A-fuzzy-extension-of-Saatys-priority-theory

FuzzySetsandSystems11(1983)229-241North-Holland229AFUZZYEXTENSIONOFSAATY’SPRIORITYTHEORYP.J.M.vanLAARHOVEN*andW.PEDRYCztDelftUniversityofTechnology,Delfi,NetherlandsReceivedSeptember1982RevisedFebruary1983Wepresentafuzzymethodforchoosingamonganumberofalternativesunderconflictingdecisioncriteria:afuzzyversionofSaaty’spairwisecomparisonmethod(1980)extendedbydeGraan(1980)andLootsma(1981).Eachratioexpressingtherelativesignificanceofapairoffactorsisdisplayedinamatrix,fromwhichsuitableweightscanbeextracted.Sincetheseratiosareessentiallyfuzzy-theyexpresstheopinionofadecision-makerontheimportanceofapairoffactors-wehaveadaptedtheabove-mentionedmethodinsuchaway,thatdecision-makersareaskedtoexpresstheiropinionsinfuzzynumberswithtriangularmembershipfunctions.Weapplythemethodattwodistinctlevels:firsttofindfuzzyweightsforthedecisioncriteria,andsecond,tofindfuzzyweightsforthealternativesundereachofthedecisioncriteria.Byasuitablecombinationoftheseresults,weobtainfuzzyscoresforthealternatives,aswellastheirsensitivities.Usingthesefuzzyscores,thedecision-makersshouldbeabletomakeachoiceforoneofthealternatives.Keywords:Fuzzynumbers,Fuzzyarithmetic,Multi-criteriadecisionanalysis.1.IntroductionInthispaperweconsidertheproblemofchoosingamonganumberofalternativesunderconflictingjudgementsmadeonthebasisofdecisioncriteria.Theaimistochoosethealternativelikelytobethebest.Weconsiderthewell-knownsituation,whereinformationisavailableontheperformanceofthealternativesunderthedecisioncriteriainquestion.ThemethodwepresenthereisafuzzyversionoftheextensionsbydeGraan[4]andLootsma[8]ofSaaty’sprioritytheory[9],whichhasbeendevelopedtoweightthesignificantfactorsinadecisionproblemviapair-wisecomparison.Eachratioexpressingtheinterrelativesignificanceofapairoffactorsisdisplayedinamatrix,fromwhichsuitableweightscanbeextracted.Itiscommonlyfeltasamajorproblemtoquantifytheabovementionedratios.Therefore,attemptshavebeenmadetotackletheproblembyusingmembershipfunctions(seee.g.BaasandKwakernaak[l]andJain[5&Foracriticalreviewof*PresentlyatPhilipsResearchLaboratories,Eindhoven,NetherlandstPresentlyatSilesianTechnicalUniversity,DepartmentofAutomaticControlandComputerScience,44-100Gliwice,Pstrowskiego16,Poland0165-0114/83/$3.0001983,ElsevierSciencePublishersB.V.(North-Holland)230P.J.M.vanLmhoven,W.Pedrycztheseandothermethods,werefertoKicker?[6].Thedilferencebetweenthesemethodsandourmethodisthreefold:(1)Weusefuzzynumberswithtriangularmembershipfunctions.Thus,calcula-tionsaresimple,sinceweonlyusesimpleoperationsintheparameters(addition,multiplication,...)whichdeterminethemembershipfunctions.(2)Weareabletohandledecisionproblems,whereeithernoinformationormultipleinformationisavailableforcertainpairsoffactors.(3)Usingtheprincipleofhierarchiccomposition,weapplyprioritytheoryontwolevels:first,byassigningweightstothesignificantperformanceordecisioncriteria,andsecond,byweighingthealternativesundereachofthecriteriaseparately.Inthenextsectionsweshallfirstdevoteourattentiontoprioritytheoryand,morespecifically,totheproblemofextractingsuitableweightsfromamatrixcontainingratiosobtainedbypair-wisecomparisonsofthedecisionfactors.Asweusefuzzyratiosinournewmethod,wemakesomeremarksonfuzzynumbersandseveraloperationsonthesefuzzynumbers.Hereinweuseasimpletriangularmembershipfunctiontoexpresssuchafuzzyratio.Next,wepresentthenewmethodandanexample,dealingwiththechoiceamonganumberofcandidatesforaprofessorshipatauniversity.ThisexampleisafuzzyversionofaproblemoriginallypresentedbyLootsma[7].2.PrioritytheoryWeconsideradecisionproblem,wherenfactors,F,,F2,...,F,,,havebeenidentified;itisourpurposetoobtainestimatesofthepositiveweightsWlsw2,***9w,,ofthesefactors,assumedtobenormalizedinthesensethatCyC1Wi=1.SupposeamatrixR=(tii)isavailable,whereririsanestimatefortherelativesignificanceofthefactorsFiandFj,i.e.forWi/Wj.WeassumethatRisareciprocalmatrix,i.e.rij’rji=1,i,j=1,2,...,n.(1)Now,thereareseveralwaysofobtainingestimatesfortheweightswl,w2,...,w,fromthematrixI?(seeSaaty[9j).AswasalreadyshownbyLootsma[8],oneofthesemethods,logarithmicregression,issuitableforextensiontothesituation,inwhichmultiplecomparisonsforpairsoffactorsareavailable.Wethere-forerestrictourattentiontothismethod.Weestimatethevectorwbythe-normalized-vectorCX,whichminimizes1OnCj-M%/Cfj))*.ij(2)Underadditionalconditions(normalization)thisresultsinestimatingwibythecorrespondingnormalizedrowmean.Suppose,however,thematrixRhasemptycells(noriravailable)orcellswithmorethanoneentry(severalrij’savailable).Suchasituationcanoccur,whenseveraldecision-makersexpresstheiropinionontherelativesignificanceofapairoffactors.InthiscaseweestimatewbytheAfuzzyextension:ofSaaty’sprioritytheory231normalizedvector,whichminimizesCi(Inrijk-lll(fXJCtj))2i-cjk=l(3)whererijk(k=1,2,...,6,)are6,estimatesforWi/Wj(SijC~IIbeequalto0,ifnocomparisonsareavailable,equaltooneorgreaterthanone,inwhichcasetherearemultiplecomparisons)andwherewehavetaken(1)intoaccount.IfweputYijk=lnrijk,weminimize&k$+l(Yijk-4+%I’bysolvingtheassociatednormalequations-%j$l%-j$laij$=iiYijk,i=l,2,...,n