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

Contents 1. Efficiency 2. Data Envelopment Analysis 3. Mathematical Models 4. Network Models 5. Research Areas

1. Efficiency

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

Measurement Parametric approach Regression analysis (Aigner-Chu) Nonparametric approach Data envelopment analysis (Charnes-Cooper-Rhodes)

Output Eff. ＝ A/A * Ave. production input output Max. production Input Eff. ＝ I * /I Parametric approach Production function

Two-input single-output Y0Y0

Input Eff. ＝ OA * /OA Input efficiency o o o X 1 X 2 O Dominated region ● ● Isoquant (Y 0 )

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 )

Example: Production function: unrestricted in sign.

2. Data Envelopment Analysis

O A ● ● ● ● ● B C D E Y X E* Production function Single-input single-output output side DMUXY A105 B20 C30 D4035 E3622 Eff /3 Non-parametric approach

Isoquant X 2 X 1 Input side DMUX1X1 X2X2 Y A B C D E Eff /5 5/6

Production transformation curve Output side DMUXY1Y1 Y2Y2 A B10040 C D10030 E Eff /4 4/5

Emrouznejad et al. (2008) Socio-economic Planning Science 42,

3. Mathematical Models

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

Linear transformation

Envelopment form (Dual of the ratio form) ● is the target on the frontier.

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 )

4. Network Models

X1kX1k X2kX2k X mk Y1kY1k Y2kY2k Y sk DMU k Conventional black box concept

CCR Ratio model

Envelopment model θ unrestricted in sign

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

Ratio model

Envelopment model

… 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.

More general case

Ratio model

Envelopment model

Parallel system

Ratio model

A network system Y 1, Y 2, Y X 1, X 2 3

Model

Efficiencies

5. Research Areas

Models I: Increasing marginal product II: Decreasing marginal product III: Negative marginal product- Congestion

Multipliers Strictly positive ： non-Archimedean number ， Absolute range Relative range (Assurance region, cone ratio)

Data type Traditional data Undesirable data Ordinal data Qualitative data Interval data Stochastic data Fuzzy data

Applications Novel application A new area A new journal Implications Special data type Derivation of multiplier restrictions

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, Chiang Kao, 2009, Efficiency decomposition in network data envelopment analysis: A relational model. European J. Operational Research 192, Chiang Kao, 2009, Efficiency measurement for parallel production systems. European J. Operational Research 196, Chiang Kao and Shiuh-Nan Hwang, 2010, Efficiency measurement for network systems: IT impact on firm performance. Decision Support Systems 48, Chiang Kao and Shiuh-Nan Hwang, 2011, Decomposition of technical and scale efficien- cies in two-stage production systems. European J. Operational Research 211, Chiang Kao, 2011, Efficiency decomposition for parallel production systems. J. Operational Research Society (accepted) (SCI) doi: /jors Chiang Kao, 2008, A linear formulation of the two-level DEA model. Omega, Int. J. Management Science 36, Chiang Kao and Shiang-Tai Liu, 2004, Predicting bank performance with financial forecasts: A case of Taiwan commercial banks. J. Banking & Finance 28,

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