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DMOR DEA

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O1O1 O2O2 O3O3 O4O4 O2O2 O7O7 O6O6 OR Variable Returns to Scale Constant Returns to Scale

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DEA example For each bank branch we have one output measure and one input measure Branch Personal transactions ('000) No. of workers Croydon12518 Dorking4416 Redhill8017 Reigate2311

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Efficiency Inputs are changes into outputs Branch Personal transactions per worker ('000) Croydon6.94 Dorking2.75 Redhill4.71 Reigate2.09

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Relative efficiency We can compare all branches relative to Croydon BranchRelative efficiency Croydon=6.94/6.94 = 100% Dorking=2.75/6.94 = 40% Redhill=4.71/6.94 = 68% Reigate=2.09/6.94 = 30%

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More outputs Branch Personal transactions ('000) Business transactions ('000) No. of workers Croydon1255018 Dorking442016 Redhill805517 Reigate231211

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Efficiency We now haev two “efficiencies”: Branch Personal transactions per worker ('000) Business transactions per worker ('000) Croydon6.942.78 Dorking2.751.25 Redhill4.713.24 Reigate2.091.09

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Graphically

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Reigate – Personal transactions per worker 2090 – Business transactions per worker 1090 – Slope 2090/1090=1.92 1.92

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Relative efficiency Relative efficiency for Reigate For Reigate = 36% For Dorking = 43%

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Relative efficiency Technical efficiency – Extended efficiency due to Koopmans, Pareto: A given entity is fully efficient, if no input and no output can be improved without worsening some other input or output. – Relative efficiency: A given entity is efficient based on the available evidence, if performance of other entities do not indicate that no input and no output can be improved without worsening some other input or output. There is no reference to prices and weights of inputs and outputs. You don’t need to establish the relation between inputs and outputs Dominating entities A desirable direction

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labor capital If we know that there is a technology which enables producing q 0 units of output using L units of labor and K units of capital according to the prodction function: Technical efficiency definition: Produce a given level of output using the minimal level of inputs labor capital Then we can measure inefficiency: e.g. suppose that entity A produces q 0 units of outputs Then OA’/OA is entity A’s efficiency A A’ O

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labor capital Production function isoquant is not known directly DEA estimates it from the data using interval-wise linear interpolation Assume that firms A, B, C, D, E, F, G, H, I all produce q 0 units of output A O C B D I H G F E E, F, G, H, I is the efficient frontier DEA approach

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DEA efficiency Efficiency of A according to DEA is OA’/OA A’ is a shadow or a phantom of A – It is a linear combination of F and G labor capital A O C B D I H G F E E, F, G, H, I is efficient frontier A’

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Branch Personal transactions per worker ('000) Business transaction s per worker ('000) Croydon6.942.78 Dorking2.751.25 Redhill4.713.24 Reigate2.091.09 A1.232.92 B4.432.23 C3.322.81 D3.702.68 E3.342.96 Adding a couple of new branches

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Adding branch F Branch F has 1000 personal transactions per worker And 6000 business transactions per worker

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Branch Personal transactions per worker ('000) Business transactions per worker ('000) Croydon6.942.78 Dorking2.751.25 Redhill4.713.24 Reigate2.091.09 A1.232.92 B4.432.23 C3.322.81 D3.702.68 E3.342.96 F1.006.00 G5.00 Adding branch G

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The efficient one does not have to win in any category

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Constant returns to scale (CCR)– primal problem

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F is efficient in a weak sense

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Constant and Variable returns to scale (CRS i VRS): decomposition into scale efficiency and pure technical efficiency

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Primal problem Multiplier model: Dual problem: Envelopment model:

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“Strong disposal” assumption – Ignores presence of nonzero slack variables – Different solutions may have nonzero slack variables or not – Therefore one uses 2 phase of the dual problem to maximize these variables (to see whether there exists a solution with nonzero slack variables)

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First and second phase of the dual problem may be written together and solved in two steps

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Model Input-oriented Output-oriented

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Example: Input oriented dual problem for P5

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Input oriented primal problem for P5

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Results

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Efficient frontier projection in input oriented model

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Efficient frontier projection in output oriented model

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Next example

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Model BBC (Variable Returns to Scale): Dual problem for DMU5 Technical efficiency for DMU5 may be reached for DMU2, which lies on the efficient frontier Variable Returns to Scale

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DMU 4 is weakly DEA efficient The same problem for DMU4 gives: 3)

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How to interpret weights? Assume that we consider an entity with efficiency less than 1 Assume that, the rest of the weights are zero Then the phantom inputs of the entity are: And the phantom outputs of the entity are:

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Linear-Programming Applications

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