Multicriteria Decision Aid Classification Methods Applied Optimization Volume 73 Series Editors: Panos M. Pardalos University of Florida, U.S.A. Donald Hearn University of Florida, U.S.A. The titles published in this series are listed at the end of this volume. Multicriteria Decision Aid Classification Methods by Michael Doumpos and Constantin Zopounidis Technical University of Crete, Department of Production Engineering and Management, Financial Engineering Laboratory, University Campus, Chania, Greece KLUWER ACADEMIC PUBLISHERS NEW YORK,BOSTON, DORDRECHT, LONDON, MOSCOW eBookISBN: 0-306-48105-7 Print ISBN: 1-4020-0805-8 ©2004 Kluwer Academic Publishers NewYork, Boston, Dordrecht, London, Moscow Print ©2002 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook maybe reproducedor transmitted inanyform or byanymeans,electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: http://kluweronline.com and Kluwer's eBookstoreat: http://ebooks.kluweronline.com To my parents Christos and Aikaterini Doumpos To my wife Kleanthi Koukouraki and my son Dimitrios Zopounidis Table of contents PROLOGUE xi CHAPTER 1: INTRODUCTIONTO THE CLASSIFICATION PROBLEM 1. Decision making problematics 1 2. The classificationproblem 4 3. Generaloutlineofclassificationmethods 6 4. Theproposedmethodologicalapproachandtheobjectivesof thebook 10 CHAPTER2: REVIEWOFCLASSIFICATIONTECHNIQUES 1. Introduction 15 2. Statistical and econometric techniques 15 2.1 Discriminant analysis 16 2.2 Logit and probit analysis 20 3. Non-parametric techniques 24 3.1 Neural networks 24 3.2 Machine learning 27 3.3 Fuzzy set theory 30 3.4 Rough sets 32 CHAPTER 3: MULTICRITERIADECISIONAID CLASSIFICATION TECHNIQUES 1. Introduction to multicriteria decision aid 39 1.1 Objectives and general framework 39 1.2 Brief historical review 40 1.3 Basic concepts 41 2. Methodological approaches 43 2.1 Multiobjective mathematical programming 45 2.2 Multiattribute utility theory 48 viii 2.3 Outrankingrelationtheory 50 2.4 Preference disaggregation analysis 52 3. MCDAtechniquesforclassificationproblems 55 3.1 Techniques based on the direct interrogation of the decision maker 55 3.1.1 The AHP method 55 3.1.2 The ELECTRETRI method 59 3.1.3 Other outrankingclassificationmethods 64 3.2 The preferencedisaggregationparadigm in classification problems 66 CHAPTER 4: PREFERENCE DISAGGREGATION CLASSIFICATION METHODS 1. Introduction 77 2. The UTADIS method 78 2.1 Criteria aggregation model 78 2.2 Model development process 82 2.2.1 General framework 82 2.2.2 Mathematicalformulation 86 2.3 Model development issues 96 2.3.1 The piece-wise linearmodeling ofmarginal utilities 96 2.3.2 Uniqueness of solutions 97 3. The multi-group hierarchical discrimination method (MHDIS) 100 3.1 Outline and main characteristics 100 3.2 The hierarchical discrimination process 101 3.3 Estimation of utility functions 105 3.4 Model extrapolation 111 Appendix: Postoptimalitytechniques forclassificationmodel development inthe UTADISmethod 113 CHAPTER 5: EXPERIMENTAL COMPARISON OF CLASSIFICATION TECHNIQUES 1. Objectives 123 2. The consideredmethods 124 3. Experimentaldesign 126 3.1 The factors 126 3.2 Datagenerationprocedure 131 4. Analysis of results 134 5. Summary of major findings 143 Appendix: Development ofELECTRE TRI classification models using apreference disaggregation approach 150 ix CHAPTER 6: CLASSIFICATIONPROBLEMSINFINANCE 1. Introduction 159 2. Bankruptcy prediction 161 2.1 Problem domain 161 2.2 Data and methodology 164 2.3 The developed models 172 2.3.1 The model oftheUTADISmethod 172 2.3.2 The model ofthe MHDIS method 174 2.3.3 The ELECTRE TRI model 176 2.3.4 The rough setmodel 178 2.3.5 The statistical models 179 2.4 Comparison ofthebankruptcy predictionmodels 181 3. Corporate credit risk assessment 185 3.1 Problem domain 185 3.2 Data and methodology 188 3.3 The developedmodels 194 3.3.1 The UTADIS model 194 3.3.2 The model ofthe MHDIS method 196 3.3.3 The ELECTRE TRImodel 199 3.3.4 The rough set model 200 3.3.5 The models of the statistical techniques 201 3.4 Comparison ofthe credit risk assessment models 202 4. Stock evaluation 205 4.1 Problem domain 205 4.2 Data and methodology 209 4.3 The developed models 215 4.3.1 The MCDAmodels 215 4.3.2 The rough setmodel 220 4.4 Comparison ofthe stock evaluation models 222 CHAPTER 7: CONCLUSIONSANDFUTUREPERSPECTIVES 1. Summary ofmainfindings 225 2. Issues for future research 229 REFERENCES 233 SUBJECT INDEX 251 Prologue Decision making problems, according to their nature, the policy of the deci- sion maker, and the overall objective of the decision, may require the choice of an alternative solution, the ranking of the alternatives from the best to the worst ones or the assignment of the considered alternatives into predefined homogeneous classes. This last type of decision problem is referred to as classification or sorting. Classification problems are often encountered in a variety of fields including finance, marketing, environmental and energy management, human resources management, medicine, etc. The major practical interest of the classification problem has motivated researchers in developing an arsenal of methods for studying such problems, in order to develop mathematical models achieving the higher possible clas- sification accuracy and predicting ability. For several decades multivariate statistical analysis techniques such as discriminantanalysis (linear and quad- ratic), and econometric techniques such as logit and probit analysis, the lin- ear probability model, etc., have dominated this field. However, the paramet- ric nature and the statistical assumptions/restrictions of such approaches have been an issue of major criticism and skepticism on the applicability and the usefulness of such methods in practice. The continuous advances in other fields including operations research and artificial intelligence led many scientists and researchers to exploit the new capabilities ofthese fields, in developing more efficient classification techniques. Among the attempts made one can mention neural networks, machine learning, fuzzy sets as well as multicriteria decision aid. Multicrite- ria decision aid (MCDA) has several distinctive and attractive features, in- volving, mainly, its decision support orientation. The significant advances in MCDA over the last three decades constitute a powerful non-parametric al- ternative methodological approach to study classification problems. Al- xii though the MCDA research, until the late 1970s, has been mainly oriented towards the fundamental aspects of this field, as well as to the development of choice and ranking methodologies, during the 1980s and the 1990s sig- nificant research has been undertaken on the study of the classification prob- lem within the MCDA framework. Following the MCDA framework, the objective of this book is to provide a comprehensive discussion of the classification problem, to review the ex- isting parametric and non-parametric techniques, their problems and limita- tions, and to present the MCDA approach to classification problems. Special focus is given to the preference disaggregation approach of MCDA. The preference disaggregation approach refers to the analysis (disaggregation) of the global preferences (judgement policy) of the decision maker in order to identify the criteria aggregation model that underlies the preference result (classification). The book is organized in seven chapters as follows: Initially, in chapter 1 an introduction to the classification problem is pre- sented. The general concepts related to the classification problem are dis- cussed, along with an outline of the procedures used to develop classification models. Chapter 2 provides a comprehensive review of existing classification techniques. The review involves parametric approaches (statistical and econometric techniques) such as the linear and quadratic discriminant analy- sis, the logit and probit analysis, as well as non-parametric techniques from the fields of neuralnetworks, machine learning, fuzzy sets, and rough sets. Chapter 3 is devoted to the MCDA approach. Initially, an introduction to the main concepts of MCDA is presented along with a panorama of the MCDA methodological streams. Then, the existing MCDA classification techniques are reviewed, including multiattribute utility theory techniques, outranking relation techniques and goal programming formulations. Chapter 4 provides a detailed description of the UTADIS and MHDIS methods, including their major features, their operation and model develop- ment procedures, along with their mathematical formulations. Furthermore, a series of issues is also discussed involving specific aspects of the functional- ity of the methods and their model development processes. Chapter 5 presents an extensive comparison of the UTADIS and MHDIS methods with a series of well-established classification techniques including the linear and quadratic discriminant analysis, the logit analysis and the rough set theory. In addition, ELECTRE TRI a well-known MCDA classifi- cation method based on the outranking relation theory is also considered in the comparison and a new methodology is presented to estimate the parame- ters of classification models developed through ELECTRE TRI. The com- parison is performed through a Monte-Carlo simulation, in order to investi-