APPLICATIONS OF THE LINGUISTIC OWA OPERATOR IN GRO

DECSAIDepartmentofComputerScienceandArticialIntelligenceApplicationsoftheLinguisticOWAOperatorinGroupDecisionMakingF.Herrera,E.Herrera-Viedma,J.L.VerdegayTechnicalReport#DECSAI-96132.September,1996ETSdeIngenieraInformatica.UniversidaddeGranada.18071Granada-Spain-Phone[phone],Fax[phone]APPLICATIONSOFTHELINGUISTICOWAOPERATORINGROUPDECISIONMAKINGF.Herrera,E.Herrera-ViedmaandJ.L.VerdegayDept.ofComputerScienceandArticialIntelligence,UniversityofGranada,18071-Granada,SpainABSTRACTAssumingagroupdecisionmakingproblemwheretheexpertsexpresstheiropinionsbymeansoflinguisticpreferencerelations,theapplicationoftheLinguisticOWAoperatorguidedbyfuzzymajorityisanalyzed.TwodierentperspectivesoftheuseoftheLOWAoperatorarepresented:(i)intheselectionprocess,toaggregateindividuallinguisticpreferencerelationsinacollectiveoneandtocalculatedierentlinguisticchoicedegreesofthealternatives,and(ii)intheconsensusreachingprocesstoobtainthelinguisticconsensusmeasures.Inallcases,theconceptoffuzzymajorityisrepresentedbymeansofafuzzylinguisticquantierusedtoobtaintheweightsthattheLOWAoperatorneedsinitsaggregationway.1INTRODUCTIONTherearemanyproblemsinwhichthesolutionsdependonthesynthesisofinformationsuppliedbydierentsources(e.g.thestrategicplanning,thedia-gnosis,thecapitalinvestment,etc.).WhenthenalsolutionconsistsofmakingadecisiontheymaybemodeledasaparticularcaseofaGroupDecisionMaking(GDM)problem.AGDMproblemmaybedenedasadecisionsituationinwhich(i)therearetwoormoreexperts,eachofthemcharacterizedbyhisownperceptions,attitudes,motivations,andpersonalities,(ii)whorecognizetheexistenceofa12Chapter1commonproblem,and(iii)attempttoreachacollectivedecision.Itisusuallysolvedinaprocessoftwophases:1.consensusphase,whichattemptstoachievethemaximumpossiblecon-sensusdegreeaboutnaldecision,and2.selectionphase,whichattemptstomakethebestdecisionaccordingtotheexperts’opinions.AnyGDMmodelmustinclude(i)acomprehensiverepresentationofexperts’opinions,and(ii)adatafusiontechniquethatcombinesthoseindividualopin-ionsandallowstoreachthenaldecision.Futhermore,theymustbeabletodealwiththefuzzinessofhumanjudgements.Todoso,manyauthorshaveusedtheFuzzySetsTheorytomodeltheimpreciseinformation[12].Inafuzzyframework,fuzzypreferencerelationsaretheclassicalrepresenta-tionusedtoprovidetheexperts’preferences.Thesemayhaveanumericalorlinguisticnature.Intherstcase,itisassumedthattheexpertsareabletoexpresstheirpreferenceswithexactnumericalvalues(e.g.,numbersinthe[0,1]interval)[11].Inthesecondcase,itissupposedthattheexpertscannotestimatetheirpreferenceswithexactnumericalvalues,andthen,theyuselinguisticas-sessmentsinsteadofnumericalvalues(e.g.,linguistictermsofapre-establishedlabelset)[4].WeassumeaGDMmodelinwhichtheexpertsprovidetheirpreferencesbymeansofthelinguisticpreferencerelationsandwepresenthowtheapplicationoftheLinguisticOWA(LOWA)operatorguidedbyfuzzymajority[3,7]con-tributestosolvetheGDMproblem.WeshowhowtheLOWAoperatorcanbeappliedfromtwodierentperspectives:1.Intheselectionphase:Itisappliedtoaggregatetheindividuallinguisticpreferencerelationsinacollectiveoneandtoderivedierentlinguisticchoicedegreesofthealtern-ativesfromthatcollectivelinguisticpreferencerelation.2.Intheconsensusphase:Itisusedtodeterminesomelinguisticconsensusmeasuresaccordingtothemajorityexperts’preferences.Todoso,thepaperisstructuredasfollows:Section2presentsthebasicele-mentsusedtodevelopourstudy,i.e.,thetypeoflabelsetusedtoprovideApplicationsoftheLinguisticOWAOperatorinGDM3theopinionsandtheconsideredGDMproblem;Section3showstheLOWAoperatorguidedbyfuzzymajority;Section4analyzesthetwoapplicationsoftheLOWAoperatorintheresolutionprocessofconsideredGDMproblem;andnally,someconclusionsarepointedout.2PRELIMINARIESWeuselabelsetswithanoddcardinal,representingthemiddletermanas-sessmentofapproximately0.5,withtherestofthetermsbeingplacedsym-metricallyarounditandthelimitofgranularity11ornomorethan13.Thesemanticoftheelementsinthelabelsetisgivenbyfuzzynumbersdenedonthe[0,1]interval,whicharedescribedbymembershipfunctions.Becausethelinguisticassessmentsarejustapproximateonesgivenbytheexperts,wecanconsiderthatlineartrapezoidalmembershipfunctionsaregoodenoughtocap-turethevaguenessofthoselinguisticassessments,sinceitmaybeimpossibleorunnecessarytoobtainmoreaccuratevalues.Thisrepresentationisachievedbythe4-tuple,(ai;bi;i;i),thersttwoparemetersindicatetheintervalinwhichthemembershipvalueis1;thethirdandfourthparametersindicatetheleftandrightwidths.Moreover,thetermset,S=fs0;:::;sTg;musthavethefollowingcharacteristics:1)Thesetisordered:sisjifij.2)Thereisthenegationoperator:Neg(si)=sjsuchthatj=T-i.3)Maximizationoperator:Max(si;sj)=siifsisj.4)Minimizationoperator:Min(si;sj)=siifsisj.Example1.Forexample,thisisthecaseofthefollowingtermset:MAMaximum(1;1;0:25;0)VMVeryMuch(:75;:75;:15;:25)MUMuch(:6;:6;:1;:15)MMedium(:5;:5;:1;:1)LLittle(:4;:4;:15;:1)VLVeryLittle(:25;:25;:25;:15)MIMinimum(0;0;0;:25)Inwhatfollows,weshallusethissetofsevenlabelsinallexamples.4Chapter1Then,alinguisticsettingoftheGDMproblemisasfollows[5]