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DataEnvelopmentAnalysis(DEA)
数据包络分析Over4000publishedarticlesfrom1978to2008DataEnvelopmentAnalysis(DEADEAWhatitisConceptsFarrellmeasuresofEfficiencytechnicalallocativescaleRunningDEADangersofDEADEAWhatitisDataEnvelopmentAnalysis(DEA)Thismethodisaspecialbutimportantapplicationoflinearprogramming.DEAiswidelyappliedtoevaluatetheperformanceofDecisionMakingUnits(DMUs)suchasprivatefirms,publicagencies,schools,andhospitals.DataEnvelopmentAnalysis(DEAImportantFeatures(1)Efficiency(Productivity)Management(2)RelativeMeasurevs.AbsoluteMeasure(SocialScience)(NaturalScience)(3)Multi-DimensionalAnalysisvs.Cost/BenefitAnalysisImportantFeatures(1)RelativeComparison(2)MultipleInputsandOutputs(3)EfficiencyMeasurement(0%-100%)(4)AvoidtheSpecificationErrorbetweenInputsandOutputs(5)Production/CostAnalysisImportantFeatures(1)RelativeComparisonImportaDataEnvelopmentAnalysis(DEA)AmathematicaloptimisationtechniquewheretheunknownvariablesaretheweightsWorkthroughsomeexamplestoexplainStartwithasimpleexampleof8similarorganisations(decisionmakingunitsDMUs)producingtwooutputsusingasingleinputDataEnvelopmentAnalysis(DEATable1.
SingleInputandSingleOutputCaseConceptsSalesEmployee01234567896543210Figure1.
ComparisonsofBranchStoresEfficiencyFrontierABCDFGEHTable1.SingleInputandSingSalesEmployee0123456789Figure2.
RegressionLinevs.FrontierLineEfficiencyFrontierABCDFGEHRegressionLineB:thebeststoreTheefficiencyoftheotherstoresaredeterminedby1=B>E>D>C>H>A=G>F=0.46543210SalesEmployee012Table2.EfficiencySalesEmployee0123443210Figure3.
ImprovementofStoreAA1ABA2Table2.EfficiencySalesEmployTable3.
TwoInputsandOneOutputCaseFigure4.
TwoInputsandOneOutputCaseProductionPossibilitySetEfficientFrontiersAEDFCHIGBTable3.TwoInputsandOneOu012345012345Employee/SalesArea/SalesFigure5.
ImprovementofStoreAADEA1P012345012345Employee/SalesAreaTable4.OneInputandTwoOutputsCaseFigure6.
OneInputandTwoOutputsCaseProductionPossibilitySetDMUInputOut1Out2O1/IO2/I145450451012555281769.63.2375525525774601925763.29.6510010010001106552453304.5675516516533890450360549603603006510563641966.53.511611534152.56.812603301205.52Table4.OneInputandTwoOut谁最有效？谁最有效？W1=1,W2=0
W1=1,W2=0W1=1,W2=0W1=1,W2=0W1=0,W2=1W1=0,W2=1W1=0,W2=1W1=0,W2=1W1=0.5,W2=0.5W1=0.5,W2=0.5W1=0.5,W2=0.5W1=0.5,W2=0.5W1=0.75,W2=0.25W1=0.75,W2=0.25W1=0.75,W2=0.25W1=0.75,W2=0.25W1=0.25,W2=0.75W1=0.25,W2=0.75W1=0.25,W2=0.75W1=0.25,W2=0.75样本与实际SolidlinesconnectinngDMUs1,2,3,4,5Representsanefficiencyfrontierorproductionpossibilitycurve.Contrastwith“theoreticalfrontier”(brokencurve)–DEAefficiencyisrelativetothesample–butisourbestestimate样本与实际SolidlinesconnectinngDEfficienciesforallDMUs
mustnotexceed100%10.0w1+1.0w2<=1unit19.6w1+3.2w2<=1unit27.0w1+7.0w2<=1unit33.2w1+9.6w2<=1unit41.0w1+10.0w2<=1unit54.5w1+6.0w2<=1unit63.2w1+3.2w2<=1unit75.0w1+4.0w2<=1unit86.0w1+5.0w2<=1unit96.5w1+3.5w2<=1unit102.5w1+6.8w2<=1unit115.5w1+1.5w2<=1unit12w1,w2>=0EfficienciesforallDMUs
must生产可能集合的形成生产可能集合的形成Max10w1+1w2St10w1+1w2=1(onlyactiveconstraint)AndallotherconstraintsinactiveAsolution:w1=0.1,w2=010w1+1w2=1.0(100%efficient)DMU1Max10w1+1w2DMU1DMU6Max4.5w1+6.0w2St7.0w1+7.0w2=1(activeconstraint3)3.2w1+9.6w2=1(activeconstraint4)AndallotherconstraintsinactiveUniquesolution:w1=0.05803,w2=0.08484.5w1+6.0w2=0.77PeersareDMUs3and4DMU6Max4.5w1+6.0w2DMU6DMU6DEA
softwareFreesoftwareEMS–HolgerScheelDEA---FreedownloadDEAP–TimCoelli–FreedownloadOthersoftwareSaitech–Cooper,SeifordandTonesoftwareFrontierAnalysisWarwickDEAEMQC-FareandGrosskopf中文MaxDEADEAsoftwareFreesoftwareLPFormulationTable5.HospitalCaseEfficiencyofthejthDMULPFormulationTable5.HospitaEfficiencyofthejthDMUEfficiencyofthejthDMUNow,assumingthatHospital(B)isevaluatedbyDEA,wehaveMaxs.t.Now,assumingthatHospital(BMaxs.t.Maxs.t.EfficiencyofthekthDMUMaxs.t.EfficiencyofthekthDMUMaxs基本问题的一般表述DMU1DMU2DMUnv1x11x12x1nv2X21x22X2n……………………vmxm1xm2xmnu1y11y12y1nu2y12y12y1n……………………usys1ys2ysn决策单元评估指标……投入产出……………………………………基本问题的一般表述DMU1DMU2DMUnv1x11x12x生产率与效率分析lecture6数据包络分析CCRmodelCCRmodelR:AReferenceSetR:AReferenceSetPrimalProblemPrimalProblemDualProblemDualProblemSecond-stage:SlackSecond-stage:SlackReferenceSet:ReferenceSet:Table1.6:2inputs–1outputCaseExampleProblemTable1.6:2inputs–1outpuPrimalProblemPrimalProblemDualProblemDualProblem0DFEFigure1.7:EfficiencyofDMUAAA1CEfficiencyFrontier0DFEFigure1.7:Efficiencyof生产率与效率分析lecture6数据包络分析Table1.7:ResultsofDEAReferenceSetDMUEfficiencyA0.8330.1670.1670.833000B0.7270.1820.0910.727000C10.2000.2001000D10.2500.1251000E10.50001000F10112.000Software:EMSResultsTable1.7:ResultsofDEARefeBCCmodelVariableReturnstoScaleBCCmodelVariableReturnstoSCCRmodelCCRmodelBCCmodelBCCmodelDualProblemBCCmodel:DualProblemBCCmodel:(C)0OutputInputEfficiencyFrontierofCCRmodelFigure2.1:EfficiencyFrontierandProductionPossibilitySet(A)(B)EfficiencyFrontierofBCCmodelbacde(C)0OutputInputEfficiencyFronCCR有效与BCC有效OutputInputFrontierofCCRmodelFrontierofBCCmodelbacdPABRCCR有效与BCC有效OutputInputFrontierCostAnalysisCostAnalysis0EfficiencyFrontierofCCRmodelEfficiencyFrontierofBCCmodelEcdekgjihbfP0EfficiencyFrontierofCCRmoDualProblemDualProblemAnEconomicInterpretationDuetoMichaelFarrell(1957)Technicalefficiency=OB/OACapitalLabourMincombinationsrequired(isoquant)ABOOtherthingsequal=outputAnEconomicInterpretationDueAnEconomicInterpretation
(Outputmaximisationorientation)Technicalefficiency=OB/OAOutput1(e.g.calls)Output2(e.g.lines)DCOOtherthingsequal=inputsMaxcombinationsachieveable(productionfrontier)AnEconomicInterpretation
(OuEconomicInterpretation(3)Theisoquantandproductionfrontierarenotknowndirectly,butmightbeestimatedfromknowndata,usingpiecewiseinterpolationCapitalLabourMincombinationsrequired(isoquant)ABOEFGHJKLBisanartificialobservation-acombinationofFandGEconomicInterpretation(3)TheAllocativeefficiencyDependsonknowingpricesAE=mincost/actualcost=OD/OBCapitalLabourMincombinationsrequired(isoquant)ABOCEfficientIsocostlineDAllocativeefficiencyDependsoScaleefficiencyOutputInputOMAPQRT.E.=PR/PAS.E.=PQ/PRT.&S.E=PQ/PAThisisinputorientation.Whataboutoutputorientation?ScaleefficiencyOutputInputOMARunningDEAPurpose-builtsoftwareExcel/SolvermacrosOrganisedataforinputIdentifyinputs,outputsandnoncontrollablesRunInterpretRunningDEAPurpose-builtsofHowreliableisDEA?Dependsonwhetherfrontiercanbepopulatedbyefficientfirms:numberofobservationsnumberofdimensionsclosenesstofrontierofenoughfirmsdistributionofvariablesHowreliableisDEA?DependsonDangersofDEA(1)OutliersappearefficientCapitalLabourMincombinationsrequired(isoquant)ABOEFGHJKLBisanartificialobservation-acombinationofFandGDangersofDEA(1)OutliersappeDangersofDEA(2)TechnicalefficiencyisnoteconomicefficiencyOutput2(e.g.meterreading)Maxcombinationsachieveable(productionfrontier)DCOBistechnicallyefficientbuteconomicallyinefficientOutput1(e.g.energy)BIsovaluelinesDangersofDEA(2)TechnicaleffDangersofDEA(3)Dilemma:toincludeornottoincludevariablesInclude=>spuriouslyefficientExclude=>spuriouslyinefficientNowell-establishedstatisticaltestforinclusion/exclusionDangersofDEA(3)Dilemma:toDataEnvelopmentAnalysis(DEA)
数据包络分析Over4000publishedarticlesfrom1978to2008DataEnvelopmentAnalysis(DEADEAWhatitisConceptsFarrellmeasuresofEfficiencytechnicalallocativescaleRunningDEADangersofDEADEAWhatitisDataEnvelopmentAnalysis(DEA)Thismethodisaspecialbutimportantapplicationoflinearprogramming.DEAiswidelyappliedtoevaluatetheperformanceofDecisionMakingUnits(DMUs)suchasprivatefirms,publicagencies,schools,andhospitals.DataEnvelopmentAnalysis(DEAImportantFeatures(1)Efficiency(Productivity)Management(2)RelativeMeasurevs.AbsoluteMeasure(SocialScience)(NaturalScience)(3)Multi-DimensionalAnalysisvs.Cost/BenefitAnalysisImportantFeatures(1)RelativeComparison(2)MultipleInputsandOutputs(3)EfficiencyMeasurement(0%-100%)(4)AvoidtheSpecificationErrorbetweenInputsandOutputs(5)Production/CostAnalysisImportantFeatures(1)RelativeComparisonImportaDataEnvelopmentAnalysis(DEA)AmathematicaloptimisationtechniquewheretheunknownvariablesaretheweightsWorkthroughsomeexamplestoexplainStartwithasimpleexampleof8similarorganisations(decisionmakingunitsDMUs)producingtwooutputsusingasingleinputDataEnvelopmentAnalysis(DEATable1.
SingleInputandSingleOutputCaseConceptsSalesEmployee01234567896543210Figure1.
ComparisonsofBranchStoresEfficiencyFrontierABCDFGEHTable1.SingleInputandSingSalesEmployee0123456789Figure2.
RegressionLinevs.FrontierLineEfficiencyFrontierABCDFGEHRegressionLineB:thebeststoreTheefficiencyoftheotherstoresaredeterminedby1=B>E>D>C>H>A=G>F=0.46543210SalesEmployee012Table2.EfficiencySalesEmployee0123443210Figure3.
ImprovementofStoreAA1ABA2Table2.EfficiencySalesEmployTable3.
TwoInputsandOneOutputCaseFigure4.
TwoInputsandOneOutputCaseProductionPossibilitySetEfficientFrontiersAEDFCHIGBTable3.TwoInputsandOneOu012345012345Employee/SalesArea/SalesFigure5.
ImprovementofStoreAADEA1P012345012345Employee/SalesAreaTable4.OneInputandTwoOutputsCaseFigure6.
OneInputandTwoOutputsCaseProductionPossibilitySetDMUInputOut1Out2O1/IO2/I145450451012555281769.63.2375525525774601925763.29.6510010010001106552453304.5675516516533890450360549603603006510563641966.53.511611534152.56.812603301205.52Table4.OneInputandTwoOut谁最有效？谁最有效？W1=1,W2=0
W1=1,W2=0W1=1,W2=0W1=1,W2=0W1=0,W2=1W1=0,W2=1W1=0,W2=1W1=0,W2=1W1=0.5,W2=0.5W1=0.5,W2=0.5W1=0.5,W2=0.5W1=0.5,W2=0.5W1=0.75,W2=0.25W1=0.75,W2=0.25W1=0.75,W2=0.25W1=0.75,W2=0.25W1=0.25,W2=0.75W1=0.25,W2=0.75W1=0.25,W2=0.75W1=0.25,W2=0.75样本与实际SolidlinesconnectinngDMUs1,2,3,4,5Representsanefficiencyfrontierorproductionpossibilitycurve.Contrastwith“theoreticalfrontier”(brokencurve)–DEAefficiencyisrelativetothesample–butisourbestestimate样本与实际SolidlinesconnectinngDEfficienciesforallDMUs
mustnotexceed100%10.0w1+1.0w2<=1unit19.6w1+3.2w2<=1unit27.0w1+7.0w2<=1unit33.2w1+9.6w2<=1unit41.0w1+10.0w2<=1unit54.5w1+6.0w2<=1unit63.2w1+3.2w2<=1unit75.0w1+4.0w2<=1unit86.0w1+5.0w2<=1unit96.5w1+3.5w2<=1unit102.5w1+6.8w2<=1unit115.5w1+1.5w2<=1unit12w1,w2>=0EfficienciesforallDMUs
must生产可能集合的形成生产可能集合的形成Max10w1+1w2St10w1+1w2=1(onlyactiveconstraint)AndallotherconstraintsinactiveAsolution:w1=0.1,w2=010w1+1w2=1.0(100%efficient)DMU1Max10w1+1w2DMU1DMU6Max4.5w1+6.0w2St7.0w1+7.0w2=1(activeconstraint3)3.2w1+9.6w2=1(activeconstraint4)AndallotherconstraintsinactiveUniquesolution:w1=0.05803,w2=0.08484.5w1+6.0w2=0.77PeersareDMUs3and4DMU6Max4.5w1+6.0w2DMU6DMU6DEA
softwareFreesoftwareEMS–HolgerScheelDEA---FreedownloadDEAP–TimCoelli–FreedownloadOthersoftwareSaitech–Cooper,SeifordandTonesoftwareFrontierAnalysisWarwickDEAEMQC-FareandGrosskopf中文MaxDEADEAsoftwareFreesoftwareLPFormulationTable5.HospitalCaseEfficiencyofthejthDMULPFormulationTable5.HospitaEfficiencyofthejthDMUEfficiencyofthejthDMUNow,assumingthatHospital(B)isevaluatedbyDEA,wehaveMaxs.t.Now,assumingthatHospital(BMaxs.t.Maxs.t.EfficiencyofthekthDMUMaxs.t.EfficiencyofthekthDMUMaxs基本问题的一般表述DMU1DMU2DMUnv1x11x12x1nv2X21x22X2n……………………vmxm1xm2xmnu1y11y12y1nu2y12y12y1n……………………usys1ys2ysn决策单元评估指标……投入产出……………………………………基本问题的一般表述DMU1DMU2DMUnv1x11x12x生产率与效率分析lecture6数据包络分析CCRmodelCCRmodelR:AReferenceSetR:AReferenceSetPrimalProblemPrimalProblemDualProblemDualProblemSecond-stage:SlackSecond-stage:SlackReferenceSet:ReferenceSet:Table1.6:2inputs–1outputCaseExampleProblemTable1.6:2inputs–1outpuPrimalProblemPrimalProblemDualProblemDualProblem0DFEFigure1.7:EfficiencyofDMUAAA1CEfficiencyFrontier0DFEFigure1.7:Efficiencyof生产率与效率分析lecture6数据包络分析Table1.7:ResultsofDEAReferenceSetDMUEfficiencyA0.8330.1670.1670.833000B0.7270.1820.0910.727000C10.2000.2001000D10.2500.1251000E10.50001000F10112.000Software:EMSResultsTable1.7:ResultsofDEARefeBCCmodelVariableReturnstoScaleBCCmodelVariableReturnstoSCCRmodelCCRmodelBCCmodelBCCmodelDualProblemBCCmodel:DualProblemBCCmodel:(C)0OutputInputEfficiencyFrontierofCCRmodelFigure2.1:EfficiencyFrontierandProductionPossibilitySet(A)(B)EfficiencyFrontierofBCCmodelbacde(C)0OutputInputEfficiencyFronCCR有效与BCC有效OutputInputFrontierofCCRmodelFrontierofBCCmodelbacdPABRCCR有效与BCC有效OutputInputFrontierCostAnalysisCostAnalysis0EfficiencyFrontierofCCRmodelEfficiencyFrontierofBCCmodelEcdekgjihbfP0EfficiencyFrontierofCCRmoDualProblemDualProblemAnEconomicInterpretationDuetoMichaelFarrell(1957)Technicalefficiency=OB/OACapitalLabourMincombinationsrequired(isoquant)ABOOtherthingsequal=outputAnEconomicInterpretationDueAnEconomicInterpretation
(Outputmaximisationorientation)Technicalefficiency=OB/OAOutput1(e.g.calls)Output2(e.g.lines)DCOOtherthingsequal=inputsMaxcombinationsachieveable(productionfrontier)AnEconomicInterpretation
(OuEconomicInterpretation(3)
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