20140506——CCR-DEA-MODEL

1CCRDEAMODELWilliamW.Cooper,LawrenceM.SeifordandJoeZhu.DATAENVELOPMENTANALYSIS----History,ModelsandInterpretations.2CCRDEAMODEL•AssumethattherearenDMUstobeevaluated.EachDMUconsumesvaryingamountsofmdifferentinputstoproducesdifferentoutputs.•Specifically,DMUjconsumesamountxijofinputiandproducesamountyrjofoutputr.•Assumethatxij≥0andyrj≥0andfurtherassumethateachDMUhasatleastonepositiveinputandonepositiveoutputvalue.Reference:WilliamW.Cooper,LawrenceM.SeifordandJoeZhu.DATAENVELOPMENTANALYSIS--History,ModelsandInterpretations3DEAEvaluationSystemweightsweightsInputsOutputsDMU’sys1ys2ysnsusyrjrDMUjur4•AsintroducedbyCharnes,Cooper,andRhodes,theratioofoutputstoinputsisusedtomeasuretherelativeefficiencyoftheDMUj=DMUotobeevaluatedrelativetotheratiosofallofthej=1,2,…,nDMUj.•WecaninterprettheCCRconstructionasthereductionofthemultiple-output/multiple-inputsituation(foreachDMU)tothatofasingle‘virtual'outputand‘virtual’input.•ForaparticularDMUtheratioofthissinglevirtualoutputtosinglevirtualinputprovidesameasureofefficiencythatisafunctionofthemultipliers.CCRDEAMODEL5•Inmathematicalprogrammingparlance,thisratio,whichistobemaximized,formstheobjectivefunctionfortheparticularDMUbeingevaluated,sothatsymbolicallywhereitshouldbenotedthatthevariablesaretheur'sandthevi'sandtheyro'sandxio'saretheobservedoutputandinputvalues,respectively,ofDMUo,theDMUtobeevaluated.•Ofcourse,withoutfurtheradditionalconstraints(1.1)isunbounded.CCRDEAMODEL6CCRDEAMODEL•Asetofnormalizingconstraints(oneforeachDMU)reflectstheconditionthatthevirtualoutputtovirtualinputratioofeveryDMU,includingDMUj=DMUo,mustbelessthanorequaltounity.Themathematicalprogrammingproblemmaythusbestatedas7DEAEvaluationSystemweightsweightsInputsOutputsDMU’sys1ys2ysnsusyrjrDMUjurvixij8CCRDEAMODEL•Remark:Afullyrigorousdevelopmentwouldreplaceur,vi≥0with•whereεisanon-Archimedeanelementsmallerthananypositiverealnumber.•Thisconditionguaranteesthatsolutionswillbepositiveinthesevariables.•Italsoleadstotheε0in(1.6)which,inturn,leadstothe2ndstageoptimizationoftheslacksasin(1.10).9CCRDEAMODEL•Theaboveratioform(1.2)yieldsaninfinitenumberofsolutions;if(u*,v*)isoptimal,then(αu*,αv*)isalsooptimalforα0.•However,thetransformationdevelopedbyCharnesandCooper(1962)forlinearfractionalprogrammingselectsarepresentativesolution[i.e.,thesolution(u,v)forwhich=1]andyieldstheequivalentlinearprogrammingprobleminwhichthechangeofvariablesfrom(u,v)to(μ,ν)isaresultoftheCharnes-Coopertransformation,10•Charnes-Coopertransformation↓CCRDEAMODEL11CCRDEAMODEL•for(1.3)theLPdualproblemis12•Thislastmodel,(1.4),issometimesreferredtoasthe“Farrellmodel”becauseitistheoneusedinFarrell(1957).•IntheeconomicsportionoftheDEAliteratureitissaidtoconformtotheassumptionof“strongdisposal”becauseitignoresthepresenceofnon-zeroslacks.•IntheoperationsresearchportionoftheDEAliteraturethisisreferredtoas“weakefficiency.”CCRDEAMODEL13•Byvirtueofthedualtheoremoflinearprogrammingwehavez*=θ*.Henceeitherproblemmaybeused.Onecansolvesay(1.4),toobtainanefficiencyscore.•Becausewecansetθ=1andλ*k=1withλ*k=λ*0andallotherλ*j=0,asolutionof(1.4)alwaysexists.•Moreoverthissolutionimpliesθ*≤1.•Theoptimalsolution,θ*,yieldsanefficiencyscoreforaparticularDMU.CCRDEAMODEL14•TheprocessisrepeatedforeachMDUji.e.,solve(1.4),with(Xo,Yo)=(Xk,Yk),where(Xk,Yk)representvectorswithcomponentsxik,yrkand,similarly(Xo,Yo)hascomponentsxok,yok.•DMUsforwhichθ*1areinefficient,whileDMUsforwhichθ*=1areboundarypoints.•Arealltheboundarypointsefficient?(Q,B,D)CCRDEAMODEL15•CCRDEAMODEL16•CCRDEAMODELD？17•Someboundarypointsmaybe“weaklyefficient”becausewehavenon-zeroslacks.•Thismayappeartobeworrisomebecausealternateoptimamayhavenon-zeroslacksinsomesolutions,butnotinothers.•However,wecanavoidbeingworriedeveninsuchcasesbyinvokingthefollowinglinearprograminwhichtheslacksaretakentotheirmaximalvalues.CCRDEAMODEL18CCRDEAMODEL•wherewenotethechoicesofsi−andsr+donotaffecttheoptimalθ*whichisdeterminedfrommodel(1.4).Copy19•Thesedevelopmentsnowleadtothefollowingdefinitionbaseduponthe“relativeefficiency”definition1.2whichwasgiveninsection1above.•Definition1.3(DEAEfficiency):TheperformanceofDMUoisfully(100%)efficientifandonlyifboth(i)θ*=1and(ii)allslackssi−*=sr+*=0.•Definition1.4(WeaklyDEAEfficient):TheperformanceofDMUoisweaklyefficientifandonlyifboth(i)θ*=1and(ii)si−*≠0and/orsr+*≠0forsomeiandrinsomealternateoptima.CCRDEAMODEL20•Definition1.1(Efficiency–ExtendedPareto-KoopmansDefinition):Full(100%)efficiencyisattainedbyanyDMUifandonlyifnoneofitsinputsoroutputscanbeimprovedwithoutworseningsomeofitsotherinputsoroutputs.•Definition1.2(RelativeEfficiency):ADMUistoberatedasfully(100%)efficientonthebasisofavailableevidenceifandonlyiftheperformancesofotherDMUsdoesnotshowthatsomeofitsinputsoroutputscanbeimprovedwithoutworseningsomeofitsotherinputsoroutputs.CCRDEAMODEL21•Itistobenotedthattheprecedingdevelopmentamountstosolvingthefollowingproblemintwosteps:wherethesi−andsr+areslackvariablesusedtoconverttheinequalitiesin(1.4)toequivalentequations.Hereε0isaso-callednon-Archimedeanelementdefinedtobesmallerthananypositi