THERE IS NO MATH_省略_IERARCHY PROCESS_Thomas L. SAATY

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.Fuzzifyingwillleavetheeigenvectorveryneartowhatitwasinthefirstplaceandisthereforenotonlyapoorpracticebutl