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vii TABLE OF CONTENTS CHAPTER 1 2 TITLE PAGE DECLARATION ii DEDICATION iii ACKNOWLEDGMENT iv ABSTRACT v ABSTRAK vi TABLE OF CONTENTS vii LIST OF TABLES x LIST OF FIGUERS xii LIST OF SYMBOLS xiv LIST OF APPENDIXES xvii INTRODUCTION 1 1.1 Introduction 1 1.2 Research Background 1 1.3 Motivation for Research 4 1.4 Problem Statements 4 1.5 Research Objectives 5 1.6 Research Questions 5 1.7 Significant of the Study 5 1.8 Scope of the Study 6 1.9 Research Methodology 6 1.10 Thesis Organization 6 LITRATURE REVIEW 9 2.1 Introduction 9 2.2 Data Envelopment Analysis 9 viii 3 4 5 6 2.3 Common DEA Models 11 2.4 Super-efficiency DEA Models 15 2.5 Allocation DEA Models 17 2.6 Non-controllable Data 19 2.7 Integer-valued DEA Models 20 2.8 Software for Solving DEA Models 22 2.9 Conclusion 22 ARASH METHOD IN DEA 23 3.1 Introduction 23 3.2 The Shortcomings in DEA Definitions and Techniques 23 3.3 Arash Method (AM) 27 3.4 AM and CF 36 3.5 AM and Non-controllable Data 38 3.6 Non-linear AM 42 3.7 Conclusion 49 KOUROSH METHOD IN DEA 50 4.1 Introduction 50 4.2 Kourosh Model (KM) 50 4.3 KM and Revenue-efficiency 55 4.4 Combining Kourosh and Arash Methods 56 4.5 KAM and Profit-efficiency 60 4.6 Conclusion 64 INTEGER ARASH METHOD 65 5.1 Introduction 65 5.2 Integer DEA Axioms 65 5.3 Integer Arash Model 66 5.4 Examining Integer AM 74 5.5 Conclusion 78 IMPROVED KOUROSH AND ARASH MODEL 79 6.1 Introduction 79 6.2 The Efficient Tape 79 ix 7 6.3 The Targets Regions and Sensitivity Index 81 6.4 Numerical Example 87 6.5 Conclusion 93 SUMMARY AND CONCLUSION 95 7.1 Summary of the Research 95 7.2 Conclusion 96 7.3 Suggestions for Future Research 96 REFERENCES Appendix A 97 102 x LIST OF TABLES TABLE NO. TITLE PAGE 2.1 4 DMUs with two inputs and one output 11 2.2 Some of the previous common CRS DEA Models 14 2.3 Some of the common super-efficiency CRS DEA Models 16 2.4 Some of the allocation CRS DEA Models 19 2.5 Two non-controllable DEA models 19 2.6 Some mixed integer DEA models 21 3.1 The example of 3 DMUs with NFD 24 3.2 The example of 3 DMUs with clearer NFD 25 3.3 The example of 5 DMUs with NFD 26 3.4 The AM scores for DMUs in Tables 3.1, 3.2 and 3.3 32 3.5 17 airports with four inputs and three outputs 33 3.6 The score of conventional DEA models and AM in CRS 34 3.7 The score of conventional DEA models and AM in VRS 35 3.8 Three DMUs with two inputs and two outputs 37 3.9 The differences between 0-AM and CF model scores 38 3.10 The -AM scores by changing 3.11 The example of 12 DMUs with one non-controllable input 39 3.12 The scores of CCR, NCN and -AM (Equation 3.4) 40 3.13 The outcome of 0.00001-AM (Equation 3.5) and 3.14 The slacks of 0.00001errors in input values by AM 3.15 12 DMUs with two inputs and two outputs 43 3.16 The example of 6 DMUs 45 3.17 The example of 18 DMUs 47 3.18 The linear AM scores 48 3.19 The SBM and Non-linear AM scores 48 >0 38 . 40 41 xi 4.1 5 DMUs with one input and two outputs 54 4.2 The differences between 0-KM and RF model scores 56 4.3 The -KM scores by changing 4.4 The example of four DMUs 4.5 The scores of linear and non-linear AM, KM and KAM 60 4.6 The example of 3 DMUs 62 4.7 The scores of PF models and -KAM in VRS 62 4.8 The scores of PF models and -KAM in CRS 63 4.9 The scores of -KAM in CRS 64 5.1 The example of 6 DMUs 67 5.2 The efficiency score and proposed target by integer 0-AM 75 5.3 The outcome of integer 0.02-AM 75 5.4 The outcome of integer 0.00001-AM 77 6.1 The example of five DMUs. 87 6.2 The 0.1-KAM outcomes. 88 6.3 The example of 19 DMUs. 89 6.4 The outcomes of 10 -KAM CRS 91 6.5 >0 The outcomes of 10 -KAM VRS 56 59 91 xii LIST OF FIGURES FIGURE NO. TITLE PAGE 2.1 Radial Approach 12 2.2 Non-Radial Approach 12 2.3 AP method to rank technically efficient DMUs 17 2.4 Cost-efficiency 18 3.1 Three technically efficient DMUs 24 3.2 AP method for evaluating B 24 3.3 Five technically efficient DMUs 26 3.4 AP method to evaluate A 26 3.5 An epsilon error in data of A 28 3.6 Non-radial approach in AM 28 5.1 The integer and real inputs space 68 5.2 2% errors by the scale of ’s inputs 68 5.3 3% errors by the scale of ’s inputs 69 5.4 5% errors by the scale of ’s inputs 69 5.5 6% errors by the scale of ’s inputs 69 5.6 8% errors by the scale of ’s inputs 69 5.7 8% errors by the scale of ’s inputs 70 5.8 10% errors by minimum input scales 70 5.9 The example of eight DMUs 71 5.10 Arash Method outcomes 71 6.1 Production function and VRS PPS 80 6.2 Production function and CRS/VRS PPS 80 6.3 Production frontier and efficient tape 81 6.4 A region of target in efficient tape 81 6.5 An -higher PPS 82 xiii 6.7 T 6.8 T The regions of targets by 0.1-KAM 88 6.9 The regions of targets by 0.25-KAM 88 6.10 10 -KAM CRS/VRS sensitivity indexes 92 6.6 6.11 6.12 frontiers in inputs space 83 frontiers in outputs space 83 The outcomes of 0.1-KAM CRS 92 The outcomes of 0.1-KAM VRS 93 xiv LIST OF SYMBOLS DEA - Data envelopment analysis DMU - Decision making unit CCR - Charnes, Cooper and Rhodes model ADD - Additive model SBM - Slack-based measure model BCC - Banker, Charnes and Cooper model ERM - Enhanced Russell measure model AP - Andersen and Petersen model LP - Linear programming CF - Cost-efficiency model RF - Revenue-efficiency model PF - Profit-efficiency model NCN - Non-controllable model PPS - Production possibility set CRS - Constant returns to scale VRS - Variable returns to scale NFD - Near and far data AM - Arash Method/model KM - Kourosh Method/model KAM - Kourosh and Arash Method/model - Number of DMUs - Number of DMU inputs - Number of DMU outputs - Index of DMUs - Index of DMU inputs - Index of DMU outputs - Index of specific DMU whose efficiency is being assessed - Observed value of input of DMU xv Observed value of output - Multiplier used to compute linear combination of DMUs’ inputs/outputs - Cf Rf Pf of DMU - Non-negative slack or potential reduction of input of DMU Non-negative slack or potential increase of output of DMU Positive specified weight or price for input of DMU Positive specified weight or price for output of DMU Optimal multipliers to identify the reference sets for a DMU Optimal slack to identify an excess utilization of input of DMU Optimal slack to identify a shortage utilization of output Target of input of DMU after evaluation Target of output of DMU after evaluation of DMU - Optimal technical efficiency score in input-oriented approach of CCR - Optimal technical efficiency score in output-oriented approach of CCR - Optimal technical efficiency score of SBM - Optimal technical efficiency score of input-oriented SBM - Optimal technical efficiency score of ERM - Optimal super-efficiency score of SBM - Optimal super-efficiency score of -norm - Optimal cost-efficiency score - Optimal revenue-efficiency score - Optimal profit-efficiency score - A positive real number - A positive real number - The nonnegative integer numbers set - The nonnegative real numbers set - The positive real numbers set - The empty set - A set of corresponding indexes of integer inputs - A set of corresponding indexes of integer outputs - A set of corresponding indexes of controllable data - A set of corresponding indexes of non-controllable data - A nonnegative real number corresponding to input - A nonnegative real number corresponding to output - A very small positive real number - For all xvi - There is at least - Very less than - A very small positive real number - A very small number in - A very small positive real number corresponding to input - A very small positive real number corresponding to output - Optimal efficiency score of AM with degree of freedom - Optimal efficiency score of KM with degree of freedom - Optimal efficiency score of KAM with degree of freedom - The CRS PPS - The CRS PPS with degree of freedom - {0} The efficient tape with 2 diagonals Optimistic efficient target of input of DMU after evaluation Optimistic efficient target of output of DMU after evaluation Pessimistic efficient target of input of DMU after evaluation of DMU after evaluation - Pessimistic efficient target of output -AM - Arash model with -KM - Kourosh model with -KAM - Kourosh and Arash model with degree of freedom degree of freedom degree of freedom xvii LIST OF APPENDICES APPENDIX A TITLE The accepted, proceedings published, presented in conferences and submitted papers PAGE 102