U
NIVERSITÉ
P
ARIS
IX
D
AUPHINE
U
NIVERSIDADE
T
ÉCNICA DE
L
ISBOA
I
NSTITUTO
S
UPERIOR
T
ÉCNICO
T
HÈSE
En vue de l’obtention en cotutelle des grades académiques :
DOCTEUR EN INFORMATIQUE (Spécialité : Recherche Opérationnelle) et DOCTEUR EN INGÉNIERIE ET MANAGEMENT INDUSTRIEL.
M
ULTIPLE
C
RITERIA
D
ECISION
A
IDING FOR
S
ORTING
P
ROBLEMS
:
C
ONCEPTS
,
M
ETHODOLOGIES
,
AND
A
PPLICATIONS
Juscelino
A
LMEIDA
D
IAS
J
URY
Directeurs de thèse : José Rui DE MATOS FIGUEIRA
Professeur à l’Instituto Superior Técnico, Universidade Técnica de Lisboa, Portugal Bernard Michel ROY
Professeur Émérite à l’Université Paris Dauphine, France Rapporteurs : Manuel António CERQUEIRA DA COSTA MATOS
Professeur à la Faculté d’Ingénierie, Université de Porto, Portugal PatricePERNY
Professeur au Laboratoire d’Informatique de Paris 6, Université Pierre et Marie Curie, France
Suffrageants : Ana Paula FERREIRA DIAS BARBOSA PÓVOA
Professeur à l’Instituto Superior Técnico, Universidade Técnica de Lisboa, Portugal VincentMOUSSEAU
Professeur au Laboratoire de Génie Industriel, École Centrale de Paris, France Paulo VASCONCELOS DIAS CORREIA
Professeur à l’Instituto Superior Técnico, Universidade Técnica de Lisboa, Portugal
Les Universit´es n’entendent donner aucune approbation ni improbation aux opinions ´emises dans les th`eses : ces opinions doivent ˆetre consid´er´ees
Abstract
Multiple Criteria Decision Aiding for Sorting Problems: Concepts, Methodologies, and Applications
Our thesis deals with sorting problems following a constructive approach. The aim is to assign objects of a decision, evaluated on multiple criteria, to a set of categories. Each category is pre-defined to receive these objects, which will be or might be processed in the same way. Our research provides a taxonomy framework, in which ten different types of sorting problems is defined with a practical usefulness. Two sorting methods are proposed within this taxon-omy, called Electre Tri-C and Electre Tri-nC. These methods deal with decision aiding contexts, where the set of categories is completely ordered. Each category is characterized by a single typical reference action and several ones, respectively. The assignment procedure is based on a descending rule and an ascending rule, which must be used conjointly. Our research also provides a segmenting description algorithm for analyzing the relationship between the assign-ing conditions of decision aidassign-ing sortassign-ing models (e.g. Electre Tri-C) and the preferences of decision makers, including an analysis of incoherencies and incompatibilities, without making use of an optimization model. The main research results are validated by two real-world appli-cations (assisted reproduction and agro-environmental risk), which are modeled with medical and environmental experts, respectively.
Keywords:
Multiple criteria decision aiding, Constructive approach, Sorting problems, Characteristic refer-ence actions, Electre Tri-C, Segmenting description algorithm.
Resumo
Problemas de Classificac¸˜ao em Apoio `a Decis˜ao com Crit´erios M´ultiplos: Conceitos, Metodologias e Aplicac¸˜oes
A nossa investiga¸c˜ao centra-se em problemas de classifica¸c˜ao seguindo uma abordagem constru-tivista. O objectivo ´e afectar objectos de decis˜ao, avaliados com crit´erios m´ultiplos, em cate-gorias. Cada categoria ´e pr´e-definida para agrupar estes objectos, que ir˜ao ser ou dever˜ao ser sujeitos ao mesmo tratamento. A nossa pesquisa fornece uma taxonomia, onde definimos dez tipos de problemas de classifica¸c˜ao com interesse pr´atico. Dois m´etodos de classifica¸c˜ao s˜ao propostos no ˆambito desta taxonomia, denominados Electre Tri-C e Electre Tri-nC. Estes m´etodos enquadram-se em contextos de apoio `a decis˜ao, onde as categorias s˜ao completamente ordenadas. Cada categoria ´e caracterizada, respectivamente, por uma ac¸c˜ao de referˆencia t´ıpica ou m´ultiplas ac¸c˜oes t´ıpicas. O procedimento de afecta¸c˜ao ´e composto por uma regra descendente e uma regra ascendente, que devem ser usadas conjuntamente. A nossa pesquisa fornece tamb´em um algoritmo de descri¸c˜ao segmentada para analisar a rela¸c˜ao entre as condi¸c˜oes de afecta¸c˜ao de modelos de classifica¸c˜ao em apoio `a decis˜ao (e.g. Electre Tri-C) e as preferˆencias dos decisores, incluindo uma an´alise de incoerˆencias e incompatibilidades, sem aux´ılio de um modelo de optimiza¸c˜ao. Os principais resultados da investiga¸c˜ao s˜ao validados por duas aplica¸c˜oes reais (reprodu¸c˜ao assistida e riscos agro-ambientais), que foram modelizadas, respectivamente, com especialistas m´edicas e agro-ambientais.
Palavras-chave:
Apoio `a decis˜ao com crit´erios m´ultiplos, Abordagem construtivista, Problemas de classifica¸c˜ao, Ac¸c˜oes de referˆencia caracter´ısticas, Electre Tri-C, Algoritmo de descri¸c˜ao segmentada.
R´
esum´
e
Probl`emes de Tri en Aide `a la D´ecision `a Crit`eres Multiples :
Concepts, M´ethodologies et Applications
Cette th`ese prend appui sur les probl`emes de tri suivant une approche constructive. Le but c’est d’affecter des objets de d´ecision, ´evalu´es selon plusieurs crit`eres, `a un ensemble de cat´egories. Chaque cat´egorie est d´efinie a priori pour recevoir ces objets, qui seront ou doivent ˆetre trait´es de la mˆeme fa¸con. Cette recherche fournit une taxonomie selon laquelle dix diff´erents types de probl`emes de tri sont d´efinis avec un int´erˆet pratique. Deux m´ethodes de tri sont propos´ees en tenant compte cette taxonomie, appel´ee Electre Tri-C et Electre Tri-nC. Ces m´ethodes sont appropri´ees aux contextes d´ecisionnels o`u l’ensemble de cat´egories est compl`etement ordonn´e. Chaque cat´egorie est caract´eris´ee par des action de r´ef´erence typiques. La proc´edure d’affectation est bas´ee sur l’utilisation conjointe d’une r`egle descendante et d’une r`egle ascendante. Nos travaux fournissent ´egalement un algorithme de description segment´ee pour analyser la liai-son entre les conditions d’affectation d’une m´ethode de tri, telle qu’Electre Tri-C, et les pr´ef´erences du d´ecideur, en incluant une analyse de ces incoh´erences et incompatibilit´es du mod`ele de tri, sans avoir besoin de bˆatir un mod`ele d’optimisation. Les principaux r´esultats de notre recherche sont valid´es par deux applications r´eelles (procr´eation m´edicale assist´e et risque agro-environnemental), qui ont ´et´e mod´elis´ees en interaction respectivement avec des experts m´edicaux et environnementaux.
Mots-cl´es :
Aide multicrit`ere `a la d´ecision, Approche constructive, Probl`emes de tri, Action caract´eristiques de r´ef´erence, Electre Tri-C, Algorithme de description segment´ee.
To my family with all my love, especially to: Mama Papa Angy Neury Rosy Edyr Zezinha Elsa Jacky Djila L´ıgia Denny Arlindo Cardo Titi Didi Din Olivio Pedro by Juss, JAD
Acknowledgements
My supervisors and dear friends: Prof. Jos´e Rui Figueira Prof. Bernard Roy
Jury of the preliminary and final discussions: Prof. Manuel Matos (Universit´e de Porto, Portugal) Prof. Patrice Perny (Universit´e Pierre et Marie Curie, France) Prof. Ana P´ovoa (Universidade T´ecnica de Lisboa, Portugal) Prof. Vincent Mousseau (´Ecole Centrale de Paris, France) Prof. Paulo Correia (Universidade T´ecnica de Lisboa, Portugal)
Financial support: Funda¸c˜ao para a Ciˆencia e a Tecnologia, Portugal (Grant SFRH/BD/22985/2005) COST Action Number IC0602 Funda¸c˜ao Calouste Gulbenkian, Portugal (Grant 109475)
My host institutions: Instituto Superior T´ecnico, Universidade T´ecnica de Lisboa, Portugal LAMSADE, Universit´e Paris-Dauphine, France
My friends and colleagues: Afonso Zinga Prof. Clara Murteira (FEUC) Dominique Fran¸cois (LAMSADE) Nat´alia Almeida (IST) Prof. Cristina Bazgan (LAMSADE) Prof. Jos´e Figueiredo (IST) S´onia Sousa (IST) S´onia Toubaline (LAMSADE) Teresa Marques (IST) Tommi Tervonen Vera Teixeirinha (IST)
Contents
Abstract v
Resumo vii
R´esum´e ix
List of Figures xx
List of Tables xxii
General introduction
1
I
Generalities
7
1 Multiple criteria decision aiding 9
1.1 Introduction . . . 10
1.2 Brief review of the literature . . . 14
1.2.1 General statistics on MC-SC2 . . . 14
1.2.2 Several scientific approaches . . . 17
1.2.3 MCDA applications . . . 19
1.3 Conclusions . . . 20
2 Taxonomy of sorting problems 23 2.1 Introduction . . . 24
2.2 Decision aiding sorting contexts . . . 25
2.2.1 Structure of the categories . . . 25
2.2.2 Interaction with the decision makers . . . 26
2.2.3 Constraints on the size of the categories . . . 27
2.3.1 Totally ordered sorting problems . . . 28
2.3.2 Partially ordered sorting problems . . . 29
2.3.3 Unordered sorting problems . . . 30
2.4 Conclusions . . . 31
II
New developments in sorting problems
33
3 Electre Tri-C 35 3.1 Introduction . . . 363.2 Concepts, definitions, and notation . . . 38
3.3 Problem statement . . . 41
3.3.1 Designing the categories . . . 41
3.3.2 Structural requirements . . . 43
3.4 The Electre Tri-C method . . . 45
3.4.1 Assignment procedure . . . 46
3.4.2 Justification of Electre Tri-C . . . 48
3.4.3 Properties of Electre Tri-C . . . 57
3.5 A numerical example . . . 67
3.6 Conclusions . . . 71
4 Electre Tri-nC 73 4.1 Introduction . . . 74
4.2 Problem statement . . . 75
4.2.1 Additional concepts, definitions, and notation . . . 75
4.2.2 Designing the categories . . . 77
4.2.3 Structural requirements . . . 78
4.3 The Electre Tri-nC method . . . 80
4.3.1 Assignment procedure . . . 81
4.3.2 Justification of Electre Tri-nC . . . 83
4.3.3 Properties of Electre Tri-nC . . . 90
4.4 Modifying the definition of the categories . . . 97
4.5 A numerical example . . . 102
4.6 Conclusions . . . 106
CONTENTS xvii
5.1 Introduction . . . 108
5.2 The concept of variable thresholds . . . 110
5.2.1 General definition of variable thresholds . . . 110
5.2.2 Variable thresholds as affine functions . . . 115
5.3 Practical comments on variable thresholds . . . 117
5.4 Variable thresholds in sorting problems . . . 119
5.5 Conclusions . . . 121
6 On the comparison of some sorting methods 123 6.1 Introduction . . . 124
6.2 A comparison within our taxonomy framework . . . 126
6.3 Electre Tri-C versus Electre Tri-B . . . 127
6.4 Electre Tri-nC versus some sorting methods . . . 130
6.5 Computational and aggregation issues . . . 133
6.5.1 Binary relations and parameters . . . 133
6.5.2 Partial concordance indices . . . 135
6.5.3 Partial discordance indices . . . 136
6.5.4 Aggregating partial indices . . . 136
6.5.5 Credibility indices . . . 136
6.6 Formulating assignment rules . . . 137
6.7 Conclusions . . . 138
7 A segmenting description algorithm for sorting problems 141 7.1 Introduction . . . 142
7.2 Definition of the systems of inequalities . . . 146
7.2.1 Initial system of inequalities . . . 147
7.2.2 Adding inequalities on the weights . . . 148
7.2.3 Adding inequalities from assignment examples . . . 148
7.3 The segmenting description (DS) algorithm . . . 152
7.4 A numerical example . . . 157
7.4.1 Applying the DS algorithm to C0(X) . . . 157
7.4.2 Applying the DS algorithm to C1(X) . . . 160
7.4.3 Applying the DS algorithm to C3(X) . . . 161
7.5 Incoherence analysis with the DS algorithm . . . 163
7.5.2 The DS-C algorithm . . . 164
7.6 Advantages of the segmenting description . . . 169
7.7 Conclusions . . . 171
8 Structuring approach for size-constrained sorting problems 173 8.1 Introduction . . . 174
8.2 Concepts, definitions, and notation . . . 175
8.3 Motivating examples . . . 176
8.4 New approaches for sorting problems . . . 183
8.4.1 Unique decision’s approach . . . 183
8.4.2 Sequential decision’s approach . . . 185
8.5 Conclusions . . . 187
III
MCDA applications
189
9 A DSS prototype implemented in MS Excel 191 9.1 Basic assumptions . . . 1929.2 Basic interface . . . 192
9.3 Implementing options . . . 196
9.4 Conclusions . . . 199
10 An assisted reproduction application 201 10.1 Introduction . . . 202
10.2 The decision aiding sorting model . . . 203
10.2.1 Modeling the set of criteria . . . 203
10.2.2 Modeling the set of categories . . . 205
10.2.3 Modeling the imperfect knowledge of the data and the arbitrariness 205 10.2.4 Modeling the role of the criteria . . . 208
10.3 Assignment results and discussion . . . 210
10.4 Conclusions . . . 212
11 An environmental risk application 215 11.1 Introduction . . . 216
11.2 The decision aiding sorting model . . . 217
11.2.1 Identifying agricultural risk of erosion . . . 217
CONTENTS xix
11.2.3 Relationship between the GIS and the MCDA model . . . 221
11.2.4 Modeling the set of categories . . . 224
11.2.5 Modeling the imperfect knowledge of the data and the arbitrariness 226 11.2.5.1 Managing the slope thresholds’ values . . . 226
11.2.5.2 Managing the experts’ qualitative statement . . . 227
11.2.5.3 Numerical values assigned to the thresholds . . . 229
11.2.6 Modeling the role of the criteria . . . 230
11.2.7 Choosing a credibility level . . . 231
11.3 Assignment results and discussion . . . 231
11.3.1 Electre Tri-C assignment results . . . 232
11.3.2 Spatial visualization of the assignment results . . . 232
11.3.3 Discussion and validation of the model . . . 235
11.4 Conclusions . . . 237
Conclusions and future research
239
Appendices 249
A Long r´esum´e en fran¸cais 251
B List of defined objects 259
C An alternative selecting function 263
D The variable thresholds 265
E The comparison of sorting methods 267
F The segmenting description algorithm 271
G The assisted reproduction application 277
H The environmental risk application 281
1 Relationship between the obtained results and our objectives . . . 4
2.1 Diagram of the taxonomy framework . . . 28
2.2 Hierarchic set of categories . . . 30
3.1 On merging the categories . . . 45
3.2 On splitting the categories . . . 45
3.3 Descending rule selecting process . . . 47
3.4 Ascending rule selecting process . . . 48
3.5 The sum selecting function (3.6) . . . 51
3.6 The min selecting function (3.7) . . . 53
3.7 Relationship between λ and Γ(a) . . . 66
3.8 Representation of the characteristic actions . . . 69
5.1 Representation of Ig, Qg, and Pg (max) . . . 112
5.2 Representation of Ig, Qg, and Pg (min) . . . 112
5.3 Representation of the indifference zone . . . 120
6.1 Definition of boundary actions . . . 128
7.1 DSC and ASC pre-selected category . . . 150
7.2 Traceable tree of an equivalent system . . . 156
7.3 Traceable tree of an equivalent system to C1(X) . . . 165
8.1 Classical representation of a decision aiding sorting model . . . 174
8.2 Representation of a unique decisions’ context . . . 184
8.3 Representation of a sequential decisions’ context . . . 186
9.1 Relational interface schema . . . 193
9.2 Parameters interface sheet . . . 194
LIST OF FIGURES xxi
9.4 Set of Actions (A) interface sheet . . . 195
9.5 ETRIC Results interface sheet . . . 196
9.6 Algorithm implementation’s options . . . 197
9.7 Descending rule’s implementation . . . 198
9.8 Ascending rule’s implementation . . . 198
10.1 Illustration of the variable thresholds on criterion g1 . . . 207
11.1 Localization of the two watersheds . . . 218
11.2 Qualitative statement on g2 . . . 220
11.3 Qualitative statement on g3 . . . 220
11.4 Qualitative statement on g4 . . . 221
11.5 Qualitative statement on g5 . . . 221
11.6 The GIS general structure and the links to the MCDA model . . . 222
11.7 Land use in the watershed of Violettes . . . 223
11.8 Numerical coding values on g2 . . . 228
11.9 Numerical coding values on g3 . . . 228
11.10Numerical coding values on g4 . . . 229
11.11Numerical coding values on g5 . . . 229
11.12Spatial visualization of the assignment results . . . 233
1.1 Types of publications analyzed . . . 15 1.2 Nr. of publications per year . . . 15 1.3 List of keywords . . . 16 1.4 MCDA applications . . . 19 2.1 Brief review of literature on sorting methods . . . 31 3.1 Counter-example (sum function) . . . 52 3.2 Outranking credibility indices (counter-example) . . . 52 3.3 Summary results for the descending rule . . . 62 3.4 Criteria and parameters . . . 67 3.5 Potential actions . . . 68 3.6 Characteristic actions . . . 68 3.7 Outranking credibility (potential actions) . . . 69 3.8 Comparison between potential actions and characteristic actions . . . 70 3.9 Electre Tri-C assignment results . . . 70 4.1 Criteria and parameters . . . 103 4.2 Potential actions . . . 103 4.3 Characteristic actions . . . 104 4.4 Outranking credibility (potential actions) . . . 104 4.5 Assignment results . . . 105 6.1 Sorting methods and our taxonomy framework . . . 126 6.2 Sorting methods, binary relations, and parameters . . . 134 6.3 Assignment rules using a membership degree . . . 137 6.4 Assignment rules using a membership degree with a credibility level . . . . 138 7.1 Inferring the parameters for a sorting model . . . 143 7.2 Partial concordance indices . . . 157
LIST OF TABLES xxiii
7.3 DS: Renaming the variables . . . 158 7.4 Step 1: Initial system of inequalities, C0(X) . . . 158
7.5 A solution of the DS system . . . 159 7.6 Partial concordance indices (action a9) . . . 162
7.7 Initial system of inequalities, C0(X) . . . 164
7.8 Applying the DS-C algorithm to C1(X) . . . 168
10.1 The ART set of criteria . . . 204 10.2 Performance of the characteristic actions/couples . . . 205 10.3 Assigning discriminating thresholds’ values . . . 206 10.4 Discriminating thresholds on the criterion g1 . . . 207
10.5 Discriminating thresholds on the criterion g2 . . . 208
10.6 Discriminating thresholds on the criterion g6 . . . 208
10.7 The weights according to the embryologist . . . 209 10.8 Veto thresholds on the criterion g1 . . . 209
10.9 Choice of the credibility level, λ . . . 210 10.10Summary of the assignment results . . . 211 11.1 Four categories of risk in environmental analysis . . . 224 11.2 Performance of the characteristic reference plots . . . 225 11.3 Assigning discriminating thresholds’ values . . . 229 11.4 On providing the weights of the criteria . . . 230 11.5 Summary of the assignment results . . . 232 11.6 Choice of the credibility level, λ . . . 235 11.7 Validation of the credibility level, λ . . . 235 E.1 On computing partial concordance indices in sorting methods . . . 268 E.2 On computing partial discordance indices in sorting methods . . . 269 E.3 On aggregating partial indices in sorting methods . . . 270 E.4 On computing credibility indices in sorting methods . . . 270 F.1 Applying the DS algorithm to C0(X) . . . 272
F.2 Applying the DS algorithm to C1(X) . . . 273
F.3 Applying the DS algorithm to C1(X) . . . 273
F.4 Applying the DS algorithm to C3(X) . . . 274
F.6 Applying the DS algorithm to C1(X) . . . 275
G.1 Performances of the couples . . . 278 G.2 Comprehensive outranking credibility indices . . . 279 G.3 Electre Tri-C versus ART assignment results . . . 280 H.1 Performances of the agricultural plots (Violettes) . . . 282 H.2 Comprehensive outranking credibility indices . . . 283 H.3 Electre Tri-C versus environmental expert’s assignment results . . . 285
General introduction
Living in a global competitive market, taking good and advisable decisions is not based on insights. Knowing a set of powerful decision aiding tools allows to describe, to select, to rank, or to assign to groups the objects of a decision according to a particular decision aiding context.
How to assess the degree of severity of a breast cancer patient, according to several symptoms? In a region where mining was a crucial activity several decades ago, how sensitive is a certain zone of that region to collapses and subsidence, based on a set of relevant attributes? Several European Union programs deal with the problem of allocating funds to projects. What is the subset of projects chosen for an automatic allocation of funds, which projects should be rejected and which ones should be re-analyzed for possible funding in the future, since each project is analyzed in several aspects? In a big distribution corporate company with a large number of retail stores, evaluated according to several characteristics, what are the underperforming and the outperforming retail stores? Employees in a big oil extraction company are rewarded for their performance, indeed they have different distinctive skills and productivity degrees, but how can we identify those to whom we should assign a high bonus, those to whom we should give a medium bonus, and those with no bonus at all?
This kind of questions frequently arise in real-world decision aiding problems, both in private and public organizations. They are related with a broad scope or areas as medical diagnosis, territory planning, resource allocation, personnel policy and management, and many others. There is a basic common feature to all of the above situations; their very nature is multidimensional. All the actions or alternatives (patients, zones, projects, retail stores, and employees) are simultaneously evaluated under a set of relevant and frequently heterogeneous and conflicting dimensions or criteria (built up from symptoms, attributes, aspects, characteristics, or skills and productivity levels). Actions and criteria are what
we consider as the basic data of a multiple criteria decision aiding (MCDA) evaluation model (see Roy and Bouyssou, 1993; Bouyssou et al., 2006). But, this is not the only common feature of the above decision aiding situations; all of them require the assignment of actions to distinct groups: patients should be assigned to their most adequate severity risk degree groups, zones should be assigned to risk collapse and subsidence groups, and so on. The term “groups” can be replaced by “categories”, “classes”, or “clusters”. In what follows, we will only use the term “categories”, when there is no risk of misunder-standing. Assigning actions to categories is an old activity and has becoming recently of the uttermost importance for “solving” many decision aiding problems with a strong impact in the life, evolution, and management of modern organizations, mainly due to the increasing of a fierce competition among organizations and the complexity of social systems.
According to Doumpos and Zopounidis (2002b), there are different kinds of classifi-cation problems. When the categories are defined a posteriori and the assignment of the actions to them is done using a similarity measure, we deal with clustering techniques, where the categories can be called clusters. The categories can also be known a priori, i.e. defined before assigning the actions. In such a case, two types of classification problems are usually distinguished: nominal classification problems and ordinal classification prob-lems. A nominal classification problem consists of assigning a set of actions to a finite, pre-defined, and unordered set of categories. An ordinal classification problem consists of assigning a set of actions to a finite, pre-defined, and completely ordered set of categories. One of the main application field of decision aiding in the sorting problematic is related with the case, where the categories are defined a priori. One of the most distinctive features of sorting problematic is the absolute judgment character, in which each action is considered to be independent of the remaining ones when assigning it to a certain category. It means that the assignment of an action to a certain category only takes into account the intrinsic evaluation of such an action on all the criteria, and neither depends nor influences the category to which another action should be assigned.
In our thesis, one of our contributions is the definition of several types of decision aiding sorting problems, within a general framework, in order to avoid the dichotomy between “ordinal” and “nominal”, which can be a source of misunderstanding. These types of sorting problems will be analyzed within a constructive approach. The aim is to co-construct a decision aiding sorting model through an interaction process between the analyst and the decision makers.
General introduction 3
between the research results and the objectives as well as the classical presentation of the structure of the thesis.
Research scientific objectives
Our research is focused in multiple criteria decision aiding field, more precisely it deals with sorting problems. Our scientific objectives have been defined as follows:
1
to propose a taxonomy for decision aiding sorting problems based on multiple criteria for structuring the field of practical applications;
2
to make a preliminary analysis of the so-called size-constrained sorting problems, while keeping the intrinsic character of the sorting problematic;
3
to propose new decision aiding sorting methods for the case where the completely ordered set of categories is characterized by typical reference actions;
4
to compare different decision aiding sorting methods with respect to the theoretical and practical issues;
5
to look for a new approach to deal with the disaggregation/aggregation paradigm for decision aiding sorting models without making use of an optimization model;
6
to implement a prototype of a Decision Support System, based on the new concepts and the new decision aiding sorting methods proposed in our thesis, to deal with practical applications.
Structure of the thesis
The relationship between our research results, taking into account the scientific objectives, is presented in Figure 1. There are six chapters, which provide the main research results and four chapters, which provide complementary research results. Three of these chapters present our preliminary results. The two chapters regarding the MCDA applications are used for validating the main research results.
The chapters presented in Figure 1 are organized into three parts. After some gener-alities, the proposed developments in sorting problems and two MCDA applications are analyzed.
TX EC DS CP NC TC PE 1 3 5 4 3 2 6
Main research results: TX: Taxonomy (Chapter 2) EC: Electre Tri-C (Chapter 3) DS: Segmenting description (Chapter 7) NC: Electre Tri-nC (Chapter 4)
AP: MCDA applications (Chapters 10 and 11)
Complementary research results: TC: Sized-category (Chapter 8) VT: Variable thresholds (Chapter 5) CP: Comparison (Chapter 6)
PE: Prototype in MS Excel (Chapter 9) Preliminary research results
Figure 1: Relationship between the obtained results and our objectives
Part I contains two chapters mainly related with the presentation of the MCDA field. Chapter 1 presents some main concepts and definitions regarding the research field, which are relevant for the following chapters, as well as a brief review of the literature according to the main approches, which are currently used for sorting problems. Chapter 2 provides a taxonomy framework for structuring the chosen field of research in order to clearly show the needs of research and promising avenues for future research. In these introductory chapters no sorting approaches are described in order to quickly arrive at the main part of this research.
Part II contains six chapters and it is the core business of our research. There are three main chapters (Chapters 3, 4, and 7) and three complementary ones (Chapters 5, 6, and 8). Chapter 3 is devoted to the Electre Tri-C method, a new sorting method based on a single typical reference action for characterizing each one of the completely ordered set of categories. According to our taxonomy framework, Electre Tri-C belongs to the T CU type of sorting problems. Chapter 4 is devoted to the Electre Tri-nC method, which must be considered as a generalization of Electre Tri-C. In such a case, each category is characterized by several typical reference actions. Let us notice that there
General introduction 5
are some overlapping with Chapter 3, but we intend to provide a self-containing chapter. Chapter 7 provides a segmenting description algorithm for analyzing a set of parameters of a decision aiding sorting model, including the weights of the criteria and the chosen credibility level. The aim is to analyze the compatibility of the set of characteristic reference actions and the compatibility of a set of assignment examples provided by the decision makers, without making use of an optimization model.
The three complementary chapters are relevant to the consistency of our research. Chapter 5 provides an additional analysis of the sorting problems, which must be modeled by using the variable thresholds. The concept of variable thresholds is very complex and perhaps difficult to understand. This is a simple contribution for making all the main ideas together and for facilitating the modeling framework of sorting problems. Chapter 6 provides a comparison framework applied to several sorting methods, taking into account our taxonomy for sorting problems and the main differences and similarities between the proposed sorting methods and the existing ones. Our contribution for sorting problems seems original after analyzing the main related sorting methods. Chapter 8 is devoted to a general framework for sorting problems when they are presented with size-contraints with respect to the definition of the set of categories. Due to time contraints, this chapter is shortened, but the main idea remain very useful for future research.
Part III contains two main chapters related with two real-world decision aiding sort-ing models, and another complementary chapter related with a prototype of a Decision Support System (DSS). Thus, Chapter 9 presents the prototype of the DSS, which has been implemented in MS Excel. Chapter 10 is devoted to an assisted reproduction appli-cation, which has been modeled within the Electre Tri-C framework. This real-world MCDA application is a result of a collaboration with a medical experts’ team. The aim is to assign a patient waiting for an assisted reproductive technology treatment to a cate-gory, which represents the number of embryos to be transferred back to the uterus of a woman. Chapter 11 provides an environmental risk application, which has also been modeled within the Electre Tri-C framework. This real-world MCDA application is a result of a collaboration with an environmental experts’ team. The aim is to assign agri-cultural plots to a category, which represents the risk level on the stream of the studied area. Such risks may cause damages in the reproduction habitat of the salmonid fishes.
Seven appendices are also presented after the general conclusions and the avenues for future research. These conclusions are going to be presented chapter per chapter. The appendices provide complementary research results and the sorting data, and the assignment results concerning the MCDA applications.
Part I
Chapter
1
Multiple criteria decision aiding
Real-world complex problems are commonly modeled by a set of powerful tools using several points of view of the problem due to its multidimensional vary nature. These points of view can be modeled to play an appropriate role within a multiple criteria paradigm. This chapter presents several key concepts and definitions, which are useful to our thesis framework (Section 1.1) and it provides a brief review of the literature (Section 1.2), including some statistics regarding a sample of 600 analyzed references dealing with multiple criteria sorting, classification, and clustering problems, denoted MC-SC2, a list of several scientific approaches for sorting problems, as well as some MCDA applications. Last section provides our concluding remarks.
1.1
Introduction
In this section we will only present some key concepts and their definitions, which are useful for our thesis framework, i.e. multiple criteria decision aiding for sorting problems. Let us notice that some other key concepts and definitions will be introduced in the next chapters, where they are really needed. This option allows to provide a more friendly lecture of our thesis.
Definition 1.1 (Decision aiding).
Decision aiding is the activity of the person who, through the use of explicit but not necessarily completely formalized models, helps to obtain elements of responses to the questions posed by a stakeholder in a decision process. These elements work towards clarifying the decision and usually towards recommending, or simply favoring, a behavior that will increase the consistency between the evolution of the process and this stake-holder’s objectives and value system (Roy, 2005).
The term “decision aiding” is used instead of “decision support”, “decision making”, or “decision analysis” to avoid any simplistic assimilation. Among the stakeholder, we can identify the decision makers as one of the main actor of the decision aiding process. This process is usually assisted by an analyst. The term “decision makers” represents those in whose name or for whom the decision aiding must be given and the “analyst” represents a facilitator of the decision aiding process, which must perform her/his role in interaction with the decision makers.
Two conceptions of decision aiding have been analyzed by Roy (2010). The so-called Anglo-Saxon conception, which is primarily positivist, and the so-called European tion, which is is primarily constructivist. Our thesis mainly belongs to the latter concep-tion, in which the decision aiding is given within a co-constructive approach. For more details about this approach, see also Tsouki`as (2007); Roy and Vanderpooten (1997, 1996). Definition 1.2 (Potential action).
A potential action, denoted a, is an object of a decision, which is possible to be imple-mented or has some interest within the decision aiding process. The set of potential actions is usually denoted A (Roy, 2005).
The set of potential actions can be completely known a priori or it may appear progres-sively during the decision aiding process. The term alternative is a particular case of potential action, where two alternatives cannot be implemented conjointly. In practice
1.1 Introduction 11
the potential actions can be patients waiting for a medical treatment, credit demand files, risk zones, candidates for a job, environmental measures, or R&D projects.
The concept of criterion plays a fundamental role in our thesis framework. This concept can be defined in several detailed ways, taking into account the multidimensional vary nature of the sorting problems, which allows to incorporate several fundamental points of view of the problem, as follows.
Definition 1.3 (Criterion).
A criterion, usually denoted g, is a tool co-constructed for evaluating and comparing potential actions according to a point of view, which must be (as far as it is possible) well-defined. Therefore, g(a) is the performance of the potential action a on the criterion g (Roy, 2005).
Definition 1.4 (Criterion).
A criterion is a function g, defined on [the set of actions], taking its values in a totally ordered set, and representing the decision maker’s preferences according to some point of view, such that g : [the set of actions] → Xg, where Xg is a totally ordered set (Vincke,
1992).
In our thesis framework, based on the role played by each criterion on a decision aiding sorting model, several key features must be associated to it, such as: the preference scale, the preference direction, the discriminating thresholds, the intrinsic weight, and, additionally, the veto threshold.
According to Roy (2005), a scale of a criterion is a totally ordered set of degrees, or scores of evaluation, which is characterized by a number, a verbal statement, or a pictograph. This idea is also clearly expressed in Definition 1.4. Thus, the scale of a criterion must be preference ordered. There are two major types of preference scale as follows:
- qualitative scale: when each degree is defined by a qualitative statement, which can have several different meaning taking into account the decision aiding context; - quantitative scale: when each degree is defined by a numerical value with a clear
and unique meaning in the decision aiding context.
The preference direction of a criterion allows to express the preferences of the decision makers between all the pair of degrees of the scale. There are two mutually exclusive cases:
- increasing preference direction, or max: if the preferences increase when the perfor-mances increase too;
- decreasing preference direction, or min: if the preferences increase when the perfor-mances decrease.
The discriminating thresholds allows to model the imperfect knowledge of the data on the computation of the performances of the potential actions as well as the arbitrariness that affects the definition of each criterion. These thresholds are usually called indifference threshold, denoted qg, and preference threshold, denoted pg. Let us notice that such
thresholds will be analyzed in Chapters 3, 5, and 6. Based on these two thresholds, three managing models are usually distinguished (Roy and Vincke, 1984):
- true-criterion model: if pg = qg = 0;
- quasi-criterion model: if pg = qg >0;
- pseudo-criterion model: if pg >qg >0.
The weight of the criterion, denoted wg, and the veto threshold, denoted vg, play two
different role. The weight of the criterion is a coefficient of relative importance and it must be interpreted as a voting power of the criterion. The weight of a criterion has an intrinsic character regarding the modeling framework, where our thesis belongs. This means that the assigned numerical values are independent from the performances of the actions. The veto threshold is used for taking into account critical difference on the performances of two actions, before an overall conclusion statement. However, this threshold is not always required to play such a role.
The assignment of the potential actions to the categories is based on the evaluation of each one according to a coherent set of at least two criteria (see Definition 1.5). However, due to the role played by the criteria and taking into account the Electre framework, where our thesis is mainly inserted, it is required to have at least three criteria for exclud-ing any possibility of dictatorship of one of the criterion and for givexclud-ing the possibility of a criterion to play an appropriate role according to the preferences of the decision makers. Definition 1.5 (Coherent set of criteria).
The set of criteria, denoted F , is called coherent if and only if it fulfills the following three requirements (Roy, 2005):
1.1 Introduction 13
- exhaustiveness: when all the relevant points of view are taken into account for an appropriate evaluation of the objects of the decision;
- cohesiveness: when the aggregating preferences of the decision makers are consistent in comparison to their partial preferences regarding each criterion;
- non-redundancy: when there is no criterion considered as redundant, or if a criterion is deleted from the set of criteria, the two above requirements are not fulfilled anymore.
None of the requirements from Definition 1.5 implies the independence of the criteria. But, such criteria are usually defined to be independent for simplifying the modeling framework. Nevertheless, if the dependence of the criterion, also called interaction between the criteria, is meaningfulness then some works have been proposed for taking such a dependence into account (see, for instance, Figueira et al., 2009; Montano Guzm´an, 2003). Taking into account the manner in which the decision aiding is envisaged, four general problematics have been defined by Roy (1985):
- description problematic, denoted P.δ: the aim is to describe the main data of the decision aiding process. It deals with a description procedure;
- choice problematic, denoted P.α: the aim is to select a small number of “good” actions in such a way that a single action can be chosen, using a relative comparison approach. It deals with a selection procedure;
- ranking problematic, denoted P.γ: the aim is to rank the set of actions for obtaining a complete or partial order on such a set, using a relative comparison approach. It deals with a classifying procedure;
- sorting problematic, denoted P.β: the aim is to assign each action to one or more categories, which are defined a priori, using an absolute comparison approach. It deals with an assignment procedure.
Additional problematics have also been proposed, mainly based on the four above ones, as follows:
- design problematic (Belton and Stewart, 2002): the aim is to search for, to identify, or to create new object of a decision;
- choice of k from m actions problematic, denoted P.k/m (Bana e Costa, 1992b): the aim is to select the k best actions;
- successive choice of k from m actions problematic, denoted P.αxk (Bana e Costa, 1992b): the aim is to select the k best actions successively;
- portfolio problematic (Belton and Stewart, 2002): the aim it to select a subset of actions among a largest set of possibility taking into account several key features; - typologie problematic (Henriet, 2000): the aim is to built a set of prototypes for
characterizing a set of categories, which is defined a posteriori.
- ordinal sorting problematic (Bana e Costa, 1992a): is the same as the sorting prob-lematic, but only applied when the set of categories is completely ordered.
- nominal sorting problematic (Bana e Costa, 1992a): is the same as the sorting problematic, but only applied when the set of categories is unordered.
A relevant part of the co-construction interactive process between the analyst and the decision makers is performed by using the description procedure. The whole data regarding our thesis framework is composed the set of actions (A), the set of criteria (F ), the set of reference actions (B), the discriminating thresholds (qg and pg), the weights
of the criteria (wg), the veto thresholds (vg), and so on. Our thesis deals with sorting
problems, our contributions belong to sorting problematic.
1.2
Brief review of the literature
This section presents a brief review of the literature on MCDA, especially for dealing with sorting, classification, and clustering problems, denoted MC-SC2 (see also our website, which has been updated over the last years, http://mcsc2.ist.utl.pt).
1.2.1
General statistics on MC-SC2
This section provides a statistical overview of the research direction applied to MC-SC2 problems, which has been produced by gesBib, a useful database tool, based on 600 analyzed references.
1.2 Brief review of the literature 15
The number of distinct authors of the analyzed sample is 904, which research has been made in 44 countries, 235 cities, 331 institutions, and 207 universities. The average number of authors per publication is almost 2.5.
What kind of publications has been analyzed? More that 80% of the publications are journal articles. This is due to the availability of the PDF files of these publications on Internet following the two institutions’ subscriptions, where our research has been developed (see Table 1.1 for more details). These analyzed publications are available into 172 scientific journal or publishers, in which the European Journal of Operational Research was the main source of the references, with 22% of the journal articles.
Table 1.1: Types of publications analyzed
Nr. of pages
Type Nr. publication Total Average St. deviation Journal articles 491 7770 16 8 Books 14 5483 392 314 PhD dissertation 13 2842 219 68 Technical report 40 1344 34 18 Incollection 24 605 25 13 Master dissertation 3 494 165 91 Inproceedings 10 114 11 5 Miscellaneous 5 44 9 6 Total 600
Table 1.2: Nr. of publications per year
Year Nr. Publications Year Nr. Publications Year Nr. Publications
2010 10 2000 58 1990 4 2009 17 1999 40 1989 2 2008 16 1998 36 1987 2 2007 30 1997 29 1985 2 2006 25 1996 22 1984 1 2005 46 1995 8 1983 1 2004 43 1994 8 1980 1 2003 51 1993 1 1977 1 2002 86 1992 6 1971 3 2001 47 1991 4
The analyzed publications belong to a period of 29 years (see Table 1.2). The majority of them have been published after 1995. The choice of such publications is due to the facility of access and relevant changes of the online availability rules of the publications by the master publishers.
Table 1.3 presents the list of the most used keywords associated with the publications, after a pre-processing operation. This table can allow to present the research direction in the literature. We tried to divide these keywords into six classes, while indicating the positioning of our thesis.
Table 1.3: List of keywords
Class Keyword Our thesis deals with ... 1 Multiple criteria decision aiding X
Multiple attribute decision making . Multiple objective optimization . Multivariate statistical analysis .
2 Classification X Sorting X Clustering . Decision rules . 3 Rough sets . Fuzzy sets . Neural networks . Pattern recognition . Machine learning . Discriminant analysis . 4 Preference elicitation X Disaggregation/aggregation X Axiomatic measurement . 5 Preference modelling X Concordance / discordance X Outranking relations X
6 Decision support systems X
Let us notice that in the pre-processing operation applied to the list of keywords (see Table 1.3 ), for instance, multiple criteria decision aiding has been retained instead of the following used keywords: decision aid, decision aiding, decision analysis,
1.2 Brief review of the literature 17
MCDA, MCDM, multi criteria analysis, multicriteria, multicriteria analysis, multicrite-ria decision, multicritemulticrite-ria decision aid, multicritemulticrite-ria decision aiding, multicritemulticrite-ria deci-sion analysis, multicriteria decideci-sion making, multicriteria decideci-sion problems, multicrite-ria evaluation, multicritemulticrite-ria group decision making, multicritemulticrite-ria method, multicriterion aggregation procedure, multicriterion decision aid method, multicriterion decision making, multiple criteria, multiple criteria aid, multiple criteria analysis, multiple criteria deci-sion aid, multiple criteria decideci-sion aiding, multiple criteria decideci-sion analysis, and multiple criteria decision making.
1.2.2
Several scientific approaches
In this section, we will list several approaches to deal with sorting problems, which can be used conjointly. These approaches are related with multiple criteria decision aiding (the master framework of our thesis), multiple attribute decision making (based on a set of attributes, in which each one is not necessarily characterized by an ordered preference scale), multiple objective optimization (based on the optimization of several objective functions for characterizing a decision aiding problem with a set of decision variables and several constraints for obtaining efficient solutions, non-dominated solutions, or Pareto optimal solutions), and multivariate statistical analysis (based on statistical approaches).
Clustering approach. The aim is to characterize general similarities between the actions. A set of categories, usually called clusters is defined a posteriori, which is not necessarily ordered. Despite several works dealing with the clustering approach, we are interested in multiple criteria clustering (see Fern´andez et al., 2010; De Smet and Montano Guzm´an, 2004; Bisdorff, 2002; Zahir, 2002; Won and Kim, 1997; Wong and Lane, 1983).
Decision rules approach. Decision rules are composed by a set of verbal statements of a decision tree or induced from data, i.e., for instance, based on rough sets approximations. Each rule takes the form “if [condition(s)] then [category(ies)]”. Several works have been proposed to deal with sorting problems (see Blaszczynski et al., 2007; Dombi and Zsiros, 2005; Greco et al., 2004; Azibi and Vanderpooten, 2002; S lowi´nski et al., 2002; Tsumoto, 1998).
Rough sets approach. The aim of the rough sets is to obtain a formal approximation of a set, by using a lower set and an upper set. Several works have been proposed to
extend this concept to MCDA and, especially, for dealing with sorting problems (see Dembczy´nski et al., 2009; Greco et al., 2002, 2001, 2000).
Fuzzy sets approach. The aim of the fuzzy sets is to associated a membership degree with each element that belong to a set. The membership degree can be assessed gradually in the range [0, 1] by an appropriate membership function. The fuzzy set theory have been used in several works to deal with sorting problems (see Hoffmann et al., 2007; Amo et al., 2004; Shen and Chouchoulas, 2002; Sarkar, 2002).
Neural networks approach. A neural network, or artificial neural network, is usually an adaptive model, roughly similar to biological neural networks. This model can change its structure during the learning phase in order to find patterns in the data. Several works have been proposed to deal with sorting problems (see West et al., 2005; Sohn and Dagli, 2004; Agarwal et al., 2001; ¨Ostermark, 2000, 1999; Zadeh and Nassery, 1999).
Pattern recognition approach. The aim of pattern recognition is to learn a cate-gory of the pattern induced from the data, or based on a set of recognized knowledge. Several works have been proposed to deal with sorting problems (see Salappa et al., 2005; Ma´ndziuk and Shastri, 2002; Kundu and Martinsek, 1998; Maglaveras et al., 1998).
Machine learning approach. Machine learning deals with algorithms for recogniz-ing complex patterns automatically in order to take intelligent decisions based on data. Several works have been proposed to deal with sorting problems (see S´anchez et al., 2002; Ishibuchi et al., 2001; Dreiseitl et al., 2001; Galindo and Tamayo, 2000; Garreli i Guiu et al., 1999).
Discriminant analysis approach. Discriminant analysis is used in statistics and machine learning to identify a relationship between the main features or discriminate two or more categories of objects. Several works have been proposed to deal with sorting problems (see Glen, 2003; Lam and Moy, 2003, 2002; Major and Ragsdale, 2000).
Axiomatic measurement. Several studies have been proposed for an axiomatic anal-ysis of the sorting models in order to provide some axiomatic foundations and some key properties (see Bouyssou and Pirlot, 2009; Bouyssou and Marchant, 2007b,a; Marchant, 2007; M¨unnich, 2006; Bouyssou and Marchant, 2005; Greco et al., 2004; Fortemps and
1.2 Brief review of the literature 19
Pirlot, 2004; Dubois et al., 2003; S lowi´nski et al., 2002; Marichal, 2000; Tsouki`as and Vincke, 1995).
Several studies have been made for providing a literature review in sorting problems. These studies cover several scientific approaches based on several methodologies (see Figueira et al., 2010; Behzadian et al., 2010; Liberatore and Nydick, 2008; Zopounidis and Doumpos, 2002a,b). As for a brief review of the literature for some decision aiding sorting methods, see Chapters 2 and 6. As for a review of the literature concerning the preference elicitation, following the disaggregation/aggregation paradigm, see Chapter 7.
1.2.3
MCDA applications
Several real-world MCDA applications have been modeled over the last decades for taking into account multiple criteria, especially for dealing with sorting problems. Table 1.4 presents an illustrate sample as such applications, including the decision aiding technique (see also Table 2.1, in Chapter 2).
Table 1.4: MCDA applications
Application Reference Decision aiding technique Cancer care West et al. (2005) Neural networks
Belacel (2000) Proaftn Belacel and Boulassel (2000) Proaftn Climate change Diakoulaki and Hontou (2003) Electre Tri Cropping systems Arondel and Girardin (2000) Electre Tri Dangerous material transport Costa et al. (2004) Electre Tri Economy and finance Beynon and Peel (2001) Rough sets
Doumpos et al. (2001) Utadis Dimitras et al. (1995) Electre Tri Electricity market Mavrotas et al. (2003) Electre Tri Energy management and planning Diakoulaki et al. (1999) Utadis Land-use suitability assessment Joerin et al. (2001) Electre Tri
Joerin and Musy (2000)
Medical diagnosis Belacel and Boulassel (2004) Procftn Belacel and Boulassel (2001) Proaftn Belacel et al. (2001) Proaftn National priorities Georgopoulou et al. (2003) Electre Tri Non-financial performance Andr´e and Roy (2007) Electre Tri
Table 1.4 – continued from previous page
Application Reference Decision aiding technique Public transport ticket system Mousseau et al. (2001b) Electre Tri
Risk of nanomaterials Tervonen et al. (2009b) Smaa-Tri Seismic signal Zadeh and Nassery (1999) Neural networks Skills accreditation system Siskos et al. (2007) Electre Tri Stock portfolio selection Xidonas et al. (2009b) Electre Tri
Xidonas et al. (2009a) Electre Tri, Electre III Pendaraki et al. (2004) Utadis
Doumpos et al. (2000) M.h.Dis Tourism industry Roget and Gonz´alez (2005) Electre Tri Water resources management Raju et al. (2000) Electre Tri Zoning risk analysis Merad et al. (2004) Electre Tri
1.3
Conclusions
This chapter presented several key concepts and definitions, which are useful for the remaining chapters, such as the decision aiding, the decision maker, the analyst, the potential action, the criterion and related key features, the coherent set of criteria, and the sorting problematic.
As for the review of the literature, we firstly presented some statistics about 600 analyzed references, in which a list of keywords allows to identify some research direc-tions. Our thesis follows a multiple criteria decision aiding paradigm for dealing with the so-called classification and sorting. In Chapter 2, we will present a general taxonomy framework for sorting problems for identifying several types of sorting problems and fields of practical applications. Some of the existing sorting methods are associated with such a taxonomy framework. According to our research objectives, new sorting methods will be proposed in Chapter 3 and 4, for dealing with the case, where the pre-defined and completely ordered set of categories is characterized by typical reference actions.
The preference elicitation based on the disaggregation/aggregation paradigm will be briefly review in Chapter 7 and we will proposed a new approach for analyzing a set of parameters of a decision aiding sorting model, without using an optimization model, as is the case of the current used techniques.
1.3 Conclusions 21
the assignment of an action is based on the credibility of outranking relations instead of a similarity relations, which is currently used in the existing sorting methods based on typical reference actions.
Several scientific alternative approaches have been listed for dealing with sorting prob-lems, such as clustering, decision rules, rough sets, fuzzy sets, neural networks, pattern recognition, machine learning, discriminant analysis, and axiomatic measurement.
Finally, a list of MCDA applications was presented for showing the practical interest of this research field.
Chapter
2
Taxonomy of sorting problems
The aim of this chapter is to present a taxonomy for multiple criteria decision aiding for sorting problems in order to highlight different types of sorting problems as well as the fields of practical applications. Section 2.1 is devoted to the justification of our taxonomy framework. Section 2.2 presents three levels of analysis of the decision aiding contexts, which must be taking into account when modeling a sorting problem, including the structure of the categories, the interaction with the decision makers, and the constraints on the size of the categories. Section 2.3 provides our taxonomy framework for classifying the sorting problems into several types, including the totally ordered sorting problems, the partially ordered sorting problems, and the unordered sorting problems. Last section provides our concluding remarks.
2.1
Introduction
A classification problem can have a multiple attribute and/or a multiple criteria charac-ter. In the latter case, the objects of a decision, or the elements that contribute to the decision (i.e. actions, alternatives, candidates, options, ...) are evaluated according to a coherent set of criteria (see Roy, 1985, pp. 223-332) in order to get the basic data of the problem. In such a case, the scale associated with each criterion is preference ordered and the obtained classification models are also based on the preferences of the decision makers. Consequently, two different decision makers can have different preferences and the output of the decision aiding process can also be different in this particular issue of classification problems. The aim is to help the decision maker to make well justified classification deci-sions. In the case of multiple attribute classification problems, the actions are evaluated according to a set of general attributes, which are not necessarily preference ordered and the preferences of the decision makers are not incorporated in the analysis. The aim is to detect and characterize general similarities. In what follows, only the multiple criteria classification problems are analyzed, which is the main objective of our thesis.
In several works, the multiple criteria classification problems are divided in two groups: supervised classification problems and unsupervised classification problems (see Perny, 1998; Henriet, 2000). As for unsupervised classification problems, the categories, or an equivalent designation, are only known/defined a posteriori. Multiple criteria classifica-tion based on examples and multiple criteria clustering belong to this group. For an analysis of a taxonomy of clustering procedures see Cailloux et al. (2007). In supervised classification problems, the categories are defined a priori. These categories can be defined by decision rules (see Tsumoto, 1998; Greco et al., 2001, 2002; Dombi and Zsiros, 2005; Blaszczynski et al., 2007), assignment examples, and standard references (see Yu, 1992; Roy and Bouyssou, 1993; Perny, 1998; Belacel and Boulassel, 2004).
Henriet (2000, Chap. 2) proposed a taxonomy of classification methods for supervised classification problems in which the methods were divided into explicit and implicit meth-ods. The explicit methods are mainly based on decision rules and the implicit methods are based on examples or standard references already assigned to the categories.
The classification problems based on standard references will be called here multiple criteria decision aiding sorting problems, and they will be the object of our analysis. In terms of the terminology used, there are several concepts to deal with sorting problems. The perspective of analysis, which provides an idea of what is expected to be done with respect to the objects of a decision, is so-called ordinal sorting problematic when the
2.2 Decision aiding sorting contexts 25
set of categories are completely ordered, and nominal sorting problematic when the set of categories are unordered. The concepts of nominal classification, nominal sorting, and classification are often used for dealing with unordered categories. The concepts of ordinal classification, ordinal sorting, and sorting are often used for dealing with a completely ordered set of categories. The term “ordinal” can be a source of misunderstanding. For instance, is it related with the completely ordered set of categories or to the completely preference ordered of the criteria scales? Similarly, the term “nominal” can also be a source of some misunderstanding. For instance, is it related with the unordered set of categories or even a type of scale called “nominal”? The criteria really play a role of a criteria or a role of an attribute? Furthermore, the dichotomy between “ordinal” and “nominal” is not well suited to deal with “mixte categories” of sorting problems. In such a case, there are some categories where it is not possible to establish an order. In our taxonomy framework, we propose to use only sorting problems and several types of sorting problems.
2.2
Decision aiding sorting contexts
In decision aiding sorting problems the categories are supposed to be defined a priori through a co-construction process between the analyst and the decision maker with respect to a particular decision aiding context. This context can also provide some constraints on the size of the categories. In general, the definition of the categories is related with the way in which the actions that are going to be assigned to each category should further be processed. This further analysis must be the same for all actions assigned to the same category (at least in a first step).
The combination of three dimensions, or levels of analysis, of the decision aiding contexts leads to the definition of several types of sorting problems. The first level contains the structure of the categories. The second level is related with the interaction with decision makers in order to define reference actions. The third level concerns the possible definition of some constraints on the size of the categories.
2.2.1
Structure of the categories
The structure of the categories is related with the decision aiding context. After a chosen option of this level, the problem must be analyzed accordingly. In this first level of the analysis a T P N structure formulation can be defined as follows:
T represents a totally ordered categories formulation. This means that there is a complete order on the set of categories.
P represents a partially ordered categories formulation. This means that there is a partial order on the set of categories.
N represents an unordered categories formulation. This means that there is no order on the set of categories.
When the decision aiding context allows to formulate a sorting problem with unordered categories, the criteria can effectively play an important role on the definition of such cate-gories and on the assignment process. Therefore, several sets of weights can be associated with the set of criteria. This means that the same criterion does not necessarily have the same weight in order to assign an action to two different categories. The set of criteria can be used conjointly with a set of attributes. In this work, only the role of the criteria is taken into account. It should be noticed that the unordered categories formulation can be associated to the case where there is an implicit order but it is not possible to define it in an ordinal way.
2.2.2
Interaction with the decision makers
The interaction with the decision makers is a useful tool to obtain or to infer the required basic input and parameters from one of the main actors of the decision aiding process. In this process one must obtain a coherent set of criteria, the performances of actions (potential actions, reference actions, or assignment examples) and the related parameters (indifference and preference thresholds, weights of criteria, veto thresholds, credibility level, utility levels, and so on).
In this taxonomy framework, let us consider only the interaction with respect to the definition of categories in the decision aiding context. This second level must take into account the T P N structure and it can be defined as a BC categorization formulation as follows:
B represents decision bounds or the definition of boundary actions. It includes boundary actions by considering one or several reference actions per cate-gory or additive utility levels.
C represents typical decision or the definition characteristic actions. It includes characteristic actions, prototypes, or typical actions by considering one or several reference actions per category.
2.3 Modeling different types of sorting problems 27
An assignment rule within a categorization of type B may be formulated as follows: an action judged between the boundaries of a category must be assigned to that category. More precisely, an action which is preferred to at least one lower boundary action and not preferred to any upper boundary action, reflecting the boundaries of a category, must be assigned to that category (see Perny, 1998). According to several existing works, an assignment rule within a categorization of type C may be formulated as follows: an action which is judged similar, indifferent, or roughly equivalent to at least one prototype of a category must be assigned to that category (see Belacel and Boulassel, 2004).
2.2.3
Constraints on the size of the categories
One of the main features of the sorting problems is related with the absolute evaluation of each action to be assigned. According to the intrinsic value of the potential actions, for instance, all of them can be assigned to the same category or a category may remain empty with respect to the assignment of such actions. Moreover, the decision aiding context can provide insights to establish or not some constraints in the definition of the categories in which the number of actions to assign to each of them is bounded. It should be noticed that such constraints are not related with the preferences of the decision makers, but to the nature of the decision aiding sorting context.
The assignment of an action may depend on its intrinsic value while taking into account the remaining actions to be assigned. Therefore, the third level of analysis of the decision aiding context includes the constraints on the size of categories and it can be define as an SU assignment formulation as follows:
S represents an assignment with sized categories formulation. This means that at least the size of one category is bounded.
U represents an assignment with unsized categories formulation. This means that there is no constraint on the size of any category.
2.3
Modeling different types of sorting problems
The combination of the three levels of analysis of the decision aiding context as defined in the previous sections provides twelve different types of sorting problems (see Figure 2.1) which are divided into three groups: totally ordered sorting problems, partially ordered sorting problems, and unordered sorting problems.
Decision aiding sorting problems Totally ordered categories decision Bounds Unsized categories T BU Sized categories T BS Typical decisions Unsized categories T CU Sized categories T CS Partially ordered categories decision Bounds Unsized categories P BU Sized categories P BS Typical decisions Unsized categories P CU Sized categories P CS Unordered categories decision Bounds Unsized categories N BU Sized categories N BS Typical decisions Unsized categories N CU Sized categories N CS
Research direction in the literature Main research in our thesis Preliminary research in our thesis
Figure 2.1: Diagram of the taxonomy framework
2.3.1
Totally ordered sorting problems
Several alternative concepts have been addressed to deal with sorting approaches when the categories are completely ordered within the so-called ordinal sorting problematic: ordinal classification, ordinal sorting, sorting, and so on. Taking into account our taxonomy, four types of sorting problems are proposed when the categories are completely ordered:
2.3 Modeling different types of sorting problems 29
with constraints on the size of categories.
T BU Totally ordered sorting problems based on a decision bounds formula-tion without constraints on the size of categories.
T CS Totally ordered sorting problems based on a typical decision formulation with constraints on the size of categories.
T CU Totally ordered sorting problems based on a typical decision formulation without constraints on the size of categories.
The outranking sorting methods proposed over the last fifteen years to deal with totally ordered sorting problems should fall into the T BU type (see Table 2.1). For a comparison of such sorting methods regarding the methodological and practical issues, see Chapter 6.
2.3.2
Partially ordered sorting problems
The dichotomy between “ordinal” and “nominal” is not well suited to deal with this group of sorting problems. In such a case, there are some categories where it is not possible to establish an order. Taking into account our taxonomy, four types of sorting problems are proposed when the categories are partially ordered:
P BS Partially ordered sorting problems based on a decision bounds formula-tion with constraints on the size of categories.
P BU Partially ordered sorting problems based on a decision bounds formu-lation without constraints on the size of categories.
P CS Partially ordered sorting problems based on a typical decision formula-tion with constraints on the size of categories.
P CU Partially ordered sorting problems based on a typical decision formula-tion without constraints on the size of categories.
In this group of sorting problems, the types P BS and P BU seem more difficult to model. In such cases, an hierarchic set of categories (see Figure 2.2) can be proposed and the set of criteria can be modeled in order to define the lower and the upper boundary actions for each category. Furthermore, it is also possible to define (q − 1) completely ordered categories and only one category not related with the others ones, which can be used for the unclassified actions.
