Modeling-Undesirable-Factors-in-Efficiency-Evaluat

InterfaceswithOtherDisciplinesModelingundesirablefactorsinefficiencyevaluation:CommentRolfF€area,b,*,ShawnaGrosskopfbaDepartmentofAgricultureandResourceEconomics,OregonStateUniversity,Corvallis,OR97331-3612,USAbDepartmentofEconomics,OregonStateUniversity,Corvallis,OR97331-3612,USAReceived23September2002;accepted30January2003AbstractInarecentpaperinthisjournalSeifordandZhu[Eur.J.Operat.Res.142(2002)16]proposeusingdatatranslationasameansofintegratingundesirableoutputsandinputsintodataenvelopmentanalysis(DEA)models.By
reversingthebadoutputs,theycanbereadilyincludedinthestandardDEAoutputincreasingmodel,andtheusualscalingwillactuallyreducethebadoutputs.Thisrecognizesthefactthatincreasesinthoseoutputsandinputsarenotdesirable,yetdoesnotresorttotheusualadhoctreatmentofundesirableoutputsasinputs.Herewesuggestanalternativeapproachwhichallowsustoexplicitlymodelajointenvironmentaltechnologyandgaugeperformanceintermsofincreasedgoodoutputanddecreasedundesirableoutput.ThisisachievedbyadoptingadirectionaldistancefunctionwhichmaybeestimatedusingtheusuallinearprogrammingtechniquesemployedinDEA.
2003ElsevierB.V.Allrightsreserved.Keywords:DEA;Undesirableoutputs;Datatranslation;DirectionaldistancefunctionInarecentpaperinthisjournalSeifordandZhu(2002)proposeusingdatatranslationasameansofintegratingundesirableoutputsandin-putsintodataenvelopmentanalysis(DEA)models.By
reversingthebadoutputs,theycanbereadilyincludedinthestandardDEAoutputincreasingmodel,andtheusualscalingwillactuallyreducethebadoutputs.Thisrecognizesthefactthatin-creasesinthoseoutputsandinputsarenotdesir-able,yetdoesnotresorttoadhoctreatmentofundesirableoutputsasinputs,asinHailuandVeeman(2001)forexample.Weprefertheirap-proachtotheadhocinputapproach,butwouldliketosuggestanalternativeapproachwhichal-lowsustoexplicitlymodelajointenvironmentaltechnologyandgaugeperformanceintermsofincreasedgoodoutputanddecreasedundesirableoutput.Thatis,wedistinguishbetweenweakdis-posability–whichisapropertyofajointenvi-ronmentaltechnology–andthechoiceofdirectioninwhichperformanceismeasured.LetPðxÞdenotethesetofdesirableoutputsy2RMþandundesirableoutputsu2RKþwhichareproduciblefromtheinputvectorx2RNþ.1*Correspondingauthor.Address:303BallardExtensionHall,OregonStateUniversity,Corvallis,OR97331-3612,USA.Tel.:[phone];fax:[phone].E-mailaddress:[e-mail](R.F€are).1Weconsideronly
goodinputshere.Thecaseofundesir-ableinputsislefttothereader.0377-2217/$-seefrontmatter
2003ElsevierB.V.Allrightsreserved.doi:10.1016/S0377-2217(03)00191-7EuropeanJournalofOperationalResearch157(2004)242–245firstthattheoutputsetsPðxÞ,x2RNþareclosedandboundedsetsandthatinputsarefreelydisposable,asinSeifordandZhu(2002).2WesaythatPðxÞisanenvironmentaloutputset1.ifoutputsareweaklydisposable,3i.e.,ifðy;uÞ2PðxÞand05h51thenðhy;huÞ2PðxÞ,and2.ifgoodandbadoutputsarenull-joint,4i.e.,ðy;uÞ2PðxÞ;u¼0impliesthaty¼0.FollowingF€areandPrimont(1995),Weakdisposabilityofoutputs,tosomeex-tent,allowsforthepossibilitythatoneoftheoutputsisabad.Forexample,ifoutput1iselectricityproducedbyburningcoalandoutput2issulfurdioxideemissionsthen...weakdisposabilityimpliesthata10%reduc-tioninsulfurdioxideemissionsispossibleifaccompaniedbya10%reductionintheout-putofelectricity,holdingtheinputvectorconstant.Alternatively,ifregulationsareimposedre-strictingemissionsofsulfurdioxide,targetemis-sionsmaybeachievedbydivertingsomeofthegiveninputvectortothe
cleanupofthoseemis-sions.Thisimpliesthatlessinputwouldbeavail-ablefortheproductionofelectricity,resultinginasimultaneousdeclineingoodandbadoutput.Thenull-jointnesspropertydescribesthebasicpollutionproblem,namelythatwhenweproducegoodoutputs(giventechnology),thebadoutputsarebyproductsoftheproductionprocess.Ifwecouldproducethegoodoutputwithouttheby-products,therewouldbenopollutionproblem.Incontrasttoourtwoassumptions,SeifordandZhumaintaintheassumptionofstrongdispos-abilityofalloutputs,i.e.,ifðy;uÞ2PðxÞthenðy0;u0Þ2PðxÞ;ðy0;u0Þ5ðy;uÞ:Thisdiffersfromtheweakdisposabilityprop-ertyaboveinthatnonproportionalreductionsineitheroutputarefeasible.Theyalsodonotimposenull-jointness.Tocomparethetwo,considerthefollowingdata(Table1).ThetranslatedundesirableoutputislabelleduTandiscalculatedas3u;3isthesmallestpositiveintegerlargerthanthemaximumobservedundesirableoutput.Ifweplotthesedataandapplytheassociateddisposabilityproperties,wearriveatthefollowingillustrationofthetwotechnologies(Fig.1).InthefigurethetranslatedSeiford–Zhudataaredenotedwithprimes;theresultingtechnologyistheboxgoingthroughc0anda0,andincludingtherelevantsegmentsoftheaxes.Thetechnologysatisfyingweakdisposabilityandnull-jointnessisboundedby0cbandtheverticalsegmentfrompointbdowntotheu-axisandtherelevantsectionoftheu-axis.Null-jointnessimpliesthatproduc-tionalongthey-axisisnotfeasible.Thenextquestionishowtomodeltheenvi-ronmentaloutputsetinaDEAframework.Lettherebek¼1;...;Kobservations,thenwecanwriteourenvironmentaloutputsetas2SeeF€areandPrimont(1995)fordetails.3Shephard(1970).4ShephardandF€are(1974).Table1Exampledat