生产率与效率分析lecture6-数据包络分析1

1、Data Envelopment Analysis (DEA)数据包络分析Over 4000 published articlesfrom 1978 to 20081DEAWhat it isConceptsFarrell measures of Efficiencytechnical allocativescaleRunning DEADangers of DEA 2Data Envelopment Analysis (DEA)This method is a special but important application of linear programming.DEA is wid

2、ely applied to evaluate the performance of Decision Making Units (DMUs) such as private firms, public agencies, schools, and hospitals.3Important Features(1) Efficiency (Productivity) Management(2) Relative Measure vs. Absolute Measure (Social Science) (Natural Science)(3) Multi-Dimensional Analysis

3、 vs. Cost / Benefit Analysis4(1) Relative Comparison(2) Multiple Inputs and Outputs(3) Efficiency Measurement (0%-100%)(4) Avoid the Specification Error between Inputs and Outputs(5) Production/Cost AnalysisImportant Features5Data Envelopment Analysis (DEA)A mathematical optimisation technique where

4、 the unknown variables are the weightsWork through some examples to explainStart with a simple example of 8 similar organisations (decision making units DMUs) producing two outputs using a single input6Table 1. Single Input and Single Output CaseConceptsSalesEmployee0 1 2 3 4 5 6 7 8 96543210Figure

5、1. Comparisons of Branch StoresEfficiencyFrontierABCDFGEH7SalesEmployee0 1 2 3 4 5 6 7 8 9Figure 2. Regression Line vs. Frontier LineEfficiencyFrontierABCDFGEHRegression LineB: the best storeThe efficiency of the other stores are determined by1 = B E D C H A = G F = 0.465432108Table 2. EfficiencySal

6、esEmployee0 1 2 3 443210Figure 3. Improvement of Store AA1ABA29Table 3. Two Inputs and One Output CaseFigure 4. Two Inputs and One Output CaseProduction Possibility SetEfficient FrontiersAEDFCHIGB10012345012345Employee/SalesArea/SalesFigure 5. Improvement of Store AADEA1P11Table 4. One Input and Two

7、 Outputs CaseFigure 6. One Input and Two Outputs CaseProduction Possibility SetDMUInputOut1Out2O1/IO2/I145450451012555281769.63.2375525525774601925763.29.6510010010001106552453304.5675516516533890450360549603603006510563641966.53.511611534152.56.812603301205.5212谁最有效？13W1=1 , W2=0 14W1=1 , W2=0 15W1

8、=0 , W2=1 16W1=0 , W2=1 17W1=0.5 , W2=0.5 18W1=0.5 , W2=0.5 19W1=0.75 , W2=0.25 20W1=0.75 , W2=0.25 21W1=0.25 , W2=0.75 22W1=0.25 , W2=0.75 23样本与实际Solid lines connectinng DMUs 1,2,3,4,5 Represents an efficiency frontier or production possibility curve. Contrast with “theoretical frontier” (broken cu

9、rve) DEA efficiency is relative to the sample but is our best estimate24Efficiencies for all DMUsmust not exceed 100%10.0 w1 + 1.0 w2 =1 unit 19.6 w1 + 3.2 w2 = 1 unit 27.0 w1 + 7.0 w2 = 1 unit 33.2 w1 + 9.6 w2 = 1 unit 41.0 w1 + 10.0 w2 = 1 unit 54.5 w1 + 6.0 w2 = 1 unit 63.2 w1 + 3.2 w2 = 1 unit 7

10、5.0 w1 + 4.0 w2 = 1 unit 86.0 w1 + 5.0 w2 =1 unit 96.5 w1 + 3.5 w2 = 1 unit 102.5 w1 + 6.8 w2 = 1 unit 115.5 w1 + 1.5 w2 = 025生产可能集合的形成26Max 10w1 + 1 w2St 10 w1 + 1 w2 = 1 (only active constraint)And all other constraints inactiveA solution: w1 =0.1, w2 =010w1 + 1w2 = 1.0 (100% efficient)DMU127DMU6M

11、ax 4.5 w1 + 6.0 w2St 7.0 w1 + 7.0 w2 = 1 (active constraint 3)3.2 w1 + 9.6 w2 = 1 (active constraint 4)And all other constraints inactiveUnique solution: w1 =0.05803, w2 =0.08484.5 w1 + 6.0 w2 = 0.77Peers are DMUs 3 and 428DMU629DEA softwareFree software EMS Holger Scheel DEA - Free downloadDEAP Tim

12、 Coelli Free download Other software Saitech Cooper, Seiford and Tone softwareFrontier AnalysisWarwick DEAEMQC - Fare and Grosskopf中文 MaxDEA30LP FormulationTable 5. Hospital CaseEfficiency of the j th DMU31Efficiency of the j th DMU32Now, assuming that Hospital (B) is evaluated by DEA, we haveMaxs.t

13、.33Maxs.t.34Efficiency of the k th DMUMaxs.t.35基本问题的一般表述DMU1DMU2DMUnv1x11x12x1nv2X21x22X2nvmxm1xm2xmnu1y11y12y1nu2y12y12y1nusys1ys2ysn决策单元评估指标投入产出36CCR model37R : A Reference Set38Primal Problem39Dual Problem40Second-stage: Slack41Reference Set:42Table 1.6 : 2 inputs 1 output CaseExample Problem43Pr

14、imal Problem44Dual Problem450DFEFigure 1.7 : Efficiency of DMU AAA1CEfficiency Frontier46Table 1.7 : Results of DEAReference SetDMUEfficiencyA0.8330.1670.1670.833000B0.7270.1820.0910.727000C10.2000.2001000D10.2500.1251000E10.50001000F10112.000Software: EMS Results47BCC modelVariable Returns to Scale

15、48CCR model49BCC model50Dual ProblemBCC model:51(C)0OutputInputEfficiency Frontier of CCR modelFigure 2.1 : Efficiency Frontier and Production Possibility Set (A)(B)Efficiency Frontier of BCC modelbacde52CCR有效与BCC有效OutputInputFrontier of CCR model Frontier of BCC modelbacdPABR53Cost Analysis540Effic

16、iency Frontier of CCR modelEfficiency Frontier of BCC modelEcdekgjihbfP55Dual Problem56An Economic InterpretationDue to Michael Farrell (1957)Technical efficiency = OB/OACapitalLabourMin combinationsrequired (isoquant)ABOOther things equal = output57An Economic Interpretation(Output maximisation ori

17、entation)Technical efficiency = OB/OAOutput 1(e.g. calls)Output 2(e.g. lines)DCOOther things equal = inputsMax combinations achieveable (production frontier)58Economic Interpretation (3)The isoquant and production frontier are not known directly, but might be estimated from known data, using piecewi

18、se interpolationCapitalLabourMin combinationsrequired (isoquant)ABOEFGHJKLB is an artificial observation - a combination of F and G59Allocative efficiencyDepends on knowing pricesAE = min cost/actual cost = OD/ OBCapitalLabourMin combinationsrequired (isoquant)ABOCEfficient Isocost lineD60Scale effi

19、ciencyOutputInputOMAPQRT.E. = PR/PAS.E. = PQ/PRT.& S.E = PQ/PAThis is input orientation. What about output orientation?61Running DEAPurpose - built softwareExcel/Solver macrosOrganise data for inputIdentify inputs, outputs and noncontrollablesRunInterpret 62How reliable is DEA?Depends on whether frontier can be populated by efficient firms:number of observationsnumber of dimensionscloseness t