A Sequential Selection Process in Group Decision M

DECSAIDepartmentofComputerScienceandArticialIntelligenceASequentialSelectionProcessinGroupDecisionMakingwithaLinguisticAssessmentApproachF.Herrera,E.Herrera-Viedma,J.L.VerdegayTechnicalReport#DECSAI-94103February,1994InternationalJournalofInformationSciences80(1995)1-17.ETSdeIngenieraInformatica.UniversidaddeGranada.18071Granada-Spain-Phone[phone],Fax[phone]ASequentialSelectionProcessinGroupDecisionMakingwithaLinguisticAssessmentApproachF.Herrera,E.Herrera-Viedma,J.L.VerdegayDept.ofComputerScienceandArticialIntelligenceUniversityofGranada,18071-Granada,Spaine-mail:herrera,viedma,[e-mail]quentialSelectionProcessinGroupDecisionMakingunderlinguisticassessmentsispresented,whereasetoflinguisticpreferencerelationsrepresentsindividualspreferences.Acollectivelinguisticpreferenceisobtainedbymeansofadenedlinguisticorderedweightedaveragingoperatorwhoseweightsarechosenaccordingtotheconceptoffuzzymajority,speciedbyafuzzylinguisticquantier.Thenwedenetheconceptsoflinguisticnondominance,linguisticdom-inanceandstrictdominancedegreesaspartsoftheSequentialSelectionProcess.Thesolutionalternative(s)isobtainedbyapplyingtheseconcepts.Keywords:Groupdecisionmaking,linguisticlabels,linguisticpreferences,fuzzyma-jority,fuzzylinguisticquantiers,linguisticnondominancedegree,linguisticdominancedegree,strictdominancedegree.1.IntroductionAgroupdecisionmakingprocesscanbedenedasadecisionsituationinwhich(i)therearetwoormoreindividualsthatdierintheirpreferences(valuesystems)buthavethesameaccestoinformation,eachofthemcharacterizedbyhisorherownperceptions,attitudes,motivations,andpersonalities,(ii)whorecognizetheexistenceofacommonproblem,and(iii)attempttoreachacollectivedecision[2].Theuseofpreferencerelationsisnormalingroupdecisionmaking.Moreover,sincehumanjudgmentsincludingpreferencesareoftenvague,fuzzysetsplaysanimportantroleindecisionmaking.Severalauthorshaveprovidedinterestingresultsongroupdecisionmakingorsocialchoicetheorywiththehelpoffuzzysets.Theyhaveproventhatfuzzysetsprovidedamoreexibleframeworkforthediscussionofgroupdecisionmaking[8,10,11,12,13,14,16,18].InafuzzyenvironmentitissupposedthereexistsanitesetofalternativesX=fx1;:::;xngaswellasanitesetofindividualsN=f1;:::;mg,andeachindividualk2NprovideshisfuzzypreferencerelationonX,i.e.,pkXxX,andPk(xi;xj)denotesthedegreeofpreferenceofalternativexioverxj,Pk(xi;xj)2[0;1].Inthisframeworktomakedecisionsconsiststochoiceoneormorealternativesofmentionedalternativessetaccordingtoindividuals’fuzzypreferencerelations.Sometimes,however,anindividualcouldhavevagueinformationabouttheprefer-encedegreeofalternativexioverxjandcannotestimatehispreferencewithanexactnumericalvalue.Thenamorerealisticapproachmaybetouselinguisticassessmentsinsteadofnumericalvalues,i.e.,bysupposingthatthevariables(preferencerelations)whichparticipateintheproblemareassessedbymeansoflinguisticterms[7,20,4,6,9,15].Ascaleofcertaintyexpressions(linguisticallyassessed)wouldbepresentedtotheindividual,whocouldthenuseittodescribehisdegreeofcertaintyinapreference.Assumingasetofalternativesordecisions,thebasicquestionishowtorelatetodif-ferentdecisionschemata.Accordingto[3]thereare(atleast)twopossibilities,agroupselectionprocessandaconsensusprocess.Therst,acalculationofsomemeanvaluedecisionschemaofasetofdecisions,wouldimplythechoiceofanalgebraicconsensusasamappingl:DxD!D,whereasthesecond,themeasureofdistancebetweenschemata,couldbecalledtopologicalconsensusinvolvingamappingk:DxD!L,whereLisacompletelattice.Here,weshallfocusontherstpossibility,fordevelopingaGroupChoiceProcessunderlinguisticpreferences.Assumingasetoflinguisticpreferencesrepresentingindividualspreferences,thegroupchoiceprocessdevelopsaccordingtothefollowingscheme:alinguisticorderedweightedaveraging(LOWA)operatorisdenedforlinguisticlabels,basedontheor-deredweightedaveragingoperator[19],andtheconvexcombinationoflinguisticlabels[5].Theconceptsoffuzzymajority,representedbymeansoflinguisticquantiersandtheLOWAoperator,areusedinordertoobtainacollectivelinguisticpreference.Fi-nally,asequentialselectionprocessactingonthecollectivepreferencerelationisdenedaccordingtothefollowingtwosteps:1.-Usingtheconceptofnondominatedalternatives[17]fordeninganondominancelinguisticdegreeandobtainingthesetofmaximalnondominatedalternativesfromthecollectivelinguisticpreference.2.-Deningtheconceptsoflinguisticdominancedegreeandstrictdominancede-greewithlinguisticlabels,andapplyingthemtothesetofmaximalnondominatedalternativesforobtainingthebestalternatives.Graphically,FormalRepresentationExpertsLinguisticSetCollectiveLinguistictheLinguisticPreferencesAlternativesSolutionAlternativesStrictDominanceDegreeTermSetofMaximalConceptFuzzyMajorityLinguisticFuzzyQuantifiersLinguisticAveragingLabelsConvexCombinationofLinguisticWeightedOrderedLinguisticDominanceDegreePreferenceAveragingOrderedWeightedSEQUENTIALSELECTIONPROCESSAggregationofNon-dominatedNon-dominatedFig.1.GroupChoiceProcessTheaimofthispaperistopresentthegroupchoiceprocessusingasequen