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1FUZZYAGGREGATIONOFNUMERICALPREFERENCESMichelGrabisch,SergeiA.Orlovski,RonaldR.YagerAbstract:Theproblemofaggregatingnumericalvaluesisaddressedinthischapter.The rstpartdealswiththeaggregationofcriteriaintoasingleone.Propertieswhicharesuitableforthiscasearepresented,togetherwiththemostcommonaggregationoperators.Aspecialsectionisdevotedtoorderedweightedaveraging(OWA)operators,andfuzzyintegrals.Then,relationbe-tweenproperties,linksbetweenoperators,characterizationofsomeoperatorsarepresented.AnimportantsectionisdevotedtothebehavioralanalysisofOWAoperatorsandfuzzyintegrals,linkingvaluesofparameterswiththeatti-tudeofthedecisionmaker.Thelastsectionofthis rstpartisconcernedwiththeproblemofidenti cationofoperatorsinapracticalproblem,akeyissueineveryapplication.Thesecondpartisdevotedtospecialaspectsinaggregationofpreferences,inamultiattributecontext.TheParetoprincipleisexplained,andthentheagreement-discordanceprinciple,whichisthebasisofELECTREIIIandIV,isaddressed.1.1THEAGGREGATIONPROBLEMThemainobjectofchapter2isconcernedwithaggregation.Itcomesnaturallyafterchapter1,dealingwiththemodellingofpreference,andbeforechapter3,ofwhichconcernisthechoiceproblem,i.e.howtoexploitaggregatedprefer-ences.12Aggregationreferstotheprocessofcombiningvalues(numericalornonnumerical)intoasingleone,sothatthe nalresultofaggregationtakesintoaccountinagivenfashionalltheindividualaggregatedvalues.Inordertoclarifythisde nition,weelaborateonthewordsvaluesandfashion.Indecisionmaking,valuestobeaggregatedaretypicallypreferenceorsat-isfactiondegrees.ApreferencedegreetellstowhatextentanalternativeAispreferredtoanalternativeB(seechapter1),andthusisarelativeappraisal.Bycontrast,asatisfactiondegreeexpressestowhatextentagivenalternativeissatisfactorywithrespecttoagivencriterion,orakindofdistancetoapro-totypewhichmayrepresenttheidealalternativeforthedecisionmaker.Itisanabsoluteappraisal.Inamoregeneralwayanddependingontheapplica-tion,valuestobeaggregatedcanbecon dencedegreesinthefactthatagivenalternativeistrue,expert'sopinions,similaritydegrees,etc.Valuescanbelongtoeitherasinglenumericalornonnumericalscale,buttheexistenceofaweakorderrelationonthesetofallpossiblevaluesistheminimalrequirementwhichhastobesatis edinordertoperformaggregation.Thus,nominalscales,i.e.enumeratedvalueswithoutanystructure,suchasfcabbage,turnip,horseradish,cucumbergarenotsuitable.Atypicalexampleofnonnumericalscaleisalinguisticscaleoftenusedinfuzzysettheory,suchasfsmall,medium,largeg,orfunsuitable,acceptable,average,good,excellentg.Numericalscalescanbeofordinalorcardinaltype.Onanordinalscale,numbershavenoothermeaningthande ninganorderrelationonthescale,anddistancesordi erencesbetweenvaluescannotbede ned.Thisisthecasewhenpeopleareaskedtoevaluateaproductonanintegerscalefrom1to5,whereforexample1correspondsto\verybadand5to\excellent.Here,wehavenoreasontosaythatthedistancebetween1(\verybad)and2(\bad)isthesamethanthedistancebetween2and3(\medium).Ontheotherhand,weareallowedtode neoperationsotherthancomparisononacardinalscale.Hereagain,thenatureoftheoperationdependsonthekindofscale:intervalscales,wherethepositionofthezeroisamatterofconvention(e.g.thetemperatureexpressedontheCelsiusscale)allowtoperformonlylinearoperations,sothatforexamplemultiplicationisnotallowed,whileratioscales,whereatruezeroexists(e.g.theKelvinscalefortemperature)allowtoperformratiosandmultiplication.Wedon'tfurtherdetailthesemeasurementtheoreticalaspects,andtheinterestedreaderisaskedtorefertothemonographsofKrantzetal.[Krantzetal.,1971]andRoberts[Roberts,1979]foracompletetreatmentofthequestion.Anotherdistinctionamongvaluesisrelatedwithprecision.Precisevaluesaresimplyelementsofthescale,whileimprecisevaluesaresubsets(fuzzyornot)ofthescale,whichmeansthatthetrue(butunknown)valueliessomewhere3inthissubset.Imprecisevaluescanbeconsideredonanytypeofscales,eithernumericalornonnumerical.Section1.2.9willbrieyaddressthisissue.Oncevaluesarede nedwecanaggregatethemandobtainanewvaluede nedonthesamescale,butthiscanbedoneinmanydi erentwaysaccordingtowhatisexpectedfromtheaggregationoperation,whatisthenatureofthevaluestobeaggregated,andwhatkindofscalehasbeenused.Weelaborateonthesethreepoints.Aggregationoperatorscanberoughlydividedintothreeclasses,eachpos-sessingverydistinctbehaviororsemantic.Conjunctiveoperatorscombinevaluesasiftheywererelatedbyalogical\andoperator.Thatis,theresultofcombinationishighifandonlyifallthevaluesarehigh.Triangularnormsarethesuitableoperatorsfordoingconjunctiveaggregation.Ontheotherhand,disjunctiveoperatorscombinevaluesasan\oroperator,sothattheresultofcombinationishighifsome(atleastone)valuesarehigh.Themostcommondisjunctiveoperatorsaretriangularconorms.Betweenconjunctiveanddisjunc-tiveoperators,thereisroomforathirdcategory,namelyaveragingoperators.Theyarelocatedbetweenminimumandmaximum,whicharetheboundsofthet-normandt-conormfamilies.Averagingoperatorshavethepropertytobecompensative,thatis,lowvaluescanbecompensatedbyhighvalues,sothattheresultofcombinationwillbemedium.Thereareofcourseotherfamiliesofoperatorswhichdonot tin