*947*
*62*
*18MB*

*English*
*Pages [324]*
*Year 2021*

- Author / Uploaded
- Alireza Alinezhad
- Seyyed Hamed Mirtaleb
- Javad Khalili

*Table of contents : ContentsPrefaceAbout the AuthorsChapter 1Review of Data Envelopment Analysis Models 1.1. Introduction 1.2. Methods of Efficiency Measurement 1.2.1. Parametric Methods 1.2.2. Non-Parametric Methods 1.3. Data Envelopment Analysis (DEA) 1.4. Production Possibility Set (PPS) 1.5. Production Technology 1.6. DEA Models 1.6.1. CCR Model 1.6.2. BCC Model 1.6.3. Free Disposal Hull (FDH) Model 1.6.4. Radial Model 1.6.5. Directional Distance Function (DDF) Model 1.6.6. Additive Model 1.6.7. Non-Controllable Variable (NCN) Model 1.6.8. Undesirable Inputs/Outputs Model 1.6.9. Generalized DEA Model 1.6.10. Window Analysis Model 1.6.11. Malmquist Index 1.6.12. Slacks-Based Measure (SBM) Model 1.6.13. Assurance Region (AR) Model 1.6.14. Bounded Variable (BND) Model 1.6.15. Categorical Variable (CAT) Model 1.6.16. Bilateral Comparison Model 1.6.17. Enhanced Russell Measure (ERM) Model 1.6.18. Congestion Model 1.6.19. Cost and New Cost Efficiency Models 1.6.20. Revenue and New Revenue Efficiency Models 1.6.21. Profit and New Profit Efficiency Models 1.6.22. Ratio Efficiency Model 1.6.23. Scale Elasticity Model 1.6.24. Epsilon-Based Measure (EBM) Model 1.6.25. Weak Disposability Model 1.6.26. Metafrontier Model 1.6.27. Dynamic DEA Model 1.6.28. Network DEA Model 1.6.29. Context Dependent Model 1.6.30. Bootstrapping Model ConclusionChapter 2DEA Excel Software 2.1. Introduction 2.2. Software Presentation 2.2.1. Settings 2.2.2. BCC and CCR Models 2.2.3. Additive Model 2.2.4. NCN Model 2.2.5. Undesirable Input/Output Model 2.3. Running the DEA Excel Software ConclusionChapter 3DEA Frontier Software 3.1. Introduction 3.2. Software Presentation 3.2.1. Envelopment Model 3.2.2. Multiplier Model 3.2.3. SBM Model 3.2.4. Measure-Specific Model 3.3. Running the DEA Frontier Software ConclusionChapter 4DEA Solver Software 4.1. Introduction 4.2. Software Presentation 4.2.1. Models 4.2.2. Data File Preparation 4.2.2.1. CCR, BCC, IRS, DRS, GRS, SBM, FDH, EBM, Scale Elasticity, and Congestion Models 4.2.2.2. AR Model 4.2.2.3. AR Global Model 4.2.2.4. Super-Efficiency Model 4.2.2.5. NCN and NDSC Models 4.2.2.6. BND Model 4.2.2.7. Bilateral, System and CAT Models 4.2.2.8. Cost Efficiency and New Cost Efficiency Models 4.2.2.9. Revenue Efficiency and New Revenue Efficiency Models 4.2.2.10. Profit Efficiency and New Profit Efficiency Models 4.2.2.11. Malmquist-Radial and Malmquist, Window Analysis Models 4.2.2.12. Scale Elasticity Model 4.2.2.13. Congestion Model 4.2.2.14. Undesirable Outputs Model 4.2.2.15. Weighted SBM Model 4.2.2.16. Hybrid Model 4.2.2.17. Network DEA (Network SBM) Model 4.2.2.18. Dynamic DEA (Dynamic SBM) Model 4.2.2.19. Dynamic and Network SBM (DNSBM) Model 4.2.2.20. Non-Convex Model 4.2.2.21. Resampling Model 4.2.2.22. DDF Model 4.2.3. Types of Results for Worksheets 4.2.3.1. Score 4.2.3.2. Projection 4.2.3.3. Weight 4.2.3.4. Weighted Data 4.2.3.5. Summary 4.2.3.6. Slack 4.2.3.7. Graph 1 4.2.3.8. Graph 2 4.2.3.9. RTS 4.2.3.10. Malmquist k 4.2.3.11. Scale Elasticity 4.2.3.12. Congestion 4.2.3.13. Resample Summary 4.2.3.14. Window k 4.2.3.15. Decomposition 4.2.3.16. SASvsCRSvsVRS 4.3. Running the DEA Solver Software 4.3.1. Score 4.3.2. Projection 4.3.3. Weight 4.3.4. Weighted Data 4.3.5. Summary 4.3.6. Slack 4.3.7. Graph 1 4.3.8. Graph 2 4.3.9. RTS 4.3.10. Malmquist k 4.3.11. Scale Elasticity 4.3.12. Congestion 4.3.13. Resample Summary 4.3.14. Window k 4.3.15. Decomposition 4.3.16. SASvsCRSvsVRS ConclusionChapter 5DEAP Software 5.1. Introduction 5.2. Software Presentation 5.2.1. Executable File (DEAP.EXE) 5.2.2. Data File (EG-dta.txt) 5.2.3. Instruction File (EG-ins.txt) 5.2.4. Results File (EG-out.txt) 5.3. Running the DEAP Software ConclusionChapter 6EMS Software 6.1. Introduction 6.2. Software Presentation 6.2.1. Data File 6.2.2. Final Results 6.3. Running the EMS Software ConclusionChapter 7Frontier Analyst Software 7.1. Introduction 7.2. Software Presentation 7.2.1. Toolbar 7.2.1.1. Home 7.2.1.2. Source Data 7.2.1.3. Analysis 7.2.1.4. Result 7.2.1.5. Graph 7.2.1.6. Report 7.2.1.7. Translation 7.2.2. Data Entry 7.2.2.1. Transfer Data from Excel 7.2.2.2. Transfer Data from the Clipboard 7.2.2.3. Link to External Database 7.2.2.4. Use SPSS Software Information 7.2.2.5. Import Data from the Text File 7.2.2.6. Enter Data Manually 7.2.2.7. Import Data via Web Files 7.3. Running the Frontier Analyst Software 7.3.1. Potential Improvement 7.3.2. Reference Comparison 7.3.3. Input/Output Contributions 7.3.4. Reference Contributions 7.3.5. Efficiency Scores 7.3.6. Problem Solving Process Diagram 7.3.7. X-Y Plot 7.3.8. Efficiency Plot 7.3.9. Efficient Frontier Plot ConclusionChapter 8Pioneer Software 8.1. Introduction 8.2. Software Presentation 8.2.1. Software Models 8.2.2. Data Entry 8.2.3. Final Results 8.3. Running the Pioneer Software ConclusionChapter 9MaxDEA Software 9.1. Introduction 9.2. Software Presentation 9.2.1. Toolbar 9.2.1.1. Prepare Data 9.2.1.2. Run Model 9.2.1.2.1. Envelopment Models 9.2.1.2.1.1. Measuring Efficiency Based on Distance 9.2.1.2.1.2. Efficiency Measurement Based on Orientation 9.2.1.2.1.3. Efficiency Measurement Based on the RTS 9.2.1.2.1.4. Efficiency Measurement Based on Frontier 9.2.1.2.1.5. Advanced Models 1 9.2.1.2.1.6. Advanced Models 2 9.2.1.2.1.7. Results for Envelopment Models 9.2.1.2.1.8. Options 9.2.1.2.2. Multiplier Models 9.2.1.2.2.1. Measuring Efficiency Based on Distance 9.2.1.2.2.2. Efficiency Measurement Based on Orientation 9.2.1.2.2.3. Efficiency Measurement Based on the RTS 9.2.1.2.2.4. Advanced Models 1 and 2 9.2.1.2.2.5. Bootstrapping Model 9.2.1.2.2.6. Results 9.2.1.2.2.7. Options 9.2.1.3. Final Results 9.2.1.4. Frontier Graphs 9.2.1.5. Help 9.2.1.6. Software Language Setting 9.2.1.7. Editing 9.2.1.8. Filtering and Sorting 9.3. Running the MaxDEA Software ConclusionChapter 10MDeap Software 10.1. Introduction 10.2. Software Presentation 10.2.1. Toolbar 10.2.1.1. File 10.2.1.2. Edit 10.2.1.3. Unit 10.2.1.4. Variable 10.2.1.5. Model 10.2.1.6. View 10.2.1.7. Help 10.3. Running the MDeap Software ConclusionChapter 11PIM Software 11.1. Introduction 11.2. Software Presentation 11.2.1. Toolbar 11.2.1.1. File 11.2.1.2. View 11.2.1.3. Data 11.2.1.4. Run 11.2.1.5. Models 11.2.1.5.1. Model Selection 11.2.1.5.2. Final Results 11.2.1.6. Help 11.2.1.7. Project Explorer 11.2.2. Data Entry 11.3. Running the PIM Software ConclusionChapter 12DEAOS Web-Based Software 12.1. Introduction 12.2. Software Presentation 12.2.1. The Main Tools 12.2.2. Data Entry 12.2.3. Problem Solving 12.2.4. Final Results 12.3. Running the DEAOS Software ConclusionChapter 13Software Packages Comparison 13.1. Introduction 13.2. Software Packages Comparison ConclusionReferencesIndexBlank PageBlank Page*

MATHEMATICS RESEARCH DEVELOPMENTS

INTRODUCTION AND COMPARISON OF DATA ENVELOPMENT ANALYSIS SOFTWARE PACKAGES

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MATHEMATICS RESEARCH DEVELOPMENTS Additional books and e-books in this series can be found on Nova’s website under the Series tab.

MATHEMATICS RESEARCH DEVELOPMENTS

INTRODUCTION AND COMPARISON OF DATA ENVELOPMENT ANALYSIS SOFTWARE PACKAGES ALIREZA ALINEZHAD SEYYED HAMED MIRTALEB AND

JAVAD KHALILI

Copyright © 2021 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. We have partnered with Copyright Clearance Center to make it easy for you to obtain permissions to reuse content from this publication. Simply navigate to this publication’s page on Nova’s website and locate the “Get Permission” button below the title description. This button is linked directly to the title’s permission page on copyright.com. Alternatively, you can visit copyright.com and search by title, ISBN, or ISSN. For further questions about using the service on copyright.com, please contact: Copyright Clearance Center Phone: +1-(978) 750-8400 Fax: +1-(978) 750-4470 E-mail: [email protected].

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Published by Nova Science Publishers, Inc. † New York

To: Our families

CONTENTS Preface

ix

About the Authors

xi

Chapter 1

Review of Data Envelopment Analysis Models

1

Chapter 2

DEA Excel Software

41

Chapter 3

DEA Frontier Software

49

Chapter 4

DEA Solver Software

61

Chapter 5

DEAP Software

101

Chapter 6

EMS Software

139

Chapter 7

Frontier Analyst Software

149

Chapter 8

Pioneer Software

171

Chapter 9

MaxDEA Software

189

Chapter 10

MDeap Software

227

Chapter 11

PIM Software

239

Chapter 12

DEAOS Web-Based Software

267

Chapter 13

Software Packages Comparison

285

viii

Contents

References

291

Index

303

PREFACE Performance measurement is done by various methods, one of which is Data Envelopment Analysis (DEA). Due to the ability of DEA models to meet practical requirements, extensive research can be conducted in the fields of mathematics, management, economics, and engineering. Therefore, during recent years, the use of this method has been considered with significant growth among researchers. DEA evaluates the performance of Decision Making Units (DMUs) by using linear programming. Since linear programming should be solved for each DMU, performance measurement for a large number of DMUs is difficult and time-consuming. For this purpose, various software packages have been designed and developed to address these problems in recent decades. As a result, the need to use different software in this area has increased day by day as different DEA models have grown among engineering communities and various organizations. Most of this software includes basic models and advanced models. Each of this software is designed for different purposes and has different features and applications in DEA research. Researchers can select the appropriate software based on the required model and results. The main objectives behind writing this book are to introduce, express the advantages and disadvantages of each of these software packages, as well as their comparison.

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Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

First, the basic concepts and models which are available in software packages are reviewed. Then, 11 DEA software packages are reviewed to know how to use them, and finally, the software is compared according to different criteria. The present book can be used by students and professors of mathematics, industrial engineering, and management, as well as used by researchers, experts, and managers who study and analyze the performance and productivity in different areas. We gratefully acknowledge those who have contributed to the compilation of this book, and it is hoped that this book would be useful for readers, researchers, and managers. Alireza Alinezhad Seyyed Hamed Mirtaleb Javad Khalili

ABOUT THE AUTHORS

Alireza Alinezhad is an Iranian researcher who received his B.Sc. degree in Applied Mathematics from Iran University of Science and Technology, M.Sc. degree in Industrial Engineering from Tarbiat Modarres University, and Ph.D. degree in Industrial Engineering, from Islamic Azad University, Science and Research Branch. He is currently an Associate Professor in the Department of Industrial Engineering, Islamic Azad University, Qazvin Branch, Iran. His researches are including Data Envelopment Analysis (DEA), Multiple Criteria Decision Making (MCDM), and quality engineering and management.

xii

About the Authors

Seyyed Hamed Mirtaleb has been educated at the Qazvin Islamic Azad University, Iran, where he received his B.Sc. and M.Sc. degrees in Industrial Engineering in 2011, 2015, respectively. The title of his master thesis is "Development of control charts in terms of uncertainty using multi-objective decision making and data envelopment analysis". His research interests are primarily in the areas of Data Envelopment Analysis (DEA), Multiple Criteria Decision Making (MCDM), and quality engineering and management.

Javad Khalili has M.Sc. in Industrial Engineering from the Islamic Azad University of Qazvin. He received his B.Sc. degree in the field of Industrial Engineering - Industrial Production in 2012 and his M.Sc. degree in the field of Industrial Engineering - System Management and Productivity from Islamic Azad University of Qazvin, Iran, in 2017. His master's thesis is entitled

About the Authors

xiii

"Performance evaluation in aggregate production planning by integrated approach of DEA and MADM under uncertain condition". His researches are including Multiple Criteria Decision Making (MCDM), Data Envelopment Analysis (DEA), Supply Chain Management (SCM), and production planning.

Chapter 1

REVIEW OF DATA ENVELOPMENT ANALYSIS MODELS 1.1. INTRODUCTION In general, efficiency means the degree and quality of attaining the desired set of goals. Efficiency is a part of productivity and is defined in various ways. Nonetheless, in a simple term, it involves the output-input ratio in a system. On the other hand, efficiency is considered as the ratio of the minimum possible cost to the realized cost to provide a certain amount of output in comparison with other units. Scale efficiency, technical efficiency, allocative efficiency are various types of efficiency, all of which are used to maximize production with a certain cost or minimize cost with a certain level of production and their results are to maximize the profit of the Decision Making Unit (DMU).

1.2. METHODS OF EFFICIENCY MEASUREMENT There are different methods of measuring efficiency, which can be categorized into parametric and non-parametric methods.

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Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

1.2.1. Parametric Methods Parametric methods were first presented in economics with the production function simultaneously. In this method, a specific production function is estimated by using various statistical and econometric techniques, and then efficiency can be determined by using this function [1].

1.2.2. Non-Parametric Methods These methods do not require recognizing and estimating the production function and there is no limitation on the number of inputs and outputs. The main difference between the non-parametric and parametric methods is that the parametric methods calculate the efficiency of the units by estimating the production function. However, non-parametric methods form efficient frontier and consider the distance from DMU to the efficient frontier to calculate efficiency.

1.3. DATA ENVELOPMENT ANALYSIS (DEA) Data Envelopment Analysis (DEA) was first introduced in 1978 by Charnes et al. [2]. DEA is used as a non-parametric method to calculate the performance of units [3]. This method separates efficient units from inefficient units, examines and recognizes the causes of inefficiency. DEA performs optimizations on individual observations on the limit points of each DMU and measures the optimal performance of each DMU. In other words, the DEA focuses on optimizing individual observations and examines the status of each observation. In fact, the amount of efficiency is calculated for each of the observed DMUs, which may be above or below the efficiency limit. The DEA could examine the function of units against competitors, and decide for a better future based on the results [4].

Review of Data Envelopment Analysis Models

3

Furthermore, DMUs are units which convert inputs to outputs. Each DMU𝑗 , 𝑗 = 1, … , 𝑛, uses 𝑥𝑖𝑗 , 𝑖 = 1, … , 𝑚, inputs to generate 𝑦𝑟𝑗 , 𝑟 = 1, … , 𝑠, outputs [5]. Input is a factor by which efficiency decreases by increasing it and maintaining all other factors, while efficiency increases by decreasing it and maintaining other factors [6]. The output is a factor by which the efficiency increases by increasing it and maintaining all other factors, while efficiency decreases by decreasing it and maintaining other factors. The advantages of DEA over parametric methods are as follows: • • • • •

Focus on each observation versus the population mean Use of multiple inputs and outputs Simultaneously Compatibility with exogenous variables No need to know the type of distribution function Ability to use inputs and outputs with different measurement scales

1.4. PRODUCTION POSSIBILITY SET (PPS) Production Possibility Set (PPS) can be introduced equivalent to DMUs with the ability to be created, which is defined by Eq. (1) [7]. T = {(𝑥̅ , 𝑦̅) ∈ R𝑚 × R𝑠 |y̅ can be produced from x̅}

(1)

After introducing the PPS, its frontier, which actually represents the maximum output for a combination of input patterns, can be considered as a production function. This frontier is called the efficient frontier and the efficiency of each DMU relative to the feasible units is calculated on this frontier. The efficient frontier covers the whole set of production possibilities, and the DEA is adapted from this theme. Since the set does not have the real PPS, an experimental production set is possible to be created by using the principles of the subject.

Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

4

1.5. PRODUCTION TECHNOLOGY The relationship between input and output of production, service, or enterprise system can be expressed as Eq. (2). 𝑦𝑗 = 𝑓(𝜆𝑥1𝑗 , 𝜆𝑥2𝑗 , ⋯ , 𝜆𝑥𝑚𝑗 ); 𝜆, ℎ > 1

(2)

In production technology, the concept of Return To Scale (RTS) is an economic concept and indicates the amount of output which can be obtained from input changes. The types of production technology can be summarized in Table 1 and Figure 1. Table 1. Types of production technology [8, 9] Points bc oa , ab cd , de

Attribute ℎ=𝜆 ℎ𝜆

RTS Constant Return to Scale (CRS) Increasing Return to Scale (IRS) Decreasing Return to Scale (DRS)

Figure 1. Different types of RTS [8, 9].

If a DMU exhibits all three CRS, IRS, and DRS behaviors over a time period, its RTS is a Variable Return to Scale (VRS).

Review of Data Envelopment Analysis Models

5

1.6. DEA MODELS To evaluate the efficiency of DMUs, two perspectives can be used by focusing on inputs and outputs. In the input-oriented model, a unit is inefficient if it is possible to reduce any of the inputs without increasing the other inputs or reducing any of the outputs [10]. The input-oriented approach occurs when inputs are controlled. In addition, regarding an output-oriented model, a unit is inefficient if it is possible to increase each output without increasing one input or decreasing another. The output-oriented approach occurs when the outputs are controlled [11]. It is considered an efficient unit if neither of the two mentioned factors is met. The efficiency of less than one for one unit means that the linear combination of other units can produce the same amount of output using fewer inputs. In the following, the classic DEA models are introduced according to input-oriented and output-oriented approaches. Nowadays, this method is widely used to evaluate efficiency in various fields. Various models have been proposed to calculate efficiency; each of them is briefly reviewed.

1.6.1. CCR Model CCR model was provided by Charans, Cooper, and Rhodes in 1978 [12]. The CCR multiplier model (primal) with the CRS and input-oriented is defined as Eq. (3) [10]. 𝑠

𝑚𝑎𝑥 𝑧𝑝 = ∑ 𝑢𝑟 𝑦𝑟𝑝 𝑠

𝑟=1

𝑚

𝑠. 𝑡: ∑ 𝑢𝑟 𝑦𝑟𝑗 − ∑ 𝑣𝑖 𝑥𝑖𝑗 ≤ 0 ; 𝑗 = 1, … , 𝑝, … , 𝑛 𝑟=1

𝑖=1

(3)

Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

6

𝑚

∑ 𝑣𝑖 𝑥𝑖𝑝 = 1 𝑖=1

𝑢𝑟 , 𝑣𝑖 ≥ 0 ; 𝑖 = 1, … , 𝑚 , 𝑟 = 1, … , 𝑠 Moreover, the CRS and output-oriented of the CCR multiplier model are in the form of Eq. (4) [10]. 𝑚

𝑚𝑖𝑛 𝑧𝑝 = ∑ 𝑣𝑖 𝑥𝑖𝑝 𝑚

𝑖=1

(4)

𝑠

𝑠. 𝑡: ∑ 𝑣𝑖 𝑥𝑖𝑗 − ∑ 𝑢𝑟 𝑦𝑟𝑗 ≥ 0 ; 𝑗 = 1, … , 𝑝, … , 𝑛 𝑠

𝑖=1

𝑟=1

∑ 𝑢𝑟 𝑦𝑟𝑝 = 1 𝑟=1

𝑢𝑟 , 𝑣𝑖 ≥ 0; 𝑖 = 1, … , 𝑚 , 𝑟 = 1, … , 𝑠 The Eq. (5), with CRS and output-oriented, is related to the CCR envelopment model (dual) [10]. 𝑚𝑖𝑛 θ𝑝

(5)

𝑛

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 ≤ θ𝑝 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 ≥ 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1

𝜆𝑗 ≥ 0 , θ𝑝 𝑓𝑟𝑒𝑒 ; 𝑗 = 1, … , 𝑛 The CCR envelopment model, with the CRS and output-oriented, is in relation to Eq. (6) [10].

Review of Data Envelopment Analysis Models 𝑚𝑎𝑥 ∅𝑝

7 (6)

𝑛

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 ≤ 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 ≥ ∅𝑝 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1

𝜆𝑗 ≥ 0 , ∅𝑝 𝑓𝑟𝑒𝑒 ; 𝑗 = 1, … , 𝑛

1.6.2. BCC Model In 1984, Banker, Charnes, and Cooper introduced the BCC model [13]. The BCC multiplier model is introduced by the VRS and the input-oriented in Eq. (7) [14]. 𝑠

𝑚𝑎𝑥 𝑧𝑝 = ∑ 𝑢𝑟 𝑦𝑟𝑝 + 𝑢′ s

𝑟=1

(7)

m

𝑠. 𝑡: ∑ 𝑢𝑟 𝑦𝑟𝑗 − ∑ 𝑣𝑖 𝑥𝑖𝑗 +𝑢' ≤ 0 ; 𝑗 = 1, … , 𝑝, … , 𝑛 𝑚

r=1

i=1

∑ 𝑣𝑖 𝑥𝑖𝑝 = 1 𝑖=1

𝑢𝑟 , 𝑣𝑖 ≥ 0 , 𝑢′ 𝑓𝑟𝑒𝑒 ; 𝑖 = 1, … , 𝑚 , 𝑟 = 1, … , 𝑠 Also, the BCC multiplier model with the VRS and output-oriented is in the form of Eq. (8) [14]. 𝑚

𝑚𝑖𝑛 𝑧𝑝 = ∑ 𝑣𝑖 𝑥𝑖𝑝 + 𝑣 ′ 𝑖=1 𝑚

𝑠

𝑠. 𝑡: ∑ 𝑣𝑖 𝑥𝑖𝑗 − ∑ 𝑢𝑟 𝑦𝑟𝑗 + 𝑣 ′ ≥ 0 ; 𝑗 = 1, … , 𝑝, … , 𝑛 𝑖=1

𝑟=1

(8)

Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

8 𝑠

∑ 𝑢𝑟 𝑦𝑟𝑝 = 1 𝑟=1

𝑢𝑟 , 𝑣𝑖 ≥ 0 , 𝑣 ′ 𝑓𝑟𝑒𝑒 ; 𝑖 = 1, … , 𝑚 , 𝑟 = 1, … , 𝑠 The Eq. (9), with the VRS and input-oriented, is related to the BCC envelopment model [14, 15]. 𝑚𝑖𝑛 θ𝑝

(9)

𝑛

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 ≤ θ𝑝 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 ≥ 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1 𝑛

∑ 𝜆𝑗 = 1 𝑗=1

𝜆𝑗 ≥ 0 , 𝜃𝑝 𝑓𝑟𝑒𝑒 ; 𝑗 = 1, … , 𝑛 The VRS and output-oriented of the BCC envelopment model are proposed in Eq. (10) [16]. 𝑚𝑎𝑥 ∅𝑝 𝑛

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 ≤ 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 ≥ ∅𝑝 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1 𝑛

∑ 𝜆𝑗 = 1 𝑗=1

𝜆𝑗 ≥ 0 , ∅𝑝 𝑓𝑟𝑒𝑒 ; 𝑗 = 1, … , 𝑛

(10)

Review of Data Envelopment Analysis Models

9

1.6.3. Free Disposal Hull (FDH) Model Free Disposal Hull (FDH) model was presented by Deprins in 1984 [17]. The difference between the CCR model and FDH is that the FDH model does not limit itself to convexity, and this FDH orientation seems attractive. It is easier to remove the convexity condition since it is difficult to find a justification for establishing the convexity condition in the PPS. Therefore, 𝜆𝑗 , 𝑗 = 1, … , 𝑛, can participate in the formation of the virtual unit (the share of participation of each unit is one or zero). This problem can be expressed in the case of VRS and input-oriented as mixed programming in Eq. (11) [18]. 𝑚𝑖𝑛 θ𝑝

(11)

𝑛

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 ≤ θ𝑝 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 ≥ 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1 𝑛

∑ 𝜆𝑗 = 1 𝑗=1

𝜆𝑗 ∈ {0,1} ; 𝑗 = 1, … , 𝑛 The VRS and output-oriented approaches of the FDH model is defined in the form of Eq. (12) [18]. 𝑚𝑎𝑥 ∅𝑝 𝑛

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 ≤ 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 ≥ ∅𝑝 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1

(12)

10

Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili 𝑛

∑ 𝜆𝑗 = 1 𝑗=1

𝜆𝑗 ∈ {0,1} ; 𝑗 = 1, … , 𝑛 In model (12), the components of 𝜆𝑗 are constrained to bivalent and θ is a free and continuous variable.

1.6.4. Radial Model Anderson and Petersen provided the radial model in 1993 [19, 20]. Basic DEA models do not allow the comparison of these units easily due to the lack of full rankings between efficient units because all efficient DMUs are assigned as an efficient score of one in these models, and the need to rank efficient units and maintain the inefficiency of inefficient units is inevitable. In the AP (Anderson-Petersen) model, the unit under consideration is removed from the evaluation, which causes the efficiency score assigned to the efficient units to be equal and larger to one in the AP model, and the ranking between the efficient units takes place as well. The only difference between the supper-efficiency model and the standard models is the addition of conditions 𝑗≠𝑝 to the constraints, where DMU𝑝 is not considered in the reference set. For instance, the radial CCR multiplier model, with the CRS and input-oriented approaches, is in the form of Eq. (13) [21]. 𝑠

𝑚𝑎𝑥 𝑧𝑝 = ∑ 𝑢𝑟 𝑦𝑟𝑝 𝑚

𝑟=1

𝑠. 𝑡: ∑ 𝑣𝑖 𝑥𝑖𝑝 = 1 𝑖=1

(13)

Review of Data Envelopment Analysis Models 𝑠

11

𝑚

∑ 𝑢𝑟 𝑦𝑟𝑗 − ∑ 𝑣𝑖 𝑥𝑖𝑗 ≤ 0 ; 𝑗 = 1, … , 𝑛 , 𝑗 ≠ 𝑝 𝑟=1

𝑖=1

𝑢𝑟 , 𝑣𝑖 ≥ 0 ; 𝑖 = 1, … , 𝑚 , 𝑟 = 1, … , 𝑠 n

Regarding VRS envelopment models, a constraint

𝜆𝑗 = 1 should

j= 1 j p

be added to the model (13). The radial BCC envelopment model is proposed by the VRS and the input-oriented approaches by Eq. (14) [21]. 𝑚𝑖𝑛 θ𝑝

(14)

𝑛

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 ≤ θ𝑝 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑗≠𝑝 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 ≥ 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1 𝑗≠𝑝 𝑛

∑ 𝜆𝑗 = 1 𝑗=1 𝑗≠𝑝

𝜆𝑗 ≥ 0 , 𝜃𝑝 𝑓𝑟𝑒𝑒; 𝑗 = 1, … , 𝑛 , 𝑗 ≠ 𝑝

1.6.5. Directional Distance Function (DDF) Model In 1996, Chambers et al. introduced the Directional Distance Function (DDF) model [22]. In this model, it is assumed that there are n units of DMU with m input and s output. i input and r output are considered as 𝑥𝑖𝑗 and 𝑦𝑟𝑗 , respectively. The DDF model allows the analyst to choose a direction to move the inefficient point across the frontier. It is also assumed (𝑔 𝑥 , 𝑔 𝑦 ) ∈ R𝑚+𝑠 with (𝑔 𝑥 , 𝑔 𝑦 ) ≠ 0 to be in the desired direction + for (𝑥𝑝 , 𝑦𝑝 ). In this case, the DDF model can be written as Eq. (15) [23].

12

Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili ̅ T (𝑥𝑝 , 𝑦𝑝 ; 𝑔 𝑥 , 𝑔 𝑦 ) = 𝛽 𝑚𝑎𝑥 D

(15)

𝑛

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 + 𝛽𝑔𝑖𝑥 ≤ 𝑥𝑖𝑝 ; 𝑖 = 1, ⋯ , 𝑚 𝑗=1 𝑛 𝑦

∑ 𝜆𝑗 𝑦𝑟𝑗 − 𝛽𝑔𝑟 ≥ 𝑦𝑟𝑝 ; 𝑟 = 1, ⋯ , 𝑠 𝑗=1 𝑛

∑ 𝜆𝑗 = 1 𝑗=1

𝜆𝑗 ≥ 0 ; 𝑗 = 1, … , 𝑛 In Eq. (15), G = (𝑔 𝑥 , 𝑔 𝑦 ) makes the projection of (𝑥𝑝 , 𝑦𝑝 ) on the efficient frontier with the (−𝑔 𝑥 , 𝑔 𝑦 ) direction. Charnes et al. expressed the DDF standard model as Eq. (16) by choosing the direction of (𝑔 𝑥 , 𝑔 𝑦 ) = (−𝑥𝑖𝑝 , 𝑦𝑟𝑝 ) [24]. ̅ T (𝑥𝑝 , 𝑦𝑝 ; 𝑔 𝑥 , 𝑔 𝑦 ) = 𝛽 𝑚𝑎𝑥 D

(16)

𝑛

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 ≤ (1 − 𝛽)𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 ≥ (1 − 𝛽)𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1 𝑛

∑ 𝜆𝑗 = 1 𝑗=1

𝜆𝑗 ≥ 0 ; 𝑗 = 1, … , 𝑛

1.6.6. Additive Model The additive model was presented in 1985 by Charnes et al. [25]. In input-oriented models, while maintaining the number of outputs at a given level, it acts proportionally as much as possible to reduce the number of

Review of Data Envelopment Analysis Models

13

inputs and increases the outputs proportionally by maintaining the amount of input in output-oriented models. In the additive model, the reduction of inputs and the increase of outputs are considered, simultaneously. This model in the envelopment form is as Eq. (17) [26]. 𝑚

𝑚𝑎𝑥

𝑧𝑝 = ∑ 𝑠𝑖− 𝑖=1 𝑛

𝑠

+ ∑ 𝑠𝑟+

(17)

𝑟=1

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 + 𝑠𝑖− =𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 − 𝑠𝑟+ = 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1

𝜆𝑗 , 𝑠𝑖− , 𝑠𝑟+ ≥ 0 ; 𝑖 = 1, … , 𝑚 , 𝑟 = 1, … , 𝑠 , 𝑗 = 1, … , 𝑛

1.6.7. Non-Controllable Variable (NCN) Model In 1986, Banker and Morey provided a model of DEA with noncontrollable input/output [27, 28]. One of the most important features of DEA methods is that the inputs and outputs can be controlled, but, in reality, the inputs and outputs of the system may be dependent due to environmental factors or factors beyond its control. This type of data is called non-controllable. The Eq. (18) is related to the model with noncontrollable inputs [29]. 𝑚

𝑚𝑖𝑛 Z =

θ𝑝 – 𝜀 (∑ 𝑠𝑖− 𝑖=1

s

+ ∑ s+r ) r=1

𝑛

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 + 𝑠𝑖− = θ𝑝 𝑥𝑖𝑝 ; 𝑖 ∈ CI 𝑗=1 𝑛

∑ 𝜆𝑗 𝑥𝑖𝑗 + 𝑠𝑖− = 𝑥𝑖𝑝 ; 𝑖 ∉ CI 𝑗=1

(18)

14

Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 − 𝑠𝑟+ = 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1

𝜆𝑗 ≥ 0 , 𝑠𝑖− , 𝑠𝑟+ ≥ 0 ; 𝑖 = 1, … , 𝑚 , 𝑟 = 1, … , 𝑠 , 𝑗 = 1, … , 𝑛 Eq. (19) is related to the models with non-controllable outputs [29]. 𝑚

𝑚𝑎𝑥 𝑔 =

𝑠

∅𝑝 + 𝜀 (∑ 𝑠𝑖− 𝑖=1

+ ∑ 𝑠𝑟+ )

(19)

𝑟=1

𝑛

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 + 𝑠𝑖− = 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 − 𝑠𝑟+ = ∅𝑝 𝑦𝑟𝑝 ; 𝑟 ϵ CO 𝑗=1 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 − 𝑠𝑟+ = 𝑦𝑟𝑝 ; 𝑟 ∉ CO 𝑗=1

𝜆𝑗 ≥ 0 , 𝑠𝑖− , 𝑠𝑟+ ≥ 0 ; 𝑖 = 1, … , 𝑚 , 𝑟 = 1, … , 𝑠 , 𝑗 = 1, … , 𝑛

1.6.8. Undesirable Inputs/Outputs Model In 1989, Fare et al. introduced the model with undesirable inputs/outputs [30]. Traditional DEA models are based on the premise that reducing inputs and increasing outputs improves performance. In practice, it should be noted that organizations do not always seek to maximize outputs and minimize input, and inputs or outputs can be desirable (good) and undesirable (bad). For example, the number of defective goods is an undesirable output which should be reduced to improve performance. Eq. (20) is related to the model with undesirable inputs [31]. 𝑚

𝑚𝑖𝑛 θ𝑝 – ε

(∑ 𝑠𝑖− 𝑖=1

𝑠

+ ∑ 𝑠𝑟+ ) 𝑟=1

(20)

Review of Data Envelopment Analysis Models

15

𝑛

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 + 𝑠𝑖− = θ𝑝 𝑥𝑖𝑝 ; 𝑖 ∈ DI 𝑗=1 𝑛

∑ 𝜆𝑗 𝑥 ′ 𝑖𝑗 + 𝑠𝑖− = θ𝑝 𝑥 ′ 𝑖𝑝 ; 𝑖 ∉ DI 𝑗=1 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 − 𝑠𝑟+ = 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1

𝜆𝑗 ≥ 0 , 𝑠𝑖− , 𝑠𝑟+ ≥ 0 ; 𝑖 = 1, … , 𝑚 , 𝑟 = 1, … , 𝑠 , 𝑗 = 1, … , 𝑛 In Eq. (20), there is 𝑥 ′ 𝑖𝑝 = 𝑥 𝑚𝑎𝑥 𝑖𝑗 − 𝑥𝑖𝑝 , 𝑖 ∉ DI. Likewise, Eq. (21) is related to the model with undesirable outputs [31].

𝑚

𝑚𝑎𝑥 𝑔 = 𝑛

∅𝑝 + 𝜀 (∑ 𝑠𝑖− 𝑖=1

𝑠

+ ∑ 𝑠𝑟+ )

(21)

𝑟=1

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 + 𝑠𝑖− = 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 − 𝑠𝑟+ = ∅𝑝 𝑦𝑟𝑝 ; 𝑟 𝜖 DO 𝑗=1 𝑛

∑ 𝜆𝑗 𝑦 ′ 𝑟𝑗 − 𝑠𝑟+ = ∅𝑝 𝑦 ′ 𝑟𝑝 ; 𝑟 ∉ DO 𝑗=1

𝜆𝑗 ≥ 0 , 𝑠𝑖− , 𝑠𝑟+ ≥ 0 ; 𝑖 = 1, … , 𝑚 , 𝑟 = 1, … , 𝑠 , 𝑗 = 1, … , 𝑛 Where 𝑦 ′ 𝑟𝑝 = 𝑦 𝑚𝑎𝑥 𝑟𝑗 − 𝑥𝑟𝑝 , 𝑟 ∉ DO.

1.6.9. Generalized DEA Model Generalized DEA model was provided by Zhan-xin in 2002 [32, 33]. The purpose of the generalized DEA model is to separate the "evaluated

16

Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

unit set" from the reference set. Basic DEA models evaluate the associated efficiency between DMUs, and the reference set is calculated as the same unit set, while the generalized DEA is a comparison between the evaluated unit and the reference set. Eq. (22) shows the generalized state of the inputoriented BCC envelopment model [34]. 𝑚𝑖𝑛 θ𝑝

(22)

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 ≤ θ𝑝 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗∈T

∑ 𝜆𝑗 𝑦𝑟𝑗 ≥ 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗∈T

𝜆𝑗 ≥ 0 ; 𝑗 = 1, … , 𝑛 , 𝑗 ∈ T, 𝑖 ∈ V Where T represents the reference set and V indicates the evaluated unit set.

1.6.10. Window Analysis Model Window analysis model was presented in 1984 by Charnes et al. [35, 36]. This method solves the problem of insufficient observations in time assessments by enabling the combination of observations in time and cross-sectional series [37]. This method works based on the moving average and is useful for finding the performance trends of a unit over time. Each unit in a different period is treated as an independent unit. In this case, the performance of a unit in a particular period is evaluated against the performance of that unit in other periods, in addition to the performance of other units. This situation increases the number of periods studied in the analysis, which is useful when studying small-scale samples. Determining the width of the window (d) or the number of time periods represents a range of concurrent analyzes and includes only observations of a time period to cross-sectional analyzes of observations from all studied periods [38]. The choice of window analysis imposes constraints on the internal analysis of the firm structure. On the other hand, the flexibility of the

Review of Data Envelopment Analysis Models

17

window analysis model is low compared to the models which can measure efficiency by assuming the VRS and are usually estimated by assuming the CRS. If efficiency measurement is proposed to be based on the CRS, according to the scale efficiency as well as VRS, efficiency is not measurable and assumed constant. The window analysis model makes it possible to observe the trend of firm efficiency change over time. This feature can be used to evaluate the performance of firms to increase productivity. In this method, the relationship 𝑑 ≤ 𝑝 is true and the number of windows can be obtained based on 𝑤 = 𝑝 − 𝑑 + 1. Also, the number of DMUs in each window is equal to nd. To show this model, it is assumed that n units of DMU exist in t, 𝑡 = 1, … , T, time period and all of them use m inputs to produce s outputs. Therefore, the sample contains the observed T × N, and the unit n in period 𝑛 𝑛 𝑛 t (DMU𝑡𝑛 ) has one vector of m dimensions of the inputs (X 1𝑡 , X 2𝑡 , ⋯ , X 𝑚𝑡 ) 𝑛 𝑛 𝑛 and is one vector of s dimensions of the outputs Y1𝑡 , Y2𝑡 , ⋯ , Y𝑠𝑡 . The window starts from time K (1≤K ≤T) and has width W (1≤ W≤ T−K) which is shown by KW and has N ×W observations. The matrix of inputs and outputs for window analysis can be shown in the vectors of Eq. (23), respectively [39]. 1 2 N 1 N 1 N X KW = (X K , XK , ⋯ , XK , X K + 1, ⋯ , X K + 1, X K + W, ⋯ , X K + W) (23)

YKW = (YK1 , YK2 , ⋯ , YKN , YK1 + 1, ⋯ , YKN + 1, YK1 + W, ⋯ , YKN + W) The input-oriented window analysis model for DMU𝑡𝑛 as the CRS is written as Eq. (24) [40]. 𝑚𝑖𝑛 θ𝑝

(24)

𝑠. 𝑡: 𝜆 𝑥𝐾𝑊 ≤ θ𝑝 𝑥 𝜆𝑦𝐾𝑊 ≥ 𝑦

𝑡

𝑡

; 𝑡 = 1, … , T

; 𝑡 = 1, … , T

𝜆𝑗 ≥ 0 ; 𝑗 = 1, … , 𝑛

18

Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

1.6.11. Malmquist Index This index was introduced by Malmquist in 1953 [41, 42]. The Malmquist index calculates the efficiency changes of a DMU between two time periods, which is defined as the product of Catch up (C) and Frontier shift (F) terms by the Eq. (25) [43]. 2

1

MI = C × F = [

𝛿 ((𝑥𝑝 , 𝑦𝑝 ) ) 1

𝛿 1 ((𝑥𝑝 , 𝑦𝑝 ) )

2

×

2

𝛿 ((𝑥𝑝 , 𝑦𝑝 ) ) 1 ] 𝛿 2 ((𝑥𝑝 , 𝑦𝑝 ) )

1 2

(25)

The term of efficiency change refers to the amount of DMU effort to improve efficiency, while the term frontier shift reflects the change in DMU efficient frontier by considering two time periods. In other words, the MPI method is considered as the effect of efficiency change or technical efficiency change, as well as frontier or technological shift [44]. The values of C and F are also determined by using Eq. (26) [45].

𝐶=

𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 𝑜𝑓(𝑥2𝑝,𝑦2𝑝 ) 𝑤𝑖𝑡ℎ 𝑟𝑒𝑠𝑝𝑒𝑐𝑡 𝑡𝑜 𝑝𝑒𝑟𝑖𝑜𝑑 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 𝑜𝑓(𝑥1𝑝,𝑦1𝑝 ) 𝑤𝑖𝑡ℎ 𝑟𝑒𝑠𝑝𝑒𝑐𝑡 𝑡𝑜 𝑝𝑒𝑟𝑖𝑜𝑑 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 2

=

𝛿 2 ((𝑥𝑝 , 𝑦𝑝 ) ) 1

𝛿 1 ((𝑥𝑝 , 𝑦𝑝 ) )

𝐹 = √𝜑1 𝜑2 𝜑1 =

𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 𝑜𝑓(𝑥1𝑝,𝑦1𝑝 ) 𝑤𝑖𝑡ℎ 𝑟𝑒𝑠𝑝𝑒𝑐𝑡 𝑡𝑜 𝑝𝑒𝑟𝑖𝑜𝑑 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 𝑜𝑓(𝑥1𝑝,𝑦1𝑝 ) 𝑤𝑖𝑡ℎ 𝑟𝑒𝑠𝑝𝑒𝑐𝑡 𝑡𝑜 𝑝𝑒𝑟𝑖𝑜𝑑 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 1

=

𝜑2 =

𝛿 1 ((𝑥𝑝 , 𝑦𝑝 ) ) 1

𝛿 2 ((𝑥𝑝 , 𝑦𝑝 ) ) 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 𝑜𝑓(𝑥2𝑝,𝑦2𝑝 ) 𝑤𝑖𝑡ℎ 𝑟𝑒𝑠𝑝𝑒𝑐𝑡 𝑡𝑜 𝑝𝑒𝑟𝑖𝑜𝑑 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 𝑜𝑓(𝑥2𝑝,𝑦2𝑝 ) 𝑤𝑖𝑡ℎ 𝑟𝑒𝑠𝑝𝑒𝑐𝑡 𝑡𝑜 𝑝𝑒𝑟𝑖𝑜𝑑 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛

(26)

Review of Data Envelopment Analysis Models

19

2

=

𝛿 1 ((𝑥𝑝 , 𝑦𝑝 ) ) 2

δ2 ((𝑥𝑝 , 𝑦𝑝 ) ) 1

𝐹=[

1

𝛿 ((𝑥𝑝, 𝑦𝑝) ) 1

𝛿2 ((𝑥𝑝, 𝑦𝑝) )

2

1

×

𝛿 ((𝑥𝑝, 𝑦𝑝 ) ) 2

1 2

]

𝛿2 ((𝑥𝑝, 𝑦𝑝 ) )

To calculate the Malmquist index in the input-oriented, it is required to solve four programming problems based on Eq. (27) [45]. D𝑡 (𝑥𝑝𝑡 , 𝑦𝑝𝑡 ) = 𝑚𝑖𝑛 θ𝑝

(27)

𝑛 𝑡 𝑡 𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 ≤ θ𝑝 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛 𝑡 𝑡 ∑ 𝜆𝑗 𝑦𝑟𝑗 ≤ 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1

𝜆𝑗 ≥ 0 ; 𝑗 = 1, … , 𝑛 𝑡 𝑡 In Eq. (27), 𝑥𝑖𝑗 and 𝑦𝑟𝑗 are the input and output matrixes for period t,

respectively. Then, the efficiency of the problem for DMU𝑝 in time of 𝑡 + 1 (D𝑡+1 (𝑥𝑝𝑡+1 , 𝑦𝑝𝑡+1 )) D

𝑡

(𝑥𝑝𝑡+1 , 𝑦𝑝𝑡+1 )

is

obtained.

Furthermore,

the

value

of

for DMU𝑝 , which is the distance of DMU𝑝 at the time of

𝑡 + 1 with t frontier, is obtained by using the linear programming problem (28) [45]. D𝑡 (𝑥𝑝𝑡+1 , 𝑦𝑝𝑡+1 ) = 𝑚𝑖𝑛 θ𝑝 𝑛 𝑡 𝑡+1 𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 ≤ θ𝑝 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛 𝑡 𝑡+1 ∑ 𝜆𝑗 𝑦𝑟𝑗 ≤ 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1

𝜆𝑗 ≥ 0 ; 𝑗 = 1, … , 𝑛

(28)

20

Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili Similarly, D𝑡+1 (𝑥𝑝𝑡 , 𝑦𝑝𝑡 ) is calculated based on Eq. (29). D𝑡+1 (𝑥𝑝𝑡 , 𝑦𝑝𝑡 ) = 𝑚𝑖𝑛 θ𝑝

(29)

𝑛 𝑡+1 𝑡 𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 ≤ θ𝑝 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛 𝑡+1 𝑡 ∑ 𝜆𝑗 𝑦𝑟𝑗 ≤ 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1

𝜆𝑗 ≥ 0 ; 𝑗 = 1, … , 𝑛

1.6.12. Slacks-Based Measure (SBM) Model In 2001, Tone developed the Slacks-Based Measure (SBM) [46]. This model is one of the types of additive models and provides a scalar number as efficiency for each unit. Further, the objective function is stable concerning the unit change. That is, efficiency will not change if instead of 𝑥𝑖𝑗 and 𝑦𝑟𝑗 to be put 𝑘𝑗 𝑥𝑖𝑗 and 𝑐𝑗 𝑦𝑟𝑗 , respectively (𝑘 and 𝑐 are positive numbers). The non-oriented and CRS approach of the SBM model is defined as Eq. (30) [47]. m

1 1−𝑚

𝑠𝑖− 𝑥𝑖𝑝

1 1+𝑠

s

𝑠𝑟+ 𝑦𝑟𝑝

𝑚𝑖𝑛 𝑝 =

i =1

r =1

𝑛

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 + 𝑠𝑖− = 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 − 𝑠𝑟+ = 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1

𝜆𝑗 , 𝑠𝑖− , 𝑠𝑟+ ≥ 0 ; 𝑖 = 1, … , 𝑚 , 𝑟 = 1, … , 𝑠 , 𝑗 = 1, … , 𝑛

(30)

Review of Data Envelopment Analysis Models

21

The SBM multiplier model is proposed as Eq. (31) [47]. 𝑠

𝑚

𝑚𝑎𝑥 ∑ 𝑢𝑟 𝑦𝑟𝑝 − ∑ 𝑣𝑖 𝑥𝑖𝑝 𝑟=1 𝑠

(31)

𝑖=1 𝑚

𝑠. 𝑡: ∑ 𝑢𝑟 𝑦𝑟𝑗 − ∑ 𝑣𝑖 𝑥𝑖𝑗 ≤ 0 ; 𝑗 = 1, … , 𝑛 𝑟=1

𝑖=1

1 1 𝑣𝑖 ≥ [ ] ; 𝑖 = 1, … , 𝑚 𝑚 𝑥𝑖𝑝 m

1−

s

𝑣𝑖 𝑥𝑖𝑝 +

i= 1

𝑢𝑟 ≥

r=1

𝑠

𝑢𝑟 𝑦𝑟𝑝

1 [ ] ; 𝑟 = 1, … , 𝑠 𝑦𝑟𝑝

𝑣𝑖 , 𝑢𝑟 ≥ 0 ; 𝑖 = 1, … , 𝑚 , 𝑟 = 1, … , 𝑠 Eq. (32), with the CRS and input-oriented, is related to the SBM model [47]. 𝑚

1 𝑠𝑖− 𝑚𝑖𝑛 𝑝 = 1 − ∑ 𝑚 𝑥𝑖𝑝

(32)

𝑖=1

𝑛

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 + 𝑠𝑖− = 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 − 𝑠𝑟+ = 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1

𝜆𝑗 , 𝑠𝑖− , 𝑠𝑟+ ≥ 0 ; 𝑖 = 1, … , 𝑚 , 𝑟 = 1, … , 𝑠 , 𝑗 = 1, … , 𝑛 Likewise, the output-oriented and CRS of the SBM model will be in the form of Eq. (33) [48].

22

Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili 1

𝑚𝑖𝑛 𝑝 = 1 1+𝑠

(33)

s

r= 1

𝑠𝑟+ 𝑦𝑟𝑝

𝑛

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 + 𝑠𝑖− = 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 − 𝑠𝑟+ = 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1

𝜆𝑗 , 𝑠𝑖− , 𝑠𝑟+ ≥ 0 ; 𝑖 = 1, … , 𝑚 , 𝑟 = 1, … , 𝑠 , 𝑗 = 1, … , 𝑛 The weighted SBM model is defined as Eq. (34) [47]. 1 1−𝑚 𝑚𝑖𝑛 𝑝 = 1 1+𝑠

m

i= 1 s

r=1

𝑤𝑖− 𝑠𝑖− 𝑥𝑖𝑝 (34) 𝑤𝑟+ 𝑠𝑟+ 𝑦𝑟𝑝

𝑛

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 + 𝑠𝑖− = 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 − 𝑠𝑟+ = 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1 𝑚

∑ 𝑤𝑖− = 1 𝑖=1 𝑠

∑ 𝑤𝑟+ = 1 𝑟=1

𝜆𝑗 , 𝑠𝑖− , 𝑠𝑟+ ≥ 0 ; 𝑖 = 1, … , 𝑚 , 𝑟 = 1, … , 𝑠 , 𝑗 = 1, … , 𝑛 Thus, the super-efficiency SBM model in the non-oriented can be defined as Eq. (35) [49].

Review of Data Envelopment Analysis Models 1 1−𝑚 𝑚𝑖𝑛 𝑝 = 1 1+𝑠

m

i=1 s

r=1

23

𝑠𝑖− 𝑥𝑖𝑝 (35) 𝑠𝑟+ 𝑦𝑟𝑝

𝑛

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 + 𝑠𝑖− = 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑗≠𝑝 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 − 𝑠𝑟+ = 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1 𝑗≠𝑝

𝜆𝑗 , 𝑠𝑖− , 𝑠𝑟+ ≥ 0 ; 𝑖 = 1, … , 𝑚 , 𝑟 = 1, … , 𝑠 , 𝑗 = 1, … , 𝑛 , 𝑗 ≠ 𝑝 Meanwhile, the generalized SBM model is proposed in the form of Eq. (36) [50]. 1 1−𝑚 𝑚𝑖𝑛 𝑝 = 1 1+ 𝑠

m

i=1 s

r=1

𝑠𝑖− 𝑥𝑖𝑝 (36) 𝑠𝑟+ 𝑦𝑟𝑝

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 + 𝑠𝑖− = 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗∈T

∑ 𝜆𝑗 𝑦𝑟𝑗 − 𝑠𝑟+ = 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗∈T

𝜆𝑗 , 𝑠𝑖− , 𝑠𝑟+ ≥ 0 ; 𝑖 = 1, … , 𝑚 , 𝑟 = 1, … , 𝑠 , 𝑗 = 1, … , 𝑛 , 𝑗 ∈ T, 𝑖 ∈ V

1.6.13. Assurance Region (AR) Model Thompson et al. provided the Assurance Region (AR) model in 1986 [51, 52]. Since weights are considered greater than or equal to zero in the

Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

24

CCR model, some input and output weights may be zero in the optimal solution. In this case, some inputs and outputs are ignored in the efficiency evaluation. Another issue is the equality of the value of all inputs and outputs, for example, the difference between a specialist doctor and a nurse or a simple staff member should be considered in evaluating a hospital. All of these issues led to the development of the AR model. Furthermore, due to the large ratio of weights to the problem, constraints are imposed to prevent the weights from being zero. Some researchers have considered the bounds as Eq. (37) on the input and output weights [53]. 𝑙1,𝑖 ≤

𝑣𝑖 𝑢𝑟 ≤ 𝑟1,𝑖 , L1,𝑟 ≤ ≤ R 1,𝑟 ; 𝑖 = 1, … , 𝑚 , 𝑟 = 1, … , 𝑠 𝑣1 𝑢1

(37)

This model is the same as the basic model (Type I model in chapter 9), which 𝑟1,𝑖 and 𝑙1,𝑖 are the upper and lower bounds for input weights (𝑣𝑖 ), R 1,𝑟 and L1,𝑟 are the upper and lower bounds for output weights (𝑢𝑟 ), respectively. The general case (Type II model in chapter 9) is also as an Eq. (38) [53]. 𝑥1 m

i=1

L𝑖 𝑥𝑖𝑝

≤

𝑣1 𝑥1

≤

m

𝑣𝑖 𝑥𝑖𝑝

i=1

𝑥1

(38)

m

U𝑖 𝑥𝑖𝑝

i=1

1.6.14. Bounded Variable (BND) Model In 1997, Cooper and Tone introduced the Bounded Variable (BND) model [54]. This model is used in non-controllable and Non-Discretionary (NDSC) models where upper and lower bounds constraints must be defined. Thus, constraints of Eq. (39) are defined for non-discretionary variables [55]. N

N

N

N

L𝑝𝑥 ≤ X N 𝜆 ≤ U𝑝 𝑥 , L𝑝𝑦 ≤ Y N 𝜆 ≤ U𝑝 𝑦

(39)

Review of Data Envelopment Analysis Models N

25

N

Where, (L𝑝𝑥 , U𝑝 𝑥 ) is the upper and lower bounds of non-discretionary N

N

inputs and (L𝑝𝑦 , U𝑝 𝑦 ) is the upper and lower bounds non-discretionary outputs [55]. Eq. (40) is related to the input-oriented BND model. 𝑚𝑖𝑛 θ𝑝

(40)

𝑛 C C 𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 ≤ θ𝑝 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛 C C ∑ 𝜆𝑗 𝑦𝑖𝑗 ≤ 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1 N L𝑝𝑥 N L𝑝𝑦

N

≤ X N 𝜆 ≤ U𝑝 𝑥 N

≤ Y N 𝜆 ≤ U𝑝 𝑦

𝜆𝑗 ≥ 0 ; 𝑗 = 1, ⋯ , 𝑛 In addition, the output-oriented BND model is defined as Eq. (41) [55]. 𝑚𝑖𝑛 𝜂𝑝

(41)

𝑛 C C 𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 ≤ 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛 C C ∑ 𝜆𝑗 𝑦𝑖𝑗 ≤ 𝜂𝑝 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1 N L𝑝𝑥 N L𝑝𝑦

N

≤ X N 𝜆 ≤ U𝑝 𝑥 N

≤ Y N 𝜆 ≤ U𝑝 𝑦

𝜆𝑗 ≥ 0 ; 𝑗 = 1, ⋯ , 𝑛

1.6.15. Categorical Variable (CAT) Model Banker and Morey presented the Categorical Variable (CAT) model in 1986 [56]. This condition is related to non-controllable categorical

26

Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

variables and the decision maker cannot choose a category level [57]. In some cases, the decision maker does category selection. Algorithm (1) assumes that there exists l level (1≤l≤L) and the reference sets and projected points on the frontier with different levels in a category or higher category must be obtained. First, one of the DEA models is selected. Algorithm (1) is then executed for the ℎ = 𝑙, 𝑙 + 1, ⋯ , L step [58]. o

Algorithm (1)

Step 1: Organize a DMU set of h-level and higher and DMU𝑝 to be evaluated. Step 2: A. If DMU𝑝 is efficient, the third step is done; B. If DMU𝑝 is inefficient, the points of the set are recorded on the frontier, and if ℎ = L, the third step is executed, otherwise, h is replaced by h+1 and the first stage is implemented. Step 3: Examine the reference set, reference point, and evaluated categorical level from the second step and choose the most appropriate point and categorical level for DMU𝑝 .

1.6.16. Bilateral Comparison Model In 1981, Charans et al. provided the bilateral comparison model [59]. This model is used to compare a DMU in two different categories. It is proposed that each DMU in category A (or B) is evaluated according to the DMUs in the opposite group, i.e., B (or A). These internal comparisons can

Review of Data Envelopment Analysis Models

27

lead to more differences between the two groups. Therefore, this idea can be expressed in the model (42) for each DMU, 𝑎 𝜖 A [60]. 𝑚𝑖𝑛 θ

(42)

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 ≤ θ𝑥𝑎 ; 𝑖 = 1, … , 𝑚 𝑗∈B

∑ 𝜆𝑗 𝑦𝑟𝑗 ≥ 𝑦𝑎 ; 𝑟 = 1, … , 𝑠 𝑗𝜖B

𝜆𝑗 ≥ 0 ; 𝑗 = 1, … , 𝑛 , ∀ 𝑗𝜖𝐵

1.6.17. Enhanced Russell Measure (ERM) Model Pastor et al. introduced the Enhanced Russell Measure (ERM) Model in 1999 [61, 62]. In this model, the limitations of radial measurement which cover some inefficient inputs and outputs are ignored and only weak efficiency is measured according to Eq. (43) [63]. m

R(𝑥𝑝 , 𝑦𝑝 ) = 𝑚𝑖𝑛

i=1 s

r=1

θ𝑖 𝑚 (43) ∅𝑟 𝑠

𝑛

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 ≤ θ𝑖 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 ≥ ∅𝑟 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1

𝜆𝑗 ≥ 0 , 0 ≤ θ𝑖 ≤ 1 , ∅𝑟 ≥ 1 ; 𝑖 = 1, … , 𝑚 , 𝑟 = 1, … , 𝑠 , 𝑗 = 1, … , 𝑛

28

Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

1.6.18. Congestion Model Tone and Saho proposed the congestion model in 2004 [64], where an increase (decrease) in one or more inputs causes a decrease (increase) in one or more outputs. Therefore, a DMU is said to be congested when P𝑐𝑜𝑛𝑣𝑒𝑥 is highly efficient and has a state in P𝑐𝑜𝑛𝑣𝑒𝑥 which more outputs can be obtained using fewer resources at one or more inputs. It is initially assumed that DMU (𝑥𝑝 , 𝑦𝑝 ) in P𝑐𝑜𝑛𝑣𝑒𝑥 set has strong efficiency. The linear programming (44) is defined for this DMU [65]. 𝑠

1 𝑠𝑟+ 𝑚𝑎𝑥 ∑ 𝑠 𝑦𝑟𝑝 𝑛

(44)

𝑟=1

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 + 𝑠− 𝑖 = 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 − 𝑠+ 𝑟 = 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1

𝜆𝑗 , 𝑠𝑖− , 𝑠𝑟+ ≥ 0 ; 𝑖 = 1, … , 𝑚 , 𝑟 = 1, … , 𝑠 , 𝑗 = 1, … , 𝑛 s

A two-stage process is used to solve the model (44). First,

r =1

𝑠𝑟+ 𝑦𝑟𝑝

is

maximized by maintaining the value of the objective function at the ∗

optimal level, and the optimal vector is equal to 𝜆∗ , 𝑠𝑖−∗ and 𝑠𝑟+ . ∗

In the first case, 𝑠𝑟+ = 0 and no congestion is observed in (𝑥𝑝 , 𝑦𝑝 ). That is, a decrease in inputs cannot increase every output, and in the latter ∗

∗

+ case, there is 𝑠+ 𝑟 ≠ 0 and 𝑠𝑟 is non-zero [65]. Therefore, congestion is ∗

∗ + detected in (𝑥𝑝 , 𝑦𝑝 ) in terms of optimal values (𝜆∗ , 𝑠− ̅𝑝 𝑖 , 𝑠𝑟 ), and 𝑥̅𝑝 𝑦 و

values are defined as Eq. (45). ∗

∗ 𝑥̅𝑝 = 𝑥𝑝 − 𝑠− ̅𝑝 = 𝑦𝑝 +, 𝑠+ 𝑟 ; 𝑖 = 1, … , 𝑚 , 𝑟 = 1, … , 𝑠 𝑖 ,𝑦

(45)

Review of Data Envelopment Analysis Models

29

1.6.19. Cost and New Cost Efficiency Models Cost efficiency model was first provided in 1951 by Debreu [66] and then developed in 1957 by Farrell as Eq. (46) [67]. 𝑚

𝑚𝑖𝑛 ∑ 𝐶𝑖𝑝 𝑥𝑖

(46)

𝑖=1 𝑛

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 ≤ 𝑥𝑖 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 ≥ 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1

𝜆𝑗 ≥ 0 ; 𝑗 = 1, … , 𝑛 Furthermore, Tone introduced the new cost efficiency model in 2002 in accordance with Eq. (47) [68, 69]. ̅𝜆 , 𝑦 ≤ Y𝜆 , 𝜆 ≥ 0} P𝑐 = {(𝑥̅ , 𝑦)|𝑥̅ ≥ X ̅ = (𝑥̅1 , ⋯ , 𝑥̅𝑛 ) X 𝑥̅𝑗 = (𝑐1𝑗 𝑥1𝑗 , ⋯ , 𝑐𝑚𝑗 𝑥𝑚𝑗 )

(47)

𝑚𝑖𝑛 𝑒𝑥̅ = 𝑒𝑥̅𝑝∗ ̅𝜆 𝑠. 𝑡: 𝑥̅ ≥ X 𝑦𝑝 ≤ Y𝜆 𝜆≥0 In the model (47), e is a row vector with elements one. The efficiency level of the new cost efficiency model is obtained from Eq. (48). 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 =

𝑒𝑥̅𝑝∗ 𝑒𝑥̅𝑝

(48)

In the cost efficiency model, fixed cost is considered for C𝑝 [70], while the new model seeks to find the optimal solution of 𝑥̅ ∗ for output

Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

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generation of 𝑦𝑝 . This model is similar to the cost efficiency model and only some of the data sets were modified as follows: • • •

"Input" values changed to "input cost × input" values; The cost of each input is equal to one; The cost efficiency model is applied for the new data set.

1.6.20. Revenue and New Revenue Efficiency Models In 1951, Debreu presented the revenue efficiency model, and Farrell developed the model as Eq. (49) [67]. 𝑚

𝑚𝑎𝑥 ∑ P𝑖𝑝 𝑦𝑟

(49)

𝑖=1 𝑛

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 ≤ 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 ≥ 𝑦𝑟 ; 𝑟 = 1, … , 𝑠 𝑗=1

𝜆𝑗 ≥ 0 ; 𝑗 = 1, … , 𝑛 Also, Tone proposed the new revenue efficiency model in 2002. Eq. (50) shows the new revenue efficiency model [68, 69]. ̅𝜆 , 𝜆 ≥ 0} P𝑐 = {(𝑥, 𝑦̅)|𝑥 ≥ X𝜆 , 𝑦̅ ≤ Y 𝑌̅ = (𝑦̅1 , ⋯ , 𝑦̅𝑛 ) 𝑦̅𝑗 = (𝑝1𝑗 𝑦1𝑗 , ⋯ , 𝑝𝑠𝑗 𝑦𝑠𝑗 ) 𝑚𝑎𝑥 𝑒𝑦̅ = 𝑒𝑦̅𝑝∗ 𝑠. 𝑡: 𝑥𝑝 ≥ X𝜆 𝑦̅ ≤ ̅ Y𝜆 𝜆≥0

(50)

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31

In model (50), e is a row vector with elements one. The efficiency of the new revenue efficiency model is calculated through Eq. (51).

𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 =

𝑒𝑦 ̅𝑝 𝑒𝑦 ̅ ∗𝑝

(51)

In the revenue efficiency, fixed revenue is considered for P𝑝 [70], while the new model seeks to find the optimal solution 𝑦̅ ∗ for input generation 𝑥𝑝 . This model is similar to the revenue efficiency model and only some of the data sets have been changed as follows: • • •

"Output" values have changed to "output revenue × output" values; The revenue of each output is considered equal to one; The revenue efficiency model is applied for the new data set.

1.6.21. Profit and New Profit Efficiency Models Profit efficiency model was provided by Debreu in 1951 [66] and developed by Farrell in the form of Eq. (52) [67]. 𝑚

𝑚

𝑚𝑎𝑥 ∑ P𝑖𝑝 𝑦𝑟 – ∑ C𝑖𝑝 𝑥𝑖 𝑖=1

𝑖=1

𝑛

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 ≤ 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 ≥ 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1

𝜆𝑗 ≥ 0 ; 𝑗 = 1, … , 𝑛

(52)

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In 2002, Tone presented the new profit efficiency model as Eq. (53) [68, 69]. P𝑐 = {(𝑥̅ , 𝑦̅)|𝑥̅ ≥ ̅ X𝜆 , 𝑦̅ ≤ ̅ Y𝜆 , 𝜆 ≥ 0}

(53)

̅ Y = (𝑦̅1 , ⋯ , 𝑦̅𝑛 ) 𝑦̅𝑗 = (𝑝1𝑗 𝑦1𝑗 , ⋯ , 𝑝𝑠𝑗 𝑦𝑠𝑗 ) ̅ X = (𝑥̅1 , ⋯ , 𝑥̅𝑛 ) 𝑥̅𝑗 = (𝑐1𝑗 𝑥1𝑗 , ⋯ , 𝑐𝑚𝑗 𝑥𝑚𝑗 ) 𝑚𝑎𝑥 𝑒𝑦̅ − 𝑒𝑥̅ = 𝑒𝑦̅𝑝∗ − 𝑒𝑥̅𝑝∗ ̅𝜆 𝑠. 𝑡: 𝑥̅ ≥ X ̅𝜆 𝑦̅ ≤ Y 𝜆≥0 In model (53), e is a row vector with elements one. The efficiency level of the new profit efficiency model is obtained from Eq. (54) [70]. 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 =

𝑒𝑦̅𝑝 − 𝑒𝑥̅𝑝 𝑒𝑦̅𝑝∗ − 𝑒𝑥̅𝑝∗

(54)

Model (53) is similar to the profit efficiency model and only some of the data sets changed as follows: • • • •

"Input" values changed to "input cost × input" values; "Output" values changed to "output revenue × output" values; The cost of each input and output is equal to one; The profit efficiency model is applied for the new data set.

1.6.22. Ratio Efficiency Model Ratio efficiency model was introduced by Debreu in 1951 [66] and developed by Farrell in 1957 according to Eq. (55) [67].

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33

m

𝑚𝑎𝑥

P𝑖𝑝 𝑦𝑟

i=1

(55)

m

C𝑖𝑝 𝑥𝑖

i=1 𝑛

𝑠. 𝑡: ∑ 𝜆𝑗 𝑥𝑖𝑗 ≤ 𝑥𝑖𝑝 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 ≥ 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1

𝜆𝑗 ≥ 0 ; 𝑗 = 1, … , 𝑛

1.6.23. Scale Elasticity Model Banker and Thrall provided the scale elasticity model in 1992 [71, 72]. This model calculates the scale elasticity in multiple input/output environments, which is obtained by the ratio of Marginal Product (MP) to Average Product (AP) and is called the degree of scale elasticity. In the single input/output, if output y is generated by input x, the scale elasticity 𝜌 is defined as Eq. (56) [73]. 𝑑𝑥 MP 𝑑𝑦 𝜌= = 𝑥 AP 𝑦

(56)

The scale elasticity in the multiple input/output is similarly considered in Eq. (56) [74]. In this case, compared to the single input/output model, the scale elasticity of efficient DMUs in the PPS is calculated. In the envelopment model, this scale is calculated by using Eq. (57). 𝜌 = 1 + 𝑤∗

(57)

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w ∗ is not uniquely determined and there are multiple optimal solutions for 𝑤 ∗. The upper (𝑤) or lower (𝑤) bound should be defined to solve linear programming (58) [74]. 𝑤 (𝑤) = 𝑚𝑎𝑥(𝑚𝑖𝑛) 𝑤 𝑠

(58)

𝑚

𝑠. 𝑡: ∑ 𝑢𝑟 𝑦𝑟𝑗 − ∑ 𝑣𝑖 𝑥𝑖𝑗 = 𝑒𝑤 ; 𝑗 = 1, … , 𝑝, … , 𝑛 𝑠

𝑟=1

𝑖=1

∑ 𝑢𝑟 𝑦𝑟𝑝 = 1 𝑟=1

𝑢𝑟 , 𝑣𝑖 ≥ 0 , 𝑤 𝑓𝑟𝑒𝑒 ; 𝑖 = 1, … , 𝑚 , 𝑟 = 1, … , 𝑠 Therefore, the upper and lower bounds of the scale elasticity are calculated as Eq. (59) [74]. 𝜌̅ = 1 + 𝑤 {𝜌 = 1 + 𝑤

(59)

1.6.24. Epsilon-Based Measure (EBM) Model Tone and Tsutsui introduced the Epsilon-Based Measure (EBM) model in 2001 [75, 76]. The radial and non-radial models have advantages and disadvantages regarding the ratio of changes in inputs to outputs. In order to try to solve this shortcoming, the EBM model was proposed based on Epsilon, which incorporates the features of the radial and non-radial models in an integrated framework. The input-oriented EBM model under CRS is defined as Eq. (60) [77]. 𝑚 ∗

γ = 𝑚𝑖𝑛 θ𝑝 − 𝜀𝑥 ∑ 𝑖=1

𝑤𝑖 − 𝑠𝑖 − 𝑥𝑖𝑝

(60)

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35

𝑛

𝑠. 𝑡: θ𝑝 𝑥𝑖𝑝 − ∑ 𝜆𝑗 𝑥𝑖𝑗 − 𝑠𝑖 − = 0 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛

∑ 𝜆𝑗 𝑦𝑟𝑗 ≥ 𝑦𝑟𝑝 ; 𝑟 = 1, … , 𝑠 𝑗=1

𝜆𝑗 , 𝑠𝑖 − ≥ 0 ; 𝑖 = 1, … , 𝑚 , 𝑗 = 1, … , 𝑛 m

Where,

𝑤𝑖− = 1 and 𝜀𝑥 are a key parameter in which the radial and

i =1

non-radial slack variables are integrated and the parameters 𝜀𝑥 and 𝑤𝑖 − determine the amount of efficiency.

1.6.25. Weak Disposability Model Shephard presented the weak disposability model in 1974 [78]. Model (61) limits inputs excess (input slack) and outputs shortfall (output slack) variables of input/output to zero with weak disposability [79]. 𝑚𝑖𝑛 θ𝑝

(61) 𝑛

𝑠 𝑠 𝑠. 𝑡: θ𝑝 𝑥𝑖𝑝 − ∑ 𝜆𝑗 𝑥𝑖𝑗 − 𝑠− 𝑖 = 0 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛 𝑤 𝑤 θ𝑝 𝑥𝑖𝑝 − ∑ 𝜆𝑗 𝑥𝑖𝑗 = 0 ; 𝑖 = 1, … , 𝑚 𝑗=1 𝑛 𝑠 𝑠 ∑ 𝜆𝑗 𝑦𝑟𝑗 − ∅𝑝 𝑦𝑟𝑝 − 𝑠+ 𝑟 = 0 ; 𝑟 = 1, … , 𝑠 𝑗=1 𝑛 𝑤 𝑤 ∑ 𝜆𝑗 𝑦𝑟𝑗 − ∅𝑝 𝑦𝑟𝑝 = 0 ; 𝑟 = 1, … , 𝑠 𝑗=1

+ 𝜆𝑗 , 𝑠− 𝑖 , 𝑠𝑟 ≥ 0 ; 𝑖 = 1, … , 𝑚 , 𝑟 = 1, … , 𝑠 , 𝑗 = 1, … , 𝑛

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Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

In model (61), s and w indicate strong and weak disposability, respectively.

1.6.26. Metafrontier Model Battese et al. provided the metafrontier model in 2004 [80]. The Eq. (62) was first defined [81]. 𝑦𝑖𝑗 = 𝑒 −U𝑖𝑗 ×

𝑒 𝑥𝑖𝑗𝛽𝑗 𝑒

𝑥𝑖𝑗 𝛽 ∗

× 𝑒 𝑥𝑖𝑗𝛽

∗ +V 𝑖𝑗

; 𝑖 = 1, … , 𝑚 , 𝑗 = 1, … , 𝑛

(62)

In Eq. (62), both U𝑖𝑗 and V𝑖𝑗 variables have normal distribution and 𝛽𝑗 is a linear parameter which is related to the input variable. Then, according to 𝑥𝑖𝑗 𝛽 ∗ ≥ 𝑥𝑖𝑗 𝛽𝑗 , 𝛽 ∗ is a linear parameter that is defined for the metafrontier function and the Technical Efficiency (TE) is determined as Eq. (63) [81]. TE𝑖𝑗 =

𝑒

𝑦𝑖𝑗 𝑥𝑖𝑗 𝛽 ∗ +𝑉𝑖𝑗

= 𝑒 −𝑈𝑖𝑗 ; 𝑖 = 1, … , 𝑚 , 𝑗 = 1, … , 𝑛

(63)

The Technology Gap Ratio (TGR) is also expressed according to Eq. (64) [81].

TGR 𝑖𝑗 =

𝑒 𝑥𝑖𝑗𝛽𝑗

∗ ; 𝑖 = 1, … , 𝑚 , 𝑗 = 1, … , 𝑛 𝑒 𝑥𝑖𝑗𝛽 Thus, the efficiency is obtained in Eq. (65) [81].

(64)

E𝑖𝑗 = TE𝑖𝑗 × TGR 𝑖𝑗 ; 𝑖 = 1, … , 𝑚 , 𝑗 = 1, … , 𝑛

(65)

1.6.27. Dynamic DEA Model Dynamic DEA model was presented in 1996 by Fare and Grosskopf [82.83]. In the traditional DEA model, the analysis is performed on a set of

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37

basic DMUs over a time period, and the performance of each DMU is compared to other units. Dynamic DEA is a relatively new approach which can overcome the shortcomings of basic DEA models [84]. In the dynamic DEA model, the analysis is performed on a DMU over several time periods. In other words, a DMU is evaluated and its efficiency or inefficiency is determined over different time periods. The difference between dynamic models and other models is related to the use of the outputs of each DMU as DMU data in any other period [85, 86]. Therefore, it is possible to compare the dynamic DMU with its past performance and the application of this type of model becomes more prominent. Figure 2 shows the general structure of a dynamic DEA model. Input t

Input t+1

Period t

Carry-over t

Period t+1

Output t

Carry-over t+1

Output t+1

Figure 2. Structure of the dynamic DEA model [87].

The structure of the dynamic DEA model is such that each DMU has its inputs or outputs, and continuous or intermediate activities are considered during the analysis.

1.6.28. Network DEA Model Farr and Groscope presented the network DEA model in 1996 [88, 89]. The basic DEA models consider production technology as a black box in which input is converted to output during a process [90]. Chen and Zhou introduced the network DEA model by providing a two-stage process in 2004 [91] and presenting a supply chain model in 2014 [92]. Some DMUs

Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

38

are made up of several sections or stages which create a network of subprocesses. THE network DEA model is used for evaluating the efficiency of this type of unit. On the other hand, the output of some units is the input of other units. Therefore, the basic DEA models cannot be used to measure the relative efficiency of the DMU in this case. The results of the network DEA model are related to the evaluation of the following items: • • • • •

The efficiency of each unit separately; The total efficiency of each unit; The minimum, maximum, and average of total efficiency in each section; The optimal value of intermediate inputs and outputs for inefficient units; The optimal value of free inputs and outputs for inefficient units.

1.6.29. Context Dependent Model Context dependent model was provided by Seiford et al. in 2003 [93, 94]. The results of this model are more accurate than inefficient units due to the evaluation of its relative attractiveness and specific performance of a DMU set at an efficient level relative to the efficiency threshold. The relevant model is in the form of Eq. (66) [95]. Ω∗𝑞 (𝑑) = 𝑚𝑎𝑥 𝛺𝑞 (𝑑) 𝜆𝑗 ,Ω𝑞 (𝑑)

𝑠. 𝑡.

(66)

𝜆𝑗 𝑦𝑟𝑗 ≥ Ω𝑞 (𝑑)𝑦𝑟𝑞 ; 𝑟 = 1, … , 𝑠

∑ 𝑗∈𝐹(𝐸 𝑙𝑜+𝑑+1 )

∑

𝜆𝑗 𝑥𝑖𝑗 ≥ 𝑥𝑖𝑞 ; 𝑖 = 1, … , 𝑚

𝑗∈𝐹(𝐸 𝑙𝑜+𝑑+1 )

𝜆𝑗 ≥ 0 , 𝑗 ∈ F(E 𝑙𝑜 +𝑑+1 ) ; 𝑗 = 1, … , 𝑛 , 𝑙 = 1, … , L , 𝑑 = 1, … , L − 𝑙𝑜

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39

In Eq. (66), E 𝑙𝑜 , 𝑙𝑜 ∈ {𝑙 = 1, … , L − 1} is the set of evaluation of the texts and DMU𝑞 = {𝑥𝑞 , 𝑦𝑞 } is a level of this set.

1.6.30. Bootstrapping Model The major disadvantage of DEA is that it is not possible to apply statistical inference due to this deterministic nature. To mitigate this disadvantage, Simar and Wilson in 1989 introduced the bootstrapping model as a tool for extracting the sensitivity of DEA scores towards the randomness which is attributed to the distribution of efficiency [96, 97]. This model is a statistical resampling method, which has proven to be effective for statistical inference in complex problems. The most important step in applying the bootstrapping model is to identify the 𝜌 method or data generation process correctly from the sampled data from the population. The main idea of the bootstrapping model is to estimate the distribution of estimated samples by using the experimental distribution of estimates obtained from bootstrapping. The bootstrapping algorithms were introduced by Simar (S), Simar and Wilson (SW), and Lothgren-Tambour (LT) based on the identical model of data generation procedure [98], where it is assumed that at a certain level of the outputs, the amount of input to construct the generated sample for data generation is obtained from random radial deviations of the same curve of the input set. Each input in the observed sample is represented as input-output X = ((𝑥𝑖 , 𝑦𝑖 ), 𝑖 = 1, ⋯ , 𝑛) in Eq. (67) [99]. (𝑥𝑖 , 𝑦𝑖 ) = (

𝑥 𝜕 (𝑥𝑖 |𝑦𝑖 ) , 𝑦𝑖 ) ;𝑖 = 1, ⋯ , 𝑛 𝜃𝑖

(67)

Where, 𝑥 𝜕 (𝑥𝑖 |𝑦𝑖 ) ∈ ∂X(𝑦𝑖 ) is an unobserved point (on the efficient frontier) which is the radial equivalent of the firm (𝑥𝑖 , 𝑦𝑖 ) on the efficient frontier ∂X(𝑦), which is compared to that point to calculate the efficiency of the firm's geometric location. It is assumed that the actual efficiency values are taken from a similar distribution such as 𝜃𝑖 ∼ F𝜃 , 𝑖 = 1, ⋯ , 𝑛.

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Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

Thus, the data generation process is an idea conditioned on the input and output ratios and the random elements of the production process are fully represented by the random input-oriented efficiency index. The main idea of bootstrapping simulation is to imitate the data generation process. Resampling data in LT and SW algorithms are generated in two steps, based on the ratios of inputs and outputs. In the first step, the input frontier curve is estimated by using the observed sample, and then, using the input frontiers and the generated efficiency values obtained from some distribution estimates F𝜃 . The input values are generated by repeating the data generation process calculated in Eq. (67). This step in the algorithm proposed by LT is based on a simple resampling of the empirical distribution of estimated efficiency values which is used to generate efficiency. Unlike the LT method, the SW algorithm in the first step uses a moderated resampling process, and this algorithm is according to the compatibility argument. In the second step, the efficiency of the bootstrapping model is estimated by calculating the radial distance of the generated efficient frontier with the assumed sample from the proposed input values (in the LT algorithm) or the main input values (in the SW algorithm).

CONCLUSION Efficiency is considered as a part of productivity and includes the output-input ratio in a system. Efficiency evaluation is possible by parametric and non-parametric methods. DEA is regarded as one of the appropriate and efficient tools in the field of productivity and efficiency, which is used as a non-parametric method to calculate the efficiency of DMUs. Charens et al. first introduced DEA in 1978, which evaluates the efficiency of DMUs based on inputs and outputs. Today, the use of DEA is expanding rapidly and is used in the evaluation of various organizations such as banks, hospitals, and refineries. So far, many methods of DEA have been presented. In this chapter, after introducing the DEA, 30 different methods of DEA reviewed.

Chapter 2

DEA EXCEL SOFTWARE 2.1. INTRODUCTION DEA Excel is one of the software under Microsoft Excel to solve DEA models, including the super-efficiency model that can be used to solve problems with 200 DMUs and 20 inputs/outputs. To this aim, this software must be added to the "add-ins" in the Excel software settings.

2.2. SOFTWARE PRESENTATION DEA Excel software environment according to Figure 1 has different parts and models, which are introduced.

Figure 1. DEA Excel software environment.

42

Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

2.2.1. Settings The "settings" window (Figure 2) contains some commands such as language, range of tolerance, parameters, and problem title. In this window, the software language, the range of tolerance, and epsilon can be changed. At the bottom of this window, the input data is also normalized.

Figure 2. DEA Excel software settings window.

2.2.2. BCC and CCR Models In the input data and model specification window, according to Figure 3, the basic information and specifications of BCC and CCR models are determined .In order to specify each section, the range of data associated with each item must be selected from the menu of each section. After defining the data, the "solve" button is used to solve the problem.

DEA Excel Software

43

Figure 3. Input data for the BCC and CCR models.

2.2.3. Additive Model This model is designed to estimate overall efficiency. As shown in Figure 4, the window of this model is similar to the BCC and CCR models, except that the super-efficiency and two-stage optimization model is removed from this case, which is only expressed in the output-oriented.

Figure 4. Input data for additive model.

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Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

2.2.4. NCN Model After defining the data for the NCN model, the name of the noncontrollable inputs/outputs must also be specified. In Figure 5, the names of the inputs (similar to the outputs) are specified.

Figure 5. Input data for non-controllable inputs.

2.2.5. Undesirable Input/Output Model The window of this model is similar to the NCN model, and the names of the undesirable inputs/outputs must be defined.

2.3. RUNNING THE DEA EXCEL SOFTWARE According to Table 1, 12 hospitals (DMU) with the number of doctors and the nurses as inputs and the number of inpatients and outpatient as outputs are evaluated. After solving the model, results are saved in various worksheets as follows:

DEA Excel Software • • • • • • • • •

45

Input-oriented CCR/BCC model (CRS-I-1) Output-oriented CCR/BCC models (CRS-O-1) Additive models (ACRS-I-1) Input-oriented non-controllable model (UCRS-I-1) Output-oriented non-controllable model (UCRS-O-1) Input-oriented undesirable model (UDRS-I-1) Output-oriented undesirable model (UDRS-O-1) Input-oriented super-efficiency model (CRS-IS-1) Output-oriented super-efficiency model (CRS-OS-1) Table 1. Inputs and outputs for evaluating hospitals

DMU A B C D E F G H I J K L

Input Doctor 151 131 160 168 158 255 135 206 244 268 306 284

Nurse 20 19 25 27 22 55 33 31 30 50 53 38

Output Inpatient 100 150 160 180 94 230 220 152 190 250 260 250

Outpatient 90 50 55 72 66 90 88 80 100 100 147 120

For instance, for the input-oriented CCR/BCC model (CRS-I-1), the initial values and efficiency scores for each DMU are shown individually for the BCC and CCR models, which for DMUA are shown in Figure 6. As the worksheet name implies, the model is resolved in CRS and input-oriented approaches. In general, the naming of each worksheet is determined by the type of model, the RTS, and its orientation so that the worksheet (CRS-I-2) provides general information about DMUs and inputs/outputs (Figure 7).

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Figure 6. DMUA evaluation in the CRS-I-1 worksheet.

Figure 7. General information in the CRS-I-2 worksheet.

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CONCLUSION DEA Excel software is one of the simplest DEA software that has a simple user interface with 7 different models. This software runs in Microsoft Excel, which is similar to software such as DEA Solver and DEA Frontier in this respect. In DEA Excel software, after defining the model and defining the data, the efficiency scores of DMUs are determined separately so that, in addition to efficiency scores, the values of slack and virtual variables for the inputs and outputs of each DMUs are specified.

Chapter 3

DEA FRONTIER SOFTWARE 3.1. INTRODUCTION DEA Frontier software was presented and developed by Joe Zhou [100] and is used as an engine to solve DEA models under Microsoft Excel. This software contains different models such as measure-specific and network DEA models to evaluate DMUs. Further, this is simple to use since it has comprehensive reports. In order to run this software, it should be installed in Excel.

3.2. SOFTWARE PRESENTATION DEA Frontier software incorporates the various models shown in Figure 1. In the following, software models are introduced.

3.2.1. Envelopment Model As shown in Figure 2, after defining the data, model selection, RTS, and orientation, the model is performed and the results are saved in the "efficiency", "target" and "slack" worksheets.

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Figure 1. DEA Frontier software environment.

Figure 2. Settings of envelopment model.

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3.2.2. Multiplier Model After defining the data, choosing the model, RTS, and orientation as shown in Figure 3, the model is performed and the results are saved in the "efficiency report" worksheet.

Figure 3. Multiplier model setting window.

3.2.3. SBM Model In the setting window of the SBM model (Figure 4), the RTS and applying the weight to the model can be specified.

Figure 4. Settings of the SBM model.

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After solving the model, the reports are saved in "efficiency target" and "slack report" worksheets.

3.2.4. Measure-Specific Model The data entry and specification of the measure-specific model are defined as shown in Figure 5 and the results are saved in "efficiency", "slack" and "target" worksheets after solving the model.

Figure 5. Measure-specific model settings.

3.3. RUNNING THE DEA FRONTIER SOFTWARE In the 97-2000 or 2003 versions of Microsoft Excel, after opening the "file" tab, the "macro security" section should be selected from "tools". In the "macro security", after choosing the "medium" mode to run the software, the Excel file returns. In the "tools" tab, selecting the "solver" command will open the "solver parameters" window, and then the "DEA" tab will be created in the toolbar. In the 2007-2016 versions of Excel software, by choosing the "macro settings" in the "trust center” settings, two options (1) and (2) should be activated as shown in Figure 6.

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Figure 6. Frontier DEA activation in the 2007-2016 versions.

In the "add-ins" tab, the "solver add-ins" option (Figure 7) is activated. The "solver" command is available in the "data" tab.

Figure 7. Add-ins settings.

Figure 8 shows the format of the datasheet. To run the software (Figure 9), the data for the evaluation of hospitals are used.

Figure 8. Format of the datasheet.

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Figure 9. Data entry for evaluating hospitals.

The results of solving the envelopment model are saved in the "efficiency" worksheet including efficiency scores, RTS, lambda values, and reference DMUs, as shown in Figure 10.

Figure 10. Results of envelopment model in the efficiency worksheet.

Also, final reports of the envelopment model in the "target" worksheet, which show the amount of changes to become an efficient unit for each variable, are shown in Figure 11.

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Figure 11. Final reports of envelopment model in the target worksheet.

The slack worksheet also shows the values of inputs excess and outputs shortfall (Figure 12).

Figure 12. Results of envelopment model in the slack worksheet.

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Figure 13. Multiplier model results.

The output of the multiplier model in the VRS is saved in the "efficiency report" worksheet as shown in Figure 13. After solving the SBM model with CRS, the results are saved in the "efficiency target" worksheet (Figure 14).

Figure 14. SBM model results in the efficiency target worksheet.

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After solving the SBM model, the final reports are saved in the "slack report" worksheet. Figure 15 shows the efficiency scores, the type of RTS, lambda values, and the reference DMUs.

Figure 15. Final reports of SBM model in the slack report worksheet.

The results of measure-specific model are shown in Figure 16 by considering the number of nurses.

Figure 16. Results of measure-specific model in the target worksheet.

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On the other hand, the results of the measure-specific model can be observed in the "efficiency" worksheet (Figure 17).

Figure 17. Results of measure-specific model in the efficiency worksheet.

Further, Figure 18 shows the reports of measure-specific model in the "slack" worksheet.

Figure 18. Final reports of measure-specific model in the slack worksheet.

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CONCLUSION DEA Frontier software, like most DEA software such as DEA Excel, runs in the Microsoft Excel. The number of different models with comprehensive outputs in different worksheets is considered as one of the advantages of this software. Due to the implementation of the software in Excel, it is possible for users to easily enter data and save the results. Therefore, DEA Frontier has become one of the high usage software among academic and experimental researches.

Chapter 4

DEA SOLVER SOFTWARE 4.1. INTRODUCTION DEA Solver software is designed by the Japanese-American software group "SAITECH" and is developed in two student and professional types [101, 102]. This software runs under Microsoft Excel and several versions of this software have been provided to contain different DEA models so far.

4.2. SOFTWARE PRESENTATION In this section, DEA Solver software including models, data types, and results worksheets is introduced.

4.2.1. Models In this software, DEA models are named "I or O" and "C or V or GRS", where I and O indicate the input-oriented and output-oriented. C, V, and GRS show the CRS, VRS, and General Returns to Scale (GRS),

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respectively. For example, "AR-I-C" represents the input-oriented AR model in the CRS. In some cases, "I or O" or "C or V" is removed. For instance, "CCR-I" indicates the input-oriented CCR model in the CRS. The GRS model has constraints L ≤ λ1 + ⋯ + λN ≤ U which its upper and lower bounds are equal to U ≥ 1 and L ≤ 1, respectively. While performing one of the models, the GRS is selected; the L and U values should be entered manually through the message box on the display. As shown in Figure 1, U = 1.2 and L = 0.8 are considered.

Figure 1. Determining the upper and lower bounds in the GRS.

As shown in Table 1, DEA models are classified into 4 categories. Table 1. Classification of models Category Radial

Non-radial and oriented

Non-radial and Non-oriented

Radial and Non-radial

Cluster or Model CCR, BCC, IRS, DRS, AR, ARG, NCN, NDSC, BND, CAT, SYS, Super Radial, Bilateral, Scale Elasticity, Congestion, Window, Malmquist Radial, FDH, Non-convex Radial, Resampling (Super) Radial, DDF Model SBM Oriented, Super SBM-oriented, Malmquist, Weighted SBM Oriented, Network SBM-oriented, Dynamic SBM-oriented, Dynamic and Network SBMoriented, Non-convex SBM-I(O), Resampling (Super)SBM-oriented, SBM Max-Oriented Cost, New Cost, Revenue, New Revenue, Profit, New Profit, Ratio, SBM Nonoriented, Super SBM Non-oriented, Malmquist-C (V, GRS), Undesirable Outputs, Weighted SBM ,Network SBM Non-oriented, Dynamic SBM Nonoriented, Dynamic and Network SBM Non-oriented, Non-convex SBM Nonoriented, Resampling (Super) SBM Non-oriented, SBM Max Non-oriented Hybrid, EBM

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The student version of this software covers 28 models. In this version, problems are solved with a maximum of 50 DMUs. In Table 2, the models in the student version are stated. Table 2. Classification of models in the student version Cluster CCR

Model CCR-I, CCR-O

BCC IRS DRS GRS AR NCN NDSC BND CAT SYS (Different Systems) SBM Oriented

BCC-I, BCC-O IRS-I, IRS-O DRS-I, DRS-O GRS-I, GRS-O AR-I-C, AR-I-V, AR-I-GRS, AR-O-C, AR-O-V, AR-O-GRS NCN-I-C, NCN-I-V, NCN-O-C, NCN-O-V NDSC-I-C, NDSC-I-V, NDSC-I-GRS, NDSC-O-C, NDSC-O-V, NDSC-O-GRS BND-I-C, BND-I-V, BND-I-GRS, BND-O-C, BND-O-V, BND-O-GRS CAT-I-C, CAT-I-V, CAT-O-C, CAT-O-V SYS-I-C, SYS-I-V, SYS-O-C, SYS-O-V

SBM-Non-oriented Weighted SBM Super SBM Oriented Super SBM Nonoriented Super Radial Cost New Cost Revenue New-Revenue Profit New-Profit Ratio (Revenue/Cost) Bilateral Window Malmquist Radial FDH

SBM-I-C, SBM-I-V, SBM-I-GRS, SBM-O-C, SBM-O-V, SBM-O-GRS, SBMAR-I-C, SBM-AR-I-V,SBM-AR-O-C, SBM-AR-O-V SBM-C, SBM-V, SBM-GRS, SBM-AR-C, SBM-AR-V Weighted SBM-C, Weighted SBM-V, Weighted SBM-I-C, Weighted SBM-I-V, Weighted SBM-O-C, Weighted SBM-O-V Super SBM-I-C, Super SBM-I-V, Super SBM-I-GRS, Super SBM-O-C, Super SBM-O-V, Super SBM-O-GRS Super SBM-C, Super SBM-V, Super SBM-GRS Super-CCR-I, Super-CCR-O, Super-BCC-I, Super-BCC-O Cost-C, Cost-V, Cost-GRS New Cost-C, New Cost-V, New Cost-GRS Revenue-C, Revenue-V, Revenue-GRS New Revenue-C, New Revenue-V, New Revenue-GRS Profit-C, Profit-V, Profit-GRS New Profit-C, New Profit-V, New Profit-GRS Ratio-C, Ratio-V Bilateral-CCR-I, Bilateral-BCC-I, Bilateral-SBM-C, Bilateral-SBM-V Window-I-C, Window-I-V, Window-I-GRS, Window-O-C, Window-O-V, Window-O-GRS Malmquist-Radial-I-C, Malmquist-Radial-I-V, Malmquist-Radial-I-GRS, Malmquist- Radial-O-C, Malmquist Radial-O-V, Malmquist-Radial-O-GRS FDH

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Cluster Malmquist Non-radial

Scale Elasticity Congestion Undesirable Outputs Hybrid Network SBM(Oriented) Network SBM (Non-oriented) Dynamic SBM(Oriented) Dynamic SBM(Non-oriented) EBM(Oriented) EBM(Non-oriented) Dynamic Network SBM(Oriented) Dynamic Network SBM (Nonoriented) Non-convex SBM Non-convex Radial Resampling (Super) Radial Directional Distance (Oriented) Directional Distance (Nonoriented) SBM Max Super SBM Max SBM Bounded Negative Data SBM Negative Data SBM Max Negative Data MalmquistSBM

Model Malmquist-I-C, Malmquist-I-V, Malmquist-I-GRS, MalmquistO-C, Malmquist-O-V, Malmquist-O-GRS, Malmquist-C, Malmquist-V, Malmquist-GRS Elasticity-I, Elasticity-O Congestion Bad Output-C, Bad Output-V, Bad Output-GRS, Non-separableC, Non-separable-V, Non-separable-GRS Hybrid-C, Hybrid-V, Hybrid-I-C, Hybrid-I-V, Hybrid-O-C, Hybrid-O-V Network SBM-I(O)-C(V) Network SBM-C(V) Dynamic SBM-I(O)-C(V) Dynamic SBM-C(V) EBM-I(O)-C(V) EBM-C(V) DNSBM-I(O)-C(V) DNSBM-C(V) Non-convex SBM-I(O, Non-oriented) Non-convex-Radial-I(O) Resample Triangular,Resample Triangular Historical, Resample Past Present, Resample Past Present Future DD-I(O)-C(V), SuperDD-I(O)-C(V) DD-C(V), Super DD-C(V) SBM Max-I-C, SBM Max-I-V, SBM Max-O-C, SBM Max-OV, SBM Max-C, Super SBM Max (Oriented), Super SBM Max (Non-oriented) SBM Bounded (Oriented), SBM Bounded (Non-oriented) Negative Data SBM -I(O)-V, Negative Data SBM-V, Negative Data Super SBM-V Negative Data SBM Max-I(O)-V, Negative Data SBM Max- V Negative Data Malmquist-SBM-I(O)-V, Negative Data Malmquist-SBM- V

The latest professional version of this software is presented in 2018. In this version, in addition to the student version models, various other models are expressed as well and these models can be seen in Table 3.

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Each of these models has been added to the student version of the software in several updates.

4.2.2. Data File Preparation Before running the software, the data of each problem must be saved in an Excel workbook based on the specific structure for each model. The following items should be considered in this format: ✓ The first row of cells should contain the problem name, input, and output names. In cell (A1), the name of the problem and other cells (B1, C1, etc.) are defined as the names of the inputs and outputs. ✓ The worksheet name in which the problem information is entered must be saved in "DAT". ✓ In order to differentiate inputs and outputs, their specific abbreviations should be used, which are stated in Table 4. Table 4. Abbreviations of inputs and outputs Heading I O IN

Description Input Output Non-controllable or Nondiscretionary input

ON

Non-controllable or Nondiscretionary output

IB OB LB

Bounded input Bounded output Lower bound of bounded variable Upper bound of bounded variable Unit cost of input

UB C

Model All models All models NCN NDSC DD NCN NDSC DD BND BND BND

Example (I)Employee (O)Sales (IN)Population

BND

(UB)Doctor

Cost, New Cost, Profit, New Profit, Ratio

(C)Manager

(ON)Area

(IB)Doctor (OB)Attendance (LB)Doctor

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Heading P

Description Unit price of output

OBad IS INS OSGood ONSGood ONSBad IR INR OR ONR D LB LG LF LN CB CG CF CN

Bad (undesirable) output Separable output Non-separable input Separable good output Non-separable good output Non-separable bad output Radial input Non-radial input Radial output Non-radial output Division Bad link Good Link Free Link Fixed Link Bad Carry-over Good Carry-over Free Carry-over Fixed Carry-over

Model Revenue, New Revenue, Profit, New Profit, Ratio Bad Output model DD Non-separable model Non-separable model Non-separable model Non-separable model Non-separable model Hybrid model Hybrid mode Hybrid mode Hybrid mode Network SBM, DNSBM Network SBM, DNSBM Network SBM, DNSBM Network SBM, DNSBM Network SBM, DNSBM Dynamic SBM, DNSBM Dynamic SBM, DNSBM Dynamic SBM, DNSBM Dynamic SBM, DNSBM

Example (P)Laptop (OBad)NOX (IS)Labor (INS)Fuel (OS)Outpatient (ONSGood)GNP (ONSBad)CO2 (IR)Labor (INR)Nurse (OR)Machine (ONR) (D)Division 1 (LB)Link12 (LG)Link13 (LF)Link14 (LN)Link15 (CB)Debt (CG)Profit (CF)Capital (CN)Area

✓ In the data worksheet, according to Figure 2, the second and other rows in the first column includes the DMUs name. Other columns are related to inputs and outputs values.

Figure 2. General structure of data entry in Excel.

✓ In some cases, it is necessary to adjust the parameters of the models, it is followed based on Table 5.

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Table 5. Settings of model parameters Defaults L = 0.8 , U = 1.2 1 and 1 (Ratio of weights= 1:1) 0.2 L=1,U=1

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Description In GRS Model, L (lower bound) and U(upper bound) Total weights to good outputs and bad outputs Upper bound of expansion rate of separable good outputs Upper (U) and lower (L) bounds to the radial input (reduction) factor θ and the radial output (expansion) factor η Number of replicas

Model GRS Bad Output Non-separable EBM-C/EBM-V

Resampling

In the following, the data format for different models of DEA Solver software is presented.

4.2.2.1. CCR, BCC, IRS, DRS, GRS, SBM, FDH, EBM, Scale Elasticity, and Congestion Models In these models, the data file is defined based on the format mentioned in the previous section. For instance, in the congestion model, the evaluation problem data of 12 hospitals with 2 inputs and 2 outputs are entered as shown in Figure 3.

Figure 3. Data entry in the congestion model.

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DMUs with inappropriate data are excluded as inappropriate units and listed in the "summary" worksheet. Therefore, the following symbols are considered: • • • •

𝑥 𝑚𝑎𝑥(𝑥 𝑚𝑖𝑛) = The maximum or minimum input value of DMU 𝑦 𝑚𝑎𝑥(𝑦 𝑚𝑖𝑛) = The maximum or minimum output value of DMU 𝐶𝑜𝑠𝑡 𝑚𝑎𝑥(𝐶𝑜𝑠𝑡 𝑚𝑖𝑛) = The maximum or minimum cost of DMU 𝑃𝑟𝑖𝑐𝑒 𝑚𝑎𝑥(𝑃𝑟𝑖𝑐𝑒 𝑚𝑖𝑛) = The maximum or minimum price of DMU

All DMUs with non-positive input (𝑥𝑚𝑎𝑥 ≤ 0) in CCR, BCC-I, IRS, DRS, GRS, SBM, and Elasticity-I models are excluded from the calculations. The use of zero and negative values are allowed when there is at least one positive value in the inputs. All DMUs are excluded from the calculations with non-positive input in SBM, Elasticity-O, and BCC-O models. Further, DMUs are excluded from the calculations with a non-positive input or a negative input in the FDH model.

4.2.2.2. AR Model Figure 4 shows an example of the data format for the AR model. Also, in addition to defining input and output, constraints which are related to the AR model should be considered. In this example, the constraints should be denoted in row 15 and 16 after "one blank row" at 14 (after the information associated with the last DMU, row 14 must be empty in order to separate the data set and the AR constraints). The weight related to the number of doctors and nurses is displayed with 𝑣(1) and 𝑣(2), the ratio of "(I) Doctor" to "(I) Nurse" should not be less than 1 and not more than 5. Therefore, the first constraint is considered as 1 ≤ 𝑣(1)⁄𝑣2) ≤ 5. Likewise, the weights related to the number of outpatients and inpatients are indicated with 𝑢(1) and 𝑢(2), the ratio of "(O) Outpatient" and "(O) Inpatient" weights should not be less than 0.2 and more than 0.5. As a result, the second constraint is considered as 0.2 ≤ 𝑢(1)⁄𝑢(2) ≤ 0.5. It should be noted that the limit between input and output weights should be considered. In addition, in the AR model, any

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DMU with non-positive values at the input or output is removed from the comparison group due to AR constraints.

Figure 4. Data format in the AR model.

4.2.2.3. AR Global Model Based on the data format in the AR global model (Figure 5), instead of restricting the virtual coefficients, frontiers are imposed on the potential input (output) relative to the total potential input (output). For example, the total virtual input of the number of doctors is expressed as the number of doctors × v(1), and the total virtual input of the number of nurses is expressed as (number of nurses × v(2) + number of doctors × v(1). Therefore, Eq. (1) is considered. L≤

𝑣(1) × Doctor ≤ U ; L = 0.5, U 𝑣(1) × Doctor + 𝑣(2) × Nurse = 0.8 (1)

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In this model, any DMU with non-positive values at the input or output is removed from the comparison group based on AR constraints.

Figure 5. Data format in the AR global model.

4.2.2.4. Super-Efficiency Model In most DEA models, DMU has the best performance with the efficiency of unity. In most cases, the number of efficient units is more than one. Thus, DMUs can be ranked with super-efficiency models. In these models, there are two clusters; radial and non-radial. The non-radial state is calculated based on the SBM model and has a similar structure. Therefore, there will be 9 SBM models in the super-efficiency approach including super-SBM-IC, super-SBM-IV, super-SBM-I-GRS, super-SBM-OC, super-SBM-OV, and super-SBM-O-GRS models are oriented and the super-SBM-C, super-SBM-V, and super-SBM-GRS

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models are non-oriented. In the super-efficiency model, DMUs are removed from the calculations with non-positive input.

4.2.2.5. NCN and NDSC Models The data format in the NCN and NDSC models is similar to the CCR model (Figure 2). In these models, the abbreviations (as shown in Table 4) are considered as follows: ▪ ▪

IN: For non-controllable or non-discretionary inputs ON: For non-controllable or non-discretionary outputs

Moreover, negative inputs and outputs are converted to zero by the software. Also, DMUs with non-positive controllable input are removed from the calculations and considered as unsuitable units.

4.2.2.6. BND Model Bounded inputs or outputs are expressed by (IB) and (OB). Therefore, the columns headed by the titles (LB) and (UB) represent the upper and lower bounds, respectively. The (LB) and (UB) columns are placed after the columns (IB) and (OB) columns. If the number of doctors and patients is considered as bounded input and output, the input data format is as shown in Figure 6. In this model, the upper and lower bounds should be specified in the data range.

Figure 6. Data format in the BND model.

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4.2.2.7. Bilateral, System and CAT Models The format of these models is similar to the CCR model, and only in the last column of these models (after the last output), there is an integer (Figure 7) which indicates the category, system, or group. For the CAT model, the number 1 is assigned to the DMUs under the most difficult environment or with the most severe competition, and the number 2 is associated with a unit with a lower degree of hardness which indicates with the symbol (cat.) (It is suggested that the numbers start and continue from 1). Similarly, integers start with one for the different System (SYS) models and are denoted by the symbol (Sys.). For the bilateral model, the DMUs must be divided into two groups, denoted by 1 and 2. All DMUs are removed from the calculations with non-positive input in the SYS and CAT models. If there is at least one positive value in the inputs, the use of zero and negative values are allowed. If in one group all the values of one input are zero, and even if all the values of the other input are positive, it is no longer possible to make a comparison between the two groups.

Figure 7. Data entry in the CAT model.

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4.2.2.8. Cost Efficiency and New Cost Efficiency Models Unit cost columns should be stated in the titles of inputs along with (C). If the input has no cost columns, it means that the cost is zero. Furthermore, DMUs with 𝑥 𝑚𝑎𝑥 ≤ 0, 𝑥 𝑚𝑖𝑛 < 0, 𝑐𝑜𝑠𝑡 𝑚𝑎𝑥 ≤ 0, and 𝑐𝑜𝑠𝑡 𝑚𝑖𝑛 < 0 are removed from the calculations. 4.2.2.9. Revenue Efficiency and New Revenue Efficiency Models In these models, DMUs are removed from the calculations with a nonpositive input, a non-positive output, or a negative output. Unit price columns should also be expressed as (P) in the titles of the outputs. If the output is priceless, it means that its price is zero. On the other hand, DMUs with 𝑝𝑟𝑖𝑐𝑒 𝑚𝑎𝑥 ≤ 0 or 𝑝𝑟𝑖𝑐𝑒 𝑚𝑖𝑛 < 0 are removed from the comparison group. 4.2.2.10. Profit Efficiency and New Profit Efficiency Models These models can be used to combine cost efficiency and revenue efficiency models, which have cost columns for inputs and price columns for outputs (Figure 8). Also, DMUs are removed with 𝑐𝑜𝑠𝑡 𝑚𝑎𝑥 ≤ 0 or 𝑐𝑜𝑠𝑡 𝑚𝑖𝑛 < 0.

Figure 8. Data format in the profit efficiency model.

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4.2.2.11. Malmquist-Radial and Malmquist, Window Analysis Models Figure 9 is an example with three time periods in which (A1) cell represents the name of the problem and cell (B1) represents the first time period. In other words, the first row contains time periods, and the data and name of the DMUs are similar to the general format of writing data in a worksheet. It is worth noting that each time period is placed at the upper left corner of the sheet and input/output has the same names throughout the time periods. There are no restrictions on the Window-I-C, Window-I-V, and Window-O-C models, however, DMUs are considered as a unit with zero. In addition, there are no limits for the Window-O-V model, and the DMU with 𝑦𝑚𝑎𝑥 ≤ 0 as the unit with zero efficiency is considered to form the efficiency score matrix.

Figure 9. Data entry in the Window analysis, Malmquist, and Malmquist-radial models.

4.2.2.12. Scale Elasticity Model This model is categorized as CRS, IRS, and DRS, and their data format is similar to the BCC or CCR models (Figure 2). All DMUs with nonpositive output are removed from the calculations. 4.2.2.13. Congestion Model This model detects congested DMUs and finds the sources of congestion quantitatively. The data format is the same as the CCR model

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(Figure 2). In this model, DMUs are removed from the calculations with non-positive output.

4.2.2.14. Undesirable Outputs Model The undesirable outputs model is divided into two models of simple bad (undesirable) outputs and non-separable good and bad outputs. In the bad outputs model, the outputs are divided into good (desirable) and bad (undesirable). Both types of outputs have no integer relation to the NonSeparable model where reduction of bad outputs inevitably reduces good outputs, and the good outputs are the same as normal outputs (O), and bad outputs are considered as "OBad". As shown in Figure 10, "rehospital" is a bad output in the evaluation of 12 hospitals. Also, DMUs are removed from the calculations with non-positive inputs, and the software replaces a very small positive number (10−8 ) instead of non-positive numbers in the inputs or outputs.

Figure 10. Data format in the undesirable output model.

In the non-separable model, many of the bad output are related to good outputs and inputs. Therefore, decreasing bad outputs will decrease the good outputs as well as some inputs. In order to consider the data in this model, the input(s) and output(s) should be considered as follows:

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76 • • • • •

Non-separable good output (ONS Good) Separable good output (OS Good) Non-separable bad output (ONS Bad) Separable input (IS) Non-separable input (INS)

In the undesirable output model, DMUs are removed from the calculations with non-positive inputs or outputs, and a very small number (10−8 ) is assigned to the non-positive values of each input or output.

4.2.2.15. Weighted SBM Model In the weighted SBM model, inputs and outputs are weighted, and these weights are written in rows after the main data set with one blank row. For example, as shown in Figure 11, row 15 is assigned to the input weights of "weight I", and row 16 is assigned to the output weights of "weight O".

Figure 11. Data entry in the weighted SBM model.

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4.2.2.16. Hybrid Model This model has two kinds of (inputs/outputs) radial and non-radial. Radial inputs and outputs are shown as (IR) and (OR) and non-radial inputs and outputs are shown as (INR) and (ONR), respectively. 4.2.2.17. Network DEA (Network SBM) Model As shown in Figure 12, each part of these models should have at least one input and one output, and all divisions should be connected by links. The data format of this model should be defined in the relevant worksheet as follows: ✓ In cell (B1), the title of the problem must be stated; ✓ The division should be considered as (D) in the second row. Thus, the first division is a set in the cell (B2) and the associated weight is placed to the right of the cell after the group name (e.g., cell C2). Weights are normalized by software (all cell weights are taken into account, even if the cells are empty); ✓ Under the division name, inputs and outputs to the division are denoted by the symbols (I) and (O) in the third row. (IN) and (ON) are also used for "fixed input" and "fixed output"; ✓ After defining the previous steps, the link from this division is expressed from the next column (e.g., cell D3). "Link" has the heading and the destination division in the parenthesis, the following 4 links are used: • (LF) for free link • (LG) for good (desirable) link • (LB) for bad (undesirable) link • (LN) for fixed (non-discretionary) link ✓ Naming this section is "(link name) (category destination) (link type)" such as (Link12) (Div2) (LF); ✓ The data associated with each column, i.e., the input and output values or the links values for DMUs are determined from the 4th row (this case is the main part of the network data);

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Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili ✓ Each division must have at least one input or output and all divisions must be connected by links, where the circular structure is permitted. ✓ In this model, zero and negative data are replaced by a small positive number and not counted in efficiency.

Figure 12. Data format in the network DEA model.

4.2.2.18. Dynamic DEA (Dynamic SBM) Model The data format in the Excel worksheet for this model is as follows: ✓ The first row contains information related to the time periods and cells (B1) and (C1) represent the "term" and its weights. The weights are relative and the software is normalized them accordingly. If the cell is associated with empty weights, the weights associated with all periods are qualified; ✓ The title of the problem is given in cell (B2); ✓ The third row contains information related to input, output, and carry-over, and its symbols are defined as follows: ▪ IN: Indicates ordinary input and fixed non-discretionary input ▪ O and ON: Indicates ordinary output and fixed non-discretionary output ✓ The transition from one period to the next, after defining a section of each input and output, is expressed in the next column (e.i. cell (D3)). The classification of carry-overs is similar to the previous model;

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✓ In the fourth row, the name of the DMUs and the values associated with the inputs and outputs are stated. As shown in Figure 13, "sub nurses" will be the input (normal) or fixed input (non-discretionary) in the evaluation of 12 hospitals. In this model, zero and negative data are replaced by a small positive number and are not counted in efficiency.

Figure 13. Data entry in dynamic DEA (Dynamic SBM) model.

4.2.2.19. Dynamic and Network SBM (DNSBM) Model In this model, each division must have at least one input or output and all divisions must be connected. The data format in the Excel worksheet is as follows: ✓ The first row contains period time information, and cells (B1) and (C1) represent the "term" and its weights. The software normalizes the weights relative to the data. If the cell associated with the weights is empty, the weights associated with all periods are considered. ✓ The division is named with (D) which should be mentioned in the second row. Thus, the first division is expressed in the cell (B2). The weights associated with each division are located on the right of the division name cell such as cell (C2). ✓ The third row contains information related to input, output, link, and carry-over. The abbreviations associated with this model are as follows: ▪ IN: Indicates ordinary input and fixed non-discretionary input

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O and ON: Indicates ordinary output and fixed nondiscretionary output ✓ After defining the input and output titles, the links and carry-overs are specified. The link titles include (LF), (LG), (LB), and (LN), and similarly, four types of carry-overs (CF), (CG), (CB), and (CN) are considered. ✓ From the fourth row, the name of the DMUs and its observed values are stated. In this model, zero and negative data are replaced by a small positive number and are not counted in efficiency.

4.2.2.20. Non-Convex Model The data format in this model is similar to the CCR model. However, the difference is that the cluster column is added in this model. Figure 14 shows an example of the non-convex model. If the cluster column is empty, the DMUs are divided into several clusters by the software depending on the RTS. In this model, it is recommended to use clustering methods.

Figure 14. Data format in the non-convex model.

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In the non-convex (priority)-SBM-I(O) model, the "priority" of input and output after the data set is shown in Figure 15. If the priorities are on "slack", this option can be used. For instance, the input of the number of doctors has the highest priority and their order of priority is the number of doctors and the number of nurses.

Figure 15. Data format in non-convex(priority)-SBM-I(O).

4.2.2.21. Resampling Model This model is divided into "resample triangular", "resample triangular historical", "resample past present" and "resample past present future" models. In all of these models, with variation on the radial and non-radial measures along with variations on the input-oriented, output-oriented, and non-oriented models, the efficiency and super-efficiency scores are calculated [103, 104]. If the number of DMUs is relatively small compared to the number of inputs and outputs, it is recommended to apply non-radial or SBM models instead of super-radial or super-SBM models. Therefore, the super-efficiency scores in these cases increase dramatically. In the "resample triangular" model, the upper and lower bounds of the input and output error rates are required. For instance, as shown in Figure 16,

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the upper bound of the first input is 3.4% and the lower bound is 2% (these numbers are based on the knowledge of experts and their experience).

Figure 16. Data entry in the resample triangular model.

In the "resample triangular historical" model, the upper and lower error bounds are obtained by using the historical data. Like Figure 7, which is related to the years 2017-2019, 2017 and 2018 are considered as past and 2019 as the current year, and the upper and lower error bounds for 2019 are obtained. Then, the super-efficiency scores are calculated for the present period and the data variations for the last periods are estimated. The data format in the "resample past present" model is similar to the "resample triangular historical" model. In the previous model, the upper and lower error rates were calculated for each DMU, which varies between DMUs. In this model, the input and output variations are calculated separately for each DMU. In this model, input and output for the next year (2020) are predicted and efficiency scores are calculated with a confidence interval. In these conditions, predictive models such as Lucas weight and the average of trend are used.

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4.2.2.22. DDF Model As shown in Figure 17, this model has two types of inputs and three types of outputs as follows: ▪

Directional Inputs (I): This input is related to 𝑥𝑖𝑗(I) (𝑖 = 1, ⋯ , I , 𝑗 = (I)

(I)

1, ⋯ , 𝑛) expressing the directional vector 𝑥𝑖𝑗 = 𝑑𝑖𝑗 . (IN)

▪

Non-directional inputs (IN): This input is related to 𝑥𝑖𝑗

▪

1, ⋯ , IN , 𝑗 = 1, ⋯ , 𝑛) and has no directional effect on the efficiency score which can be left empty; Directional (Good) output (O): This output represents the (O)

(O)

directional vector 𝑦𝑖𝑗 = 𝑑𝑖𝑗

(𝑖 =

(O)

and related values with 𝑦𝑖𝑗 (𝑖 =

1, ⋯ , O , 𝑗 = 1, ⋯ , 𝑛); ▪

(ON)

(ON)

Non-directional outputs (ON): This output represents 𝑦𝑖𝑗 vectors

and

related

values

with

(ON)

𝑦𝑖𝑗

= 𝑑𝑖𝑗

(𝑖 = 1, ⋯ , ON , 𝑗 =

1, ⋯ , 𝑛) and has no directional effect on the efficiency score that the associated class can be left empty; ▪

(OBad)

Undesirable Output (OBad): This output indicates the 𝑦𝑖𝑗 (OBad) 𝑑𝑖𝑗

directional vector and associated values with

(OBad) 𝑦𝑖𝑗

1, ⋯ , OBad , 𝑗 = 1, ⋯ , 𝑛) that this class can also be empty.

Figure 17. Data format in the DDF model.

=

(𝑖 =

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This model is divided into input-(DD-I-C (V)), output-(DD-O-C (V), and non-oriented (DD-C (V)) types. There are CRS and VRS which can be used to calculate efficiency and super-efficiency scores.

4.2.3. Types of Results for Worksheets In DEA Solver software, there are various worksheets to display the results, which is a brief description of the worksheets.

4.2.3.1. Score The score worksheet contains the DMU efficiency score, reference set, lambda values for each DMU in the reference set, and unit ranking. 4.2.3.2. Projection In this worksheet, the projection of each DMU is shown on the efficient frontier, which is analyzed by the model. In other words, it shows the exact amount and percentage changes of each input and output for inefficient units (if they want to be in the efficient frontier). 4.2.3.3. Weight The optimal weights 𝑣(𝑖) and 𝑢(𝑟) for inputs and outputs are defined in the weight worksheet. 𝑣(0) and 𝑢(0) are related to 𝜆1 + 𝜆2 + ⋯ + 𝜆𝑛 ≥L and 𝜆1 + 𝜆2 + ⋯ + 𝜆𝑛 ≤ U, respectively. In the BCC model, L = U = 1 and 𝑢(0) represent the value of the dual variable for this constraint. 4.2.3.4. Weighted Data Weighted data worksheet represents the values of the optimal weights of inputs and outputs (𝑥𝑖𝑗 𝑣(𝑖) and 𝑦𝑖𝑗 𝑢(𝑟)) for all DMUs. 4.2.3.5. Summary This worksheet shows the statistical data and a summary report of the obtained results, which is displayed in all models. At the beginning of this worksheet, it is possible to see a summary of the problem information

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including the number of DMUs, inputs, outputs, and RTS. Statistical information associated with the problem is stated separately. At the bottom of the page is about the number of efficient and inefficient units, as well as the number of repetitions and the time required to solve the problem.

4.2.3.6. Slack The input excess (s − ) and output shortfall (𝑠 + ) values for each DMU are shown in this worksheet. 4.2.3.7. Graph 1 This worksheet shows a chart for DEA scores which can be redesign by using the graph functions of Excel. 4.2.3.8. Graph 2 Efficiency scores are displayed in ascending order in this worksheet. 4.2.3.9. RTS In BCC, AR-I-V, and AR-O-V models, the RTS characteristic is saved in the RTS worksheet. RTS characteristic for inefficient DMUs is projected on the efficient frontier. 4.2.3.10. Malmquist k Results related to efficiency change indicators, frontier change, and Malmquist, which are within 𝑘 = 1, ⋯ , L, are presented in the "Malmquist k" worksheets. In each worksheet, the form related to the calculation of indicators is as follows. 4.2.3.11. Scale Elasticity The format of the input data in the scale elasticity model is similar to the CCR model. This worksheet is only for "elasticity-O". The inefficient units of the BCC problem are projected on the efficient frontier points and their elasticity is calculated based on the projected points.

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4.2.3.12. Congestion The format of the input data is similar to the CCR model and the results are related to congestion and scale diseconomies. 4.2.3.13. Resample Summary Resample summary worksheets are used for the resampling models. In these models, 95%, 80%, and 60% confidence interval worksheets are created separately with the corresponding charts. 4.2.3.14. Window k This worksheet is only for window analysis models in the range of 𝑘 = 1, ⋯ , L (length of time periods in the data). In this worksheet, there are usually two sets of information associated with "variations through window" and "variations by term". "Variations through window" values are calculated as the average score per row, and "variations by term" represents the average score per column. 4.2.3.15. Decomposition In Hybrid models, SBM, there are bad (undesirable) and non-separable outputs of decomposition worksheet, which includes the inefficiency factor for each input or output. 4.2.3.16. SASvsCRSvsVRS This worksheet is created in non-convex models and provides results such as efficiency scores, means, standard deviations of scores, and DMU rankings in various RTS.

4.3. RUNNING THE DEA SOLVER SOFTWARE After defining the data in the Excel worksheet, DEA Solver software can be run. To perform the calculations, the steps in Figure 18 must be performed, and after choosing the model (step 1), the data file is defined

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(step 2). Then, the software runs (step 3) and the run ends (step 4) after saving the final file.

Figure 18. Software running steps.

Based on the data of 12 hospitals (Table 1 in chapter 2) and explanations about the data entry of each model, after running the software and performing calculations, the results are saved in selected worksheets in the Excel file. Each worksheet has different information about the results of the selected problem, which are described above.

4.3.1. Score As shown in Figure 19, information is provided including the DMU efficiency score, reference set, λ values for each DMU in the reference set, and the DMU ranking for the CCR-I problem.

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Figure 19. Score worksheet.

4.3.2. Projection As displayed in Figure 20, the exact amount and percentage changes of each input and output for inefficient units are specified.

Figure 20. Projection worksheet.

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4.3.3. Weight The 𝑣(𝑖) and 𝑢(𝑟) values are shown in this worksheet (Figure 21).

Figure 21. Weights worksheet.

4.3.4. Weighted Data As shown in Figure 22, this worksheet represents the values of the optimal weights of inputs and outputs.

Figure 22. Weighted data worksheet.

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4.3.5. Summary As shown in Figure 23, statistical information is obtained from the data and a summary of the results such as the number of DMUs, inputs, outputs, RTS, and information about the number of efficient and inefficient units, as well as the number of repetitions and the time required to solve the problem are reported.

Figure 23. Summary worksheet.

4.3.6. Slack Slack worksheet input excess (s − ) and output shortfall (s + ) values as well as efficiency scores for each DMU (Figure 24).

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Figure 24. Slack worksheet.

4.3.7. Graph 1 As shown in Figure 25, this worksheet graphically shows efficiency scores.

Figure 25. Graph 1 worksheet.

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4.3.8. Graph 2 Efficiency scores are shown in ascending order in this worksheet (Figure 26).

Figure 26. Graph 2 worksheet.

4.3.9. RTS In the BCC, AR-I-V, and AR-O-V models, the RTS characteristic is saved in the RTS worksheet (Figure 27).

Figure 27. RTS worksheet.

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4.3.10. Malmquist k Based on the information in Figure 9, the Malmquist index calculations in three time periods are as shown in Figure 28.

Figure 28. Malmquist k worksheet.

4.3.11. Scale Elasticity The data entry in the scale elasticity model is similar to the CCR model. Therefore, based on the evaluation of hospitals with two outputs and two inputs, the results are as shown in Figure 29.

Figure 29. Scale elasticity worksheet.

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4.3.12. Congestion The data format is similar to the CCR model. The summary of this example for the congestion model is shown in Figure 30, in which units F and I are congested.

Figure 30. Congestion worksheet.

4.3.13. Resample Summary These worksheets can be observed in the resampling models. As shown in Figure 31, the results are saved in the "resample summary" worksheet.

Figure 31. Resample summary worksheet.

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4.3.14. Window k This worksheet is only for the window analysis model and range of 𝑘 = 1, ⋯ , L (length of time periods in the data). For instance, according to Figure 9, the evaluation of hospitals is considered and the reports for the third time period are presented in Figure 32.

Figure 32. Window worksheet.

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4.3.15. Decomposition The results of hybrid, SBM, bad (undesirable), and non-separable outputs models can be displayed, and also includes the inefficiency factor for each input or output. Due to the evaluation of hospitals, the results are as Figure 33.

Figure 33. Decomposition worksheet.

4.3.16. SASvsCRSvsVRS This worksheet is created in non-convex models and the results are shown in Figure 34.

Figure 34. SASvsCRSvsVRS worksheet.

Congestion ✓

✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

New Cost Efficiency

Profit Efficiency

New Profit Efficiency

Revenue Efficiency

New Revenue Efficiency

Revenue vs Cost (ratio)

Non-controllable

Dynamic SBM (Non-oriented)

Dynamic SBM (Oriented)

EBM Non-oriented

EBM Oriented

✓

✓

✓ ✓

Summary

Cost Efficiency

Score

✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

Rank

Congestion

Graph 1

CCR

Graph 2

✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

Projection

Categorical Varable

Weight

✓ ✓ ✓

Weightdata

Bounded Varable

Slack

✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

RTS

Bilateral

✓

Solution

✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

Decomposition

BCC

Division Score

✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

Elasticity

Assurance Region Global

Projection Input/ Output

Assurance Region

Model

Worksheet

Table 6. Worksheets of models in DEA Solver Projection Carry Over (Free)

✓

Projection Carry Over (Bad)

✓

DEA Solver Software 97

Window k

Up Side

Down Side

CI60%

CI80%

CI95%

Resample Summary

Resample Score

Last and Future Priods

Total Slack

SAS Projection

SAS vs VRS vs CRS

Scale Cluster Efficiency

SAS Score

Scale Elasticity

Malmquist k

Decomposition

Elasticity

✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

Scale Elasticity

Super Radial

Super SBM Non-oriented

Super SBM oriented

FDH

Undesirable Outputs

Weighted SBM

✓

✓

✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

SBM Oriented

✓

✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

✓ ✓ ✓

✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

✓ ✓

SBM Non-oriented

Summary

✓ ✓ ✓ ✓ ✓ ✓

Score

✓

Rank

✓

Graph 1

Resampling Tringular Historical

Graph 2

✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

Projection

Resampling

Weight

Decreasing RTS

Weightdata

✓ ✓

Slack

✓ ✓ ✓ ✓ ✓ ✓

RTS

Non-discritionary

Congestion

✓ ✓ ✓ ✓ ✓

Solution

✓

Division Score

Non-convex- SBM

Projection Input/Output

✓

Projection Carry Over(Bad)

✓

Scale Elasticity

✓

SAS Score

Non-convex-Radial

✓

Malmquist k

✓

Scale Cluster Efficiency

Network SBM (oriented)

Projection Carry Over(Free)

✓ ✓

SAS vs VRS vs CRS

✓

SAS Projection

✓

Total Slack

✓

Last and Future Priods

Network SBM (Non-oriented)

Resample Score

Malmquist

✓

Resample Summary

✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

CI95%

✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

CI80%

Increasing RTS

CI60%

✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

Down Side

Hybrid

Up Side

Generalized RTS

Model

Worksheet

Table 6. (Continued)

98 Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

Window k

Projection Link (as Output) Projection Link (as Input) Projection Link (Free)

Summary

Model

✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

✓ ✓ ✓ ✓ ✓

✓

Score

✓

Catch Up (Generation)

Dynamic etwork (Oriented)

Frontier Shift (Generation)

✓

✓

Malmquist (Generation)

✓

Catch Up (Transmission)

✓

Frontier Shift (Transmission)

✓

Malmquist (Transmission)

✓

Catch Up (Distribution)

✓

Frontier Shift (Distribution)

Dynamic etwork (Non-oriented)

Malmquist (Distribution)

Worksheet

Table 6. (Continued)

Malmquist Overall

✓

CmltMalmquist (Generation)

✓

✓

✓

CmltMalmquist (Distribution) CmltMalmquist (Transmission)

✓

CmltMalmquist Overall

✓

✓

Division Score

✓

Projection Input/Output

✓

Projection Carry Over (Free)

✓

Projection Carry Over (Good)

✓

Projection Carry Over (Bad)

✓

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CONCLUSION DEA Solver software contains a wide range of different DEA models and each update can increase the number of them. This software runs in Microsoft Excel and has a special place among users in terms of simplicity in application and comprehensive and complete reports. As shown in Table 6, this software has result worksheets specific to the respective models, and the information of each model is saved in special worksheets. To date, this software has been used in various studies, articles, and books. Therefore, DEA Solver can be considered as one of the most popular software in DEA.

Chapter 5

DEAP SOFTWARE 5.1. INTRODUCTION The Centre for Efficiency and Productivity Analysis (CEPA) of the Queensland University has developed a variety of software packages for research, consulting, training, and efficiency analysis, one of which is called Data Envelopment Analysis Program (DEAP) [105]. This software is designed by Tim Coelli, a member of the center, which runs under the "MS-DOS" system.

5.2. SOFTWARE PRESENTATION In order to run DEAP software, four files are necessary and sufficient, which are reviewed.

5.2.1. Executable File (DEAP.EXE) To run DEAP software, the executable file "DEAP.EXE" is necessary and the software environment can be seen in Figure 1. In the last line of it,

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the file name must be entered to save the problem information. The results of solving the problem based on the selected model are saved in several sections.

Figure 1. DEAP software environment.

5.2.2. Data File (EG-dta.txt) A list of inputs and outputs is first entered in software such as Excel, and this data must be displayed in order from left to right across the file. In the data file, the output columns are listed first followed by the input columns. Then, the data is saved as text files. Likewise, there should be no text in these files and the names of variables and DMUs should be removed. On the other hand, because of the MS-DOS system limitations, file names must satisfy certain restrictions according to the following: ✓ File names must be up to 12 characters; ✓ Three characters after (.) moreover, eight characters before (.) (xxxxxxxx.xxx).

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Also, the data file must have a "txt" file extension and be saved with the desired title in the folder of the software.

5.2.3. Instruction File (EG-ins.txt) The instruction file is as shown in Figure 2 and is named "EG1ins.txt".

Figure 2. Instruction file.

As shown in Figure 2, the following items can be determined for each problem: • • • • • • • • •

Input data file name (DATA FILE NAME) Software report file name (OUTPUT FILE NAME) Number of DMUs (NUMBER OF FIRMS) Number of time periods Outputs (NUMBER OF OUTPUTS) Inputs (NUMBER OF INPUTS) Input-oriented or output-oriented (input-oriented model (0) and output-oriented model (1)) RTS (CRS (0) and VRS (1)) Model (based on the numbers in Table 1).

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Table 1. Determining the model in DEAP software Model DEA (MULTI-STAGE) COST-DEA MALMQUIST-DEA DEA (1-STAGE) DEA (2-STAGE)

Number 0 1 2 3 4

This file is saved after defining the data and selecting the model.

5.2.4. Results File (EG-out.txt) The results of solving the problem are saved in several sections in the output file "EG-out.txt" according to the selected model. In the initial section, similar to Figure 3, general information about the problem and the type of model selected can be observed.

Figure 3. Result file.

The results of various models are different, but the result file is as follows: • •

efficiency scores Slack variables for outputs

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Slack variables for inputs Table of peers and related weights Summary of the peer number for each DMU The main input value to be on the efficient frontier The main value of the output to be on the efficient frontier Scores of each DMU

5.3. RUNNING THE DEAP SOFTWARE Based on the data of the evaluation of 12 hospitals with two inputs and outputs (Table 1 in chapter 2), as shown in Figure 4, the data file is formed as "EG1-dta.txt".

Figure 4. Data entry for evaluating hospitals.

The results of solving the single-stage model with the CRS (Figure 2) are as follows:

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Results from DEAP Version 2.1 Instruction file = EG1-ins.txt Data file = eg1-dta.txt Input oriented DEA Scale assumption: CRS Single-stage DEA - residual slacks presented EFFICIENCY SUMMARY: firm te 1 1.000 2 1.000 3 0.883 4 1.000 5 0.763 6 0.835 7 0.902 8 0.796 9 0.960 10 0.871 11 0.955 12 0.958 mean 0.910 SUMMARY OF OUTPUT SLACKS: firm output: 1 1 0.000 2 0.000 3 0.000 4 0.000 5 0.000

2 0.000 0.000 0.000 0.000 0.000

DEAP Software 6 7 8 9 10 11 12

0.000 0.000 0.000 0.000 0.000 0.000 0.000

mean

0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000

0.000

SUMMARY OF INPUT SLACKS: firm input: 1 2 1 0.000 0.000 2 0.000 0.000 3 1.218 0.000 4 0.000 0.000 5 0.000 0.000 6 12.112 0.000 7 0.000 3.349 8 0.000 0.000 9 0.000 27.206 10 6.032 0.000 11 7.320 0.000 12 0.000 12.601 mean

2.224

3.596

SUMMARY OF PEERS: firm 1 2 3

peers: 1 2 2 4

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108 4 5 6 7 8 9 10 11 12

Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili 4 2 2 2 2 2 4 1 2

1 4 4 1 1 4 1 4 1

SUMMARY OF PEER WEIGHTS: (in same order as above) firm peer weights: 1 1.000 2 1.000 3 0.900 0.139 4 1.000 5 0.057 0.579 0.153 6 0.200 1.111 7 1.294 0.259 8 0.013 0.387 0.618 9 0.835 0.647 10 1.389 11 0.860 0.967 12 1.235 0.647 PEER COUNT SUMMARY: (i.e., no. times each firm is a peer for another) firm peer count: 1 6 2 7 3 0 4 6

DEAP Software 5 6 7 8 9 10 11 12

0 0 0 0 0 0 0 0

SUMMARY OF OUTPUT TARGETS: firm output: 1 1 100.000 2 150.000 3 160.000 4 180.000 5 94.000 6 230.000 7 220.000 8 152.000 9 190.000 10 250.000 11 260.000 12 250.000

2 90.000 50.000 55.000 72.000 66.000 90.000 88.000 80.000 100.000 100.000 147.000 120.000

SUMMARY OF INPUT TARGETS: firm input: 1 1 20.000 2 19.000 3 20.850 4 27.000 5 16.797 6 33.800 7 29.765

2 151.000 131.000 141.233 168.000 120.633 212.867 208.612

109

110 8 9 10 11 12

Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili 24.686 28.812 37.500 43.300 36.412

164.045 207.129 233.333 292.260 259.529

FIRM BY FIRM RESULTS: Results for firm: 1 Technical efficiency = 1.000 PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 100.000 0.000 0.000 100.000 output 2 90.000 0.000 0.000 90.000 input 1 20.000 0.000 0.000 20.000 input 2 151.000 0.000 0.000 151.000 LISTING OF PEERS: peer lambda weight 1 1.000 Results for firm: 2 Technical efficiency = 1.000 PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 150.000 0.000 0.000 150.000 output 2 50.000 0.000 0.000 50.000 input 1 19.000 0.000 0.000 19.000 input 2 131.000 0.000 0.000 131.000 LISTING OF PEERS: peer lambda weight 2 1.000

DEAP Software Results for firm: 3 Technical efficiency = 0.883 PROJECTION SUMMARY: variable original radial slack value movement movement output 1 160.000 0.000 0.000 output 2 55.000 0.000 0.000 input 1 25.000 -2.932 -1.218 input 2 160.000 -18.767 0.000 LISTING OF PEERS: peer lambda weight 2 0.900 4 0.139

111

projected value 160.000 55.000 20.850 141.233

Results for firm: 4 Technical efficiency = 1.000 PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 180.000 0.000 0.000 180.000 output 2 72.000 0.000 0.000 72.000 input 1 27.000 0.000 0.000 27.000 input 2 168.000 0.000 0.000 168.000 LISTING OF PEERS: peer lambda weight 4 1.000 Results for firm: 5 Technical efficiency = 0.763 PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 94.000 0.000 0.000 94.000 output 2 66.000 0.000 0.000 66.000

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Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

input 1 22.000 input 2 158.000 LISTING OF PEERS: peer lambda weight 2 0.057 1 0.579 4 0.153

-5.203 -37.367

Results for firm: 6 Technical efficiency = 0.835 PROJECTION SUMMARY: variable original radial value movement output 1 230.000 0.000 output 2 90.000 0.000 input 1 55.000 -9.088 input 2 255.000 -42.133 LISTING OF PEERS: peer lambda weight 2 0.200 4 1.111 Results for firm: 7 Technical efficiency = 0.902 PROJECTION SUMMARY: variable original radial value movement output 1 220.000 0.000 output 2 88.000 0.000 input 1 33.000 -3.235 input 2 235.000 -23.039 LISTING OF PEERS: peer lambda weight 2 1.294 1 0.259

0.000 0.000

16.797 120.633

slack movement 0.000 0.000 -12.112 0.000

projected value 230.000 90.000 33.800 212.867

slack projected movement value 0.000 220.000 0.000 88.000 0.000 29.765 -3.349 208.612

DEAP Software

113

Results for firm: 8 Technical efficiency = 0.796 PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 152.000 0.000 0.000 152.000 output 2 80.000 0.000 0.000 80.000 input 1 31.000 -6.314 0.000 24.686 input 2 206.000 -41.955 0.000 164.045 LISTING OF PEERS: peer lambda weight 2 0.013 1 0.387 4 0.618 Results for firm: 9 Technical efficiency = 0.960 PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 190.000 0.000 0.000 190.000 output 2 100.000 0.000 0.000 100.000 input 1 30.000 -1.188 0.000 28.812 input 2 244.000 -9.664 -27.206 207.129 LISTING OF PEERS: peer lambda weight 2 0.835 1 0.647 Results for firm: 10 Technical efficiency = 0.871 PROJECTION SUMMARY: variable original radial slack value movement movement output 1 250.000 0.000 0.000

projected value 250.000

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Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

output 2 100.000 input 1 50.000 input 2 268.000 LISTING OF PEERS: peer lambda weight 4 1.389

0.000 -6.468 -34.667

0.000 -6.032 0.000

100.000 37.500 233.333

Results for firm: 11 Technical efficiency = 0.955 PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 260.000 0.000 0.000 260.000 output 2 147.000 0.000 0.000 147.000 input 1 53.000 -2.380 -7.320 43.300 input 2 306.000 -13.740 0.000 292.260 LISTING OF PEERS: peer lambda weight 1 0.860 4 0.967 Results for firm: 12 Technical efficiency = 0.958 PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 250.000 0.000 0.000 250.000 output 2 120.000 0.000 0.000 120.000 input 1 38.000 -1.588 0.000 36.412 input 2 284.000 -11.870 -12.601 259.529 LISTING OF PEERS: peer lambda weight 2 1.235 1 0.647

DEAP Software

115

The software output in the two-stage model with CRS (data file titled "EG2-dta.txt") is as follows: Results from DEAP Version 2.1 Instruction file = EG2-ins.txt Data file = eg2-dta.txt Input oriented DEA Scale assumption: CRS Two-stage DEA method EFFICIENCY SUMMARY: Firm te 1 1.000 2 1.000 3 0.883 4 1.000 5 0.763 6 0.835 7 0.902 8 0.796 9 0.960 10 0.871 11 0.955 12 0.958 mean 0.910 SUMMARY OF OUTPUT SLACKS: firm output: 1 1 0.000 2 0.000

2 0.000 0.000

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Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

3 4 5 6 7 8 9 10 11 12

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

mean

0.000

0.000

SUMMARY OF INPUT SLACKS: firm input: 1 1 0.000 2 0.000 3 1.218 4 0.000 5 0.000 6 12.112 7 0.000 8 0.000 9 0.000 10 6.032 11 7.320 12 0.000 mean

2.224

2 0.000 0.000 0.000 0.000 0.000 0.000 3.349 0.000 27.206 0.000 0.000 12.601 3.596

DEAP Software SUMMARY OF PEERS: firm peers: 1 1 2 2 3 4 2 4 4 5 2 4 1 6 4 2 7 1 2 8 4 1 2 9 2 1 10 4 11 1 4 12 2 1 SUMMARY OF PEER WEIGHTS: (in same order as above) firm 1 2 3 4 5 6 7 8 9 10 11 12

peer weights: 1.000 1.000 0.139 0.900 1.000 0.057 0.153 0.579 1.111 0.200 0.259 1.294 0.618 0.387 0.013 0.835 0.647 1.389 0.860 0.967 1.235 0.647

117

118

Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

PEER COUNT SUMMARY: (i.e., no. times each firm is a peer for another) firm peer count: 1 6 2 7 3 0 4 6 5 0 6 0 7 0 8 0 9 0 10 0 11 0 12 0 SUMMARY OF OUTPUT TARGETS: firm output: 1 1 100.000 2 150.000 3 160.000 4 180.000 5 94.000 6 230.000 7 220.000 8 152.000 9 190.000 10 250.000 11 260.000 12 250.000

2 90.000 50.000 55.000 72.000 66.000 90.000 88.000 80.000 100.000 100.000 147.000 120.000

DEAP Software SUMMARY OF INPUT TARGETS: firm input: 1 1 20.000 2 19.000 3 20.850 4 27.000 5 16.797 6 33.800 7 29.765 8 24.686 9 28.812 10 37.500 11 43.300 12 36.412

2 151.000 131.000 141.233 168.000 120.633 212.867 208.612 164.045 207.129 233.333 292.260 259.529

FIRM BY FIRM RESULTS: Results for firm: 1 Technical efficiency = 1.000 PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 100.000 0.000 0.000 100.000 output 2 90.000 0.000 0.000 90.000 input 1 20.000 0.000 0.000 20.000 input 2 151.000 0.000 0.000 151.000 LISTING OF PEERS: peer lambda weight 1 1.000

119

120

Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

Results for firm: 2 Technical efficiency = 1.000 PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 150.000 0.000 0.000 150.000 output 2 50.000 0.000 0.000 50.000 input 1 19.000 0.000 0.000 19.000 input 2 131.000 0.000 0.000 131.000 LISTING OF PEERS: peer lambda weight 2 1.000 Results for firm: 3 Technical efficiency = 0.883 PROJECTION SUMMARY: variable original radial value movement output 1 160.000 0.000 output 2 55.000 0.000 input 1 25.000 -2.932 input 2 160.000 -18.767 LISTING OF PEERS: peer lambda weight 4 0.139 2 0.900

slack projected movement value 0.000 160.000 0.000 55.000 -1.218 20.850 0.000 141.233

Results for firm: 4 Technical efficiency = 1.000 PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 180.000 0.000 0.000 180.000 output 2 72.000 0.000 0.000 72.000

DEAP Software input 1 27.000 input 2 168.000 LISTING OF PEERS: peer lambda weight 4 1.000

0.000 0.000

0.000 0.000

121 27.000 168.000

Results for firm: 5 Technical efficiency = 0.763 PROJECTION SUMMARY: Variable original radial slack projected value movement movement value output 1 94.000 0.000 0.000 94.000 output 2 66.000 0.000 0.000 66.000 input 1 22.000 -5.203 0.000 16.797 input 2 158.000 -37.367 0.000 120.633 LISTING OF PEERS: peer lambda weight 2 0.057 4 0.153 1 0.579 Results for firm: 6 Technical efficiency = 0.835 PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 230.000 0.000 0.000 230.000 output 2 90.000 0.000 0.000 90.000 input 1 55.000 -9.088 -12.112 33.800 input 2 255.000 -42.133 0.000 212.867 LISTING OF PEERS: peer lambda weight 4 1.111 2 0.200

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Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

Results for firm: 7 Technical efficiency = 0.902 PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 220.000 0.000 0.000 220.000 output 2 88.000 0.000 0.000 88.000 input 1 33.000 -3.235 0.000 29.765 input 2 235.000 -23.039 -3.349 208.612 LISTING OF PEERS: peer lambda weight 1 0.259 2 1.294 Results for firm: 8 Technical efficiency = 0.796 PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 152.000 0.000 0.000 152.000 output 2 80.000 0.000 0.000 80.000 input 1 31.000 -6.314 0.000 24.686 input 2 206.000 -41.955 0.000 164.045 LISTING OF PEERS: peer lambda weight 4 0.618 1 0.387 2 0.013 Results for firm: 9 Technical efficiency = 0.960 PROJECTION SUMMARY: variable original radial slack projected value movement movement value

DEAP Software output 1 190.000 output 2 100.000 input 1 30.000 input 2 244.000 LISTING OF PEERS: peer lambda weight 2 0.835 1 0.647

0.000 0.000 -1.188 -9.664

0.000 0.000 0.000 -27.206

123 190.000 100.000 28.812 207.129

Results for firm: 10 Technical efficiency = 0.871 PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 250.000 0.000 0.000 250.000 output 2 100.000 0.000 0.000 100.000 input 1 50.000 -6.468 -6.032 37.500 input 2 268.000 -34.667 0.000 233.333 LISTING OF PEERS: peer lambda weight 4 1.389 Results for firm: 11 Technical efficiency = 0.955 PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 260.000 0.000 0.000 260.000 output 2 147.000 0.000 0.000 147.000 input 1 53.000 -2.380 -7.320 43.300 input 2 306.000 -13.740 0.000 292.260

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Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

LISTING OF PEERS: peer lambda weight 1 0.860 4 0.967 Results for firm: 12 Technical efficiency = 0.958 PROJECTION SUMMARY: variable original radial slack value movement movement output 1 250.000 0.000 0.000 output 2 120.000 0.000 0.000 input 1 38.000 -1.588 0.000 input 2 284.000 -11.870 -12.601 LISTING OF PEERS: peer lambda weight 2 1.235 1 0.647

projected value 250.000 120.000 36.412 259.529

The results for the multi-stage model with the VRS and input-oriented (data file titled "EG3-dta.txt") are as follows: Results from DEAP Version 2.1 Instruction file = EG3-ins.txt Data file = eg3-dta.txt Input oriented DEA Scale assumption: VRS Slacks calculated using multi-stage method

DEAP Software EFFICIENCY SUMMARY: firm crste vrste scale 1 2 3 4 5 6 7 8 9 10 11 12

1.000 1.000 0.883 1.000 0.763 0.835 0.902 0.796 0.960 0.871 0.955 0.958

1.000 1.000 0.896 1.000 0.882 0.939 1.000 0.799 0.989 1.000 1.000 1.000

1.000 1.000 0.985 1.000 0.866 0.889 0.902 0.997 0.971 0.871 0.955 0.958

drs irs drs drs drs drs drs drs drs

mean 0.910 0.959 0.949 Note: crste = technical efficiency from CRS DEA vrste = technical efficiency from VRS DEA scale = scale efficiency = crste/vrste Note also that all subsequent tables refer to VRS results SUMMARY OF OUTPUT SLACKS: firm output: 1 1 0.000 2 0.000 3 0.000 4 0.000 5 36.000 6 0.000 7 0.000

2 0.000 0.000 2.333 0.000 0.000 2.000 0.000

125

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Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

8 9 10 11 12

0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000

mean 3.000 0.361 SUMMARY OF INPUT SLACKS: firm input: 1 1 0.000 2 0.000 3 0.729 4 0.000 5 0.000 6 8.213 7 0.000 8 0.000 9 0.000 10 0.000 11 0.000 12 0.000

2 0.000 0.000 0.000 0.000 0.327 0.000 0.000 0.000 20.891 0.000 0.000 0.000

mean 0.745 1.768 SUMMARY OF PEERS: firm 1 2 3 4 5 6 7 8

peers: 1 2 4 2 4 2 1 4 10 7 12 1 11 4

DEAP Software 9 2 1 12 10 10 11 11 12 12 SUMMARY OF PEER WEIGHTS: (in same order as above) firm peer weights: 1 1.000 2 1.000 3 0.333 0.667 4 1.000 5 0.600 0.400 6 0.286 0.714 7 1.000 8 0.016 0.371 0.007 0.605 9 0.160 0.293 0.547 10 1.000 11 1.000 12 1.000 PEER COUNT SUMMARY: (i.e., no. times each firm is a peer for another) firm peer count: 1 3 2 3 3 0 4 3 5 0 6 0 7 0 8 0 9 0

127

128 10 11 12

Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili 1 1 2

SUMMARY OF OUTPUT TARGETS: firm output: 1 1 100.000 2 150.000 3 160.000 4 180.000 5 130.000 6 230.000 7 220.000 8 152.000 9 190.000 10 250.000 11 260.000 12 250.000

2 90.000 50.000 57.333 72.000 66.000 92.000 88.000 80.000 100.000 100.000 147.000 120.000

SUMMARY OF INPUT TARGETS: firm input: 1 1 20.000 2 19.000 3 21.667 4 27.000 5 19.400 6 43.429 7 33.000 8 24.764 9 29.680 10 50.000 11 53.000 12 38.000

2 151.000 131.000 143.333 168.000 139.000 239.429 235.000 164.560 220.507 268.000 306.000 284.000

DEAP Software FIRM BY FIRM RESULTS: Results for firm: 1 Technical efficiency = 1.000 Scale efficiency = 1.000 (crs) PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 100.000 0.000 0.000 100.000 output 2 90.000 0.000 0.000 90.000 input 1 20.000 0.000 0.000 20.000 input 2 151.000 0.000 0.000 151.000 LISTING OF PEERS: peer lambda weight 1 1.000 Results for firm: 2 Technical efficiency = 1.000 Scale efficiency = 1.000 (crs) PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 150.000 0.000 0.000 150.000 output 2 50.000 0.000 0.000 50.000 input 1 19.000 0.000 0.000 19.000 input 2 131.000 0.000 0.000 131.000 LISTING OF PEERS: peer lambda weight 2 1.000 Results for firm: 3 Technical efficiency = 0.896 Scale efficiency = 0.985 (drs) PROJECTION SUMMARY:

129

130

Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

variable

original value output 1 160.000 output 2 55.000 input 1 25.000 input 2 160.000 LISTING OF PEERS: peer lambda weight 4 0.333 2 0.667

radial slack projected movement movement value 0.000 0.000 160.000 0.000 2.333 57.333 -2.604 -0.729 21.667 -16.667 0.000 143.333

Results for firm: 4 Technical efficiency = 1.000 Scale efficiency = 1.000 (crs) PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 180.000 0.000 0.000 180.000 output 2 72.000 0.000 0.000 72.000 input 1 27.000 0.000 0.000 27.000 input 2 168.000 0.000 0.000 168.000 LISTING OF PEERS: peer lambda weight 4 1.000 Results for firm: 5 Technical efficiency = 0.882 Scale efficiency = 0.866 (irs) PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 94.000 0.000 36.000 130.000 output 2 66.000 0.000 0.000 66.000 input 1 22.000 -2.600 0.000 19.400

DEAP Software input 2 158.000 LISTING OF PEERS: peer lambda weight 2 0.600 1 0.400

-18.673

-0.327

131 139.000

Results for firm: 6 Technical efficiency = 0.939 Scale efficiency = 0.889 (drs) PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 230.000 0.000 0.000 230.000 output 2 90.000 0.000 2.000 92.000 input 1 55.000 -3.359 -8.213 43.429 input 2 255.000 -15.571 0.000 239.429 LISTING OF PEERS: peer lambda weight 4 0.286 10 0.714 Results for firm: 7 Technical efficiency = 1.000 Scale efficiency = 0.902 (drs) PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 220.000 0.000 0.000 220.000 output 2 88.000 0.000 0.000 88.000 input 1 33.000 0.000 0.000 33.000 input 2 235.000 0.000 0.000 235.000 LISTING OF PEERS: peer lambda weight 7 1.000

132

Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

Results for firm: 8 Technical efficiency = 0.799 Scale efficiency = 0.997 (drs) PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 152.000 0.000 0.000 152.000 output 2 80.000 0.000 0.000 80.000 input 1 31.000 -6.236 0.000 24.764 input 2 206.000 -41.440 0.000 164.560 LISTING OF PEERS: peer lambda weight 12 0.016 1 0.371 11 0.007 4 0.605 Results for firm: 9 Technical efficiency = 0.989 Scale efficiency = 0.971 (drs) PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 190.000 0.000 0.000 190.000 output 2 100.000 0.000 0.000 100.000 input 1 30.000 -0.320 0.000 29.680 input 2 244.000 -2.603 -20.891 220.507 LISTING OF PEERS: peer lambda weight 2 0.160 1 0.293 12 0.547

DEAP Software Results for firm: 10 Technical efficiency = 1.000 Scale efficiency = 0.871 (drs) PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 250.000 0.000 0.000 250.000 output 2 100.000 0.000 0.000 100.000 input 1 50.000 0.000 0.000 50.000 input 2 268.000 0.000 0.000 268.000 LISTING OF PEERS: peer lambda weight 10 1.000 Results for firm: 11 Technical efficiency = 1.000 Scale efficiency = 0.955 (drs) PROJECTION SUMMARY: variable original radial slack projected value movement movement value output 1 260.000 0.000 0.000 260.000 output 2 147.000 0.000 0.000 147.000 input 1 53.000 0.000 0.000 53.000 input 2 306.000 0.000 0.000 306.000 LISTING OF PEERS: peer lambda weight 11 1.000 Results for firm: 12 Technical efficiency = 1.000 Scale efficiency = 0.958 (drs) PROJECTION SUMMARY: variable original radial slack projected value movement movement value

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134

output 1 250.000 output 2 120.000 input 1 38.000 input 2 284.000 LISTING OF PEERS: peer lambda weight 12 1.000

0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

250.000 120.000 38.000 284.000

According to the data in Figure 9 in chapter 4, the results of the inputoriented Malmquist model (data file titled "EG4-dta.txt") are as follows: Results from DEAP Version 2.1 Instruction file = EG4-ins.txt Data file = eg4-dta.txt Input oriented Malmquist DEA DISTANCES SUMMARY year = firm no.

1 2 3 4 5 6 7 8

1 crs te rel to tech in yr vrs ******************* te t-1 t t+1 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.940 1.000 1.000 0.833 1.000 0.778 0.852 1.000

0.447 0.590 0.630 0.554 1.024 0.630 0.751 1.044

0.940 1.000 1.000 0.833 1.000 0.778 0.852 1.000

DEAP Software 9 10 11 12

0.000 0.000 0.000 0.000

0.900 1.000 0.802 1.000

0.940 0.944 0.818 0.983

0.929 1.000 0.867 1.000

mean 0.000

0.925

0.780

0.933

year = firm no.

1 2 3 4 5 6 7 8 9 10 11 12

2 crs te rel to tech in yr vrs ******************** te t-1 t t+1 2.212 2.167 1.906 1.574 2.019 1.410 1.217 1.407 1.476 1.354 1.229 0.967

1.000 1.000 1.000 0.868 1.000 0.888 0.784 1.000 1.000 0.984 0.959 0.943

0.780 0.756 0.730 0.659 1.105 0.759 0.755 1.120 1.096 1.045 1.041 1.056

1.000 1.000 1.000 0.868 1.000 0.888 0.784 1.000 1.000 1.000 1.000 1.000

mean 1.578 0.952 0.909 0.962 year = 3 firm crs te rel to tech in yr vrs no. ******************** te t-1 t t+1 1 2

1.411 1.364

1.000 1.000

0.000 0.000

1.000 1.000

135

Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

136 3 4 5 6 7 8 9 10 11 12

1.056 1.129 1.472 0.994 1.199 1.072 1.093 1.137 0.953 0.883

0.768 0.833 1.000 0.755 0.954 0.972 1.000 1.000 0.980 0.965

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.767 0.833 1.000 0.755 0.954 0.972 1.000 1.000 1.000 1.000

mean 1.147

0.936

0.000

0.940

[Note that t-1 in year 1 and t+1 in the final year are not defined] MALMQUIST INDEX SUMMARY year =

2

firm effch techch pech 1 2 3 4 5 6 7 8 9 10 11 12

1.064 1.000 1.000 1.041 1.000 1.141 0.920 1.000 1.111 0.984 1.195 0.943

2.156 1.916 1.740 1.652 1.404 1.401 1.327 1.161 1.189 1.207 1.121 1.021

1.064 1.000 1.000 1.041 1.000 1.141 0.920 1.000 1.076 1.000 1.153 1.000

sech tfpch 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.032 0.984 1.037 0.943

2.294 1.916 1.740 1.720 1.404 1.599 1.221 1.161 1.321 1.188 1.340 0.963

DEAP Software mean 1.030 1.405 1.031 0.999 1.448 year = 3 firm effch techch pech sech tfpch 1 2 3 4 5 6 7 8 9 10 11 12

1.000 1.000 0.768 0.960 1.000 0.850 1.216 0.972 1.000 1.016 1.022 1.024

1.344 1.343 1.372 1.335 1.154 1.241 1.142 0.992 0.999 1.034 0.946 0.904

1.000 1.000 0.767 0.960 1.000 0.850 1.216 0.972 1.000 1.000 1.000 1.000

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.016 1.022 1.024

1.344 1.343 1.053 1.283 1.154 1.056 1.390 0.965 0.999 1.051 0.967 0.925

mean 0.980 1.139 0.975 1.005 1.116 MALMQUIST INDEX SUMMARY OF ANNUAL MEANS year effch techch pech

sech tfpch

2 1.030 1.405 1.031 0.999 1.448 3 0.980 1.139 0.975 1.005 1.116 mean 1.005 1.265 1.003 1.002 1.271 MALMQUIST INDEX SUMMARY OF FIRM MEANS firm effch techch pech

sech tfpch

1 1.031 1.703 1.031 1.000 1.756 2 1.000 1.605 1.000 1.000 1.605

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Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

138 3 4 5 6 7 8 9 10 11 12

0.876 1.000 1.000 0.985 1.058 0.986 1.054 1.000 1.105 0.982

1.545 1.485 1.273 1.319 1.231 1.073 1.090 1.118 1.030 0.961

0.876 1.000 1.000 0.985 1.058 0.986 1.038 1.000 1.074 1.000

1.000 1.000 1.000 1.000 1.000 1.000 1.016 1.000 1.029 0.982

1.354 1.485 1.273 1.299 1.302 1.058 1.148 1.118 1.139 0.944

mean 1.005 1.265 1.003 1.002 1.271 [Note that all Malmquist index averages are geometric means]

CONCLUSION DEAP software was provided by Tim Coelli from the Queensland University. This software runs under the "MS-DOS" system and contains four different models with input-oriented or output-oriented, as well as the CRS and VRS. The advantages of this software include simple application and the results of solving the models. Due to the mentioned advantages, this software has the potential for development in the future and can be used by researchers, experts, and managers in addition to competing with other DEA software packages.

Chapter 6

EMS SOFTWARE 6.1. INTRODUCTION Efficiency Measurement System (EMS) software runs under the "Windows 9x/NT" and uses the "LP Solver" system to calculate the efficiency scores [106]. There is no limitation to the number of DMUs, inputs, and outputs in this software. Preparing the data is considered the first step in calculating the efficiency of this software. Data are defined based on the standard format of this software in a Microsoft Excel or text file.

6.2. SOFTWARE PRESENTATION Data must be defined in an Excel or text file to be entered into the software. In the following, the format of input and output data is examined.

6.2.1. Data File Similar to Figure 1, the input data in Excel is defined as follows:

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Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili ✓ All information is defined in a worksheet and saved in the "*.xls" format; ✓ The name of the worksheet is saved as "data"; ✓ The name of the DMUs are defined in the first column; ✓ The name of the inputs and outputs are defined in the first line, respectively; ✓ The name of the inputs are defined by the string {I} and the name of the outputs are defined by the string {O};

Figure 1. Data entry in Excel file.

✓ To define non-controllable variables, {IN} is used for inputs and {ON} for outputs; ✓ Weight restrictions are defined as W(𝑝, 𝑞) ≥ 0, p is the input weights vector and q indicates the output weight vector (shadow price), and is similar to Figure 2 for inputs and outputs and are saved in the "weights" worksheet.

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Figure 2. Weight restrictions in Excel.

In addition, as shown in Figure 3, the data in the text files are defined as follows: ✓ The data are saved in a worksheet with a "* .txt" format; ✓ The columns should be separated by a space from each other (exactly one space between the two columns should be considered); ✓ The name of the DMUs are defined in the first column; ✓ The name of the inputs and outputs are defined in the first line, respectively;

Figure 3. Data format in a text file.

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✓ The name of the inputs are defined by the string {I} and the name of the outputs are defined by the string {O}; ✓ To define non-controllable variables, the string {IN} is used for inputs and {ON} for outputs. ✓ Weight restrictions similar to an Excel file are defined as W(𝑝, 𝑞) ≥ 0 and the columns are separated by a space according to Figure 4.

Figure 4. Weight restrictions in the text file.

6.2.2. Final Results The results table consists of several parts as follows: ▪

▪ ▪

▪

Name of DMUs: The name of the DMUs in the "technology" column (settings section) are displayed and the values of the removed DMUs from the "evaluation" column are displayed without efficiency points, Efficiency score: Indicates the efficiency of each DMU; Weights: This column shows the shadow prices {W} or the actual input and output {V} based on the format selected from the "DEA" menu; Benchmarking: For inefficient DMUs, reference DMUs with λ values are shown in brackets. For efficient DMUs, the number of times which are selected as the reference DMU indicates the inefficient DMUs;

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Slack variable values (S) or factor (F): Slack variable values {S} show inputs and outputs for radial and additive models and factor {F} for maximum and minimum mean, respectively.

6.3. RUNNING THE EMS SOFTWARE After defining the data in the Excel or text files, the data file can be defined as follows in the software: ✓ To specify the data file, as shown in Figure 5, the "load data" is selected from the "file" tab;

Figure 5. Choosing the data file.

✓ If needed to define the weights, the "load weight restriction" is selected from the "file" tab (Figure 6);

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Figure 6. Defining the weight restrictions.

✓ Based on Figure 7, the output can be specified by using the "format" command in the "DEA" tab;

Figure 7. Definition of results format.

In the "format" setting window (Figure 8), the "decimals" is for the number of decimal places. "Pure weights" and the multiplication of weights in the input and output values (virtual inputs/outputs) are shown in the "display weights" section.

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Figure 8. Format setting window.

After defining the data, the "run model" command in the "DEA" tab (Figure 9) is used for determining the model and its characteristics.

Figure 9. Run model window.

Figure 10. Execution window.

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Also, the problem solving strategy, the type of RTS, distance, orientation, and the super-efficiency score can be specified. As shown in Figure 10, in the "options" tab, the effect of DMUs on efficiency values can be investigated. For this purpose, the desired DMUs are selected and the "clear" option is used to remove the DMUs so that the mentioned effects should be examined. If window analysis and Malmquist models are used, the data should be defined in different time periods, the format of them which is defined in the data file as follows: DMU 1 T 0 DMU 2 T 0 ⋮

DMU n T 0 DMU 1 T 1 ⋮

DMU n T 1 ⋮

DMU n T t

Figure 11. Results of hospitals performance evaluation.

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For the window analysis model, the number of periods and the width of the window should be specified, and the number of periods should also be specified for the Malmquist index. After making the required adjustments, according to the evaluation of hospitals (Table 1 in chapter 2), the results in the CRS, radial, and inputoriented are shown in Figure 11.

CONCLUSION In DEAP software, the efficiency of DMUs can be evaluated based on the defined model. Software input data can be defined in the Excel and text files, and the results have different sections such as efficiency scores, output weights, and values of slack variables. This software has limited models compared to other DEA software, but the existence of a benchmarking option in the model results allows to be determined a reference set for DMUs and allows more detailed examination of units to make appropriate decisions to improve efficiency.

Chapter 7

FRONTIER ANALYST SOFTWARE 7.1. INTRODUCTION Frontier Analyst software was introduced by the Banxia Group [107]. This software is a Windows-based efficiency analysis tool which evaluates the performance of organization units by using the DEA.

7.2. SOFTWARE PRESENTATION Here, different parts of Frontier Analyst software are introduced.

7.2.1. Toolbar Software facilities are available through a toolbar that provides access to the most common facilities, and individual windows have specialized tools which are activated when used. As shown in Figure 1, the toolbar has four main sections.

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Figure 1. Frontier Analyst software toolbar.

• • • •

Section (1) can be used to start a new project, open projects, save, convert, print, and close the program; In section (2), it is possible to the quick access toolbar and the required tools are arranged in different groups; Section (3) use for software settings; All the software features can be accessed through the tabs in section 4.

In the following, the software toolbar is introduced.

7.2.1.1. Home As shown in Figure 2, this option consists of three sections: "clipboard", "configuration" and "information".

Figure 2. Home tab.

In the "clipboard" section, data can be moved, as well as entering the data as text. The "configuration" section is related to software updates, and the "information" provides general information about the software and related help.

7.2.1.2. Source Data In order to edit the data, tools in Figure 3 can be used to make the changes, which consist of five parts.

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Figure 3. Source data tab.

▪

Insert: In this option, input/output variables and DMUs are added. The "unit" and "variable" are selected to add new DMU(s), and input/output, respectively (Figure 4).

Figure 4. Defining input/output variables and DMUs.

▪ ▪

▪ ▪

Unit: The DMUs can be removed or disabled (inactive/active). Input/output: The type and amount of input or output variables are determined in this section. Furthermore, to change the values of variables, the desired cell must be selected and entered a new value, and remove or disable any of the input or output variables; it can be done as before. Visibility: Variables can be removed from analysis and DMUs can be disabled. Navigation: A variable or DMU with a specific value can be searched.

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7.2.1.3. Analysis As shown in Figure 5, the data analysis tab consists of four different sections.

Figure 5. Analysis tab.

▪ ▪ ▪

Table copy: In this case, the information of the project can be copied and saved in the related format. Summary: Based on the "toggle traffic lights", details and efficiency results are displayed Analysis: The data are analyzed by the selected model. The first decision to run the model is to choose the mode of minimizing inputs or maximizing outputs. To select one of these two modes, it is enough to choose the minimum or maximum type from the "DEA options" (Figure 6). In the next step, the type of RTS selected. Here, the CRS and VRS are defined, and BCC and CCR models are used for the VRS and CRS.

In DEA, the operation is not performed if a cell has a value of zero. If the software shows an error indicating the value of zero, the "replace all with one value" option should be selected from the menu and the zero value should be specified (the alternative value should be smaller than the other values in the data set). By choosing the "DEA options", advanced DEA models can be used in this software. For this purpose, the "optimization mode" is common to both basic and advanced models. The "variable configuration" tab contains general information about the input and output variables. Also, in this case, variables can be categorized (Figure 7(, and necessary changes such as enabling or disabling the variable, determining the type of variable and alternative value for the variable with zero, and the required instructions in the structure of variables can be applied.

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Figure 6. Analysis options window.

Figure 7. Window of variables configuration.

To determine the role of key variables and their importance, the weights of each can be defined in the "weight control" tab, which should not exceed 100 (Figure 8).

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Figure 8. Window of weight control.

Figure 9. Data management window.

In the "data management" tab, categorical control over a time period based on indicators such as age, region (use a categorical variable to segment the data), and different times (use a categorical variable analysis over time, Malmquist) are controlled. Additionally, in the standard mode, all variables are used in the analysis. However, if a special case is included in the variables, the "use only data selected below" must be selected, and a new condition can be created by choosing the "filter" option (Figure 9).

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As shown in Figure 9, the variable must be selected to create a new condition. Then, due to the relevant list, the type of constraint is selected and their value is specified in the "empty" section. From the "press the bottom to add a new condition" option, a new condition can be added. Likewise, if there is a relationship between the defined conditions, the terms AND, OR, NOT AND, and NOT OR are used to determine the type of relationship. ▪

Advanced setting: As shown in Figure 10, different types of models presented in this software.

Figure 10. Advanced setting window.

Here, the efficiency can be calculated in standard or super-efficiency models, as well as displaying the "traffic lights" and its range. In the default mode, efficient units (with a value of 100) are in green, units in the efficiency range between 100-90 are shown with yellow light, and units with less than 90% efficiency are shown in red. If the allocative model is considered, the values related to the cost of each unit and the combination of variables in the initial data of the problem with the suffix "-cost" should be added to the input or output variables. ▪

Data layout: In this option, the layout of the data can be arranged and the model is examined by using the "analyze now" after making the necessary changes and finalizing the data.

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7.2.1.4. Result This tab is divided into two parts "tools" and "window" (Figure 11).

Figure 11. Results tab.

In the "tools" section, the "unit details" window details are related to each DMU, such as the potential improvement, reference comparison, input/output contributions, and reference contributions. Efficiency values, analysis process diagram, X-Y plot, efficiency plot, and efficient frontier plot are also displayed in this case. In the "window" section, settings related to window display modes are made.

7.2.1.5. Graph In this tool, different types of display can be set, as well as settings for printing and copying (Figure 12).

Figure 12. Graph tab.

7.2.1.6. Report As shown in Figure 13, in this tool, the necessary settings for displaying results and reports can be considered. In the "report" tab, the results are saved and printed, and the required information type is specified by using the menu. Also, the results can be saved as an Excel file. In the "output" option, the size of the results display is determined, and in the "data options" section, the order of results displaying is arranged alphabetically and the efficiency values are ranked. Sample reports can be edited in the "template" section.

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Figure 13. Report tab.

7.2.1.7. Translation In this tab, the language is selected and a new translation section is created by the user, and the desired section is translated based on the initial definition of the language (Figure 14).

Figure 14. Translation tab.

7.2.2. Data Entry Data entry is possible in two methods. In the first method, by using the "open project" command, shows the existing projects, and then, by choosing the desired project, the data will be loaded into the "data viewer" window (Figure 15).

Figure 15. Data entry window.

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Figure 16. New project definition.

In the second method, a new project must be defined. Therefore, according to Figure 16, by choosing the "new project" command, the new project is defined.

7.2.2.1. Transfer Data from Excel First, after specifying the data range in the Excel file, the "use current selection from Excel" command (Figure 17) is selected, in which case there is no need to change the options in the next windows.

Figure 17. Window of transferring data from Excel.

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7.2.2.2. Transfer Data from the Clipboard As shown in Figure 18, the data (in each file) should be copied to a text file and should be copied by using the edit tab in the toolbar. Then, "past data from clipboard" must be selected. Furthermore, there is no need to change the options in the next windows.

Figure 18. Data transferring from the clipboard.

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7.2.2.3. Link to External Database An external database can be used to enter the data into the software (Figure 19) and the data are called by connecting to a database.

Figure 19. Data entry from the external database.

7.2.2.4. Use SPSS Software Information SPSS software data can also be used to enter the data and this way can be called similar to data transfer from Excel. 7.2.2.5. Import Data from the Text File As shown in Figure 20, the data is defined through a text file.

Figure 20. Data transferring from the text file.

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7.2.2.6. Enter Data Manually By choosing the "type into data editor" command, the name and type of the input/output variables can be specified (Figure 21).

Figure 21. Input and output variables definition.

In this software, four types of variables are considered as follows: • • • •

Controllable input/output Non-controllable input/output (non-discretionary) Output Text/date (this information is not analyzed and is only used to filter DMUs).

After defining the input and output variables, the DMUs should be defined. Then, the data is displayed in the "data viewer" window (Figure 22) and the values associated with each input and output must be entered in the corresponding cell.

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Figure 22. Manual data entry window.

7.2.2.7. Import Data via Web Files Similar to previous data entry methods, this method also imports files with the "XML" extension. As shown in Figure 23, the information of web files can be entered into the software.

Figure 23. Web file data entry window.

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7.3. RUNNING THE FRONTIER ANALYST SOFTWARE Based on the information of the evaluation of 12 hospitals (Table 1 in chapter 2), the results obtained in the Frontier Analyst software are presented in nine sections.

7.3.1. Potential Improvement Here, the basic information for each unit is shown graphically and other graphs show the reasoning of this graph. Figure 24 shows the rate of changes (percentage) of the inputs/outputs of a DMU until it becomes an efficient unit.

Figure 24. Rate of changes of the selected DMU.

The actual values of the variables and the number of their changes can also be seen from the "table view" option (Figure 25).

Figure 25. Actual values and the rate of change of the variables.

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7.3.2. Reference Comparison In this case, the values of the selected DMU variables are compared to reference units in the form of graphs, and inefficient units are evaluated in relation to efficient units (Figure 26).

Figure 26. Comparison of selected DMU variables with reference unit.

7.3.3. Input/Output Contributions As shown in Figure 27, the role of each input or output in determining the total efficiency is shown graphically, which is used to validate efficiency values in some cases.

Figure 27. Contribution of inputs and outputs in total efficiency.

7.3.4. Reference Contributions This section shows the number of times that an efficient unit (reference set for DMU) appears in an inefficient set, as shown in Figure 28.

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Figure 28. Number of inefficiencies of an efficient reference unit.

7.3.5. Efficiency Scores As shown in Figure 29, in the “score” window, the results of solving the problem according to each variable are presented in four different sections.

Figure 29. Graphical results of the efficiency scores.

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Figure 29. (Continued).

▪ ▪ ▪ ▪

Condition: The efficiency values in traffic light colors (1); Summary graph/overall: Displays the rate of changes to achieve efficiency in each of the inputs/outputs (2); Summary graph-by input/output: The rate of change in each input or output over a time period (3); Distribution: Displays the number of efficient units in terms of efficiency ranges (4) (the numbers shown at the top of each bar chart indicate the number of DMUs).

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7.3.6. Problem Solving Process Diagram Using the "diagram tool" option, the problem model can be viewed as Figure 30.

Figure 30. Diagram of the problem solving process.

7.3.7. X-Y Plot By using this plot (Figure 31), it is possible to determine the correlation of one variable with another variable, and one of the variables is temporarily removed from the analysis if there is a strong correlation. Also, the effect of removing that variable in the problem can be observed by reanalyzing the problem. For example, the correlation between the two input variables of doctor and nurse is equal to 0.87. If the correlation number is closer to zero, it means that an increase in the value of one variable will not affect the other variable. In addition, the negative correlation between the two variables indicates the high values of one variable against the low values of the other variable.

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Figure 31. Y-X plot.

7.3.8. Efficiency Plot As shown in Figure 32, the efficiency plot is drawn based on input and output variables and can be useful for identifying characteristics of efficient and inefficient units.

Figure 32. Efficiency plot.

7.3.9. Efficient Frontier Plot Frontier plot allows the efficient frontier to be viewed in two dimensions and is useful for learning or explaining DEA concepts. This plot can be drawn in the problem of output maximization with one output

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and two inputs, and the problem of input minimization with one input and two outputs. On the other hand, the additional inputs and outputs must be disabled in problems with larger dimensions.

Figure 33. Efficient frontier Plot.

As shown in Figure 33, in this plot, the units which are on the efficient frontier are shown in red and the other units are shown in blue. Further, to place one of the inefficient units on the efficient frontier, it is enough to choose a unit to draw a line from the origin to the efficient frontier. The point of intersection of this line with the efficient frontier is marked with a red point, which is the target point for converting an inefficient unit to an efficient unit.

CONCLUSION Frontier Analyst software is considered as one of the well-known software in the DEA. The latest version of this software was released in 2015, which shows the continuous improvement of the software. The advantages of the software are included the different models, graphical display of results, and ease of entering the data. In this software, there are various tools for adjusting inputs/outputs and final report, which allows the user to achieve extensive results from problem analysis after initial settings.

Chapter 8

PIONEER SOFTWARE 8.1. INTRODUCTION Pioneer software was designed by Mcloud and Barr in collaboration with the Southern Methodist University [100]. This software was designed under the Windows system and C++ programming language and has a suitable user interface.

8.2. SOFTWARE PRESENTATION

Figure 1. Pioneer software environment.

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The text file is the input and output of Pioneer software and is based on the C++ programming language. As shown in Figure 1, the software environment has three main parts, which are discussed.

8.2.1. Software Models The software has 4 models as follows: • • • •

CCR model with the CRS BCC model with the VRS Non-Decreasing Returns to Scale (NDRS) model Non-Increasing Returns to Scale (NIRS) model

In Pioneer software, in addition to the two approaches of outputoriented and input-oriented, the following four scales are defined for the models: • • • •

No scaling Geometric mean Average value Maximum value

There are other options in the model settings, as well. The superefficiency values and determining the corresponding levels, the use of the Hierarchical Decomposition (HDEA) option, and the values of the relevant parameters (Landa, beta, and block size) are other options in the settings. It should be noted that the HDEA option is used for reducing computing time.

8.2.2. Data Entry Data are defined in the software through a text file. All files follow a special rule which follows the first line as the following rules:

Pioneer Software ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

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File names (maximum 30 characters) Number of inputs Number of outputs Number of DMUs Number of restrictions Inputs name (in one line) Outputs name (in one line) Name of DMUs (values of each unit are separated by a space) Name of restrictions (values of each restriction are separated by a space).

8.2.3. Final Results The results of the problem solving are saved in a text file according to the selected model. These reports include efficiency scores, reference efficient units, and inefficient units, as well as data for each DMU.

8.3. RUNNING THE PIONEER SOFTWARE The data of hospitals with 2 inputs and 2 outputs (Table 1 in chapter 2) is considered. The data are considered based on the following structure: Hospitals 2 2 12 2 Doctor Nurse Outpatient Inpatient Hosp.A Hosp.B

20 19

151 131

100 150

90 50

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Hosp.C Hosp.D Hosp.E Hosp.F Hosp.G Hosp.H Hosp.I Hosp.J Hosp.K Hosp.L Dummy01 Dummy02

25 27 22 55 33 31 30 50 53 38 1 1 0 0

160 168 158 255 235 206 244 268 306 284 0 0

160 180 94 230 220 152 190 250 260 250 1 1

55 72 66 90 88 80 100 100 147 120

For instance, NDRS and NIRS models were solved in the inputoriented and geometric mean scale. The summary report of the NDRS model is as follows: PIONEER V.2 Data Envelopment Analysis Summary Report: Hospitals Model: NDRS Scaling: Geometric Mean Orientation: Input DMUs: 12 Time: 0.016 seconds Name Hosp.A Hosp.B Hosp.C Hosp.D Hosp.E Hosp.F Hosp.G

1 0.9058 1 1 0.9357 0.8456 0.7877 0.8304

Pioneer Software Hosp.H Hosp.I Hosp.J Hosp.K Hosp.L Efficient:

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0.6444 0.7345 0.8147 0.742 0.7989 2

The complete report for solving the NDRS model is as follows: PIONEER V.2 Data Envelopment Analysis Report: Hospitals Model: NDRS Scaling: Geometric Mean Orientation: Input Levels: 1 DMUs: 12 Inputs: 2 Outputs: 2 Time: 0.016 seconds ########################################################### Summary Results for Hosp.A Level: 1 Status: Inefficient Reference Set Value

Variables Observed Outpatient(O) 100 Inpatient(O) 90 Doctor(I) 20 Nurse(I) 151

Efficiency Rating: 0.905826 Multipliers Value Slack Outpatient(O) 0 Inpatient(O) 0 Doctor(I) 0.73147 Nurse(I) 0.73147 Ideal 0 0 0 0

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########################################################### Summary Results for Hosp.B Level: 1 Status: Weakly Efficient Efficiency Rating: 1 Reference Set Value Multipliers Value Slack Hosp.C 1 Outpatient(O) 1.1819 Inpatient(O) 0 Doctor(I) 0 Nurse(I) 1.5737 Variables Observed Ideal Outpatient(O) 150 160 Inpatient(O) 50 55 Doctor(I) 19 25 Nurse(I) 131 160 ########################################################### Summary Results for Hosp.C Level: 1 Status: Weakly Efficient Efficiency Rating: 1 Reference Set Value Multipliers Value Hosp.B 1 Outpatient(O) 1.1081 Inpatient(O) 0 Doctor(I) 1.1102e-016 Nurse(I) 1.2885 Variables Observed Ideal Outpatient(O) 160 150 Inpatient(O) 55 50 Doctor(I) 25 19 Nurse(I) 160 131

Slack

########################################################### Summary Results for Hosp.D Level: 1 Status: Inefficient Reference Set Value

Efficiency Rating: 0.93571 Multipliers Value

Slack

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Variables Observed Outpatient(O) 180 Inpatient(O) 72 Doctor(I) 27 Nurse(I) 168

Outpatient(O) Inpatient(O) Doctor(I) Nurse(I) Ideal 0 0 0 0

177 0.92163 0 0.078199 1.2271

########################################################### Summary Results for Hosp.E Level: 1 Status: Inefficient Reference Set Value

Variables Observed Outpatient(O) 94 Inpatient(O) 66 Doctor(I) 22 Nurse(I) 158

Efficiency Rating: 0.84557 Multipliers Value Slack Outpatient(O) 0 Inpatient(O) 0 Doctor(I) 0.68281 Nurse(I) 0.68281 Ideal 0 0 0 0

########################################################### Summary Results for Hosp.F Level: 1 Status: Inefficient Reference Set Value

Variables Observed

Efficiency Rating: 0.78771 Multipliers Value Slack Outpatient(O) 0.60719 Inpatient(O) 0 Doctor(I) 0.45032 Nurse(I) 0.80846 Ideal

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Outpatient(O) Inpatient(O) Doctor(I) Nurse(I)

230 90 55 255

0 0 0 0

########################################################### Summary Results for Hosp.G Level: 1 Status: Inefficient Reference Set Value

Variables Outpatient(O) Inpatient(O) Doctor(I) Nurse(I)

Observed 220 88 33 235

Efficiency Rating: 0.83045 Multipliers Value Slack Outpatient(O) 0.66923 Inpatient(O) 0 Doctor(I) 0.45723 Nurse(I) 0.45723 Ideal 0 0 0 0

########################################################### Summary Results for Hosp.H Level: 1 Status: Inefficient Reference Set Value

Variables Observed Outpatient(O) 152 Inpatient(O) 80 Doctor(I) 31 Nurse(I) 206

Efficiency Rating: 0.6444 Multipliers Value Outpatient(O) 0.75162 Inpatient(O) 0 Doctor(I) 0.022946 Nurse(I) 1.0008 Ideal 0 0 0 0

Slack

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########################################################### Summary Results for Hosp.I Level: 1 Status: Inefficient Reference Set Value

Variables Observed Outpatient(O) 190 Inpatient(O) 100 Doctor(I) 30 Nurse(I) 244

Efficiency Rating: 0.73451 Multipliers Value Slack Outpatient(O) 0.68538 Inpatient(O) 0 Doctor(I) 0.46826 Nurse(I) 0.46826 Ideal 0 0 0 0

########################################################### Summary Results for Hosp.J Level: 1 Status: Inefficient Reference Set Value

Variables Observed Outpatient(O) 250 Inpatient(O) 100 Doctor(I) 50 Nurse(I) 268

Efficiency Rating: 0.81468 Multipliers Value Slack Outpatient(O) 0.57774 Inpatient(O) 0 Doctor(I) 0.28773 Nurse(I) 0.76924 Ideal 0 0 0 0

########################################################### Summary Results for Hosp.K Level: 1 Status: Inefficient Reference Set Value

Efficiency Rating: 0.74205 Multipliers Value Slack

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Variables Observed Outpatient(O) 260 Inpatient(O) 147 Doctor(I) 53 Nurse(I) 306

Outpatient(O) Inpatient(O) Doctor(I) Nurse(I) Ideal 0 0 0 0

0.50599 0 0.20294 0.67372

########################################################### Summary Results for Hosp.L Level: 1 Status: Inefficient Reference Set Value

Variables Observed Outpatient(O) 250 Inpatient(O) 120 Doctor(I) 38 Nurse(I) 284

Efficiency Rating: 0.79891 Multipliers Value Slack Outpatient(O) 0.56656 Inpatient(O) 0 Doctor(I) 0.38708 Nurse(I) 0.38708 Ideal 0 0 0 0

The summary report of the NIRS model is as follows: PIONEER V.2 Data Envelopment Analysis Summary Report: Hospitals Model: Extremal NIRS Scaling: Geometric Mean Orientation: Input DMUs: 12 Time: 0.016 seconds

Pioneer Software Name Hosp.A Hosp.B Hosp.C Hosp.D Hosp.E Hosp.F Hosp.G Hosp.H Hosp.I Hosp.J Hosp.K Hosp.L Efficient:

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1 0.6039 1 1 1 0.5299 1 1 0.6479 0.8318 1 1 1 8

The complete report of solving the NIRS model is as follows: PIONEER V.2 Data Envelopment Analysis Report: Hospitals Model: NIRS Scaling: Geometric Mean Orientation: Input Levels: 1 DMUs: 12 Inputs: 2 Outputs: 2 Time: 0.015 seconds ########################################################### Summary Results for Hosp.A Level: 1 Status: Inefficient Reference Set Value

Efficiency Rating: 0.603884 Multipliers Value Slack Outpatient(O) 1.0706 Inpatient(O) 0

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Variables Observed Outpatient(O) 100 Inpatient(O) 90 Doctor(I) 20 Nurse(I) 151

Doctor(I) Nurse(I) Ideal 0 0 0 0

0.73147 0.73147

########################################################### Summary Results for Hosp.B Level: 1 Status: Weakly Efficient Efficiency Rating: 1 Reference Set Value Multipliers Value Slack Hosp.C 1 Outpatient(O) 1.1819 Inpatient(O) 0 Doctor(I) 0 Nurse(I) 1.5737 Variables Observed Ideal Outpatient(O) 150 160 Inpatient(O) 50 55 Doctor(I) 19 25 Nurse(I) 131 160 ########################################################### Summary Results for Hosp.C Level: 1 Status: Weakly Efficient Efficiency Rating: 1 Reference Set Value Multipliers Value Hosp.B 1 Outpatient(O) 1.1081 Inpatient(O) 0 Doctor(I) 1.1102e-016 Nurse(I) 1.2885 Variables Observed Ideal Outpatient(O) 160 150

Slack

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55 25 160

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50 19 131

########################################################### Summary Results for Hosp.D Level: 1 Status: Weakly Efficient Efficiency Rating: 1 Reference Set Value Multipliers Value Slack Outpatient(O) 1.3015 Inpatient(O) 0 Doctor(I) 0 Nurse(I) 1.2271 Variables Observed Ideal Outpatient(O) 180 0 Inpatient(O) 72 0 Doctor(I) 27 0 Nurse(I) 168 0 ########################################################### Summary Results for Hosp.E Level: 1 Status: Inefficient Reference Set Value

Variables Outpatient(O) Inpatient(O) Doctor(I) Nurse(I)

Observed 94 66 22 158

Efficiency Rating: 0.52989 Multipliers Value Slack Outpatient(O) 0.99941 Inpatient(O) 0 Doctor(I) 0.68281 Nurse(I) 0.68281 Ideal 0 0 0 0

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########################################################### Summary Results for Hosp.F Level: 1 Status: Weakly Efficient Efficiency Rating: 1 Reference Set Value Multipliers Value Slack Outpatient(O) 1.2097 Inpatient(O) 0 Doctor(I) 0 Nurse(I) 0.80846 Variables Observed Ideal Outpatient(O) 230 0 Inpatient(O) 90 0 Doctor(I) 55 0 Nurse(I) 255 0 ########################################################### Summary Results for Hosp.G Level: 1 Status: Weakly Efficient Efficiency Rating: 1 Reference Set Value Multipliers Value Slack Outpatient(O) 1.1031 Inpatient(O) 0 Doctor(I) 0.36616 Nurse(I) 0.54089 Variables Observed Ideal Outpatient(O) 220 0 Inpatient(O) 88 0 Doctor(I) 33 0 Nurse(I) 235 0 ########################################################### Summary Results for Hosp.H Level: 1 Status: Inefficient Reference Set Value

Efficiency Rating: 0.6479 Multipliers Value

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Outpatient(O) 1.0614 Inpatient(O) 0 Doctor(I) 0.017499 Nurse(I) 1.0008 Ideal 0 0 0 0

########################################################### Summary Results for Hosp.I Level: 1 Status: Inefficient Reference Set Value Hosp.J 1

Variables Observed Outpatient(O) 190 Inpatient(O) 100 Doctor(I) 30 Nurse(I) 244

Efficiency Rating: 0.83182 Multipliers Value Outpatient(O) 1.1167 Inpatient(O) 0 Doctor(I) 0.46826 Nurse(I) 0.46826 Ideal 250 100 50 268

Slack

########################################################### Summary Results for Hosp.J Level: 1 Status: Weakly Efficient Efficiency Rating: 1 Reference Set Value Multipliers Value Hosp.I 1 Outpatient(O) 0.94504 Inpatient(O) 0 Doctor(I) 0 Nurse(I) 0.76924 Variables Observed Ideal

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Outpatient(O) Inpatient(O) Doctor(I) Nurse(I)

250 100 50 268

190 100 30 244

########################################################### Summary Results for Hosp.K Level: 1 Status: Weakly Efficient Efficiency Rating: 1 Reference Set Value Multipliers Value Slack Outpatient(O) 1.2746 Inpatient(O) 0 Doctor(I) 0 Nurse(I) 0.67372 Variables Observed Ideal Outpatient(O) 260 0 Inpatient(O) 147 0 Doctor(I) 53 0 Nurse(I) 306 0 ########################################################### Summary Results for Hosp.L Level: 1 Status: Weakly Efficient Efficiency Rating: 1 Reference Set Value Multipliers Value Slack Outpatient(O) 0.92307 Inpatient(O) 0 Doctor(I) 0.38708 Nurse(I) 0.38708 Variables Observed Ideal Outpatient(O) 250 0 Inpatient(O) 120 0 Doctor(I) 38 0 Nurse(I) 284 0

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CONCLUSION Pioneer software was provided by Mcloud and Barr and sponsored by the Southern Methodist University. Data entry and final reports of this software, similar to DEAP software, are done through a text file, and the results can be easily received after determining the model from the four types of models and the relevant orientations. The software is written based on C++ programming language, and simplicity and proper user’s interface are the advantages of this software.

Chapter 9

MAXDEA SOFTWARE 9.1. INTRODUCTION MaxDEA software is one of the most powerful and professional DEA software packages, which runs under Microsoft Access. This software includes a wide range of new models and has the following features [108]: • • •

• • • • • •

High speed in optimized design and parallel computing Use of a multi-core processor Solve the bootstrapping model with 50 times speed, 162 times the game cross-efficiency model and other models with 4 times in a dual-core central processing unit Parallel computing in three automatic (usually for bootstrapping and cross-efficiency models), normal and passive modes No need to install, user-friendly interface, and ease of data entry Integrated storage of data and settings related to the models in an Access file No limitation on the number of DMUs and most comprehensive DEA models Running the multiplier models simultaneously Provide all the possible combination of DEA models

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Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili Additionally, the available models in software can be seen in Table 1. Table 1. Models of MaxDEA software

Category Distance

Orientation

RTS Frontier Malmquist index Cluster

Customized reference Bootstrapping Other

Model Radial, Maximum distance to frontier (SBM), Minimum distance to the weak efficient frontier, Minimum distance to the strong efficient frontier, DDF, Weighted additive model, Hybrid distance (Radial and SBM), Hybrid distance (EBM), Cost/Revenue/Profit/Ratio models Input-oriented, Output-oriented, Non-orientation, Modified input-oriented, Modified output-oriented, Non-orientation (Input-prioritized), Non-orientation (output-prioritized), Non-orientation (generalized prioritized) CRS, VRS, NDRS, NIRS, GRS Convex (CRS, VRS, NIRS, NDRS, and GRS), FDH non-convex (CRS, VRS, NIRS, NDRS, GRS(, ERH non-convex, FRH non-convex Adjacent Malmquist, Fixed Malmquist, Global Malmquist, Fixed window Malmquist, Adjacent window Malmquist, Sequential Malmquist Self-benchmarking, Cross-benchmarking, Dpwnward- benchmarking, Upwardbenchmarking, Lower adjacent benchmarking, Upper adjacent benchmarking, Window benchmarking Variable benchmarking, Fixed benchmarking, Minimum efficiency Bootstrap of DEA, Bootstrap of Malmquist Window, Metafrontier DEA, Dynamic DEA, Network DEA, Supper-efficiency, FDH, Bounded input/output, Undesirable output, Weak disposability, Context dependent, Non-discretionary input/output, Parallel, Weak disposability, Restricted multiplier, Preference (weighted), Fuzzy DEA, Cross efficiency, Restricted projection

9.2. SOFTWARE PRESENTATION This section introduces the full version of MaxDEA software.

9.2.1. Toolbar As shown in Figure 1, the software toolbar consists of eight different tabs.

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Figure 1. MaxDEA toolbar.

9.2.1.1. Prepare Data The first part of the toolbar is "prepare data". In this software, data can be imported from Excel, Access, "dBas" and text files. Figure 2 shows the data format in Excel.

Figure 2. Data format in Excel.

Also, the name of the DMUs must be unique. If numbers are used in the name of DMUs, it should be "DMU001, DMU002,… ". A periodic column (integers) must be added for each DMU of dynamic, window, and Malmquist models. In this software, the use of negative input/output values is allowed. By default, the use of these data applies to VRM models (for example, the non-radial SBM model). After defining the data, the related settings can be made by using the "define data" window (Figure 3). The problem name and the input/output variable can be changed by using the "field type" column to specify the input/output variable or DMU information. The "not defined" option should be disabled if some inputs or outputs are removed from the model. On the other hand, the "subDMU", "cluster", and "intermediate" options are used for the parallel, cluster, and network models, respectively.

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Figure 3. Data definition.

9.2.1.2. Run Model In this software, envelopment or multiplier models can be selected based on the required results. In addition, the envelopment model should be selected when results such as efficiency scores, reference evaluation, target variable values, and deficiencies are required, while a multiplier model should be selected when results such as efficiency scores and input/output weights coefficients are required. The multiplier model contains basic DEA models which are suitable for beginners. Envelopment models are noted in the related window, as well. It is sufficient to choose the desired model if the basic CCR and BCC models are considered in both input/output-oriented. 9.2.1.2.1. Envelopment Models As shown in Figure 4, after choosing the "envelopment model" tab, the associated window opens, which consists of nine sections.

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Figure 4. Envelopment models settings.

9.2.1.2.1.1. Measuring Efficiency Based on Distance This part covers radial, non-radial, hybrid, and financial measures, which include various subsections (Figure 4). ▪ ▪ ▪ ▪

▪

▪

Radial: This type of efficiency measurement is associated with the CCR and BCC models. Maximum distance to the frontier: This measurement option is used in ERM and SBM models. Minimum distance to the weak efficient frontier: It is noted as the less effort for an inefficient unit to reach the efficient frontier. Minimum distance to the strong efficient frontier: Calculations are performed according to the minimum distances of the units from the strong efficient frontier. DDF model: This function is a generalized approach of the radial model and includes evaluated DMU vectors (X 0 , Y0 ), the mean of all DMUs vector (1, ⋯ ,1), RDM, customized DMUs, and specific DMU. Weighted additive model: There are nine types of weights in this software:

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• •

The additive model with weights (1, ⋯ , 1) The normalized weighted additive model

•

Weights 𝑥 and 𝑦

•

Average weights 𝑥̅ and 𝑦̅

• •

Range Adjusted Measurement (RAM) Frontier measurement with adjusted bounds (Bounded Adjusted Measurement (BAM)) Efficiency measurement based on directional slack functions Customized weights (DMUs) Customized weights (special DMU)

• • • ▪

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1

0

1 0

1

𝑜

1 𝑜

Hybrid distance (EBM): Epsilon-based efficiency calculations are often considered as a suggested method for the dependency index. In this software, three methods can be used to determine the amount of epsilon and weight. Hybrid distance (SBM and radial models): This option is a combination of radial and non-radial models. First, the type of input and output distance must be defined, the "define" option is used, and in the relevant window, the radial and non-radial modes are selected (Figure 5). Likewise, if all inputs/outputs are selected radially, the model is radial, and when inputs/outputs are selected non-radially, the model is SBM.

Figure 5. Hybrid SBM and radial model settings.

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Cost/revenue/profit/ratio efficiency models: These models define the financial aspects of DEA models by considering the input/output price. These models have two types. In type I models, the original input/output values are used in the constraints (standard mode), while the cost/revenue values of input/output are used in the constraints in type II. In order to run these models, the "define price" option must be selected (Figure 6).

Figure 6. Determining the cost/revenue/profit/ratio efficiency models.

In determining the type of input/output at the beginning of the problem, the "not defined" option should be selected for the columns associated with costs or revenues (Figure 7).

Figure 7. Cost data definition window.

Furthermore, for the cost efficiency model, the amount of cost for inputs should be specified. The amount of revenue for inputs should be determined for the revenue efficiency model. On the other hand, efficiency values in the profit efficiency model might be negative. Regarding the profit and ratio efficiency model, values should be set for variables and outputs, and these values are non-discretionary for bounded or noncontrollable inputs or outputs.

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9.2.1.2.1.2. Efficiency Measurement Based on Orientation In this case, in addition to the usual models (input-oriented, outputoriented, and non-oriented), the new models are added (Figure 8). The significance of the new orientations is that the modified inputoriented and modified output-oriented super-efficiency models to overcome infeasibility problems in the basic super-efficiency models.

Figure 8. Window of orientation-based efficiency measurement.

9.2.1.2.1.3. Efficiency Measurement Based on the RTS As shown in Figure 9, five types of RTS are considered. In GRS models, the upper and lower bounds of λ must be defined.

Figure 9. Settings window of efficiency measurement based on RTS.

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In general, the upper and lower bounds for other models are as shown in Table 2. Table 2. Upper and lower bounds of the models RTS CRS VRS NIRS NDRS

GRS L 0 1 0 1

U +∞ 1 1 +∞

The "scale efficiency" or scale effect (CRS score⁄VRS score) is a special case in the problem that can be calculated as a scale efficiency or scale effect for the results of solving the model in both CRS and VRS. 9.2.1.2.1.4. Efficiency Measurement Based on Frontier Figure 10 shows the efficiency measurement based on the frontier in the following: • • • •

Convex model (CRS, VRS, NIRS, NDRS, and GRS) FDH non-convex model (CRS, VRS, NIRS, NDRS, and GRS) ERH non-convex model FRH non-convex model

Figure 10. Efficiency measurement based on the frontier.

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9.2.1.2.1.5. Advanced Models 1 As shown in Figure 11, Advanced Model 1 is defined in 16 sections.

Figure 11. Advanced models 1.

▪

Evaluation based on benchmarking: In this case, cluster models are categorized based on the characteristics of the DMUs. In general, there are seven types of cluster models according to the relationship between clusters, which are evaluated and clustered. A column representing the clusters should be added to the cluster of these models. Each DMU belongs to only one cluster. The sequence does not make sense for clustering DMUs. Likewise, similar DMUs can be placed in different time periods and different clusters. In the self-benchmarking model, each DMU is evaluated in its own cluster. In fact, it is a method of evaluating batches of DMUs in different time periods and one cluster is evaluated in each period (Figure 12).

In the cross-benchmarking model, each DMU is evaluated according to the cluster selected as the reference set. For instance, as shown in Figure 13, it is assumed that all DMUs are classified into three clusters. First,

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DMUs are evaluated based on cluster (1), then cluster (2), and finally cluster (3).

Cluster 3

Cluster 3

Cluster 2

Cluster 2

Cluster 1

Cluster 1

Figure 12. Self-benchmarking cluster model. Cluster 3

Cluster 3

Cluster 2

Cluster 2

Cluster 1

Cluster 1

Figure 13. Cross-benchmarking cluster model.

In the downward-benchmarking model, the DMUs in a cluster with the same cluster and one cluster lower than themselves (as a reference set) are evaluated. For example, as shown in Figure 14, if DMUs are classified into three clusters, DMUs in the cluster (1) with the same cluster, DMUs in the cluster (2) with the clusters (1) and (2), and cluster (3) with cluster (1), (2) and (3) are evaluated. In the upward-benchmarking model, the existing DMUs are different from the previous model and are evaluated in a cluster with the same cluster and a higher cluster (Figure 15).

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Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili Cluster 3

Cluster 3

Cluster 2

Cluster 2

Cluster 1

Cluster 1

Figure 14. Downward-benchmarking cluster model. Cluster 3

Cluster 3

Cluster 2

Cluster 2

Cluster 1

Cluster 1

Figure 15. Upward-benchmarking cluster model.

In the lower-adjacent benchmarking model, single-cluster DMUs with their lower-adjacent cluster are evaluated as a reference set. For instance, as shown in Figure 16, DMUs in the cluster (2) is evaluated with cluster (1). Also, the DMUs in the cluster (3) are evaluated with cluster (2) as the reference set. The upper-adjacent benchmarking model is different from the previous model, and the single-cluster is evaluated with the top-adjacent cluster as a reference set (Figure 17). In the window benchmarking model, the DMUs in a cluster are evaluated with DMUs in a window cluster that includes its own cluster. By default, the offset value is zero, and the window cluster is composed of the own cluster and a number of adjacent clusters. It is assumed that all DMUs are classified into five clusters. If the values window-width = 2 and offset = 0, Figure 18 is obtained.

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Figure 16. Lower-adjacent benchmarking cluster model. Cluster 3

Cluster 3

Cluster 2

Cluster 2

Cluster 1

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Figure 17. Upper-adjacent benchmarking cluster model.

In Figure 18, DMUs in the first cluster are evaluated with the first cluster as the reference set only, while the associated window width is incomplete. Additionally, DMUs in the second cluster with clusters (1) and (2), the third cluster with clusters (2) and (3), the fourth cluster with clusters (3) and (4), and the fifth cluster with clusters (4) and (5) are evaluated. Now, Figure 19 is obtained if the window-width = 2 and offset = 1. Cluster 5

Cluster 5

Cluster 4

Cluster 4

Cluster 3

Cluster 3

Cluster 2

Cluster 2

Cluster 1

Cluster 1

Figure 18. Cluster model of window benchmarking with width = 2 and offset = 0.

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Cluster 5

Cluster 5

Cluster 4

Cluster 4

Cluster 3

Cluster 3

Cluster 2

Cluster 2

Cluster 1

Cluster 1

Figure 19. Cluster model of window benchmarking with width = 2 and offset = 1.

As shown in Figure 19, DMUs in the first cluster with clusters (1) and (2), DMUs in the second cluster with clusters (2) and (3), the third cluster with clusters (3) and (4), the fourth cluster with clusters (4) and (5), as well as DMUs in the fifth cluster with only itself are evaluated as the reference set while its related window width is not complete. ▪

Customized variable benchmarking model: The reference set of this model can be determined using the "define" option (Figure 20).

Figure 20. Customized variable benchmarking model setting.

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The super-efficiency model: Removing DMU0 from the reference set in the super-efficiency model is considered as the only difference between the super-efficiency model and the standard efficiency model. In this model, the efficiency score becomes more than one. In some cases, a linear programming model is infeasible for some DMUs. To calculate the above supper-efficiency, the distance function and RTS should be selected first, and then the "supper-efficiency" option should be activated. Network DEA model: These models consist of several nodes which are connected to each other. Each node may (or may not) have its own input and output, which is called direct inputs/outputs in network models. Also, at least one node is linked to another node by an intermediate input/output which are outputs of one node can be simultaneously the outputs of another node. Network DEA models calculate the efficiency and overall efficiency for each node in a systematic framework. Examples of network DEA models can be seen in Figure 21.

Input

Input1

Input1

Node1

Node1

Node1

Output1

Intermediate1

Node 2

Intermediate1

Intermediate1

Input 2

Input 3

Input 2

Node 3

Node 2

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Intermediate 2 Input 3

Intermediate 2

Node 3

Output

Net-shape

Node 2

Output 2

Intermediate 2

chain-shape

Intermediate 3

Node 3

Output 3

chain-shape

Figure 21. Examples of network DEA models.

This software is a powerful tool for solving network models and has provided a variety of options for these models. Therefore, two options are

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considered for the network model; in the first option, the intermediate type is selected from 4 types: free, fixed, non-increasing, and non-decreasing. In the second option, the values of 𝜆 are the same for all nodes by choosing the "use identical lambda". In order to run a network model, the inputs and outputs are defined and then the indirect inputs/outputs (intermediate) are determined. In the model running step, the "network" mode is activated and the same value of 𝜆 is determined for all nodes by choosing the "use identical lambda". Then, the indirect inputs/outputs (intermediate) is defined. After specifying the intermediate, the network sub-processes are defined according to Figure 22, which includes defining the network nodes, and the weights of the nodes, assigning the inputs/outputs, and intermediates to the respective nodes. Finally, the model is run.

Figure 22. Node definition in the network DEA model.

▪

NDSC Model: This model is a special case of the generalized/bounded model. In order to run these models, the status of input/output discretion should first be set. For this

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purpose, it is enough to use the "define" (the values of slack variables rate are based on percentages). "Discretionary" indicates complete control and the "non-discretionary" option indicates limited control (Figure 23). Some special cases in the valley of these models are as follows: • Non-controllable radial model with non-discretionary inputs/outputs and slack movement range = 0% • non-discretionary radial model with non-discretionary inputs/outputs and slack movement range = 100% • Non-controllable non-radial model (SBM) with nondiscretionary input/output and slack movement range = 0% • Non-discretionary radial model (SBM) with non-discretionary input/output and slack movement range = 100%

Figure 23. Window of defining the degree of discretion.

▪

Bounded inputs/outputs model: To run this model, the upper and lower bounds of inputs and outputs should be defined by using the "define" (Figure 24). The lower bound should be less than or equal to the original value and the upper bound should be greater than or equal to the original value.

Figure 24. Window of defining the boundaries of inputs and outputs.

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If all inputs or outputs are set to the "full discretion" in the model, it is equivalent to the normal model. In addition, the NDSC model becomes a bounded model considering the Eq. (1).

{

Lower bound = Original Value × (1 − Degrees of Discretion) Original Value Upper bound = (1 − Degrees of Discretion) ▪

▪ ▪

▪

▪

▪

(1)

Weak disposability model: As mentioned in chapter 1, this model limits the slack of input/output with weak disposability to zero and is not applicable to being the non-radial SBM model. Modified SBM (MSBM) model: In this model, the distance should be adjusted according to the maximum distance to the frontier. Preference (weighted) model: Here, input and output weights in SBM models are determined based on their relative importance. To do this, the set of weights must be defined in the "define". The user is free to choose the values of the weights, and this model is equivalent to normal models if all the weights are equal. Restricted projection model: In this case, constraints are added based on the ratios of the output or input targets, and the constraint is limited similar to multiplier models. Context dependent model: As mentioned in chapter 1, the context dependent model is based on evaluating the specific performance of a DMU set at a functional efficiency level relative to the efficient frontier, and more accurate performance results are obtained by evaluating the relative attractiveness of efficient units compared to inefficient units. Fuzzy DEA model: Traditional DEA models require accurate and definite amounts of data, while the observed values for inputs or outputs may be imprecise in the real world. The imprecise data are expressed at a limited bound as fuzzy data. The application of the fuzzy theory set in DEA is divided into tolerance, α-level, fuzzy rating, and probabilistic methods. The α-level approach is the most common fuzzy DEA model. In this method, the main purpose of

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converting the DEA fuzzy model to two parametric models is based on the upper and lower bounds of the α membership function of efficiency scores. Here, the DEA model with fuzzy input/output is converted into two standard DEA models so that the upper and lower bounds of the efficiency scores are obtained for all DMUs, respectively. To run these models, the upper and lower fuzzy output/input bounds should first be defined. Although fuzzy data has upper and lower bounds, one variable should be added for each fuzzy input or output. The value of this variable should be between the upper and lower bounds (like the average of the upper and lower bounds). The variable is not used in the fuzzy model and is expressed only to facilitate the definition of data. Metafrontier model: As mentioned in chapter 1, the efficiency in this model is obtained based on the TGR and relative efficiency method. Non-concave metafrontier model: This model uses dissimilar and inhomogeneous technologies compared to the original metafrontier model and has good compatibility. Undesirable outputs model: If there are bad outputs, the model is undesirable. In order to improve the condition, the bad outputs are adjusted in the opposite direction to the good outputs, which should be considered to have more good outputs. In this software, undesirable outputs can be used for radial, DDF, SBM, and hybrid models. Radial models with undesirable outputs are the basis DDF and specific DDF model. Inseparable good and bad outputs model: In this model, bad outputs are inseparable from good outputs. Reducing bad outputs leads to the reduction of good outputs. In this regard, the SBM-NS and NSOverall models are considered. However, in this software, inseparable models are expressed by considering options such as orientation, discretion, weak disposability based on linear programming. Furthermore, the increasing upper bound for the outputs in the software is calculated as Eq. (2).

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Upper bound = Original Value/(1 − Degrees of Discretion)

(2)

This model has the following features: • • •

The total amount of good output remains unchanged; Slack variables of inseparable inputs are as inefficiency; Slack variables of inseparable bad outputs are treated as efficiency.

9.2.1.2.1.6. Advanced Models 2 As shown in Figure 25, these models consist of three parts.

Figure 25. Settings window of advanced models 2.

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Malmquist index (productivity analysis): As already mentioned in chapter 1, the Malmquist index evaluates all the factors of efficiency changes of each DMU in two time periods. In this software, nine types of Malmquist models are considered. Window analysis model: In the window analysis model, the "window width" should be defined. If the results are calculated separately for each year, the window width index should be considered equal to one. The Malmquist index and window analysis model are similarly expressed for multiplier models.

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Dynamic DEA model: In the initial data of this model, the links between periods should be considered and each DMU should have only one observation in each period. It is worth noting that in this software, the first period intermediate has the role of a link between the first and second period, the second period intermediate plays the role of a link between the second and third period, and the intermediate is not considered in the third period. The dynamic model is also a special case of the network model. Therefore, the results of all dynamic models can be obtained through the network model. Bootstrapping model: In the settings window of the bootstrapping model (Figure 26), the various parameters and modes of this model can be defined.

Figure 26. Settings window of the bootstrapping model.

9.2.1.2.1.7. Results for Envelopment Models The required information from the software results can be selected in the "result" window of Figure 27 (similarly expressed for multiple models).

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Figure 27. Settings window of the envelopment model results.

9.2.1.2.1.8. Options In this section, items such as the number of decimal places, efficiency score, computation of slack variables, and time to solve the problem are defined and similarly expressed for multiplier models (Figure 28). The purpose of solving the input-oriented model in the single-stage method is to calculate the slack variable values. Furthermore, a two-stage method should be selected if the amount of slack variables or super-efficiency is required.

Figure 28. Options for envelopment and multiplier models.

9.2.1.2.2. Multiplier Models As shown in Figure 29, the multiplier model window is displayed by choosing the "multiplier model", which consists of eight tabs.

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Figure 29. Multiplier models window.

9.2.1.2.2.1. Measuring Efficiency Based on Distance In this case, the efficiency is calculated in both radial and direct distance methods (Figure 29). 9.2.1.2.2.2. Efficiency Measurement Based on Orientation This measurement method is expressed for input-oriented, outputoriented, and non-oriented. 9.2.1.2.2.3. Efficiency Measurement Based on the RTS This is similar to envelopment models, except that the scale elasticity has been added to represent the quantitative estimate of RTS (Figure 30).

Figure 30. Window settings for efficiency measurement based on the RTS.

9.2.1.2.2.4. Advanced Models 1 and 2 Some of the advanced models 1 and 2 are described in the envelopment models section and new models are described below.

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Cross-efficiency model: As shown in Figure 31, the mean crossefficiency is normally calculated. If the number of DMUs is not more than 252, the cross matrix is shown in the results. It is worth noting that several optimal solutions are usually obtained in these models. Therefore, the cross-efficiency score is not unique. Thus, two-stage methods should be used to solve these problems. Also, the two-stage methods available in the software include the maximum-minimum balanced method (blanket benevolent and blanket aggressive), the maximum-minimum cross-efficiency (blanket benevolent and blanket aggressive), and maximumminimum cross-efficiency (blanket benevolent and blanket aggressive). The cross-efficiency model can be combined with the global Malmquist model.

Figure 31. Settings window of cross-efficiency model.

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Two-stage network model: In this model, a two-stage production system is defined and all outputs (intermediates) are transferred from the first stage to the second stage. This model does not only show the efficiency score but also receives the amount of efficiency for each stage. In order to run these models, the intermediates should be defined after choosing the model (Figure 32). Common weights model: In basic DEA models, DMUs evaluate maximum efficiency by using their weights. This type of evaluation prevents DMUs from being full ranked, and the results

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become acceptable to them. To solve this problem, the common weighted model with the concept of DMU satisfaction in relation to the set of common weights is presented. In this method, a set of weights is produced for DMUs which maximizes the lowest degree of satisfaction among DMUs, and it can be assured that the set of weights produced is unique and that the degree of satisfaction of DMUs represents only the Pareto optimal solution.

Figure 32. Two-stage network model.

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Minimum efficiency model: Unlike the basic models, which seek to achieve maximum efficiency based on the desired weights, the minimum efficiency of DMUs is obtained with respect to the desired weights of inputs and outputs in this model by considering the limitation that the maximum efficiency of all DMUs is one.

Alireza Alinezhad, Seyyed Hamed Mirtaleb and Javad Khalili

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Restricted multiplier model: As shown in Figure 33, in this software, two types of multiplier models are considered, which are shown as type I and type II. In these models, a new restriction on the ratio of input/output weights has been added to the model (also known as the AR model).

Figure 33. Multiplier model selection.

In the type I model, the constraints on the ratio of input/output weights are added as an Eq. (3). L1