A CONSENSUS MODEL FOR GROUP DECISION MAKING WITH HESITANT FUZZY INFORMATION by IJFLS

International Journal of Fuzzy Logic Systems (IJFLS) Vol.12, No.1/2/3/4, October 2022

ACONSENSUSMODELFORGROUPDECISIONMAKING WITHHESITANTFUZZYINFORMATION

Abstract. Thisarticlepresentsamoreimprovedconsensus-basedmethodfor dealingwithmulti-persondecisionmaking(MPDM)thatuseshesitantfuzzy preferencerelations(HFPRís)thatarenítintheusualformat.Weproposed aLukasiewicztransitivity(TL-transitivity)-basedtechniqueforestablishing normalisedhesitantfuzzypreferencerelations(NHFPRís)atthemostessentiallevel,afterthat,amodelbasedonconsensusisconstructed.Afterthat,a transitiveclosureformulaiscreatedinordertobuildTL -consistenthesitant fuzzypreferencerelations(HFPRís)andsymmetricalmatrices.Afterwards, aconsistencyanalysisisperformedtodeterminethedegreeofconsistencyof thedatagivenbythedecisionmakers(DMs),asaresult,theconsistency weightsmustbeassignedtothem.Aftercombiningconsistencyweightsand preset(predeÖned)priorityweights,theÖnalpriorityweightsvectorofDMs isobtained(ifthereareany).Theconsensusprocessdetermineseitherdata analysisandselectionofasuitablealternativeshouldbedonedirectlyorexternally.TheenhancementprocessaimstoimprovetheDMísconsensusmeasure, despitetheimplementationofanindicatorforlocatingsluggishpoints,inthe circumstancethatanunfavorableagreementisachieved.Finally,acomparisoncasedemonstratestherelevanceande§ectivenessoftheproposedsystem. Theconclusionsindicatethatthesuggestedstrategycanprovideinsightinto theMPDMsystem.

1. Introduction

Decision-makingisanessentialcomponentofhumanexistence.Severalofthem urgedecision-makerstomake"rational"or"great"judgments[1]basedonasetof criteriaforassessingparticularpossibilities.WhenselectingapreferencespossibilityinpracticaldomainswithasetofcriteriathatvariesandmaybeclassiÖed,there isfrequentlyacriteraconáict[2].Multi-criteriadecisionsupportapproachesare oftenusedfordecisionmakerstoevaluatechoiceoptions.Onewaydecisionsupport isimplementedisasameanstoevaluatetheuseofenvironmental,economic,and ecologicalfactors.

Multi-persondecisionmaking,alsoreferredtoasMPDM,isanimportantmethod forachievingoptimalchoiceresultsinmodernsociety.However,givenvaryinglevels ofexpertiseandinterpretation,DMíscontributionstotheevaluationwillvarydependingontheirabilities.Thisthereforepresentsachallengewhentryingtocome togetherandagreeonaconclusion.Thisisakeychallengeinthedecision-making processasitbeginswithassessing,giventhatdecisionsaretypicallybestsuited tomanydi§erentlevelsofconsideration.Oneofthedi¢cultiesinvolvedwithconsensusdecision-makingisachievingunanimousandacceptableoutcomes.Sincea

Keywordsandphrases. Fuzzypreferencerelation(FPR),Hesitantfuzzypreferencerelation (HFPR),AdditiveTransivity.

DOI : 10.5121/ijfls.2022.12401

SYEDA MIFZALAH BUKHARI, ATIQ-UR REHMAN AND MARIA BIBI Department of Basic Sciences, University of Engineering and Technology Taxila, Pakistan

GDMisaconsensus-drivenprocess,variousstrategiesforreachingane§ectiveconsensushavebeenpositedandinvestigated,resultinginanexcessivepercentageof data.Zhangetal.[3]havedevelopedacost-e§ectivemodelformaximumsupport degreeconsensusprocessing,whichcanalsobeappliedforreluctantinformation andlinguisticjudgmentswherethedegreeofcertaintyislow.Lietal.[4]proposed ane§ectiveinteractivetechniqueforachievingconsensusatalowandunpredictable cost,Öxingcommunicationproblemsbetweenhumanparticipants.Aprobabilistic hesitantfuzzypreferencerelationisatypeoffamilyofdecisionproblemsinwhich eachmemberhasitsownassociatedrateofgreaterorlesspreference.LiandWang [5]presentedanautomaticiterativeapproachtoobtainaconsensuslevel.Tianand Wangetal.andZhangetal.[6,7]developedconsensus-reachingmodelsofhuman behaviour,asdidHerrera-Viedmaetal.[8],whostudiedmodelanalysisinfuzzy environments.

Theyusedaconsensusmodelwithheterogeneouslarge-scaleGDMwithsatisfactionandindividualconcernstoexaminehowindividualsinteractwithoneanother. Thehesitantfuzzypreferencerelation(HFPR)conceptcreatedbyXiaandXu [9]iscurrentlybeingemployedinMPDMasane¢cientandsimpleapproachfor exchangingalternativefactsacrossgroupsofDMís.Theconsensus-buildingmechanism,whichisbasedonpreferencerelations,isregularlyutilizedasaapparatus inMPDMstrategies.Thefuzzypreferencerelation(HFPR)conceptmadebyXia andXuisasofnowbeingusedasanproÖcientandbasicinstrumentforagather ofDMstotradeelectiveactualities.ManyacademicshaveexploredtheHFPR suggestedin[9]inthecontextofGDM[12,13,14],butsigniÖcantdisagreements persist,forexample,whileo§eringthedecisiondegreetowhichanalternative x1 is superiortoanotheralternative x2 foragroupofthreeDMs.IfnoneoftheDMsare abletoagreeontheirappraisal,thepreferreddegreeof x1 to x2 canbeasetmade upoftheircombinedjudgementsexpressedasthehesitantfuzzyelement(HFE). TheHFPRhastheimmediatebeneÖtofprovidingtheDMswithasetofvalues displayingtheoutcomesoftheevaluation.However,theHFEssupplyalimited amountofcomponents,whichmightmakereachinganagreementdi¢cult.Most consensusmodels,forexample,arefocusedoncalculatingthedistancebetweentwo HFPRs,anddeterminingane§ectivedistancebetweenthemisquitechallenging [15].

Manyscholarshavecreatedvariouswaysforobtainingthepriorityweightvector, aswellasconsensusreachingmodels.Zhuetal.[16]weretheÖrsttoproposeaand bnormalisationprocedures.Thenormalization-basedstrategydemandedthatany twoHFEshavethesameamountofelements.Thisstrategywashighlyregarded byanumberofdecision-makingacademics.ZhangandWu[17]developedgoal programmingmodelsfortheincompletehesitantmultiplicativepreferencerelation (HFPR)andestimatedthepriorityweightvectorsusingaandbnormalisation approaches.Mengetal.[18]developedauniqueconsistencytechniqueforhesitant multilicativepreferencerelations,whichyieldedthehesitantfuzzypriorityweight vector.Zhangetal.[19]establishedvariouspreferencerelationsbasedonq-rung orthopairfuzzysetsandstudiedamethodforextractingpriorityweightsfromsuch relations.Sincemanyresearchershavebeencompelledtousethesetwoprocedures indeÖnitely,aplethoraofothernormalisationmethodshavebeendeveloped.For example,Xuetal.[20]createdaconsensusmodelfortacklingwaterallocation managementproblemsusinganadditiveconsistency-basednormalisingtechnique.

International Journal of Fuzzy Logic Systems (IJFLS) Vol.12, No.1/2/3/4, October 2022

International Journal of Fuzzy Logic Systems (IJFLS) Vol.12, No.1/2/3/4, October 2022

Becauseoftheirlimitedcompetenceandexperience,DMsmaystruggletoconstruct entirepreferenceconnectionsduringpairedjudgmentsonalternatives.Approaches thataidinthemanagementofHFPRswithinsu¢cientinformationarerequired.

Zhangetal.[21]suggestedatechniqueforguessingmissingHFPRcomponents. KhalidandBeg[22]suggestedanalgorithmfortheDMsthatmakesuseofahesitant upperboundcondition.InpairedevaluationsofDMíspreferences,theconsistency offuzzypreferencerelations(HFPR)iscrucialinthedecision-makingprocess. ThedistancemetricbetweennormalisedHFPRandconsistentHFPRiscrucialin determiningtheconsistencydegreeofHFPR.Zhuetal.[23]developedaregression approachandmethodologyfortransformingHFPRintoafuzzypreferencerelation withthemaximumlevelofconsistency.

Inthispaper,wedeÖneadditivereciprocitywhichisane§ectwaytocalculatetheDMísintheMPDMproblems.Additivereciprocityismoresu¢cientans easymethodtodeterminetheunknownandmissingvalues.Thisworkproposesa consensus-basedsolutionfordealingwiththeMPDMproblemutilisingconsistent HFPRs.Theauthorspresentanenhancedapproachforsuggestingagreementin groupdecisionmakingbasedonTL-consistencyinthecontextofHFPRs.They alsosuggestatechniqueforacceleratingtheexecutionofahigherconsensuslevel onaneasypath.WhenanFPRhasmissingpreferences,theproposedtechnique estimatesmoreappropriateandconsistentvalues.Theattributeofconsistencyis linkedtothetransitivityproperty,forwhichnumerousrelevantformsorconditions havebeenproposed.IntheÖrstphase,weusetheTL-transitivitycharacteristic toassessthemissingpreferencesofIFPRs.Then,weproposemodiÖedconsistency matricesofexpertsthatmustmeetTL-consistency.Theexpertsareassigneda levelofrelevancebasedonaccuracyweightscombinedwithconÖdenceweights.

AnewapproachfornormalisingtheHFPRsisdevelopedusingadditivereciprocityandthenextendedtotheMPDMissueutilisingconsistencyandconsensus metrics.Section4includesacomparisonexampletoevaluatethee¢cacyofthe suggestedstrategy.Thistextisarrangedasfollows:inSection2,somefundamental deÖnitionsarepresentedtoassistthereaderinunderstandingthework;inSection 3,anovelprocessisproposedanditse¢ciencyisevaluated.Section5compares theÖndingsachievedusingoursuggestedapproachtothosefoundintheliterature. TheÖnalparthassomeconclusions.

2. Preliminaries

AfuzzysetisanotionestablishedbyL.A.Zadeh[24]in1965todescribehow anitemismoreorlesstiedtoacertaincategorytowhichwedesiretoadapt.Some basicinformationisprovidedinthisparttoassisteveryoneunderstandthetopic better.

DeÖnition1. FuzzySet [24]:ThefuzzysetisdeÖnedasfollows:If X isadiscourse universeand x isaspeciÖcelementof X,thenafuzzyset A deÖnedon X canbe expressedasacollectionoforderedpairs A = f(x; A(x));x 2 Xg Aspartofthe membershipfunction A : X ! [0; 1],thedegreetowhich x relatesto A isdenoted as A(x)

DeÖnition2. HesitantFuzzySet [25]:Let X representauniversalset.AnHFS on X isdeÖnedasafunctionthatreturnsasubsetof [0; 1] whenappliedto X.The membershipdegreeof x 2 X isrepresentedbythisvalue,whichisreferredtoasthe hesitantfuzzyelement(HFE).

International Journal of Fuzzy Logic Systems (IJFLS) Vol.12, No.1/2/3/4, October 2022

DeÖnition3. FuzzyPreferenceRelation [26]:Ifthesetofalternatives A = fA1;A2;::::;Ang exists,thenthefuzzypreferencerelation U on A A withthe requirement uij 0;uij + uji =1 exists,iscalled additivereciprocity [27];i;j = 1; 2;::::;n; thedegreetowhichtheoption Ai ispriortothealternative Aj isdenoted by uij :Ai and Aj ,representedby Ai Aj areindi§erenceat uij =0 5 ; 0 uij < 0 5 denotesthat Aj ispreferableover Ai whichisindicatedby Aj Ai; thelower thevalueof uij ,thebetter,thestrongerthepreferencefor Aj over Ai,thebetter; 0 5 <uij 1 denotesthat Ai ispreferableto Aj ,whichissymbolisedby Ai Aj , thegreater uij ísvalue,thehigherthepreferencefor Ai over Aj ,thebetter.Itshould beemphasisedthatinafuzzypreferencerelation,thevalue uij isanumberbetween 0 and 1.

DeÖnition4. Hesitantfuzzypreferencerelation [28]:let A = fA1;A2;::::;Ang isa Öxedset,thenamatrix H =(hij )n n @ A A presentsahesitantfuzzypreference relation H on A,where hij = fhij j =1; 2;::::lhij g isanHFEreáectingallthe possibledegreestowhichAi ispreferredoverAj ,in hij hij isthe thelement, thenumberofvaluesin hij isrepresentedby lhij .Furthermore, hij mustmeetthe followingrequirements:

8 < : hij + hji =1;i;j =1; 2;:::;n hii = f0:5g;i =1; 2;::::;n lhij = lhji ;i;j =1; 2;:::;n

DeÖnition5. IncompleteFuzzyPreferenceRelation [11]:If U =(uij )n n isa preferencerelation,thenUisreferredtoasaincompletefuzzypreferencerelation, iftheDMisunabletoprovidesomeofitselements,itisrepresentedbytheunknown numberx,andtheDMwillbeabletoo§ertheremaining.

DeÖnition6. AdditiveTransitivity:if 8 i;k;j 2 X holds,then R isknownas additivetransitive:

R(x;z)= R(x;y)+ R(y;z) 0:5

DeÖnition7. ukasiewiczTransitivity:UissaidtobeTL-transitiveifitholds

8 i;k = j 2f1; 2;:::;ng : uik max(uij + ujk 1; 0):

DeÖnition8. ConsistentFuzzyPrefrenceRelation:AFuzzyPrefrenceRelationU issaidtobeTL-transivity(consistent),iffor 8 i;k = j 2f1; 2;:::;ng : uik max(uij + ujk 1; 0)(TL-transivity)issatisÖed.

3. Methodology

Throughoutthissection,weo§eredasuperiortechniquefordealingwithMPDM issuesusingHFPRs,whichincludedthefollowingphases:normalisation,consistencymeasures,consensusmeasures,consensusenhancingprocess,allocatingpriorityweightstoDMs,andselectionprocess.

3.1. Normalization. Withinthispart,itisproposedthatanewapproachfornormalisingHFPRsbedeveloped,becauseforanytwohesitantfuzzypreferencevalues (HFPVs), hij and hlm, jhij j6= jhlmj for i;j;l;m 2f1; 2;::::;ng inthemajority ofcases,Setsofpairwisecomparisonswiththeircardinalitiesatthe ijthand lmth placesarerepresentedby jhij j & jhlmj.Weuseadditivereciprocitytodiscoverthe unknownpreferencesofIFPRsintheÖrststage.Regardingthenormalisingofthe

9 = ;

International Journal of Fuzzy Logic Systems (IJFLS) Vol.12, No.1/2/3/4, October 2022

givenHFPRs,weproposeanewmethodforcalculatingthecomponentsthatwill beaddedtoHFPVs.Otherintermediatevaluescannotbeaddedtotheestimated elementsinceitisjustthemaximumorlowestentryofHFPV.Thesuggestedtechniqueisbasedonadditivereciprocity,inwhichwegeneratetheincompletefuzzy preferencerelation(s)IFPRsusingunknownpreferencevaluesforthecomponents tobeadded.

3.2. DetermineMissingValues. An IFPR basedonTL-consistencycanonly becompletedifeachoptionamongtheknownpreferencevaluesisinvestigatedat leastonce.Asaresult,thesystemmustrequestthattheexpertcreateasu¢cient numberofpreferences,witheachpossibilitybeingexaminedatleastonceinorder forthe IFPR tobecomeacomplete FPR.Thesequenceinwhichthemissing preferencevaluesaremeasuredhasane§ectontheÖnaloutcome.Forthepurpose ofconstructingtheunknownormissingpreferencevaluesinan IFPRR =(rij )n n, thebelowmentionedsetsareusedtodenotethealternativeíspairsforknownand unknownormissingpreferencevalues

(3.2) where rij 2 [0; 1] arethepreferencevaluesof ai over aj , rij + rji =1=) rii = 0 5 8 i 2f1; 2;::::;ng.Asaresult,basedonTL-transitivity rik max (rik + rkj 1; 0),toestimatetheunknownpreferencevalue rij ofalternative ai overalternative aj ,thefollowingsetcanbedeÖned.

(3.3) Eij = fk = i;jj(i;k) 2 Ke; (k;j) 2 Ke and (i;j) 2 Ueg for i;j;k 2f1; 2; 3;:::;ng.UsingtheEquation(3.3),theÖnalvalueof rij iscalculatedasfollows:

(3.4) rij = avek2Eij (max(rik + rkj 1; 0));if jEij j6=0 0 5;otherwise and

(3.5) rji =1 rij (additivereciprocity)

Finally,theentireFPR R =(rij )n n isproduced.Now,twoadditionalsetsof knownandunknownelements, Ke and Ue,aredeÖnedasfollows:

(3.6) Ke = Ke [f(i;j)g; and Ue = Ue f(i;j)g

Asaresult,anormalisedhesitantfuzzypreferencerelation(NHFPR) H =(hij )n n with hij + hji =1;hij = fhij j =1; 2;:::; jhij jg for jhij j = jhjij

isconstructed.Becauseofthegrowthincomplexity,thereareseveraldecisionmakingproceduresthatoccurinmulti-personsituationsintherealworldand becauseoftheunpredictabilityofthesocioeconomicenvironment,itismoredi¢cult forasingledecisionmakercanassessalloftheinterconnectedcomponentsofa decision-makingchallenge.

jrij isknownvalueg; (3.1) Ue = f(i;j)jrij isunknownvalueg;

Ke = f(i;j)

International Journal of Fuzzy Logic Systems (IJFLS) Vol.12, No.1/2/3/4, October 2022

3.3. ConsistencyAnalysis. Someconsistencymeasurements,suchastheamount ofconsistencybetweentwoalternatives,thedegreeofconsistencyamongalternatives,andHFPRíslevelofconsistency,arespeciÖedinthissubsection.Theterm consistencyindex(CI)referstothedegreeofconsistencywithavaluebetween 0 and 1.Assume Hq istheHFPRforthedecisionmaker Dq(1 q l),thenafter receivingtheNHFPR H q,Thefollowingtransitiveclosureformulamaybeused toobtainTL-consistentHFPR H q:

(3.7) ~ h q ij =max k=i;j(h q ij ; max(h q ik ;h q kj 1; 0)) ; ~ h q ij + ~ h q ji =1

where h q ij = fhij j =1; 2;:::; jhij jg.Afteranalysingthedistancebetweenthem inthefollowingmanner,wecancalculatetheHFPR H q consistencylevelbased onitssimilaritytothecorrespondingTL-consistency H q

1. TL consistencyindex(TLCI)forapairofalternativesrankedasfollows:

(3.8) TLCI(h q ij )=1 1

ij j jhij j X =1

d( h q ij ; h q ij )

where d( h q ij ; h q ij ) denotesthedistanceobtainedby d( h q ij ; h q ij )= j h q ij h q ij j Thegreaterthelevelof TLCI(h q ij ),themoreconsistent h q ij isincomparison totheotherHFPVswhenitcomestotheoptions ai and aj

2. TLCI foralternativesai,1 i n,asfollows:

with TLCI(ai) 2 [0; 1].If TLCI(ai)=1,thenthealternative ai preferencevaluesare totallyconsistent;ontheotherhand,thelower TLCI(ai),themoreinconsistency thereisinthesepreferencevalues.

3.Finally, TLCI againstNHFPRtheaverageoperatorisusedtoassess H q:

TLCI(H q) 2 [0; 1].If

H q istotallyconsistent,if

LCI(H q) isless, H q ismoreinconsistent.Equation(3.10)assignsDM Dq to theconsistencyindex,howevertheglobalconsistencyindex(CI) maybecalculated usingtheaverageoperatorandisprovidedas:

CI 2 [0; 1].The TLCI iscalculatedinthreesteps,eachcomprisingEquations(3.8)-(3.10),ExpertswhoproducedtheHFPRwithbetterconsistencyindices shouldbegivenmoreweights.Asaresult,thefollowingrelationshipmaybeused toassignconsistencyweightstoexperts: (3.12)

(3.9) TL

n X j=1;j=i (TLCI (h q ij )+ TLCI(h q ji ))

CI(ai)=

(3.10) TLCI

H q)= 1 n n X i=1 TLCI(ai)

H q

,NHFPR

(3.11)

l l X q=1 TLCI(H q)

TLCI(H q) l X q=1 TLCI(H q)

(

with

CI(

)=1

CI = 1

with

(Dq)=

with Cw(Dq) 2 [0; 1]& l X q=1

Cw(Dq)=1

3.4. ConsensusAnalysis. Somelevelsforestimatingglobalconsensusamong decisionmakersaredeÖnedinthissubsection.Itiscriticaltodeterminetheamount ofconsensusamongdecisionmakersafterexaminingNHFPRs H q;q =1; 2;:::;l Afteraggregatingthesimilaritymatrices Sqr =(sqr ij )n n foreachpairofdecision makers (Dq;Dr)(q =1;

similaritymatrixthatmaybebuiltasfollows:

; 2;:::; jhij jg Thefollowinglevelsareusedtodeterminethedegreeofglobalconsensusamongdecision-makers:

1.Thedegreeofconsensusonapairofalternatives

2.Thedegreeofconsensusonalternatives

3.Thedegreeofconsensusontherelationrepresentedby CR isspeciÖedatthe thirdleveltodeterminetheglobalconsensusdegreeamongallDMs: (3.16)

Ifallspecialistsreachalevelofglobalconsensus,acomparisonwiththethresholdconsensusdegree isrequired,thenatureoftheproblemistypicallypredetermined.If CR isachieve,itmeansthattherehasbeenasu¢cientlevel ofconsensusobtained,andtheprocessofdecision-makingmaybegin.Asidefrom that,theconsensusdegreeisunstable,andexpertsarebeingpressedtoaltertheir preferences.

3.5. EnhancementMechanism. Theenhancementmechanismactsasamoderatorintheconsensus-buildingprocess,providingdecisionmakerswithdetailed informationtohelpthemimprovetheirresults.Incaseofinadequateconsensus level,wemustdeterminetheplaceswherepreferencevalueswillbeupdated,so thatdecision-makerscanattainanappropriatelevelofconsensusInthisregard, thefollowingisadeÖnitionofanidentiÖer:

(3.17) Iq = f(i;j)j cdij <CR & hq ij isaknownvalueg

OncethepositionshavebeenestablishedusinganidentiÖer,theenhancementmechanismrecommendsthattherelevantDM Dq augmenttheelement hq ij ofHFPV hq ij , ifitislessthanthemeanvalue h q ij;ave oftheparticipantísopinions,ortodeclineif thevalueismorethanthemean,andtoremainthesameifthevalueisequaltothe

1; r = q +1;:::;l)

S =(sij )n n isacollective

(3.13)

=( 2 l(l 1) l 1 X q=1 l X r=q+1 (1 1 jhij j jhij j X =1 d( h q ij ;h r ij )))n n where 1 1 jhij j jhij j X =1 d( h q ij ;h r ij )= sqr ij & d( h q ij ;h r ij )= j h q ij h r ij j; f = 1

2;:::;l

S =(sij )n n

(ai;aj ) (3.14) cdij = sij

ai,asstatedby CDi: (3.15) CDi = 1 2(n 1) n X j=1;j=i (sij + sji)

CR = 1 n n

i=1

International Journal of Fuzzy Logic Systems (IJFLS) Vol.12, No.1/2/3/4, October 2022

International Journal of Fuzzy Logic Systems (IJFLS) Vol.12, No.1/2/3/4, October 2022

mean.Toautomaticallyimprovetheconsensus,theDMswouldnothavetogive theirupdatedcomponentsintheautomatedprocedure.Insuchacase,thenew element hq ij;new maybeevaluatedusingthefollowingequation,newfor cdij <CR:

(3.18) hq ij;new = hq ij +(1 )h q ij;ave where 2 [0; 1] isknownastheoptimizationparameter.Thenewevaluatedvalues willundoubtedlybeclosertothemeanvaluesthanthepreviousones,andsothe degreeofconsensuswillincrease.

3.6. RatingofDecisionMakers. ToanalysedecisionmakersíÖnalpriorityevaluations,emergingconsistencyweightsandpredeÖnedpriorityweightsareemployed, whichareasfollows:

(3.19) w(Dq)= wq Cw(Dq) l X q=1

wq Cw(Dq) where wq; 1 q l; denotesthedecisionmakeríspredeÖnedpriorityweights,while l X q=1

w(Dq)=1: IftheDMsdonotusepredeÖnedpriorityweights,then,astheÖnal priorityrating,theirconsistencyweightswillbeused.

3.7. AggregatedNHFPR. Onaregularbasis,thepreferencelevelassociated witheachDMmaybeweighteddi§erently.Afterreviewingdecisionmakeríspriority ratings,theirviewsaretobeconsolidatedintoaglobalone.Usingtheweighted averageoperator,wecreatethecollectiveconsistentNHFPR H c asfollows: (3.20) H c =(h c ij )n n =(

3.8. RankingofAlternatives. WhentheDMsattainanappropriatedegreeof consensus,theprocessofrankingthealternativesbeginsandthebestoneischosen. Therankingvalue v(ai) foralternative ai; (i =1; 2;:::n); isdeÖnedinthiscontext, asfollows:

4. ComparativeExample

Abankwantstoputaparticularamountofmoneyintothebestoption.To reducetherisksofmakingjudgmentsinthishighlycompetitiveandfast-paced industry,thecompanyísleaderenliststheassistanceofagroupofexpertsinthe decision-makingprocessinthehopesofreachingaconsensus.Thefollowingisa panelwiththreeoptions:

a1 istheautomobileindustry,

a2 isthefoodindustry,

a3 isthecomputerindustry

w(Dq) ~ h q ij )n n;for1 i n; 1 j n:

l X q=1

(3.21) v(ai)= 2 n(n 1) n X j=1;j=i ( 1 jh c ij j jh c ij j X =1 h c ij ) with n X i=1 v(ai)=1

International Journal of Fuzzy Logic Systems (IJFLS) Vol.12, No.1/2/3/4, October 2022

;q =1; 2; 3 fromthreeconsultingdepartmentsistaskedwithassessingthethreeoptions. ai;i =1; 2; 3:Followingpairwisecomparisons,theDMs

Threespecialistshavebeenconsulted,D

; 2; 3, respectivelywiththresholdconsensuslevel

ConsistencyAnalysis:

TheHFPRísconsistencylevelsasstatedbydecisionmakersweremeasuredusing expressions(3.7)ñ(3.12).

Dq;q =1

o§erthefollowingHFPRs H q;q =1

=0 8 H 1 = 2 6 6 4 f0 5gf0 3; 0 5gf0 5; 0 6; 0 7gf0 4g f0 7; 0 5gf0 5gf0 4; 0 6gf0 6; 0 7g f0 5; 0 4; 0 3gf0 6; 0 4gf0 5gf0 6; 0 8g f0 6gf0 4; 0 3gf0 4; 0 2gf0 5g 3 7 7 5 H 2 = 2 6 6 4 f0 5gf0 7; 0 6gf0 8; 0 6gf0 4g f0:3; 0:4gf0:5gf0:8; 0:7; 0:5gf0:4; 0:3g f0:2; 0:4gf0:2; 0:3; 0:5gf0:5gf0:7; 0:6g f0:6gf0:6; 0:7gf0:3; 0:4gf0:5g 3 7 7 5 H 3 = 2 6 6 4 f0 5gf0 7; 0 6; 0 4gf0 6gf0 4; 0 3g f0 3; 0 4; 0 6gf0 5gf0 5gf0 6g f0 4gf0 5gf0 5gf0 9; 0 5; 0 4g f0 6; 0 7gf0 4gf0 1; 0 5; 0 6gf0 5g 3 7 7 5 Normalization: Equation(3.1)ñ(3.6)wereutilisedtogenerateNHFPRsinordertonormalisethe provideddata H 1 = 2 6 6 4 f0 5gf0 3; 0 5; 0 9gf0 5; 0 6; 0 7gf0 4; 0 3; 0 4g f0 7; 0 5; 0 1gf0 5gf0 4; 0 6; 0 6gf0 6; 0 7; 0 3g f0 5; 0 4; 0 3gf0 6; 0 4; 0 4gf0 5gf0 6; 0 8; 0g f0 6; 0 7; 0 6gf0 4; 0 3; 0 7gf0 4; 0 2; 1gf0 5g 3 7 7 5 H 2 = 2 6 6 4 f0 5gf0 7; 0 6; 0 1gf0 8; 0 6; 0 6gf0 4; 0 1; 0 4g f0 3; 0 4; 0 9gf0 5gf0 8; 0 7; 0 5gf0 4; 0 3; 0 2g f0 2; 0 4; 0 4gf0 2; 0 3; 0 5gf0 5gf0 7; 0 6; 0 6g f0 6; 0 9; 0 6gf0 6; 0 7; 0 8gf0 3; 0 4; 0 4gf0 5g 3 7 7 5 H 3 = 2 6 6 4 f0 5gf0 7; 0 6; 0 4gf0 6; 0; 0 6gf0 4; 0 3; 0g f0 3; 0 4; 0 6gf0 5gf0 5; 0; 0 2gf0 6; 0; 0 6g f0 4; 1; 0 4gf0 5; 1; 0 8gf0 5gf0 9; 0 5; 0 4g f0:6; 0:7; 1gf0:4; 1; 0:4gf0:1; 0:5; 0:6gf0:5g 3 7 7 5

; 2; 3

h q ij =max k=i;j(h q ij ; max(h q ik ;h q kj 1; 0)) ; h q ij + h q ji =1

H 1 = 2 6 6 4 f0 5gf0 3; 0 5; 0 9gf0 5; 0 6; 0 7gf0 4; 0 4; 0 4g f0:7; 0:5; 0:1gf0:5gf0:4; 0:6; 0:6gf0:6; 0:7; 0:3g f0:5; 0:4; 0:3gf0:6; 0:4; 0:4gf0:5gf0:6; 0:8; 0g f0:6; 0:6; 0:6gf0:4; 0:3; 0:7gf0:4; 0:2; 1gf0:5g 3 7 7 5 H 2 = 2 6 6 4 f0 5gf0 7; 0 6; 0 2gf0 8; 0 6; 0 6gf0 5; 0 2; 0 4g f0 3; 0 4; 0 8gf0 5gf0 8; 0 7; 0 5gf0 5; 0 3; 0 3g f0 2; 0 4; 0 4gf0 2; 0 3; 0 5gf0 5gf0 7; 0 6; 0 2g f0 5; 0 8; 0 6gf0 5; 0 7; 0 7gf0 3; 0 4; 0 8gf0 5g 3 7 7 5 ~ H 3 = 2 6 6 4 f0 5gf0 7; 0 6; 0 4gf0 6; 0; 0 6gf0 5; 0 3; 0g f0 3; 0 4; 0 6gf0 5gf0 5; 0; 0 2gf0 6; 0; 0 6g f0 4; 1; 0 4gf0 5; 1; 0 8gf0 5gf0 9; 0 5; 0 4g f0:5; 0:7; 1gf0:4; 1; 0:4gf0:1; 0:5; 0:6gf0:5g 3 7 7 5 (i).InNHFPRs H q , q =1; 2; 3; theconsistencymeasuresofpairsofalternatives are: 2 1110 96673 2 10 966710 9333 9667110 9333 1110 8667 86671 3 7 7 5 (ii). TLCI alternatives a1;a2;a3; & a4 are: TLCI(a1)=(0:9889; 0:9667; 0:9889);TLCI(a2)=(1; 0:9667; 1) TLCI(a3)=(1; 0 9556; 1) ;TLCI(a3)=(0 9889; 0 9111; 0 9889) (iii). TLCI againstNHFPRs H q are: TLCI(H 1)=0:9945;TLCI(H 2)=0:95; TLCI(H 3)=0 9945

(CI) maybecalculatedusing(3.11)asfollows: CI =0 9797

D1;D2 andD3 as follows: International Journal of Fuzzy Logic Systems (IJFLS) Vol.12, No.1/2/3/4, October 2022 10

Theglobalconsistencyindex

Now,using(3.12),estimatetheconsistencyweightsoftheDMs

Cw(D1)=0 3384;Cw(D2)=0 3232 Cw(D3)=0 3384 Now l X q=1 Cw(Dq)=1 Consensusmeasures: s 12 ij = 2 6 6 4 10 56670 86670 9333 0 566710 80 7667 0:86670:810:7 0:93330:76670:71 3 7 7 5 ;s 13 ij = 2 6 6 4 10 66670 73330 8667 0 666710 63330 6667 0:73330:633310:6667 0:86670:66670:66671 3 7 7 5 s 23 ij = 2 6 6 4 10:90:73330:8 0 910 56670 7 0 73330 566710 8333 0 80 70 83331 3 7 7 5 Similaritymatrix S = 2 6 6 4 10 71110 77780 8667 0 711110 66670 7111 0 77780 666710 7333 0 86670 71110 73331 3 7 7 5 (i).Consensusdegreeonapairofalternatives (ai;aj ) usingthesimilaritymatrix S : cdij = sij ;i;j =1; 2; 3; 4 (ii).Consensusdegreeonalternatives (ai) CD1 =0 7852;CD2 =0 6963; CD3 =0 7259;CD4 =0 7704; (iii).Consensusdegreeontherelation CR =0 7445 FinalweightsofDMs: Theconsistencyweights Cw(Dq);q =1; 2; 3 willbeutilisedastheDMsÖnal weightsbyusing(3.19),becausenopre-determinedweightsareinvolved.Asaresult, w(D1)=0:3384;w(D2)=0:3232 w(D3)=0:3384; International Journal of Fuzzy Logic Systems (IJFLS) Vol.12, No.1/2/3/4, October 2022 11

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CollectiveNHFPRconstruction: Afterapplying(3.20),wegetthecollectiveNHFPR

Theequation(3.21)isusedtogettheÖnalrankingorderofthealternativesafter analysingtherankingvalues.

)=1 Asaresult,thepreferredorderofalternativesis

Asaresult,DMsmustaltertheirpreferenceswiththehelpof(3.17).Thefollowing arethemeanvaluesoftheexpertíspreferences:

TheidentiÖer(3.17)nowo§ersthefollowingsetofoptionsforimprovingrelevant preferencevalues.

theirpreferencerelationsasaresult:

H c H c = 2 6 6 4 f0:5gf0:55; 0:56; 0:51gf0:63; 0:39; 0:63gf0:47; 0:3; 0:27g f0 45; 0 44; 0 49gf0 5gf0 57; 0 43; 0 43gf0 56; 0 33; 0 38g f0 37; 0 61; 0 37gf0 43; 0 57; 0 57gf0 5gf0 73; 0 63; 0 15g f0 53; 0 7; 0 73gf0 44; 0 67; 0 62gf0 27; 0 37; 0 85gf0 5g 3 7 7 5

FinalRankingofalternatives:

v(a1)=0 2394;v(a2)=0 2267;v(a3)=0 2461& v(a4)=0 2878 n X i=1 v

a4 a3 a1 a2 Enhancementmechanism: As 0 7445= CR< (given =0 8)

(

Have = 2 6 6 4 f0 5gf0 57; 0 57; 0 47gf0 63; 0 4; 0 63gf0 4; 0 23; 0 27g f0:43; 0:43; 0:53gf0:5gf0:57; 0:43; 0:43gf0:53; 0:33; 0:37g f0:37; 0:6; 0:37gf0:43; 0:57; 0:57gf0:5gf0:73; 0:63; 0:33g f0:6; 0:77; 0:73gf0:47; 0:67; 0:63gf0:27; 0:37; 0:67gf0:5g 3 7 7 5

Iq = f(1; 2); (2; 1); (2; 3); (3; 2) ; (2; 4); (4; 2); (3; 4) ; (4; 3)g:

AssumethattheDMsusing (3

18) werepleasedwiththesuggestionsandimproved

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Equation(3.1)ñ(3.6)wereutilisedtogenerateNHFPRsinordertonormalisethe provideddata

ConsistencyAnalysis:

TheHFPRísconsistencylevelsasstatedbydecisionmakersweremeasuredusing expressions(3.7)ñ(3.12).

H 1 new = 2 6 6 4 f0 5gf0 435; 0 535gf0 5; 0 6; 0 7gf0 4g f0 565; 0 465gf0 5gf0 485; 0 515gf0 565; 0 515g f0 5; 0 4; 0 3gf0 515; 0 485gf0 5gf0 665; 0 715g f0 6gf0 435; 0 485gf0 335; 0 285gf0 5g 3 7 7 5 H 2 new = 2 6 6 4 f0 5gf0 635; 0 585gf0 8; 0 6gf0 4g f0 365; 0 415gf0 5gf0 685; 0 565; 0 465gf0 465; 0 315g f0 2; 0 4gf0 315; 0 435; 0 535gf0 5gf0 715; 0 615g f0:6gf0:535; 0:685gf0:285; 0:385gf0:5g 3 7 7 5 H 3 new = 2 6 6 4 f0:5gf0:635; 0:585; 0:435gf0:6gf0:4; 0:3g f0:365; 0:415; 0:565gf0:5gf0:535gf0:565g f0 4gf0 465gf0 5gf0 815; 0 565; 0 365g f0 6; 0 7gf0 435gf0 185; 0 435; 0 635gf0 5g 3 7 7 5

Normalization:

H 1 new = 2 6 6 4 f0 5gf0 435; 0 54; 0 19gf0 5; 0 6; 0 7gf0 4; 0 18; 0 4g f0 56; 0 46; 0 81gf0 5gf0 48; 0 51; 0 51gf0 56; 0 515; 0 21g f0 5; 0 4; 0 3gf0 52; 0 49; 0 49gf0 5gf0 665; 0 715; 0g f0:6; 0:82; 0:6gf0:44; 0:485; 0:79gf0:335; 0:285; 1gf0:5g 3 7 7 5 H 2 new = 2 6 6 4 f0:5gf0:6; 0:59; 0:1gf0:8; 0:6; 0:6gf0:4; 0:11; 0:4g f0:4; 0:41; 0:9gf0:5gf0:68; 0:565; 0:465gf0:46; 0:31; 0:17g f0 2; 0 4; 0 4gf0 32; 0 435; 0 535gf0 5gf0 73; 0 62; 0 62g f0 6; 0 89; 0 6gf0 54; 0 69; 0 83gf0 28; 0 38; 0 38gf0 5g 3 7 7 5 H 3 new = 2 6 6 4 f0 5gf0 6; 0 585; 0 44gf0 6; 0; 0 6gf0 4; 0 3; 0g f0 4; 0 415; 0 56gf0 5gf0 535; 0; 0 18gf0 565; 0; 0 565g f0 4; 1; 0 4gf0 465; 1; 0 82gf0 5gf0 8; 0 56; 0 365g f0 6; 0 7; 1gf0 435; 1; 0 435gf0 2; 0 44; 0 635gf0 5g 3 7 7 5

h q ij =max k=i;j(h q ij ; max(h q ik ;h q kj 1; 0)) ; h q ij + h q ji =1

International Journal of Fuzzy Logic Systems (IJFLS) Vol.12, No.1/2/3/4, October 2022

~ H 1 new = 2 6 6 4 f0 5gf0 5; 0 54; 0 2gf0 5; 0 6; 0 7gf0 4; 0 31; 0 4g f0 5; 0 46; 0 8gf0 5gf0 485; 0 52; 0 52gf0 57; 0 51; 0 21g f0 5; 0 4; 0 3gf0 515; 0 48; 0 48gf0 5gf0 665; 0 715; 0g f0 6; 0 69; 0 6gf0 43; 0 49; 0 79gf0 335; 0 285; 1gf0 5g 3 7 7 5 H 2 new = 2 6 6 4 f0:5gf0:64; 0:6; 0:2gf0:8; 0:6; 0:6gf0:4; 0:215; 0:4g f0:36; 0:4; 0:8gf0:5gf0:69; 0:54; 0:465gf0:44; 0:315; 0:24g f0:2; 0:4; 0:4gf0:31; 0:43; 0:535gf0:5gf0:72; 0:62; 0:62g f0 6; 0 78; 0 6gf0 53; 0 685; 0 76gf0 28; 0 38; 0 38gf0 5g 3 7 7 5 H 3 new = 2 6 6 4 f0 5gf0 64; 0 59; 0 44gf0 6; 0; 0 6gf0 414; 0 3; 0g f0 36; 0 41; 0 56gf0 5gf0 535; 0; 0 2gf0 565; 0; 0 565g f0 4; 1; 0 4gf0 465; 1; 0 8gf0 5gf0 8; 0 57; 0 385g f0 585; 0 7; 1gf0 435; 1; 0 435gf0 2; 0 43; 0 615gf0 5g 3 7 7 5 (i).InNHFPRs H q,q=1,2,3,theconsistencymeasuresofpairsofalternatives are: 10 9710 965 97110 9767 1111 976711 3 7 7 5 ; 40:99510:99331 5 (ii). TLCI alternatives a1;a2;a3 and a4 are: TLCI(a1)new =(0 985; 0 9783; 0 9983);TLCI(a2)new =(1; 0 9822; 0 9978) TLCI(a3)new =(1; 1; 0 9953) ;TLCI(a4)new =(0 985; 0 9806; 0 9961) (iii). TLCI againstNHFPRs H q are: TLCI(H 1)new =0 9925;TLCI(H 2)new =0 9852; TLCI(H 3)new =0 9969 Theglobalconsistencyindex (CI) maybecalculatedusing(3.11)asfollows: CInew =0 9915

D1;D2 andD3 as follows: Cw(D1)new =0:3337;Cw(D2)new =0:3312 Cw(D3)new =0:3351

Now,using(3.12),estimatetheconsistencyweightsoftheDMs

Now l X q=1 Cw(Dq)new =1 Consensusmeasures: s 12 ijnew = 2 6 6 4 10 90 86670 9767 0 910 90 8867 0 86670 910 745 0 97670 88670 7451 3 7 7 5 ;s 13 ijnew = 2 6 6 4 10 8350 73330 8267 0 83510 70 71 0 73330 710 7783 0 82670 710 77831 3 7 7 5 ; s 23 ijnew = 2 6 6 4 10 90170 73330 8033 0:901710:66670:73 0:73330:666710:8667 0:80330:730:86671 3 7 7 5 Similaritymatrix Snew = 2 6 6 4 10:87890:77780:8689 0:878910:75560:7756 0 77780 755610 7967 0 86890 77560 79671 3 7 7 5 (i).Consensusdegreeonapairofalternatives (ai;aj ) basedonthesimilarity matrix S : cdij = sij ;i;j =1; 2; 3; 4 (ii).Consensusdegreeonalternatives (ai) CD1 new =0 8419;CD2 new =0 8034; CD3 new =0:7767;CD4 new =0:8137; (iii).Consensusdegreeontherelation CRnew =0:8089 FinalweightsofDMs: Theconsistencyweights Cw(Dq);q =1; 2; 3 willbeutilisedastheDMsÖnal weightsbyusing(3.19),becausenopre-determinedweightsareinvolved.Asaresult, w(D1)new =0 3337;w(D2)new =0 3312; w(D3)new =0:3351; CollectiveNHFPRconstruction: Afterapplying(3.20),wegetthecollectiveNHFPR H c .

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FinalRankingofalternatives:

Theequation(3.21)isusedtogettheÖnalrankingorderofthealternativesafter analysingtherankingvalues.

5. Comparison

BycomparingourresultstothoseofXuetal.[29]basedonconsistencymeasure, consensusmeasure,andÖnalranking,weclearlyverifythesuggestedtechnique.AccordingtoXuetal.[29],theinitialconsistencylevels,consensuslevel,andultimate rankingofalternativesareasfollows: CR =0 7554;a4 a3 a2 a1 respectively.

Additivereciprocityisincorporatedintooursuggestedstrategytoevaluatethe HFPVísunidentiÖedcomponentsthroughoutthenormalisationprocessandcreate theconsistentHFPRsinaccordance.

TABLE:TheresultswerecomparedoftheXuetal.resultsandthesuggested method

6. Conclusions

Aconsensus-basedtechniquefordealingwiththeMPDMproblemusingconsistentHFPRsisproposedinthispaper.Inthisregard,thenotionofHFPRswas derivedfromtheworkofXu[20],andweproposeane§ectiveadditivereciprocitybasedstrategyfornormalisingHFPRs.

Theaboveexampledemonstratesastep-by-stepprocedurefornormalisingthe HFPR.Followingaconsistencystudy,DMswereassignedconsistencyweights.To carrygreaterweightintheaggregationprocess,DMswithahighlevelofconsistencyshouldbegivenlargerweights.Furthermore,animprovementMechanism isprovidedtoaidintheimplementationofahigherdegreeofagreementona straightforwardpath.WhentheDMshavereachedanadequatelevelofconsensus,

H c new = 2 6 6 4 f0 5gf0 57; 0 56; 0 28gf0 631; 0 4; 0 64gf0 4; 0 28; 0 26g f0 43; 0 44; 0 72gf0 5gf0 57; 0 36; 0 39gf0 52; 0 27; 0 34g f0 369; 0 6; 0 36gf0 43; 0 64; 0 61gf0 5gf0 73; 0 62; 0 33g f0 6; 0 72; 0 74gf0 48; 0 73; 0 66gf0 27; 0 38; 0 67gf0 5g 3 7 7 5

v(a1)new =0 2234;v(a2)new =0 2244;v(a3)new =0 2605 and v(a4)new =0 2917 n X i=1 v(ai)=1:Asaresult,thepreferredorderofalternativesis a4 a3 a2 a1

RankingOrder 4 a3 a2 a1 a3 a1 a2

4 a3 a2 a1

AfterEnhancement

International

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.12, No.1/2/3/4, October 2022

thetechniquemovesontotheselectionstep,whichincludesmethodsforaggregatingandrankingtodeterminethebestsolution.Acomparativecaseiscreatedto highlightthefeasibilityande¢cacyoftheproposedmethod.TheÖndingsprovide usabetterunderstandingoftheMPDMprocess.

7. FutureWork

Themainfutureworkofthisproblemisthatwecanextendthisproblemby utilizingdualhesitantfuzzysetandbyapplyingaggregationoperatorwecanmake thisproblemmorecompact.Thisproblemcanalsobeextendedbyusingtriangular fuzzynumber.

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Currentaddress :DepartmentofMathematics,COMSATSUniversityIslamabadatLahore, Lahore5400,Pakistan

E-mailaddress : smbukhari735@gmail.com

DepartmentofMathematics,COMSATSUniversityIslamabadatLahore,Lahore 5400,Pakistan

E-mailaddress : atiqurehman@cuilahore.edu.pk

DepartmentofBasicSciences,UniversityofEngineeringandTechnologyTaxila, Pakistan

E-mailaddress : mariabibi782@gmail.com