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DMOR

DMOR. DEA. Variable Returns to Scale. O 7. O 2. O 6. OR. O 3. O 4. O 2. Constant Returns to Scale. O 1. DEA example. For each bank branch we have one output measure and one input measure. Efficiency. Inputs are changes into outputs. Relative efficiency.

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DMOR

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  1. DMOR DEA

  2. VariableReturns to Scale O7 O2 O6 OR O3 O4 O2 ConstantReturns to Scale O1

  3. DEA example • For each bank branch we have one outputmeasure and one inputmeasure

  4. Efficiency • Inputsarechangesintooutputs

  5. Relativeefficiency • We cancompareallbranchesrelative to Croydon

  6. More outputs

  7. Efficiency • We nowhaevtwo“efficiencies”:

  8. Graphically

  9. Reigate • Personaltransactions per worker2090 • Business transactions per worker1090 • Slope2090/1090=1.92 1.92

  10. Relativeefficiency • Relativeefficiency for Reigate • ForReigate = 36% • ForDorking = 43%

  11. Relativeefficiency • Technicalefficiency • Extendedefficiencydue to Koopmans, Pareto: • A givenentityisfullyefficient, ifno input and no outputcan be improvedwithoutworseningsomeotherinputoroutput. • Relativeefficiency: • A givenentityisefficientbased on theavailableevidence, ifperformance of otherentities do not indicatethatno input and no outputcan be improvedwithoutworseningsomeotherinputoroutput. • Thereis no reference to prices and weights of inputs and outputs. • Youdon’tneed to establishtherelationbetweeninputs and outputs Dominatingentities A desirabledirection

  12. If we knowthatthereis a technology whichenables • producing q0 units of output • using L units of labor and K units of capital according to theprodctionfunction: capital Technicalefficiencydefinition: Produce a givenlevel of outputusingtheminimallevel of inputs labor capital • Then we canmeasureinefficiency: • e.g. supposethatentity A produces q0 units of outputs • ThenOA’/OAisentityA’sefficiency A A’ O labor

  13. DEA approach E, F, G, H, I istheefficientfrontier capital C E • Productionfunctionisoquantis not knowndirectly • DEA estimatesitfromthe data usinginterval-wiselinearinterpolation • Assumethatfirms A, B, C, D, E, F, G, H, I allproduce q0 units of output A B F D G H I O labor

  14. DEA efficiency E, F, G, H, I isefficientfrontier capital C E A B F D A’ G H I O labor • Efficiency of A according to DEA isOA’/OA • A’ is a shadowor a phantom of A • Itis a linearcombination of F and G

  15. Adding a couple of newbranches

  16. AddingbranchF • BranchF has1000 personaltransactions per worker • And 6000 business transactions per worker

  17. Addingbranch G

  18. Theefficient one does not have to win inanycategory

  19. Constantreturns to scale (CCR)– primal problem

  20. F isefficientin a weaksense

  21. Constant and Variablereturns to scale(CRS i VRS): decompositionintoscaleefficiency and puretechnicalefficiency

  22. Primal problem Multiplier model: • Dual problem: Envelopment model:

  23. “Strongdisposal”assumption • Ignorespresence of nonzeroslackvariables • Differentsolutionsmayhavenonzeroslackvariablesor not • Therefore one uses 2 phase of the dual problem to maximizethesevariables (to seewhetherthereexists a solutionwithnonzeroslackvariables)

  24. First and secondphase of the dual problem may be writtentogether and solvedintwosteps

  25. Model Input-oriented Output-oriented

  26. Example: Inputoriented dual problem for P5

  27. Inputorientedprimal problem for P5

  28. Results

  29. Efficientfrontierprojectionininputoriented model

  30. Efficientfrontierprojectioninoutputoriented model

  31. Nextexample

  32. Model BBC (VariableReturns to Scale): Dual problem for DMU5 VariableReturns to Scale Technicalefficiency forDMU5 may be reached forDMU2, whichlies on theefficientfrontier

  33. DMU 4 isweakly DEA efficient 3) The same problem for DMU4 gives:

  34. How to interpretweights? • Assumethat we consider an entitywithefficiency less than 1 • Assumethat, therest of theweightsare zero • Thenthephantominputs of theentityare: • And thephantomoutputs of theentityare:

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