DEA models with undesirable inputs and outputs

AnnOperRes(2010)173:177–194DOI10.1007/s10479-009-0587-3DEAmodelswithundesirableinputsandoutputsW.B.Liu·W.Meng·X.X.Li·D.Q.ZhangPublishedonline:15July2009©SpringerScience+BusinessMedia,LLC2009AbstractDataEnvelopmentAnalysis(DEA)modelswithundesirableinputsandoutputshavebeenfrequentlydiscussedinDEAliterature,e.g.,viadatatransformation.Thesestud-ieswerescattedintheliterature,andoftenconfinedtosomeparticularapplications.InthispaperwepresentasystematicinvestigationonmodelbuildingofDEAwithouttransferringundesirabledata.Wefirstdescribethedisposabilityassumptionsandanumberofdifferentperformancemeasuresinthepresenceofundesirableinputsandoutputs,andthendiscussdifferentcombinationsofthedisposabilityassumptionsandthemetrics.ThisapproachleadstoaunifiedpresentationofseveralclassesofDEAmodelswithundesirableinputsand/oroutputs.KeywordsDataEnvelopmentAnalysis·Undesirableinputsandoutputs·ExtendedStrongDisposability1IntroductionSincetheintroductionoftheDEAin1978,ithasbeenwidelyusedinefficiencyanalysisofmanybusinessandindustryapplications.Excellentliteraturesurveyscanbefoundin,forinstance,Seiford(1996)andCooperetal.(2004).Themostwell-knownDEAmodelsaretheCCRmodel(Charnesetal.1978),theBCCmodel(Bankeretal.1984),theAdditivemodel(Charnesetal.1985),andtheConeRatiomodel(Charnesetal.1989).TheseDEATheauthorswishtoexpresstheirsincerethankstotherefereesforhisorherconstructivereviewsandsuggestions,whichleadtosignificantimprovementsofthispaper.W.B.Liu()KentBusinessSchool,UniversityofKent,Canterbury,CT27PE,UKe-mail:[e-mail]stration,EastChinaNormalUniversity,Shanghai,200062,ChinaX.X.Li·D.Q.ZhangInstituteofPolicyandManagement,ChineseAcademyofSciences,Beijing,100080,China178AnnOperRes(2010)173:177–194modelswereallformulatedfordesirableinputsandoutputs.However,therefrequentlyexistundesirableinputsand/oroutputsinrealapplications.InDEAliterature,therealreadyexistedmuchresearchconcerningapplicationswithun-desirableinputsand/oroutputs.Someoftheexistingapproachesarebrieflysummarizedasfollows:Anintuitivereactionistoapplysometransformations.Themostacceptableoneisf(U)=−U;thesocalledtheADDapproachsuggestedbyKoopmans(1951).Thentheundesirableinputsoroutputswillbecomedesirableafterthistransformation.However,thedatamaythenbecomenegative,anditisnotstraightforwardtodefineefficiencyscoresfornegativedata.Scheel(2001)attemptedthistaskbuthisdefinitionsseemedtobecompli-cated.Translationf(U)=−U+βisanotherwidelyusedone(e.g.AliandSeiford1990;Pastor1996;Scheel2001;SeifordandZhu2002).Howeveritiswell-knownthatnotonlyrankingbutalsoclassificationmaydependonβ.Anothertransformationisthemultiplica-tiveinverse:f(U)=1/U(e.g.GolanyandRoll1989;Lovelletal.1995).Beinganonlineartransformation,itsbehaviorsareevenmorecomplicated(Scheel1998).Thushowtoprop-erlyselectasuitabletransformationisverymuchcase-dependent.Theapproachesbasedondata-transformationmayunexpectedlyproduceadverseresultsasdiscussedinLiuandSharp(1999).Therealsoexistmanyapproachesthatcanavoiddatatransformation.Forexample,onemayregardundesirableinputsasdesirableoutputs,orundesirableoutputsasdesirablein-puts,seeLiuandSharp(1999)foraninitialattempttoformulatethismethod.Thisapproachisanattractivemethodinstudyingoperationalefficiencyduetoitssimplicityandelegance,althoughitchangesthephysicalinput-outputrelationship.ItsstartingpointisthatefficientDMUswishtominimizedesirableinputsandundesirableoutputs,andtomaximizedesir-ableoutputsandundesirableinputs.Ifoneonlywishestoinvestigateoperationalefficiencyfromthispointofview,thereisnoneedtodistinguishbetweeninputsandoutputs,butonlyminimumandmaximum.Wewillfurtherextendthisapproachinthisworkanddiscussitsrelationshipwithotherapproaches.RelatedtoADD,thereareseveralrecentpapersonDEAmodelshandlingnegativedata(butdesirable)withdirectionaldistancefunctions,suchas,FäreandGrosskopf(2004)(whereaweakfreedisposabilitywasused),SilvaPortelaetal.(2004),andYu(2004).AsshowninSeifordandZhu(2005),suchapproachesarecloselyrelatedtotheweightedadditivemodelsinordertomeasureproportionalgainsfromtheslacks,whicharealwayspositive.Itisimportanttorealizethattheadditivemodelsareabletohandlenegativedata.Aftersomesimplemodifications,theycaneasilyhandleundesirablecasesaswell.Thesewillbediscussedlateron.Ourinvestigationpaysmoreattentiontotheoreticalaspectsoftheseissues.OurmainideaistoexaminethemwithinthegeneralframeworkproposedinLiuetal.(2006),wheresomeessentialbuildingblocksofaDEAmodelwereidentifiedandillustrated.Theprinci-palobjectiveofthispaperistodiscussthefreedisposabilityassumptionsandtodescribeanumberofdifferentperformancemeasuresormetricfunctionswithattractivepropertiesinthepresenceofundesirableinputsandoutputsandnegativedata.Wewilldiscussdifferentcombinationsofthefreedisposabilityassumptionsandthemetrics.ThisapproachleadstoaunifiedpresentationforsomeexistingDEAmodelswithundesirableinputsand/orout-puts.Wealsodiscusssomerelationshipamongtheexistingapproaches.WeundertaketheseinvestigationsbyutilizingsomeoftheideasusedinLiuetal.(2006).Theplanofthispaperisasfollows:InSect.2wediscussdisposabilitya