生产率与效率分析lecture6-数据包络分析1

DataEnvelopmentAnalysis(DEA)数据包络分析Over4000publishedarticlesfrom1978to2008DEA•Whatitis•Concepts•FarrellmeasuresofEfficiency–technical–allocative–scale•RunningDEA•DangersofDEADataEnvelopmentAnalysis(DEA)Thismethodisaspecialbutimportantapplicationoflinearprogramming.DEAiswidelyappliedtoevaluatetheperformanceofDecisionMakingUnits(DMUs)suchasprivatefirms,publicagencies,schools,andhospitals.ImportantFeatures(1)Efficiency(Productivity)Management(2)RelativeMeasurevs.AbsoluteMeasure(SocialScience)(NaturalScience)(3)Multi-DimensionalAnalysisvs.Cost/BenefitAnalysis•(1)RelativeComparison•(2)MultipleInputsandOutputs•(3)EfficiencyMeasurement(0%-100%)•(4)AvoidtheSpecificationErrorbetweenInputsandOutputs•(5)Production/CostAnalysisImportantFeaturesDataEnvelopmentAnalysis(DEA)•Amathematicaloptimisationtechniquewheretheunknownvariablesaretheweights•Workthroughsomeexamplestoexplain•Startwithasimpleexampleof8similarorganisations(decisionmakingunitsDMUs)producingtwooutputsusingasingleinputStoreABCDEFGHEmployee23345568Sale13234235Sale/Employee0.510.6670.750.80.40.50.625Table1.SingleInputandSingleOutputCaseConceptsEmployee01234567896543210Figure1.ComparisonsofBranchStoresEfficiencyFrontierABCDFGEHEmployee0123456789Figure2.RegressionLinevs.FrontierLineEfficiencyFrontierABCDFGEHRegressionLineB:thebeststoreTheefficiencyoftheotherstoresaredeterminedby1ofemployeeperSalesothersofemployeeperSales0B1=BEDCHA=GF=0.46543210StoreABCDEFGHEfficiency0.510.6670.750.80.40.50.625Table2.EfficiencyEmployee0123443210Figure3.ImprovementofStoreAA1ABA2StoreABCDEFGHIEmployee1x47842565.56FloorArea2x33124242.52.5Saley111111111Table3.TwoInputsandOneOutputCaseFigure4.TwoInputsandOneOutputCaseProductionPossibilitySetEfficientFrontiersAEDFCHIGB0123450123456789Employee/SalesArea/Sales012345012345Employee/SalesArea/SalesFigure5.ImprovementofStoreAOAOPAofEfficiencyADEA1PTable4.OneInputandTwoOutputsCaseFigure6.OneInputandTwoOutputsCaseProductionPossibilitySetDMUInputOut1Out2O1/IO2/I145450451012555281769.63.2375525525774601925763.29.6510010010001106552453304.5675516516533890450360549603603006510563641966.53.511611534152.56.812603301205.52谁最有效？W1=1,W2=0W1=1,W2=0W1=0,W2=1W1=0,W2=1W1=0.5,W2=0.5W1=0.5,W2=0.5W1=0.75,W2=0.25W1=0.75,W2=0.25W1=0.25,W2=0.75W1=0.25,W2=0.75样本与实际SolidlinesconnectinngDMUs1,2,3,4,5Representsanefficiencyfrontierorproductionpossibilitycurve.Contrastwith“theoreticalfrontier”(brokencurve)–DEAefficiencyisrelativetothesample–butisourbestestimateEfficienciesforallDMUsmustnotexceed100%•10.0w1+1.0w2=1unit1•9.6w1+3.2w2=1unit2•7.0w1+7.0w2=1unit3•3.2w1+9.6w2=1unit4•1.0w1+10.0w2=1unit5•4.5w1+6.0w2=1unit6•3.2w1+3.2w2=1unit7•5.0w1+4.0w2=1unit8•6.0w1+5.0w2=1unit9•6.5w1+3.5w2=1unit10•2.5w1+6.8w2=1unit11•5.5w1+1.5w2=1unit12•w1,w2=0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)DMU1DMU6•Max4.5w1+6.0w2•St7.0w1+7.0w2=1(activeconstraint3)•3.2w1+9.6w2=1(activeconstraint4)•Andallotherconstraintsinactive•Uniquesolution:w1=0.05803,w2=0.0848•4.5w1+6.0w2=0.77•PeersareDMUs3and4DMU6DEAsoftware•Freesoftware–EMS–HolgerScheelDEA---Freedownload–DEAP–TimCoelli–Freedownload•Othersoftware–Saitech–Cooper,SeifordandTonesoftware–FrontierAnalysis–WarwickDEA–EMQC-FareandGrosskopf–中文MaxDEALPFormulationHospitalABCDEFGHIJKLDoctors201925272255333130505338Nurses151131160168158255235206244268306284Outpatients10015016018094230220152190250260250Inpatients9050557266908880100100147120Table5.HospitalCasemjmjjsjsjjijirjrxvxvxvyuyuyuxvyu22112211m1is1rEfficiencyofthejthDMU:TheamountofthethoutputofthethDMU:TheamountofthethinputofthethDMU:Theweightassignedtothethoutput:Theweightassignedtothethinputrjijriyrjxijurvimjmjjsjsjjijirjrxvxvxvyuyuyuxvyu22112211m1is1rEfficiencyofthejthDMUNow,assumingthatHospital(B)isevaluatedbyDEA,wehave128438120250113119501501151209010013119501502121212121212121vvuuvvuuvvuuvvuuMaxs.t..0,0,0,028438120250131195015015120901001131195015021212121212121212121vvuuvvuuvvuuvvuuvvuuMaxs.t.EfficiencyofthekthDMU.0v,0u)n~1j(0xvyu1xvyuirm1iijis1rrjrm1iikis1rrkrMaxs.t.基本问题的一般表述DMU1DMU2DMUnv1x11x12x1nv2X21x22X2n……………………vmxm1xm2xmnu1y11y12y1nu2y12y12y1n……………………usys1ys2ysn决策单元评估指标……投入产出……………………………………1.1,212222111211212222111211snssnnmnmmnnyyyyyyyyyYxxxxxxxxxXnsnmsjjjijmjjjijyyyjiyxxxjix,,,DMUththeofinputththeofamountThe:,,,DMUththeofinputththeofamountThe:n),2,1,j(UnitMakingDecisionthjThe:DMU2121j0,,,,0,,,2.1,,2,11subjecttoMaximize212111111122112211smmjmjsjsjmiikisiikimkmkkskskkuuuvvvnjxvxvyuyuxvyuxvxvxvyuyuyu:Theweightassignedtothethoutput12:Theweightassignedtothethinput12riurr,,svii,,mCCRmodel0,,,,0,,,1subjecttoMaximize212111111111smmjmjsjsjmkmkskskuuuvvvxvxvyuyuxvxvyuyunjxvyujRmiijisrrjrk,,1,:1*1****θ,u,v:SolutionOptimalAnR:AReferenceSet0and010subjecttoMaximize1111rimiikisrrjr