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Performance Evaluation: Network Data Envelopment Analysis 高 強 國立成功大學工業與資訊管理學系 於 中山大學企業管理系 100 年 11 月 5 日
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Contents 1. Efficiency 2. Data Envelopment Analysis 3. Mathematical Models 4. Network Models 5. Research Areas
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1. Efficiency
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Definition Output point of view (Actual output produced)/(Maximal output can be produced) Input point of view (Minimal input required)/(Actual input used) Technically Efficient Production T. Koopmans : A feasible input/output vector where it is technologically impossible to increase any output (and/or reduce any input) without simultaneously reducing another output (and/or increasing any other input). => Pareto optimality
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Measurement Parametric approach Regression analysis (Aigner-Chu) Nonparametric approach Data envelopment analysis (Charnes-Cooper-Rhodes)
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Output Eff. = A/A * Ave. production input output Max. production Input Eff. = I * /I Parametric approach Production function
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Two-input single-output Y0Y0
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Input Eff. = OA * /OA Input efficiency o o o X 1 X 2 O Dominated region ● ● Isoquant (Y 0 )
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Single-input two-output Output Eff.=OA/OA* O 1 * O 1 A A * O 2 * O 2 ● ● ● ● Y 1 Y 2 O Dominated region o o o Product transformation curve (X 0 )
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Example: Production function: unrestricted in sign.
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2. Data Envelopment Analysis
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O A ● ● ● ● ● B C D E Y X E* Production function Single-input single-output output side DMUXY A105 B20 C30 D4035 E3622 Eff. 1 1 1 1 2/3 Non-parametric approach
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Isoquant X 2 X 1 Input side DMUX1X1 X2X2 Y A205100 B303100 C502100 D404100 E483100 Eff. 1 1 1 4/5 5/6
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Production transformation curve Output side DMUXY1Y1 Y2Y2 A1002050 B10040 C1005020 D10030 E1004012 Eff. 1 1 1 3/4 4/5
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Emrouznejad et al. (2008) Socio-economic Planning Science 42, 151-157
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3. Mathematical Models
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Ratio form Input i and output r of DMU j: (X ij, Y rj ) DMU k chooses most favorable multipliers u r,v i to calculate E k
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Linear transformation
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Envelopment form (Dual of the ratio form) ● is the target on the frontier.
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Constant RTS Variable RTS Variable returns-to-scale Technical Eff. = A/A *, Scale Eff. = A * /A 0, Aggregate Eff.=A/A 0 = (A/A * )×(A * /A 0 )
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4. Network Models
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X1kX1k X2kX2k X mk...... Y1kY1k Y2kY2k Y sk...... DMU k Conventional black box concept
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CCR Ratio model
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Envelopment model θ unrestricted in sign
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Two-stage series system Z pj : Intermediate product p of DMU j Process 1 X1kX1k X2kX2k X mk...... DMU k Process 2 Y1kY1k Y2kY2k Y sk...... Z1kZ1k Z2kZ2k Z qk...... System
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Ratio model
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Envelopment model
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… h l Z p (l) p=1,…,q … t Z p (t) p=1,…,q XiXi i=1,…,m YrYr r=1,…,s General case System efficiency is the product of the h process efficiencies.
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More general case
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Ratio model
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Envelopment model
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Parallel system
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Ratio model
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A network system Y 1, Y 2, Y 3 1 2 X 1, X 2 3
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Model
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Efficiencies
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5. Research Areas
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Models I: Increasing marginal product II: Decreasing marginal product III: Negative marginal product- Congestion
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Multipliers Strictly positive : non-Archimedean number , 10 -5 Absolute range Relative range (Assurance region, cone ratio)
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Data type Traditional data Undesirable data Ordinal data Qualitative data Interval data Stochastic data Fuzzy data
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Applications Novel application A new area A new journal Implications Special data type Derivation of multiplier restrictions
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References Chiang Kao and Shiuh-Nan Hwang, 2008, Efficiency decomposition in two-stage data envelopment analysis: An application to non-life insurance companies in Taiwan. European J. Operational Research 185, 418-429. Chiang Kao, 2009, Efficiency decomposition in network data envelopment analysis: A relational model. European J. Operational Research 192, 949-962. Chiang Kao, 2009, Efficiency measurement for parallel production systems. European J. Operational Research 196, 1107-1112. Chiang Kao and Shiuh-Nan Hwang, 2010, Efficiency measurement for network systems: IT impact on firm performance. Decision Support Systems 48, 437-446. Chiang Kao and Shiuh-Nan Hwang, 2011, Decomposition of technical and scale efficien- cies in two-stage production systems. European J. Operational Research 211, 515-519. Chiang Kao, 2011, Efficiency decomposition for parallel production systems. J. Operational Research Society (accepted) (SCI) doi:10.1057/jors.2011.16. Chiang Kao, 2008, A linear formulation of the two-level DEA model. Omega, Int. J. Management Science 36, 958-962. Chiang Kao and Shiang-Tai Liu, 2004, Predicting bank performance with financial forecasts: A case of Taiwan commercial banks. J. Banking & Finance 28, 2353-2368.
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Thank You
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