]
DM2 [((0.40,0.35,0.50),0.6,0.1,0.2), ((0.6,0.55,0.68),0.7,0.15,0.18)]
DM3 [((0.40,0.35,0.50),0.6,0.1,0.2), ((0.6,0.55,0.68),0.7,0.15,0.18)]
DM4[((0.6,0.55,0.68),0.7,0.15,0.18), ((0.90,0.85,0.90),0.7,0.2,0.2)]
[((0.61,0.56,0.7),0.8,0.15,0.13), ((0.70,0.65,0.80),0.9,0.2,0.1)]
[((0.61,0.56,0.7),0.8,0.15,0.13), ((0.70,0.65,0.80),0.9,0.2,0.1)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.61,0.56,0.7),0.8,0.15,0.13)]
[((0.53,0.48,0.6),0.7,0.1,0.15), ((0.68,0.63,0.77),0.87,0.17,0.1)]
[((0.6,0.55,0.65),0.78,0.13,0.13), ((0.70,0.65,0.80),0.9,0.2,0.1)]
[((0.57,0.52,0.63),0.73,0.1,0.13), ((0.67,0.62,0.73),0.83,0.13,0.1)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.6,0.55,0.65),0.78,0.13,0.13)]
[((0.6,0.55,0.65),0.78,0.13,0.13), ((0.70,0.65,0.80),0.9,0.2,0.1)]
[((0.57,0.52,0.63),0.73,0.1,0.13), ((0.67,0.62,0.73),0.83,0.13,0.1)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.6,0.55,0.65),0.78,0.13,0.13)]
[((0.57,0.52,0.63),0.73,0.1,0.13), ((0.67,0.62,0.73),0.83,0.13,0.1)]
[((0.6,0.55,0.65),0.78,0.13,0.13), ((0.70,0.65,0.80),0.9,0.2,0.1)]
[((0.57,0.52,0.63),0.73,0.1,0.13), ((0.67,0.62,0.73),0.83,0.13,0.1)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.6,0.55,0.65),0.78,0.13,0.13)]
A1
A2
A3
DM1[((0.6,0.55,0.65),0.78,0.13,0.13), ((0.70,0.65,0.80),0.9,0.2,0.1)]
DM2 [((0.57,0.52,0.63),0.73,0.1,0.13), ((0.67,0.62,0.73),0.83,0.13,0.1)]
DM3 [((0.40,0.35,0.50),0.6,0.1,0.2), ((0.6,0.55,0.65),0.78,0.13,0.13)]
DM4[((0.57,0.52,0.63),0.73,0.1,0.13), ((0.67,0.62,0.73),0.83,0.13,0.1)]
DM1[((0.6,0.55,0.65),0.78,0.13,0.13), ((0.70,0.65,0.80),0.9,0.2,0.1)]
DM2 [((0.57,0.52,0.63),0.73,0.1,0.13), ((0.67,0.62,0.73),0.83,0.13,0.1)]
DM3 [((0.40,0.35,0.50),0.6,0.1,0.2), ((0.6,0.55,0.65),0.78,0.13,0.13)]
DM4[((0.57,0.52,0.63),0.73,0.1,0.13), ((0.67,0.62,0.73),0.83,0.13,0.1)]
DM1[((0.61,0.56,0.7),0.8,0.15,0.13), ((0.70,0.65,0.80),0.9,0.2,0.1)]
DM2 [((0.61,0.56,0.7),0.8,0.15,0.13), ((0.70,0.65,0.80),0.9,0.2,0.1)]
DM3 [((0.40,0.35,0.50),0.6,0.1,0.2), ((0.61,0.56,0.7),0.8,0.15,0.13)]
DM4 [((0.53,0.48,0.6),0.7,0.1,0.15), ((0.68,0.63,0.77),0.87,0.17,0.1)]
[((0.57,0.52,0.63),0.73,0.1,0.13), ((0.67,0.62,0.73),0.83,0.13,0.1)]
[((0.57,0.52,0.63),0.73,0.1,0.13), ((0.67,0.62,0.73),0.83,0.13,0.1)] C7C8C9
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.46,0.41,0.55),0.65,0.1,0.18)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.46,0.41,0.55),0.65,0.1,0.18)]
[((0.46,0.41,0.55),0.65,0.1,0.18), (0.65,0.60,0.70),0.8,0.1,0.1)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.46,0.41,0.55),0.65,0.1,0.18)]
[((0.6,0.55,0.65),0.78,0.13,0.13), ((0.70,0.65,0.80),0.9,0.2,0.1)]
[((0.57,0.52,0.63),0.73,0.1,0.13), ((0.67,0.62,0.73),0.83,0.13,0.1)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.6,0.55,0.65),0.78,0.13,0.13)]
[((0.57,0.52,0.63),0.73,0.1,0.13), ((0.67,0.62,0.73),0.83,0.13,0.1)]
[((0.61,0.56,0.7),0.8,0.15,0.13), ((0.70,0.65,0.80),0.9,0.2,0.1)]
[((0.61,0.56,0.7),0.8,0.15,0.13), ((0.70,0.65,0.80),0.9,0.2,0.1)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.61,0.56,0.7),0.8,0.15,0.13)]
[((0.53,0.48,0.6),0.7,0.1,0.15), ((0.68,0.63,0.77),0.87,0.17,0.1)]
[((0.6,0.55,0.65),0.78,0.13,0.13), ((0.70,0.65,0.80),0.9,0.2,0.1)]
[((0.57,0.52,0.63),0.73,0.1,0.13), ((0.67,0.62,0.73),0.83,0.13,0.1)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.6,0.55,0.65),0.78,0.13,0.13)]
[((0.57,0.52,0.63),0.73,0.1,0.13), ((0.67,0.62,0.73),0.83,0.13,0.1)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.54,0.49,0.63),0.73,0.13,0.15)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.54,0.49,0.63),0.73,0.13,0.15)]
[((0.54,0.49,0.63),0.73,0.13,0.15), ((0.70,0.65,0.80),0.9,0.2,0.1)]
[((0.45,0.43,0.57),0.65,0.1,0.17), ((0.68,0.63,0.75),0.85,0.15,0.1)]
[((0.61,0.56,0.7),0.8,0.15,0.13), ((0.70,0.65,0.80),0.9,0.2,0.1)]
[((0.61,0.56,0.7),0.8,0.15,0.13), ((0.70,0.65,0.80),0.9,0.2,0.1)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.61,0.56,0.7),0.8,0.15,0.13)]
[((0.53,0.48,0.6),0.7,0.1,0.15), ((0.68,0.63,0.77),0.87,0.17,0.1)]
Table3(continued).
A4
DM1[((0.6,0.55,0.65),0.78,0.13,0.13), ((0.70,0.65,0.80),0.9,0.2,0.1)]
DM2 [((0.57,0.52,0.63),0.73,0.1,0.13), ((0.67,0.62,0.73),0.83,0.13,0.1)]
DM3 [((0.40,0.35,0.50),0.6,0.1,0.2), ((0.6,0.55,0.65),0.78,0.13,0.13)]
DM4[((0.57,0.52,0.63),0.73,0.1,0.13), ((0.67,0.62,0.73),0.83,0.13,0.1)]
A5 DM1[((0.6,0.55,0.65),0.78,0.13,0.13), ((0.70,0.65,0.80),0.9,0.2,0.1)]
DM2 [((0.57,0.52,0.63),0.73,0.1,0.13), ((0.67,0.62,0.73),0.83,0.13,0.1)]
DM3 [((0.40,0.35,0.50),0.6,0.1,0.2), ((0.6,0.55,0.65),0.78,0.13,0.13)]
DM4 [((0.57,0.52,0.63),0.73,0.1,0.13), ((0.67,0.62,0.73),0.83,0.13,0.1)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.54,0.49,0.63),0.73,0.13,0.15)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.54,0.49,0.63),0.73,0.13,0.15)]
[((0.54,0.49,0.63),0.73,0.13,0.15), ((0.70,0.65,0.80),0.9,0.2,0.1)]
[((0.45,0.43,0.57),0.65,0.1,0.17), ((0.68,0.63,0.75),0.85,0.15,0.1)]
[((0.59,0.54,0.65),0.75,0.1,0.13), ((0.65,0.60,0.70),0.8,0.1,0.1)]
[((0.59,0.54,0.65),0.75,0.1,0.13), ((0.65,0.60,0.70),0.8,0.1,0.1)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.59,0.54,0.65),0.75,0.1,0.13)]
[((0.59,0.54,0.65),0.75,0.1,0.13), ((0.65,0.60,0.70),0.8,0.1,0.1)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.46,0.41,0.55),0.65,0.1,0.18)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.46,0.41,0.55),0.65,0.1,0.18)]
[((0.46,0.41,0.55),0.65,0.1,0.18), (0.65,0.60,0.70),0.8,0.1,0.1)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.46,0.41,0.55),0.65,0.1,0.18)]
[((0.53,0.48,0.6),0.7,0.1,0.15), (0.65,0.60,0.70),0.8,0.1,0.1)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.53,0.48,0.6),0.7,0.1,0.15)]
[((0.40,0.35,0.50),0.6,0.1,0.2), ((0.53,0.48,0.6),0.7,0.1,0.15)]
[((0.53,0.48,0.6),0.7,0.1,0.15), (0.65,0.60,0.70),0.8,0.1,0.1)]
Table4: Aggregationresult C1C2C3
A1 [((0.31,0.7,1),0.87,0.2,0.12), ((0.58,0.83,1),0.85,0.15,0.13)]
A2 [((0.22,0.64,1),0.84,0.16,0.11), ((0.39,0.75,0.1),0.8,0.2,0.15)]
A2 [((0.43,0.77,1),0.85,0.18,0.13), ((0.74,0.88,1),0.87,0.12,0.12)]
A4 [((0.2,0.62,0.99),0.83,0.13,0.1), ((0.23,0.64,1),0.88,0.18,0.1)]
A5 [((0.23,0.64,1),0.86,0.15,0.1), ((0.43,0.77,1),0.9,0.15,0.1)]
[((0.29,0.63,0.97),0.81,0.12,0.11), ((0.46,0.74,0.99),0.78,0.18,0.15)]
[((0.29,0.63,0.97),0.81,0.12,0.11), ((0.46,0.74,0.99),0.78,0.18,0.15)]
[((0.47,0.74,0.99),0.75,0.16,0.16), ((0.67,0.84,0.99),0.71,0.2,0.2)]
[((0.15,0.49,0.93),0.71,0.11,0.15), ((0.27,0.61,0.97),0.84,0.15,0.11)]
[((0.23,0.49,0.89),0.72,0.12,0.25), ((0.52,0.73,0.97),0.76,0.18,0.16)]
[((0.25,0.52,0.91),0.75,0.16,0.15), ((0.53,0.73,0.97),0.79,0.2,0.15)]
[((0.44,0.68,0.95),0.8,0.15,0.13), ((0.66,0.81,0.98),0.74,0.2,0.18)]
[((0.2,0.46,0.88),0.71,0.14,0.16), ((0.36,0.61,0.95),0.86,0.19,0.11)]
[((0.39,0.64,0.95),0.84,0.16,0.11), ((0.55,0.75,0.97),0.8,0.2,0.15)] C4C5C6
A1 [((0.43,0.6,0.88),0.81,0.11,0.1), ((0.46,0.62,0.9),0.85,0.15,0.1)]
A2 [((0.31,0.49,0.82),0.73,0.13,0.15), ((0.46,0.62,0.91),0.87,0.18,0.11)]
A2 [((0.51,0.66,0.93),0.89,0.2,0.11), ((0.61,0.74,0.95),0.81,0.2,0.15)]
A4 [((0.24,0.42,0.77),0.65,0.1,0.18), ((0.36,0.54,0.84),0.75,0.1,0.13)]
A5 [((0.25,0.43,0.78),0.65,0.12,0.19), ((0.52,0.68,0.92),0.73,0.18,0.18)]
[((0.34,0.67,0.98),0.78,0.13,0.13), ((0.56,0.79,0.99),0.73,0.18,0.18)]
[((0.31,0.41,0.69),0.65,0.11,0.18), ((0.48,0.57,0.81),0.8,0.15,0.13)]
[((0.37,0.46,0.74),0.71,0.14,0.16), ((0.53,0.61,0.86),0.86,0.19,0.11)]
[((0.59,0.66,0.89),0.89,0.2,0.11), ((0.69,0.74,0.91),0.81,0.2,0.15)]
[((0.4,0.49,0.75),0.73,0.13,0.15), ((0.54,0.62,0.86),0.87,0.18,0.11)]
[((0.39,0.49,0.73),0.71,0.11,0.15), ((0.53,0.61,0.83),0.84,0.15,0.11)]
[((0.49,0.49,0.66),0.71,0.1,0.15), ((0.59,0.59,0.73),0.79,0.1,0.11)]
[((0.42,0.42,0.6),0.65,0.1,0.18), ((0.54,0.54,0.7),0.75,0.1,0.13)]
[((0.61,0.6,0.75),0.81,0.11,0.1), ((0.63,0.62,0.78),0.85,0.15,0.1)]
[((0.48,0.49,0.65),0.71,0.11,0.15), ((0.61,0.61,0.77),0.84,0.15,0.11)]
[((0.48,0.49,0.65),0.71,0.11,0.15), ((0.61,0.61,0.77),0.84,0.15,0.11)]
CMC,2021,vol.66,no.3
Table4(continued).
C7C8C9
A1 [((0.57,0.49,0.56),0.71,0.11,0.15), ((0.7,0.61,0.7),0.84,0.15,0.11)]
A2 [((0.57,0.49,0.56),0.71,0.11,0.15), ((0.7,0.61,0.7),0.84,0.15,0.11)]
A2 [((0.58,0.49,0.59),0.73,0.13,0.15), ((0.71,0.62,0.74),0.87,0.18,0.11)]
A4 [((0.57,0.49,0.56),0.71,0.11,0.15), ((0.7,0.61,0.7),0.84,0.15,0.11)]
A5 [((0.57,0.49,0.56),0.71,0.11,0.15), ((0.7,0.61,0.7),0.84,0.15,0.11)]
[((0.55,0.37,0.37),0.61,0.1,0.2), ((0.65,0.46,0.45),0.69,0.1,0.16)]
[((0.67,0.49,0.47),0.71,0.11,0.15), ((0.78,0.61,0.61),0.84,0.15,0.11)]
[((0.67,0.49,0.49),0.73,0.13,0.15), ((0.79,0.62,0.66),0.87,0.18,0.11)]
[((0.59,0.41,0.41),0.65,0.11,0.18), ((0.74,0.57,0.58),0.8,0.15,0.13)]
[((0.68,0.49,0.48),0.71,0.1,0.15), ((0.76,0.59,0.56),0.79,0.1,0.11)]
[((0.76,0.49,0.38),0.71,0.11,0.15), ((0.85,0.61,0.53),0.84,0.15,0.11)]
[((0.68,0.41,0.32),0.65,0.11,0.18), ((0.82,0.57,0.5),0.8,0.15,0.13)]
[((0.76,0.49,0.4),0.73,0.13,0.15), ((0.85,0.62,0.59),0.87,0.18,0.11)]
[((0.64,0.37,0.29),0.61,0.1,0.2), ((0.73,0.46,0.36),0.69,0.1,0.16)]
[((0.69,0.42,0.32),0.65,0.1,0.18), ((0.8,0.54,0.43),0.75,0.1,0.13)]
Table5: Normalizedmatrix C1C2C3C4C5C6C7C8C9
A1[0.48,0.66][0.69,0.59][0.05,0.53][0.52,0.60][0.00,0.00][0.37,0.56][0,0][0,0][0.76,0.88]
A2[0.08,0.19][0.69,0.59][0.26,0.61][0.29,0.59][0.21,0.21][0,0][0,0][0.94,0.12][-2.44,0.64]
A3[1,1][1.00,1][1,1][1,1][1,1][1,1][1,1][1,1.204[1,1]
A4[0,0][0,0][0,0][0,0][0.315,0.3147][0.3304,0.8185][0,0][0.62,0.97][0.0000,0.0000]
A5[0.2,0.48][0.78,0.77][0.92,0.75][0.01,0.50][0.28,0.28][0.33,0.82][0,0][1,1][-2.14,0.45]
Toevaluatetheweightsofthecriteria,BWMisapplied.Decision-makersdefinetheproductqualityas themostdesiredcriterionandaccuracyin fillingtheorderastheleastpreferredcriterion.Accordingtothe importanceratingscale,best-to-other,andothers-to-worstvectorsweredeterminedasin Tabs.6 and 7.After applyingtheBWMmodel,theweightvectorresultedispresentedin Tab.8.TheresultoftheBWMtoweight thesetofcriteriashowsthattheproductquality(C1)criterionisthehighestwithweight0.315,andthelowest onehastheaccuracyin fillingtheorder(C9)withweight0.027.Therestofthecriteriaarearrangedas follows:supplierexpenditureemergencyorder(C2)withweight0.192,technologyservice:problemsolving(C4)withweight0.128,supplieradequatelytestnewproducts(C3)withweight0.096, technologyservice:responsiveness(C5)withweight0.077,thesupplierprovidesnoticeofproduct problems(C7)withweight0.064,thesupplierprovidestechnicalassistance(C6)withweight0.055,and consistencyofdeliveredproduct(C8)withweight0.048.
BasedontheweightvectordeterminedbyBWM,theweightedmatrix(Tab.9)wascalculatedas Eq.(18) shows.Accordingto Eqs.(19), (20),roughBAAcanbecalculated,asshownin Tab.10.
Table6: Best-to-othersvector Best-to-OthersC1C2C3C4C5C6C7C8C9 C10.10.20.40.30.50.70.60.80.9
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CMC,2021,vol.66,no.3
Table7: Other-to-worstvector
Others-to-WorstC9 C10.9 C20.8 C30.6 C40.7 C50.5 C60.3 C70.4 C80.2 C90.1
Table8: Weightvector C1C2C3C4C5C6C7C8C9 Weights0.3150.1920.0960.1280.0770.0550.0640.0480.027
Table9: Weightedmatrix C1C2C3C4C5C6C7C8C9
A1[0.46,0.52][0.32,0.31][0.1,0.15][0.19,0.20][0.08,0.08][0.08,0.09][0.06,0.06][0.05,0.05][0.05,0.05]
A2[0.34,0.37][0.32,0.31][0.1,0.15][0.16,0.20][0.09,0.09][0.06,0.06][0.06,0.06][0.09,0.09][-0.04,0.04]
A3[0.63,0.63][0.38,0.38][0.1,0.19][0.26,0.26][0.15,0.15][0.11,0.11][0.13,0.13][0.09,0.09][0.05,0.05]
A4[0.32,0.32][0.19,0.19][0.1,0.10][0.13,0.13][0.10,0.10][0.07,0.07][0.06,0.06][0.08,0.08][0.03,0.03]
A5[0.38,0.47][0.34,0.34][0.1,0.17][0.13,0.19][0.10,0.10][0.07,0.07][0.06,0.06][0.10,0.10][-0.03,-0.03]
Table10: BAAmatrix
C1C2C3C4C5C6C7C8C9
BAA[0.4128, 0.4489] [0.3020, 0.2982] [0.1329, 0.1476] [0.169,0.192][0.102,0.102][0.075, 0.075] [0.074, 0.074] [0.079, 0.079] [0.038, 0.038]
ThedistanceofthealternativesfromtheroughBBAisobtainedusing Eq.(21),andtherankingofthe alternativesbasedontherangeiscalculatedaccordingto Eq.(22);theresultsareexpressedin Tab.11.As Fig.4 shows,thethirdalternativeisthebestforthisprivatehealthcarecompany,whilealternative4isthe leastpreferredone.Byapplyingtheproposedapproachinasupplierselectionprobleminthehealthcare industry,theresultsshowthat:
Alternatives Rank
0.4
0.2
0.6 -0.4-0.200.20.40.60.8
A1 A2
0
A3 A4
-0.2
-0.4
A5 -0.6
Figure4: Rankingofthealternatives
Table11: Evaluatingresultsofalternatives
C1C2C3C4C5C6C7C8C9SumRank
A1[0.0472, 0.0711] [0.0180, 0.0118] [ 0.0319, 0.0006] [0.026, 0.013] [ 0.025, 0.025] [0.000, 0.000] [ 0.010, 0.010] [ 0.031, 0.031] [0.009, 0.009] [0.0023, 0.0382] 2
A2[ 0.0728, 0.0789] [0.0180, 0.0118] [ 0.0119, 0.0064] [ 0.004, 0.011] [ 0.009, 0.009] [ 0.020, 0.020] [ 0.010, 0.010] [0.014, 0.014] [ 0.077, 0.077] [ 0.1798, 0.0061] 4
A3[0.2172, 0.1811] [0.0780, 0.0818] [0.0591, 0.0444] [0.087, 0.064] [0.052, 0.052] [0.035, 0.035] [0.054, 0.054] [0.014, 0.014] [0.016, 0.016] [0.6123, 0.5422] 1
A4[ 0.0928, 0.1289] [ 0.1120, 0.1082] [ 0.0369, 0.0516] [ 0.041, 0.064] [ 0.001, 0.001] [ 0.002, 0.002] [ 0.010, 0.010] [ 0.001, 0.001] [ 0.011, 0.011] [ 0.3077, 0.3777] 5
A5[ 0.0328, 0.0211] [0.0380, 0.0418] [0.0511, 0.0204] [ 0.039, 0.000] [ 0.003, 0.003] [ 0.002, 0.002] [ 0.010, 0.010] [0.017, 0.017] [ 0.069, 0.069] [ 0.0497, 0.0163] 3
Decision-makersandexpertsinthehealthcaresectordefinedninecriteriathatcontrolsupplier selectiondecisionsamong fivesuppliers.UsingtheBest-Worstmethod,wefoundthattheproduct quality(C1)criterionhasthehighestweight(0.315),andthelowestonehastheaccuracyin filling theorder(C9)(0.027).
Theorderoftheweightvectorisasfollows:C1>C2>C4>C3>C5>C7>C6>C8>C9.Oneof themainfeaturesofBWMistheconsistencyratiothatmeasureshowconsistentisthecomparison. Theconsistencyratiois0.0129,whichmeansthatthemodeliscompatible.
UsingtheMABACmethod,thebestsupplierforthecompanybasedonthepreviouslyestablishedset ofcriteriaisthethirdsupplier,andtheleastpreferredisthefourthsupplier.The firstandthethird supplierareintheupperapproximationarea,whilethesecond,thefourth,andthe fifthareinthe lowerapproximationarea,asshownin Fig.4.Therankingofthesuppliersonthiscasestudyisas follows:A3>A1>A5>A2>A4.As Fig.4 shows,thealternativeA3islocatedattheupper corneroftheupperapproximationarea.followedbythealternativeA1atthelowercornerofthe area,whilethealternativeA4islocatedattheendofthelowerapproximationarea,soitisthe mostanti-idealalternativeandattheendoftheranking.
Thisstudysupportshealthcaremanagersinlookingattheproblemofselectingtheoptimalsupplierin moredetailandtakingintoaccountthefactorofuncertaintytowhichthedecision-makermaybe exposedtointheevaluationprocessesinasignificantway.
2768 CMC,2021,vol.66,no.3
6Conclusions
Supplierselectionisamajorsupplychainproblemthathasalotofuncertaininformation,whichleadsto adifficultdecision-makingprocess.Anintegratedapproachforthemanipulationofuncertaintyinthe supplierselectionprocessbasedonRNsandplithogenicsettheorywasapplied.Firstly,wediscussedthe conceptofplithogenicsetanditsaggregationoperatorthatimprovetheaggregationwithhigh considerationofuncertainty.Then,weconsideredthetransformationofevaluationvaluesintorough numberstocovertheupperandlowerevaluationofthedecision.Thirdly,theBWMwasappliedto evaluateandconcludetheweightofthecriteria.Finally,weusedtheMABACmethodtorankthesetof alternatesuppliersbasedonthedefinedcriteriaweightthatwascomputedbytheBWM.
Ourapproachprovidesanexceptionallevelofconsiderationofuncertainty.Firstly,thegroupdecision evaluationispresentedasroughnumbersthatobserveupperandlowerapproximationlimitsofthedecision whichhandlethediversityofDMs’ judgments.Secondly,theassessmentoftheDMswascombinedusing plithogenicaggregationoperation,whichconsidersthecontradictiondegreetoensuremoreaccurate aggregatedresults.Thirdly,BWMwasappliedtoidentifytheweightvectorofthecriteria,whichwas consideredasausefulandstraightforwardpairwisecomparisonmethod.Theconsistencyratioofthe BWMevaluationwascomputedtoevaluatehowconsistentistheevaluation.Finally,theMABAC methodwasappliedtoassessthesetofalternatesuppliersusingcriteriaweightsresultedfromBWM.
Somedeficiencyofthisstudyisthatofnothighlightingthepriorityofdecision-makers,inordertogeta comprehensiveviewoftheproblemandtogetthebestevaluationthattakesintoaccounttheweightsand priorityofdecision-makersaccordingtotheproblem.Thisstudyalsoneedstomakecomparisonsofthe resultsfoundthroughothermethods.Ourrecommendationforthefurtheruseofthisapproachisto employitindecision-makingproblemsfromdifferent fields.Moreover,theweightvectorofthecriteria couldbecomputedbyvariousMCDMmethods,insteadofBWM.Also,theplithogenicsetoperators shouldaddanadvantagetootherdecisionmakingapproaches.
FundingStatement: Theauthor(s)receivednospecificfundingforthisstudy.
ConflictsofInterest: Theauthorsdeclarethattheyhavenoconflictsofinteresttoreportregardingthe presentstudy.
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