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JSystSciSystEng(Dec2006)15(4):457-464ISSN:1004-3756(Paper)1861-9576(Online)DOI:10.1007/s11518-006-5021-7CN11-2983/N©SystemsEngineeringSocietyofChina&Springer-Verlag2006THEREISNOMATHEMATICALVALIDITYFORUSINGFUZZYNUMBERCRUNCHINGINTHEANALYTICHIERARCHYPROCESSThomasL.SAATYUniversityofPittsburgh,Pittsburgh,PA15260saaty@katz.pitt.edu()AbstractFuzzylogichasdifficultyproducingvalidanswersindecision-making.Absentaretheoremstoprovethatitworkstoproduceresultsalreadyknownthatarebeingestimatedwithjudgmentsbytransformingsuchjudgmentsnumerically.ThenumericalrepresentationofjudgmentsintheAHPisalreadyfuzzy.Makingfuzzyjudgmentsmorefuzzydoesnotleadtoabettermorevalidoutcomeanditoftenleadstoaworseone.ThecompatibilityindexoftheAHPisusedtoillustratehowtheanswersobtainedbyfuzzifyingAHPjudgmentsdonotproducebetterresultsthandirectderivationoftheprincipaleigenvector.Otherauthorswhodidexperimentswithgivendataindecisionmakingquotedintheconclusionssectionofthepaper,haveobservedthatfuzzysetsgivesthepoorestanswersamongallmethodsusedtoderivebestdecisions.Keywords:AnalyticHierarchyProcess,fuzzylogic,compatibilityindex,validity1.IntroductionInthisbriefnoteweshowthatalthoughfuzzythinkingmaybeusefulinsomeengineeringpracticeswheresocalledcrispnumbersareobtainedfromcountingorfromusingscalesofmeasurement,itishighlycontroversialandnotveryusefultouseitindecisionmakingwherethenumbersassignedtojudgmentsarealreadyfuzzyandmakingthemmorefuzzydoesnothelpproducemorevalidoutcomes.Notonlycanthisbeshownwithexamples,butwhatmakesourassertionmoretothepointistheexistenceofmathematicaltheorywhichprovesthatwhenthejudgmentsofadecisionmatrixasintheAnalyticHierarchyProcess(AHP)(Saaty,2004)areperturbedbyasmallamountasonedoesinfuzzysimulation,theresultingeigenvectorisalsoperturbedbyasmallamount.Ifajudgmentispoorinrepresentingtheunderlyingsituationwithaknownnumericaloutcome,theeigenvectorwillnotbecompatiblewiththeknownoutcome.Fuzzifyingwillleavetheeigenvectorveryneartowhatitwasinthefirstplaceandisthereforenotonlyapoorpracticebutleavesuswithlittleconfidencethatithelpswhenwedonotknowtheanswer.Fuzzysetpracticehasbecomeaselfdefeatingnumbercrunchingenterpriseinordertopublishpapers.ItlooksforanynumberstomanipulateinordertoseemmodernandsophisticatedwithoutregardtowhetheritissacrificingmathematicalrigorandvalidityandThereisNoMathematicalValidityforUsingFuzzyNumberCrunchingintheAnalyticHierarchyProcessJOURNALOFSYSTEMSSCIENCEANDSYSTEMSENGINEERING458alsoscientifictruth.Weareconvincedthatwhateverthepurposeofintroducingfuzzyconceptstohelponethinkaboutmulti-valuedlogic,theintentwasnevertoapplyitblindlytoallnumbersencounteredinpracticeparticularlythosethatariseindecisionmakingbecausejudgmentsbytheirverynaturearealreadyfuzzy.2.PerturbationTheAHPusesafundamentalscaleofabsolutenumberstoindicatethedominanceofonecriterionoveranotherwithrespecttoahighergoaloranalternativewithrespecttoanotherwithregardtoacommonattributeorconditionthattheysatisfy.Thesmallerelementisusedastheunitandthelargeroneisestimatedasamultipleofthatunit.Ithasbeenshownthatthisfundamentalscalecanbederivedfromthebasictheoryofstimulusresponse(Saaty,2004)andthatitleadstothefollowingvaluesthatcorrespondtoverbalexpressionsofdominance:acriterioncomparedwithitselfisalwaysassignedthevalue1sothemaindiagonalentriesofthepairwisecomparisonmatrixareall1.Thenumbers3,5,7,and9correspondtotheverbaljudgments“moderatelymoredominant”,“stronglymoredominant”,“verystronglymoredominant”,and“extremelymoredominant”(with2,4,6,and8forcompromisebetweenthepreviousvalues).Reciprocalvaluesareautomaticallyenteredinthetransposeposition.Oneisallowedtointerpolatevaluesbetweentheintegers,ifdesired.Withoutgreatdetail,wegiveabriefsummaryofwhatJ.H.Wilkinson(1965)proved:Toafirstorderapproximation,perturbationw1intheprincipalrighteigenvectorw1duetoaperturbationAinthepairwisecomparisonnbynmatrixA=[aij]whereAisconsistent,thatisaij=aik/ajk,foralli,j,k=1,…,nisgivenby:TT11(())njijijjj=2w=vAw/-vwwλλ∆∆∑HereTindicatestransposition.Theeigenvectorw1isinsensitivetoperturbationinA,if1)thenumberoftermsnissmall,2)iftheprincipaleigenvalue1λisseparatedfromtheothereigenvalues,hereassumedtobedistinct(otherwiseaslightlymorecomplicatedargumentcanbemade)and,3)ifnoneoftheproductsTjivwofleftandrighteigenvectorsissmallbutifoneofthemissmall,theyareallsmall.However,T11vw,theproductofthenormalizedleftandrightprincipaleigenvectorsofaconsistentmatrixisequaltonthatasanintegerisneververysmall.Ifnisrelativelysmallandtheelementsbeingcomparedarehomogeneous,noneofthecomponentsofw1isarbitrarilysmallandcorrespondingly,noneofthecomponentsofT1visarbitrarilysmall.Theirproductcannotbearbitrarilysmall,andthuswisinsensitivetosmallperturbationsoftheconsistentmatrixA.Theconclusionisthatnmustbesmall,andonemustcomparehomogeneouselements.Itistheneasytoseethatfuzzifyingtheentriesofamatrixdoesnotautomaticallyimplythattheprioritiesrepresentedbyitsprincipaleigenvectorgiveamoreaccuraterepresentationofthejudgmentsused.Examplestoshowthatonecannotrelyonwholesalesmallimprovementsinconsistencytoobtaincloserresultstoknownva