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IEEETRANSACTIONSONFUZZYSYSTEMS,VOL.XX,NO.Y,MONTH20060AConsensusModelforGroupDecisionMakingwithIncompleteFuzzyPreferenceRelationsE.Herrera-Viedma,S.Alonso,F.Chiclana,andF.HerreraAbstract—Twoprocessesarenecessarytosolvegroupdecisionmakingproblems:aconsensusprocessandaselectionprocess.Theconsensusreachingprocessisnecessarytoobtainafinalsolutionwithacertainlevelofagreementbetweentheexperts;andtheselectionprocessisnecessarytoobtainsuchafinalsolution.In[19],wepresentaselectionprocesstodealwithgroupdecisionmakingproblemswithincompletefuzzyprefe-rencerelations,whichusesconsistencymeasurestoestimatetheincompletefuzzypreferencerelations.Inthispaperwepresentaconsensusmodel.Themainnoveltyofthisconsensusmodelisthatofbeingguidedbybothconsensusandconsistencymeasures.Also,theconsensusreachingprocessisguidedautomatically,withoutmoderator,throughbothconsensusandconsistencycriteria.Todothat,afeedbackmechanismisdevelopedtogenerateadviceonhowexpertsshouldchangeorcompletetheirpreferencesinordertoreachasolutionwithhighconsensusandconsistencydegrees.Ineachconsensusround,expertsaregiveninformationonhowtochangetheirpreferences,andtoestimatemissingvaluesiftheircorrespondingpreferencerelationisincomplete.Additionally,aconsensusandconsistencybasedinducedorderedweightedaveragingoperatortoaggregatetheexperts’preferencesisintroduced,whichcanbeusedinconsensusmodelsaswellasinselectionprocesses.Themainimprovementsofthisconsensusmodelisthatitsupportsthemanagementofincompleteinformationanditallowstoachieveconsistentsolutionswithagreatlevelofagreement.IndexTerms—GroupDecisionMaking,FuzzyPreferenceRe-lations,Consensus,Aggregation.I.INTRODUCTIONGroupdecisionmaking(GDM)problemsconsistinfindingthebestalternative(s)fromasetoffeasiblealternativesX={x1,...,xn}accordingtothepreferencesprovidedbyagroupofexpertsE={e1,...,em}.Duetotheirapparentmeritswhenaggregatingexperts’preferencesintogrouppreferences[20],[22],[38],weassumethatexpertsprovidefuzzypreferencerelations[6],[14],[22],[26],[32],[36].AdifficultythathastobeaddressedwhendealingwithrealGDMproblemsisthelackofinformation.Indeed,theremaybecaseswhereanexpertwouldnotbeabletoefficientlyexpressanykindofpreferencedegreebetweentwoormoreoftheavailableoptions.Thismaybeduetoanexpertnotpossessingapreciseorsufficientlevelofknowledgeofpartoftheproblem,orbecausethatexpertisunabletodiscriminatethedegreetowhichsomeoptionsarebetterthanothers.ExpertsinthesesituationswouldrathernotguessthoseEnriqueHerrera-Viedma,SergioAlonsoandFranciscoHerreraarewiththeDepartmentofComputerScienceandArtificialIntelligence,Univer-sityofGranada,18071Granada,Spain(email:viedma@decsai.ugr.es;her-rera@decsai.ugr.es;salonso@decsai.ugr.es).FranciscoChiclanaiswiththeCentreforComputationalIntelligence,SchoolofComputing,DeMontfortUniversity,LeicesterLE19BH,UK(email:chiclana@dmu.ac.uk).preferencedegreesandasaconsequencetheymightprovideincompleteinformation[1],[9],[19],[27],[28],[40].Usually,GDMproblemsarefacedbyapplyingtwodifferentprocessesbeforeafinalsolutioncanbegiven[5],[16],[18],[24]:1)theconsensusprocessand2)theselectionprocess.Theconsensusprocessreferstohowtoobtainthemaximumdegreeofconsensusoragreementbetweenthesetofexperts.Usually,theconsensusprocessisguidedbyahumanfigurecalledmoderator[16],[23],[24].Theselectionprocessobtainsthefinalsolutionaccordingtothepreferencesgivenbytheexperts.Itinvolvestwodifferentsteps[17],[33]:aggregationofindividualpreferencesandexploitationofthecollectivepreference.Clearly,itispreferablethattheexpertshadachievedahighlevelofconsensusconcerningtheirpreferencesbeforeapplyingtheselectionprocess.In[1],[19]weintroduceaselectionprocesstodealwiththeGDMproblemswithincompletefuzzypreferencerelations.Inthisselectionprocesswepresentaconsistencybasedprocedurewhichisabletoestimateallmissingvaluesfromtheknownpreferences.Inthispaper,wefocusontheconsensusprocess.Intheliterature,wecanfindmanyapproachestomodeltheconsensusprocessesinGDM[3]–[5],[7],[10],[11],[16],[18],[23]–[25],[29],[35],[37],[46].Mostoftheseapproachesuseonlyconsensusmeasurestocontrolandguidetheconsensusprocess.Ifaconsensusprocessisseenasatypeofpersuasionmodel[8],thenothercriteriacouldbeusedtoguidetheconsensusreachingprocessesasitcouldbe,forexample,thecooperationorconsistencycriterion.Afirstapproachtoconsensususingaconsistencycriterioncanbefoundin[12],althoughpreferencerelationswereassumedtobecomplete.Also,inthecontextoftheanalyticalhierarchyprocess(AHP)[34],consistencyhasbeenusedinGDM[2],[39].TheaimofthispaperistopresentaconsensusmodelforGDMproblemswithincompletefuzzypreferencerelations.Thisconsensusmodelwillnotonlybebasedonconsensusmeasuresbutalsoonconsistencymeasures.Asin[16],weusetwokindsofconsensusmeasurestoguidetheconsensusreachingprocesses,consensusdegrees(toevaluatetheagree-mentofalltheexperts)andproximitydegrees(toevaluatetheagreementbetweentheexperts’individualpreferencesandthegrouppreference).Tocomputethem,firstly,allmissingvaluesoftheincompletefuzzypreferencerelationsareestimatedusingtheconsistencybasedestimationprocedurepresentedin[19].Afterwards,someconsistencymea
本文标题:Basis for a Consensus Model in Group Decision Maki
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