Data Envelopment Analysis. Second Edition

Transcription

1 Data Envelopment Analysis Second Edition

2 DATA ENVELOPMENT ANALYSIS A Comprehensive Text with Models, Applications, References and DEA-Solver Software Second Edition WILLIAM W. COOPER University of Texas at Austin, U.S.A. LAWRENCE M. SEIFORD University of IVIichigan, U.S.A. KAORU TONE National Graduate Institute for Policy Studies, Japan bprmger

3 William W. Cooper University of Texas, USA Kaoru Tone National Graduate Institute for Policy Studies, Japan Lawrence M. Seiford University of Michigan, USA Library of Congress Control Number: ISBN-10: (HB) ISBN-10: (e-book) ISBN-13: (HB) ISBN-13: (e-book) Printed on acid-free paper by Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science-i-Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights springer.com

4 To Ruth, Bev and Michiko

5 Contents List of Tables List of Figures Preface xvii xxi xxv 1. GEIMERAL DISCUSSION Introduction Single Input and Single Output Two Inputs and One Output Case One Input and Two Outputs Case Fixed and Variable Weights Summary and Conclusion Problem Supplement for Chapter BASIC CCR MODEL Introduction Data The CCR Model From a Fractional to a Linear Program Meaning of Optimal Weights Explanatory Examples Example 2.1 (1 Input and 1 Output Case) Example 2.2 (2 Inputs and 1 Output Case) Illustration of Example Summary of Chapter Selected Bibliography Problem Supplement for Chapter CCR MODEL AND PRODUCTION CORRESPONDENCE Introduction Production Possibility Set The CCR Model and Dual Problem The Reference Set and Improvement in Efficiency 47

6 ii DATA ENVELOPMENT ANALYSIS 3.5 Theorems on CCR-Efficiency Computational Aspects of the CCR Model Computational Procedure for the CCR Model Data Envelopment Analysis and the Data Determination of Weights (=Multipliers) Reasons for Solving the CCR Model Using the Dual {DLPo) Example The Output-Oriented Model An Extension of the Two Phase Process in the CCR Model Discretionary and Non-Discretionary Inputs Summary of Chapter Notes and Selected Bibliography Related DEA-Solver Models for Chapter Problem Supplement for Chapter 3 71 ALTERNATIVE DEA MODELS Introduction The BCC Models The BCC Model The Output-oriented BCC Model The Additive Model The Basic Additive Model Translation Invariance of the Additive Model A Slacks-Based Measure of Efficiency (SBM) Definition of SBM Interpretation of SBM as a Product of Input and Output Inefficiencies Solving SBM SBM and the CCR Measure The Dual Program of the SBM Model Oriented SBM Models A Weighted SBM Model Decomposition of Inefficiency Numerical Example (SBM) A Hybrid Measure of Efficiency (Hybrid) A Hybrid Measure Decomposition of Inefficiency Comparisons with the CCR and SBM Models An Illustrative Example Russell Measure Models Summary of the Basic DEA Models Summary of Chapter Notes and Selected Bibliography Appendix: Free Disposal Hull (FDH) Models Related DEA-Solver Models for Chapter Problem Supplement for Chapter 4 120

7 Contents IX RETURNS TO SCALE Introduction Geometric Portrayals in DEA BCC Returns to Scale CCR Returns to Scale Most Productive Scale Size Further Considerations Relaxation of the Convexity Condition Decomposition of Technical Efficiency Scale Efficiency Mix Efficiency An Example of Decomposition of Technical Efficiency An Example of Returns to Scale Using a Bank Merger Simulation Background Efficiencies and Returns to Scale The Effects of a Merger Summary Additive Models Multiplicative Models and "Exact" Elasticity Summary of Chapter Appendix: FGL Treatment and Extensions Related DEA-Solver Models for Chapter Problem Supplement for Chapter MODELS WITH RESTRICTED MULTIPLIERS Introduction Assurance Region Method Formula for the Assurance Region Method General Hospital Example Change of Efficient Frontier by Assurance Region Method On Determining the Lower and Upper Bounds Another Assurance Region Model Cone-Ratio Method Polyhedral Convex Cone as an Admissible Region of Weights Formula for Cone-Ratio Method A Cone-Ratio Example How to Choose Admissible Directions An Application of the Cone-Ratio Model Negative Slack Values and Their Uses A Site Evaluation Study for Relocating Japanese Government Agencies out of Tokyo Background The Main Criteria and their Hierarchy Structure Scores of the 10 Sites with respect to the 18 Criteria Weights of the 18 Criteria by the 18 Council Members (Evaluators) Decision Analyses using Averages and Medians 201

8 X DATA ENVELOPMENT ANALYSIS Decision Analyses using the Assurance Region Model Evaluation of "Positive" of Each Site Evaluation of "Negative" of Each Site Uses of "Positive" and "Negative" Scores Decision by the Council Concluding Remarks Summary of Chapter Notes and Selected Bibliography Related DEA-Solver Models for Chapter Problem Supplement for Chapter NON-DISCRETIONARY AND CATEGORICAL VARIABLES Introduction Examples Non-controllable, Non-discretionary and Bounded Variable Models Non-controllable Variable (NCN) Model An Example of a Non-Controllable Variable Non-discretionary Variable (NDSC) Model Bounded Variable (BND) Model An Example of the Bounded Variable Model DBA with Categorical DMUs An Example of a Hierarchical Category Solution to the Categorical Model Extension of the Categorical Model Comparisons of Efficiency between Different Systems Formulation Computation of Efficiency Illustration of a One Input and Two Output Scenario Rank-Sum Statistics and DEA Rank-Sum-Test (Wilcoxon-Mann-Whitney) Use of the Test for Comparing the DEA Scores of Two Groups Use of the Test for Comparing the Efficient Frontiers of Two Groups Bilateral Comparisons Using DEA An Example of Bilateral Comparisons in DEA Evaluating Efficiencies of Different Organization Forms Summary of Chapter Notes and Selected Bibliography Related DEA-Solver Models for Chapter Problem Supplement for Chapter ALLOCATION MODELS Introduction Overall Efficiency with Common Prices and Costs Cost Efficiency Revenue Efficiency Profit Efficiency An Example 261

9 Contents XI 8.3 New Cost Efficiency under Different Unit Prices A New Scheme for Evaluating Cost Efficiency Differences Between the Two Models An Empirical Example Extensions Decomposition of Cost Efficiency Loss due to Technical Inefficiency Loss due to Input Price Inefficiency Loss due to Allocative Inefficiency Decomposition of the Actual Cost An Example of Decomposition of Actual Cost Summary of Chapter Notes and Selected Bibliography Related DEA-Solver Models for Chapter Problem Supplement for Chapter DATA VARIATIONS Introduction Sensitivity Analysis Degrees of Freedom Algorithmic Approaches Metric Approaches Multiplier Model Approaches Statistical Approaches Chance-Constrained Programming and Satisficing in DEA Introduction Satisficing in DEA Deterministic Equivalents Stochastic Efficiency Summary of Chapter SUPER-EFFICIENCY MODELS Introduction Radial Super-efficiency Models Non-radial Super-efficiency Models Definition of Non-radial Super-efficiency Measure Solving Super-efficiency Input/Output-Oriented Super-efficiency An Example of Non-radial Super-efficiency Extensions to Variable Returns-to-Scale Radial Super-efficiency Case Non-radial Super-efficiency Case Summary of Chapter Notes and Selected Bibliography Related DEA-Solver Models for Chapter Problem Supplement for Chapter

10 Xli DATA ENVELOPMENT ANALYSIS 11. EFFICIENCY CHANGE OVER TIME Introduction Window Analysis An Example Application Analysis Malmquist Index Dealing with Panel Data Catch-up Effect Frontier-shift Effect Malmquist Index The Radial Ml The Non-radial and Slacks-based Ml The Non-radial and Non-oriented Ml Scale Efficiency Change Illustrative Examples for Model Comparisons Concluding Remarks Summary of Chapter Notes and Selected Bibliography Related DEA-Solver Models for Chapter SCALE ELASTICITY AND CONGESTION Introduction Scale Elasticity in Production Congestion Strong Congestion Weak Congestion Summary of Degree of Scale Economies and Congestion Illustrative Examples Degree of Scale Economies and Strong Congestion Weak vs. Strong Congestion Summary of Chapter Notes and Selected Bibliography Related DEA-Solver Models for Chapter Problem Supplement for Chapter UNDESIRABLE OUTPUTS MODELS Introduction An SBM with Undesirable Outputs An Undesirable Output Model Dual Interpretations Returns-to-scale (RTS) Issues Imposing Weights to Inputs and/or Outputs Non-separable 'Good' and 'Bad' Output Model Illustrative Examples Separable Bad Outputs Models An Example with Both Separable and Non-separable Inputs/Outputs 375

11 Contents Xlil 13.5 Comparisons with Other Methods Summary of Chapter Related DEA-Solver Models for Chapter ECONOMIES OF SCOPE AND CAPACITY UTILIZATION Introduction Economies of Scope Definition Checking for Economies of Scope Checking a Virtual Merger Comparisons of Business Models Illustrative Examples Capacity Utilization Fixed vs. Variable Input Resources Technical Capacity Utilization Price-Based Capacity Utilization Measure Long-Run and Short-Run Capacity Utilization Illustrative Examples Summary of Chapter Notes and Selected Bibliography Related DEA-Solver Models for Chapter Problem Supplement for Chapter A DEA GAME Introduction Formulation Coalition and Characteristic Function Solution Coalition and Individual Contribution The Shapley Value DEA min Game Summary of Chapter Notes and Selected Bibliography Problem Supplement for Chapter MULTI-STAGE USE OF PARAMETRIC AND NON-PARAMETRIC MODELS Introduction OLS Regressions Modification and Extensions Stochastic Frontier Analysis and Composed Error Models DEA and Regression Combinations Multi-stage DEA-Regression Combinations and its Application to Japanese Banking Introduction The Multistage Framework An Application to Japanese Banking 433

12 xiv DATA ENVELOPMENT ANALYSIS Discussion Summary of This Case Study Summary of Chapter Notes and Selected Bibliography 439 Appendices 443 A-Linear Programming and Duality 443 A.l Linear Programming and Optimal Solutions 443 A.2 Basis and Basic Solutions 443 A.3 Optimal Basic Solutions 444 A.4 Dual Problem 445 A.5 Symmetric Dual Problems 446 A.6 Complementarity Theorem 447 A.7 Farkas' Lemma and Theorem of the Alternative 448 A.8 Strong Theorem of Complementarity 449 A.9 Linear Programming and Duality in General Form 451 B-Introduction to DEA-Solver 454 B.l Platform 454 B.2 Installation of DEA-Solver 454 B.3 Notation of DEA Models 454 B.4 Included DEA Models 456 B.5 Preparation of the Data File 456 B.5.1 The CCR, BCC, IRS, DRS, GRS, SBM, Super-Efficiency, Scale Elasticity, Congestion and FDH Models 456 B.5.2 The AR Model 457 B.5.3 The ARC Model 458 B.5,4 The NCN and NDSC Models 459 B.5.5 The BND Model 460 B.5.6 The CAT, SYS and Bilateral Models 460 B.5.7 The Cost and New-Cost Models 461 B.5.8 The Revenue and New-Revenue Models 462 B.5.9 The Profit, New-Profit and Ratio Models 462 B.5.10 The Window and Malmquist Models 462 B.5.11 The Hybrid Model 463 B.5.12 Weighted SBM Model 464 B.5.13 The Bad Outputs Model 465 B.5.14 The Non-separable Outputs Model 465 B.6 Starting DEA-Solver 466 B.7 Results 466 B.8 Data Limitations 473 B.8.1 Problem Size 473 B.8.2 Inappropriate Data for Each Model 474 B.9 Sample Problems and Results 475 B.IO Summary 475 B.10.1 Models that Require Numbers to be Supplied through Keyboard 475

13 Contents XV B.10.2 Summary of Headings to Inputs/Outputs 475 C-Bibliography 477 index 479 Index 483

14 List of Tables 1.1 Single Input and Single Output Case EfBciency Two Inputs and One Output Case One Input and Two Outputs Case Hospital Case Comparisons of Fixed vs. Variable Weights Optimal Weights for Hospitals A and B Example Results of Example Example Results of Example Primal and Dual Correspondences Example Results of Example Problem for Phase III Process CCR-Score, Reference Set, Slacks and % Change State-mandated Excellence Standards on Student Outcomes Non-Discretionary Inputs Worksheets Containing Main Results Data and Scores of 5 Stores Optimal Weights and Slacks CCR-projection in Input and Output Orientations Primal and Dual Correspondences in BCC Model Data and Results of Example Data and Results of CCR and SBM A Comparison: Hybrid, CCR and SBM Measures of Inefficiency: (Hybrid) Summary of Model Characteristics Decomposition of Efficiency Score Decomposition of Technical Efficiency Data of 11 Regional and 9 City Banks* 157

15 XVlil DATA ENVELOPMENT ANALYSIS 5.3 EfRciencies and Returns to Scale Weights, Slacks and Projections Efficiency of Projected and Merged Banks Results of Input-oriented/Output-oriented BCC Cases Data for 14 Hospitals Efficiency and Weight of 14 Hospitals by CCR Model Efficiency and Weight of 14 Hospitals with Assurance Region Method Efficiency of 14 Hospitals by CR (Cone-Ratio) and CCR Models Number of Bank Failures (through ) Inputs and Outputs CCR and Cone-Ratio Efficiency Scores (1984, 1985)* Printout for Cone-Ratio CCR Model - Interstate Bank of Fort Worth, Scores (Sij) of 10 Sites (A-J) with respect to 18 Criteria (Cl- C18) Statistics of Weights assigned the 18 Criteria (C1-C18) by 18 Council Members Averages and Medians of Scores of the 10 Sites Data for Public Libraries in Tokyo Efficiency of Libraries by CCR and NCN Data of 12 Japanese Baseball Teams in Projection of Attendance by CCR and Bounded Models Categorization of Libraries Nine DMUs with Three Category Levels Comparisons of Stores in Two Systems Comparisons of Two Systems Example of Bilateral Comparisons Sample Data for Allocative Efficiency Efficiencies Comparison of Traditional and New Scheme Data for 12 Hospitals New Data Set and EfRciencies Decomposition of Actual Cost Data for a Sensitivity Analysis Initial Solutions Results of 5% Data Variations OLS Regression Estimates without Dummy Variables Stochastic Frontier Regression Estimates without Dummy Variables OLS Regression Estimates without Dummy Variables on DEAefficient DMUs Stochastic Frontier Regression Estimates without Dummy Variables on DEA-efficient DMUs 297

16 LIST OF TABLES xix 10.1 Test Data Andersen-Petersen Ranking* Non-radial Super-efRciency Data for Super-efRciency Super-efRciency Scores under Variable RTS Super-efRciency Scores under Constant RTS Window Analysis: 56 DMUs in U.S. Army Recruitment Battalions 3 Outputs - 10 Inputs Example Input-oriented Scores Catch-up, Frontier-shift and Malmquist Index Example Example Comparisons Example Results by the Non-oriented Non-radial Model Example Example BCC-0 Results Congestion Results Separable Bad Outputs Case: Data set Separable Bad Outputs Case: Results Non-Separable Inputs/Outputs Case: Data Set Non-Separable Inputs/Outputs Case: Decomposition of Inefficiency Data for Groups 1 and 2 as Speciahzed and Group 3 as Diversified Firms Twenty Virtual Diversified Firms Efficiency Score and Degree of Economies of Scope Data for 12 Hospitals Technical Capacity Utilization Price-Based Data Set Profits and Losses Comparisons of Current and Maximum Profits Score Matrix Division of Reward based on Fixed Weights Optimal Rewards with Optimal Weights Normalized Score Matrix Coalition and Characteristic Function (1) Coalition and Characteristic Function (2) Each Member's Marginal Contribution to Coalitions The Shapley Value Division of Reward based on Shapley Value Single Players' Values for the Min Game Case CoaHtions' Values for the Min Game Case (1) 416

17 DATA ENVELOPMENT ANALYSIS Coalitions'Values for the Min Game Case (2) The Shapley Value for the Min Game Case Data of 3 Shops Single Players'Values of the Market Arcade Game Coalitions'Values of the Market Arcade Game The Shapley Value of the Market Arcade Game Characteristic Function Values subject to the AR Constraints The Shapley Value for the AR Case Stochastic Frontier Estimation Results Comparison of the Initial and Final Efficiency Scores 437 A.l Symmetric Primal-Dual Problem 447 A.2 General Form of Duality Relation 452 B.l Window Analysis by Three Adjacent Years 469 B.2 Sample Data of Scale Elasticity 471 B.3 Models Requiring Numbers Through Keyboard 475 B.4 Headings to Inputs/Outputs 476

18 List of Figures 1.1 Comparisons of Branch Stores Regression Line vs. Frontier Line Improvement of Store ^ Two Inputs and One Output Case Improvement of Store A One Input and Two Outputs Case Improvement Excel File "HospitalAB.xls" Example Example Region P The Case of DMU A A Scatter Plot and Regression of DEA and Engineering Efficiency Ratings Envelopment Map for 29 Jet Engines A Comparative Portrayal and Regression Analysis for 28 Engines (Ah Data Except for Engine 19) Production Possibility Set Example Production Frontier of the CCR Model Production Frontiers of the BCC Model The BCC Model The Additive Model Translation in the BCC Model Translation in the Additive Model FDH Representation Translation in the CCR Model Returns to Scale Returns to Scale: One Output-One Input Most Productive Scale Size The IRS Model 150

19 ii DATA ENVELOPMENT ANALYSIS 5.5 The DRS Model The GRS Model Scale Efficiency Merger is Not Necessarily Efficient Efficiency Scores with and without Assurance Region Constraints Assurance Region Convex Cone generated by Two Vectors The Hierarchical Structure for the Capital Relocation Problem Positives and Negatives of the 10 Sites One Input Exogenously Fixed, or Non-Discretionary DMUs with Controllable Category Levels Comparisons between Stores using Two Systems Bilateral Comparisons-Case Bilateral Comparisons-Case Case Case Controllable Input and Output Categorization (1) Categorization (2) Technical, Allocative and Overall Efficiency Decomposition of Actual Cost Stable and Unstable DMUs A Radius of Stability The Unit Isoquant Spanned by the Test Data in Table Catch-up Scale Efficiency Change Three DMUs in Two Periods Scale Elasticity Congestion Diversified and Specialized Firms Diversified Firms D and Virtual Diversified Firms V Firms with Two Different Scopes Price-Based Capacity Utihzation Decomposition of the Maximum Profit Price-Based Capacity Utilization with a Variable Output Downward Trend in Mean Measured Efficiency by Static DEA Model Comparison of Mean Efficiency between Initial and Final Estimates 438 B.l Sample.xls in Excel Sheet 457 B.2 Sample-AR.xls in Excel Sheet 458 B.3 Sample-ARG.xls in Excel Sheet 459 B.4 Sample-NCN (NDSC).xls in Excel Sheet 459 B.5 Sample-BND.xls in Excel Sheet 460

20 LIST OF FIGURES xxili B.6 Sample-CAT.xls in Excel Sheet 461 B.7 Sample-Cost (New-Cost).xls in Excel Sheet 461 B.8 Sample-Revenue(New-Revenue).xls in Excel Sheet 462 B.9 Sample-Window(Malmquist).xls in Excel Sheet 463 B.IO Sample-Hybrid.xls in Excel Sheet 464 B.ll Sample-Weighted SBM.xls in Excel Sheet 464 B.12 Sample-BadOutput.xls in Excel Sheet 465 B.13 Sample-NonSeparable.xls in Excel Sheet 466 B.14 A Sample Score Sheet 467 B.15 A Sample Graph2 468 B.16 Variations through Window 469 B.17 Variations by Term 470 B.18 Sample Malmquist Index 470 B.19 Malmquist Index (Graph) 471 B.20 Worksheets "Elasticity" 472 B.21 Worksheets "Congestion" 472 B.22 Worksheets "Decomposition" 473