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- 1. Année Académique 2008-2009 Faculté des Sciences Appliquées T HÈSE soutenue à l’Université Libre de Bruxelles En vue de l’obtention du grade académique de DOCTEUR EN SCIENCES de L’INGENIEUR On the use of multicriteria ranking methods in sorting problems Philippe Nemery de Bellevaux Directeur de thèse : Prof. Philippe Vincke - Université Libre de Bruxelles Encadrement de thèse : Prof. Yves De Smet - Université Libre de Bruxelles Jury de thèse : Prof. Hugues Bersini - Université Libre de Bruxelles Prof. Denis Bouyssou - Université Paris-Dauphine Prof. Bertrand Mareschal - Université Libre de Bruxelles Prof. Marc Pirlot - Faculté Polytechnique de Mons
- 2. "Suis-je plus ou moins ceci ou cela qu’une plante ou qu’un chimpanzé ? La réponse est impossible et absurde. Car toute hiérarchie suppose une unidimensionnalité. Et cela, c’est l’une de mes principales préoccupations: lequel de nous deux est supérieur à l’autre? Eh bien, cela dépend en quoi. Dès qu’il y a une seule caractéristique, il y a une réponse. Mais dès qu’il y a deux caractéristiques, il n’existe plus de réponse. Par conséquent, dire "je suis plus complexe qu’un chimpanzé, parce que mon cerveau compte plus de neurones" est possible, comme il serait possible de dire bien d’autres choses puisqu’il existe beaucoup d’autres critères de performance." Albert Jacquard et Axel Kahn dans "L’avenir n’est pas écrit", p.28
- 3. Acknowledgment The realization of this thesis is a long-term labour whose outcome is certainly due to the contribution of several outstanding people. We would like to thank these persons ; not only for their effective contribution to this work but also for the patience they have showed during these four years of research. After graduating, starting immediately by doing research was in our case, somewhat disconcerting. Finding a research direction is not always easy, especially when "so many" doors are open. Nevertheless, we would like to thank sincerely and gratefully Professor Marie-Ange Remiche and Professor Philippe Vincke for giving us the opportunity of doing research without imposing us any subject nor direction. Thanks to them for the faith they have shown. We wish to thank Professors Hugues Bersini, Denis Bouyssou, Yves De Smet, Bertrand Mareschal and Marc Pirlot for accepting to be part of the jury. Moreover, having Professors Hugues Bersini, Yves De Smet, Bertrand Mareschal, Marc Pirlot and Philippe Vincke during our yearly accompaniment committees, certainly had a positive and fruitful contribution to the achievement of this thesis. We would like to thank Professor Yves De Smet and Professor Marie-Ange Remiche for being present these four years and supporting our moods during the difﬁculties encountered when researching. They have permit us to take some distance from our work and to keep two feet on the ground. We dedicate a special and warm regard to Professor Denis Bouyssou for his pertinent remarks and words of encouragement, when we really needed them. We learned a lot from Professor Betrand Mareschal, both scientiﬁcally and personally, while working together on industrial multicriteria decision problems. Above all, he always has good advices for choosing wine. All the members of the Service des Mathématiques de la Gestion are unforgettable. We wish to thank Olivier Cailloux, Aurélie Casier, Yves De Smet, Quantin Hayez, Claude Lamboray and Karim Lidouh for their judicious remarks and for the interesting conversations we had all along v
- 4. Acknowledgment this work. Thank you all for having read (and corrected !) some parts of our work. Moreover, Catherine Berard, Vinciane de Wilde, Rose-Marie Brynaert and Vanessa Palacios Perez have been a very reliable and professional adminstrative staff. Their support and kindness during these years were incredible. We would like to thank all the members of the LAMSADE (Wassila, Sonia, Nicolas, Guilaume, etc.), at the Université Paris-Dauphine for having welcomed and introduced us to la vie parisienne. Special and warm thanks to Professor Vincent Mousseau, Professor Alexis Tsoukias and Wassila Ouerdane. Moreover, our stay at Paris has been possible given the ﬁnancial supports of the Cost IC0602 Action (STMS). We are grateful to Iryna Yevseyeva, Claude, Michael, Pierre, Laurent, Slobodan, Nico and Thomas for having read some parts of this work and for their judicious and pertinent remarks. We can not forget the students from our faculty which permitted us to have the necessary hindsight. Teaching courses, exercises and working with the students of our faculty, was a real pleasure. Thanks to them for having accepted us like we are and for the human contacts we have (had). In particular, we are grateful to Laurent Huenaerts and Pierre Janssen for their kind collaboration and their concrete contributions to some proofs of this work. Thanks to all our friends, comrades and homemates for being what they are for us. Last but not least, we would like to thank our family. Particularly, our parents and our little sister for having supported us and our mood during these years and for accepting our decision to do research. Was it so terrible ? Besides, we are immensely thankful for Camille, who was present during these last years and who gave us the necessary love, happiness and encouragements for achieving our goal. Although this work will certainly not change the world nor the science, it has changed us, our opinions and our insight towards research and towards multicriteria decision aid. That is, in our point of view, the most important achievement. So, might a reader hesitate to do research, we have only one advice: Just try it ! vi
- 5. vii
- 6. Contents Acknowledgment v Introduction 11 Resumé 15 Publications and Conferences 17 Notations 19 I State of the A rt 21 1 Introduction to classiﬁcation problems 23 1.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.2 Classiﬁcation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.2.1 1.3 2 k-Nearest Neighbors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Need of preference information . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Introduction to Multicriteria Decision Aid 33 2.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2 The actions, the criteria, . . . and the problems . . . . . . . . . . . . . . . . . . . 35 2.2.1 The set of actions A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2.2 The set of attributes F . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2.3 The set of criteria G . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2.4 The different types of MCDA problems . . . . . . . . . . . . . . . . . . 37 2.3 The Pareto dominance relation . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4 Preference Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ix
- 7. Contents 2.5 Consistent family of criteria and preferential independency . . . . . . . . . . . . 39 2.6 Pair-wise comparisons between actions based on outranking relations . . . . . . 41 2.6.1 The valued outranking degree S(a, b) . . . . . . . . . . . . . . . . . . . 41 2.6.1.1 2.6.1.2 Global concordance degree CS (b, a) . . . . . . . . . . . . . . 42 2.6.1.3 Partial discordance degree d S (b, a) . . . . . . . . . . . . . . . 42 j 2.6.1.4 2.6.2 Partial concordance degree cS (b, a) . . . . . . . . . . . . . . . 41 j The outranking degree S(b, a) . . . . . . . . . . . . . . . . . . 43 Preference degree π(a, b) . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.6.2.1 2.6.2.2 3 Uni-criterion preference degree P j (a, b) . . . . . . . . . . . . 46 Global preference degree π(a, b) . . . . . . . . . . . . . . . . 47 Some multicriteria ranking methods 49 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2 Multi Attribute Utility Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2.1 3.3 The additive model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Outranking methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3.1 Electre III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.3.1.1 3.3.1.2 Qualiﬁcation of an action . . . . . . . . . . . . . . . . . . . . 58 3.3.1.3 Computation of the pre-orders O1 and O2 . . . . . . . . . . . . 58 3.3.1.4 Partial pre-order O . . . . . . . . . . . . . . . . . . . . . . . . 60 3.3.1.5 Model assumptions and some properties . . . . . . . . . . . . 64 3.3.1.6 3.3.2 Preference relation between two actions . . . . . . . . . . . . 57 Rank Reversal phenomenon . . . . . . . . . . . . . . . . . . . 64 Promethee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.3.2.1 Entering, leaving and net ﬂows . . . . . . . . . . . . . . . . . 65 3.3.2.2 The Gaia plane and the Walking Weigths . . . . . . . . . . . . 70 3.3.2.3 Model assumptions and some properties . . . . . . . . . . . . 72 3.3.2.4 Rank Reversal phenomenon . . . . . . . . . . . . . . . . . . . 74 3.3.2.5 Some extensions of the Promethee methodology . . . . . . . . 74 3.4 3.5 4 Other multicriteria ranking methods . . . . . . . . . . . . . . . . . . . . . . . . 75 How to choose a multicriteria ranking method ? . . . . . . . . . . . . . . . . . . 76 Some multicriteria sorting methods 81 4.1 4.2 x Introduction to sorting problems . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Properties of sorting methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
- 8. Contents 4.3 Sorting based on indifference indexes . . . . . . . . . . . . . . . . . . . . . . . 88 4.3.1 Indifference Index I (a, b) . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.3.1.1 4.3.1.2 Partial discordance index d Ij (a, b) . . . . . . . . . . . . . . . . 91 4.3.1.3 4.3.2 Partial indifference degree cIj (a, b) . . . . . . . . . . . . . . . 89 Global indifference index I (a, b) . . . . . . . . . . . . . . . . 92 PROAFTN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.3.2.1 4.3.2.2 Assignment rules . . . . . . . . . . . . . . . . . . . . . . . . 93 4.3.2.3 4.4 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Sorting based on similarity indexes . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.4.1 Similarity Index SI(a,b) . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.4.1.1 4.4.1.2 4.4.2 Partial Similarity Index SI j (a, b) . . . . . . . . . . . . . . . . 95 Global Similarity Index SI(a,b) . . . . . . . . . . . . . . . . . 96 TRINOMFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.4.2.1 4.4.2.2 Assignment rules . . . . . . . . . . . . . . . . . . . . . . . . 97 4.4.2.3 4.5 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Sorting based on MAUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.5.1 UTADIS: UTilités Additives DIScriminantes . . . . . . . . . . . . . . . 98 4.5.1.1 4.5.2 MHDIS: Multi-group Hierarchical DIScrimination method . . . . . . . . 100 4.5.2.1 4.6 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Sorting based on outranking relations . . . . . . . . . . . . . . . . . . . . . . . 101 4.6.1 Electre-Tri with limiting proﬁles . . . . . . . . . . . . . . . . . . . . . . 101 4.6.1.1 4.6.1.2 Assignment rules . . . . . . . . . . . . . . . . . . . . . . . . 103 4.6.1.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.6.1.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.6.1.5 4.6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Graphical illustration . . . . . . . . . . . . . . . . . . . . . . 110 Trichotomic Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.6.2.1 4.6.2.2 4.6.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Comparison with Electre-Tri . . . . . . . . . . . . . . . . . . 114 nTomic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.6.3.1 4.6.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Filtering Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 xi
- 9. Contents 4.6.4.1 4.6.4.2 4.6.5 Filtering by strict preference . . . . . . . . . . . . . . . . . . 118 Filtering by indifference . . . . . . . . . . . . . . . . . . . . . 123 PairClass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.6.5.1 4.6.5.2 Assignment Rules . . . . . . . . . . . . . . . . . . . . . . . . 126 4.6.5.3 4.7 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 II F lowS ort: a ﬂow-based sorting method 133 5 137 Notation and conditions 5.1 5.2 6 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Flow-based assignment procedures 6.1 Limiting proﬁles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.1.1 6.2 Strongly preferred limiting proﬁles . . . . . . . . . . . . . . . . . . . . 143 Central proﬁles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.2.1 6.3 141 Strongly preferred central proﬁles . . . . . . . . . . . . . . . . . . . . . 149 Inﬂuence of the preference parameters . . . . . . . . . . . . . . . . . . . . . . . 150 6.3.1 6.3.2 7 Limiting proﬁles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Central proﬁles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Analysis of some properties of F lowS ort 159 7.1 Coherence of the net-ﬂow assignment rule . . . . . . . . . . . . . . . . . . . . . 159 7.2 Property of monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 7.3 Property of weak homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 7.4 Properties of category conformity . . . . . . . . . . . . . . . . . . . . . . . . . 160 7.5 Relationship between Cφ− and Cφ+ . . . . . . . . . . . . . . . . . . . . . . . . . 162 7.6 Relationship between the assignments with limiting proﬁles and central proﬁles . 163 7.7 Property of weak stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 7.7.1 7.7.2 7.8 Negative ﬂow assignment rules . . . . . . . . . . . . . . . . . . . . . . . 166 Positive ﬂow assignment rules . . . . . . . . . . . . . . . . . . . . . . . 167 Strong stability conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 7.8.1 xii Inﬂuence of Condition 6.1.1 on the stability. . . . . . . . . . . . . . . . . 173
- 10. Contents 8 Comparison between F lowS ort and some existing sorting methods 8.1 177 Comparison between F lowS ort and Electre-Tri . . . . . . . . . . . . . . . . . . 177 8.1.1 8.1.2 Impact of a simultaneous comparison . . . . . . . . . . . . . . . . . . . 184 8.1.4 9 Intuitive comparison with Electre-Tri . . . . . . . . . . . . . . . . . . . 182 8.1.3 8.2 Empirical comparison with Electre-Tri . . . . . . . . . . . . . . . . . . . 177 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Comparison with the UTADIS model . . . . . . . . . . . . . . . . . . . . . . . 186 I nterval and F uzzy F lowS ort 191 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 9.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 9.3 Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 9.4 I nterval F lowS ort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 9.4.1 9.4.2 9.5 Limiting proﬁles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Central proﬁles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 F uzzy F lowS ort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 9.5.1 Fuzzy numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 10 S oftware and applications 211 11 Conclusions 219 III 223 Outranking based sorting methods 12 Electre-Tri-Central 227 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 12.2 Assignment rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 12.3 Properties of Electre-Tri-Central . . . . . . . . . . . . . . . . . . . . . . . . . . 232 12.4 Illustrative example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 12.5 Relationship between Electre-Tri-Central and PROAFTN . . . . . . . . . . . . . 236 12.6 Deﬁning a category by several reference proﬁles. . . . . . . . . . . . . . . . . . 239 12.7 Comparison with ELECTRE-TRI-C . . . . . . . . . . . . . . . . . . . . . . . . 242 12.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 13 Partially ordered categories 245 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 xiii
- 11. Contents 13.2 Assignment rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 13.3 Illustrative example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 13.4 Particular subproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 13.4.1 Completely non-ordered categories . . . . . . . . . . . . . . . . . . . . 255 13.4.2 Completely ordered categories . . . . . . . . . . . . . . . . . . . . . . . 257 14 Conclusion Part III 259 Conclusion 261 Bibliography 263 A Proof of Propositions 5.2.1 - 7.7.4 275 A.1 Proof of proposition 5.2.1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 A.2 Proof of proposition 7.1.1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 A.3 Proof of proposition 7.2.1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 A.4 Proof of proposition 7.3.1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 A.5 Proof of proposition 7.3.2: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 A.6 Proof of proposition 7.4.1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 A.7 Proof of proposition 7.4.2: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 A.8 Proof of proposition 7.4.3: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 A.9 Proof of proposition 7.5.1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 A.10 Proof of proposition 7.6.1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 A.11 Proof of propositions 7.7.1-7.7.4: . . . . . . . . . . . . . . . . . . . . . . . . . . 285 B Interval and Fuzzy FlowSort: proofs 287 B.1 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 B.2 Proof proposition 9.3.1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 B.3 Proof proposition 9.4.1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 B.4 Proof proposition 9.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 B.5 Proof proposition 9.4.3: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 B.6 Proof proposition 9.4.4: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 C Data of the application of chapter 10 297 D Proofs of Part III 299 D.1 Proof Proposition 12.3.2: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 xiv
- 12. Contents D.2 Proof Proposition 12.3.1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 D.3 Link between PROAFTN and Electre Tri . . . . . . . . . . . . . . . . . . . . . . 301 xv
- 13. List of Figures 1.1 Representation of the classiﬁcation model in classiﬁcation problem taken from [Doumpos and Zopounidis, 2002], p.7. . . . . . . . . . . . . . . . . . . . . . . . 25 1.2 Representation of a classiﬁcation procedure fG which assigns each action ai to none, one or several categories of the set C = {C1 , . . . ,CK }. . . . . . . . . . . . . 26 1.3 Representation of the k-NN with thwo attributes g1 , g2 and with k = 1 and k = 3. 1.4 Illustration of the utilization of attributes. . . . . . . . . . . . . . . . . . . . . . 31 1.5 Illustration of the utilization of criteria. . . . . . . . . . . . . . . . . . . . . . . . 31 1.6 Comparison of an action a to 4 different reference proﬁles ri , (i = 1, 2, 3, 4). . . . 32 1.7 Comparison of an action a to 4 different reference proﬁles ri , (i = 1, 2, 3, 4). . . . 32 2.1 Representation of the partial concordance index cS (b, a). . . . . . . . . . . . . . 42 j 2.2 Representation of the partial discordance index d S (b, a). . . . . . . . . . . . . . 43 j 2.3 Outranking graph of A where a → b ⇔ aSb ; a ↔ b ⇔ aI b ; a b ⇔ aJ b and © ⇔ cI c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.4 Preference function of type 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.1 Representation of the complete pre-order of the set A by using the MAUT theory. 52 3.2 Pair-wise linear marginal utility functions. . . . . . . . . . . . . . . . . . . . . . 53 3.3 Concave, linear and convex linear marginal utility functions. . . . . . . . . . . . 54 3.4 Outranking graph of A where a → b ⇔ aSb, a ↔ b ⇔ aI b; a b ⇔ aJ b and © ⇔ cI c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.5 The partial pre-order O of A obtained with Electre III . . . . . . . . . . . . . . . 61 3.6 The partial pre-order O of A obtained by "reducing" the outranking graph. . . . . 62 3.7 Representation of the O1 ,O2 and O3 rankings as well as the concordance matrix obtained with the Electre-III demo software [Lamsade, 2008] for the Example 3.2. 63 3.8 Representation of the O1 ,O2 and O3 rankings when suppressing action a4 from A : illustration of the rank reversal phenomenon. [Lamsade, 2008]. . . . . . . . . 65 29 1
- 14. List of Figures 3.9 Chart representation of the ﬂows. . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.10 Complete ranking of A on basis of the positive ﬂows (φ+ ). . . . . . . . . . . . . 69 3.11 Complete ranking of A on basis of the negative ﬂows (φ− ). . . . . . . . . . . . . 69 3.12 Complete Promethee I ranking of A . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.13 Complete Promethee II ranking of A . . . . . . . . . . . . . . . . . . . . . . . . 69 3.14 Gaia plane with δ = 72.8%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.15 Complete ranking of A on basis of the Gaia-plane. . . . . . . . . . . . . . . . . 71 3.16 Representations of the proﬁles of the uni-criterion net-ﬂows of actions a3 and a5 . 71 3.17 Representations of the proﬁles of the uni-criterion net-ﬂows of actions a3 and a2 . 72 3.18 Representations of the ﬂows of the actions of set A 4. . . . . . . . . . . . . . . . 74 3.19 Representation of the ρ-relation. . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.1 Representation of the different classiﬁcation problems on the basis of the different relations between the predeﬁned groups. . . . . . . . . . . . . . . . . . . . . 82 4.2 Properties of sorting procedures according to the sorting problem where particular sorting procedures may verify classical properties (which is represented by "→"). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.3 Representation of the partial indifference index cIj (a, b). . . . . . . . . . . . . . . 89 4.4 Representation of the partial indifference index cIj (a, b). . . . . . . . . . . . . . . 90 4.5 Representation of the partial discordance index d Ij (a, b). . . . . . . . . . . . . . . 91 4.6 Representation of the performances of the reference proﬁles r1 and r2 and the ˙ ˙ actions a1 and a4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.7 Representation of some similarity functions for the computation of SI j (a, b). . . . 96 4.8 Representation of the UTADIS sorting model. . . . . . . . . . . . . . . . . . . . 98 4.9 Representation of the classiﬁcation paradigm taken from [Doumpos and Zopounidis, 2002] p.83. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.10 Illustration of completely ordered categories deﬁned by limiting proﬁles. . . . . . 102 4.11 Representation of preference relation between the limiting proﬁles: rk r j ⇔ r j ← rk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.12 Representation of reduced preference relation between the limiting proﬁles: rk r j ⇔ r j ← . . . ← rk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.13 Reduced "optimistic -graph": x y ⇔ x → y . . . . . . . . . . . . . . . . . . . 105 4.14 Reduced "pessimistic S-graph": : xSy ⇔ x y . . . . . . . . . . . . . . . . . . 105 4.15 Representation of the performances of the limiting proﬁles r1 , r2 , r3 , r4 . . . . . . 106 4.16 Illustration of the paradox of Condorcet where a → b means that a b. . . . . . 109 4.17 Assignment of any point (x,y) of the plan with the " -optimistic" (right) and "S-pessimistic" (S) rules when q = 0 and p = 0, and with w1 = w2 = 0.5. . . . . 111 2
- 15. List of Figures 4.18 Assignment of any point (x,y) of the plan when considering the “union” of the optimistic and pessimistic result and where q = 0 and p = 0, and with w1 = w2 = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.19 Representation of the sets R2 , R3 for the deﬁnition of C1 ,C2 and C3 . . . . . . . 112 4.20 Assignment rules of Trichotomic Segmentation based on the decision tree. . . . . 113 4.21 Assignment rules of Trichotomic Segmentation based on the decision tree when | R2 |=| R3 |= 1 and when s = t = s = t = λ. . . . . . . . . . . . . . . . . . . 115 4.22 Representation of the goodness D j (ai ) and badness d j (ai ) functions. . . . . . . . 116 4.23 Representation of the goodness and badness plan. . . . . . . . . . . . . . . . . . 117 4.24 Representation of the limiting proﬁles deﬁning the ordered categories . . . . . . 119 4.25 Illustration of the paradox of Condorcet where a → b means that aPb. . . . . . . 122 4.26 Representation of the uni-criterion preference and indifference relation. . . . . . 124 4.27 Representation of the preference relation computed in the PairClass procedure. . 127 4.28 Illustration of the case where we deﬁne proﬁles as limiting one. . . . . . . . . . . 128 5.1 Representation of K completely ordered categories by limiting proﬁles . . . . . . 138 5.2 Representation of K completely ordered categories by central proﬁles . . . . . . 138 6.1 Representation of the complete ranking obtained by computing the positive ﬂows. This leads to the Cφ+ -assignment. . . . . . . . . . . . . . . . . . . . . . . 141 6.2 A ﬂow and category representation with limiting proﬁles. . . . . . . . . . . . . . 143 6.3 Representation of the limiting proﬁles and the actions to be assigned. . . . . . . . 144 6.4 Flow-diagram for a1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6.5 A ﬂow and category representation with central proﬁles. . . . . . . . . . . . . . 149 6.6 Flow-diagram for a1 6.7 Assignment of any point (x,y) of the plan with the positive, negative and net ﬂows when q = 0 and p = 0, w1 = w2 = 0.5: identical assignments in the 3 cases. 153 6.8 Assignment of any point (x,y) of the plan with the net ﬂows when q = 0.05 and p = 0.075, w1 = w2 = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.9 Assignment of any point (x,y) of the plan with the positive ﬂows when q = 0.05 and p = 0.075, w1 = w2 = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 6.10 Assignment of any point (x,y) of the plan with the negative ﬂows when q = 0.05 and p = 0.075, w1 = w2 = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.11 Assignment of any point (x,y) of the plan with the positive, negative and net ﬂows when q = 0 and p = 0, w1 = w2 = 0.5. . . . . . . . . . . . . . . . . . . . 156 6.12 Assignment of any point (x,y) of the plan with the positive (left) and negative (right) ﬂows when q = 0 and p = 0, w1 = w2 = 0.5 (right) and by choosing the worst category in case of equality. . . . . . . . . . . . . . . . . . . . . . . . . . 156 3
- 16. List of Figures 6.13 Assignment of any point (x,y) of the plan with the positive ﬂows when q = 0.05 and p = 0.075, w1 = w2 = 0.5 (right) and by choosing the worst category in case of equality (left). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.14 Assignment of any point (x,y) of the plan with the negative ﬂows when q = 0.05 and p = 0.075, w1 = w2 = 0.5 (right) and by choosing the worst category in case of equality (left). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.15 Assignment of any point (x,y) of the plan with the net ﬂows when q = 0.05 and p = 0.075, w1 = w2 = 0.5 (right) and by choosing the worst category in case of equality (left). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 7.1 7.2 Illustration of the relationship between the assignments with limiting proﬁles and centroids: case I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 7.3 Illustration of the relationship between the assignments with limiting proﬁles and centroids: case II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 7.4 Representation of the relationship when deﬁning reference proﬁles as either central (upper-ﬁgure) or limiting (lower-ﬁgure) proﬁles. . . . . . . . . . . . . . . . 165 7.5 Representation of the suppression (lower case) or addition (upper case) of a ’worse’ category when using the negative ﬂows: weak stability. . . . . . . . . . 167 7.6 Representation of the suppression (lower case) or addition (upper case) of a ’better’ category when using the negative ﬂows: stability. . . . . . . . . . . . . . . . 168 8.1 Comparison of F lowS ort and Electre-Tri : different scenarios. . . . . . . . . . . 178 8.2 Representation of the example of 8.1.2: assignments obtained with F lowS ort which can not be obtained with Electre-Tri. . . . . . . . . . . . . . . . . . . . . 183 8.3 Illustration of an assignment with Electre-Tri: III . . . . . . . . . . . . . . . . . 184 8.4 Illustration of an assignment with Electre-Tri: I . . . . . . . . . . . . . . . . . . 185 8.5 Illustration of an assignment with Electre-Tri: II . . . . . . . . . . . . . . . . . . 185 8.6 Representation of the assignment rule of UTADIS . . . . . . . . . . . . . . . . . 186 8.7 Representation of the ﬂows values of the limiting proﬁles . . . . . . . . . . . . . 187 8.8 Representation of the ﬂows values of the limiting proﬁles . . . . . . . . . . . . . 188 8.9 Representation of the ﬂows values of the limiting proﬁles . . . . . . . . . . . . . 188 9.1 Illustration of interval performances of the reference proﬁles and action a on one criteria, where the stars and the bullet represent the mean values of the intervals. . 192 9.2 Illustration of the performances of limiting proﬁles deﬁned by intervals. . . . . . 194 9.3 Illustration of the positive ﬂow intervals of the reference proﬁles under the conditions 9.3.1-9.3.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 9.4 4 Illustration of particular situation where π(ai , r2 ) ≤ γ and π(r2 , ai ) ≤ γ with γ = 0.5162 ˙ ˙ Illustration of the assignment rules when working with the positive ﬂow intervals. 196
- 17. List of Figures 9.5 Illustration of the performances of reference proﬁles. . . . . . . . . . . . . . . . 200 9.6 Illustration of the interval ﬂow-diagram for action a1 . . . . . . . . . . . . . . . . 202 9.7 Representation of a fuzzy interval x and its parameters xu , xl , α, β. . . . . . . . . . 204 9.8 Illustration of the fuzzy performances of the the actions of R1 on criterion 1 in scenario 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 9.9 Illustration of the fuzzy performances of the actions of R1 on criterion 1 in scenario 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 10.1 Screen-shot of the software when encoding the preference parameters. . . . . . . 213 10.2 Screen-shot of the software when encoding the performances of the actions to be sorted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 10.3 Screen-shot of the software representing the evaluations of action a1 with respect to the reference proﬁles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 10.4 Screen-shot of the software representing the positive and negative ﬂow-plane for action a1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 10.5 Screen-shot of the software representing all the actions assigned to C1 according to net ﬂow assignment rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 10.6 Screen-shot of the software representing the distribution of the actions into the categories according to the positive and negative ﬂow-plane for action a1 . . . . . 216 10.7 Screen-shot of the software representing the assignments of all the actions according to the different assignment rules. . . . . . . . . . . . . . . . . . . . . . . 217 11.1 Representation of the ρ-relation. . . . . . . . . . . . . . . . . . . . . . . . . . . 222 12.1 The reduced "optimistic S-graph": xSy ⇔ x → y . . . . . . . . . . . . . . . . . . 229 12.2 The reduced “pessimistic S-graph”: xSy ⇔ x y . . . . . . . . . . . . . . . . 230 12.3 The reduced "optimistic S-graph" and "pessimistic S-graph"when a is indifferent to more than one central proﬁle. . . . . . . . . . . . . . . . . . . . . . . . . . . 233 ˙ 12.4 Example of categories deﬁned by central proﬁles (R = {r1 , r2 , r3 }) and limiting ˙ ˙ ˙ proﬁles (R = {r1 , r2 , r3 , r4 }) and the actions a2 and a3 . . . . . . . . . . . . . . . 234 12.5 Representation of the performances of the central proﬁles r3 , r2 and r1 ; the ˙ ˙ ˙ limiting proﬁles r4 , r3 , r2 , r1 and the actions a2 and a3 . . . . . . . . . . . . . . . 235 12.6 Situation I and II with ∀i = 1, 2 : pi = qi = 0, w1 = w2 . . . . . . . . . . . . . . 239 ˙ ˙ ˙ ˙ 12.7 Case IV= Copt (a) = C2 and Cpess (a) = C2 . . . . . . . . . . . . . . . . . . . . . 240 ˙ ˙1 ˙ ˙1 12.8 Case XII: Copt (a) = C2 and Cpess (a) = C3 . . . . . . . . . . . . . . . . . . . . . 242 13.1 Representation of partially ordered categories. . . . . . . . . . . . . . . . . . . . 246 13.2 Example of partially ordered reference proﬁles in the optimistic and pessimistic reduced "S-graph" where r1 = rI and r1 = rN . . . . . . . . . . . . . . . . . . . 247 ˙1 ˙ ˙1 ˙ 5
- 18. List of Figures 13.3 Case I: Example of the optimistic and pessimistic reduced "S-graph" with a: ˙ ˙1 ˙ ˙2 Copt (a) = C1 and Cpess (a) = C2 . . . . . . . . . . . . . . . . . . . . . . . . . . 248 13.4 Representation of the performances of the central proﬁles r1 , r2 , r2 and r3 . . . . 249 ˙1 ˙1 ˙2 ˙1 ˙ ˙1 ˙ ˙1 13.5 Case II: Copt (a) = C1 and Cpess (a) = C3 . . . . . . . . . . . . . . . . . . . . . . 252 ˙ ˙2 ˙1 ˙ ˙1 13.6 Case III: Copt (a) = C2 ∪ C2 and Cpess (a) = C3 . . . . . . . . . . . . . . . . . . . 252 ˙ ˙2 ˙ ˙2 13.7 Case IV= Copt (a) = C2 and Cpess (a) = C2 . . . . . . . . . . . . . . . . . . . . . 252 ˙ ˙2 ˙1 ˙ ˙2 ˙1 13.8 Case V: Copt (a) = C2 ∪ C2 and Cpess (a) = C2 ∪ C2 . . . . . ˙ ˙2 ˙1 ˙ ˙1 13.9 Case VI: Copt (a) = C2 ∪ C2 and Cpess (a) = C2 . . . . . . . . ˙ ˙1 ˙ ˙1 13.10 Case VII: Copt (a) = C3 and Cpess (a) = C3 . . . . . . . . . ˙ ˙1 ˙ ˙1 13.11 Case VIII: Copt (a) = C1 and Cpess (a) = C3 . . . . . . . . . 13.12 Case 13.13 Case 13.14 Case 13.15 Case ˙ ˙1 ˙ ˙1 IX: Copt (a) = C1 and Cpess (a) = C1 . . . . ˙ ˙1 ˙ ˙1 X: Copt (a) = C3 and Cpess (a) = C3 . . . . ˙ ˙1 ˙ ˙1 ˙2 XI: Copt (a) = C1 and Cpess (a) = C2 and C2 ˙ ˙1 ˙ ˙1 XII: Copt (a) = C2 and Cpess (a) = C3 . . . . . . . . . . . . . . 252 . . . . . . . . . . . 253 . . . . . . . . . . . 253 . . . . . . . . . . . 253 . . . . . . . . . . . . . . . . . 254 . . . . . . . . . . . . . . . . . 254 . . . . . . . . . . . . . . . . . 254 . . . . . . . . . . . . . . . . . 255 13.16 Representation of the ideal and nadir reference proﬁle in case of nominal classiﬁcation problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 6
- 19. List of Tables 2.1 Evaluation matrix with preferentially independent criteria. . . . . . . . . . . . . 40 2.2 The performance evaluation matrix . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.3 The binary relations between the actions of A with λ = 0.6. . . . . . . . . . . . . 45 2.4 The binary relations between the actions of A with λ = 0.7. . . . . . . . . . . . . 45 3.1 The performance evaluation matrix of A . 3.2 The binary relations between the actions of A with λ = 0.9. . . . . . . . . . . . . 60 3.3 Evaluation matrix of the 9 candidates . . . . . . . . . . . . . . . . . . . . . . . . 62 3.4 Preference parameters of the Electre III method. . . . . . . . . . . . . . . . . . . 63 3.5 Evaluation of the performances of the actions of A . . . . . . . . . . . . . . . . . 67 3.6 Preference parameters of the Promethee method. . . . . . . . . . . . . . . . . . 67 3.7 Unicriterions net ﬂows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.8 Positive, negative, net ﬂows and ranking of A . . . . . . . . . . . . . . . . . . . . 68 3.9 Stability intervals at different levels (in %). . . . . . . . . . . . . . . . . . . . . 72 . . . . . . . . . . . . . . . . . . . . . 60 3.10 Input-Output matrix: an extract of Table 7 in [Guitouni et al., 1999]. . . . . . . . 78 3.11 Input-Output matrix after application of the propagation rules: an extract of Table 8 in [Guitouni et al., 1999]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.1 Evaluation of the performances of the central reference proﬁles. . . . . . . . . . 94 4.2 Evaluation of the performances of the actions of A . . . . . . . . . . . . . . . . . 94 4.3 Assignments of the actions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.4 Resume of the assignment results when using the Electre-Tri rules. . . . . . . . . 106 4.5 Evaluation of the performances of the actions of A . . . . . . . . . . . . . . . . . 107 4.6 Evaluation of the performances of the limiting proﬁles of R . . . . . . . . . . . . 107 4.7 Pair-wise comparisons between the actions and the limiting proﬁles ri , ∀ j = 1, . . . , 4: outranking degrees and preference relations. . . . . . . . . . . . . . . . 108 7
- 20. List of Tables 4.8 Assignment of the actions according to the different procedures. . . . . . . . . . 108 4.9 Resume of the assignment results when using the Electre-Tri rules. . . . . . . . . 110 4.10 Assignment of the actions according to the different procedures: Copt ,Cpess and CT S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.11 Pair-wise comparisons between the actions and the limiting proﬁles ri , ∀ j = 1, . . . , 4: valued preference relations. . . . . . . . . . . . . . . . . . . . . . . . . 121 4.12 Assignment of the actions according to the different procedures: Copt , Cpess and CFP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.13 Resume of the assignment results when using the Electre-Tri rules. . . . . . . . . 123 4.14 Preference degrees between action a and the reference examples. . . . . . . . . . 129 4.15 Comparison of different sorting methods on the basis of their properties. The following abbreviations are used: PO: partially ordered, CO: completely ordered, CNO: completely not ordered ; CP: central proﬁle, LP: limiting proﬁle ; RelDeg.: Relation or Degree, Ind: Indifference, Sim.: Similarity, Out.: Outranking ; IY: Yes, N: No ; W-S: Weak or Strong ; /: out of subject . . . . . . . . . . . . . . 132 6.1 The performances of the reference proﬁles. . . . . . . . . . . . . . . . . . . . . 145 6.2 The different thresholds and weights. . . . . . . . . . . . . . . . . . . . . . . . . 145 6.3 The performances of the actions to be sorted. . . . . . . . . . . . . . . . . . . . 145 6.4 The preference degrees between the reference proﬁles and the actions. . . . . . . 146 6.5 Computation of the different ﬂow values. . . . . . . . . . . . . . . . . . . . . . 146 6.6 The assignments of the actions according to Electre-Tri and F lowS ort. . . . . . 147 6.7 The performances of the reference proﬁles. . . . . . . . . . . . . . . . . . . . . 150 6.8 The different thresholds and weights. . . . . . . . . . . . . . . . . . . . . . . . . 150 6.9 The preference degrees between the reference proﬁles and the alternatives. . . . . 151 6.10 The ﬂow-values of the alternatives. . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.1 7.2 The different thresholds and weights. . . . . . . . . . . . . . . . . . . . . . . . . 172 7.3 Computation of the different ﬂow values when considering the set R . . . . . . . 172 7.4 Computation of the different ﬂow values when considering the set R . . . . . . . 173 7.5 Number of instability occurrences in presence of preferred and strongly preferred limiting proﬁles when working with the positive ﬂows. . . . . . . . . . . . . . . 175 8.1 Comparison between F lowS ort and Electre-Tri in the case of limiting proﬁles. . 179 8.2 8 The performances of the reference proﬁles. . . . . . . . . . . . . . . . . . . . . 172 Comparison between F lowS ort and Electre-Tri in the case of limiting proﬁles: analysis of the assignments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
- 21. List of Tables 8.3 Comparison between F lowS ort and Electre-Tri in the case of limiting proﬁles verifying Condition 6.1.1: analysis of the assignments. . . . . . . . . . . . . . . 181 8.4 Evaluation of the performances of the actions of A of example 8.1.2 . . . . . . . 182 8.5 Evaluation of the limiting proﬁle rl . . . . . . . . . . . . . . . . . . . . . . . . . 182 8.6 The preference degrees between the rl and the actions. . . . . . . . . . . . . . . 183 9.1 Interval evaluations of the reference proﬁles on the different criteria. . . . . . . . 200 9.2 The different thresholds and weights. . . . . . . . . . . . . . . . . . . . . . . . . 200 9.3 The preference matrix of the reference proﬁles. . . . . . . . . . . . . . . . . . . 201 9.4 The performances of the actions to be sorted. . . . . . . . . . . . . . . . . . . . 201 9.5 The preference degrees between the reference proﬁles and the actions. . . . . . . 201 9.6 Computation of the different ﬂow values. . . . . . . . . . . . . . . . . . . . . . 202 9.7 Results of the assignments of the actions according to the different rules. . . . . . 203 9.8 Computation of the different ﬂow values for am . . . . . . . . . . . . . . . . . . . 203 2 9.9 Computation of the different fuzzy ﬂow values for R1 in scenario 1. . . . . . . . 208 9.10 Computation of the different fuzzy ﬂow values for R1 in scenario 2. . . . . . . . 208 9.11 Computation of the different fuzzy ﬂow values for R2 in scenario 1. . . . . . . . 209 9.12 Computation of the different fuzzy ﬂow values for R2 in scenario 2. . . . . . . . 209 9.13 Results of the assignments of the action a2 according to the different rules in scenario 1, scenario 2 and when working directly with crisp evaluations. . . . . . 209 10.1 The limiting proﬁles of the 4 categories of different suppliers. . . . . . . . . . . . 212 10.2 The preference parameters associated to the 10 criteria of evaluation. . . . . . . . 212 ˙ 12.1 Evaluation of the performances of the central reference proﬁles of R . . . . . . . 234 12.2 Pair-wise comparisons between the actions and the central proﬁles r j , ∀ j = 1, 2, 3. 235 ˙ 12.3 Assignment of the actions according to the different procedures (with central and limiting proﬁles). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 ˙ 12.4 Assignment results when deﬁning C2 by two reference proﬁles r2 and r2 . . . . . . 241 ˙1 ˙2 13.1 Evaluation of the performances of the central reference proﬁles. . . . . . . . . . 249 13.2 Pair-wise comparisons between the actions and the reference proﬁles. . . . . . . 250 13.3 Classiﬁcation result of the actions according to respectively the Optimistic and Pessimistic version as well as the two PROAFTN assignment rules. . . . . . . . 250 B.1 The preference matrix of the reference proﬁles in case of scenario 1. . . . . . . . 287 B.2 The preference matrix of the reference proﬁles in case of scenario 2. . . . . . . . 287 B.3 The preference degrees between the reference proﬁles and the actions in scenario 1.288 9
- 22. B.4 The preference degrees between the reference proﬁles and the actions in scenario 2.288 B.5 Computation of the different fuzzy ﬂow values for R1 in scenario 1. . . . . . . . 289 B.6 Computation of the different fuzzy ﬂow values for R1 in scenario 2. . . . . . . . 289 B.7 Computation of the different fuzzy ﬂow values for R2 in scenario 1. . . . . . . . 290 B.8 Computation of the different fuzzy ﬂow values for R2 in scenario 2. . . . . . . . 290 C.1 The performances of the suppliers to be sorted. . . . . . . . . . . . . . . . . . . 297 C.2 Flows of the suppliers with respect to the reference proﬁles and the corresponding assignments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 10
- 23. Introduction The subject of this PhD thesis is the use of multicriteria ranking methods in sorting problems. In sorting problems, a person, called decision maker, wants to assign an object, called action, to predeﬁned classes. On the other hand, a multicriteria ranking method is a method which ranks the actions from the best to the worst one while taking into account several (often conﬂicting) criteria. Sorting problems are known since the Antiquity. For instance, in the fourth century before Christ, the ancient Greek philosopher Epicurus sorted the human desires into two classes: vain desires (e.g. the desire of immortality) and natural desires (e.g. the desire of pleasure). This sorting was supposed to help people in ﬁnding a peaceful mood. Nowadays, sorting problems come up naturally in our daily life. A doctor for example will diagnose a patient on the basis of his symptoms. Based on his examination, he will assign the patient to a known pathology-class in order to prescribe the appropriate treatment. In enterprizes, projects are often sorted into priority-based categories. Recently, a study [Observador, 31th March 2008] showed that over 20 million Brazilians have moved from the lower social categories ("D" and "E") to category "C", the ﬁrst tier of the “middle class”, and are now active consumers due to an increase in legal employment. Hurricanes or cyclones are sorted into one of the 5 Safﬁr-Simpson categories based on their wind speed, superﬁcial pressure and tide-hight. Sorting aims thus to regroup actions with similar behaviors or characteristics for description, organizational or predictive purposes. The possible caused damage of cyclones can be evaluated using the Safﬁr-Sympson categories and the necessary protective measures can be taken. The decision maker thus deﬁnes the classes relatively to the consequences that will be given to the actions belonging to a same class. In this work, we emphasize problems where the decision maker expresses a preference relation among the classes. For instance, the human resources department might sort candidates in a recruitment process into promising people or into unadapted persons. Obviously, the human resources manager prefers the ﬁrst category. Analogously, projects of the highest priority class are considered to be of high potential for the company and will thus receive an immediate ﬁnancing and the necessary human support. Furthermore, we will consider that the expressed preference relation among categories might be partial. 11
- 24. Introduction In addition, we will suppose that the decision maker is able to deﬁne the categories by means of some norms or representative elements, called "reference proﬁles". The human resource manager is likely to have a clear idea of the proﬁle of a promising manager. The actions to be sorted will be compared to these proﬁles in order to determine their assignment. This comparison can be based either on a similarity index or on preference relations. Let us remark that the comparison of an action to the reference proﬁles is indeed independent from the other actions. Clearly, the diagnosis of a patient for instance does not affect the examination of another patient1 . Similarly, ranking problems arise frequently in our daily life. A company may want to evaluate its suppliers in order to develop special partnerships with the most promising ones. The father (mother) of a family might be interested in the best energy-supplier for his (her) house. Besides, no student can ignore the existence of university rankings. We have all been confronted once to the delicate task of ordering actions from the best to the worst one. However, it might be difﬁcult to obtain a complete order. We are then is presence of a partial order. The ranking problem is often somehow awkward and difﬁcult since it usually (unfortunately) involves conﬂicting points of view or criteria. Most of the time, no best candidate or supplier exists. Methods of the multicriteria decision aid ﬁeld, help a decision maker in this decision process by proposing step stones and techniques to ﬁnd a (compromise) solution. In ranking methods, the actions are thus compared pairwise or by means of, for example, an aggregated score. This constitutes a major difference with sorting methods since the actions to be sorted, are not compared to each other. Besides, nothing ensures the decision maker that the best ranked action is actually well suited for his problem. For instance, all candidates, even the best candidate, may not be adapted for a speciﬁc job. On the other hand, if an action is assigned to the best category, the decision maker might be sure that the action answers his needs. Obviously, this depends on a good a priori deﬁnition of the categories. This is another distinguishing feature between a ranking and a sorting method. In this dissertation, we will analyze the applicability of a multicriteria ranking method to assign a set of actions to predeﬁned categories. An action will be pairwise compared to the reference proﬁles by computing outranking or preference relations. On the basis of these comparisons, an action will be ranked with respect to the reference proﬁles. The assignment of the action will be deduced from its relative position with respect to solely the reference proﬁles. 1 We 12 are considering that the patients are not considered yet as examples of a training set.
- 25. Introduction Besides developing new sorting methods, we are interested in the properties these ranking-based sorting methods present. Are there some conditions to be imposed on, for example, the reference proﬁles ? Moreover, what are the differences with some existing methods ? Is there a speciﬁc reason or need to tackle sorting problems by ranking methods instead of sorting methods ? Is there an advantage in deﬁning the categories by means of criteria, even when there is no order on the categories ? Can existing partially ranking methods be used in problems where the categories are partially ordered ? In this work, we try to give a ﬁrst answer to these questions and to some emerging questions. This thesis is divided into 3 parts. Part I (Chapter 1 - Chapter 4) contains a review of the literature on multicriteria ranking and sorting methods. Our main contributions can be found in Part II (Chapter 5 - Chapter 11) and Part III (Chapter 12 and Chapter 13). In particular, Chapter 1 is devoted to a brief introduction to the classiﬁcation problem and the need of taking preference information into account. In Chapter 2, some aspects of the multicriteria decision ﬁeld are presented whereas Chapter 3 is devoted to the description of some well known multicriteria ranking methods. In Chapter 4, we propose a deep analysis of some multicriteria sorting methods which use reference proﬁles. The properties of these methods are compared as well as their approach. This leads to Tab.4.15 where the methods are compared. In Part II we use the Promethee ranking method for sorting problems where the categories are completely ordered. This leads to the F lowS ort method which uses preference relations. In Chapter 5, we precise the used notations and the conditions of the model. In Chapter 6 we deﬁne the assignment rules. The properties of F lowS ort are analyzed in Chapter 7. The F lowS ort method is compared theoretically and empirically to Electre-Tri and UTADIS in Chapter 8. Furthermore, F lowS ort is extended in Chapter 9 to the case where the parameters of the model are not precisely deﬁned. Finally, we present in Chapter 10 the implementation of F lowS ort in a user friendly software, developed by students. First conclusions on this approach are given in Chapter 11. In Part III we ﬁrst present a slightly modiﬁed version of Electre-Tri for the cases where the categories are deﬁned by central proﬁles instead of limiting ones. Moreover, this permits us to compare in Chapter 12 an outranking-based approach to similarity or indifference based sorting methods. Finally, in Chapter 13 we propose a ﬁrst investigation to treat problems where the categories are partially ordered. The particular problems where the categories are completely ordered and completely non-ordered are analyzed as well. In Conclusion, we discuss the most interesting aspects which still deserve to be further investigated. Finally, the proofs of the propositions as well as the numerical data of some examples can be found in the Appendix. 13
- 26. Resumé Notre thèse est consacrée à l’étude des méthodes de rangements multicritères dans le cadre de la problématique de tri. Dans un problème de tri une personne, appelée décideur, désire assigner un objet, appelé action, à des catégories prédéﬁnies. Des problèmes de tri surgissent régulièrement dans la vie de tous les jours. Par exemple, un médecin ausculte son patient et sur base des symptômes observés, il assigne son patient à une catégorie de pathologies. Ainsi, le médecin peut prescrire un traitement approprié. Par ailleurs, on catégorise les cyclones tropicaux en fonction de leur vitesse, pression superﬁcielle et de la hauteur de marée. En fonction de la catégorie du cyclone, des dégâts éventuels peuvent être prédits et des mesures de protection adéquates devront être prises. Dans un problème de tri, un décideur regroupe ainsi les actions qu’il considère similaires, à des ﬁns descriptives, organisationnelles ou préventives. Nous supposerons en outre que le décideur exprime une relation de préférence entre les classes préalablement déﬁnies. D’autre part, les méthodes de rangement permettent de ranger les actions de la meilleure à la moins bonne. Nul étudiant ne peut nier l’existence des " rankings " d’universités. Une société ordonne les candidats à l’issu d’un entretien d’embauche. Une société désire par ailleurs établir des partenariats avec les fournisseurs les plus performants. Nous sommes tous confrontés à cette tâche délicate de ranger les actions de la meilleure à la moins bonne. Les méthodes d’aide à la décision proposent des techniques permettant à un décideur d’obtenir un rangement d’actions. L’objectif de cette thèse est d’étudier la possibilité de résoudre des problèmes de tri à l’aide de méthodes de rangement. L’approche adoptée est de ranger une action particulière par rapport à des normes ou proﬁls déﬁnissant les catégories. L’assignation de l’action sera dès lors basée sur sa position dans ce rangement particulier. Quelles sont les hypothèses nécessaires pour un tel modèle ? Ces méthodes présentent-elles un biais ou ont-elles d’autres avantages par rapport aux méthodes de tri existantes? Est-il préférable de modéliser les catégories à l’aide de critères même si celles-ci ne présentent pas de relation de préférence ? Dans cette thèse nous donnerons des premièrs éléments de réponse en développant de nouvelles méthodes de tri basées sur des méthodes de rangement existantes. 15
- 27. Resumé 16
- 28. Publications and Conferences The research presented in this PhD thesis has lead to several publications in peer-reviewed journals and proceedings. Ph. Nemery, "A multicriteria sorting method for partially ordered categories", Proceedings of the doctoral work shop of EUROMOT 2008 - The Third European Conference on Management of Technology, "Industry-University Collaborations in Techno Parks", Nice Mareschal, B. and De Smet, Y. and Nemery, Ph.: "Rank reversal in the PROMETHEE II method: some new results", to appear in the procedeeings of the IEEE International Conference on Industrial Engineering and Engeneering Management Nemery, Ph. and Lamboray, Cl. : "FlowSort : a ﬂow-based sorting method with limiting and central proﬁles", TOP (Ofﬁcial Journal of the Spanish Society of Statistics and Operations Research), 16, 90-113, 2008 Nemery, Ph. and Lamboray, Cl. : "FlowSort : a sorting method based on ﬂows" in Proceedings of the ORP3 Conference, Guimarães, Portugal, 2007, p. 45-60 Besides, two papers are currently under review (both in 4OR). Two working papers have been published in proceedings of conferences, without peer-review: Ph. Nemery: "An outranking-based sorting method for partially ordered categories", DIMACS, Workshop and Meeting of the COST Action ICO602, Paris, Université Paris Dauphine, 28-31 October 2008 Cailloux, O. and Lamboray, Cl. and Nemery, Ph.: "A taxonomy of clustering procedures" in Proceedings of the 66th Meeting of the EWG on MCDA, Marrakech, Maroc, 2007 17
- 29. Publications and Conferences Most of the research results has also been presented in various conferences and seminars: Nemery, Ph.: "On the use of outranking relations in all classiﬁcation problems", Cost IC0602 International Doctoral School ; Algorithmic Decision Theory: MCDA, Data Mining and Rough Sets ; Session 2008 : April 11-16, 2008, Troina, Italy Nemery, Ph.: "FlowSort: a sorting method for group-decision making", Multiple Criteria Sorting Workshop, February 19, 2008, Université Paris Dauphine, invited speaker Nemery, Ph. and Janssen, P.: "A sorting method under uncertainty: extensions of FlowSort", ORBEL 22, Brussels, Belgium, 2008 Nemery, Ph.: "Extensions of the FlowSort sorting method for group decision-making", MCDM 2008 - 19th International Conference on Multiple Criteria Decision Making, Auckland, New-Zealand, 2008 Nemery, Ph.: "Resolving sorting problems with ranking methods" Cost IC0602 International Doctoral School, Han-sur-Lesse, Belgium, 2007 (pdf) Nemery, Ph. and Lamboray, Cl. : "FlowSort : a ﬂow-based sorting method with limiting and central proﬁles" ORP3, Guimarães, Portugal, 2007 Nemery, Ph. and Lamboray, Cl. : "FlowSort: a sorting method based on ﬂows: some extensions" 22nd EUROPEAN CONFERENCE on Operational Research, Prague, Czech Republic, 2007 Casier, A and De Smet, Y and Mareschal, B and Nemery Ph.: "About the interpretation of unicriterion net ﬂows in the PROMETHEE method" 22nd EUROPEAN CONFERENCE on Operational Research, Prague, Czech Republic, 2007 Nemery, Ph. and Lamboray, Cl. and Huenaerts, L: "FlowSort : a sorting method based on ﬂows" ORBEL 21, Luxembourg, January 2007 De Smet, Y. and Nemery Ph.: "The sorting problem based on disjunctive categories : a ﬁrst investigation", EURO 2006 Conference, Reykjavik, Iceland, July 2006 18
- 30. Notations • A = {a1 , . . . , an }: a set of n actions • F = { f1 , . . . , fq }: a set of q attributes • G = {g1 , . . . , gq }: a set of q criteria • Ω = {ω1 , . . . , ωq }: the set of weights associated to the q attributes or criteria • C = {C1 , . . . ,CK }: a set of K classes or categories • R = {r1 , . . . , rm }: a set of m reference proﬁles • CX (ai ): the set of classes or categories to which ai is assigned, according to procedure X • S(ai , a j ): the outranking degree of action ai over a j • π(ai , a j ): the preference degree of action ai over a j • Pk (ai , a j ): the uni-criterion preference degree of action ai over a j on criterion k • cS (ai , a j ): the partial outranking concordance degree of ai over a j on criterion k k • cI (ai , a j ): the partial indifference concordance degree of ai over a j on criterion k k S • dk (ai , a j ): the partial outranking discordance degree of ai over a j on criterion k I • dk (ai , a j ): the partial indifference discordance degree of ai over a j on criterion k • | {.} |: the cardinality of the set {.} • N: the set of natural numbers 19
- 31. Part State of the A rt 21
- 32. 1 Introduction to classiﬁcation problems In this chapter we give a brief introduction to the classiﬁcation paradigm and differentiate several existing grouping problems (clustering problems, ordinal and nominal classiﬁcation problems). Some well-known "classical" classiﬁcation methods are brieﬂy described. The need for taking into account preference information into the classiﬁcation method is furthermore intuitively proposed. 1.1 General Introduction Generally, we may deﬁne "to classify" by organizing data into groups which share common characteristics. Grouping problems have been extensively studied in the literature and are commonly encountered in various application ﬁelds such as health care, biology, ﬁnance, marketing, agriculture, etc. [Richard et al., 2001], [Doumpos and Zopounidis, 2002]. Many terms can be found such as problems of classiﬁcation, segmentation, discriminant analysis, ﬁltering, clustering, etc. Nevertheless, two major families of problems are usually distinguished: the supervised and the unsupervised grouping problems. In unsupervised groupings problems, there is no a priori information available about the groups (which are often called clusters in this context). The purpose is precisely to elicit a structure in a given data set. Generally, the aim is, in this context, to obtain different clusters of objects where objects of a same cluster are "similar" and objects of different clusters "dissimilar". The similitude notion is often expressed in terms of object proximity, distance, similarity or dissimilarity measures, etc. One might for instance consider a marketing problem where the aim is to discover similar customer behaviors in the retail industry which permits to detect different types of clients. In biology for instance, scientists regroup species of organisms, according to (for example) shared physical characteristics. This leads to a taxonomy of the species. In discriminant analysis, the most famous example is the Fisher’s Iris data set analysis. On the basis of four features (length and width of sepal and petal), three species (ﬂowers) of Iris may be determined (iris setosa, iris versicolor and iris virginica). Among the most common unsupervised grouping or clustering procedures, one may cite the 23
- 33. Introduction to classiﬁcation problems K-means, hierarchical, ﬁnite mixture densities algorithms, etc. [Hartigan, 1975], [Oliver et al., 1996],[Jain and R., 1998],[Jain et al., 1999],[Doreian et al., 2005a]. The reader will ﬁnd a survey of clustering procedures in [Hartigan, 1975],[Jain et al., 1999],[Cailloux et al., 2007],[Nemery, 2006]. On the other hand, the groups may be deﬁned a priori. C. Zopounidis and M. Doumpos deﬁne the (supervised) classiﬁcation problem as follows: "Classiﬁcation refers to the assignment of a ﬁnite set of alternatives into predeﬁned groups [Doumpos and Zopounidis, 2002]." The purpose is thus not to discover or elicit the groups, but to label objects according to the deﬁnition of the groups, called classes. We may think for instance about the medical diagnosis problem where a new patient has to be assigned to a known pathology-class based on a set of symptoms. In information science, documents are classiﬁed to one or more classes based on their contents. In Australia, the Ofﬁce of Film and Literature Classiﬁcation is a government funded organization which classiﬁes all ﬁlms that are released for public exhibition. There exists different classes such as the E-class (ﬁlms exempted of classiﬁcation, e.g. documentaries), the G-class (general ﬁlms with a content which is very mild in impact), the PG-class (ﬁlms for which parental guidance is recommended), the M-class (ﬁlms recommended for mature audience), etc. [Wikipedia, 2008]. A new ﬁlm will classiﬁed to one of these classes. In this work, we focus our interest on supervised classiﬁcation problems and we use abusively the term classiﬁcation instead of supervised classiﬁcation. The general idea behind classiﬁcation is thus to predict the class membership of a set of new objects on the basis of assignment rules. Most of the classiﬁcation methods, proposed for the development of classiﬁcation models, exploit the knowledge that is provided through the a priori deﬁnition of the groups. The model may be extracted from a set of classiﬁed examples, referred to as the the training sample or reference set and noted R = {r1 , . . . , rm }. This set consists of a collection of pairs: (an object, a class). The class label will be denoted by y, taking its values in the discrete set C = {C1 , . . . ,CK } where K is the number of classes. The objects, called actions or alternatives, are described by means of independent variables, noted g1 , . . . , gq . The set of variables is denoted by G . Henceforth, the independent variables will be referred to as criteria or attributes. The attributes, such as properties or characteristics, deﬁne a nominal description of the actions (e.g. a color, a measure, etc.) and allows to express (or measure) if two actions are similar. On the other hand, a criterion deﬁnes an ordinal description enabling to specify if an action is preferred over another1 . Each action of the training sample will be considered as a vector consisting of the performances of the action on each variable: r j = [g1 (r j ), . . . , gq (r j )]. The goal of the classiﬁcation model is to develop an application fG which maps any action, deﬁned by the vector of independent variables g, to the dependent variable, y, its classiﬁcation laˆ bel, where y ∈ C. Formally, we may represent the model as follows: G → C : g → y = fG (g) = C. This is illustrated in Fig.1.1 taken from [Doumpos and Zopounidis, 2002]. Let us remark, that an object may, according to some classiﬁcation procedures, be assigned to none, one or several 1A 24 more precise deﬁnition of the criteria concept will be given further in this work.
- 34. 1.1. General Introduction classes ( fG is thus not an application anymore). Figure 1.1 — Representation of the classiﬁcation model in classiﬁcation problem taken from [Doumpos and Zopounidis, 2002], p.7. The development of such a model is done such that the a priori classiﬁcation of the elements ˆ of the training sample (C), corresponds as much as possible to the estimated classiﬁcation (C). If the model performs "well" (i.e. if there is a high classiﬁcation adequacy), the model can be used for assigning a set of new objects, noted A = {a1 , . . . , an }, described by G to one or several categories. This is illustrated in Fig. 1.2 where ai actually represents g(ai ), ∀i = 1, . . . , n. In classiﬁcation problems, classes are predeﬁned, designed or conceived relatively to the treatment or the consequences that will be given to the actions belonging to a same group [Roy and 25
- 35. Introduction to classiﬁcation problems 9 C 6 A a1 • a •2 fG aj • a •i . z . . s a •3 a •n . 8. . C1 E Cj . . . CK 7 Figure 1.2 — Representation of a classiﬁcation procedure fG which assigns each action ai to none, one or several categories of the set C = {C1 , . . . ,CK }. Bouyssou, 1993]. Actions assigned to a same group, will thus be investigated, treated, used, etc. in a same manner. For instance, all the patients, assigned to a common known pathology-class, may receive a similar treatment. All the documents of a same class may geographically be regrouped in a library. A decision maker confronted to a classiﬁcation problem, needs to choose an assignment procedure adapted to his problem. The choice of this procedure is obviously crucial and may be inﬂuenced by several factors. Indeed the way of deﬁning the classes, the properties that the method should fulﬁll but also the meaning given to classes have to be considered before opting for a particular method. In the literature we usually distinguish classiﬁcation and sorting problems. C. Zopounidis and M. Doumpos mentioned that "Classiﬁcation refers to the case where the groups are deﬁned in a nominal way. On the contrary, sorting (a term which is used by multicriteria decision aiding researchers) refers to the case where the groups are deﬁned in an ordinal way, starting from those including the most preferred actions to those including the least preferred actions [Zopounidis and Doumpos, 2002a]." In this work we consider a nominal classiﬁcation problem as follows. The decision maker deﬁnes the classes such that he considers the actions belonging to different classes as dissimilar or not enough similar. Moreover, if he deﬁnes the classes by representative elements, he considers them as not similar (different, dissimilar). This opinion may be based on attributes providing a description of the classes or the representative elements. On the other hand, several authors consider sorting problems when the classes (called categories in this context) are deﬁned in an ordinal way: the categories are completely ordered from the best 26
- 36. 1.2. Classiﬁcation Methods to the worst [Doumpos and Zopounidis, 2004a]. As an example we may cite the different hotel categories where a four-star hotel is considered to be better than a one-star hotel. Chemicals may be sorted in different categories corresponding to different dangerousness. The revised Annex II Regulations for the control of pollution by noxious liquid substances includes a new four-category categorization system for noxious and liquid substances [Imo-Org, 2008]. In these previous examples, the categories are completely ordered. Let us remark, that in Chapter 4, we present a more exhaustive view of the different classiﬁcation problems. This chapter is devoted to nominal classiﬁcation methods. In Section 1.2 we describe brieﬂy some classical classiﬁcation methods. Furthermore, in Section 1.3, we present intuitively the needs for taking into account preference information in the classiﬁcation model. 1.2 Classiﬁcation Methods Methodologies for addressing classiﬁcation problems have been developed from a variety of research areas, such as statistics and econometrics, artiﬁcial intelligence, operations research, etc. Usually we distinguish two main families of classiﬁcation methods: the parameter-based and parameter-free techniques. In the former case, the classiﬁcation problem is addressed by statistical and econometric techniques using statistical assumptions on the data set. Among others, one may cite the linear discriminant analysis and the quadratic discriminant analysis (based on a priori probability distributions) [Fisher, 1939], the linear probability model, the logit and probit analysis (based on the development of a non-linear function measuring the group-membership probability) [Berkenson],[Bliss, 1934], etc. However, these techniques have been severely criticized for their statistical assumptions [Altman et al., 1981]. On the other hand, in parameter-free techniques no statistical assumptions are made. The methods will adjust themselves according to the characteristics of the data [Zopounidis and Doumpos, 2002b]. One may cite among others the neural networks [Culloh and Pitts, 1943; Zadeh and Nassery, 1999], machine learning [Goldberg, 1989], decision trees [Quinlan, 1986], fuzzy set theories [Zadeh, 1965], rough sets [Pawlak, 1984a], k-nearest neighbors [Fix and Hodges, 1951; Han and Kamber, 2001], etc. The reader may ﬁnd more information about these methods in [J.B.Mac Queen, 1966; Rulon et al., 1967; Gower and Legendre, 1986; Wallace and Dowe, 1994; Batagelj and Ferligoj, 1998; Lortie and Rizzo, 1999; Zopounidis and Doumpos, 2002b; Doumpos and Zopounidis, 2002].In the next section, we will brieﬂy describe one of these methods, namely the k-nearest neighbor. Nevertheless, previous classiﬁcation methods do not incorporate decision maker’s preferences. As we will in Section 1.3, this can play a crucial role in the assignment results. 27
- 37. Introduction to classiﬁcation problems 1.2.1 k-Nearest Neighbors This method has been initially introduced by [Fix and Hodges, 1951] and its mathematical properties have been given by [Hart, 1967]. Practical applications have been discussed by [Fukugana and Hummels, 1987]. In this method, the assignment of a action ai is based on the proximity of ai to the actions of the training set. The proximity is usually expressed by means of a distance, a (dis-) similarity measure or a proximity measure. Generally, a dissimilarity measure has the following properties [Doreian et al., 2005b] : ∀xi , x j ∈ A ∪ R : d (xi , x j ) → ℜ with 1. d (xi , xi ) = 0 2. d (xi , x j ) ≥ 0, non-negativity 3. d (xi , x j ) = d (x j , xi ), symmetry When the following conditions are also satisﬁed, the dissimilarity measure is called a distance : 4. d (xi , x j ) = 0 ⇒ xi = x j 5. ∀xz : d (xi , x j ) ≤ d (xi , xz ) + d (xz , x j ), triangle inequality For numerical data we can use the L p distance : d (xi , x j ) = xi − x j (1.1) p with 1/p q xi − x j p p = ( ∑ |gk (xi ) − gk (x j )| ) (1.2) k =1 where 1 ≤ p < ∞. The higher the values for p, the bigger the importance attached to the differences. For p=2, we ﬁnd the well-known Euclidean distance ; the Manhattan distance corresponds when p=1 and when p = ∞ it is equal to the maximum of absolute difference in coordinates. On the other hand, different similarity indexes can be deﬁned for numerical attributes such for example the similarity, the cosine and the Dice coefﬁcient as well as the distance exponent which are respectively given by the following formulas : s(xi , x j ) = 1/(1 + d (xi , x j )) (1.3) T scos (xi , x j ) = xi x j / xi . x j (1.4) T sDice = 2xi x j /( xi 2 + x j 2) (1.5) α (1.6) sexp = exp(− xi − x j 28 )
- 38. 1.2. Classiﬁcation Methods For categorical data, similarity measures can also be deﬁned [Everitt, 1993; Dubes; Jain et al., 1999]. Assuming binary attributes with values α, β = ±, let dαβ be a number of attributes having outcomes α in xi and β in x j . We can then deﬁne respectively the Rand (Eq.1.7) and Jaccard (Eq.1.8) indices : d++ + d−− R(xi , x j ) = (1.7) d++ + d+− + d−+ + d−− J (xi , x j ) = d++ d++ + d+− + d−+ (1.8) where d++ corresponds thus to the number of attributes for which xi and x j have the same response. From the training set, constituted by a set of training actions and their labels, the subset of the k (with k ∈ N) nearest training actions to ai (called the k neighbors) is extracted. The action ai is assigned to the class which is the most represented among the k neighbors. Let us consider Fig.1.3 where two classes, C1 and C2 , are deﬁned by the training set R = {r1 , . . . , r6 } and where {r1 , . . . , r3 } are representatives of C1 and {r4 , . . . , r6 } of C2 . Based on for instance the Euclidean distance, if we ﬁx k = 1, ai will be assigned to C2 (since d (ai , r5 ) < d (ai , r j ) with j = 5). On the other hand, if k = 3, ai will be assigned to C1 . g2 r6 r3 k=1 s ai r1 r2 r4 r5 k=3 g1 Figure 1.3 — Representation of the k-NN with thwo attributes g1 , g2 and with k = 1 and k = 3. Obviously, the assignment of ai depends on the number k and on the used distance or proximity measure. The determination of the appropriate number of neighbors is thus a crucial issue which has been addressed by [Bezdek, 1991]. Moreover, it can be useful to weight the contributions of the neighbors, so that the nearer neighbors contribute more to the average than the more distant ones. This extension can also be found in [Bezdek, 1991]. 29
- 39. Introduction to classiﬁcation problems A main advantage of this method lies indeed in its simplicity. Moreover, no assumption is needed on the data and some interesting optimality features have been proven in [Bezdek, 1991]. Nevertheless, it suffers from the drawback of the need of a high memory-space [Belacel, 2000a]. 1.3 Need of preference information In this section we brieﬂy motivate the need of taking preference information into account. This section aims not to formally deﬁne some concepts but rather to give a ﬁrst intuitive approach to the reader. As pointed out in previous section, classical classiﬁcation methods used statistical assumptions on the data, distances, similarity measures, etc. for assigning the actions to the categories. The used measures of the model are most of the time symmetrical or do not consider preference information. For simplicity reasons we consider in this section "preference information" as information on the basis of which a decision maker might express a preference of an action on another action. For instance, when comparing two actions a and b on the basis of the price, a client might prefer action a on b if the price-value of a is less than the price-value of b. The client aims to minimize the price. On the other hand, a vendor might prefer b since he would like to maximize his proﬁt. Minimizing or maximizing the values of the characteristics of the actions permits to establish an order on the set of characteristic values and thus to express a preference2 . To illustrate intuitively the impact of taking into account this preference information let us consider Fig.1.4 and 1.5. In the former case, we use so-called attributes ( f1 , f2 ) for describing the actions whereas in the latter case, so-called criteria (g1 , g2 ) where we suppose that the features of the objects have to be maximized. The main difference between attributes and criteria lies in the fact that we associate preference information to the features describing the actions (e.g. a decision maker’s preference orientation). Attributes and criteria will be precisely deﬁned in Chapter 2. In the ﬁrst case, we might use a similarity relation (or a distance) to compare the objects a1 , a2 , a3 , a4 and b. In the ﬁrst ﬁgure, when working with the attributes f1 and f2 , we can notice that all the points of the circle (with b as midpoint of the circle) are at the same Euclidean distance to point b. We can thus consider, that they are all similar or dissimilar. Let us now consider that the decision maker considers that both criteria have to be maximized. We can thus notice that a1 is, on both criteria, better than a2 , a3 , a4 , b. We will say that that a1 is preferred to a2 since it is better than a2 on both criteria. This will be note as follows: a1 a2 . We have thus moreover that a1 a j , with j = 2, 3, 4, and a1 b. Actions a2 , a3 and b are analogously preferred to a4 . On the other hand, the decision maker might not be able to compare 2 In 30 the next chapter, we deﬁne more formally the concept of preference information.
- 40. 1.3. Need of preference information a2 to a3 since a2 is better than a3 on criterion g2 and a3 is better on criterion g1 . Actions a2 and a3 are thus considered as incomparable (which is noted as follows: a3 J a2 ). f2 g2 T q a1 a2q q a1 a2q J b q q b J q a4 qa q 3 a4 f1 Figure 1.4 — Illustration of the utilization of attributes. qa 3 E g1 Figure 1.5 — Illustration of the utilization of criteria. We can thus remark that there exists three different "sub-zones" on the circle (with b as midpoint of the circle) although all points are at the same distance: the points "preferred by" b, represented in red (with for example a4 ), a zone of points which are incomparable to b (the brawn zone) and points which are preferred to b, represented in green (with for example a1 ). We can thus notice, that the fact of taking into account preference information, permits to reﬁne or precise the comparisons between the actions. The aim of sorting procedures is exactly to take into account this granularity introduced by the preference information. As illustration of this, let us consider the following basic classiﬁcation problem (represented in Fig.1.6) where four categories, noted Ci , (∀i = 1, . . . , 4), have been deﬁned by some typical representatives elements. These elements are called proﬁles and noted ri , ∀i = 1, . . . , 4. If we use a similarity relation or a distance, we can conclude from the left ﬁgure of Fig.1.6 that action a compares it-self in the same way to all the reference proﬁles. This can be motivated by the fact that the Euclidean distance are the same: d (ri , a) = d (r j , a), ∀i, j = 1, . . . 4). It might thus be difﬁcult to assign action a to a category rather than to another. On the contrary, if the decision maker considers that both criteria have to be maximized, he obtains the following relations: r1 a, r2 J a, r3 J a and a r4 (Fig.1.6-right). In this context, the decision maker might have his own reasons to assign a to a particular category considering the preference relations between a and the proﬁles ri . He might adopt for: • an optimistic approach by assigning a to category C1 since r4 , r3 , and r2 are not preferred to action a. • a compromise approach by assigning a to category C2 and C3 since a behaves as r2 and r3 do with respect to r1 and r4 . • an pessimistic approach by assigning a to category C4 since a is not preferred to r2 , r3 and r1 31
- 41. Introduction to classiﬁcation problems f2 g2 T q r1 r2 q q r1 r2 q J a a q q J r4 qr q 3 r4 qr q 3 f1 E g1 Figure 1.6 — Comparison of an action a to 4 different reference proﬁles ri , (i = 1, 2, 3, 4). • ... The approach will obviously be chosen by the decision maker and depend on the classiﬁcation model. Let us consider now, that the decision maker wants to minimize solely criterion g1 . We have thus other preference relations between a and the reference proﬁles (Fig.1.7-right): r2 is now preferred to a instead of incomparable, etc. The consequence of this, is that according the same adopted approach, the assignments might be different. g2 f2 T q r1 r2 q q r1 r2 q J a a q q J r4 q qr 3 r4 f1 q qr 3 g1 ' Figure 1.7 — Comparison of an action a to 4 different reference proﬁles ri , (i = 1, 2, 3, 4). We can thus conclude that the preference orientation plays a role in classiﬁcation. This will formally be deﬁned in Chapter 4 by a property of preference-orientation dependency. In this work, we will analyze classiﬁcation procedures which take preference information into account. In the next chapter, we deﬁne more formally the notions of preference, preference information, etc. 32
- 42. 2 Introduction to Multicriteria Decision Aid The aim of this chapter is to give a short introduction to the Multicriteria Decision Aid (MCDA) ﬁeld by presenting some important notions and deﬁnitions. It is of course impossible to propose an exhaustive bibliography of what has been done in almost thirty years, but it is not the purpose. Firstly, we address the motivations of "aiding" actors in making decisions in a complex world. This discipline offers, like suggested in the term-itself, an "aid". It does not necessarily "solve" a problem. As we will see, the philosophy of this young research ﬁeld can be seen as one of the logical continuations or extensions of the Operational Research (OR) branch. Nevertheless, it will certainly not supplant the Operational Research. The notions of actions, criteria, preference structures, etc. will be introduced and discussed in this chapter. The different type of problems encountered in MCDA will furthermore be brieﬂy explained. 2.1 Motivations Operational Research (OR) emerged just before the outbreak of World War II. General Pile, Commander in Chief of the Anti-Aircraft Command in Great Britain, requested scientiﬁc assistance for the coordination of the radar equipment at gun sites, which gave the slant rage and bearing an attacking bomber with some newly approaches. Concretely, radars had to be placed optimally to warn citizens of Great Britain of an eminent attack. Meanwhile, some other people became involved in problems concerning the detection of ships and submarines by the use of radar equipment in airplanes [Closkey and Trefethen, 1980; Pomerol and Barba-Romero, 1993]. Two years after the beginning of the war, Britain’s military services had acquired formally established operational research groups. The OR discipline has ﬁrst been applied on military problems (hence its name) but after a while it has known a huge expansion in the industrial world. The techniques and the developed methods have been successfully applied on several different problems (problems of industrial planning, transportation problems, scheduling problems, combinations problems, traveling 33
- 43. Introduction to Multicriteria Decision Aid problems, etc.) and are still used in many different ﬁelds. Concrete examples of applications can be for instance the determination of the shortest path between a source and destination, ﬁnding the optimal sequence of set of operations to be performed on some goods, resolving some highly constraints problems, etc. The classical OR problems were generally tackled by the modeling of the problems with a unique criterion function. The aim of the mono-criterion modeling is to obtain an optimization problem (maximization or minimization) under several constraints, which optimal solution represents the best choice. Where in the beginning, only one criterion function had to be optimized (for example the price, the distance and/or the cost), different aspects (like for example environmental, human, esthetics or power criteria) need to be considered when making a decision. Most of the time, we have to deal with conﬂicting criteria when facing a complex and global problem. For instance, when engaging a new employee, a company will have to choose between several candidates. The headhunter may be looking for an experienced candidate, with a high educational degree but who is still young. If the headhunter chooses the person with the highest experience, he will probably opt for an older candidate. Alternatively, a young candidate may present a lack of experience. The headhunter will have thus to make a compromise: no best solution exists. Moreover, when facing and comparing two alternatives or possible solutions, a decision maker can express a preference, an indifference or an incomparability. The incomparability can be due to the lack of information or to the fact that the solutions are too different. One interesting feature of MCDA is pointing out these two aforementioned situations which is difﬁcult to bring to the light when using an unique criterion-function. On the other hand, using an unique criterion-function or aggregating all the criteria to a unique and artiﬁcial value (what is done with the multi-attribute utility function) leads to the transitivity of preference and indifference. This can be severely criticized and can be refuted in the following situation. Consider 401 cups of coffee noted C0 , C1 , ..., C400 . One assumes that i the cup Ci contains exactly 1 + 100 grams of sugar. In this context, any normal person is unable to differentiate two successive cups. We have thus an indifference situation between C0 and C1 , C1 and C2 , C2 and C3 , ..., C399 and C400 . Nevertheless, like Luce explained in [Luce, 1956], it is obvious that no one will consider that C0 and C400 are indifferent to him since there is now a difference of 4 gram between both cups. The MCDA ﬁeld did not appear as the Messiah for solving all the complex problems of the world. The MCDA is an aiding tool for a multicriteria paradigm but certainly not a decision making one. There is thus a philosophical change in approaching the problem that can be found in marketing as well. In the beginning, the product was the central point in a marketing campaign. Nowadays, the client, and the associated services, are central. By the same way, the decision maker is now the central actor: the optimization function has been placed like the product, in the "back-yard". As we will see, MCDA will ﬁrst of all responsibilize the decision maker and make him aware of the aspects, the aims and/or the consequences related to the decision that he will make. This is, 34
- 44. 2.2. The actions, the criteria, . . . and the problems in our point of view, the ﬁrst purpose of the MCDA. It is certainly not to give a ﬁnal decision as different methods can propose different results. Moreover, it is necessary to explain the decision maker, the differences between the existing methods (hypothesis, advantages, disadvantages,...) and discuss with him which one may be the most appropriate for his problem. These are some of the roles of the analyst, who helps a decision maker, facing a problem. Although the decision maker is usually not conscious of it, the process of making a decision generally involves four phases. Moreover, the instant of decision can not (always) be identiﬁed. Schärlig distinguishes during this process the phases of information, conception, choice and retrospective analysis [Schärlig, 1985]. The order between these phases can of course be different and is often characterized by passages from one phase to another, depending on the progression of the consideration. The phase of information corresponds to the horizon seeing. The candidates for a solution are detected and the conceivable criteria are considered as well as. The phase of conception, on the other hand, allows to deﬁne the set of choice (i.e. the candidates to be determined) and their evaluations on the different chosen criteria. The decision making is done on the ﬁnal set of candidates, not necessarily corresponding to the initial one. The retrospective analysis is rarely done by a formal study but is certainly presented in the decision maker’s mind. 2.2 2.2.1 The actions, the criteria, . . . and the problems The set of actions A When facing a decision problem, the ﬁrst step may be to identify the different objects submitted to the decision making process. These objects can be potential decisions, projects, feasible solutions, items, units, alternatives, candidates, etc. and will be called the actions. The set of actions will be noted in the rest of this work A . A can be deﬁned in extenso (an enumeration of all the actions is thus possible: A = {a1 , ..., an }) or by comprehension ( mathematical properties or characterizations) when the set is too big or inﬁnite. As mentioned before, the decision process may be evolutive. This implicates that the actions are not always deﬁned once and for all. When the actions evolute, A is said to be evolutive. On the other hand, A is called stable when it is deﬁned a priori and will not change [Vincke, 1992]. Finally, A can be globalized, if each element excludes any other, or can be fragmented, if combinations of elements from A constitute possible issues [Vincke, 1992]. 2.2.2 The set of attributes F An attribute [Latin: attribuere ; attribut: ad- + tribuere: to allot [The Free Dictionary, 2008]] is a function f , deﬁned on A , taking its values in a set, noted V , ordered or not. It represents a feature or a characteristic inherent in or ascribed to an action [The Free Dictionary, 2008]. 35
- 45. Introduction to Multicriteria Decision Aid As several attributes will be considered, we will note f j the j-th attributes and vij = f j (ai ) the evaluation of the i-th action of A on this j-th attribute. The set of all the attributes will be noted F = { f1 , ..., fq }. 2.2.3 The set of criteria G The actions of a decision problem will be analyzed and evaluated according to the decision maker’s (DM’s) point of view and preferences. A criterion [Greek: kriterion, from krites, judge, from krinein: to separate, to judge [The Free Dictionary, 2008]] can be deﬁned as "A standard, a rule, or a test on which a judgment or decision can be based [The Free Dictionary, 2008]". Vincke deﬁnes formally a criterion as follows [Vincke, 1992]: Deﬁnition 2.1. A criterion is a function g, deﬁned on A , taking its values in a totally ordered set and representing the decision maker’s preferences according to some point of view. g : A → V where V is a totally ordered set If V is for instance the set of real-values, we suppose thus implicitly that the criterion has to be maximized. A more detailed and comprehensive deﬁnition can be found in [Roy and Bouyssou, 1993] where: A function g with real values deﬁned on A , is for a decision-maker a criterion-function or criterion ... if the decision maker recognizes the existence of an axis of signiﬁcance on which two possible actions ai and a j may be compared ... and he accepts to model this comparison as follows: g(ai ) ≥ g(a j ) ⇒ ai Sg a j ⇔ ai outranks a j on criterion g ⇔ ai is at least as good as a j on criterion g where Sg deﬁnes a binary outranking relation restricted to the signiﬁcation of criterion g. As several criteria will be considered, we will note g j the j-th criterion and eij = g j (ai ) the evaluation of the i-th action of A on this j-th criterion. An action ai will be represented by the following vector : ai ≡ [ei , ..., ei ]. Moreover, the set of all the criteria will be noted q 1 G = {g1 , ..., gq }. We will suppose, except of explicit counter-indication, that the criteria have to be maximized. As we will see in Section 2.5, the set of criteria has to respect some conditions. The complete characterization of the criterion (aspects, values, factors,...) is one of the most difﬁcult and crucial steps in a decision aiding process. Roy and Bouyssou has proposed a methodology to construct G as a set of coherent criteria. This will be presented in Section 2.5 [Roy and Bouyssou, 1993] . Let us remark that Vincke distinguishes, in [Vincke, 1992], several types of criteria such as real-criterion, quasi-criterion, pseudo-criterion and interval-criterion depending on the induced underlying preference structure. 36
- 46. 2.3. The Pareto dominance relation 2.2.4 The different types of MCDA problems When searching for an optimal solution in "traditional" problems, we model the situation such that the set of considered actions is ﬁxed once for all, such that every solution is exclusive from the others and such that solutions can be ranked incontestably from the worst to the best. However, the set of actions doesn’t necessarily fulﬁl this three characteristics. This is the reason why it is sometimes preferred to analyze the problem differently. Having the set of actions A and a set of criteria G , a decision maker may be facing different type-problems [Roy and Bouyssou, 1993]: • The choice problem : a subset of actions considered as the best according to the criterionset G , has to be chosen [the α − problem] • The sorting problem : a partition of the set A must be done with respect to some preestablished norms [the β − problem] • The ranking problem : a ranking of all the actions from the best to the worst must be realized [the γ − problem] • The description problem : a description, in an appropriate language, of the actions and their consequences has to be given [the δ − problem] Let us remark that other reference problems may be found in the literature (see for instance in [Bana e Costa, 1990],[Henriet, 2000]). Furthermore, real problems often combine simultaneously several of these problems as we can cite for example the portfolio problem, the design problem, choosing k among n actions [?], etc. Moreover, same problems may lead to different elaborations of A , G and different problematics [Vincke, 1992; Roy and Bouyssou, 1993]. Besides, as will explained later, the main idea of Part II of this work, is the using of an existing ranking method for tackling a sorting problem. To tackle these problems, we may use the Pareto dominance relation, deﬁne a preference structure, compare the actions pair-wise, etc. This will be the subject of the next sections. We will tackle the speciﬁc problems of ranking and sorting in the next chapters. 2.3 The Pareto dominance relation Let us ﬁrst, regardless of the sublying preference structure, the inter-criterion relations and the type of MCDA problem, deﬁne the following notions. Let us remind that we suppose that the criteria have to be maximized. 37

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