Data Envelopment Analysis with GAMS: A Handbook on Productivity Analysis and Performance Measurement 3031307003, 9783031307003

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International Series in Operations Research & Management Science

Ali Emrouznejad Konstantinos Petridis Vincent Charles

Data Envelopment Analysis with GAMS A Handbook on Productivity Analysis and Performance Measurement

International Series in Operations Research & Management Science Founding Editor Frederick S. Hillier, Stanford University, Stanford, CA, USA

Volume 338 Series Editor Camille C. Price, Department of Computer Science, Stephen F. Austin State University, Nacogdoches, TX, USA Editorial Board Members Emanuele Borgonovo, Department of Decision Sciences, Bocconi University, Milan, Italy Barry L. Nelson, Department of Industrial Engineering & Management Sciences, Northwestern University, Evanston, IL, USA Bruce W. Patty, Veritec Solutions, Mill Valley, CA, USA Michael Pinedo, Stern School of Business, New York University, New York, NY, USA Robert J. Vanderbei, Princeton University, Princeton, NJ, USA Associate Editor Joe Zhu, Foisie Business School, Worcester Polytechnic Institute, Worcester, MA, USA

The book series International Series in Operations Research and Management Science encompasses the various areas of operations research and management science. Both theoretical and applied books are included. It describes current advances anywhere in the world that are at the cutting edge of the field. The series is aimed especially at researchers, advanced graduate students, and sophisticated practitioners. The series features three types of books: • Advanced expository books that extend and unify our understanding of particular areas. • Research monographs that make substantial contributions to knowledge. • Handbooks that define the new state of the art in particular areas. Each handbook will be edited by a leading authority in the area who will organize a team of experts on various aspects of the topic to write individual chapters. A handbook may emphasize expository surveys or completely new advances (either research or applications) or a combination of both. The series emphasizes the following four areas: Mathematical Programming: Including linear programming, integer programming, nonlinear programming, interior point methods, game theory, network optimization models, combinatorics, equilibrium programming, complementarity theory, multiobjective optimization, dynamic programming, stochastic programming, complexity theory, etc. Applied Probability: Including queuing theory, simulation, renewal theory, Brownian motion and diffusion processes, decision analysis, Markov decision processes, reliability theory, forecasting, other stochastic processes motivated by applications, etc. Production and Operations Management: Including inventory theory, production scheduling, capacity planning, facility location, supply chain management, distribution systems, materials requirements planning, just-in-time systems, flexible manufacturing systems, design of production lines, logistical planning, strategic issues, etc. Applications of Operations Research and Management Science: Including telecommunications, health care, capital budgeting and finance, economics, marketing, public policy, military operations research, humanitarian relief and disaster mitigation, service operations, transportation systems, etc. This book series is indexed in Scopus.

Ali Emrouznejad • Konstantinos Petridis • Vincent Charles

Data Envelopment Analysis with GAMS A Handbook on Productivity Analysis and Performance Measurement

Ali Emrouznejad University of Surrey Guildford, UK

Konstantinos Petridis University of Macedonia Thessaloniki, Greece

Vincent Charles University of Bradford Bradford, UK

ISSN 0884-8289 ISSN 2214-7934 (electronic) International Series in Operations Research & Management Science ISBN 978-3-031-30700-3 ISBN 978-3-031-30701-0 (eBook) https://doi.org/10.1007/978-3-031-30701-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

In the era of artificial intelligence and data science, measuring the performance and productivity of any entity is inevitable for its success and sustainability. Data Envelopment Analysis (DEA), a mathematical programming technique that is frequently used to evaluate the performance of Decision-Making Units (DMUs) that convert multiple inputs into multiple outputs, is a promising tool in this regard. In time, DEA has indeed proven to be an excellent decision support system tool for measuring DMU efficiency and productivity. Since the seminal work of Charnes, Cooper, and Rhodes (1978), the number of journal articles has grown exponentially over the past decades, with numerous applications of DEA documented in the literature. This unique book describes how the General Algebraic Modeling System (GAMS), a computationally efficient tool to deal with complex optimisation problems, can be used to solve a range of DEA models. This book is the first of its kind to provide readers with a comprehensive reference that includes the solution codes for a range of DEA models in GAMS. The book covers theoretical background as well as application examples. The book is organised into eight chapters. It begins with an introduction to the GAMS environment, with a focus on how to build a simple model, compile it, and derive the results. More specifically, Chap. 1 covers the syntax of the GAMS formulation, the import of data, loop structures, and the presentation of the results. Chapter 2 lays the background to the DEA approach. Both envelopment and multiplier models are introduced, and these are discussed in terms of the type of frontier scale they use (i.e., CRS or VRS). Assurance regions and weight restrictions are addressed next. The topic of how resources should be allocated and used to achieve the best outcomes is of particular interest, as evidenced by the subsection on the most productive scale size. Finally, the chapter discusses super-efficiency models as an approach to dealing with situations in which there are multiple efficient units, and one needs to discriminate between them. Chapter 3 focuses on extensions of DEA models, demonstrating recent developments in DEA formulations. The models analytically described and modelled with v

vi

Preface

GAMS in this chapter concern DEA models with exogenously fixed variables and categorical variables, DEA models for handling desirable and undesirable outputs, congestion, and chance constraints. Chapter 4 is dedicated to addressing non-radial DEA models. First, the chapter looks into CRS DEA and VRS DEA and their respective GAMS formulations. Then, the range-adjusted measure (RAM) is introduced, along with its CRS and VRS variants. The chapter further addresses the issue of negative data in DEA and presents three different approaches for dealing with such data. In the context of DEA formulations, several measures have been proposed over the years for measuring the efficiency of DMUs. Chapter 5 delves into exploring allocative, cost, technical, revenue, and profit efficiency, along with their GAMS formulations. Chapter 6 focuses on special cases in DEA. More specifically, the benefit-of-thedoubt approach (widely used in the development of composite indicators) is addressed first. This is then followed by a discussion of the multi-objective DEA models. Chapter 7 addresses the topic of productivity change over a given period of time, with a focus on the Malmquist Productivity Index and the Malmquist–Luenberger Productivity Index. Lastly, concluding remarks and direction for future work are presented in Chap. 8. The chapters composing this book should be of considerable interest and provide our readers with useful information for their studies and research. We wish you informative reading! Guildford, UK Thessaloniki, Greece Bradford, UK June 2023

Ali Emrouznejad Konstantinos Petridis Vincent Charles

Contents

1

Introduction to GAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Syntax of the GAMS Model . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Compiling the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Analysing the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Importing Data into GAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Loops in GAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Presenting the Results . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 2 6 6 6 8 10

2

Introduction to Data Envelopment Analysis . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Envelopment Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 GAMS Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 The CRS DEA Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 The VRS DEA Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Projected Values and Targets . . . . . . . . . . . . . . . . . . . . . . 2.3 Multiplier Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 The CRS DEA Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 The VRS DEA Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Assurance Regions/Weight Restrictions . . . . . . . . . . . . . . . . . . . . 2.5 Most Productive Scale Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Super-Efficiency Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 13 14 15 16 18 22 23 23 26 26 30 34

3

Extensions of DEA Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Exogenously Fixed Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 DEA CRS Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 DEA VRS Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Undesirable Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 41 42 43 51 vii

viii

Contents

3.4

Congestion in DEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Congestion Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Congestion with Slack Variables . . . . . . . . . . . . . . . . . . . . Categorical Variables in DEA . . . . . . . . . . . . . . . . . . . . . . . . . . . Chance-Constrained DEA Model . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 The Modelling of Chance Constraints . . . . . . . . . . . . . . . . 3.6.2 Stochastic Efficiency in Marginal Chance-Constrained Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 60 64 68 68

4

Non-radial DEA Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Non-radial CRS DEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Non-radial VRS DEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Range-Adjusted Measure Model . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 RAM CRS DEA Model . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 RAM VRS DEA Model . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Negative Data in DEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 The Range Directional Model . . . . . . . . . . . . . . . . . . . . . . 4.5.2 The Modified Slacks-Based Model . . . . . . . . . . . . . . . . . . 4.5.3 The Semi-Oriented Radial Measure . . . . . . . . . . . . . . . . . .

75 75 75 76 80 82 85 90 90 92 95

5

Allocative, Cost, Technical, Revenue, and Profit Efficiency . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Allocative and Cost Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Data for Cost Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 GAMS Formulation for Allocative and Cost Efficiency . . . 5.3 Revenue and Technical Efficiency . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Data for Revenue Efficiency . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 GAMS Formulation for Revenue Efficiency . . . . . . . . . . . 5.4 Profit Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 GAMS Formulation for Profit Efficiency . . . . . . . . . . . . . .

103 103 103 105 105 109 109 111 111 111

6

Special Cases in DEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Benefit-of-the-Doubt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Data for the BoD Example . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 GAMS Formulation for BoD . . . . . . . . . . . . . . . . . . . . . . 6.3 Multi-objective Linear Programming in DEA . . . . . . . . . . . . . . . . 6.3.1 Data for MOLP in DEA . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Scenarios for Weights for MOLP in DEA . . . . . . . . . . . . . 6.3.3 GAMS Formulation for MOLP in DEA . . . . . . . . . . . . . .

119 119 119 119 120 120 122 122 123

7

Productivity Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Calculation of the Malmquist Productivity Index . . . . . . . . . . . . . 7.2.1 Data for the Calculation of MPI . . . . . . . . . . . . . . . . . . . .

127 127 127 128

3.5 3.6

69

Contents

. . . .

129 132 133 135

Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 General Algebraic Modelling System . . . . . . . . . . . . . . . . . . . . . . 8.3 Data Envelopment Analysis Models . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Conventional DEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Productivity Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Data Envelopment Analysis Software . . . . . . . . . . . . . . . . . . . . . . 8.5 GAMS for Data Envelopment Analysis . . . . . . . . . . . . . . . . . . . . 8.6 Conclusions and Future Developments . . . . . . . . . . . . . . . . . . . . .

141 141 141 141 142 144 145 146 147

7.3

8

ix

7.2.2 GAMS Formulation for MPI . . . . . . . . . . . . . . . . . . . . . Calculation of the Malmquist–Luenberger Productivity Index . . . 7.3.1 Data for the Calculation of MLPI . . . . . . . . . . . . . . . . . . 7.3.2 GAMS Formulation for MLPI . . . . . . . . . . . . . . . . . . . .

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

About the Author

Ali Emrouznejad (Professor and Chair in Business Analytics) is Director of the Centre for Business Analytics in Practice (CBAP) at Surrey Business School, UK. His areas of research interest include performance measurement and management, efficiency and productivity analysis as well as AI and big data. He holds an MSc in applied mathematics and received his PhD in operational research and systems from Warwick Business School, UK. Professor Emrouznejad is editor of Annals of Operations Research, associate editor of Socio-Economic Planning Sciences, department editor of OR-Spectrum and IMA Journal of Management Mathematics, and member of editorial boards or guest editor in several other scientific journals including the European Journal of Operational Research and Journal of Operational Research Society. He has published over 250 articles in top-ranked journals. He is the author of the book on “Applied Operational Research with SAS”, and editor of several other books including “Performance Measurement with Fuzzy DEA”, “Managing Service Productivity”, and “Handbook of Research on Strategic Performance Management and Measurement Using DEA”. He is also the founder of www.DEAzone. com and co-founder of Performance Improvement Management Software (PIM-DEA www.DEAsoftware. co.uk).

xi

xii

About the Author

Konstantinos Petridis is Research Assistant at the University of Macedonia. He holds a B.Sc. in Mathematics from the Aristotle University of Thessaloniki, an M.Sc. in Logistics Management from the Department of Economic Sciences, Aristotle University of Thessaloniki while his Ph.D. thesis was focused on Operational Research with application to natural resources. His research interests include the study and modelling of production and distribution of biomass and natural resources systems, supply chain management design and sustainability through the use of operational research and performance measurement techniques in healthcare and waste to energy plants. Dr. Petridis’ current works are focused on forest supply chain and production and distribution of forest products, and energy forms produced from biomass and fossil fuel co-firing. Moreover, one of the fields he is currently working on is the integration of several operational research techniques in an algorithmic way for ranking, forecasting, and decision-making with multiple criteria (multiobjective programming). His educational interests are concentrated in optimisation and simulation software emphasised in industrial production and planning and his research has been published in international journals and book chapters.

Vincent Charles, PhD, PDRF, FRSS, FHEA, FPPBA, MIScT, CMBE, is an Associate Professor of AI for Business and Director of PGR Studies at the School of Management, University of Bradford. In addition, he holds multiple visiting professorship positions across the Globe. He is an experienced researcher in the fields of AI/ML, (big) data science, and OR/MS. He has more than 25 years of teaching, research, and consultancy experience, having been a full professor and director of research for more than a decade in triple-crown business schools. He is a former Elected Member of the Senate. He holds Executive Certificates from the MIT, HBS, and IE Business School. He has published over 150 research outputs and is a recipient of many international academic honours and awards. AWS Certified Cloud Practitioner, AWS Accredited Educator, Certified Six Sigma Black Belt, and Advance HE Certified

About the Author

xiii

External Examiner, UK. He is an Associate Editor of the Journal of the Operational Research Society, Expert Systems with Applications, Machine Learning with Applications, Intelligent Systems with Applications, RAIRO—Operations Research, and Decision Analytics Journal. He further serves as a Special Issue Editor for several journals, including the Journal of Business Research, OR Spectrum, Environmental Science & Policy, Information Systems Frontiers, Government Information Quarterly, and the Business Process Management Journal. He also has industry experience as a Chief Analytics Officer and has worked on projects in the telecommunications and retail industries.

Chapter 1

Introduction to GAMS

1.1

Introduction

One of the most widely used software for mathematical programming is the General Algebraic Modeling System (GAMS). The software is user-friendly and facilitates the process of going from a mathematical statement of the problem to its solution. The main use of GAMS is for optimisation. One of the features that makes the GAMS software easy to use and popular in academic and commercial settings is that the user can provide the mathematical model while GAMS transforms it into representations required by solvers (CPLEX, BARON, and so on). GAMS can solve a wide variety of mathematical programming problems, including Linear Programming (LP), Non-Linear Programming (NLP), Integer Programming (IP), Mixed Integer Linear Programming (MILP), and Mixed Integer Non-Linear Programming (MINLP), among others. The reader can find more information regarding GAMS on the official GAMS page (https://www.gams.com/), where the latest and older versions of the software are available for download and keep track of news and announcements. A limited version of the software can be downloaded for free on the GAMS website. The purpose of this book is to present Data Envelopment Analysis (DEA) models using GAMS. The examples presented in this book consist of a number of variables that can be compiled on the free version of GAMS. The book begins with an introduction to the GAMS environment, with a focus on how to build a model, compile it, and derive the results. More specifically, in the following sections of the chapter, the import of data, the syntax of the model, loop structures, and presentation of the results will be analytically explained.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Emrouznejad et al., Data Envelopment Analysis with GAMS, International Series in Operations Research & Management Science 338, https://doi.org/10.1007/978-3-031-30701-0_1

1

2

1.2 1.2.1

1

Introduction to GAMS

The Syntax of the GAMS Model An Illustrative Example

In this section, the reader will be introduced to the syntax of a GAMS model through an illustrative example. The basic parts of a GAMS model are as follows: • Sets: refer to the indices of the variables and parameters of the model. • Scalars: refer to the fixed value of a parameter. • Parameters: refer to the parameters of the mathematical model, the parts that are pre-defined by the user. Tables are multi-dimensional parameters. • Variables: refer to the declaration of the variables as the results of the model. Variables are changed by the solver. • Equations: refer to the equations, constraints, and objective function statement. • Model: refers to the equations, constraints, and objective function of the model. • Solve: is the statement based on which the model is solved and the objective function is handled (max, min). • Display: using this command, the results of the variables and parameters are presented to the user. A typical GAMS model structure is in the following order: Sets, Parameters, Equations, Model, Solve, and Display (the Display command can be used anywhere in the model). To illustrate the GAMS formulation, let us consider a transportation problem, which is widely used in the operational research literature. The transportation model is mathematically formulated and described below. Based on this model, it is assumed that there are three production plants that produce a single product to supply four markets. The plants are denoted by i (i = 1, . . ., 3) while the markets are denoted by j ( j = 1, . . ., 4). The supply is denoted by parameter Si while demand is denoted by Dj; the variable xi, j represents the quantities to be transported from plant i to market j in order to satisfy demand and supply constraints. The objective of the model is to find the optimal allocation of quantities that minimises the overall transportation cost (whereas ci, j is the unit transportation cost from plant i to market j). The overall cost (TC) is minimised. 3

4

ci,j
xi,j

min TC = i=1 j=1

s:t: m

j=1 n i=1

xi,j ≥ Si , i = 1, . . . , 3 xi,j ≤ Dj , j = 1, . . . , 4

xi,j ≥ 0, i = 1, . . . , 3, j = 1, . . . , 4 TC free

ð1:1Þ

1.2

The Syntax of the GAMS Model

3

Table 1.1 Supply data (Si)

Plant (i) 1 2 3

Si 150 100 250

Table 1.2 Demand data (Dj)

Market ( j) 1 2 3 4

Dj 100 100 200 100

Table 1.3 Unit transportation cost data (ci,j)

i 1 2 3

j 1 0.54 0.30 0.44

2 0.32 0.78 0.19

3 0.10 0.76 0.61

4 0.29 0.27 0.97

The Linear Programming (LP) model (1.1) is an algebraic form of the transportation model. The data for model (1.1) are given in Tables 1.1, 1.2, and 1.3. Figure 1.1 summarises the corresponding GAMS formulation for the transportation model (1.1) with the data derived from Tables 1.1–1.3. In the GAMS representation of the transportation model (1.1), the first part of the syntax is the declaration of sets (indices) based on which variables, parameters, and tables will be introduced. Each index is introduced to GAMS as follows: Index (i) Description (Plants) /Plant1*Plant3/ Inside the / /, the elements of the set are stated. When Plant1*Plant3 is written, then GAMS assumes that the elements are Plant1, Plant2, and Plant3. The declaration of sets is terminated with the semicolon (;). Once the sets/indices have been determined, then the parameters (one-dimensional data structures) and tables (multidimensional data structures) can be introduced. For example, the supply parameter is declared by first writing the word Parameter and the symbol that will be used to represent the supply S(i). Anything written after the symbol S(i)is considered to be a description of this particular data structure. The same can be seen for the demand structure. The declaration of the parameters is terminated with the semicolon (;) after the last backslash (/). In the case of multidimensional data structures like the unit transportation cost (ci, j), the easiest way to introduce the data is using Table. The declaration is the same as for the parameters; however, when using Table, there is no need for / /. The data are declared by providing the elements of the first dimension as a row and of the second dimension as a column. In this case, the row is expressed by i Plants /Plant1*Plant3/ while the column is expressed by j Markets /Market1*Market4/. The declaration of the table is terminated with the semicolon (;).

4

1 Introduction to GAMS

Table 1

Table 1

Table 1

(1.1)

Fig. 1.1 Mathematical formulation of the transportation model and the corresponding GAMS formulation

1.2

The Syntax of the GAMS Model

5

Following the structure of the GAMS model, the declaration of the variables is done. Any variable that is specified in the Variables is assumed by GAMS to be a free variable. Thus, as TC represents the objective function, it is a free variable. However, the quantities to be transported from plant i to market j is a non-negative variable; thus, it is declared under Nonnegative Variables. If xi, j were strictly positive, then the variable would be listed under Positive Variables. After the declaration of parameters and variables, the constraints and objective function are declared symbolically under the Equation section of the GAMS model. The first part of the symbolical declaration of variables is to provide the names of the objective function and constraints, followed by a semicolon (;). Afterwards, the mathematical model is provided in a symbolic form to GAMS. The function SUM(i, 3

x(i,j)) is used for the summation and expresses the following term

xi,j while

i=1 3

PROD(i,x(i,j)) is used for the product and expresses the following term

i=1

xi,j . The

objective function is stated as: TotalCost.. TC=E=SUM(i,SUM(j,c(i,j)*x(i,j)));

The first part of the objective function provides information about the variable that is not summed upon. In this case, the term c(i,j)*x(i,j) is summed with respect to indices i and j, so there is no index which is not summed. The symbol =E= stands for equal (=), the symbol =L= stands for less than or equal to (≤) while the symbol =G= stands for greater than or equal to (≥). Each equation must be followed by a semicolon (;). Once the equations (objective function and constraints) are declared, the Model is stated by writing the following syntax: Model Transportation_example (The model name) /All/. The part /All/ refers to the equations that form the model. As the transportation model consists of all the constraints and the objective function, then the word All is written between the / / . If a constraint is not taken into account (for example, Supply(i)), then the syntax would be as follows: Model Transportation_example /TotalCost, Demand/;

Finally, the Solve statement is declared, where the model direction (minimisation of maximisation) of the variable of the objective function and the mathematical programming technique (Linear Programming, Non-Linear Programming, etc.) that will be used are listed. The results of the model are displayed using the Display command. Using this command, certain information can be derived about the variables; x.l (level) provides the optimal solutions of the model for variable xi, j, whereas x.m (marginal) provides the marginal values for variable xi, j.

6

1.2.2

1

Introduction to GAMS

Compiling the Model

After the model is set up, it can be compiled with the compile button , with the F9 key, or with the commands File ! Run. The model has been compiled successfully if the following message (Fig. 1.2) is returned after the compilation:

1.2.3

Analysing the Results

The results after solving the model are presented in Figs. 1.3 and 1.4. It can be seen that the results suggest that 150 units of products must be transported from Plant1 to Market3, 100 units of products from Plant2 to Market4, etc. Empty entries in this table (e.g., Plant1—Market1) stand for 0. The marginal values of the variable xi, j are shown in Fig. 1.4. The total cost of the optimal solutions, calculated based on the information presented in Figs. 1.3 and 1.5, is 135.5 monetary units.

1.3

Importing Data into GAMS

Besides typing the data structures (parameters, tables) into the GAMS model, the data can also be imported into GAMS with the following procedure. Assuming that the data (Supply, Demand, Unit Transportation Cost) are stored in an Excel spreadsheet (Fig. 1.5), then the code below can be used in order to import data into GAMS. As it can be seen from Fig. 1.5, the supply parameter lies in the range B3:C5. Similarly, the demand parameter lies in the range E3:F6 while the unit cost parameter lies in the range H3:L6. Thus, instead of stating the parameters/tables as seen in the GAMS model in Fig. 1.1, with the following commands, data can be imported into GAMS. The Excel file that contains the data must be in the same folder as the GAMS file (.gms).

Fig. 1.2 Normal compilation of the transportation example

1.3

Importing Data into GAMS

Fig. 1.3 Levels of decisions for the variable xi,j

Fig. 1.4 Marginal values of the variable xi,j

Fig. 1.5 Data for supply (Si), demand (Dj), and unit transportation cost (ci,j)

$CALL GDXXRW.EXE Chapter1_data.xls par=S rng=Sheet1!B3:C5 Rdim=1 Parameter S(i); $GDXIN Chapter1_data.gdx $LOAD S $GDXIN $CALL GDXXRW.EXE Chapter1_data.xls par=D rng=Sheet1!E3:F6 Rdim=1 Parameter D(j); $GDXIN Chapter1_data.gdx $LOAD D $GDXIN $CALL GDXXRW.EXE Chapter1_data.xls par=c rng=Sheet1!H3:L6 Cdim=1 Rdim=1 Parameter c(i,j); $GDXIN Chapter1_data.gdx $LOAD c $GDXIN

7

8

1

Introduction to GAMS

For more information regarding the import of data from Excel (or any other data format) into GAMS, the reader can visit the following link.1

1.4

Loops in GAMS

One of the most significant functions in GAMS is that the model can be solved iteratively (for different scenarios, etc.). In order to use loops in a GAMS model, the following syntax must be used: loop(index, ... );

Besides loop, there are other iterative structures in GAMS (e.g., while, repeat. . .until) that can be found in the GAMS documentation.2 In order to demonstrate the usage of loops in a GAMS model, let us consider the transportation example described above. Assume that the transportation cost is forecasted to increase from 1% to 10%. Let us calculate the corresponding total cost. In order to construct the new cost data structure for an increase from 1% to 10%, then assuming that c1sc,i,j is the new cost for the corresponding scenarios, the new cost structure is computed based on the following formula: c1sc,i,j = 1 þ

sc
ci,j 100

ð1:2Þ

sc In Eq. (1.2), sc = 1, . . ., 10. Hence, 1 þ 100 = 1:01, . . . , 1:10. The corresponding formulation in GAMS would be as follows. The new characteristic is the ORD (sc) function, which returns the order of the elements in a set. For example, Sce1 is the first element; therefore, ORD(Sce1)=1. A new dimension has been added to the new parameter, and in order to introduce the new parameter to the model, a new parameter has to be introduced for each scenario.

Set sc Scenarios /Sce1*Sce10/; Parameter c1(sc,i,j); loop(sc, c1(sc,i,j)=(1+ORD(sc)/100)*c(i,j); );

1

https://www.gams.com/help/index.jsp?topic=%2Fgams.doc%2Fuserguides%2Fuserguide%2F_ u_g__data_exchange__excel.html 2 https://www.gams.com/help/index.jsp?topic=%2Fgams.doc%2Fuserguides%2Fmccarl%2Floop. htm

1.4

Loops in GAMS

9

Also, the cost parameter has to be introduced to the objective function so that, in each iteration, the LP model should be solved for different transportation costs. The following syntax is used before the solve statement. The new parameter (assuming it is c0i,j), which initially did not have any values, now has new values for each iteration (sc = 1, . . ., 10). The corresponding LP model to GAMS syntax is presented in Fig. 1.6.

Fig. 1.6 Scenario formulation of the transportation model and the corresponding GAMS formulation using loop

10

1

Introduction to GAMS

Fig. 1.7 Results of the objective per each cost scenario loop(sc, ck(i,j)=c1(sc,i,j); Solve Transportation_example min TC using LP; res_TC(sc)=TC.l; );

1.4.1

Presenting the Results

The results of the model after solving iteratively the transportation model for various scenarios are presented in Fig. 1.7. It can be seen that the total transportation cost for scenario 1 (an increase by 1%) is 136.855 monetary units, whereas the total transportation cost for scenario 10 (increase by 10%) is 149.050 monetary units.

Chapter 2

Introduction to Data Envelopment Analysis

2.1

Introduction

The field of production economics has rapidly changed over the last decades. This rapid change is partly due to the Data Envelopment Analysis (DEA) technique, which assesses the comparative performance of a set of units based on inputs and outputs, measuring the efficiency of the transformation procedure. The inputs are consumed in order to produce outputs; thus, the fraction of outputs produced to inputs consumed is the efficiency of the transformation. Because it can easily take into account the existence of numerous inputs and multiple outputs without making any assumptions about the functional form, DEA is more popular than econometric techniques (Charles & Kumar, 2012). A typical statistical approach is characterised as a central tendency approach, and it examines producers in relation to an average producer. DEA, on the other hand, uses an extreme-point comparison and only evaluates the best producer with each producer(s). Furthermore, DEA does not impose any underlying assumptions about an inputs-and-outputs functional form like the econometric approach does. Instead, it creates its own functional form based on the inputs and outputs of several units, thus avoiding the risk of incorrectly defining the frontier technology. Also, whereas DEA easily takes into account the existence of multiple outputs, the parametric approach measures the efficiency of units producing a single output with a set of multiple inputs. Lastly, for datasets with fewer than 100 observations, the parametric approach’s decomposition of the error term into two parts—one reflecting the stochastic error and the other indicating inefficiency—is not helpful. Contrarily, DEA performs effectively with a limited sample size. The minimal sample size needed for DEA analysis is often merely two times greater than the total number of inputs and outputs (Golany & Roll, 1989; Homburg, 2001), although there are various other empirical rules of thumb available (see Banker et al., 1989; Cooper

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Emrouznejad et al., Data Envelopment Analysis with GAMS, International Series in Operations Research & Management Science 338, https://doi.org/10.1007/978-3-031-30701-0_2

11

12

2 Introduction to Data Envelopment Analysis

et al., 2007a; Dyson et al., 2001; Friedman & Sinuany-Stern, 1998; Nunamaker, 1985; Raab & Lichty, 2002). In measuring the relative efficiency of units considering their inputs and outputs, DEA determines a subset of “best practice” units, and the efficiency level of the other units is determined by comparing them to a frontier formed from the “best practice” units. Each unit is analysed independently to determine if the unit under consideration could improve its performance by boosting output and decreasing input. Performance measurement is also a valuable tool for businesses and firms with multiple branches (like banks), as it allows each business/firm or economic organisation to compare its position to an ideal benchmark. In this sense, DEA enables decision-makers to comprehend the nature of a unit’s inefficiencies by comparing it with a selected set of efficient units that share a similar profile. For instance, if an organisation consumes more inputs while producing the same amount of outputs as another unit (firm, business, organisation, etc.), then the transformation mechanism most likely needs to be changed in order for the unit to perform better or be closer to the ideal situation, the benchmark, by reducing some of its inputs. The benchmark consists of units that perform better than others. DEA helps the units involved in the analysis in improving their operations and making short and long-term policy decisions; performance measurement also serves as a policy tool in this dynamically changing economic environment in which businesses must survive and thrive. Improving a unit’s operations, such as reducing inventory and minimising lead and transportation time, results in improved service levels and lower production cost. The use of the DEA technique is important because it can be applied to any organisation, production process, etc., taking multiple inputs and outputs into account in order to extract unit efficiency. This allows for a more realistic representation of the actual operations performed by modelling the transformation procedure in greater detail and, therefore, providing a realistic performance score. DEA has been used in diverse fields, ranging from agriculture to education, healthcare, banking and finance, transportation, and regional competitiveness, just to name a few. Needless to say, ever since its inception, DEA has grown in popularity among academics and practitioners alike, as evidenced by the relatively large number of methodological and application-oriented publications (Charles et al., 2018, 2021; Emrouznejad & Yang, 2018). Interested readers can consult a collection of contributions from DEA experts in Charles and Kumar (2012) and Osman et al. (2014), among others. For ease of use, the unit (process, procedure, business, or any other economic or other organisation) will be referred to as a Decision-Making Unit (DMU); the aim of the DEA technique is to identify fully efficient DMUs by comparing each DMU’s inputs and outputs against similar DMUs. Following the problem formulation, the decision-maker (DM) may need to examine an input- or output-oriented model. In the first case, the inputs are minimised while the outputs are held constant, whereas in the output-oriented model, the outputs are maximised while the inputs are kept constant. The choice between an input and an output orientation is commonly made

2.2

Envelopment Models

13

Y2

Y2

Y1

Y1

Fig. 2.1 Efficient frontiers based on input-oriented (left) and output-oriented (right) DEA models

Table 2.1 Input (a) and output (b) oriented DEA models (a)

(b)

min θ - ε


m i=1

si- þ

s r=1

sþ r

s:t: n

j=1 n j=1

max φ þ ε


m i=1

si- þ

s r=1

sþ r

s:t: xi,j
λj þ yr,j


si-

λj - s þ r

= xi,o
θ, i = 1, ::, m = yr,o , r = 1, ::, s

n

j=1 n j=1

xi,j
λj þ si- = xi,o , i = 1, ::, m yr,j
λj - sþ r = yr,o
φ, r = 1, ::, s

λj , si- , sþ r ≥ 0, 8j, i, r

λj , si- , sþ r ≥ 0, 8j, i, r

θ free

φ free

in view of which factors are more easily controlled by the DM. The frontiers in these two cases are depicted in Fig. 2.1. Besides the type of orientation, there are several DEA models for calculating the efficiency of DMUs based on pre-determined inputs and outputs. These DEA models are the multiplier and envelopment forms, which will be extensively examined in the following sections.

2.2

Envelopment Models

The envelopment models are presented in Table 2.1. The model presented in Table 2.1a is an input-oriented model while the one presented in Table 2.1b is an output-oriented one. The objective function in the first case is to minimise the free variable θ, which measures the efficiency of each DMU; the evaluation of each

14

2 Introduction to Data Envelopment Analysis

Table 2.2 Projections to the efficient frontier under input (a) and output (b) orientation

(a) xi,j = xi,j
θ - s, i þ, yr,j = yr,j þ sr

(b) xi,j = xi,j - si- , yr,j = yr,j
φ þ sþ, r

DMU’s efficiency is conducted upon a pre-determined set of i inputs (xi, j) and r outputs (yr, j) for each DMU j. The variables λj are the peers of each DMU j; the peers are used to provide information about the proximity of the DMU under investigation to other DMUs. Both mathematical formulations (a) and (b) represent Linear Programming (LP) models, which are solved for each DMU under examination, o. If for example, DMU5 has in its reference set DMUs 2 and 6, then λ2, λ6 ≠ 0. The nonnegative variables si- and sþ r are slack variables corresponding to the inputs and outputs. A fully efficient DMU is the one with  þ θ = 1 and si- = sþ r = 0 for model (a), while for model (b), φ = 1 and si = sr = 0,   whereas θ and φ are the optimal values after solving the LP models (a) and (b) for each DMU under examination. The range of values for input efficiency is 0 ≤ θ ≤ 1 while for output-oriented efficiency is φ ≥ 1; for output-oriented models, in order to capture the degree of inefficiency of a DMU, the reciprocal is calculated, such that 0 ≤ 1/φ ≤ 1. Following the discussion about the orientation of input and output DEA models, the optimal values of efficiency variables are of great interest as the input and output projections to the efficient frontier are calculated. In order to do that, the following equations are proposed in Table 2.2.

2.2.1

Illustrative Examples

In order to provide a better understanding of the proposed models, two illustrative examples are provided for input- and output-oriented models. Assume that the efficiency of 10 companies is to be evaluated based on their supply chain characteristics. The inputs are based on production and transportation costs and on holding inventory (which is also an index for supply chain management improvement, i.e., the less inventory held, the better the supply chain is managed). The outputs are based on data regarding each company’s operations (how well each company performs in terms of satisfied demand instances) and the economic data (revenues from their economic activities) as presented in Table 2.3. The efficiency of each DMU will be investigated using an input-oriented DEA model presented in the GAMS code in Fig. 2.4.

2.2

Envelopment Models

15

Table 2.3 Data for illustrative example Production cost (K £) 0.255 0.980 0.507 0.305 0.659 0.568 0.583 0.627 0.772 0.211

DMU1 DMU2 DMU3 DMU4 DMU5 DMU6 DMU7 DMU8 DMU9 DMU10

2.2.2

Transportation cost (K £) 0.161 0.248 0.937 0.249 0.248 0.508 0.628 0.675 0.657 0.065

Holding inventory quantities (tonnes) 0.373 0.606 0.749 0.841 0.979 0.919 0.732 0.738 0.486 0.457

No of satisfied demand instances 20 6 17 2 19 17 17 10 9 2

Revenues (K £) 2.64 5.29 2.43 8.99 2.94 0.75 6.36 7.20 2.16 9.55

GAMS Implementation

This section is dedicated to modelling DEA with GAMS. The first part of a GAMS model is the statement of the sets (indices of the model), as there are two ways to solve a model in GAMS. The first is to write the model, introducing the objective function and each constraint of the mathematical programming model analytically. The second option is to write the model as an algebraic representation. The indices of the proposed model are introduced in the GAMS model as follows: Sets

j DMUs /DMU1*DMU10/ g Inputs and Outputs /ProdCost, TrnCost, HoldInv, Rev/ i(g) Inputs /ProdCost, TrnCost, HoldInv/ r(g) Outputs /SatDem, Rev/;

In correspondence with the model presented, Fig. 2.2 represents the data for inputs and outputs, wherein j stands for the DMUs that will be used for efficiency extraction. When declaring sets, there are two ways to present the elements of each set (always enclosed by / /): 1. /DMU1*DMU10/ for consecutive elements 2. /DMU1, DMU3, DMU8/ for non-consecutive elements An overall index g is introduced, which includes inputs and outputs, as well as two new subsets of g, namely i(g) for inputs and r(g) for outputs. As can be seen, the names of inputs and outputs are shortened (for instance, Production Cost) because gaps to the names of variables or set elements are prohibited in GAMS. In the GAMS model, the variables (along with their indices) are presented as follows:

16

2 ProdCost

Introduction to Data Envelopment Analysis

TrnCost

HoldInv

SatDem

Rev

20

2.64

DMU1

0.255

0.161

0.373

DMU2

0.98

0.248

0.606

6

5.29

DMU3

0.507

0.937

0.749

17

2.43

DMU4

0.305

0.249

0.841

2

8.99

DMU5

0.659

0.248

0.979

19

2.94

DMU6

0.568

0.508

0.919

17

0.75

DMU7

0.583

0.628

0.732

17

6.36

DMU8

0.627

0.675

0.738

10

7.2

DMU9

0.772

0.657

0.486

9

2.16

DMU10

0.917

0.639

0.234

8

7.3;

Fig. 2.2 Data(j,g) Data for inputs and outputs Variables efficiency objective function Theta

efficiency

Lambda(j) dual weights sminus(i) slacks assigned to inputs splus(r)

slacks assigned to inputs;

Nonnegative variables Lambda(j) sminus(i) splus(r);

After defining the model’s variables, the parameters must be defined. Parameters in GAMS are a good way to store all of the information when solving mathematical programming models iteratively (like in DEA). As a result, for each LP model solved, efficiency and peers (lambda values) as well as slack variables (corresponding to inputs and outputs) should be provided. In this example, 10 LP models are solved (the same number as the DMUs of the study). Parameters DMU_data(g) Data for inputs and outputs for each DMU eff(j) efficiency (Theta values) lamres(j,j) peers for each DMU (Lambda values) slacks(j,g) slacks for inputs and outputs;

2.2.3

The CRS DEA Model

This section entails the algebraic formulation of the DEA model, as presented in the first DEA paper by Charnes et al. (1978). Built upon Farrell’s (1957) measure, this

2.2

Envelopment Models

17

classical DEA model of Charnes et al. (1978) (so-called the CCR model) assumed a convex cone. The model became known as the constant returns-to-scale (CRS) model, with an assumption that all units are operating at an optimal scale. CRS reflects the fact that the output will change by the same proportion as the inputs (e.g., a tripling of all inputs will triple the output). The CRS DEA model can be orientated in two different ways: input-orientated (minimisation problem) or output-orientated (maximisation problem). As can be seen, in the first part, the objective function and constraints are introduced. For instance, CON1(i) stands for 8i. The objective function in the GAMS code models the objective function of the input-oriented DEA model, as presented in Table 2.1a, while the constant ε is set to 10-6. Equations OBJ objective function CON1(i) input duals CON2(r) output dual;

OBJ..

efficiency=E=Theta-1E-6*(SUM(i,sminus(i))+SUM(r,splus(r)));

CON1(i)..

SUM(j, Lambda(j)*Data(j,i))+sminus(i)=E=Theta*DMU_data(i);

CON2(r)..

SUM(j, Lambda(j)*Data(j,r))-splus(r)=E=DMU_data(r);

Model DEA_CRS input oriented DEA CRS / OBJ, CON1, CON2 /;

Because the model is solved for each DMU, a loop is required in order to solve the LP model for different parameters. The loop is shown in the following GAMS code blocks. Based on this structure, each row of the data table is substituted to the right-hand side of constraints 1 and 2. Figure 2.3 depicts this procedure as a graphical representation. IN

OUT

DMU 1

x ,IN y1,OUT

DMU 2

x2,IN y2,OUT

1

DMU 3

x3,IN y3,OUT

DMU 4

x4,IN y4,OUT

DMU 5

x5,IN y 5

,OUT

min T s.t. x1,IN  x2 ,IN  x3,IN  x4 ,IN  x1,IN d x1,IN I ˜T y1,OUT  y2 ,OUT  y3,OUT  y4 ,OUT  y5 ,OUT t y1,OUT T U UT

min T s.t. x1,IN  x2 ,IN  x3,IN  x4 ,IN  x1,IN d x2 ,IN ˜ T y1,OUT  y2 ,OUT  y3,OUT  y4 ,OUT  y5 ,OUT t y2 ,OUT

Fig. 2.3 Graphical representation

18

2

Introduction to Data Envelopment Analysis

loop(jj, DMU_data(g) = Data(jj,g); Solve DEA_CRS using LP minimizing Theta ; eff(jj)=Theta.l; slacks(jj,i)=sminus.l(i); slacks(jj,r)=splus.l(r); loop(k, Lamres(jj,k)=Lambda.l(k); ); ); Display eff, Lamres, slacks;

Finally, the results are stored in the parameters created above. As can be seen, instead of using index j in the loop, index jj is used. This is because the index/set used for looping cannot be used for summation, products and so on. To overcome this problem, the command alias(jj, j) is introduced to create a set similar to j. The second loop with respect to k, which is again a similar set to j, is introduced in order to store all of the peers (lambda values) of each DMU to a square matrix, namely Lamres. The display command allows the user to visualise all of the solutions and parameters. The implementation of the CRS DEA model using GAMS syntax is shown in Fig. 2.4. The GAMS model results are presented in Figs. 2.5, 2.6, and 2.7. As can be seen from Fig. 2.5, DMUs 1, 4, and 10 have an efficiency score of 1 and according to Fig. 2.6, those DMUs are fully efficient because their corresponding slacks are 0. In GAMS, a row with 0 values is not displayed in the results. The results regarding the peers of each DMU are presented in Fig. 2.7.

2.2.4

The VRS DEA Model

Variable returns-to-scale (VRS) is another type of frontier scale that takes into account the fact that an increase or decrease in inputs or outputs does not result in a proportional change in the outputs or inputs, respectively. In this sense, VRS reflects the fact that production technology may exhibit increasing, constant, or decreasing returns-to-scale. The VRS DEA model is also known as the BCC model, after Banker et al. (1984), who introduced a convex hull instead of a convex cone around the data. Indeed, Banker et al. (1984) extended the CRS model by relaxing the assumption of CRS to VRS. As with the CRS model, the VRS model can be orientated in two different ways: input-orientated (problem minimisation) or output-orientated (maximisation problem). Because the VRS model considers the variation of efficiency with respect to the scale of operation, it is only applicable

2.2

Envelopment Models

19

Table 2.4

Fig. 2.4 The mathematical formulation of the CRS DEA model and the corresponding GAMS formulation

20

Fig. 2.4 (continued)

2

Introduction to Data Envelopment Analysis

2.2 Envelopment Models PARAMETER eff

21

efficiency report

DMU1

1.000,

DMU2

0.806,

DMU3

0.444,

DMU4

1.000,

DMU5

DMU6

0.382,

DMU7

0.814,

DMU8

0.792,

DMU9

0.373,

DMU10 1.000

0.663

Fig. 2.5 Efficiency results (θ) for each DMU, under the CRS technology PARAMETER slacks

ProdCost

DMU2

TrnCost

Rev

0.272 0.181

0.255

DMU6

0.057

DMU7

0.181

DMU8 DMU9

HoldInv

0.522

DMU3 DMU5

for inputs and outputs

0.034

1.494

0.177 0.046

0.083

Fig. 2.6 Slack results (si- , , sþ, r ) for each DMU under the CRS technology PARAMETER Lamres

DMU1

peers for each DMU

DMU4

DMU10

DMU1

1.000

DMU2

0.223

0.460

0.077

DMU3

0.847

0.019

0.004

DMU4

1.000

DMU5

0.945

DMU6

0.850

DMU7

0.736

0.325

DMU8

0.336

0.464

DMU9

0.388

DMU10

0.050

0.205 0.293 0.156 1.000

Fig. 2.7 Peer results (λj ) for each DMU under the CRS technology

when scale effects can be demonstrated. In order to evaluate the efficiency, peers, and slack variables for DMUs under the VRS technology, the model presented in Table 2.1a is solved by adding the constraint ∑jλj = 1. The new model is presented below.

22

2

Introduction to Data Envelopment Analysis

Equations OBJ objective function CON1(i) input duals CON2(r) output dual CON3 VRS orientation;

OBJ..

efficiency=E=Theta-1E-6*(SUM(i,sminus(i))+SUM(r,splus(r)));

CON1(i)..

SUM(j, Lambda(j)*Data(j,i))+sminus(i)=E=Theta*DMU_data(i);

CON2(r)..

SUM(j, Lambda(j)*Data(j,r))-splus(r)=E=DMU_data(r);

CON3..

SUM(j, Lambda(j))=E=1;

The results of the above VRS DEA model are shown in Figs. 2.8, 2.9, and 2.10. According to Figs. 2.8 and 2.9, DMUs 1, 4, 7, and 10 are fully efficient because their corresponding slack variables are 0. The results of the peers are shown in Fig. 2.10.

2.2.5

Projected Values and Targets

As shown in Table 2.2, the projected values are computed in order to show how the inputs and outputs could be improved (decreased and increased, respectively) in order to be included in the benchmark. The parameters introduced are xproject(j,i) PARAMETER eff

efficiency report

DMU1

1.000,

DMU2

0.892,

DMU3

0.503,

DMU4

1.000,

DMU5

DMU6

0.449,

DMU7

1.000,

DMU8

0.855,

DMU9

0.658,

DMU10 1.000

Fig. 2.8 The efficiency results (θ) for each DMU, under the VRS technology

PARAMETER slacks

ProdCost

DMU2

TrnCost

HoldInv

0.310

0.004

3.000

0.257

0.150

0.040

3.000

1.890

6.412

2.262

0.563

DMU3 DMU5

slacks for inputs and outputs

SatDem

Rev

6.557

0.182

DMU6

0.067

DMU8

0.114

DMU9

0.089

0.210

Fig. 2.9 The slack results (si- , , sþ, r ) for each DMU under the VRS technology

0.666

2.3

Multiplier Models

PARAMETER Lamres

DMU1

DMU1

1.000

DMU2

0.567

DMU3

1.000

DMU4

23 peers for each DMU

DMU4

DMU7

0.375

DMU10

0.057

1.000

DMU5

0.953

DMU6

1.000

0.047

DMU7

1.000

DMU8

0.077

DMU9

0.618

0.348

0.352

0.224 0.382

DMU10

1.000

Fig. 2.10 Peer results (λj ) for each DMU under the VRS technology

and yproject(j,r), which model the projections of each input i and each output r, of each DMU j. The GAMS code for calculating projections, regardless of the technology used (CRS or VRS), is as follows: Parameters xproject(j,i), yproject(j,r); xproject(j,i)=eff(j)*Data(j,i)-slacks(j,i); yproject(j,r)=Data(j,r)+slacks(j,r);

As can be seen, the projections of each input and output are computed based on the optimal efficiency values [stored in the parameter eff(j)] and the table of slacks corresponding to inputs and outputs. The projected values of each input and output for each type of technology (CRS and VRS) are shown in Figs. 2.11 and 2.12.

2.3 2.3.1

Multiplier Models The CRS DEA Model

The dual linear programming representations of envelopment models are multiplier DEA models. In the multiplier models, weight factors are assigned to inputs and outputs; so, instead of λj, which models the peers of each DMU, vi and ur non-negative variables are introduced. Thus, the envelopment CRS DEA model is shown in Table 2.4a, followed by the corresponding multiplier model (dual), which is shown in Table 2.4b:

24

2

Introduction to Data Envelopment Analysis

CRS

PARAMETER xproject

ProdCost

TrnCost

HoldInv

DMU1

0.255

0.161

0.373

DMU2

0.268

0.200

0.488

DMU3

0.225

0.143

0.332

DMU4

0.305

0.249

0.841

DMU5

0.256

0.164

0.394

DMU6

0.217

0.137

0.317

DMU7

0.474

0.330

0.596

DMU8

0.496

0.357

0.584

DMU9

0.242

0.162

0.181

DMU10

0.917

0.639

0.234

TrnCost

HoldInv

PARAMETER xproject

VRS

ProdCost

DMU1

0.255

0.161

0.373

DMU2

0.312

0.221

0.541

DMU3

0.255

0.161

0.373

DMU4

0.305

0.249

0.841

DMU5

0.257

0.165

0.395

DMU6

0.255

0.161

0.373

DMU7

0.583

0.628

0.732

DMU8

0.536

0.463

0.631

DMU9

0.508

0.344

0.320

DMU10

0.917

0.639

0.234

Fig. 2.11 Projections for inputs (xi,j ) for each DMU under CRS and VRS technology

Regarding the multiplier DEA formulation in Table 2.4b, if μ = 0 then CRS technology is assumed, whereas, if μ is a free variable, then VRS technology is assumed. The corresponding GAMS syntax does not change in terms of indices and sets but in terms of the declaration of equations. The GAMS syntax of the CRS multiplier DEA model as described in Table 2.4b, with the data from Fig. 2.2, is shown in Fig. 2.13. As a CRS multiplier DEA model is presented, then μ = 0 is

2.3

Multiplier Models

25

PARAMETER yproject

CRS

SatDem

Rev

DMU1

20.000

2.640

DMU2

6.000

5.290

DMU3

17.000

2.430

DMU4

2.000

8.990

DMU5

19.000

2.940

DMU6

17.000

2.244

DMU7

17.000

6.360

DMU8

10.000

7.200

DMU9

9.000

2.160

DMU10

8.000

7.300

SatDem

Rev

DMU1

20.000

2.640

DMU2

12.557

5.290

DMU3

20.000

2.640

DMU4

2.000

8.990

DMU5

19.150

2.940

DMU6

20.000

2.640

DMU7

17.000

6.360

DMU8

10.000

7.200

DMU9

15.412

4.422

8.000

7.300

VRS

PARAMETER yproject

DMU10

Fig. 2.12 Projections for outputs (yr,j ) for each DMU under CRS and VRS technology

assumed and even if stated in the variable declaration, μ is not present in the mathematical formulation (objective function and constraints). The results for efficiency and dual values assigned to inputs and outputs under the CRS technology are presented in Figs. 2.14, 2.15, and 2.16.

26

2

Introduction to Data Envelopment Analysis

Table 2.4 Envelopment model (a) and its dual (multiplier) model (b) (b) VRS; but if μ = 0, then it is CRS

(a) CRS min θ s:t: n

s

max r=1

s:t:

xij
λj ≤ xio
θ

j=1 n j=1

s

r=1 m

yrj
λj ≥ yro

λj ≥ 0, j = 1, ::, n θ free

2.3.2

ur
yr,o þ μ

ur
yr,j -

m

vi
xi,j þ μ ≤ 0, j = 1, ::, n

i=1

vi
xi,o = 1

i=1

vi ≥ 0, i = 1, . . . , m ur ≥ 0, r = 1, . . . , s

The VRS DEA Model

The multiplier VRS DEA model is formulated in GAMS and the syntax is shown in Fig. 2.17. The results for efficiency and dual values assigned to inputs and outputs under the VRS technology are presented in Figs. 2.18, 2.19, and 2.20. The efficiency scores derived from the multiplier DEA model presented in Figs. 2.14 and 2.18 are the same as the efficiency scores derived from the envelopment models presented in Figs. 2.5 and 2.8, correspondingly.

2.4

Assurance Regions/Weight Restrictions

One of the advantages of the mathematical formulations of the multiplier DEA is that weights are assigned to inputs and outputs correspondingly and preferences regarding the inputs and outputs can be introduced. Preferences among inputs and outputs are introduced with the following constraints. As an example, consider Table 2.5, where the first part of the constraints for the assurance regions is non-linear while the next line is the corresponding linear transformation. Assume that the relative importance of the Transportation Cost to Production Cost and of Holding Inventory to Production Cost is given in the following regions: 0:5
vProdCost ≤ vTrnCost ≤ 4
vProdCost 2
vHoldInv ≤ vProdCost ≤ 6
vHoldInv Similar regions are assumed for Demand Satisfaction and Revenue. 2
uSatDem ≤ u Re v ≤ 6
uSatDem

2.4

Assurance Regions/Weight Restrictions

27

Table 2.14b

Fig. 2.13 Mathematical formulation of the multiplier CRS DEA model and the corresponding GAMS formulation

28

2

PARAMETER eff

Introduction to Data Envelopment Analysis

efficiency report

DMU1

1.000,

DMU2

0.806,

DMU3

0.444,

DMU4

1.000,

DMU5

DMU6

0.382,

DMU7

0.814,

DMU8

0.792,

DMU9

0.373,

DMU10 1.000

0.663

Fig. 2.14 Results for efficiency of multiplier DEA model of each DMU, under the CRS technology

Fig. 2.15 Results for dual input variable (vi ) of multiplier DEA model of each DMU, under the CRS technology

res_v

results for dual input

ProdCost

DMU1

HoldInv

1.303

1.117

3.922

DMU2 DMU3

1.972

DMU4

3.279

DMU5

Fig. 2.16 Results for dual output variable (ur ) of multiplier DEA model of each DMU, under the CRS technology

TrnCost

4.032

DMU6

1.761

DMU7

1.715

DMU8

0.673

0.783

DMU9

0.324

1.543

DMU10

0.492

2.344

res_u

results for dual output

SatDem

Rev

DMU1

0.050

DMU2

0.013

0.137

DMU3

0.012

0.102

DMU4

0.012

0.109

DMU5

0.018

0.108

DMU6

0.022

DMU7

0.011

0.098

DMU8

0.011

0.095

DMU9

0.035

0.028

DMU10

0.072

0.058

2.4

Assurance Regions/Weight Restrictions

29

Fig. 2.17 The mathematical formulation of the multiplier VRS DEA model and the corresponding GAMS formulation

30

2

Introduction to Data Envelopment Analysis

Fig. 2.17 (continued)

The corresponding constraints are modelled with the GAMS syntax and are presented in Fig. 2.21. The results for the optimal solutions of vi , ur are shown in Figs. 2.22 and 2.23. In order to cross-check whether the ratios of the optimal solutions for dual variables (vi , ur ) satisfy the constraints, the results of the ratios are presented in Table 2.6.

2.5

Most Productive Scale Size

Most Productive Scale Size (MPSS) is a unit (point) on the efficiency frontier that maximises average productivity for a given input-output mix before experiencing decreasing returns-to-scale. In other words, MPSS is a measurement that specifies how resources should be allocated and used to produce the best outcomes. In the MPSS DEA formulation, both variables θ and φ are considered in the analysis (Khodabakhshi, 2009). The formulation is shown in Table 2.7. The proposed formulation simultaneously maximises the outputs and minimises the inputs, as shown in the objective function.

2.5

Most Productive Scale Size

PARAMETER eff

31

efficiency report

DMU1

1.000,

DMU2

0.806,

DMU3

0.444,

DMU4

1.000,

DMU5

DMU6

0.382,

DMU7

0.814,

DMU8

0.792,

DMU9

0.373,

DMU10 1.000

0.663

Fig. 2.18 Efficiency results of the multiplier DEA model for each DMU, under the VRS technology Fig. 2.19 Results for the dual input variable (vi ) of the multiplier DEA model of each DMU, under the VRS technology

res_v

results for dual input

ProdCost

DMU1

TrnCost

HoldInv

1.303

1.117

3.922

DMU2 DMU3

1.972

DMU4

3.279

DMU5

4.032

DMU6

1.761

DMU7

1.715

DMU8

0.673

0.783

DMU9

0.324

1.543

DMU10

0.492

2.344

Fig. 2.20 Results for the dual output variable (ur ) of the multiplier DEA model of each DMU, under the VRS technology

res_u

SatDem

DMU1

Table 2.5 Assurance region constraints

results for dual output

0.050

DMU2

0.013

0.137

DMU3

0.012

0.102

DMU4

0.012

0.109

DMU5

0.018

0.108

DMU6

0.022

DMU7

0.011

0.098

DMU8

0.011

0.095

DMU9

0.035

0.028

DMU10

0.072

0.058

vi ≤ vU i , vo vLi
vo ≤ vi ≤ vU i
vo , i = 1, ::, m u , uLr ≤ r ≤ uU r uo L ur
uo ≤ ur ≤ uU r
uo , r = 1, ::, m vLi ≤

Rev

Preference for inputs

Preference for outputs

32

2

Introduction to Data Envelopment Analysis

Equations OBJ objective function CON1(j) CON2 CON3 CON4 CON5

Fig. 2.21 Mathematical formulation of the multiplier CRS DEA model with Assurance Region (AR) constraints and the corresponding GAMS formulation

2.5

Most Productive Scale Size

Fig. 2.22 Results for the dual input variable (vi ) of the multiplier DEA model of each DMU, under the CRS technology with Assurance Region (AR) constraints

33 res_v

ProdCost

TrnCost

HoldInv

DMU1

0.925

0.462

1.849

DMU2

0.178

0.713

1.070

DMU3

0.183

0.091

1.097

DMU4

0.335

1.341

0.670

DMU5

0.277

1.108

0.554

DMU6

0.376

0.188

0.752

DMU7

0.189

0.095

1.134

DMU8

0.185

0.093

1.113

DMU9

0.249

0.124

1.494

DMU10

0.379

0.189

2.272

Fig. 2.23 Results for the dual output variable (ur ) of the multiplier DEA model of each DMU, under the CRS technology with Assurance Region (AR) constraints

Table 2.6 Ratios for inputs dual variables

results for dual input

res_u

results for dual output

SatDem

DMU1

0.049

0.010

DMU2

0.027

0.009

DMU3

0.023

0.008

DMU4

0.026

0.009

DMU5

0.022

0.007

DMU6

0.020

0.004

DMU7

0.023

0.008

DMU8

0.023

0.008

DMU9

0.031

0.010

DMU10

0.047

0.016

vTrnCost=v

DMU1 DMU2 DMU3 DMU4 DMU5 DMU6 DMU7 DMU8 DMU9 DMU10

Rev

0.5 4 0.5 4 4 0.5 0.5 0.5 0.5 0.5

ProdCost

vProdCost=v

HoldInv

2 6 6 2 2 2 6 6 6 6

34

2

Table 2.7 Most Productive Scale Size (MPSS) DEA formulation envelopment model

Introduction to Data Envelopment Analysis max φ - θ s:t: n

xij
λj ≤ xio
θ

j=1 n j=1

yrj
λj ≥ φ
yro

λj ≥ 0, j = 1, ::, n θ free φ free

2.6

Super-Efficiency Models

In this section, super-efficiency models are presented. Many DEA applications make use of super-efficiency measures, such as ranking effective DMUs, assessing the Malmquist productivity index, and contrasting the performances of two groups (Cooper et al., 2007b). Super-efficiency models are motivated by situations in which there are multiple efficient units, and one needs to discriminate between them. Based on the mathematical formulations, the efficiency scores from these models are derived by removing the data on the DMU under evaluation from the reference set and the calculation. It should be noted that unlike in many DEA models, where the efficient units are indicated by an efficiency score of one, the scores of superefficiency models yield a value of more than or equal to 1. In this sense, DMs evaluate and rank the efficient DMUs based on their super-efficiency scores. Andersen and Petersen’s (1993) paper was the first to introduce the idea of super-efficiency models. The concept of super-efficiency can be applied to both radial and non-radial models, the mathematical formulation of which will be presented in Table 2.8.

Table 2.8 Super-Efficiency radial (a) and non-radial (b) input-oriented models (a) min θ s:t: n

xij
λj ≤ xio
θ, i = 1, . . . , m

(b) min 1 þ s:t: n

j=1

xij
λj - xio
ζi ≤ xio , i = 1, . . . , m

j=1

j≠o n

m

1
ζ m i=1 i

yrj
λj ≥ yro , r = 1, . . . , s

j≠o n

j=1 j≠o

yrj
λj ≥ yro , r = 1, . . . , s

j=1

λj ≥ 0, j = 1, ::, n

j≠o

θ free

λj ≥ 0, j = 1, ::, n ζi ≥ 0, i = 1, ::, m

2.6

Super-Efficiency Models

35

In order to ensure that the summation of all of the DMUs will be conducted except n

for the one under investigation

xij
λj , the following procedure should be

j=1 j≠o

followed. A new parameter (the name of this parameter is cc) is considered, which is dimensionless (no index) as in this parameter the order of the DMU under investigation will be assigned. For example, in the first iteration, DMU1 is examined so it is first in the set. Therefore, in the first equation, in order to guarantee that the summation will be conducted for all of the DMUs except for the one under investigation, the following should be written in the GAMS syntax: SUM(j$(ORD(j)cc), Lambda(j)*Data(j,i))

With the use of conditional operation ($), the summation is conducted for all of the DMUs except for the one which is the same as the DMU under investigation. Thus, for iteration 1 (cc = 1), in mathematical format the previous GAMS syntax 10

will be

xij
λj , for iteration 2 (cc = 2)

10

j=1

j=1

j≠1

j≠2

xij
λj , and so on. The corresponding

GAMS syntax is shown in Fig. 2.24. The results of the analysis for each DMU are shown in Fig. 2.25, while the results for the peers (λj ) of each DMU are presented in Fig. 2.26. The formulation of the super-efficiency model is shown in Fig. 2.27. The characteristics that change in terms of the previous formulation is that the objective function is defined as 1 þ m1


m

i=1

ζ i whereas the denominator (m) stands for the

cardinal of set i. In this case, the cardinal (number of elements in a set) is introduced in the GAMS syntax with CARD() function. Also, the new non-negative variable ζ i is introduced to the mathematical formulation; thus, it must be declared in the variables section.

36

2

Introduction to Data Envelopment Analysis

Fig. 2.24 The mathematical formulation of the radial super-efficiency CRS-DEA model and the corresponding GAMS formulation

2.6

Super-Efficiency Models

PARAMETER eff

37

efficiency report

DMU1

2.662,

DMU2

0.806,

DMU3

0.444,

DMU4

2.723,

DMU5

DMU6

0.382,

DMU7

0.814,

DMU8

0.792,

DMU9

0.373,

DMU10 3.095

Fig. 2.25 Super-Efficiency (θ) results for each DMU under the CRS technology

PARAMETER Lamres

DMU1

DMU1

1.000

DMU2

0.567

DMU3

1.000

DMU4

peers for each DMU

DMU4

0.375

DMU10

0.057

1.000

DMU5

0.953

DMU6

1.000

0.047

DMU7

1.000

DMU8

0.077

DMU9

0.618

DMU10

DMU7

0.348

0.352

0.224 0.382 1.000

Fig. 2.26 Results for the peers (λj ) of each DMU under the CRS technology

0.663

38

2 Introduction to Data Envelopment Analysis

cc order of DMU under investigation 'o';

Order of DMU under invesgaon for each iteraon

Equations OBJ objective function CON1(i) CON2(r) OBJ..

efficiency=E=1+(1/CARD(i))*SUM(i,z(i));

T n

CON1(i)..

SUM(j$(ORD(j)cc), Lambda(j)*Data(j,i))z(i)*DMU_data(i)=L=DMU_data(i);

¦ xij ˜ O j d xio ˜T j 1 j zo

i 1,...,m n

CON2(r)..

SUM(j$(ORD(j)cc),

Lambda(j)*Data(j,r))=G=DMU_data(r);

¦ yrj ˜ O j t yro j 1 j zo

r 1,...,s

model DEA_Super_radial_efficiency_CRS input oriented DEA CRS / OBJ, CON1, CON2 /; loop(jj, DMU_data(g) = Data(jj,g); cc=ORD(jj); Solve DEA_Super_radial_efficiency_CRS using LP minimizing Theta;

xi ,o inputs yr ,o outputs

Order of each DMU under invesgaon Solve the model in Table 2.29a by minimising the objecve funcon using LP

Fig. 2.27 The mathematical formulation of the super-efficiency CRS-DEA model and the corresponding GAMS formulation

2.6

Super-Efficiency Models

39

loop(k, Lamres(jj,k)=Lambda.l(k); );

Peers of each DMU j, O *j Efficiency values

eff(jj)= efficiency.l;

Display eff, Lamres;

Fig. 2.27 (continued)

( T * ) for each DMU j Display the parameters of the results

Chapter 3

Extensions of DEA Models

3.1

Introduction

Data Envelopment Analysis models have evolved over the years, offering formulations that correspond to instances beyond strictly measuring performance. In this chapter, extensions of DEA models will be presented, demonstrating recent developments in DEA formulations. The models that will be analytically described and modelled with GAMS in this chapter concern DEA models with exogenously fixed variables and categorical variables, DEA models for handling desirable and undesirable outputs, congestion, and chance constraints.

3.2

Exogenously Fixed Variables

DEA models with exogenously fixed variables are very important in the DEA literature as this formulation provides a realistic point of view to the efficiency measurement problem. The mathematical formulations for DEA models with exogenously fixed inputs and outputs are shown in Table 3.1. There are two types of inputs (and correspondingly outputs). Regarding inputs (and outputs), there are two sets; the ones that are controlled by decision-makers (d = {1, . . ., m} 2 Inf) and the ones that are exogenously fixed (k = {m + 1, . . ., p} 2 I f) whereas I = Inf [ I f. The same is assumed for outputs, as well. It can be seen in Table 3.1 that the exogenously fixed inputs and outputs do not contribute to the constraints regarding input and output efficiency (θ and φ). Only the slacks corresponding to non-exogenously fixed inputs and outputs are introduced in the objective function. The formulation demonstrated in Table 3.1 concerns VRS; however, under the CRS technology, the mathematical formulation changes, as in Table 3.2. It can be

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Emrouznejad et al., Data Envelopment Analysis with GAMS, International Series in Operations Research & Management Science 338, https://doi.org/10.1007/978-3-031-30701-0_3

41

42

3

Extensions of DEA Models

Table 3.1 DEA models for exogenously fixed inputs (a) and outputs (b) under the VRS technology (a)

(b)

min θ - ε 

m d=1

sd- þ

s r=1

sþ r

i=1

s:t: n

j=1 n j=1 n j=1 n

m

max φ þ ε 

sd- þ

m r=1

sþ r

s:t: n

xdj  λj þ sd- = xdo  θ, d = 1, ::, m

j=1 n

xkj  λj þ sk- = xko , k = 1, ::, p yrj 

λj - s þ r

j=1 n

= yro , r = 1, ::, s

j=1 n

λj = 1

xij  λj þ si- = xio , i = 1, ::, m ydj  λj - sþ d = φ  ydo , d = 1, ::, m ykj  λj - sþ k = yko , k = 1, ::, p λj = 1

j=1

j=1

λj , si- , sþ r ≥ 0, 8j, i, r

λj , si- , sþ r ≥ 0, 8j, i, r

θ free

φ free

Table 3.2 DEA models for exogenously fixed inputs (a) and outputs (b) under the CRS technology (a)

(b)

min θ - ε 

m d=1

sd- þ

s r=1

sþ r

s:t: n

j=1 n j=1 n j=1

max φ þ ε 

m i=1

sd- þ

m r=1

sþ r

s:t: xdj  λj þ sd- = xdo  θ, d = 1, ::, m xkj  λj þ sk- = yrj 

λj - s þ r

n

λj  xko , k = m þ 1, ::, p

j=1

= yro , r = 1, ::, s

n

j=1 n j=1 n j=1

xij  λj þ si- = xio , i = 1, ::, m ydj  λj - sþ d = φ  ydo , d = 1, ::, m ykj  λj - sþ k =

n j=1

λj , si- , sþ r ≥ 0, 8j, i, r

λj , si- , sþ r ≥ 0, 8j, i, r

θ free

φ free

λj  yko , k = m þ 1, ::, p

seen that under the CSR technology, the right-hand side is multiplied with the term n j = 1 λj .

3.2.1

DEA CRS Model

Taking the same indicative data set (as described in Chap. 2) for applying the inputoriented DEA model with exogenously fixed inputs and assuming that the Production Cost and Transportation Cost are inputs under managerial control, while the

3.2

Exogenously Fixed Variables

43

Holding Inventory is an exogenously fixed input, then the following sets are formulated in GAMS: Sets

j DMUs /DMU1*DMU10/ g Inputs and Outputs /ProdCost, TrnCost,HoldInv, SatDem,Rev/ d(g) Inputs /ProdCost, TrnCost/ k(g) Exogenously fixed inputs /HoldInv/ r(g) Outputs /SatDem, Rev/;

The index for inputs under managerial control is denoted by d (Production Cost, Transportation Cost) while the index for exogenously fixed inputs is denoted by k (Holding Inventory). In order to discriminate better between the slacks that correspond to inputs under managerial control and the ones corresponding to exogenously fixed inputs, the following variables are stated as follows. Variables efficiency objective function Theta efficiency (Theta values) Lambda(j) dual weights (Lambda values) sminus_nf(d) slacks assigned to inputs under managerial control sminus_f(k) slacks assigned to exogenously fixed inputs splus(r) slacks assigned to inputs; Nonnegative variables Lambda(j) sminus_nf(d) sminus_f(k) splus(r);

The slack variables corresponding to inputs under managerial control (sd- ) are denoted by sminus_nf(d) while the slack variables corresponding to exogenously fixed inputs (sk- ) are denoted by sminus_f(k). The mathematical formulation as described in Fig. 3.1. The results for efficiency scores, slack variables, and reference set after solving the model under the CRS technology are demonstrated in Figs. 3.2, 3.3, and 3.4. From Fig. 3.2, it can be seen that DMUs 1, 4, and 10 are efficient. Based on Fig. 3.3, which shows that the corresponding slack variables (for inputs and outputs) for DMUs 1, 4, and 10 are all 0, the DMUs are fully efficient.

3.2.2

DEA VRS Model

The corresponding mathematical formulation of the DEA model with exogenously fixed inputs is described in this section (Fig. 3.5). It can be seen that the difference between this model and the one examined under the CRS technology is that the

44

3 Extensions of DEA Models

Table 2.4

Fig. 3.1 The mathematical formulation of the CRS-DEA model with exogenously fixed inputs and the corresponding GAMS formulation

3.2

Exogenously Fixed Variables

Fig. 3.1 (continued)

45

46

3

Extensions of DEA Models xd ,o Inputs under managerial control

loop(jj,

xk ,o Exogenously fixed

DMU_data(g) = Data(jj,g);

inputs

yr ,o Outputs Table 3.2a

Display the parameters of the results

Display eff, Lamres, slacks;

Fig. 3.1 (continued)

PARAMETER eff

efficiency report

DMU1

1.000,

DMU2

0.753,

DMU3

0.439,

DMU4

1.000,

DMU5

DMU6

0.382,

DMU7

0.598,

DMU8

0.504,

DMU9

0.189,

DMU10 1.000

0.663

Fig. 3.2 The efficiency results (θ) for each DMU from the DEA model with exogenously fixed inputs under the CRS technology

n

constraint

j=1

xkj  λj þ sk- =

n j=1

λj  xko is substituted with

n j=1

λj = 1. The results

after running the model are shown in Figs. 3.6, 3.7, and 3.8. It can be seen that the fully efficient DMUs under the VRS technology are 1, 4, 7, and 10 based on Figs. 3.6 and 3.7.

3.2

Exogenously Fixed Variables

PARAMETER slacks

for inputs and outputs

ProdCost DMU2

TrnCost

HoldInv

0.483

DMU3 DMU5

47

SatDem

Rev

4.065 0.270

0.181

0.317 0.580

DMU6

0.057

0.464

DMU7

0.129

0.237

DMU8

0.103

0.088

DMU9

0.026

0.010

1.494

Fig. 3.3 Slack results (sd- , , sk- , , sþ, r ) for each DMU from the DEA model with exogenously fixed inputs under the CRS technology

PARAMETER Lamres

DMU1

peers for each DMU

DMU4

DMU1

1.000

DMU2

0.458

0.454

DMU3

0.848

0.021

DMU4

1.000

DMU5

0.945

DMU6

0.850

0.050

DMU7

0.803

0.472

DMU8

0.433

0.674

DMU9

0.439

0.111

DMU10

DMU10

1.000

Fig. 3.4 The results for the peers (λj ) of each DMU from the DEA model with exogenously fixed inputs under the CRS technology

48

3

Extensions of DEA Models

Mathemacal GAMS formulation

formulaon Sets

j 1,...,10

j DMUs /DMU1*DMU10/ g Inputs and Outputs /ProdCost, TrnCost, HoldInv, SatDem, Rev/

d(g)

Inputs under managerial control /ProdCost, TrnCost/

g

­ ProdCost , TrnCost , ½ ® ¾ ¯ HoldInv , SatDem, Rev ¿

d

^ ProdCost , TrnCost `

dg k

k(g)

Exogenously fixed inputs /HoldInv/

r

^ SatDem, Rev `

r(g) Outputs /SatDem, Rev/;

rg

alias(jj,j);

j

alias(kk,jj); Table

^ HoldInv `

kg

jj

kk 1,...,10

Data(j,g) Data for inputs and outputs

TrnCost

HoldInv

SatDem

Rev

DMU1

ProdCost 0.255

0.161

0.373

20

2.64

DMU2

0.98

0.248

0.606

6

5.29

DMU3

0.507

0.937

0.749

17

2.43

DMU4

0.305

0.249

0.841

2

8.99

DMU5

0.659

0.248

0.979

19

2.94 0.75

DMU6

0.568

0.508

0.919

17

DMU7

0.583

0.628

0.732

17

6.36

DMU8

0.627

0.675

0.738

10

7.2

DMU9

0.772

0.657

0.486

9

2.16

DMU10

0.917

0.639

0.234

8

7.3;

Variables efficiency objective function Theta

efficiency (Theta values)

Lambda(j) dual weights (Lambda values) sminus_nf(d) slacks assigned to inputs under managerial control sminus_f(k)

slacks assigned to exogenously fixed inputs

splus(r) slacks assigned to inputs;

Table 2.4

Efficiency T Oj sd sk

sr

Nonnegative Variables Lambda(j) dual weights (Lambda values) sminus_nf(d) slacks assigned to inputs under managerial control

sminus_f(k)

slacks assigned to exogenously fixed inputs

Oj t 0 sd t 0

sk t 0

Fig. 3.5 The mathematical formulation of the VRS-DEA model with exogenously fixed inputs and the corresponding GAMS formulation

3.2

Exogenously Fixed Variables

splus(r) slacks assigned to inputs; Parameters DMU_data(g) slice of data

49 sr t 0

xd ,o xk ,o yr , o

eff(j) efficiency report

Parameter for reporng the opmal efficiency values for each DMU j, T *

Lamres(j,j) peers for each DMU

Parameter for reporng the peers of each DMU j, O *j

slacks(j,g) slacks for inputs and outputs;

Parameter for reporng the slack values for each DMU j, sd,* , sk,* , sr ,*

Equations OBJ objective function CON1(d) input constraint for inputs under managerial control CON2(k) input constraint for exogenously fixed inputs CON3(r) output dual CON4

VRS constraint;

OBJ.. efficiency=E=Theta-1E6*(SUM(d,sminus_nf(d))+SUM(r,splus(r)));

s § m · efficiency T  H ˜ ¨ sd  sr ¸ ¨¨ ¸¸ r 1 ¹ ©d 1

¦

n

CON1(d).. SUM(j, Lambda(j)*Data(j,d))+sminus_nf(d)=E=Theta*DMU_data(d);

¦ xdj ˜ O j  sd

xdo ˜ T

j 1 d 1,..,m

n

CON2(k).. SUM(j, Lambda(j)*Data(j,k))+sminus_f(k)=E=DMU_data(k);

¦ xkj ˜ O j  sk

xko

j 1 k m  1,.., p n

CON3(r).. SUM(j, Lambda(j)*Data(j,r))splus(r)=E=DMU_data(r);

¦ yrj ˜ O j  sr j 1 r 1,..,s n

CON4.. SUM(j, Lambda(j))=E=1; model Exogenous_input_DEA_VRS Exogenously input oriented DEA model /OBJ, CON1, CON2, CON3, CON4/;

Fig. 3.5 (continued)

Oj ¦ j 1

1

¦

yrj

50

3

Extensions of DEA Models xd ,o Inputs under

managerial control loop(jj,

xk ,o Exogenously fixed

DMU_data(g) = Data(jj,g);

inputs yr ,o Outputs

solve Exogenous_input_DEA_VRS using LP minimizing Efficiency ; eff(jj)=Theta.l; slacks(jj,d)=sminus_nf.l(d);

Solve the model in Table 3.1a by minimising the objecve funcon using LP Efficiency values ( T * ) for each DMU j ,*

slacks(jj,k)=sminus_f.l(k);

Slack values ( sd

slacks(jj,r)=splus.l(r);

) for each DMU j

, sk,* , sr,*

loop(kk, Lamres(jj,kk)=Lambda.l(kk);

Peers of each DMU j,

);

O *j

);

Display the parameters of the results

Display eff, Lamres, slacks;

Fig. 3.5 (continued)

PARAMETER eff

efficiency report

DMU1

1.000,

DMU2

0.797,

DMU3

0.503,

DMU4

1.000,

DMU5

DMU6

0.449,

DMU7

1.000,

DMU8

0.656,

DMU9

0.330,

DMU10 1.000

0.666

Fig. 3.6 Efficiency results (θ) for each DMU from the DEA model with exogenously fixed inputs under the VRS technology

PARAMETER slacks

ProdCost

DMU2

TrnCost

0.505

DMU3 DMU5

for inputs and outputs

0.310 0.182

DMU6

0.067

DMU8

0.055

DMU9

0.056

HoldInv

SatDem

0.038

6.488

Rev

0.376

3.000

0.584

0.150

0.210

0.546

3.000

1.890

0.113

11.000

0.480

Fig. 3.7 Slack results (sd- , , sk- , , sþ, r ) for each DMU from the DEA model with exogenously fixed inputs under the VRS technology

3.3

Undesirable Outputs

PARAMETER Lamres

DMU1

51

peers for each DMU

DMU4

DMU1

1.000

DMU2

0.458

0.454

DMU3

0.848

0.021

DMU4

DMU10

1.000

DMU5

0.945

DMU6

0.850

0.050

DMU7

0.803

0.472

DMU8

0.433

0.674

DMU9

0.439

0.111

DMU10

1.000

Fig. 3.8 The results for the peers (λj ) of each DMU from the DEA model with exogenously fixed inputs under the VRS technology

3.3

Undesirable Outputs

The outputs of a production process may not always be desirable to the DM (Khoshroo et al., 2018). As a matter of fact, in real-life applications, both desirable (good) and undesirable (bad) output factors are commonly present (Charles et al., 2012). For example, in the energy sector, there are desirable outputs (e.g., power and heat) and undesirable outputs, as well (such as CO2 emissions, waste, noise, etc.). If production inefficiency exists, the undesirable outputs should be reduced to reduce the inefficiencies; in other words, the undesirable and desirable outputs should be treated differently when evaluating the performance of the production. The most common approaches or models to handle undesirable outputs in DEA are: (a) ignoring undesirable outputs altogether in the DEA model; (b) treating undesirable outputs as inputs in the production function; (c) treating undesirable outputs in the non-linear (hyperbolic) DEA model; (d) linear approximation to the non-linear DEA model; (e) linear monotone decreasing transformation to undesirable outputs in the DEA model; and (f) directional distance function approach to treat undesirable outputs in the DEA model (Charles et al., 2014). Radial and non-radial DEA models have been proposed in order to measure the efficiency under undesirable outputs. The mathematical formulation is shown in Table 3.3. The outputs are divided into two categories; desirable outputs (d = 1, . . ., m) are denoted by yds d,j while undesirable outputs (k = s + 1, . . ., p) are denoted by . The objective function aims to maximise the free variable β which measures the yund k,j inefficiency of each DMU j. A DMU is efficient if β = 0 and inefficient for any positive number of β.

52

3

Table 3.3 DEA model for handling undesirable outputs

Extensions of DEA Models

max β s:t: n

j=1 n j=1 n j=1 n

xi,j  λj ≤ xi,o , i = 1, ::, m ds yd,j  λj ≥ yds d,o þ β  yd,o , d = 1, ::, s und yk,j  λj ≤ yund k,o - β  yk,o , k = s þ 1::, p

λj = 1

j=1

λj , ≥ 0, j = 1, ::, n β≥0

Table 3.4 Data for illustrative example with desirable and undesirable outputs

DMU1 DMU2 DMU3 DMU4 DMU5 DMU6 DMU7 DMU8 DMU9 DMU10

Production cost (K £) 0.255 0.980 0.507 0.305 0.659 0.568 0.583 0.627 0.772 0.211

Transportation cost (K £) 0.161 0.248 0.937 0.249 0.248 0.508 0.628 0.675 0.657 0.065

Holding inventory quantities (tonnes) 0.373 0.606 0.749 0.841 0.979 0.919 0.732 0.738 0.486 0.457

No of satisfied demand instances 20 6 17 2 19 17 17 10 9 2

Revenues (K £) 2.64 5.29 2.43 8.99 2.94 0.75 6.36 7.20 2.16 9.55

CO2 (K tonnes) 7.3 6.63 10.3 5.4 18.9 4.1 8.91 8.2 14.2 5.1

Assume a production process that produces undesirable outputs (CO2 emissions); the data that are used in this instance are shown in Table 3.4. The GAMS formulation of this model is shown in Fig. 3.9. The results of the beta values (β) and reference set for each DMU are shown in Figs. 3.10 and 3.11. It can be seen from Fig. 3.10 that GAMS returns only the non-negative values of the β variables; DMUs 2, 3, 5, 8, and 9 are inefficient as β > 0. On the contrary, DMUs 1, 4, 6, 7, and 10 are efficient as β = 0 or 1 β = 1. The same can be concluded from the reference set, as well (Fig. 3.11).

3.3

Undesirable Outputs

53

Mathemacal GAMS formulation

formulaon Sets

j 1,...,10

j DMUs /DMU1*DMU10/ g Inputs and Outputs /ProdCost, TrnCost, HoldInv, SatDem, Rev, CO2/

i(g)

Inputs under managerial control /ProdCost, TrnCost,

g

½ ­ ProdCost , TrnCost , ¾ ® ¯ HoldInv , SatDem, Rev, CO2¿

i

^ ProdCost , TrnCost `

ig

HoldInv/

d

^ SatDem, Rev `

d(g) Desirable outputs /SatDem, Rev/;

dg

k(g) Undesirable outputs /CO2/;

kg

alias(jj,j);

j

k

alias(kk,jj);

^CO2 ` jj

kk

1,...,10

Table Data(j,g) Data for inputs and outputs

DMU1 DMU2 DMU3 DMU4 DMU5 DMU6 DMU7 DMU8 DMU9 DMU10

ProdCost 0.255 0.98 0.507 0.305 0.659 0.568 0.583 0.627 0.772 0.917

TrnCost 0.161 0.248 0.937 0.249 0.248 0.508 0.628 0.675 0.657 0.639

HoldInv 0.373 0.606 0.749 0.841 0.979 0.919 0.732 0.738 0.486 0.234

SatDem 20 6 17 2 19 17 17 10 9 8

Variables efficiency objective function Beta

inefficiency (Beta values)

Lambda(j) dual weights (Lambda values)

Rev 2.64 5.29 2.43 8.99 2.94 0.75 6.36 7.2 2.16 7.3

CO2 7.3 6.63 10.3 5.4 18.9 4.51 8.91 8.2 14.2 5.1;

Table 3.12

Efficiency

E Oj

Nonnegative Variables Beta

inefficiency (Beta values)

Lambda(j) dual weights (Lambda values) Parameters DMU_data(g) slice of data

E Oj t 0 xi ,o

ydds,o ykund ,o eff(j) Beta values

Parameter for reporng the opmal inefficiency values for each DMU j, E *

Fig. 3.9 The mathematical formulation of the DEA model with undesirable outputs and the corresponding GAMS formulation

54

3 Lamres(j,j) peers for each DMU

Extensions of DEA Models Parameter for reporng the peers of each DMU j,

O *j

Equations OBJ objective function CON1(i) input constraint CON2(d) output constraint for desirable outputs CON3(k) output constraint for undesirable outputs CON4 OBJ..

VRS constraint;

efficiency=E=Beta;

efficiency E n

CON1(i)..

SUM(j, Lambda(j)*Data(j,i))=L=DMU_data(i);

CON2(d)..

SUM(j,

¦ xi, j ˜ O j d xi,o

j 1 i 1,..,m n

¦y

Lambda(j)*Data(j,d))=G=DMU_data(d)+beta*DMU_data(d);

d,j

d

1,..,s

n

CON3(k)..

SUM(j, Lambda(j)*Data(j,k))=L=DMU_data(k)beta*DMU_data(k)

¦y

k,j

und ˜ O j d ykund ,o  E ˜ yk ,o

j 1

k

s  1,.., p

n

CON4.. SUM(j, Lambda(j))=E=1;

˜ O j t ydds,o  E ˜ ydds,o

j 1

¦O j 1

j

1

model Undesirable_output_DEA Undesirable output DEA model /OBJ, CON1, CON2, CON3, CON4/; xi ,o Inputs

loop(jj, DMU_data(g) = Data(jj,g);

solve Exogenous_input_DEA using LP minimizing Efficiency ; eff(jj)=Beta.l;

ydds,o

Desirable Outputs

ykund ,o

Undesirable Outputs

Solve the model in Table 3.11 by maximising the objecve funcon using LP Beta values ( E * ) for each DMU j

loop(kk, Lamres(jj,kk)=Lambda.l(kk); );

Peers of each DMU j,

O *j

);

Display eff, Lamres;

Fig. 3.9 (continued)

Display the parameters of the results

3.4

Congestion in DEA

55

PARAMETER eff

Beta values

DMU2 0.077,

DMU3 0.167,

DMU5 0.035,

DMU8 0.038,

DMU9 0.606

Fig. 3.10 The results of the beta values (β) for each DMU from the DEA model considering undesirable outputs PARAMETER Lamres

DMU1

DMU1

1.000

DMU2

0.391

DMU3

0.947

DMU4

peers for each DMU

DMU4

DMU6

DMU7

0.523

DMU10

0.086 0.053

1.000

DMU5

0.891

0.109

DMU6

1.000

DMU7

1.000

DMU8

0.397

DMU9

0.309

DMU10

0.529 0.305

0.075 0.386 1.000

Fig. 3.11 The results for the peers (λj ) of each DMU from the DEA model considering undesirable outputs

3.4 3.4.1

Congestion in DEA Congestion Index

Congestion is one of the most widely known and studied terms in economic science. This is because congestion is a common occurrence in real life, even if it is not something that relevant decision-makers anticipate for. For instance, excessive water irrigation of a maize yield can result in greater yield loss. In the framework of DEA, congestion was first studied by Färe and Svensson (1980), who posited that congestion occurs when increasing some of the inputs may obstruct outputs. Today, the most widely accepted definition of the term congestion is that it refers to the situation where a reduction in the inputs of a production process results in an increase in one or more of its produced outputs, or where an increase in the inputs results in a decrease in one or more of the outputs (Cooper et al., 1996b). The majority of known approaches for assessing the congestion of DMUs rely on the classic concept of congestion and the assumption that inputs and outputs fluctuate proportionally. Consequently, it is important to understand what happens if inputs are increased or decreased asymmetrically.

56

3

Extensions of DEA Models

Table 3.5 Input-oriented model under (a) strong and (b) weak disposability (a) min θ s:t: n

j=1 n j=1 n

xi,j  λj ≤ θ  xi,o , i = 1, ::, m yr,j  λj ≥ yr,o , r = 1, ::, s λj = 1

j=1

λj , ≥ 0, j = 1, ::, n θ free

(b) min θ s:t: n

j=1 n j=1 n

xi,j  λj = θ0  xi,o , i = 1, ::, m yr,j  λj ≥ yr,o , r = 1, ::, s λj = 1

j=1

λj , ≥ 0, j = 1, ::, n θ free

In order to explain congestion in DEA, the envelopment models discussed in Chap. 2 will be used. In Table 3.5, an input-oriented DEA model under strong and weak disposability is shown. The difference between the two mathematical formulations is that in the input constraint, the formulation under strong disposability (a) is inequality, while under weak disposability (b), the constraint becomes equality. As in the constraint regarding the inputs for the formulation under weak disposability (θ) the equality must hold; then, θ ≤ θ. Input congestion, C θ, θ , is defined as the result of the following fraction C θ, θ = θ=θ. If C θ, θ = 1, then there is congestion, while for C θ, θ < 1, no congestion is presented. The corresponding GAMS representation of the mathematical formulations presented in Table 3.5a, b is shown in Fig. 3.12. In Fig. 3.12, the two models are introduced as symbolic equations in the same GAMS file. Even if the equations (objective function and constraints) are written under the equation section, the models are stated in the model section. More specifically, the model, as formulated in Fig. 3.12, entails an objective function (OBJ), an input constraint (CON1_s), an output constraint (CON2_s), and a VRS constraint. The second model consists of an objective function (OBJ1), an input constraint (CON1_w), an output constraint (CON2_w), and a VRS constraint. All equations stated under the equation section must belong to a model; otherwise, an error will occur. The results of the efficiency scores for each DMU are used in the loop in order to calculate the congestion index. The results regarding the efficiency scores of the models under strong and weak disposability, as well as the values for the congestion index, are shown in the next tables. From Fig. 3.13, it can be seen that DMUs 1, 4, 7, and 10 present input congestion, while there is no input congestion for the rest of the DMUs (Figs. 3.14 and 3.15).

3.4

Congestion in DEA

57

Mathemacal GAMS formulation

formulaon Sets

j 1,...,10

j DMUs /DMU1*DMU10/ g Inputs and Outputs /ProdCost, TrnCost, HoldInv, SatDem, Rev, CO2/

i(g)

Inputs under managerial control /ProdCost, TrnCost, HoldInv/

g

­ ProdCost , TrnCost , ½ ® ¾ ¯ HoldInv , SatDem, Rev, ¿

­ ProdCost , TrnCost ½ ® ¾ ¯ HoldInv ¿ ig i

r

^ SatDem, Rev `

r(g) Desirable outputs /SatDem, Rev/;

rg

alias(jj,j);

j

alias(kk,jj);

jj

kk

1,...,10

Table Data(j,g) Data for inputs and outputs

DMU1 DMU2 DMU3 DMU4 DMU5 DMU6 DMU7 DMU8 DMU9 DMU10

ProdCost 0.255 0.98 0.507 0.305 0.659 0.568 0.583 0.627 0.772 0.917

TrnCost 0.161 0.248 0.937 0.249 0.248 0.508 0.628 0.675 0.657 0.639

Variables efficiency efficiency1

HoldInv 0.373 0.606 0.749 0.841 0.979 0.919 0.732 0.738 0.486 0.234

SatDem 20 6 17 2 19 17 17 10 9 8

Rev 2.64 5.29 2.43 8.99 2.94 0.75 6.36 7.2 2.16 7.3;

objective function for model 1 objective function for model 2

Table 2.4

Efficiency

Theta

Strong disposability efficiency

T

Theta_tilde

Weak disposability efficiency

T%

Lambda(j) dual weights (Lambda values)

Oj

Nonnegative Variables Lambda(j) dual weights (Lambda values) Parameters DMU_data(g) slice of data

Oj t 0 xi ,o Inputs yr ,o Outputs

eff_Theta(j)

Strong disposability efficiency

Parameter for reporng the

values

strong ( T ) and weak

eff_Theta_tilde(j)

Weak disposability efficiency values

disposability ( T% ) efficiency scores

Congestion(j) Congestion for each DMU

Parameter for reporng the congeson index for each



* * DMU j, C T , T%



Fig. 3.12 The mathematical formulation of the input-oriented DEA model under strong and weak disposability and congestion index calculation with corresponding GAMS formulation

58

3 Lamres_s(j,j) peers for each DMU for strong

Extensions of DEA Models Parameter for reporng the

disposability model Lamres_w(j,j) peers for each DMU for weak

peers of each DMU j,

O *j

disposability model;

under weak and strong disposability Equations OBJ_s objective function for strong disposability CON1_s(i) input constraint for strong disposability CON2_s(r) output constraint for strong disposability OBJ_w objective function for weak disposability CON1_w(i) input constraint for weak disposability CON2_w(r) output constraint for weak disposability VRS OBJ..

VRS constraint;

efficiency=E=Theta;

T

n

CON1_s(i)..

SUM(j, Lambda(j)*Data(j,i))=L=Theta*DMU_data(i);

CON2_s(r)..

SUM(j, Lambda(j)*Data(j,r))=G=DMU_data(r);

¦ xi, j ˜ O j d T ˜ xi,o

j 1 i 1,..,m n

¦y

r,j

˜ O j t yr ,o ,r 1,..,s

j 1

OBJ_w..

efficiency1=E=Theta_tilde;

T% n

CON1_w(i)..

¦ xi, j ˜ O j

SUM(j, Lambda(j)*Data(j,i))=E=Theta_tilde*DMU_data(i);

T% ˜ xi,o

j 1 i 1,..,m n

CON2_w(r)..

SUM(j, Lambda(j)*Data(j,r))=G=DMU_data(r);

¦y

r,j

˜ O j t yr ,o ,r 1,..,s

j

1

j 1 n

¦O

VRS.. SUM(j, Lambda(j))=E=1;

j 1

model Input_DEA_strong_disposability Input oriented DEA model for strong disposability /OBJ_s, CON1_s, CON2_s, VRS/; model Input_DEA_weak_disposability

Input oriented DEA model

for weak disposability /OBJ_w, CON1_w, CON2_w, VRS/;

loop(jj,

xi ,o Inputs

DMU_data(g) = Data(jj,g);

solve Input_DEA_strong_disposability using LP minimizing Theta;

Fig. 3.12 (continued)

yr ,o Outputs

Solve the model in Table 3.61a by minimising the objecve funcon using LP

3.4

Congestion in DEA

59

eff_Theta(jj)=Theta.l; solve Input_DEA_weak_disposability using LP minimizing Theta_tilde; eff_Theta_tilde(jj)=Theta_tilde.l; Congestion(jj)=eff_Theta(jj)/eff_Theta_tilde(jj);

Theta values ( T * ) for each DMU j Solve the model in Table 3.61b by minimising the objecve funcon using LP Theta lde values ( T%* ) for each DMU j Congeson index for each DMU j

);

Display the results for T * , * T%* and congeson C T , T%



Display eff_Theta, eff_Theta_tilde, congestion;



Fig. 3.12 (continued)

PARAMETER eff_Theta

Strong disposability efficiency values

DMU1

1.000,

DMU2

0.892,

DMU3

0.503,

DMU4

1.000,

DMU5

DMU6

0.449,

DMU7

1.000,

DMU8

0.855,

DMU9

0.658,

DMU10 1.000

0.666

Fig. 3.13 Efficiency results for the input-oriented DEA model under strong disposability (θ)

PARAMETER eff_Theta_tilde

Weak disposability efficiency values

DMU1

1.000,

DMU2

1.000,

DMU3

1.000,

DMU4

1.000,

DMU5

DMU6

0.509,

DMU7

1.000,

DMU8

1.000,

DMU9

0.769,

DMU10 1.000

1.000



Fig. 3.14 Efficiency results for the input-oriented DEA model under weak disposability (θ )

PARAMETER Congestion

Congestion for each DMU

DMU1

1.000,

DMU2

0.892,

DMU3

0.503,

DMU4

1.000,

DMU5

DMU6

0.882,

DMU7

1.000,

DMU8

0.855,

DMU9

0.856,

DMU10 1.000

Fig. 3.15 Congestion results, C θ, θ



0.666

60

3

Table 3.6 Input-oriented VRS envelopment model for measuring congestion

m

max i=1

Extensions of DEA Models

ζþ i

s:t: n

j=1 n j=1 n

- ,  xi,j  λj - ζ þ , i = 1, ::, m i = θ  xi,o - si

yr,j  λj = yr,o þ sþ, r , r = 1, ::, s λj = 1

j=1 si- ,

≥ ζþ i , i = 1, ::, m λj , ≥ 0, j = 1, ::, n

3.4.2

Congestion with Slack Variables

In this section, the congestion will be calculated based on slack variables. The basic advantage of computing congestion using slack variables is that not only can the congestion be detected, but it can be measured, as well. Also, there is better discrimination between congestion and other factors that affect inefficiency. The model that is applied for measuring input congestion using slack variables is presented in Table 3.6. The model takes the optimal values derived from the classical envelopment DEA model (input-oriented VRS); θ is the optimal value of the efficiency score as calculated by Table 2.1a along with a convexity condition, while the optimal slack variables assigned to inputs (si- , ) and outputs (sþ, r ) are derived from the same formulation (as stated in Sect. 3.1.4). The mathematical formulation presented in Table 3.6 is modelled in GAMS in Fig. 3.16. The optimal values of the variable ζþ i are used in order to measure the amount of congestion for each input and output.

3.4

Congestion in DEA

61

Mathemacal formulaon

GAMS formulation Sets

j 1,...,10

j DMUs /DMU1*DMU10/ g Inputs and Outputs /ProdCost, TrnCost, HoldInv, SatDem, Rev, CO2/

i(g)

Inputs under managerial control /ProdCost,

g

i

TrnCost, HoldInv/

­ HoldInv, ProdCost , TrnCost ½ ¾ ® ¿ ¯ HoldInv , SatDem, Rev

­ HoldInv, ProdCost , ½ ® ¾ ¯TrnCost ¿

ig d

^ SatDem, Rev `

r(g) Desirable outputs /SatDem, Rev/;

dg

alias(jj,j);

j

alias(kk,jj);

jj

kk

1,...,10

Table Data(j,g) Data for inputs and outputs

DMU1 DMU2 DMU3 DMU4 DMU5 DMU6 DMU7 DMU8 DMU9 DMU10

ProdCost 0.255 0.98 0.507 0.305 0.659 0.568 0.583 0.627 0.772 0.917

TrnCost 0.161 0.248 0.937 0.249 0.248 0.508 0.628 0.675 0.657 0.639

Variables efficiency

HoldInv 0.373 0.606 0.749 0.841 0.979 0.919 0.732 0.738 0.486 0.234

SatDem 20 6 17 2 19 17 17 10 9 8

Rev 2.64 5.29 2.43 8.99 2.94 0.75 6.36 7.2 2.16 7.3;

objective function for model 1

efficiency1

objective function for model 2

Table 2.4

Efficiency

Theta

Strong disposability efficiency

T

Theta_tilde

Weak disposability efficiency

T%

zeta(i)

Congestion with slack measures

]i

Lambda(j) dual weights (Lambda values)

Oj

Nonnegative Variables Lambda(j) dual weights (Lambda values)

Oj t 0

sminus(i)

slack variable assigned to input

si t 0

splus(r)

slack variable assigned to output;

sr t 0

Parameters DMU_data(g) slice of data

xi ,o Inputs yr ,o Outputs

eff_Theta(j)

res_eff_2(j)

Strong disposability efficiency

Parameter for reporng strong

values

disposability efficiency ( T ) and

results for sum of zeta variables

the sum of ] i variables ( n

¦ i 1] i ) Fig. 3.16 The mathematical formulation of the input DEA congestion using slack variables with the corresponding GAMS formulation

62

3 res_zeta(j,i)

Results for zeta variable

Lamres_s(j,j)

peers for each DMU for strong

Extensions of DEA Models

Parameter for reporng the ] i values Parameter for reporng the

disposability model

peers of each DMU j,

O *j under

strong disposability slice_Theta

slice of efficiency

slice_xproj(i)

slice for input projection

slice_yproj(r)

slice for output projection

Parameters for reporng the slack variables for inputs and

slice_sminus(i) slice for input slack

outputs, projected values for

slack_in(j,i)

results for input slack variables

inputs and outputs and slice

slack_out(j,r)

results for output slack variables

xproj(j,i)

projected values for inputs for

parameters

each DMU yproj(j,r)

projected values for outputs for each DMU;

Equations OBJ_s objective function for strong disposability CON1_s(i) input constraint for strong disposability CON2_s(r) output constraint for strong disposability OBJ_w objective function for weak disposability CON1_w(i) input constraint for weak disposability CON2_w(r) output constraint for weak disposability CON3_w(i) slack constraint VRS

VRS constraint;

OBJ..efficiency=E=Theta-1E3*(SUM(i,sminus(i))+SUM(r,splus(r)));

s §m · T  H ˜ ¨ si  sr ¸ ¨ ¸ 1 1 i r © ¹

¦

¦

n

CON1_s(i)..

SUM(j, Lambda(j)*Data(j,i))=L=Theta*DMU_data(i);

CON2_s(r)..

SUM(j, Lambda(j)*Data(j,r))=G=DMU_data(r);

¦ xi, j ˜ O j  si

T ˜ xi,o

j 1 i 1,..,m n

¦y

r,j

˜ O j  sr

yr ,o ,r 1,.., s

j 1

m

OBJ_w..

efficiency1=E=SUM(i,zeta(i));

¦] i i 1

n

CON1_w(i)..

SUM(j, Lambda(j)*Data(j,i))zeta(i)=E=slice_xproj(i);

Fig. 3.16 (continued)

¦ xi, j ˜ O j  ] i

j 1 i 1,..,m

ˆxi,o

3.4

Congestion in DEA

63 n

CON2_w(r)..

SUM(j, Lambda(j)*Data(j,r))=E=slice_yproj(r);

¦y

˜Oj

r,j

ˆyr ,o ,

j 1

r 1,..,s n

¦O

VRS.. SUM(j, Lambda(j))=E=1;

j 1

j

1

model Input_DEA_strong_disposability Input oriented DEA model for strong disposability /OBJ_s, CON1_s, CON2_s, VRS/; model Input_DEA_weak_disposability_slack

Input oriented DEA

model for weak disposability with slack measures /OBJ_w, CON1_w, CON2_w, CON3_w, VRS/;

loop(jj,

xi ,o Inputs

DMU_data(g) = Data(jj,g);

yr ,o Outputs

slack_out(jj,r)=splus.l(r);

Solve the model in Table 2.1a by minimising the objecve funcon using LP, along with the convexity condion Theta values ( T * ) for each DMU j Slack values (inputs, outputs) for each DMU j ( si,* , sr ,* )

xproj(jj,i)=eff_theta(jj)*Data(jj,i)-

Projected values of inputs and outputs for each DMU j

solve Input_DEA_strong_disposability using LP minimizing Theta;

eff_Theta(jj)=Theta.l; slack_in(jj,i)=sminus.l(i);

slack_in(jj,i); yproj(jj,r)=Data(jj,r)+slack_out(jj,r);

xˆi,o T * ˜ xi,o  si,* yˆ r ,o yr ,o  sr,*

slice_xproj(i)=xproj(jj,i); slice_yproj(r)=yproj(jj,r); slice_Theta=eff_Theta(jj);

Data used for the second DEA model (Table 2.2)

slice_sminus(i)=slack_in(jj,i); solve Input_DEA_weak_disposability_slack using LP maximizing efficiency1; res_zeta(jj,i)=zeta.l(i);

res_eff_2(jj)=SUM(i,zeta.l(i));

Solve the model in Table 3.21 by minimising the objecve funcon using LP The zeta values ( ] i* ) of inputs for each DMU j The sum of zeta values (

¦

of inputs for each DMU j

m ]* i 1 i

); Display eff_Theta, res_zeta, res_eff_2;

Display the results for T * , ] i* and

Fig. 3.16 (continued)

¦

m ]* i 1 i

)

64

3.5

3

Extensions of DEA Models

Categorical Variables in DEA

The use of categorical variables is an important extension in DEA that can improve the construction process of peer groups and incorporate “on-off” attributes, such as the presence or absence of a drive-in window in a banking network (Banker & Morey, 1986). In other words, categorical variables are frequently used to indicate the presence or absence of a characteristic. In the presence of categorical variables, an additional dimension is considered when measuring the performance of the DMUs. This additional dimension is linked to considering categorical variables that are assigned to each DMU based on different service orientations (Banker & Morey, 1986). The model that is formulated takes into account, besides inputs and outputs, a vector for specifying different service orientations for each DMU. The mathematical formulation of the model is provided in Table 3.7. In the DEA model presented above, a new index that models categorical orientations (l = 1, .., L - 1) is introduced. Also, a vector that entails information regarding the service orientation of the DMUs is introduced. Assuming that there are L different levels (with the first layer assumed to be the lowest in quality while the last one is the highest in quality), Table 3.8 is formulated. In the mathematical presentation with categorical variables, as shown in Fig. 3.17, binary variables (tl) are introduced for each categorical orientation l. The resulting formulation is a Mixed Integer Linear Programming (MILP) problem. Some new functions regarding index operations are shown in Table 3.7. The index that the objective function is summed upon L - 1; the corresponding GAMS formulation is objective=E=SUM(l$(ORD(l)=2 AND ORD(l)