Fuzzyregression-basedmathematicalprogrammingmodelforqualityfunctiondeployment
Y.CHENy,J.TANGz*,R.Y.K.FUNG§andZ.RENy
Qualityfunctiondeployment(QFD)isbecomingawidelyusedcustomer-drivenapproachandtoolinproductdesign.TheinherentfuzzinessinQFDmodellingmakesfuzzyregressionmoreappealingthanclassicalstatisticaltools.Anewfuzzyregression-basedmathematicalprogrammingapproachforQFDproductplan-ningispresented.First,fuzzyregressiontheorieswithsymmetricandnon-symmetrictriangularfuzzycoefficientsarediscussedtoidentifytherelationalfunctionsbetweenengineeringcharacteristicsandcustomerrequirementsandamongengineeringcharacteristics.Byembeddingtherelationalfunctionsobtainedbyfuzzyregression,amathematicalprogrammingmodelisdevelopedtodeterminetargetsofengineeringcharacteristics,takingintoconsiderationthefuzziness,financialfactorsandcustomerexpectationsamongthecompetitorsinproductdevelopmentprocess.TheproposedmodellingapproachcanhelpdesignteamassessrelationalfunctionsinQFDeffectivelyandreconciletradeoffsamongthevariousdegreeofcustomersatisfactionanddetermineasetofthelevelofattainmentofengineeringcharacteristicsforthenew/improvedproducttowardsahighercustomerexpectationwithindesignbudget.Thecomparisonresultsundersymmetricandnon-symmetriccasesandthesimulationanalysisaremadewhentheapproachisappliedtoaqualityimprovementproblemforanemulsificationdynamitepackingmachine.
1.Introduction
Theemergenceofaglobaleconomycharacterizedbyintenseinternationalmarketcompetitionandrapidtechnologicalchangeisforcingmanycompaniestoplacetheemphasisonnewproductsasasourceofnewsalesandprofits(TakeuchandNonaka1986).Thesecompaniesrealizethatitiscrucialfortheirsurvivaltodesignandmanufactureefficientlyproductspreferredbycustomersatacompetitivecostwithinshorttimeoverthoseofferedbycompetitors.Thekeytodothatsuccess-fullyistousecustomer-drivendesignandmanufacturingmethodology.Amongwhich,qualityfunctiondeployment(QFD)(Akao1990,HauserandClausing1998,Vairaktarakis1999)isawidelyusedcustomer-drivenapproachtobuildhighqualityintoproductduringthewholedesignandmanufacturingprocesses(Wassermann1993,MoskowitzandKim1997,Fungetal.1998).
Generally,QFDusesfoursetsofmatricescalledhouseofquality(HOQ)torelatethecustomerrequirements(CRs)toproductplanning,partsdeployment,process
RevisionreceivedJune2003.
yDepartmentofMechanicalEngineeringandzDepartmentofSystemEngineering,SchoolofInformationScience&Engineering,NortheasternUniversity(NEU),Shenyang,Liaoning110004,P.R.China.
§DepartmentofManufacturingEngineering&EngineeringManagement,CityUniversityof,83TatCheeAvenue,KongLoon,,P.R.China.
*Towhomcorrespondenceshouldbeaddressed.e-mail:jftang@mail.neu.edu.cn
InternationalJournalofProductionResearchISSN0020–73print/ISSN1366–588Xonline#2004Taylor&FrancisLtd
http://www.tandf.co.uk/journalsDOI:10.1080/002070310001619623
1010Y.Chenetal.
planningandmanufacturingoperations.ThefirstHOQmatrixwasmostfrequentlyemployedinindustries,whoseobjectiveistodeterminethetargetlevelsfortheengineeringcharacteristics(ECs)ofaproducttoachieveahigherlevelofoverallcustomersatisfaction.Itisacomplexdecisionprocesswithmultiplevariablestodeterminethetargetlevelsinpractice.Itiscurrentlyaccomplishedmainlyinasubjectiveadhocmannerorinaheuristicway,suchasprioritization-basedmethods,withtheaimofafeasibledesign,ratherthananoptimalone.Therefore,develop-mentofamoreeffectiveandreasonableprogrammingmodeltodeterminethetargetvaluesfortheECsofaproducttowardsthemaximumdegreeofcustomersatisfac-tionwithinlimitedrecoursesisusuallythefocusinthefirstHOQprocessplanning.
Wasserman(1993)formulatedtheQFDplanningprocessasalinearprogram-mingmodeltoselectthemixofdesignfeatures,toresultinthehighestlevelofcustomersatisfaction.ThemodelfocusesonprioritizingtheallocationofresourcesamongdesignfeaturesratherthandeterminingthetargetlevelsofECs.MoskowitzandKim(1997)proposedadecisionsupportprototypeforoptimizingproductdesignsbaseduponanintegratedmathematicalprogrammingformulationandsolu-tionapproach.Fungetal.(1998)suggestedafuzzyinferencemodeltofacilitatethedesigndecisionontargetvaluesforECswiththeuseofafuzzyrulebase.Tangetal.(2002)andFungetal.(2002)consideredafuzzyformulationcombinedwithagenetic-basedinteractiveapproachtoQFDplanning.TwofuzzyoptimizationmodelsfordeterminingtargetvaluesofECswithfinancialconsiderationaredevel-oped.Thesemodelsconsidernotonlytheoverallcustomersatisfaction,butalsotheenterprisesatisfactionwiththecostcommittedtotheproduct.ParkandKim(1998)presentedanintegrateddecisionmodelforselectinganoptimalsetofECsusingamodifiedHOQmodel.
MostofthosemodelsandmethodsmentionedabovewereimplicitlyassumedthattherelationshipfunctionsbetweenCRsandECsandamongECscouldbeidentifiedusingengineeringknowledge.Itwouldbedifficulttojustifythisassump-tioninageneralsituation.EspeciallywhenagivenHOQcontainslargenumberofCRsandECs,manytradeoffshavetobemadeamongthedegreesofcustomersatisfactionaswellasamongtheimplicitorexplicitrelationshipsbetweenECsandCRsandamongECs,andhenceitisdifficulttoidentifyrelationshipfunctions.Moreover,theserelationshipsaretypicallyvagueandimpreciseinpracticalsitua-tion.Thevaguenessorimpreciseness(Chanetal.1999,Fungetal.1999,Temponietal.1999,VangegasandLabib2001,ChanandWu2002)arisesmostlyfromthefactthattheCRstendtobesubjective,qualitativeandnon-technical,andtheyneedtobetranslatedintotheECs,inmorequantitativeandtechnicalterms.Further,dataavailableforproductdesignareoftenlimited,inaccurateorvagueatbest.
TheinherentfuzzinessinQFDmodellingmakesfuzzyregressionmoreappealingthanclassicalstatisticaltools.Fuzzyregressionanalysisreferstoasetofmethodsbywhichestimatesaremadeforthemodelparametersfromknowledgeaboutthevaluesofagiveninput–outputdataset.Thegoalanalysisis(1)tofindanappro-priatemathematicalmodeland(2)todeterminethebest-fittingcoefficientsofthemodelfromthegivendata(TanakaandWatada1998).Henceforth,Kimetal.(2000)proposedafuzzymulticriteriamodellingapproachforQFDplanningusingfuzzylinearmodelswithsymmetrictriangularfuzzynumbercoefficients.Inmoregeneralcases,thesymmetrictriangularcoefficientmaynotbesuitableandefficienttomodeltheserelationshipsfunctions.Moreover,theapproachistechnicallyonesidewithoutconsiderationofthedesignbudget,resultinginanunreasonableandunreliableQFD
Fuzzyregressionmodelforqualityfunctiondeployment1011
planninginpractice.Infact,costsandbudgetsforachievingtargetlevelsofECsforaproductareconstrained.Therefore,thefinancialfactorisalsoanimportantconsiderationandshouldnotbeneglectedinQFDplanning.
Inthepresentpaper,thefuzzylinearregressionwithsymmetrictriangularfuzzycoefficientsisfirstconsideredtomodeltherelationalfunctionsbetweenECsandCRsandamongECs,andthenthesymmetrictriangularfuzzycoefficientisextendedtonon-symmetrictriangularfuzzycoefficients.ThedesignbudgetandcustomerexpectationsamongthecompetitorsaretakenintoconsiderationintheQFDprod-uctplanning.Thecomparisonresultsundersymmetricandnon-symmetriccasesandthesimulationanalysisaremadewhentheproposedmodellingapproachisappliedtoaqualityimprovementproblemofanemulsificationdynamitepackingmachine.
Therestofthispaperisorganizedasfollows.Insection2,thegeneralQFDproblemisdefinedformallyandafuzzymulti-objectivemodelisformulatedandillustrated.Section3describeshowtoidentifytherelationshipsbetweenECsandCRsandamongECsusingfuzzylinearregressiontheorycombinedwithfuzzyoptimizationtheorywithsymmetrictrianglefuzzycoefficients.Fuzzylinearregres-sionwithnon-symmetricfuzzycoefficientsisdevelopedinsection4tomodeltherelationalfunctionforQFDproductplanning.Theresultsobtainedfrombothsymmetricandnon-symmetriccasesonthequalityimprovementproblemofanemulsificationdynamitepackingmachineareanalysedandcomparedinsection5.Finally,theconclusionsarepresentedinthesection6.
2.Modelformulations2.1.Problemdefinition
Todescribetheproblemclearly,thenotationusedinthefirstHOQissummarizedasfollows:CRiECjComprxjyiwiSrljcjCjCB
ithcustomerrequirement,i¼1,2,...m,jthengineeringcharacteristic,j¼1,2,...n,rthcompetitor,r¼1,2,...l,levelofattainmentoftheECj,
customerperceptionofthedegreeofsatisfactionoftheCRi,relativeimportanceoftheCRi,andarescaledsuchthat0 degreeofoverallcustomersatisfactionoftherthcompetitor,r¼1,2,...l,targetvalueofECj,j¼1,2,...n,costcoefficientofxj,j¼1,2,...n, costfunctionforachievingxj,j¼1,2,...n,totalcostofproductdevelopment,budgetofproductdevelopment. ThebasicconceptofthefirstHOQintheproductdesignistotranslatethedesiresofcustomerintoECs.Letfi(i¼1,2,...m)beafunctionalrelationshipbetweenyiandthelevelsofattainmentofECs,i.e.yi¼fi(x1,x2,xn)andgj(j¼1,2,...,n)arethefunctionalrelationshipbetweenxjandotherlevelsofattainmentofECs,i.e.xj¼gj(x1,...,xjÀ1,...,xn).TheproductdevelopmentprocessbasedonQFDistodetermineasetofx1,x2,...,xnforECsofthenew/improvedproducttomatchorexceedthedegreeofoverallcustomersatisfactionofallcompetitorsinthetargetmarketwithlimitedorganizationalresources.Itisacomplexdecisionprocesswith 1012Y.Chenetal. multiplevariables,requiringtrade-offandoptimizeallkindsofconflictscontainedinHOQ.LetSbethedegreeofoverallcustomersatisfactionfor(y1,y2,...,ym),theprocessofdeterminingattainmentleveloftargetvaluesofECsforaneworimprovedproductinQFDcanbeformulatedasanoptimizationproblemasfollows: 8 maxðSðy1,y2,...,ymÞÀmaxSrÞ>>r¼1,...l< s:t::yi¼fiðx1,x2,ÁÁÁxnÞ,i¼1,...,m,ð1Þ>x¼gðx,...,x,x,...,xÞ,j¼1,...,n>j1jÀ1jþ1n:j qeðxÞ 0,e¼1,...,pwhereqe(x)istheethorganizationalresourceconstraintandpisnumberoforganizationalresourceconstraints. 2.2.Objectivefunction ThedegreeofoverallcustomersatisfactionS(y1,y2,...,ym)canbeobtainedbyaggregatingthedegreesofcustomersatisfactionwithindividualCRs,i.e.: Sðy1,y2,...ymÞ¼ mXi¼1 wisiðyiÞ, ð2Þ wheresi(yi)isanindividualvaluefunctiononCRi.ForeachCR,anumericalvalue ofyi(i¼1,2,...,m)isassignedtoindicatethedegreeofsatisfactionofCRiincomparisonwiththecompetitors.Thisnumericalvaluecanbechosenfromaposi-tivescale[a,b](e.g.1–5).Therefore,si(yi)canbescaledinsuchawaythat max siðyminÞ¼1andcanbeconfiguredas:iÞ¼0andsiðyi . minmaxmin siðyiÞ¼yiÀyiyiÀyii¼1,2,...mð3ÞandS(y1,y2...ym)canbewrittenasfollows: Sðy1,y2,ÁÁÁymÞ¼ mXi¼1 wiyiÀ ymini . maxminyiÀyi ð4Þ andthusS(y1,y2,...ym)isalsoavaluebetween0and1,with0beingtheworstand1thebest.Hence,theobjectivefunctioncanbeexpressedas: maxS¼ 0 mXi¼1 wiyiÀ ymini . ymaxi À ymini ÀmaxSr r¼1,...,l ð5Þ 2.3.NormalizingthetargetvaluesofECs NormalizationoftargetvaluesofECsisthatchanging(lj¼1,...,n),thecurrenttargetvalueofECj,intothelevelofattainmentxj¼(j¼1,...,n),suchthat0 xj 1,(j¼1,...,n).TargetvaluesofECsforaproductcanbeclassifiedintotwocategories,i.e.positiveandnegative.Forpositiveones,theperformanceofECispositivelyproportionaltothetargetvalueofEC,andfornegativeones,theperfor-manceofECisthereversetothetargetvaluesofEC.Forexample,thedesignteamhopestoreducetheenergyrequiredtoclosethedoorofacarandincreasethewaterresistanceofthedoor.ThetwocategoriesoftargetvaluesofECscanbenormalized Fuzzyregressionmodelforqualityfunctiondeployment accordingtoequations(6)and(7),respectively: ljmaxÀlj xj¼max ljÀljminljÀljmin xj¼max ljÀljmin 1013 ð6Þð7Þ whereljmaxandljmincanbedeterminedbytheconsiderationofcompetitionrequire-mentandtechnologyfeasibility(Zhou1998).ForthefirstcategoryofECs,ljmaxismaximumtargetvalueoftheECjtomatchcompetitors’performance,andljministheminimumobtainable.Whereas,forthesecondcategoryofECs,ljminisaminimumtargetvalueoftheECjtomatchcompetitors’performance,andljmaxisthemaximumobtainable. 2.4.Costsandbudgetsconstraint Assumethattherearemultipleresourcesrequiredforsupportingthedesignofaproduct,includingdesignengineers,developmenttime,advancedequipmentandotherfacilities.Atthelevelofstrategicplanning,thesetypesofresourcescanbeaggregatedinfinancialterms.Forsimplicity,assumethatthecostfunctionCjforachievingthelevelofattainmentoftheECjisscaledlinearlytothelevelofattainmentxj,whichresultsin: CjðxjÞ¼ nXj¼1 cjxj, ð8Þ wherethecostcoefficientcjisdefinedasthecostrequiredwhentheECjisfully improved,i.e.whenoneunitoflevelofattainmentoftheECj,thecostiscj,andcanbedeterminedaccordingtotheexperienceofthedesignteamorbytesting.AssumingthatthetotaldesigncostCforaproductdevelopmentorimprovementisconstrainedtoabudgetB,itisformulatedas: C¼ nXj¼1 CjðxjÞ¼ nXj¼1 cjxj B: ð9Þ 2.5.Mathematicalmodel Takingintoaccountthecostconstraint,theQFDplanningproblemcanbeformulatedasalinearprogrammodel(LP): 8.m0Pmin>maxS¼wiðyiÀyiÞðymaxÀymin>iiÞÀmaxSr>>r¼1,...,li¼1>>>s:t::yi¼fiðx1,x2,...,xnÞ,i¼1,...,m>>> :ð10ÞnnPP>>C¼CðxÞ¼cxÀB 0jjjj>>>j¼1j¼1>>>minmax>>:yi yi yi,i¼1,...,m 0 xj 1,j¼1,...,nIfthevalueofanobjectivefunctionispositive,i.e.S>0,thenthemodelLPdeterminesaset0ofx1,x2,...,xnwithamaximumdegreeofoverallcustomersatis-faction.ElseS<0andLPdeterminesasetofx1,x2,...,xnthatminimizesthe 01014Y.Chenetal. differenceofthecustomersatisfactionfromanyofthecompetitors.Thefunctionalrelationships(fi¼1,2,...,m)andgj(j¼1,2,...,n)canbeidentifiedusingfuzzylinearregressionandfuzzyoptimizationwithsymmetrictrianglenumbercoefficients(seesection3)ornon-symmetrictrianglenumbercoefficients(seesection4). 3. ParameterestimationwithsymmetrictrianglefuzzycoefficientsConsiderafuzzylinearfunction: ~iÞ¼A~i¼fiðX,A~i0þA~i1x1þÁÁÁþA~inxn,Y ð11Þ ~iisthefuzzyoutputofthedegreeofcustomersatisfactionoftheCRi,whereY X¼ðx1,x2,...,xnÞTisthereal-valuedinputvectorofthelevelofattainmentof ~i¼ðA~i0,A~i1,...,A~inÞisavectoroffuzzynumbers.Theregressionanal-ECs,andA ysisprobleminHOQisdefinedasfollows.Givenanumberoflcrispdatapoints ~i0,A~i1,...,A~inwillbeðX1,yi1Þ,...,ðXr,yirÞ,...,ðXl,yilÞ,asetoffuzzyparametersA determinedsuchthatequation(11)bestfitsthegivendatapointsinasenseofgoodnessoffitundersomecriteria,wherebyXr¼ðx1r,...,xjr,...,xnrÞisthesetoflevelofattainmentofECsoftherthcompetitor,theelementxjristhelevelofattainmentoftheECjoftheCompr,andyiristhedegreeofcustomersatisfactionof ~ijhastriangularmembershipfunctions,itcanbeuniquelytheCRioftheCompr.IfACULUC~ij¼ðaLdefinedbyAij,aij,aijÞ.Here,aijisthelowerlimit,aijistheupperlimitandaij~ijthatsatisfies~ðaCisthecentrevalueofAAijijÞ¼1.Thesymmetryofthefuzzy ~ijleadsustoestablishthefollowingtworelations:coefficientA LU aCij¼ðaijþaijÞ=2CLUCaSij¼aijÀaij¼aijÀaij, ð12Þð13Þ S~whereaCijisthecentrevalueandaijisthespreadvalueofAij.Thecentrevalue ~ij,whilethespreadrepresentstheprecisiondescribesthemostpossiblevalueofA ~ij.Then,thesymmetricfuzzynumbercoefficientA~ijcanbeuniquelydescribedbyofA SLU~ij¼ðaC~apairofparameters,eitherAij,aijÞorAij¼ðaij,aijÞ.Inthissense,onecanalso~i¼ðA~i0,A~i1,...,A~inÞasavectorformintermsrepresentthefuzzycoefficientsetA SCSCCCCSSSS~ofaCijandaijasAi¼ðai,aiÞ,hereai¼ðai1,ai2,...,ainÞandai¼ðai1,ai2,...,ainÞ. Forsymmetrictriangularfuzzynumbers,themembershipfunctionA~ijfor~ijði¼1,2,...,m;j¼1,2,...,nÞcanbedescribedas:A 8>ÀaijÞ=aS,aCÀaS aij aC1ÀðaCijijijijij,>< SCCS,ð14ÞA~ijðaijÞ¼1ÀðaijÀaCijÞ=aij,aij aij aijþaij>>: 0,otherwise: ~iÞinequation(11)canbeexpressedasThefuzzyoutputfromthelinearmodelfiðX,A followsaccordingtotheextensionprincipleandfuzzyarithmeticonfuzzynumbers: ~iÞ¼ðfiCðXÞ,fiSðXÞÞ,~i¼fiðX,AY ð15Þ ~iÞ,wherefiCðXÞandfiSðXÞarethecentreandspreadofthefuzzylinearmodelfiðX,A respectively,andgivenas: CC fiCðXÞ¼aCi0þai1x1þÁÁÁþainxnSSfiSðXÞ¼aSi0þai1jx1jþÁÁÁþainjxnj: ð16Þð17Þ Fuzzyregressionmodelforqualityfunctiondeployment1015 ~idefinedin(11)hastheform:ThenthemembershipfunctionofY !8 nX>>C>aC!>ijxjrþai0Àyir >nn>XX>j¼1>CCSS>,axþaÀaxjrÀa1À>jriji0i0ijn>X>>SSj¼1j¼1>aþa>i0ijxjr>>>j¼1>>> !>>n>X>>C> yir aC>ijxjrþai0,>>>j¼1>>> !> ð18Þ Theaimofthefuzzyregressionmethod(Yenetal.1999)withnon-fuzzydataisto ~ÃdeterminetheparametersAijsuchthatthetotalspreadoffuzzyoutputforalldatais minimizedwhileeachindividualfuzzyoutputyir(i¼1,2,...,m;r¼1,2,...,l)isassociatedwithamembershipgreaterthanh.Itcanbeexpressedas: \"#nlXX xjrð19ÞminZ¼aSaSi0þij j¼1 r¼1 s:t::Y~iðyirÞ!h, r¼1,2,...,l,ð20Þ where0 h<1denotesthedegreeoffitnessoftheestimatedfuzzylinearmodeland issubjectivelypreselectedbyadeignteamaccordingtotheirengineeringknowledge. ~irwithadegreeAphysicalinterpretationofhisthatyirisinthesupportintervalofy ofmembershipofatleasthforalll.Theconstraint(20)isrewrittenas: ~i0hþ½A~i1hx1rþÁÁÁþ½A~inhxnr,yir2½fiðXrÞh¼½A r¼1,2,...,l, ð21Þ where[E]histheh–levelsetofafuzzynumber.Equation(21)canalsobefurther expressedas: ! ! nnXX ÀaCð1ÀhÞaSaSaCr¼1,2,...,nð22Þi0þijxjri0þijxjr!Àyir, j¼1 j¼1 and ð1ÀhÞaSi0þ nXj¼1 ! ! nX þaCaSaCijxjri0þijxjr!yir, j¼1 r¼1,2,...,n:ð23Þ 1016Y.Chenetal. Henceforth,thefuzzyregressionproblemaimingtodeterminethefunctionalrela-tionshipsfi(i¼1,2,...,m)istransformedintoalinearprogram: \"#8nlXX>SS>>minZ¼aþax>jr,i0ij>>>r¼1> j¼1! !>>nn>XX>C ai0,aij!0,j¼1,...,n: ð24Þ Similarly,onecandetermineacorrelationfunctionamongthedegreeofattainmentofatargetlevelofECsusingfuzzylinearregression.Assumethat:~jÞ¼A~j¼gjðXj,A~j0þA~j1x1þÁÁÁþA~j,jÀ1xjÀ1þA~j,jþ1xjþ1þÁÁÁþA~jnxn,X ð25Þ ~jisthefuzzyoutputofthedegreeofattainmentoftargetleveloftheECj,whereX Xj¼ðx1,...,xjÀ1,xjþ1,...,xnÞTisthereal-valuedinputvectorofthelevelofattain-~j¼ðA~j0,A~j1,...,A~j,jÀ1,A~j,jþ1,...,A~jnÞisasetofsymmetricfuzzymentofECs,andA triangularnumbertobedeterminedbysolvingthefollowinglinearprogrammingmodel: 8nlPP>xur,>aSminZ¼aS>j0þju>>u¼1k¼1>>u¼j>>1010>>>>nn>CBCPP>CBSS> aj0,aju!0,u¼1,...,jÀ1,jþ1,...n: ð26Þ Hence,thefunctionalrelationshipsfi(i¼1,2,...,m)andgj(j¼1,2,...,n)resultedfromthemodelsLP1andLP2aregivenas: nX CSCS~~i¼fiðx1,...,xnÞ¼ai0,ai0þyaij,aijxj, j¼1 nX CSCS ~jðx1,...,xjÀ1,xjþ1,...,xnÞ¼ai0,ai0þ~j¼gaju,ajuxu,x u¼1 u¼j i¼1,2,...,mð27Þ j¼1,2,...,n:ð28Þ Obviously,ifthecentrevalueisonlyconsideredandthespreadvalueisneglectedinthefuzzyregression,thefunctionalrelationshipsfi(i¼1,2,...,m)andgj(j¼1,2,...,n)aresubstitutedinamoresimpleway. Fuzzyregressionmodelforqualityfunctiondeployment1017 4.Extensiontonon-symmetricfuzzycoefficientscase ~ijcanbeItwasassumedabovethateachsymmetrictriangularfuzzycoefficientA S~ij¼ðaCuniquelydescribedbytwoparametersAij,aijÞ.However,ingeneral,these trianglesarenotsymmetric,andtheyaredescribedbythreeparametersinterms PULPR ofðaLij,aij,aijÞorðsij,aij,sijÞ.Theformergivesthepeakpointandtheleftandrightpoints,andthelattergivesthepeakandtheleft-andright-sidespread,wherebyaPijis PLP thepeakpointatwhichA~ijðaijÞ¼1.sijistheleft-sidespreadfromthepeakpointaij, LPLRUP andsRijrepresentstheright-sidespread.Sincesij¼aijÀaijandsij¼aijÀaij,onecanuseonespreadasthebasetonormalizetheotherone.ChoosesLijasthebase, R thensijcanbeexpressedas: L sRij¼kjsij, ð29Þ wherekjistheskewfactorandhaspositiverealnumbers.Theselectionofthevalues forkjisbasedontheknowledgeofthedesignproblemanddatacharacteristicsof P~ijcanbedescribedbythetripletsðsLECs.ThenAij,aij,kjÞ.Themembershipfunction ~ijhastheform:foreachA 8LPLP>1ÀðaPijÀaijÞ=sij,aijÀsij aij aij,>>< LPPLð30ÞA~iðaijÞ¼1ÀðaijÀaPijÞ=kjsij,aij aij aijþkjsij,>>>: 0,otherwise:Followingtheextensionprinciple,thefuzzymembershipfunctionfortheoutputcan beobtainedas: 8 ! nX>>P>aP>!! ijxjrþai0Àyir >nn>XX>j¼1>PL> yir,aPsL1À>ijxjrþai0Àsi0þijxjrn>X>>Lj¼1j¼1L>sþsxjr>iji0>>>j¼1>> !>>nX>>P>> aPijxjrþai0,>>>j¼1< ! nYX~iðyirÞ¼P>>yirÀaP !ijxjrþai0>>nX>j¼1>>>,aPxjrþaP1Àiji0 yir>nXL>>j¼1>>k0sLkjsijxjr>i0þ>>j¼1>>! ! >>nn>XX>PL>,> aPkjsL>ijxjrþai0þk0si0þijxjr>>>j¼1j¼1>: 0,otherwise: ð31ÞFromtheaboveexpression,onegetstheconstraintsoftheregressionas: !nX P 1ÀaPijxjrþai0Àyir j¼1 sLi0þ nXj¼1 Lsijxjr!h ð32Þ 1018and Y.Chenetal. yirÀ1À nXj¼1 ! PaPijxjrþai0 k0sLi0þ nXj¼1 kjsLijxjr !h,ð33Þ Aftersimplification,equations(32)and(33)havetheform: ð1ÀhÞsLi0þand ð1ÀhÞk0sLi0þ nXj¼1nXj¼1 ! !nX ÀaPsLaPijxjri0þijxjr!Àyir, j¼1 r¼1,2,...,l:ð34Þ þaPkjsLijxjri0þ ! nXj¼1 !aPijxjr !yir, r¼1,2,...,l: ð35Þ Now,theproblemistominimizethetotalspreadsinamulti-inputfuzzyoutput function.Thesumofthespreadisgivenby: Z¼ð1þk0ÞsLi0þ nXj¼1 \" # lXxjr:ð1þkjÞsLij r¼1 ð36Þ Henceforth,withanon-symmetrictrianglefuzzynumber,thefunctionalrelation-shipsfi(i¼1,2,...,m)canbeobtainedbysolving: \"#8nlXX>LL>xjr,>Þsþð1þkÞsminZ¼ð1þk>0ji0ij>>>j¼1r¼1>>>> ! !>>nnXX>>>ÀaP >> ! !>>nn>XX>>LL>ð1ÀhÞksþksxjrþaPaPr¼1,2,...,l,>i0iji0þijxjr!yir,0j>>>j¼1j¼1>>>>:LL si0,sij!0,j¼1,...,n: ð37Þ Ifoneselectsallthevaluesofkjtobe1,thenconstrains(34)and(35)areequivalentthoseforasymmetriccaseandtheexpressionofZbecomesPntoLPlL2ðsi0þj¼1sijr¼1xjrÞ,whichistwicethemagnitudeofthesymmetriccase.Sinceamultiplicationconstantwillnotchangetheresultoftheminimizationprocess,thenon-symmetricisreducedtothesymmetrictriangularcase,i.e.LP3isequivalenttoLP1. Fuzzyregressionmodelforqualityfunctiondeployment1019 Similarly,thefunctionalrelationshipsgj(j¼1,2,...,n)canbeobtainedbysolving: 8\"# nlXX>>xur,>ð1þkjÞsLminZ¼ð1þk0ÞsL>j0þju>>>u¼1r¼1>>u¼j>>>>>1010>>>>nn>XCBPX>CBLLP>>Àaþsxþaxs:t::ð1ÀhÞs@A!Àxjr,@A>ururj0juj0ju< >>>>0101>>>>nnX>CBPX>BCLL>>ð1ÀhÞ@k0sj0þkjsjuxurAþ@aj0þaPjuxurA!xjr,>>>u¼1u¼1>>u¼ju¼j>>>>>>:LL sj0,sju!0,u¼1,...,jÀ1,jþ1,...,n: u¼1u¼j u¼1u¼j r¼1,2,...,l, LP4 r¼1,2,...,l, ð38Þ 5.Illustratedexamplesandcomparisonanalysis 5.1.BuildingtheHOQforanemulsificationdynamitepackingmachine Toclarifytheperformanceofthemodellingapproach,apracticalproductdevel-opmentofanemulsificationdynamitepackingmachineiscitedasanexampletodemonstratetheeffectivenessoftheproposedmethod.Theresultsarepresentedandillustratedhere. Acorporationisundergoinganewtypeofanemulsificationdynamitepackingmachine.Accordingtothesurveyinthemarketplaceandthecollectionofcom-plaintsfromusers,therearefourmajorCRs,i.e.improvethequalityofpackingdynamite(CR1),increasetheefficiencyofpackingdynamite(CR2),reducethepack-ingnoise(CR3),andincreasetherigidityofthemachine(CR4).SevenECsareidentified:improvetheprecisionofthemouldingofclip(EC1),improvetheprecisionofpackingdynamite(EC2),increasethecontrolforceofpackingdynamite(EC3),improvetheefficiencyofpackingdynamite(EC4),increasethehardnessofthepressinghammer(EC5),reducethenoiseofcampowertransmission(EC6),andreducetheheightofthemachinebed(EC7).Inthemeantime,fivemaincompetitors,i.e.Comp1(ourcorporation),Comp2,Comp3,Comp4,andComp5areselected.TheweightsoffourCRsaredeterminedusingtheHierarchicalAnalyticProcess(AHP)method(Armacastetal.1994),engineeringmeasuredatahavebeencollectedfromthecompanyanditsmaincompetitorsacquiredbytesting,andcustomerperceptionofthedegreeofsatisfactionofeachCRshasbeenscaledfrom1(worst)to5(best).Theobjectiveoftheproblemistodevelopanewtypeofemulsificationdynamitepackingmachine,i.e.determinenewtargetvaluesfortheECs,tomatchorexceedthecustomerexpectationofallcompetitorsinthetargetmarketwiththelimiteddesignbudget.TheHOQofanemulsificationdynamitepackingmachineasshowninfigure1isfilledwiththesedata.ThenegativeandpositivesignonECsmeansthedesignteamhopestoreduceandincreasethetargetvaluesofECs,respectively. −−+ + + −−ECs ECECECECx11 x22 x33 4 ECECECx4x55 x66 x7 7 EC1EC2•oniECt3••alECe4rECor5•CEC6 EC7 Benchmarking informationCRsWeighs•••Relation Comp1Comp2Comp3Comp4Comp5min max CR1 y1 0.46•••3.4 4 1.9 3.7 3.6 1 5 CR2 y2 0.28•3.1 3 1.8 2.9 3.9 1 5 CR3 y3 0.162.2 3.7 4.3 1.8 3.5 1 5 CR4 y4 0.101.6 3.7 3.3 3.7 4 1 5 Unitesm-2m-2N ns-1HRC dB m •48.60 66.05 34.90 .30 67.70 S (%) Comp 111 7 58 90 55 75 1.9 Comp 28 6 65 85 50 68 1.7 gCompsin312 9 60 70 50 55 1.8 rCompere49 8 62 80 45 80 1.7 eunsaCompi510 10 65 75 55 70 1.6 genmin 6 4 55 60 40 50 1.5 EMMax 15 12 70 100 60 90 2 Figure1.Houseofqualityofanemulsificationdynamitepackingmachine. 1020Y.Chenetal.Fuzzyregressionmodelforqualityfunctiondeployment 5.2. 1021 Parameterestimationoffunctionalrelationshipswithsymmetrictrianglefuzzynumbercoefficients ThenormalizedtargetvaluesofECsaccordingtoequations(6)and(7)aregivenasfollows: 230:440:630:200:750:750:380:2060:780:750:670:630:500:550:60767 7X¼60:330:380:330:250:500:880:4067 40:670:500:470:500:250:250:6050:560:250:670:380:750:500:80 .Usingfuzzylinearregression,theparametersinthefunctionalrelationshipscanbeobtainedbysolvingLP1andLP2foraspecifiedvalueofh.Forexample,y1isassociatedwithx1,x2,andx3(figure1),LP1forf1isgivenas:8SSSS þ2:78aþ0:51aþ2:34aminZ¼a>10111213>>>>SSSCCCC>s:t::0:5aS>10þ0:22a11þ0:32a12þ0:10a13Àa10À0:44a11À0:63a12À0:20a13!À3:4>>>>SSSSCCCC>>0:5aþ0:22aþ0:32aþ0:10aþaþ0:44aþ0:63aþ0:20a>1011121310111213!3:4>>>>SSSCCCC>0:5aS>10þ0:39a11þ0:38a12þ0:34a13Àa10À0:78a11À0:75a12À0:67a13!À4>>>>SSSCCCC>>0:5aS>10þ0:39a11þ0:38a12þ0:34a13þa10þ0:78a11þ0:75a12þ0:67a13!4>>>>SSSCCCC<0:5aS 10þ0:17a11þ0:19a12þ0:17a13Àa10À0:33a11À0:38a12À0:33a13!À1:9 SSSCCCC>>0:5aS>10þ0:17a11þ0:19a12þ0:17a13þa10þ0:33a11þ0:38a12þ0:33a13!1:9>>>>SSSCCCC>0:5aS>10þ0:34a11þ0:25a12þ0:24a13Àa10À0:67a11À0:50a12À0:47a13!À3:7>>>>SSSCCCC>>0:5aS>10þ0:34a11þ0:25a12þ0:24a13þa10þ0:67a11þ0:50a12þ0:47a13!3:7>>>>SSSCCCC>0:5aS>10þ0:28a11þ0:13a12þ0:34a13Àa10À0:56a11À0:25a12À0:67a13!À3:6>>>>SSSSCCCC>>0:5aþ0:28aþ0:13aþ0:34aþaþ0:56aþ0:25aþ0:67a>1011121310111213!3:6>>:SSSS a10,a11,a12,a13!0 UsingthesoftwareMathprogram,thesolutionsetisgivenby: SSSC aS10¼0:74795,a11¼0,a12¼0,a13¼0,a10¼0:88072,CCaC11¼5:97588,a12¼À0:33350,a13¼À1:36986, wheretheminimizedvalueoftheobjectivefunctiononthespreadis0.41553.This ~10¼ð0:74795,0:88072Þ,A~11¼ð0,5:97588Þ,solutiongivesthefuzzycoefficientsasA ~12¼ð0,À0:33350Þ,andA~13¼ð0,À1:36986Þ.A Toexaminehowtheselectionofhinfluencesthevaluesofcentresandspreads~ofAij,severaldifferentvaluesforhareselected.Thecorrespondingresultsaregiven ~ijasshownintable1.OnecanseethathdoesnotchangethecentreofeachA SS butinfluencesthevaluesofa10andZ.Thesmallerhis,thesmallerarea10andZ.Similarly,f2,f3,f4andg2,g4,g6canbedeterminedasshownintable2,inwhichthenumbersinparenthesesrepresentthespreadsath¼0.5.Becausex1,x3,x5andx7arecorrelatedwithnootherECs,thereforeg1,g3,g5andg7arezero. 1022 h0.10.30.50.7 aS100.415530.534250.747951.24658 aS110000 aS120000 aS130000Table1. Y.Chenetal. aC100.880720.880720.880720.88072 aC115.975885.975885.975885.97588 aC12À0.33350À0.33350À0.33350À0.33350 aC13À1.36986À1.36986À1.36986À1.36986 Z0.415530.534250.747951.24658 Influenceofhonsolutions. y1Interceptx1x2x3x4x5x6x70.88(0.75)5.98À0.33À1.37 y20.(0.83)2.450.961.25 y31.00(0.80) y41.25(0.90) x20.01(0.26) x40.22(0.17)0.87 x60.(0.41) 0.97 4.20 4.00 À0.30 À0.87 Table2.Assessedfiandgjwithsymmetricfuzzynumbers(h¼0.5). 5.3 Parameterestimationoffunctionalrelationshipswithnon-symmetrictrianglefuzzynumbercoefficients Similarly,onecanobtainthecoefficientsinthefunctionalrelationshipsusingfuzzylinearregressionwithanon-symmetrictriangularfuzzynumber.Accordingtoaprioriknowledge,thedesignteamselectsthefollowingvaluesfork0,k1,k2andk3:k0¼1.2,k1¼1.7,k2¼1.5andk3¼1.2.TheLP3forf1isgivenastheform:8LLLminZ¼2:20sL>10þ7:51s11þ6:28s12þ5:15S13>>>>>LLLPPPP>s:t::0:5sL>10þ0:22s11þ0:32s12þ0:10s13þa10þ0:44a11þ0:63a12þ0:20a13!3:4>>>>>LLLPPPP>>0:6sL10þ0:42s11þ0:60s12þ0:19s13Àa10À0:44a11À0:63a12À0:20a13!À3:4>>>>>>>0:5sLþ0:39sLþ0:38sLþ0:34sLþaPþ0:78aPþ0:75aPþ0:67aP1011121310111213!4>>>>>LLLPPPP>>0:6sL>10þ0:75s11þ0:72s12þ0:s13Àa10À0:78a11À0:75a12À0:67a13!À4>>>>>LLLPPPP<0:5sL 10þ0:17s11þ0:19s12þ0:17s13þa10þ0:33a11þ0:38a12þ0:33a13!1:9>LLLPPPP>0:6sL>10þ0:32s11þ0:36s12þ0:32s13Àa10À0:33a11À0:38a12À0:33a13>>>>>LLLPPPP>0:5sL>10þ0:34s11þ0:25s12þ0:24s13þa10þ0:67a11þ0:50a12þ0:47a13>>>>>LLLPPPP>>0:6sL10þ0:s11þ0:48s12þ0:45s13Àa10À0:67a11À0:50a12À0:47a13>>>>>LLLLPPPP>>0:5sþ0:28sþ0:13sþ0:34sþaþ0:56aþ0:25aþ0:67a1011121310111213>>>>>LLLPPPP>>0:6sL>10þ0:s11þ0:24s12þ0:s13Àa10À0:56a11À0:25a12À0:67a13>>>:LLLL s10,s11,s12,s13!0 !À1:9!3:7!À3:7!3:6!À3:6 Fuzzyregressionmodelforqualityfunctiondeployment k01.21.51.92.81 k11.41.82.53.61 k21.21.62.72.91 k31.52.32.83.21 sL100.679950.598360.515820.393661.24658 LL sL11s12s13 1023 aP13 Z1.495 1.4951.4951.4951.495 aP100.914710.955510.996781.057860.88072 aP115.975885.975885.975885.975885.97588 aP12À0.33350À0.33350À0.33350À0.33350À0.33350 000000000000000À1.36986À1.36986À1.36986À1.36986À1.36986 Table3. k01.22.63.9111111111 k11111.62.23.8111111 k21111111.82.53.8111 k31111111111.92.73.6 Influenceofthevaluesofskewfactorssetonsolutions.sL10 LL sL11s12s13 aP100.91471 1.046931.102050.880720.880720.880720.880720.880720.880720.880720.880720.88072 aP115.975885.975885.975885.975885.975885.975885.975885.975885.975885.975885.975885.97588 aP12À0.33350À0.333450À0.333450À0.333450À0.333450À0.333450À0.333450À0.333450À0.333450À0.333450À0.333450À0.333450 aP13À1.36986À1.36986À1.36986À1.36986À1.36986À1.36986À1.36986À1.36986À1.36986À1.36986À1.36986À1.36986 Z1.4951.4951.4951.4951.4951.4951.4951.4951.4951.4951.4951.495 0.679950.415530.305280.747950.747950.747950.747950.747950.747950.747950.747950.74795000000000000000000000000000000000000 Table4.Influenceofk0andkjonsolutions. BysolvingtheaboveLP3,thefuzzycoefficientscanbeobtainedasfollows:~10¼ð0:695,0:91471,1:2Þ,A~11¼ð0,5:97588,1:7Þ,A~12¼ð0,À0:333450,1:5Þ,A ~13¼ð0,À1:36986,1:2Þ,A wheretheminimumvalueoftheobjectivefunctiononthespreadis1.495.Thecoefficientsunderdifferentvaluesofkjaregivenasshownintable3. Analysingtheaboveresults,onecanfindthatinthecaseofnon-symmetricmembershipfunctions,astheskewfactorsincrease,thespreadsL10decreasesandthecentrebecomeslarge.Besides,thevariouskjonlyhasinfluenceonaP10butnotonPPP a11,a12,a13andZ. Table4showsthespreadsandcentresatdifferentsettingsofskewfactors,whereineachsettingthreeoftheskewfactorsarefixedat1andoneofthemvaries. Itcanbeconcludedthatk0isthedominantskewfactor.Keepingk0constantandchanginganyoneofk1,k2andk3separatelyorallatthesamewillnotproduceany ~ij.variationsinA Similarly,f2,f3,f4andg2,g4,g6canalsobeassessedwithanon-symmetriccase(table5)inwhichnumbersinparenthesesrepresentthespreadsath¼0.5andg1,g3,g5andg7arezero. 5.4.Analysisofresults AssumethatthebudgetBis55units,maxr¼1,...,lS¼67:7%(figure1)andthecostcoefficientsforeachECareexpressedasfollows: c1¼20,c2¼25,c3¼10,c4¼15,c5¼5,c6¼30,c7¼8: 1024 y1Interceptx1x2x3x4x5x6x70.91(0.68)5.98À0.33À1.37 y20.58(0.75)2.450.961.25 Y.Chenetal. y31.04(0.72) y41.29(0.82) x20.02(0.23) x40.23(0.16)0.87 0.97 4.20 4.00 Table5. Assessedfiandgjwithnon-symmetricfuzzynumbers(h¼0.5). À0.30 À0.87x60.91(0.38) Substitutingtheresultsofsections5.2and5.3,respectively,theproblemLP(10)ofsection2.5isgivenas: 80 max>S¼0:12y1þ0:07y2þ0:04y3þ0:02y4À0:25À0:677>>>>>s:t::5:98x1À0:33x2À1:37x3þ0:88¼y1>>>>>>2:45x3þ0:96x4þ1:25x5þ0:¼y2>>>>>>4:20x6þ1:00¼y3>>>>>>4:00x7þ1:25¼y4>< 0:97x4þ0:01¼x2>>>>0:87x2À0:30x6þ0:22¼x4>>>>>>À0:87x4þ0:¼x6>>>>>>20x1þ25x2þ10x3þ15x4þ5x5þ30x6þ8x7À50 0>>>>>>1 yi 5,i¼1,...,4>>>: 0 xj 1,j¼1,...,7 .Table6summarizestheresultsobtainedfortheabovetwocases. Comparetheexistingdesignsofthepresentexamplecompanyanditscompeti-torswiththoseobtainedbysolvingtheLPmodelandcheckwherethepresentcompanyinitiallystandsvis-a`-visitscompetitors.ThecustomercompetitiveanalysisinformationcontainedintheHOQinfigure1indicatesthatthepresentcompany’sproductcurrentlyisweakiny1,y3andy4,andstronginy2,andhasthelowervalueofS(48.60%)amongallfivecompetitors.TheSvaluesofthedesignfromtheLPmodelinbothsymmetric(3.99%þ67.7%¼71.69%)andnon-symmetric(6.70%þ67.7%¼74.40%)casesaremuchhigherthanourcurrentSvalue(48.60%),andexceedthatofComp5,andbecomethehighestamongalloffivecompetitors. ComparedwithComp1’scurrentdesign,theLPmodelinbothcasesimprovedy1andy3bytradingoffy2andy4toachievethedegreeofcustomersatisfactionwithlimitedfinicalbudgets.ThetargetvaluesofECsweredeterminedtoachievesuchavaluetrade-offinthemostefficientway.Forexample,theresultforanon-symmetriccaseyieldasignificantlyhighery1thaninComp1’scurrentdesign(from3.4to5.0) Fuzzyregressionmodelforqualityfunctiondeployment maxS(%) Symmetric Non-symmetric 3.996.70Table6. 01025 x5x6x7y1y2y3y4x1x2x3x45.002.873.521.250.780.330.310.331.000.600.005.002.944.051.290.780.230.370.221.000.720.00SolutionstotheprogrammingmodelLP. becauseCR1isthemostimportantinalloffourCRs.Thevalueofy1ispositivelyrelativewithx1andnegativelywithx2andx3(table5).Toincreasey1,theresultingdesignimprovedthelevelofx1(from0.44to0.70),andloweredx2(from0.63to0.23)andx3(from0.20to0.02).Thevalueofx2ispositivelyrelativewithx4andnegativelywithx6(table5).Thedecreasingofx2resultsindecreasingx4(from0.75to0.22)andincreasingx6(from0.38to0.72).Thevalueofy2ispositivelyrelativewithx3,x4andx5,sinceincrementalchangesofy2resultedfromtheimprovementofx5(from0.75tothemaximumvalue,1)couldnotcompensateforthedecreasingchangesfromthedecreasingofx3andx4,y2decreased(from3.1to2.1).Thevalueofy3isonlypositivelyrelativewithx6,soitisnecessarythaty3increased(from2.2to4.05).BecauseCR4istheleastimportantinallCRs,financialbudgetswereassignedtoy1,y2andy3basedonprioritization,theresultingdesignlowersy4(from1.6to1.29).Fromtheaforementionedanalysis,onecanseethatthemodelconsidersallsuchinteractionsbetweenCRsandECsaswellasthoseamongECssimultaneously,anddeterminestheoptimallevelsofECs,accordingtowhichthetargetvaluesofECsforthenewtypeofemulsificationdynamitepackingmachinearedetermined.Accordingtotheresultsobtainedbythisapproach,thedesignteamcaneasilyassignthedesigntargettoeachECstobalancethedesignresources(financialbudget)andothertechnicalrequirementsandcompetitors.0Table6showsthatSis3.99%inthesymmetricand6.70%inthenon-symmetriccase.ThisimpliesthatifoneappliesthesymmetriccasetoQFDmodelling,systemicuncertaintiesandambiguitiescannotbemodelledsufficientlyandreasonablyusing0symmetriccoefficients,andhencethequalityofthetargetdesign(S)isunderesti-0mated(i.e.ÁS¼3.99%À6.70%¼À2.71%).Henceforth,itcanbeconcludedthatusingnon-symmetrictrianglefuzzymodellingcanencompassmoretypesofsystema-ticuncertaintiesandambiguitiesthatcannotbemodelledefficientlyusingsymmetrictrianglefuzzycoefficients,i.e.thenon-symmetriccaseismoregeneralandreasonablethanthesymmetriccase. 6.Conclusion Anewfuzzyregression-basedmathematicalprogrammingapproachforQFDwaspresentedtotakeintoconsiderationthefuzziness,financialfactorsandcustomerexpectationsamongthecompetitorsinproductdevelopmentprocess.Themodellingapproachappliesthefuzzylinearregressiontheorycombinedwithfuzzyoptimiza-tiontheorywithsymmetricornon-symmetrictriangularfuzzycoefficientstomodeltherelationalfunctionsbetweenECsandCRsandamongECs,whichismorescientificandreasonablethanusingengineeringknowledgeintraditionalQFDmethodology.Theapproachcanhelpadesignteamreconciletradeoffsamongthevariousdegreesofcustomersatisfactionanddetermineasetofthelevelofattain-mentofECsforthenew/improvedproducttosatisfyabudgetconstraintandmatchorexceedthecustomerexpectationofallcompetitorsinthetargetmarket. 1026Y.Chenetal. Simulationshowsthatthefuzzyregressionwithnon-symmetrictrianglefuzzycoeffi-cientscanencompassmoretypesofsystematicuncertaintiesandambiguitiesthatcannotbemodelledefficientlyusingsymmetrictrianglefuzzycoefficients.Theapproachcouldbeapplicabletoawidevarietyofdesignproblemswheremultipledesigncriteriaandfunctionaldesignrelationshipsareinvolvedinanuncertain,qualitativeandfuzzyway. Acknowledgements ThepaperwasjointfinanciallysupportedbytheNationalNaturalScienceFoundationofChina(NSFC70002009),theExcellentYouthTeacherProgramofMinistryofEducationofChinaandtheShenyangNaturalScienceFoundation(1020036-1-03)andpartlybyaStrategicResearchGrant(SRG)fromCityUniversityof(projectno.7001227).TheauthorsareindebtedtotheEditorandrefereesforinvaluablecommentsandsuggestionsonthepaper.References AKAO,Y.,1990,QualityFunctionDeployment:IntegratingCustomerRequirementsinto ProductDesign,trans.G.Mazur(Cambridge,MA:ProductivityPress). ARMACOST,R.L.,COMPONATION,P.,MULLENS,M.andSWART,W.,1994,AnAHPframework forprioritizingcustomerrequirementsinQFD:anindustrializedhousingapplication.IIETransactions,26,72–79. CHAN,L.K.,KAO,H.P.,NG,A.andWU,M.L.,1999,Ratingtheimportanceofcustomer needsinqualityfunctiondeploymentbyfuzzyandentropymethods.InternationalJournalofProductionResearch,37,2499–2518. CHAN,L.K.andWU,M.L.,2002,Qualityfunctiondeployment:aliteraturereview.European JournalofOperationalResearch,143,463–497. FUNG,R.Y.K.,LAW,D.S.T.andIP,W.H.,1999,Designtargetsdeterminationforinter-dependentproductattributesinQFDusingfuzzyinference.IntegratedManufacturingSystems,10,376–384. FUNG,R.Y.K.,POPPLEWELL,K.andXIE,J.,1998,Anintelligenthybridsystemforcustomer requirementsanalysisandproductattributetargetsdetermination.InternationalJournalofProductionResearch,36,13–34. FUNG,R.Y.K.,TANG,J.,TU,Y.andWANG,D.,2002,Productdesignresourceoptimization usinganon-linearfuzzyqualityfunctiondeploymentmodel.InternationalJournalofProductionResearch,40,585–599. HAUSER,J.R.andCLAUSING,D.,1988,Thehouseofquality.HarvardBusinessReview,May– June,63–73. KIM,K.J.,MOSKOWITZ,H.,DHINGRA,A.andEVANS,G.,2000,Fuzzymulticriteriamodelsfor qualityfunctiondeployment.EuropeanJournalofOperationalResearch,121,504–518.MOSKOWITZ,H.andKIM,K.J.,1997,QFDoptimizer:anovicefriendlyqualityfunction deploymentdecisionsupportsystemforoptimizingproductdesign.ComputersandIndustrialEngineering,33,1–655. PARK,T.andKIM,K.J.,1998,Determinationofanoptimalsetofdesignrequirementsusing houseofquality.JournalofOperationsManagement,16,469–581. TAKEUCHI,H.andNONAKA,I.,1986,Thenewproductdevelopmentgame.HarvardBusiness Review,January–February,137–146. TANAKA,H.andWATADA,J.,1998,Possibilisticlinearsystemsandtheirapplicationstothe linearregressionmodel.FuzzySetsandSystem,27,275–2. TANG,J.,FUNG,R.Y.K.andXU,B.,2002,Anewapproachtoqualityfunctiondeployment planningwithfinancialconsideration.ComputerandOperationsResearch,29,1447–1463. TEMPONI,C.,YEN,J.andTIAO,W.A.,1999,Houseofquality:afuzzylogic-basedrequire-mentsanalysis.EuropeanJournalofOperationalResearch,117,340–324. VAIRAKTARAKIS,G.L.,1999,Optimizationtoolsfordesignandmarketingofnew/improved productsusingthehouseofquality.JournalofOperationsManagement,17,5–663. Fuzzyregressionmodelforqualityfunctiondeployment1027 VANGEGAS,L.V.andLABIB,A.W.,2001,Afuzzyqualityfunctiondeployment(FQFD)model forderivingoptimumtargets.InternationalJournalofProductionResearch,39,99–120.WASSERMANN,G.S.,1993,OnhowtoprioritizedesignrequirementsduringtheQFDplanning process.IIETransactions,25,59–65. YEN,K.K.,GHOSHRAY,S.andROIG,G.,1999,Alinearregressionmodelusingtriangular fuzzynumbercoefficients.FuzzySetsandSystems,106,167–177. ZHOU,M.,1998,FuzzylogicandoptimizationmodelsforimplementingQFD.Computersand IndustrialEngineering,35,237–240. 因篇幅问题不能全部显示，请点此查看更多更全内容
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