thesis of nemery philippe ph_d flowsort

Année Académique 2008-2009 Faculté des Sciences Appliquées

THÈSE

soutenue à l’Université Libre de BruxellesEn 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 BruxellesProf. Denis Bouyssou - Université Paris-DauphineProf. Bertrand Mareschal - Université Libre de BruxellesProf. Marc Pirlot - Faculté Polytechnique de Mons

"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. Etcela, 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. Maisdès qu’il y a deux caractéristiques, il n’existe plus de réponse. Par conséquent, dire "je suisplus 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 critèresde performance."

Albert Jacquard et Axel Kahn dans "L’avenir n’est pas écrit", p.28

Acknowledgment

The realization of this thesis is a long-term labour whose outcome is certainly due to thecontribution of several outstanding people. We would like to thank these persons ; not only fortheir effective contribution to this work but also for the patience they have showed during thesefour years of research.

After graduating, starting immediately by doing research was in our case, somewhat disconcert-ing. 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 Remicheand Professor Philippe Vincke for giving us the opportunity of doing research without imposingus 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, BertrandMareschal and Marc Pirlot for accepting to be part of the jury. Moreover, having ProfessorsHugues Bersini, Yves De Smet, Bertrand Mareschal, Marc Pirlot and Philippe Vincke duringour yearly accompaniment committees, certainly had a positive and fruitful contribution to theachievement of this thesis.

We would like to thank Professor Yves De Smet and Professor Marie-Ange Remiche for beingpresent these four years and supporting our moods during the difficulties encountered whenresearching. They have permit us to take some distance from our work and to keep two feet onthe ground.

We dedicate a special and warm regard to Professor Denis Bouyssou for his pertinent remarksand words of encouragement, when we really needed them.

We learned a lot from Professor Betrand Mareschal, both scientifically and personally, whileworking together on industrial multicriteria decision problems. Above all, he always has goodadvices for choosing wine.

All the members of the Service des Mathématiques de la Gestion are unforgettable. We wish tothank Olivier Cailloux, Aurélie Casier, Yves De Smet, Quantin Hayez, Claude Lamboray andKarim Lidouh for their judicious remarks and for the interesting conversations we had all along

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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 havebeen a very reliable and professional adminstrative staff. Their support and kindness duringthese 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 vieparisienne. Special and warm thanks to Professor Vincent Mousseau, Professor Alexis Tsoukiasand Wassila Ouerdane. Moreover, our stay at Paris has been possible given the financial supportsof the Cost IC0602 Action (STMS).

We are grateful to Iryna Yevseyeva, Claude, Michael, Pierre, Laurent, Slobodan, Nico andThomas 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 necessaryhindsight. Teaching courses, exercises and working with the students of our faculty, was a realpleasure. 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 collaborationand 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 sisterfor having supported us and our mood during these years and for accepting our decision to doresearch. Was it so terrible ?Besides, we are immensely thankful for Camille, who was present during these last years andwho 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, ouropinions and our insight towards research and towards multicriteria decision aid. That is, in ourpoint of view, the most important achievement.So, might a reader hesitate to do research, we have only one advice: Just try it !

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Contents

Acknowledgment v

Introduction 11

Resumé 15

Publications and Conferences 17

Notations 19

I State of the Art 21

1 Introduction to classification problems 23

1.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.2 Classification Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.2.1 k-Nearest Neighbors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.3 Need of preference information . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2 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

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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 Partial concordance degree cSj(b,a) . . . . . . . . . . . . . . . 41

2.6.1.2 Global concordance degree CS(b,a) . . . . . . . . . . . . . . 42

2.6.1.3 Partial discordance degree dSj (b,a) . . . . . . . . . . . . . . . 42

2.6.1.4 The outranking degree S(b,a) . . . . . . . . . . . . . . . . . . 43

2.6.2 Preference degree π(a,b) . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.6.2.1 Uni-criterion preference degree P j(a,b) . . . . . . . . . . . . 46

2.6.2.2 Global preference degree π(a,b) . . . . . . . . . . . . . . . . 47

3 Some multicriteria ranking methods 49

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Multi Attribute Utility Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2.1 The additive model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 Outranking methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.3.1 Electre III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.3.1.1 Preference relation between two actions . . . . . . . . . . . . 57

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 Rank Reversal phenomenon . . . . . . . . . . . . . . . . . . . 64

3.3.2 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 Other multicriteria ranking methods . . . . . . . . . . . . . . . . . . . . . . . . 75

3.5 How to choose a multicriteria ranking method ? . . . . . . . . . . . . . . . . . . 76

4 Some multicriteria sorting methods 81

4.1 Introduction to sorting problems . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.2 Properties of sorting methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

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4.3 Sorting based on indifference indexes . . . . . . . . . . . . . . . . . . . . . . . 88

4.3.1 Indifference Index I(a,b) . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.3.1.1 Partial indifference degree cIj(a,b) . . . . . . . . . . . . . . . 89

4.3.1.2 Partial discordance index dIj(a,b) . . . . . . . . . . . . . . . . 91

4.3.1.3 Global indifference index I(a,b) . . . . . . . . . . . . . . . . 92

4.3.2 PROAFTN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.3.2.2 Assignment rules . . . . . . . . . . . . . . . . . . . . . . . . 93

4.3.2.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.4 Sorting based on similarity indexes . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.4.1 Similarity Index SI(a,b) . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.4.1.1 Partial Similarity Index SI j(a,b) . . . . . . . . . . . . . . . . 95

4.4.1.2 Global Similarity Index SI(a,b) . . . . . . . . . . . . . . . . . 96

4.4.2 TRINOMFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 96


4.4.2.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.5 Sorting based on MAUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.5.1 UTADIS: UTilités Additives DIScriminantes . . . . . . . . . . . . . . . 98

4.5.1.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.5.2 MHDIS: Multi-group Hierarchical DIScrimination method . . . . . . . . 100

4.5.2.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.6 Sorting based on outranking relations . . . . . . . . . . . . . . . . . . . . . . . 101

4.6.1 Electre-Tri with limiting profiles . . . . . . . . . . . . . . . . . . . . . . 101

4.6.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 101


4.6.1.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.6.1.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.6.1.5 Graphical illustration . . . . . . . . . . . . . . . . . . . . . . 110

4.6.2 Trichotomic Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.6.2.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.6.2.2 Comparison with Electre-Tri . . . . . . . . . . . . . . . . . . 114

4.6.3 nTomic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.6.3.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.6.4 Filtering Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

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4.6.4.1 Filtering by strict preference . . . . . . . . . . . . . . . . . . 118

4.6.4.2 Filtering by indifference . . . . . . . . . . . . . . . . . . . . . 123

4.6.5 PairClass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.6.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.6.5.2 Assignment Rules . . . . . . . . . . . . . . . . . . . . . . . . 126

4.6.5.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

II F lowSort: a flow-based sorting method 133

5 Notation and conditions 137

5.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.2 Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6 Flow-based assignment procedures 141

6.1 Limiting profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.1.1 Strongly preferred limiting profiles . . . . . . . . . . . . . . . . . . . . 143

6.2 Central profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.2.1 Strongly preferred central profiles . . . . . . . . . . . . . . . . . . . . . 149

6.3 Influence of the preference parameters . . . . . . . . . . . . . . . . . . . . . . . 150

6.3.1 Limiting profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

6.3.2 Central profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

7 Analysis of some properties of F lowSort 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 Negative flow assignment rules . . . . . . . . . . . . . . . . . . . . . . . 166

7.7.2 Positive flow assignment rules . . . . . . . . . . . . . . . . . . . . . . . 167

7.8 Strong stability conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

7.8.1 Influence of Condition 6.1.1 on the stability. . . . . . . . . . . . . . . . . 173

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8 Comparison between F lowSort and some existing sorting methods 177

8.1 Comparison between F lowSort and Electre-Tri . . . . . . . . . . . . . . . . . . 177

8.1.1 Empirical comparison with Electre-Tri . . . . . . . . . . . . . . . . . . . 177

8.1.2 Intuitive comparison with Electre-Tri . . . . . . . . . . . . . . . . . . . 182

8.1.3 Impact of a simultaneous comparison . . . . . . . . . . . . . . . . . . . 184

8.1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

8.2 Comparison with the UTADIS model . . . . . . . . . . . . . . . . . . . . . . . 186

9 I nterval and F uzzy F lowSort 191

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

9.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

9.3 Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

9.4 I nterval F lowSort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

9.4.1 Limiting profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

9.4.2 Central profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

9.5 F uzzy F lowSort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

9.5.1 Fuzzy numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

10 Software and applications 211

11 Conclusions 219

III Outranking based sorting methods 223

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

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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.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.4 Proof proposition 9.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293



C Data of the application of chapter 10 297

D Proofs of Part III 299

D.1 Proof Proposition 12.3.2: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

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D.2 Proof Proposition 12.3.1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

D.3 Link between PROAFTN and Electre Tri . . . . . . . . . . . . . . . . . . . . . . 301

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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 tonone, 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. 29

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 cSj(b,a). . . . . . . . . . . . . . 42

2.2 Representation of the partial discordance index dSj (b,a). . . . . . . . . . . . . . 43

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 matrixobtained 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 fromA : illustration of the rank reversal phenomenon. [Lamsade, 2008]. . . . . . . . . 65

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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 A4. . . . . . . . . . . . . . . . 74

3.19 Representation of the ρ-relation. . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.1 Representation of the different classification problems on the basis of the differ-ent relations between the predefined groups. . . . . . . . . . . . . . . . . . . . . 82

4.2 Properties of sorting procedures according to the sorting problem where partic-ular 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 dIj(a,b). . . . . . . . . . . . . . . 91

4.6 Representation of the performances of the reference profiles r1 and r2 and theactions 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 Zo-pounidis, 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 L99 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

List of Figures

4.18 Assignment of any point (x,y) of the plan when considering the “union” of theoptimistic 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 positiveflows. 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

6.7 Assignment of any point (x,y) of the plan with the positive, negative and netflows 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 andp = 0.075, w1 = w2 = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.9 Assignment of any point (x,y) of the plan with the positive flows when q = 0.05and p = 0.075, w1 = w2 = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.10 Assignment of any point (x,y) of the plan with the negative flows when q = 0.05and p = 0.075, w1 = w2 = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.11 Assignment of any point (x,y) of the plan with the positive, negative and netflows 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 theworst category in case of equality. . . . . . . . . . . . . . . . . . . . . . . . . . 156

3

List of Figures

6.13 Assignment of any point (x,y) of the plan with the positive flows when q = 0.05and p = 0.075, w1 = w2 = 0.5 (right) and by choosing the worst category in caseof equality (left). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.14 Assignment of any point (x,y) of the plan with the negative flows when q = 0.05and p = 0.075, w1 = w2 = 0.5 (right) and by choosing the worst category in caseof equality (left). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.15 Assignment of any point (x,y) of the plan with the net flows when q = 0.05 andp = 0.075, w1 = w2 = 0.5 (right) and by choosing the worst category in case ofequality (left). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

7.1 Illustration of particular situation where π(ai, r2)≤ γ and π(r2,ai)≤ γ with γ = 0.5162

7.2 Illustration of the relationship between the assignments with limiting profiles andcentroids: case I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.3 Illustration of the relationship between the assignments with limiting profiles andcentroids: case II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

7.4 Representation of the relationship when defining reference profiles as either cen-tral (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 ’bet-ter’ category when using the negative flows: stability. . . . . . . . . . . . . . . . 168

8.1 Comparison of F lowSort and Electre-Tri : different scenarios. . . . . . . . . . . 178

8.2 Representation of the example of 8.1.2: assignments obtained with F lowSortwhich 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



9.1 Illustration of interval performances of the reference profiles and action a on onecriteria, 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 con-ditions 9.3.1-9.3.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

9.4 Illustration of the assignment rules when working with the positive flow intervals. 196

4

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 inscenario 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

9.9 Illustration of the fuzzy performances of the actions of R1 on criterion 1 in sce-nario 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 besorted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

10.3 Screen-shot of the software representing the evaluations of action a1 with respectto the reference profiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

10.4 Screen-shot of the software representing the positive and negative flow-plane foraction a1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

10.5 Screen-shot of the software representing all the actions assigned to C1 accordingto net flow assignment rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

10.6 Screen-shot of the software representing the distribution of the actions into thecategories 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 ac-cording 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 L99 y . . . . . . . . . . . . . . . . 230

12.3 The reduced "optimistic S-graph" and "pessimistic S-graph"when a is indifferentto more than one central profile. . . . . . . . . . . . . . . . . . . . . . . . . . . 233

12.4 Example of categories defined by central profiles (R = r1, r2, r3) and limitingprofiles (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 ; thelimiting 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

12.8 Case XII: Copt(a) = C12 and Cpess(a) = C1

3 . . . . . . . . . . . . . . . . . . . . . 242

13.1 Representation of partially ordered categories. . . . . . . . . . . . . . . . . . . . 246

13.2 Example of partially ordered reference profiles in the optimistic and pessimisticreduced "S-graph" where r1

1 = rI and r11 = rN . . . . . . . . . . . . . . . . . . . 247

5

List of Figures

13.3 Case I: Example of the optimistic and pessimistic reduced "S-graph" with a:Copt(a) = C1

1 and Cpess(a) = C22 . . . . . . . . . . . . . . . . . . . . . . . . . . 248

13.4 Representation of the performances of the central profiles r11, r1

2, r22 and r1

3 . . . . 249

13.5 Case II: Copt(a) = C11 and Cpess(a) = C1

3 . . . . . . . . . . . . . . . . . . . . . . 252

13.6 Case III: Copt(a) = C22 ∪C1

2 and Cpess(a) = C13 . . . . . . . . . . . . . . . . . . . 252

13.7 Case IV= Copt(a) = C22 and Cpess(a) = C2

2 . . . . . . . . . . . . . . . . . . . . . 252

13.8 Case V: Copt(a) = C22 ∪C1

2 and Cpess(a) = C22 ∪C1

2 . . . . . . . . . . . . . . . . 252

13.9 Case VI: Copt(a) = C22 ∪C1

2 and Cpess(a) = C12 . . . . . . . . . . . . . . . . . . . 253

13.10 Case VII: Copt(a) = C13 and Cpess(a) = C1

3 . . . . . . . . . . . . . . . . . . . . 253

13.11 Case VIII: Copt(a) = C11 and Cpess(a) = C1

3 . . . . . . . . . . . . . . . . . . . . 253

13.12 Case IX: Copt(a) = C11 and Cpess(a) = C1

1 . . . . . . . . . . . . . . . . . . . . . 254

13.13 Case X: Copt(a) = C13 and Cpess(a) = C1

3 . . . . . . . . . . . . . . . . . . . . . 254

13.14 Case XI: Copt(a) = C11 and Cpess(a) = C1

2 and C22 . . . . . . . . . . . . . . . . . 254

13.15 Case XII: Copt(a) = C12 and Cpess(a) = C1

3 . . . . . . . . . . . . . . . . . . . . 255

13.16 Representation of the ideal and nadir reference profile in case of nominal clas-sification problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

6

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


3.1 The performance evaluation matrix of A . . . . . . . . . . . . . . . . . . . . . . 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

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 Table8 in [Guitouni et al., 1999]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.1 Evaluation of the performances of the central reference profiles. . . . . . . . . . 94


4.3 Assignments of the actions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.4 Resume of the assignment results when using the Electre-Tri rules. . . . . . . . . 106


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

List of Tables

4.8 Assignment of the actions according to the different procedures. . . . . . . . . . 108


4.10 Assignment of the actions according to the different procedures: Copt ,Cpess andCT 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 andCFP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121


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. Thefollowing abbreviations are used: PO: partially ordered, CO: completely ordered,CNO: completely not ordered ; CP: central profile, LP: limiting profile ; Rel-Deg.: 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 lowSort. . . . . . 147



6.9 The preference degrees between the reference profiles and the alternatives. . . . . 151

6.10 The flow-values of the alternatives. . . . . . . . . . . . . . . . . . . . . . . . . . 151



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 preferredlimiting profiles when working with the positive flows. . . . . . . . . . . . . . . 175

8.1 Comparison between F lowSort and Electre-Tri in the case of limiting profiles. . 179

8.2 Comparison between F lowSort and Electre-Tri in the case of limiting profiles:analysis of the assignments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

8

List of Tables

8.3 Comparison between F lowSort and Electre-Tri in the case of limiting profilesverifying 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


9.3 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 am2 . . . . . . . . . . . . . . . . . . . 203

9.9 Computation of the different fuzzy flow values for R1 in scenario 1. . . . . . . . 208




9.13 Results of the assignments of the action a2 according to the different rules inscenario 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 andlimiting profiles). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

12.4 Assignment results when defining C2 by two reference profiles r12 and r2

2. . . . . . 241

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 andPessimistic 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

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




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 correspond-ing assignments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

10

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, topredefined classes. On the other hand, a multicriteria ranking method is a method which ranksthe 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 beforeChrist, the ancient Greek philosopher Epicurus sorted the human desires into two classes: vaindesires (e.g. the desire of immortality) and natural desires (e.g. the desire of pleasure). Thissorting was supposed to help people in finding a peaceful mood.Nowadays, sorting problems come up naturally in our daily life. A doctor for example willdiagnose a patient on the basis of his symptoms. Based on his examination, he will assign thepatient to a known pathology-class in order to prescribe the appropriate treatment. In enter-prizes, 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 socialcategories ("D" and "E") to category "C", the first tier of the “middle class”, and are now activeconsumers due to an increase in legal employment. Hurricanes or cyclones are sorted into one ofthe 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 evaluatedusing the Saffir-Sympson categories and the necessary protective measures can be taken. Thedecision maker thus defines the classes relatively to the consequences that will be given to theactions belonging to a same class.

In this work, we emphasize problems where the decision maker expresses a preference relationamong the classes. For instance, the human resources department might sort candidates in arecruitment process into promising people or into unadapted persons. Obviously, the humanresources manager prefers the first category. Analogously, projects of the highest priority classare considered to be of high potential for the company and will thus receive an immediatefinancing and the necessary human support.Furthermore, we will consider that the expressed preference relation among categories might bepartial.

11

Introduction

In addition, we will suppose that the decision maker is able to define the categories by meansof some norms or representative elements, called "reference profiles". The human resourcemanager 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 assign-ment. This comparison can be based either on a similarity index or on preference relations. Letus remark that the comparison of an action to the reference profiles is indeed independent fromthe other actions. Clearly, the diagnosis of a patient for instance does not affect the examinationof another patient1.

Similarly, ranking problems arise frequently in our daily life. A company may want to evaluateits 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 tothe delicate task of ordering actions from the best to the worst one. However, it might be difficultto 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 supplierexists. Methods of the multicriteria decision aid field, help a decision maker in this decisionprocess by proposing step stones and techniques to find a (compromise) solution. In rankingmethods, the actions are thus compared pairwise or by means of, for example, an aggregatedscore. 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 suitedfor his problem. For instance, all candidates, even the best candidate, may not be adapted fora specific job. On the other hand, if an action is assigned to the best category, the decisionmaker might be sure that the action answers his needs. Obviously, this depends on a good apriori definition of the categories. This is another distinguishing feature between a ranking anda sorting method.

In this dissertation, we will analyze the applicability of a multicriteria ranking method to assigna set of actions to predefined categories. An action will be pairwise compared to the referenceprofiles by computing outranking or preference relations. On the basis of these comparisons, anaction will be ranked with respect to the reference profiles. The assignment of the action will bededuced from its relative position with respect to solely the reference profiles.

1We are considering that the patients are not considered yet as examples of a training set.

12

Introduction

Besides developing new sorting methods, we are interested in the properties these ranking-basedsorting methods present. Are there some conditions to be imposed on, for example, the referenceprofiles ? Moreover, what are the differences with some existing methods ? Is there a specificreason or need to tackle sorting problems by ranking methods instead of sorting methods ? Isthere an advantage in defining the categories by means of criteria, even when there is no order onthe categories ? Can existing partially ranking methods be used in problems where the categoriesare 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 theliterature on multicriteria ranking and sorting methods. Our main contributions can be found inPart 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 andthe need of taking preference information into account. In Chapter 2, some aspects of themulticriteria decision field are presented whereas Chapter 3 is devoted to the description of somewell known multicriteria ranking methods. In Chapter 4, we propose a deep analysis of somemulticriteria sorting methods which use reference profiles. The properties of these methods arecompared 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 arecompletely ordered. This leads to the F lowSort method which uses preference relations. InChapter 5, we precise the used notations and the conditions of the model. In Chapter 6 we definethe assignment rules. The properties of F lowSort are analyzed in Chapter 7. The F lowSortmethod is compared theoretically and empirically to Electre-Tri and UTADIS in Chapter 8.Furthermore, F lowSort is extended in Chapter 9 to the case where the parameters of the modelare not precisely defined. Finally, we present in Chapter 10 the implementation of F lowSort ina user friendly software, developed by students. First conclusions on this approach are given inChapter 11.

In Part III we first present a slightly modified version of Electre-Tri for the cases where thecategories are defined by central profiles instead of limiting ones. Moreover, this permits us tocompare in Chapter 12 an outranking-based approach to similarity or indifference based sortingmethods. Finally, in Chapter 13 we propose a first investigation to treat problems where thecategories are partially ordered. The particular problems where the categories are completelyordered and completely non-ordered are analyzed as well.

In Conclusion, we discuss the most interesting aspects which still deserve to be further investi-gated. Finally, the proofs of the propositions as well as the numerical data of some examples canbe found in the Appendix.

13

Resumé

Notre thèse est consacrée à l’étude des méthodes de rangements multicritères dans le cadre dela 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 tousles jours. Par exemple, un médecin ausculte son patient et sur base des symptômes observés, ilassigne son patient à une catégorie de pathologies. Ainsi, le médecin peut prescrire un traitementapproprié. Par ailleurs, on catégorise les cyclones tropicaux en fonction de leur vitesse, pressionsuperficielle et de la hauteur de marée. En fonction de la catégorie du cyclone, des dégâtséventuels 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 desfins descriptives, organisationnelles ou préventives. Nous supposerons en outre que le décideurexprime 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 lamoins bonne. Nul étudiant ne peut nier l’existence des " rankings " d’universités. Une sociétéordonne les candidats à l’issu d’un entretien d’embauche. Une société désire par ailleurs établirdes partenariats avec les fournisseurs les plus performants. Nous sommes tous confrontés à cettetâche délicate de ranger les actions de la meilleure à la moins bonne. Les méthodes d’aide à ladécision 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 deméthodes de rangement. L’approche adoptée est de ranger une action particulière par rapport àdes normes ou profils définissant les catégories. L’assignation de l’action sera dès lors basée sursa position dans ce rangement particulier.

Quelles sont les hypothèses nécessaires pour un tel modèle ? Ces méthodes présentent-elles unbiais ou ont-elles d’autres avantages par rapport aux méthodes de tri existantes? Est-il préférablede modéliser les catégories à l’aide de critères même si celles-ci ne présentent pas de relation depréférence ? Dans cette thèse nous donnerons des premièrs éléments de réponse en développantde nouvelles méthodes de tri basées sur des méthodes de rangement existantes.

15

Resumé

16

Publications and Conferences

The research presented in this PhD thesis has lead to several publications in peer-reviewedjournals and proceedings.

Ph. Nemery, "A multicriteria sorting method for partially ordered categories", Proceed-ings of the doctoral work shop of EUROMOT 2008 - The Third European Conferenceon Management of Technology, "Industry-University Collaborations in Techno Parks", Nice

Mareschal, B. and De Smet, Y. and Nemery, Ph.: "Rank reversal in the PROMETHEEII method: some new results", to appear in the procedeeings of the IEEE InternationalConference on Industrial Engineering and Engeneering Management

Nemery, Ph. and Lamboray, Cl. : "FlowSort : a flow-based sorting method with limiting andcentral profiles", TOP (Official Journal of the Spanish Society of Statistics and OperationsResearch), 16, 90-113, 2008

Nemery, Ph. and Lamboray, Cl. : "FlowSort : a sorting method based on flows" in Proceed-ings 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 ParisDauphine, 28-31 October 2008

Cailloux, O. and Lamboray, Cl. and Nemery, Ph.: "A taxonomy of clustering procedures" inProceedings of the 66th Meeting of the EWG on MCDA, Marrakech, Maroc, 2007

17

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 IC0602International Doctoral School ; Algorithmic Decision Theory: MCDA, Data Mining andRough Sets ; Session 2008 : April 11-16, 2008, Troina, Italy

Nemery, Ph.: "FlowSort: a sorting method for group-decision making", Multiple CriteriaSorting 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, Auck-land, New-Zealand, 2008

Nemery, Ph.: "Resolving sorting problems with ranking methods" Cost IC0602 InternationalDoctoral School, Han-sur-Lesse, Belgium, 2007 (pdf)

Nemery, Ph. and Lamboray, Cl. : "FlowSort : a flow-based sorting method with limiting andcentral profiles" ORP3, Guimarães, Portugal, 2007

Nemery, Ph. and Lamboray, Cl. : "FlowSort: a sorting method based on flows: some exten-sions" 22nd EUROPEAN CONFERENCE on Operational Research, Prague, Czech Repub-lic, 2007

Casier, A and De Smet, Y and Mareschal, B and Nemery Ph.: "About the interpretation ofunicriterion net flows in the PROMETHEE method" 22nd EUROPEAN CONFERENCE onOperational Research, Prague, Czech Republic, 2007

Nemery, Ph. and Lamboray, Cl. and Huenaerts, L: "FlowSort : a sorting method based onflows" ORBEL 21, Luxembourg, January 2007

De Smet, Y. and Nemery Ph.: "The sorting problem based on disjunctive categories : a firstinvestigation", EURO 2006 Conference, Reykjavik, Iceland, July 2006

18

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

• cSk(ai,a j): the partial outranking concordance degree of ai over a j on criterion k

• cIk(ai,a j): the partial indifference concordance degree of ai over a j on criterion k

• dSk (ai,a j): the partial outranking discordance degree of ai over a j on criterion k

• dIk(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

Part

State of the Art

21

1 Introduction to classificationproblems

In this chapter we give a brief introduction to the classification paradigm anddifferentiate several existing grouping problems (clustering problems, ordinaland nominal classification problems). Some well-known "classical" classifica-tion methods are briefly described. The need for taking into account preferenceinformation 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 commoncharacteristics. Grouping problems have been extensively studied in the literature and arecommonly encountered in various application fields such as health care, biology, finance,marketing, agriculture, etc. [Richard et al., 2001], [Doumpos and Zopounidis, 2002]. Manyterms 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 structurein a given data set. Generally, the aim is, in this context, to obtain different clusters of objectswhere 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 ordissimilarity measures, etc.One might for instance consider a marketing problem where the aim is to discover similarcustomer behaviors in the retail industry which permits to detect different types of clients. Inbiology for instance, scientists regroup species of organisms, according to (for example) sharedphysical characteristics. This leads to a taxonomy of the species. In discriminant analysis, themost famous example is the Fisher’s Iris data set analysis. On the basis of four features (lengthand width of sepal and petal), three species (flowers) of Iris may be determined (iris setosa, irisversicolor and iris virginica).Among the most common unsupervised grouping or clustering procedures, one may cite the

23

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 surveyof 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 definethe (supervised) classification problem as follows: "Classification refers to the assignment of afinite set of alternatives into predefined groups [Doumpos and Zopounidis, 2002]." The purposeis thus not to discover or elicit the groups, but to label objects according to the definition of thegroups, called classes. We may think for instance about the medical diagnosis problem wherea new patient has to be assigned to a known pathology-class based on a set of symptoms. Ininformation science, documents are classified to one or more classes based on their contents. InAustralia, the Office of Film and Literature Classification is a government funded organizationwhich classifies all films that are released for public exhibition. There exists different classessuch as the E-class (films exempted of classification, e.g. documentaries), the G-class (generalfilms with a content which is very mild in impact), the PG-class (films for which parental guid-ance 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 intereston supervised classification problems and we use abusively the term classification instead ofsupervised classification.

The general idea behind classification is thus to predict the class membership of a set of newobjects on the basis of assignment rules. Most of the classification methods, proposed for thedevelopment of classification models, exploit the knowledge that is provided through the a prioridefinition of the groups. The model may be extracted from a set of classified examples, referredto as the the training sample or reference set and noted R = r1, . . . ,rm. This set consists of acollection of pairs: (an object, a class).

The class label will be denoted by y, taking its values in the discrete set C = C1, . . . ,CK whereK is the number of classes. The objects, called actions or alternatives, are described by meansof 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 asproperties 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 criteriondefines an ordinal description enabling to specify if an action is preferred over another1. Eachaction of the training sample will be considered as a vector consisting of the performances ofthe 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 la-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, thatan object may, according to some classification procedures, be assigned to none, one or several

1A more precise definition of the criteria concept will be given further in this work.

24

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 elementsof 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 beused for assigning a set of new objects, noted A = a1, . . . ,an, described by G to one or severalcategories. 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 treat-ment or the consequences that will be given to the actions belonging to a same group [Roy and

25


C1

C j

'

&

$

%

a1

a j

ai

a2

a3

an

. . .

...

...

•

•

•

••

•

A C

CK

s

fG

-

z

Figure 1.2 — Representation of a classification procedure fG which assigns each action ai tonone, 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 beregrouped in a library.

A decision maker confronted to a classification problem, needs to choose an assignmentprocedure adapted to his problem. The choice of this procedure is obviously crucial and maybe influenced by several factors. Indeed the way of defining the classes, the properties thatthe method should fulfill but also the meaning given to classes have to be considered beforeopting for a particular method. In the literature we usually distinguish classification and sortingproblems. C. Zopounidis and M. Doumpos mentioned that "Classification refers to the casewhere the groups are defined in a nominal way. On the contrary, sorting (a term which is usedby multicriteria decision aiding researchers) refers to the case where the groups are defined inan ordinal way, starting from those including the most preferred actions to those including theleast preferred actions [Zopounidis and Doumpos, 2002a]."

In this work we consider a nominal classification problem as follows. The decision maker definesthe classes such that he considers the actions belonging to different classes as dissimilar or notenough similar. Moreover, if he defines the classes by representative elements, he considersthem as not similar (different, dissimilar). This opinion may be based on attributes providing adescription of the classes or the representative elements.

On the other hand, several authors consider sorting problems when the classes (called categoriesin this context) are defined in an ordinal way: the categories are completely ordered from the best

26

1.2. Classification Methods

to the worst [Doumpos and Zopounidis, 2004a]. As an example we may cite the different hotelcategories where a four-star hotel is considered to be better than a one-star hotel. Chemicalsmay be sorted in different categories corresponding to different dangerousness. The revisedAnnex II Regulations for the control of pollution by noxious liquid substances includes a newfour-category categorization system for noxious and liquid substances [Imo-Org, 2008]. In theseprevious examples, the categories are completely ordered.Let us remark, that in Chapter 4, we present a more exhaustive view of the different classificationproblems.

This chapter is devoted to nominal classification methods. In Section 1.2 we describe brieflysome classical classification methods. Furthermore, in Section 1.3, we present intuitively theneeds 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 ofresearch areas, such as statistics and econometrics, artificial intelligence, operations research, etc.

Usually we distinguish two main families of classification methods: the parameter-basedand parameter-free techniques. In the former case, the classification problem is addressed bystatistical and econometric techniques using statistical assumptions on the data set. Amongothers, one may cite the linear discriminant analysis and the quadratic discriminant analysis(based on a priori probability distributions) [Fisher, 1939], the linear probability model, thelogit and probit analysis (based on the development of a non-linear function measuring thegroup-membership probability) [Berkenson],[Bliss, 1934], etc. However, these techniques havebeen severely criticized for their statistical assumptions [Altman et al., 1981].

On the other hand, in parameter-free techniques no statistical assumptions are made. Themethods will adjust themselves according to the characteristics of the data [Zopounidis andDoumpos, 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 andHodges, 1951; Han and Kamber, 2001], etc.

The reader may find more information about these methods in [J.B.Mac Queen, 1966; Rulonet 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].Inthe 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


1.2.1 k-Nearest Neighbors

This method has been initially introduced by [Fix and Hodges, 1951] and its mathematicalproperties 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 ofthe training set. The proximity is usually expressed by means of a distance, a (dis-) similaritymeasure 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) 7→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 Lp distance :

d(xi,x j) = ‖xi− x j‖p (1.1)

with

‖xi− x j‖p = (q

∑k=1|gk(xi)−gk(x j)|p)

1/p

(1.2)

where 1 ≤ p < ∞. The higher the values for p, the bigger the importance attached to thedifferences. For p=2, we find the well-known Euclidean distance ; the Manhattan distancecorresponds when p=1 and when p = ∞ it is equal to the maximum of absolute difference incoordinates.

On the other hand, different similarity indexes can be defined for numerical attributes such forexample the similarity, the cosine and the Dice coefficient as well as the distance exponent whichare respectively given by the following formulas :

s(xi,x j) = 1/(1 + d(xi,x j)) (1.3)

scos(xi,x j) = xTi x j/‖xi‖.‖x j‖ (1.4)

sDice = 2xTi x j/(‖xi‖2 + ‖x j‖2) (1.5)

sexp = exp(−‖xi− x j‖α) (1.6)

28

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 havingoutcomes α in xi and β in x j. We can then define respectively the Rand (Eq.1.7) and Jaccard(Eq.1.8) indices :

R(xi,x j) =d++ + d−−

d++ + d+−+ d−+ + d−−(1.7)

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 sameresponse.

From the training set, constituted by a set of training actions and their labels, the subset of thek (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 considerFig.1.3 where two classes, C1 and C2, are defined by the training set R = r1, . . . ,r6 and wherer1, . . . ,r3 are representatives of C1 and r4, . . . ,r6 of C2. Based on for instance the Euclideandistance, if we fix k = 1, ai will be assigned to C2 (since d(ai,r5) < d(ai,r j) with j 6= 5). On theother hand, if k = 3, ai will be assigned to C1.

g1

g2

?

sai

r6

?r1

?r4

r5?

?r3

?r2

k = 3

k = 1

Figure 1.3 — Representation of the k-NN with thwo attributes g1, g2 and with k = 1 andk = 3.

Obviously, the assignment of ai depends on the number k and on the used distance or proximitymeasure. The determination of the appropriate number of neighbors is thus a crucial issue whichhas been addressed by [Bezdek, 1991]. Moreover, it can be useful to weight the contributions ofthe neighbors, so that the nearer neighbors contribute more to the average than the more distantones. This extension can also be found in [Bezdek, 1991].

29


A main advantage of this method lies indeed in its simplicity. Moreover, no assumption is neededon the data and some interesting optimality features have been proven in [Bezdek, 1991]. Nev-ertheless, 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. Thissection aims not to formally define some concepts but rather to give a first intuitive approach tothe reader.

As pointed out in previous section, classical classification methods used statistical assumptionson the data, distances, similarity measures, etc. for assigning the actions to the categories. Theused measures of the model are most of the time symmetrical or do not consider preferenceinformation.

For simplicity reasons we consider in this section "preference information" as information onthe 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 preferaction a on b if the price-value of a is less than the price-value of b. The client aims to minimizethe 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 anorder 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 usconsider Fig.1.4 and 1.5. In the former case, we use so-called attributes ( f1, f2) for describing theactions whereas in the latter case, so-called criteria (g1,g2) where we suppose that the featuresof the objects have to be maximized. The main difference between attributes and criteria liesin 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 inChapter 2.

In the first case, we might use a similarity relation (or a distance) to compare the objectsa1,a2,a3,a4 and b. In the first figure, when working with the attributes f1 and f2, we can noticethat all the points of the circle (with b as midpoint of the circle) are at the same Euclideandistance 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 ispreferred 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 areanalogously preferred to a4. On the other hand, the decision maker might not be able to compare

2In the next chapter, we define more formally the concept of preference information.

30

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 anda3 are thus considered as incomparable (which is noted as follows: a3J a2).

b

a1a2

a3a4

qqqqq

f1

f2

Figure 1.4 — Illustration of the utilization of attributes.

b

a1a2

a3a4

- g1

6

g2

J

J

qqqqq

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 midpointof the circle) although all points are at the same distance: the points "preferred by" b, representedin 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 thusnotice, that the fact of taking into account preference information, permits to refine or precisethe comparisons between the actions.

The aim of sorting procedures is exactly to take into account this granularity introduced by thepreference information. As illustration of this, let us consider the following basic classificationproblem (represented in Fig.1.6) where four categories, noted Ci, (∀i = 1, . . . ,4), have beendefined by some typical representatives elements. These elements are called profiles and notedri, ∀i = 1, . . . ,4. If we use a similarity relation or a distance, we can conclude from the left figureof Fig.1.6 that action a compares it-self in the same way to all the reference profiles. This can bemotivated 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, heobtains the following relations: r1 a, r2J a, r3J a and a r4 (Fig.1.6-right). In this context, thedecision maker might have his own reasons to assign a to a particular category considering thepreference 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 preferredto action a.

• a compromise approach by assigning a to category C2 and C3 since a behaves as r2 and r3do with respect to r1 and r4.

• an pessimistic approach by assigning a to category C4 since a is not preferred to r2, r3 andr1

31


a

r1r2

r3r4

qqqqq

f1

f2

a

r1r2

r3r4

- g1

6

g2

J

J

qqqqq

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 classificationmodel.

Let us consider now, that the decision maker wants to minimize solely criterion g1. We havethus other preference relations between a and the reference profiles (Fig.1.7-right): r2 is nowpreferred to a instead of incomparable, etc. The consequence of this, is that according the sameadopted approach, the assignments might be different.

a

r1r2

r3r4

qqqqq

f1

f2

a

r1r2

r3r4

J

J

qq

qqq

6

g1

g2

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 willformally be defined in Chapter 4 by a property of preference-orientation dependency.

In this work, we will analyze classification procedures which take preference information intoaccount. In the next chapter, we define more formally the notions of preference, preference in-formation, etc.

32

2 Introduction to MulticriteriaDecision Aid

The aim of this chapter is to give a short introduction to the Multicriteria Deci-sion Aid (MCDA) field by presenting some important notions and definitions.It is of course impossible to propose an exhaustive bibliography of what hasbeen done in almost thirty years, but it is not the purpose.Firstly, we address the motivations of "aiding" actors in making decisions ina 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 oneof 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 introducedand discussed in this chapter. The different type of problems encountered inMCDA 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 scientificassistance for the coordination of the radar equipment at gun sites, which gave the slant rage andbearing an attacking bomber with some newly approaches. Concretely, radars had to be placedoptimally to warn citizens of Great Britain of an eminent attack. Meanwhile, some other peoplebecame involved in problems concerning the detection of ships and submarines by the use ofradar 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 formallyestablished operational research groups.

The OR discipline has first been applied on military problems (hence its name) but after awhile it has known a huge expansion in the industrial world. The techniques and the developedmethods have been successfully applied on several different problems (problems of industrialplanning, transportation problems, scheduling problems, combinations problems, traveling

33

Introduction to Multicriteria Decision Aid

problems, etc.) and are still used in many different fields. Concrete examples of applications canbe for instance the determination of the shortest path between a source and destination, findingthe optimal sequence of set of operations to be performed on some goods, resolving some highlyconstraints problems, etc.

The classical OR problems were generally tackled by the modeling of the problems with aunique criterion function. The aim of the mono-criterion modeling is to obtain an optimizationproblem (maximization or minimization) under several constraints, which optimal solutionrepresents 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, estheticsor power criteria) need to be considered when making a decision. Most of the time, we have todeal with conflicting criteria when facing a complex and global problem. For instance, whenengaging a new employee, a company will have to choose between several candidates. Theheadhunter may be looking for an experienced candidate, with a high educational degree butwho is still young. If the headhunter chooses the person with the highest experience, he willprobably opt for an older candidate. Alternatively, a young candidate may present a lack ofexperience. 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 makercan express a preference, an indifference or an incomparability. The incomparability can be dueto 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 isdifficult 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 aunique and artificial value (what is done with the multi-attribute utility function) leads to thetransitivity of preference and indifference. This can be severely criticized and can be refuted inthe following situation. Consider 401 cups of coffee noted C0, C1, ..., C400. One assumes thatthe cup Ci contains exactly 1 + i

100 grams of sugar. In this context, any normal person is unableto 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 isobvious that no one will consider that C0 and C400 are indifferent to him since there is now adifference of 4 gram between both cups.

The MCDA field did not appear as the Messiah for solving all the complex problems of theworld. The MCDA is an aiding tool for a multicriteria paradigm but certainly not a decisionmaking one. There is thus a philosophical change in approaching the problem that can befound in marketing as well. In the beginning, the product was the central point in a marketingcampaign. 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 theproduct, in the "back-yard".As we will see, MCDA will first of all responsibilize the decision maker and make him aware ofthe aspects, the aims and/or the consequences related to the decision that he will make. This is,

34

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 asdifferent methods can propose different results. Moreover, it is necessary to explain the decisionmaker, 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 someof 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 decisiongenerally involves four phases. Moreover, the instant of decision can not (always) be identified.Schärlig distinguishes during this process the phases of information, conception, choice andretrospective analysis [Schärlig, 1985]. The order between these phases can of course be differentand is often characterized by passages from one phase to another, depending on the progressionof the consideration.The phase of information corresponds to the horizon seeing. The candidates for a solution aredetected and the conceivable criteria are considered as well as. The phase of conception, on theother hand, allows to define the set of choice (i.e. the candidates to be determined) and theirevaluations on the different chosen criteria. The decision making is done on the final set ofcandidates, not necessarily corresponding to the initial one. The retrospective analysis is rarelydone by a formal study but is certainly presented in the decision maker’s mind.

2.2 The actions, the criteria, . . . and the problems

2.2.1 The set of actions A

When facing a decision problem, the first step may be to identify the different objects submittedto the decision making process. These objects can be potential decisions, projects, feasiblesolutions, items, units, alternatives, candidates, etc. and will be called the actions. The set ofactions 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 orinfinite.

As mentioned before, the decision process may be evolutive. This implicates that the actions arenot always defined once and for all. When the actions evolute, A is said to be evolutive. On theother 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 com-binations 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]] isa function f , defined on A , taking its values in a set, noted V , ordered or not. It represents afeature or a characteristic inherent in or ascribed to an action [The Free Dictionary, 2008].

35


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 notedF = f1, ..., fq.

2.2.3 The set of criteria G

The actions of a decision problem will be analyzed and evaluated according to the decisionmaker’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 setand representing the decision maker’s preferences according to some point of view.

g : A 7→ 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 bemaximized.

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 cri-terion ... if the decision maker recognizes the existence of an axis of significance on which twopossible actions ai and a j may be compared ... and he accepts to model this comparison asfollows:

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 representedby the following vector : ai ≡ [ei

1, ...,eiq]. Moreover, the set of all the criteria will be noted

G = g1, ...,gq. We will suppose, except of explicit counter-indication, that the criteria have tobe 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 mostdifficult and crucial steps in a decision aiding process. Roy and Bouyssou has proposed amethodology 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 asreal-criterion, quasi-criterion, pseudo-criterion and interval-criterion depending on the inducedunderlying preference structure.

36

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 suchthat the set of considered actions is fixed once for all, such that every solution is exclusivefrom 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 reasonwhy 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 differenttype-problems [Roy and Bouyssou, 1993]:

• The choice problem : a subset of actions considered as the best according to the criterion-set G , has to be chosen [the α− problem]

• The sorting problem : a partition of the set A must be done with respect to some pre-established norms [the β− problem]

• The ranking problem : a ranking of all the actions from the best to the worst must berealized [the γ− problem]

• The description problem : a description, in an appropriate language, of the actions andtheir 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 simultane-ously several of these problems as we can cite for example the portfolio problem, the designproblem, 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 existingranking method for tackling a sorting problem.

To tackle these problems, we may use the Pareto dominance relation, define a preferencestructure, compare the actions pair-wise, etc. This will be the subject of the next sections. Wewill 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 thetype of MCDA problem, define the following notions. Let us remind that we suppose that thecriteria have to be maximized.

37


Definition 2.2. The Pareto dominance relation ≺P [De Smet, 2006] :

ai ≺P a j⇔ gk(ai) ≤ gk(a j) ∀gk ∈ G and ∃gx ∈ G | gx(ai) < gx(a j)

ai W a j⇔ gk(ai) ≤ gk(a j) , ∀gk ∈ G

This Pareto dominance relation between two actions expresses a somewhat unanimity over allthe criteria. In most cases only a few pair of couples satisfy this relation but nevertheless it caneventually leads to a reduction of the set of actions to be explored.

Definition 2.3. An action ae is said to be efficient in A:

⇔ @ ad ∈ A : ae ≺P ad

The non-efficient actions (also called the dominated actions) can be removed from A to reducethe dimensions of the problem.

Definition 2.4. The ideal point aIA in A is defined with the following evaluations:

g j(aIA) = max

a∈Ag j(a), ∀g j ∈ G

Definition 2.5. The nadir point aNA in A is defined with the following evaluations:

g j(aNA ) = min

a∈Ag j(a), ∀g j ∈ G

These points are, most of the time, virtual points: it is very seldom for an action to be the best(worst) for all the criteria. This shows us that MCDA does not "optimize" but searches for a"best" compromise according to decision maker’s preference.

Some examples of these notions will be given in next chapter (see Example 3.2).

2.4 Preference Structure

It is obvious now, that in the MCDA field, the central person is the decision maker. Beforedetailing different MCDA methods which reflect as good as possible the decision maker’scriteria and preferences, we will address the following binary relations1 which suppose onlya pairwise comparison between the actions. In this section, we will consider the preferencerelation as non-valued. Moreover, we will not incorporate explicitly the DM’s criteria. This willbe treated in the next sections.

1S is a binary relation if it is a subset of A×A

38

2.5. Consistent family of criteria and preferential independency

Confronted to just two actions ai,a j of A , we can assume that a decision maker expresses one ofthe following reactions [Vincke, 1992] (see also Section 2.6.1.4):

Indifference : ai I a j : the two actions are indifferentPreference : ai P a j : the DM has a preference for action ai

Incomparability : ai J a j: the DM can not make a decision: it is impossible for him to compare them.

These binary relations, which may be obtained as given in Section 2.6.1, are defined indepen-dently from the properties of the set of actions A . Nevertheless, they have to satisfy the followingconditions in order to be accepted as reflecting the indifference, preference, or incomparabilityfeeling of a DM: ∀ai,a j ∈ A :

aiPa j⇒ ai¬Pa j : P is asymmetricaiI ai : I is reflexiveaiI a j⇒ a jI ai : I is symmetricai¬J a j : J is irreflexiveaiJ a j⇒ a jJ ai : J is symmetric

Definition 2.6. The three relations P, I,J make up a preference structure on A , if they fulfilthe above mentioned conditions and if, given any two elements ai,a j of A , one and only one ofthe following relations is true : aiPa j, a jPai, aiIa j, aiJ a j [Vincke, 1989].

Sometimes, we can express the following statement : "ai is as least as good as a j", noted aiSa j⇔aiPa j or aiIa j (see Section 2.6.1). This enables us to define a preference structure with the Srelation and implicates directly the following statements: ∀ai,a j ∈ A :

aiPa j ⇔ aiSa j, a j¬Sai

aiI a j ⇔ aiSa j, a jSai

aiJ a j ⇔ ai¬Sa j, a j¬Sai

2.5 Consistent family of criteria and preferential independency

The problem modeling is the first step to tackle a multicriteria problem. Like in almost all kindsof problems, it is a very crucial part. Defining the actions on which a decision will be taken,selecting and determining the criteria to evaluate these actions and eliciting the decision maker’spreference according to his criteria, will definitely influence the quality contribution that theMCDA can offers.

To have a family of criteria which represents in a consistent way the decision maker’s prefer-ences, it is necessary that the criteria of G defined on A , satisfy the following axioms [Roy,1974]:

1. Axiom of exhaustibilityIf ∀gk ∈ G : gk(ai) = gk(a j) =⇒ ∀c ∈ A : (cQ a j ⇒ cQ ai) ∧ (a jQ ′c ⇒aiQ ′c) with Q ,Q ′ ∈ I,J,P,S.

39


In particular, if ∀gk ∈ G : gk(ai) = gk(a j)⇒ aiIa j.Intuitively, if two actions have the same evaluations on each criterion and if, not withstand-ing, the decision maker’s feels a preference (as slight can be) for one of these actions, thereshould be a another criterion being considered.

2. Axiom of cohesionIf two actions ai,a j are indifferent, weakening ai or strengthening a j on one or more criteriashould lead to aiIa j or aiPa j. This axiom expresses that the higher or the better an actionai on a given criterion k, the higher gk(ai). It ensures a certain coherence between the localmeaning of criterion-value and it’s influence on the global preference.

3. Axiom of non-redundancyG can not be composed of a criterion gk such that at least one of the aforementioned axiomsis violated after the removing of gk from G .

If these axioms are fulfilled, the criteria define a so-called "consistent" family of criteria.

On the other hand, it is also possible to determine a subset of criteria D that is separable from Gand that is said to be preferentially independent from G \D [Vincke, 1992]:

i f ∀a,b,c,d ∈ A |

gk(a1) = gk(a2), ∀k ∈ G \Dgk(a3) = gk(a4), ∀k ∈ G \Dgk(a1) = gk(a3), ∀k ∈Dgk(a2) = gk(a4), ∀k ∈D

=⇒ [ a1Pa2⇔ a3Pa4 ]

In other words, D is preferentially independent in G , if the preferences between actions, whichdiffer only by their values taken on the criteria from S are not dependent from the values on thecriteria from G \S .

Example 2.1. Let us consider 4 actions evaluated on 4 criteria which evaluations are given inTab.2.1. D = g3,g4 is preferentially independent in G , if the preference of a1 on a2 implies

ai g1 g2 g3 g4

a1 6 5 7 9a2 6 5 8 8a3 4 3 7 9a4 4 3 8 8

Table 2.1 — Evaluation matrix with preferentially independent criteria.

the preference of a3 on a4. The decision maker "must" thus prefer an action with evaluations forcriterion g3 and g4 equal (respectively) 7 and 9 and without regarding g1 and g2.

40

2.6. Pair-wise comparisons between actions based on outranking relations

2.6 Pair-wise comparisons between actions based on outranking re-lations

The preference structure on the set A may be obtained by pair-wise comparing the actions onthe basis of an outranking relation. The concept of outranking is due to B. Roy, who can beconsidered as its founder [Vincke, 1992].

Definition 2.7. An outranking relation [Roy, 1974] is a binary relation S defined in A such thataSb if, given what is known about the decision maker’s preferences and given the quality of thevaluations of the actions and the nature of the problem, there are enough arguments to decidethat a is at least as good as b, where there is no essential reason to refute that statement.

The outranking methods which have been proposed in the literature differ, amongst other aspects,by the way they formalize the definition of the outranking relation. There is obviously no reasonfor an outranking relation to be complete or transitive. The outranking methods proceed generallyin two phases: the building of the outranking relation (see Section 2.6.1) and the exploitation ofthese with respect to the chosen statement of the problem [Vincke, 1992].

2.6.1 The valued outranking degree S(a,b)

2.6.1.1 Partial concordance degree cSj(b,a)

For each criterion g j ∈ G the assertion "b is at least as good as a" or "b outranks a" is measuredby the partial concordance index noted cS

j(b,a). This degree is obtained as follows [Roy andBouyssou, 1993]2

cSj(b,a) =

1, if g j(b)+ p j < g j(a)g j(b)+p j−g j(a)

p j−q j, if g j(b)+ q j < g j(a) < g j(b)+ p j

0, otherwise.(2.1)

where q j, p j (satisfying p j > q j,∀ j) represent respectively the indifference and preferencethresholds as illustrated in Fig.2.1. These thresholds may be constant or depend on the per-formances of a: q j = q j(a) and p j = p j(a). The indifference threshold indicates the largestdifference between the performances of the actions on criterion g j such that they remain indif-ferent for the decision maker. The preference threshold indicates the smallest difference betweenthe performances of the actions on criterion g j such that one action is preferred to the other onewith respect to criterion g j. The stronger the confidence of the decision maker with this assertion,the higher this index. Moreover, it is always between zero and one where zero means that b doesnot outrank a and one that b is as least as good as a (on this particular criterion).

Several authors have proposed to determine these parameters from examples given by the deci-sion maker within a particular problem context. Let us refer the interested reader for the determi-nation of these parameters to [The and Mousseau, 2002; Dias et al., 2002; Dias and Mousseau,2006b,a; Damart et al., 2007].

2We will suppose, without any loss of generality, that the criteria have to be maximized.

41


6

g j(a)-

g j(b) g j(b)+ q j g j(b)+ p j

cSj (b,a)

1

0

Figure 2.1 — Representation of the partial concordance index cSj (b,a).

2.6.1.2 Global concordance degree CS(b,a)

The global concordance index CS(b,a) aggregates all the partial concordance indexes on thedifferent criteria by taking into account their corresponding weights (denoted by w j and suchthat ∑

qj=1 w j = 1) :

CS(b,a) = [ ∑j=1,...,q

w j× cSj(b,a)] (2.2)

In the case of Electre I, the weight of a criterion plays the same part as a number of votesin a voting procedure [Vincke, 1992]. In this context, the weighs may be seen as the relativeimportance attached to each criterion.

2.6.1.3 Partial discordance degree dSj (b,a)

For each criterion g j ∈ G , a measure of the discordance with the assertion "b is at least as goodas a" is traduced by the partial discordance index dS

j (b,a). This index is computed as follows:

dSj (b,a) =

1, if g j(a) > g j(b)+ v j(g j(b))0, if g j(a) ≤ g j(b)+ p jg j(a)−g j(b)−p j

v j−p j, otherwise.

(2.3)

where v j (satisfying v j > p j) represents the veto threshold for criterion g j as illustrated in Fig.2.2.This threshold defines the smallest difference between the performances of the actions on crite-rion g j such that the decision maker puts his veto on the assertion that b outranks a on criteriong j. A discordance index reaches the value 1 when there is a strong opposition (or veto) to theoutranking relation and it is equal to 0 when there is no discordance.

42


6

g j(a)-

g j(b)+ p j g j(b)+ v j

dSj (b,a)

1

0

Figure 2.2 — Representation of the partial discordance index dSj (b,a).

2.6.1.4 The outranking degree S(b,a)

The valued3 outranking degree S(b,a) (also called credibility degree) measures the credibility ofthe assertion "b outranks a" and is computed as follows 4:

S(b,a) = CS(b,a)×∏V

[1−dS

j (b,a)1−CS(b,a)

] (2.4)

where V = j ∈ G : dSj (b,a) > CS(b,a). This outranking degree, is always between zero and

one, takes thus both the concordance and the discordance degrees into account. This outrankingdegree is obviously not symmetric. Moreover, on the basis of a cutting threshold, noted as λ, itpermits, when pair-wise comparing two actions a and b, to traduce one of the following decisionmaker’s assertions:

1. aI b : a and b are indifferent ⇔ S(a,b) ≥ λ and S(b,a) ≥ λ

2. aJ b : a and b are incomparable ⇔ S(a,b) < λ and S(b,a) < λ

3. a b : a is preferred to b⇔ S(a,b) ≥ λ and S(b,a) ≤ λ

4. a≺ b : b is preferred to a⇔ S(a,b) ≤ λ and S(b,a) ≥ λ

Example 2.2. Consider the performance matrix given in Tab.2.2 of the 6 actions a1, . . . ,a6.Suppose that for all criteria, which have to be maximized, we have the following thresh-olds: ∀g j ∈ G : q j = 0.01 and p j = 0.05. Moreover, we have the following weight-vector:Ω = [0.25,0.25,0.10,0.2,0.2].

3One may find in the literature: "fuzzy" outranking degree. We will not use this term since we will use in Chapter9 fuzzy numbers.

4The reader may find a characterization on concordance in [Bouyssou and Pirlot, 2005],[Bouyssou and Pirlot,2007].

43


ai g1 g2 g3 g4 g5

a1 0.188 0.172 0.168 0.122 0.114a2 0.125 0.069 0.188 0.244 0.205a3 0.156 0.241 0.134 0.22 0.136a4 0.188 0.034 0.174 0.146 0.159a5 0.188 0.276 0.156 0.171 0.205a6 0.156 0.207 0.18 0.098 0.182

Table 2.2 — The performance evaluation matrix

Let us compute CS(a1,a2):c(a1,a2) = 1×0.25 + 1×0.25 + 0.75×0.1 + 0×0.2 + 0×0.2 = 0.5750.

The concordance matrix is thus as follows:

CS =

1.0000 0.5750 0.4900 0.7650 0.3550 0.63880.5000 1.0000 0.6188 0.7500 0.5000 0.61880.8025 0.6300 1.0000 0.7225 0.4762 0.73000.7500 0.4538 0.5500 1.0000 0.4950 0.69500.9950 0.7450 0.8050 0.9800 1.0000 0.96500.7925 0.7400 0.6500 0.6725 0.3475 1.0000

Consider now the computation of the single-criterion discordance index with ∀g j : v j = 1 andd j = 0.005. The vector d j(a1,a2) = [0, 0, 0.0151, 0.1176, 0.0864]. These values permit us tocalculate the credibility degree matrix S :S (a1,a2) = 0.5750 × [(1 − 0) × (1 − 0) × (1 − 0.0151) × (1 − 0.1176) × (1 − 0.0864)] =0.575×0.9849×0.8824×0.9136 = 0.4565.

The matrix S is thus as follows :

S =

1.0000 0.4565 0.4085 0.7195 0.2791 0.57620.4245 1.0000 0.5014 0.7063 0.3753 0.52200.7580 0.5497 1.0000 0.6659 0.4133 0.67110.6497 0.3790 0.4079 1.0000 0.3543 0.56660.9880 0.6753 0.7694 0.9672 1.0000 0.94660.7563 0.6218 0.5569 0.6260 0.2894 1.0000

If we fix the λ-threshold to 0.6 or 0.7 we obtain the binary relations between the actions given inrespectively Tab.2.3 and 2.4.

From the binary relation matrix, we obtain thus the outranking graph given in Fig.2.3 (whenconsidering λ = 0.6).

44


ai a1 a2 a3 a4 a5 a6

a1 I J ≺ I ≺ ≺a2 J I J ≺ ≺a3 J I J a4 I ≺ ≺ I ≺ ≺a5 I a6 ≺ ≺ I

Table 2.3 — The binary relations between the actions of A with λ = 0.6.

ai a1 a2 a3 a4 a5 a6

a1 I J ≺ ≺ ≺a2 J I J J Ja3 J I J ≺ Ja4 ≺ ≺ J I ≺ Ja5 J I a6 J J J ≺ I


-

1

o 7* iY

o

-

:

a4

a2

a6a5a3

a1

Figure 2.3 — Outranking graph of A where a→ b⇔ aSb ; a↔ b⇔ aI b ; a b⇔ aJ b and©⇔ cI c.

45


2.6.2 Preference degree π(a,b)

The preference degrees, which take explicitly into account the decision makers preferences,are used to compare two actions a and b. These degrees may be given directly by the decisionmaker or they can be computed by aggregating his preferences on several, eventually conflicting,criteria: the uni-criterion preference degrees.

These preference degrees are in a second step used to rank, choose, sort actions in methods suchas Promethee I, Promethee II, Pairclass, Promsort, F lowSort, etc.

2.6.2.1 Uni-criterion preference degree P j(a,b)

This degree reflects the decision maker’s preference of an action a over action b while consider-ing just one criterion g j. This preference degree is represented by a preference function Pk andis in function of the differences taken by the actions on this criterion. If g j has to be maximized,we have:

P j(a,b) = f j(g j(a)−g j(b)) (2.5)

The values taken by P j(a,b) are in the interval [0,1] and P j(a,b) = 0 if g j(a) ≤ g j(b).

The preference function P j will reflect how the preference of the decision maker increasesin function of the difference g j(a)− g j(b). The decision maker has the choice between 6types of preference functions types: the usual criterion function, the quasi-criterion function,the linear preference function, the function with thresholds, the linear criterion function withindifference and preference functions (also called the preference function of type 5) and thegaussian function. The reader may find more information on these criterion functions in [Bransand Mareschal, 2002] (p.54). Let us remark that, as pointed out in [Keyser and Peeters, 1996],the differences on the criteria must be meaningful.

6

- g j(a)−g j(b)

1

P j(a,b)

q j p j

Figure 2.4 — Preference function of type 5.

Fig.2.4 represents the preference function of type 5. The parameters to be fixed are q j and p j

representing respectively the indifference and preference thresholds.

46


Several authors have proposed to determine these parameters from examples given by the de-cision maker within a particular problem context. Let us refer the interested reader for the de-termination of these parameters to [Doumpos and Zopounidis, 2004c,b], [Frikha et al., 2004,2007].

2.6.2.2 Global preference degree π(a,b)

The global preference of an action a over an action b is computed as follows :

π(a,b) =

q∑

j=1w j×P j(a,b)

q∑

j=1w j

(2.6)

where w j represents the weight associated to each criterion g j. The weights permit to aggregateall the uni-criterion preference degrees as a marginal contribution to the global preference degree.The weights of the criteria express trade-offs between the criteria and must be expressed by thedecision maker on a ratio scale (not on a ordinal scale) [Keyser and Peeters, 1996]. If the weightsare normalized, we have:

π(a,b) =q

∑j=1

w j×P j(a,b) (2.7)

The preference degrees are obviously not symmetric and 0 ≤ π(a,b) + π(b,a) ≤ 1,∀a ∈ A .Moreover, π(a,a) = 0,∀a ∈ A .

Let us remark that the preference degree are analogous to the concordance degree from theElectre-family. Nevertheless, the indifference and preference threshold are always constant inthe Promethee methods which has his advantages (ease of use) and disadvantages (less sophisti-cated). Furthermore, no notion of discordance can be found. Moreover, it is difficult to expressa crisp preference assertion (preference, indifference and incomparability) when working withpreference degrees (on the contrary of using outranking degrees, see Section 2.6.1.4).

47

3 Some multicriteria rankingmethods

This chapter is devoted to multicriteria ranking methods. At first, some dif-ferent types of "rankings" are defined as well as some elementary rankingmethods (such as the weighted sum, the rule of Borda, etc.). Multi attributevalued methods are secondly presented and discussed. In particular, the addi-tive model. In the third part, we present the outranking methods with specialattention to Electre III and Promethee. These two methods are analyzed deeplyand some examples are proposed to illustrate some of their properties. Otherranking methods are briefly mentioned and we end this chapter by discussinga way of choosing a multicriteria ranking method in a particular decision prob-lem.

3.1 Introduction

B. Roy defines the ranking problem as follows:

Definition 3.1. La problématique de rangement: [Roy and Bouyssou, 1993], p.69:consiste à poser le problème en termes de rangement des actions de A ou de certaines d’entreelles, c’est-à-dire à orienter l’investigation vers la mise en évidence d’un classement défini surun sous-ensemble de A conçu en vue de discriminer les actions se présentant comme "suffisam-ment satisfaisantes" en fonction d’un modèle de préférences et ce compte-tenu du caractèrerévisable et/ou transitoire de A ; cette problématique prépare une forme de recommendation oude simple participation visant:- soit à indiquer un ordre partiel ou complet portant sur des classes regroupant des actionsjugées équivalentes ;- soit à proposer l’adoption d’une méthodologie fondée sur une procédure de classement (detout ou partie de A) convenant à une éventuelle utilisation répétitive et/ou automatisée.

Ph. Vincke defines the ranking problem in the following concise terms [Vincke, 1992], p.28:

49

Some multicriteria ranking methods

Definition 3.2. A multicriteria ranking problem is a situation in which, having defined a set Aof actions and a consistent family G of criteria on A , one wishes to rank the actions of A fromthe best to the worst.

The aim can thus be summarized as obtaining a partial or complete ranking of the actions of A ,where rankings can be modeled by means of a binary relation Σ. A binary relation Σ on A is asubset of the Cartesian product A×A , where ∀ai,a j ∈ A , (ai,a j) ∈ Σ means that action ai has arank at least as good as a j.

Definition 3.3. Let Σ be a binary relation defined on A [Vincke, 1992]:Σ is antisymmetric if: ∀ai,a j ∈ A , (ai,a j) ∈ Σ, and (a j,ai) ∈ Σ⇒ ai = a j

Σ is complete if: ∀ai,a j ∈ A , (ai,a j) ∈ Σ or (a j,ai) ∈ ΣΣ is reflexive if: ∀ai ∈ A , (ai,ai) ∈ ΣΣ is transitive if: ∀ai,a j ∈ A , (ai,a j) ∈ Σ and (a j,ak) ∈ Σ⇒ (ai,ak) ∈ S

Definition 3.4. A partial order is a transitive and antisymmetric binary relation. [Vincke, 1992]

Definition 3.5. A partial pre-order is a transitive binary relation. [Vincke, 1992]

Definition 3.6. A linear order is a complete, transitive and antisymmetric binary relation.[Vincke, 1992]

The following sections will completely be dedicated to a short review of some importantmethods which permit to tackle the ranking problem. It is of course impossible to describe allthe existing methods. Nevertheless, we can distinguish 3 families of methods : the multi attributeutility methods, outranking methods and interactive methods [Vincke, 1992]. They differ in theway of how they take into account the decision maker’s criteria and preferences. The last ofthese type of methods, will not be considered here, and we refer the interested reader to [Steuer,1986] for more information.

Before describing more in detail the MAUT and outranking methods, let us briefly propose thefollowing elementary ranking methods:

Weighted Sum: This is the most intuitive ranking method where a score is assigned to eachaction of A as follows:

∀ai ∈ A : s(ai) =q

∑j=1

w j×g j(ai) (3.1)

where w j is the weight of criterion g j and represents, up to a factor, a substitution ratebetween criteria [Vincke, 1992]. This score permits to obtain a complete pre-order:

∀ai,a j ∈ A : ai WS a j⇔ s(ai) ≥ s(a j)

This method assumes that all the criteria are expressed in identical units and that dif-ferences, on a criterion have the same meaning, as differences on another criterion.Moreover, a large difference on one criterion can compensate for several small ones on

50

3.2. Multi Attribute Utility Theory

other criteria 1. Moreover, since all actions are ranked from the "best" to the "worst", thecase of incomparability between two actions does not exist. As mentioned before, theincomparability of actions can sometimes point out the lack of information, the presence ofcontradictions or incertitude.

Borda: Borda’s well-known rule, orders the alternatives according to their sums of ranks theyoccupy on the different criteria. Formally, if ri

j denotes the rank of action ai on criterion g j,we may define the Borda score as follows:

∀ai ∈ A : b(ai) = bi =q

∑j=1

rij (3.2)

The Borda ranking B is the weak order defined as follows:

∀ai,a j ∈ A : ai B a j⇔ bi ≤ b j

Since Borda’s rule seems very intuitive, it has received a large attention by the scientificcommunity. A famous characterization of this rule is due to Young [Young, 1974]. Borda’srule can also be seen as a special case of so-called scoring rules. Instead of assigning 1point to the first position, 2 points to the second position, and so on, one may assign moregenerally s1 points to the first position, s2 points to the second position, and so on, as longas s1 ≤ s2 ≤ . . . ≤ sn [Lamboray, 2007]. This has as consequence that differences betweenthe ranks may compensate each other since they are summed. This method is seen as theancestor of the multiple attribute utility theory [Vincke, 1992].

Condorcet: In the Condorcet or the majority rule, an action ai is preferred to an actiona j if the number of criteria for which ai is better then a j is larger than the number ofcriteria for which a j is better than ai. This binary relation is not transitive (the paradox ofCondorcet, where there exists a circuit in the relation, is an illustration of this aspect) butcompensations between criteria have disappeared since the differences on the criteria arenot considered. This method is seen as the ancestor of the outranking methods [Vincke,1992; Vansnick, 1986].

Let us now emphasize on the MAUT and some outranking methods.

3.2 Multi Attribute Utility Theory

The Multi Attribute Utility theory is widely used by the anglo-saxons and is based on themain axiom that every decision maker tries to optimize, unconsciously or implicitly, a functionwhich aggregates all his point of views. In other words, the decision maker’s preferences maybe represented by an a priori unknown function, called the utility function U = U(g1, ...,gq)

1This is called a total compensatory behavior.

51


[Vincke, 1992].

Every action of the studied set A is evaluated on the basis of this function and receives a "utilityscore". This allows to rank, easily, all the actions from the best to the worst. This involves thatthe case of incomparability between two actions does not exist since two scores are always com-parable. As mentioned before, this has the consequence that some usefull aspects may be hidden.

'

&

$

%

a1

ai

a j

a2

a3

an

. . .

. . .

. . .

•

•

•

••

•

A

s

U(g1, . . . ,gq)

- U(a1)

6

~

*

j

*

U(a j) = U(an)

U(a2)

0

U(ai)

1

Figure 3.1 — Representation of the complete pre-order of the set A by using the MAUTtheory.

We will present in this work only the case of certainty and its most known model : theadditive model. Nevertheless, the MAUT has been extremely used and developed in the caseof uncertainty and with stochastic information (i.e. probabilities). Moreover, most of the time,the considered criteria, used in modeling the utility function, are real criteria (i.e. the inducedpreference structure is a total pre-order) as little research has been done in other cases.

3.2.1 The additive model

The most used and popular utility model is the additive model. The decision maker’s preferencesare expressed by taking into account several criteria. The different criteria are first transformed,to avoid scale problems (usually by marginal utility contributions). We may then add up thevalues for these transformed criteria. The "simple" weighted sum is thus a special case of thismodel (see Section 3.1). The general additive utility function can be written as follows:

∀ai ∈ A : U(ai) =q

∑j=1

U j(g j(ai))×w j =q

∑j=1

U′j(g j(ai)) (3.3)

with U j(g j) ≥ 0, U j(g j) non decreasing and where w j represents the weights of the different

52

3.2. Multi Attribute Utility Theory

criteria. Generally, we impose the following constraints on the set of marginal utility functionswith respect to set A . If we consider the (virtual) nadir and ideal action (see Definitions 2.4,2.5), we may impose:

Condition 3.2.1. ∀A , g j ∈ G : U j(g j(aIA)) = 1

Condition 3.2.2. ∀A , g j ∈ G : U j(g j(aNA )) = 0

Some examples of marginal utility functions are represented in Fig.3.2 and 3.3 where wesuppose that the criteria have to be maximized. The forms of the marginal utility functions aredetermined by the decision maker and correspond to different attitudes with respect to risk. Ifthe decision maker estimates that small differences on low criteria performances are significant,he will opt for concave functions (risk-averse attitude). On the other hand, if he considers smalldifferences on high performances as important, he will opt for convex functions (risk-pronebehavior) [Doumpos and Zopounidis, 2002]. The linear functions represent a risk-neutralattitude [Doumpos and Zopounidis, 2002].

-

6

U′i (g j)

g j

w j

0

Figure 3.2 — Pair-wise linear marginal utility functions.

Let us remark that this is a method of total aggregation in order to compute trade-off betweencriteria that respects the axioms of comparability, reflexivity, transitivity of choices, continuityand dominance [Beuthe and Scannella, 1997]. Moreover, the set of criteria must form a consistentfamily of criteria.

The preferences expressed on each criterion and the global preferences need to construct a totalpre-order. This implicates that the additive model uses criteria that are preferentially indepen-dent (see Section 2.5) [Vincke, 1992]. Actually, if we consider Γ, a subset of criteria, we have∀a,b,c,d ∈ A :

• g j(a) = g j(b), ∀ j ∈ G\Γ

• g j(c) = g j(d), ∀ j ∈ G\Γ

53


-

6

U′i (g j)

g j

w j

0

Figure 3.3 — Concave, linear and convex linear marginal utility functions.

• g j(a) = g j(c), ∀ j ∈ Γ

• g j(b) = g j(d), ∀ j ∈ Γ

This implies that :

U(a)−U(b) = ∑j∈G

[U j(g j(a))−U j(g j(b))] = ∑j∈G

[U j(g j(c))−U j(g j(d))] = U(c)−U(d)

And thus we have that :

a b⇔U(a)−U(b) > 0⇔U(c)−U(d) > 0⇔ c d

There exist two different methods to construct the utility function U :

Direct Method : All the functions U j will be estimated to construct directly U . The decisionmaker will evaluate the parameters by answering direct questions about his preferences.Several questioning procedure exist [Vincke, 1989] and Fishburn proposes in [Fishburn,1973] some methods which can englobe probabilities, utilization of compensations betweenU j, etc.The analyst will try to estimate analytically this function by asking "good" questions to thedecision maker. Nevertheless, two fundamental problems are inherent to this approach :

• Which properties should the decision maker’s preference have, to be analytically es-timable?

• How to construct these functions? How to determine its parameters?

Indirect Method : These methods will estimate U with global judgements expressed by thedecision maker on a learning set E . This can eventually be done by giving the rankingof the actions of E and elicit parameters which have to be in respect to this ranking.Jacquet-Lagreze and Siskos are on the basis of the UTA method which uses linear programs

54

3.3. Outranking methods

to elicit these parameters [UTA]. We refer the interested reader to [Krantz et al., 1971;Beuthe and Scannella, 1997; Tervore et al., 2004] for other methods which permit to inferthe parameters.

Without2 having explained in detail the aforementioned methods, we have to stress the followingconsiderations. These family of method enables to build an complete pre-order of the actions,which is a rich result, specially, compared to the poor ones obtained with the Pareto dominance.Nevertheless, the amount information needed to build such models (lots of questions to beasked to the decision maker) is very important and the hypothesis of this theory (existence,construction, additivity, etc. of U) are very strong. Such rich result is on the other hand notalways needed (e.g. we may only want to obtain the best action). These considerations have ledto develop a mid-way between the dominance relation and the multi-attribute utility theory : theoutranking methods [Vincke, 1992].

3.3 Outranking methods

Although outranking methods are different, most of them have some common characteristics.The outranking methods generally proceed in two phases: first, the building of the outrankingrelation and then, the exploitation of these with respect to the chosen statement of the problem[Vincke, 1992].

Outranking methods exploit pair-wise comparison between actions (outranking relations,preference degrees), instead of using absolute scores such in the MAUT. They lead to a transi-tive, partial or complete pre-order of the set A which we will note ℜ(A). Usually, a rankingprocedure ρ : A → ℜ(A) transforms the binary outranking relations S between any orderedpair (a,b) ∈ A×A , into a transitive binary relation Σ. Often, the ranking procedures takes intoaccount global information (such as how they behave with respect to the other actions) in ℜ(A)such that ∀(a,b) ∈ A×A we have one of the following statements [Roy and Bouyssou, 1993]:- aΣb∧¬(aΣb): a has a better rank than b- aΣb∧bΣa: a and b are indifferent- ¬(aΣb)∧¬(bΣa): a and b are incomparable.

The pair-wise comparisons S are most of the times not transitive since for example cycles mayoccur. As a consequence, it may happen that the order in the final (global) ranking does not corre-spond to these pair-wise (local) comparisons [Mareschal et al., 2008]. This phenomenon is calledpair-wise rank reversal and can be formally defined as follows ([Mareschal et al., 2008],[Roy andBouyssou, 1993] p.358):

Definition 3.7. pair-wise rank reversal occurs when aSb but where b Σ a in ℜ(A): i.e. a outranksb pair-wise but b has a better rank than a in the final ranking in ℜ(A).

2This section is inspired from [Vincke, 1992].

55


Moreover, since the global ranking depends directly on the set of actions A , the order may varywhen the initial set A is altered by adding or removing an action: this phenomenon is called"rank reversal". If we note Ax = A \x or Ax = A ∪x, we may define it formally as follows:

Definition 3.8. rank reversal occurs whenever the relative ranking of two actions in the globalranking ℜ(Ax) or ℜ(Ax) is reversed, with respect to ℜ(A), when a third action is removed oradded to the initial set A .

We may thus have that: ℜ(A) 6= ℜ(Ax), or ℜ(A) 6= ℜ(Ax) where ℜ(A) 6= ℜ(Ax) means thus∃(ai,a j) ∈ A ,Ax : [aiΣa j] in ℜ(A) ∧ [¬(aiΣa j)] in ℜ(Ax).

Let us remark, that for any ranking method based on pair-wise comparisons which is minimalconsistent (i.e. if | A |= 2, ℜ(A) respects the pair-wise comparisons) we have the followingproperty [Mareschal et al., 2008]:

Proposition 3.3.1. For any minimal consistent ranking method: the ranking method does notsuffer from the rank reversal phenomenon if and only if the ranking method does not suffer frompair-wise rank reversal.

The interested reader may find more information about ranking irregularities in [Mareschal et al.,2008], [Roy and Bouyssou, 1993], [Keyser and Peeters, 1996], [X.Wanga and E. Triantaphyl-loub, 2006]. In the next sections we present briefly the particular outranking methods Electre-IIIand Promethee.

3.3.1 Electre III

Bernard Roy can be considered as the father of the family of the methods Electre I, ElectreII, Electre III, Electre Tri, etc. exploiting an outranking relation. Electre I (ELEction et ChoixTRaduisant la REalité) is a method aiding a decision maker for the α− problems and aimsto obtain a subset of actions K such that all the other actions, not belonging to this subset,are outranked by at least one action of K . On the other hand, Electre II and III are used forγ− problems, i.e. to rank a set of actions.

Whereas in Electre I and II, only true criteria can be handled, Electre III takes indifference andpreference thresholds into account.

Electre III 3 [Roy, 1978] is a well-established partial ranking method which has been appliedsuccessfully in many real-world applications [Tervore et al., 2004; Karagiannidis and Mous-siopoulos, 1997; Hokkanen and Salminen, 1996; Kangas and Pykäläinen, 2001; Karagiannidisand Papadopoulos, 2008]. It permits to pair-wise compare a set of alternatives A , on the basis ofoutranking degrees (see Section 2.6.1), and to obtain a partial pre-order of this set A .

3This section is inspired from [Tervore et al., 2004].

56


These valued outranking relations, are said to be less sensitive to data variations and parameters[Vincke, 1992]. The method requires as input a coherent set of criteria (noted G = g1, . . . ,g j)and the associated preference information such as weights (∀g j ∈ G : w j), thresholds (indif-ference, preference and veto thresholds ; noted respectively ∀g j ∈ G : q j, p j, v j) and finally, acutting level (λ).

Electre III is based on two phases:

1. At first, an outranking relation for any ordered pair (ai,a j) ∈ A×A is computed (see Sec-tion 2.6.1).

2. The second phase consists of exploiting this relation by the ascending and descending dis-tillation procedures which produce respectively two complete pre-orders O1 and O2. Thefinal partial pre-order O obtained by computing the intersection of O1 and O2. The dis-tillation procedures are based on the preference relations between the actions and on thequalifications of the actions, defined in the next section.

3.3.1.1 Preference relation between two actions

Let us remark that in Section 2.6.1.4, we have shown that on the basis of the fuzzy outrankingdegrees S(ai,a j) and S(a j,ai), a decision maker may express a preference, indifference and in-comparability relation between two actions ai and a j. These relations were obtained on the basisof a constant cutting level λ. Nevertheless, since the arbitrariness induced by the computation ofthese indexes ([Tervore et al., 2004], p.5), we may not compare these outranking degrees betweenthemselves. In other words, if S(a,b) > S(c,d), it does not mean necessarily that the outrankingdegree of action a over b is more credible than the one of c over d. For this reason, a cuttingfunction L(λ) (also called a discrimination threshold) may be defined as follows:

L(λ) = α + β×λ (3.4)

where α and β are two parameters. B. Roy and D. Bouyssou advise in [Roy and Bouyssou,1993] to fix these values as follows: α =−0.15 and β = 0.3. If λ ∈ [0,1] and if [S(a,b) = λ andS(c,d) = λ−η with η > L(λ)], we may say that aSb is strictly more credible than cSd.

The preference relation between two actions, is, in Electre III, defined as follows:

a3 b ⇔ S(a,b)−S(b,a) > L(λ) and S(a,b) > λ (3.5)

Let us remark that the preference in relation in Electre III (i.e. 3), is slightly different from theone defined in 2.6.1.4 that will be noted ET . The preference relation ET , used in for instanceElectre-Tri (see Section 4.6.1) has been defined as follows:

aET b⇔ S(a,b) ≥ λ and S(b,a) ≤ λ

Nevertheless, If we fix α = 0 and β = 0 in Eq.3.4, we have that L(λ) = 0,∀λ, and thus thataET b ⇒ a3 b:

57


S(a,b) ≥ λ and S(b,a) ≤ λ

⇒ S(a,b) ≥ S(b,a)⇒ a3 bWe can thus notice that under these conditions the ET -relation is stronger than the 3-relation.

In the rest of this work, we suppose that stands actually for ET .

3.3.1.2 Qualification of an action

The qualification score Q (a) of each action of A is based on the strength P(a) and the weaknessF(a) of an action a and is defined as follows:

Q (a) = P(a)−F(a) (3.6)

P(a) =| b ∈ A : a3 b | (3.7)

F(a) =| b ∈ A : b3 a | (3.8)

3.3.1.3 Computation of the pre-orders O1 and O2

The descending distillation procedure, which computes the first complete pre-order O1 of the ac-tions of A , may be intuitively explained as follows (the detailed algorithm is given in Alg.3.1). Itpartitions A into completely ordered equivalency classes C1, . . . ,Cm, where C1 is the best equiva-lency class and Cm the worst. C1 is obtained by distilling A , i.e. by extracting the best actions onthe basis of their qualification. C2 is distilled from the remaining actions: A \C1, and so on. Eachclass C j is composed of ex aequo (ties) of the set A in the complete pre-order O1. The scheme ofthe procedure may be explained as follows:

• Initially, we start with the working set D0 = A . We distill this set D0 by taking the "best"elements according to their qualification, and obtain class C1.

• Then, we distill the new set D1 = D0 \C1 = A \ (C1), and obtain C2.

• . . .

• We distill the set D j = D j−1 \C j = A \ (C1∪ . . .∪C j), and obtain C j+1.

• . . .

• The procedures finishes when Dm+1 = /0 or when A \ (C1∪ . . .∪Cm+1) = /0

The “distillation”-phase depends at each step on the (highest) qualification-score of the remain-ing actions (i.e. on the value of their outranking degrees) in the distillation set [Schärlig, 1985].

The ascending distillation procedures is analogous, but instead of starting with the best equiva-lency class, we begin by distilling the worst one and obtain finally O2. The formal algorithm isgiven in Alg.3.2.

58


Algorithm 3.1 — The descending distillation algorithm which gives O1 ([Tervore et al., 2004])

l=0 ; A0 = A ; C0 = /0

while Al 6= /0 dok=0 ; Dk = 0 ; λk = max(a,b)∈Al

S(a,b) with a 6= bwhile | Dk |= 1 or λk = 0 do

λk+1 = max(a,b)∈DKS(a,b) with a 6= b and S(a,b) > λk−L(λk)

if S(a,b) > λk−L(λk),∀a ∈ Dk thenλk+1 = 0

end ifComputation of the qualifications Q(a),∀a ∈ Dk

Qmax = maxa∈DkQ(a);Dk+1 = a ∈ Dk : Q(a) = Qmaxk=k+1

end whileCl+1 = Dk ; Al+1 = Al \Cl+1 ; l=l+1

end while

Algorithm 3.2 — The ascending distillation algorithm which gives O2 ([Tervore et al., 2004])

l=0 ; A0 = A ; E0 = /0

while Al 6= /0 dok=0 ; Dk = Al ; λk = max(a,b)∈Al

S(a,b) with a 6= bwhile | Dk |= 1 or λk = 0 do

λk+1 = max(a,b)∈DKS(a,b) with a 6= b and S(a,b) > λk−L(λk)

if S(a,b) > λk−L(λk),∀a ∈ Dk thenλk+1 = 0

end ifComputation of the qualifications Q(a),∀a ∈ Dk

Qmin = mina∈DkQ(a);Dk+1 = a ∈ Dk : Q(a) = Qmink=k+1

end whileEl+1 = Dk ; Al+1 = Al \El+1 ; l=l+1

end while

59


3.3.1.4 Partial pre-order O

The partial pre-order O is defined as the intersection of the complete pre-orders O1 and O2. Inthe final partial pre-order, the following relations hold [Tervore et al., 2004]:

aO b⇐⇒ (aO1 b∧aO2 b)∨ (aI O1b∧aO2 b)∨ (aO1 b∧aI O2b)

aI Ob⇐⇒ (aI O1b∧aI O2b)

aJ Ob⇐⇒ (aO1 b∧bO2 a)∨ (bO1 a∧aO2 b)

Example 3.1. Let us first consider the following small example where 4 actions have to be rankedwith the partial ranking method Electre III. The 4 actions are evaluated according to 2 criteria,which have to be maximized. We will suppose that these criteria are true-criteria (i.e. ∀ j = 1,2 :q j = p j = 0 and with v j = 0). The performance matrix of A is given in Tab.3.1. The binaryrelations (based on the values of the credibility matrix S ) are given in Tab.3.2 and representedin Fig.3.4 when fixing λ > 0.5. After the ascending and descending distillation procedures, weobtain the partial ranking represented in Fig.3.5. The results were obtained by using the freedemo version of Electre III, available at [Lamsade, 2008].

ai g1 g2

a1 1 1a2 0 0.5a3 0.5 0a4 0 0

Table 3.1 — The performance evaluation matrix of A .

S =

1 1 1 10 1 0.5 10 0.5 1 10 0 0 1

ai a1 a2 a3 a4

a1 I a2 ≺ I J a3 ≺ J I a4 ≺ ≺ ≺ I


We can thus remark from Fig.3.5, that Electre III leads to a complete pre-order although a moreintuitive partial ranking could be expected on the basis of the outranking graph in Fig.3.4. If

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a1

a4

a2

a3

/

I

Figure 3.4 — Outranking graph of A where a→ b⇔ aSb, a↔ b⇔ aI b; a b⇔ aJ b and©⇔ cI c.

a4

a2,a3

a1

?

?

O = O1 = O2

Figure 3.5 — The partial pre-order O of A obtained with Electre III

we represent only the transitive outranking relations, we obtain the partial pre-order given inFig.3.6 (what we will call the reduced outranking graph). Actually, the Electre III method cannot always clearly differentiate the indifference and incomparability between two actions, in thefinal ranking [Roy and Bouyssou, 1993], p.423. This can nevertheless be accepted by the decisionmaker since a2 and a3 behave similarly with respect to the actions a1 and a4.

Example 3.2. Let us consider the following concrete example inspired from [Pomerol andBarba-Romero, 1993]. To engage a new employee, a company will consider 6 potentialapplicants. These applicants will be evaluated according to 5 quantitative criterions. For theselast two, the headhunter will express his total pre-order by the mean of a note between 0 and 10.

In our problem, the set of actions A is stable and defined by extension. The candidates will bedenoted as follows: A = a1, . . . ,a6. The considered criteria are :

61


a1

a4

a2

a3

/

I

Figure 3.6 — The partial pre-order O of A obtained by "reducing" the outranking graph.

1. the educational degree expressed in years

2. the professional experience expressed in years

3. the age expressed in years

4. the appreciation resulting from the interview (a score between 0 and 10)

5. the results from a psycho-technical test (a score between 0 and 10)

Except from criterion 3, the criteria have to be maximized. The evaluation matrix is given intable 3.3. The preference parameters are given in Tab.3.6.

ai g1 g2 g3 g4 g5

MAX MAX MIN MAX MAXa1 6 5 28 5 5a2 4 2 25 10 9a3 5 7 35 9 6a4 6 1 27 6 7a5 6 8 30 7 9a6 5 6 26 4 8

Table 3.3 — Evaluation matrix of the 9 candidates

There is no dominant action. All the candidates are efficient actions. The ideal point aIA is in our

case a fictitious candidate with the following characteristics : aIA ≡ [6,8,25,10,9].

The concordance matrix is equal to the credibility matrix since we do not use the veto-thresholds.The ascending and descending rankings, O1 and O2, as well as the partial ranking O are givenin Fig.3.7. Let us remark that these results have been obtained by using the free demo version ofElectre III which can be found in [Lamsade, 2008].

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Table 3.4 — Preference parameters of the Electre III method.

g1 g2 g3 g4 g5

wi 0.25 0.25 0.1 0.2 0.2qi 0 0 0 0 0pi 1 1 1 1 1

Figure 3.7 — Representation of the O1,O2 and O3 rankings as well as the concordance matrixobtained with the Electre-III demo software [Lamsade, 2008] for the Example 3.2.

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3.3.1.5 Model assumptions and some properties

Electre III can be said to be "neutral" since it does not discriminate actions in their ranking onthe basis of their label or their given name [Bouyssou and Perny, 1992; Bouyssou, 1992].

Moreover, when two actions are compared similarly to any other action of the set A (in termsof outranking degrees: S(a,x) = S(b,x);S(x,a) = S(x,b) and S(a,b) = S(b,a)∀x ∈ A ), theywill be considered as globally indifferent. This is often called the non-discriminatory propertyof a ranking method. Let us first remark that, as mentioned in [Roy and Bouyssou, 1993], p.423,Electre III, in its initial version, does not permit to treat in a different way the incomparabilityand the indifference between two actions (see Example 3.1 where a2 and a3 are ex-aequo,although actions a2 and a3 are incomparable). For that purpose a modified version has beenproposed which can be found in [Roy and Bouyssou, 1993]. Let us remark that Promethee doesnot permit it neither.

Moreover, [Perny, 1992] and [Gabrel, 1990] have point out that Electre III does not enjoys theproperty of monotonicity since the rankings do not respond "in the right direction" to a modifi-cation of performances of the actions.

3.3.1.6 Rank Reversal phenomenon

As pointed out in Section 3.3, it may happen that the pair-wise comparison between actions arenot respected in the final ranking. This has been called the pair-wise rank reversal. Since theElectre III method aggregates the pair-wise comparisons in a final ranking, it may suffer fromthis drawback too: in O1 of Example 3.2 we may notice that a6 has a lower rank than a4 (seeFig.3.7 and 3.8) although we have that S(a4,a6) < S(a6,a4).

Since pair-wise rank reversal may happen, the addition or suppression of an action of the set Amay lead to a rank reversal phenomenon (see Proposition 3.3.1): when computing the O1,O2 andO3 of A4 (i.e. A \a4) we may remark that these are altered. In the complete pre-order O1(A4)we have that a3 has a better rank than a2 whereas a2 and a3 are now incomparable in O(A4)(see Fig.3.7 and 3.8).

Let us mention in this context the work [X.Wanga and E. Triantaphylloub, 2006] where somecomputational experiments on randomly generated decision problems were executed to test theperformance of the Electre III method and an examination of some real-life case studies is alsodiscussed. The results of these examinations show that the rates of ranking irregularities wererather significant in both the simulated decision problems and the real-life cases studied in thispaper [X.Wanga and E. Triantaphylloub, 2006].

3.3.2 Promethee

Promethee I and II (Preference Ranking Organization Method for Enrichment Evaluations,[Brans and Vincke, 1985], [Mareschal and Mertens, 2003], [Mareschal et al., 2008]) are two well

64


Example 3.3.

Figure 3.8 — Representation of the O1,O2 and O3 rankings when suppressing action a4 fromA : illustration of the rank reversal phenomenon. [Lamsade, 2008].

known ranking methods which have been used in many practical problems [Brans and Mareschal,1991; Figueira et al., 2004; Brans et al., 1986; D’Avignon and Mareschal, 1989; Mareschal andMertens, 1990; Laukkanen et al., 2002; Dreschler, 2004].

Like the Electre methods, the Promethee methods are based on a pair-wise comparison of theactions, leading to valued preference degrees (see Section 2.6.2). These preference degrees are,in a second phase, exploited by the computation of flows. These flows lead to a partial andcomplete ranking of the set of actions: the Promethee I and Promethee II ranking.

3.3.2.1 Entering, leaving and net flows

Once all the ordered pairs of A ×A have been compared, by the computation of the preferencedegrees, we obtain the preference matrix Π where (Π)i, j = πi, j = π(ai,a j). This matrix is thenused to obtain a complete pre-order. The main idea is to analyze how an action ai is preferredto all the other actions (i.e. how ai outranks the other actions) and inversely, how the others arepreferred to ai (i.e. how ai is being outranked).

65


For that purpose, the entering and leaving flows (also often called respectively positive and neg-ative flows) are computed as follows:

φ+(ai) =

1| A | −1 ∑

a j∈Aπ(ai,a j) (3.9)

φ−(ai) =

1| A | −1 ∑

a j∈Aπ(a j,ai) (3.10)

These two outranking flows (scores) permit to have two complete pre-orders of the actions of A .The intersection of this two pre-orders lead to the Promethee I partial pre-order :

ai outranks a j : aiΣa j ⇔

φ+(ai) > φ+(a j)φ−(ai) < φ−(a j)

φ+(ai) > φ+(a j)φ−(ai) < φ−(a j)

⇒ ¬(a jΣai)

φ−(ai) > φ−(a j)φ+(ai) < φ+(a j)

⇒ ¬(aiΣa j)

aiJ a j

Let us remark that the constraint on the preference degrees 0 ≤ π(ai,ai) + π(a j,ai) ≤ 1 has asconsequence that the Promethee I partial ranking is not necessarily complete [Bouyssou, 1992].

To obtain a complete pre-order on the basis of the positive and negative flows, a decision makermay use the net flows, which are defined as follows:

φ(ai) = φ+(ai)−φ

−(ai) (3.11)

The intuition behind the net flows is that this score aggregates both the "being preferred to"character and the "being preferred by" character of an action ai. Behind this intuition, [Mareschalet al., 2008] have given a more theoretical definition to the net flows. When comparing this globalscore to the pair-wise preference degrees, it is proven that the net flow is the centered scorethat minimizes the sum of the squared deviations between the global ranking and the pair-wisecomparisons of the actions.

Let us finally remark that we may compute the uni-criterion net flows of an action ai as follows:

φk(ai) =1

| A | −1 ∑a j∈A

[Pk(ai,a j)−Pk(a j,a j)] (3.12)

This gives thus an idea of the performance on the k-th criterion of ai with respect to all the otheractions of A . The order of these uni-criterion net flows always corresponds to the order of theevaluations on this criterion (i.e. there can not be a rank reversal phenomenon). Nevertheless,this is not a "simple" scale transformation from g j to φ j. The uni-criterion net flow values arealways between -1 and 1. The preference parameters given by the decision maker (such as p and

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Table 3.5 — Evaluation of the performances of the actions of A .

ai g1 g2 g3 g4 g5

a1 6 4 28 5 5a2 4 2 25 10 9a3 5 7 35 9 6a4 6 1 27 6 7a5 6 8 30 7 9a6 5 6 26 4 8

Table 3.6 — Preference parameters of the Promethee method.

g1 g2 g3 g4 g5

wi 0.25 0.25 0.1 0.2 0.2qi 0 0 0 0 0pi 1 1 1 1 1

q) have a direct impact on the scores. If for example, p is chosen rather small (with preferencefunction 5), we lose the proportionality of the original scale. On the other hand, if p is chosenrather high, the differences between the flows become smaller (than on the original scale) butwithout getting the extreme values of -1 and 1 4. Moreover, the scale transformation dependsdirectly on the number of actions. Finally, we may interpret the uni-criterion net flows as atransformation based on a weighted sum of pair-wise comparisons. All these assertions caneasily be experimentally verified.

The decision maker must be aware of this transformation, since we have furthermore:

φ(ai) =q

∑k=1

wk×φk(ai) (3.13)

which permits to consider the net flows of an action as a kind of weighted sum (see Eq.3.3, 52).

Example 3.4. Let us consider the same example as in Example 3.2 where 6 candidates have tobe ranked in order to select the best person for a job. The candidates are evaluated on the basis of5 criteria: the educational degree expressed in years (g1), the professional experience expressedin years (g2), the age expressed in years (g3), the appreciation resulting from the interview (ascore between 0 and 10) (g4), the results from a psycho-technical test (a score between 0 and 10)(g5). All the criteria, expect of criteria 3, have to be maximized. The performances are remindedin Tab.3.5.

The preference parameters are given in Tab.3.6. We obtain the preference matrix Π and theuni-criterion flows are given in Tab.3.7.

4This may have a direct impact on the signification of the weights.

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Table 3.7 — Unicriterions net flowsai φ1 φ2 φ3 φ4 φ5

a1 0,60 -0,20 -0,20 -0,60 -1,00a2 -1,00 -0,60 1,00 1,00 0,80a3 -0,40 0,60 -1,00 0,60 -0,60a4 0,60 -1,00 0,20 -0,20 -0,20a5 0,60 1,00 -0,60 0,20 0,80a6 -0,40 0,20 0,60 -1,00 0,20

Table 3.8 — Positive, negative, net flows and ranking of A .

ai φ+ φ− φ rank

a1 0.33 0.57 -0.24 6a2 0.51 0.45 0.06 2a3 0.45 0.5 -0.05 3a4 0.37 0.53 -0.16 5a5 0.7 0.16 0.54 1a6 0.4 0.55 -0.15 4

Π =

0 0.5 0.35 0.25 0.1 0.450.5 0 0.5 0.75 0.3 0.50.6 0.5 0 0.45 0.2 0.450.5 0.25 0.55 0 0.1 0.450.65 0.5 0.8 0.65 0 0.90.55 0.5 0.3 0.55 0.1 0

The leaving, entering and net flows (obtained with the preference matrix Π or the uni-criterionflows) as well as the rank of the actions are given in Tab.3.8. The induced ranking on the basis ofthe leaving and entering flows are given in respectively Fig.3.10 and 3.11. One may notice thatthe complete rankings are not the same: actions a4 and a6 have a different rank. This leads tothe partial ranking of Promethee I represented in Fig.3.12. The complete Promethee II rankingis given in Fig.3.13.

Let us remark that, although the Promethee I partial ranking is identical to the Electre III partialranking; the positive and negative flow-rankings are not the same as the O1 and O2 rankings(see Example 3.2, Fig.3.7). This is normal since the qualification (see Section 3.6) reflects boththe preferences and weaknesses of an action with respect to the other actions. In Promethee I,we use the preferences and weaknesses independently from each other. On the contrary, the netflows take both into account. One may wonder about considering both aspects independentlyfrom each other when comparing actions since all (most) aspects should be considered.

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Figure 3.9 — Chart representation of the flows.

a1 a4 a6 a3 a2 a5

Figure 3.10 — Complete ranking of A on basis of the positive flows (φ+).

a1 a6 a4 a3 a2 a5

Figure 3.11 — Complete ranking of A on basis of the negative flows (φ−).

a4

a6

a3 a2 a5

I

a1/

o

Figure 3.12 — Complete Promethee I ranking of A .

a1 a4 a6 a3 a2 a5

Figure 3.13 — Complete Promethee II ranking of A .

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3.3.2.2 The Gaia plane and the Walking Weigths

Complementary to these outranking methods, [Mareschal, 1991] has proposed a way to constructinteractively, with the decision maker, the preferences functions Pk, and to analyze the sensitivityof the results in function of the weights wk. The Weight Stability Intervals (or Walking Weights)give for each criterion the limits within each weight might be modified without altering thePromethee II ranking. It is a useful tool when the decision maker wants to perform a sensitivityanalysis and to generate alternative sets of weights.

Moreover, [Brans and Mareschal, 2002] has proposed a method to explore and structure adecision problem. A visual representation is given in a plane called the Gaia plane, where Gaiastands fort Geometrical Analysis for Interactive Assistance. The interested reader may find moreinformation in [Mareschal and Mertens, 1990], [Brans and Mareschal, 2002].

The GAIA [Brans and Mareschal, 2002] plane provides a comprehensive view of the decisionproblem (see Example 3.5):

• the involved criteria (or grouped categories of criteria) are represented by axes and theirorientation permit to detect conflicts between criteria (green axis in Fig.3.14),

• the decision actions are represented by symbols which position informs the decision makerabout their strong and weak features with respect to the different criteria (blue points inFig.3.14),

• a "Π" decision axis representing the weighting of the criteria and their compromise (redaxis in Fig.3.14).

Let us remark that the quantity of preserved information when projecting on the Gaia plane isexpressed by the δ-quantity ([Brans and Mareschal, 2002], p.93).

Example 3.5. The Gaia-map is represented in Fig.3.14. If we use the projections on the Π-stickto rank the actions (represented in Fig.3.15), we can notice that in this case the ranking of Gaiais not same the Promethee II ranking: actions a3, a2 and a1,a4 are permuted.

From the Gaia-plane, a decision maker may notice that actions a5 and a3 have a global iden-tical behavior (since they are very close in the Gaia plane). This is confirmed by the profile-representation of their uni-criterions flows. Moreover, these actions are very different from actiona2 (see Fig.3.16 and 3.17).

In Tab.3.9 we have given the stability intervals associated to the different criteria for differentlevels of the Promethee II ranking where the k-th corresponds to the first k actions in the ranking.

All this features are implemented in the "Decision Lab" software available on-line: [Brans andMareschal, 2002], p173; [Brans and Mareschal, 2000]. A review of its features has been givenin [Geldermann and Zhang, 2001].

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Figure 3.14 — Gaia plane with δ = 72.8%.

a4 a1 a6 a2 a3 a5

Figure 3.15 — Complete ranking of A on basis of the Gaia-plane.

Figure 3.16 — Representations of the profiles of the uni-criterion net-flows of actions a3 anda5.

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Figure 3.17 — Representations of the profiles of the uni-criterion net-flows of actions a3 anda2.

Table 3.9 — Stability intervals at different levels (in %).

level 1 level 2 level 3 level 4 level 5min Max min Max min Max min Max min Max

g1 5.6 100 5.6 34 5.6 26.1 22 26.1 22.6 26.1g2 5.6 100 5.6 37.4 19.7 32 19.7 23.2 19.7 23.2g3 0 100 4.8 30.8 15.3 30.8 17.6 30.8 17.6 27.3g4 0 100 7.7 100 7.7 20.9 7.7 19.1 10.7 19.1g5 0 100 0 100 12.2 100 16.3 100 16.3 100

3.3.2.3 Model assumptions and some properties

Several authors have studied some theoretical properties of the Promethee I and Promethee IIranking methods [Bouyssou and Perny, 1992],[Bouyssou, 1992],[Keyser and Peeters, 1996]. Letus point out in this section the major characteristics given to these ranking methods.

Let us first remark, that if we consider that the preference degree π(a,b) are not resulting froman aggregation of several criteria, but corresponds to a percentage of voters considering thataction a is preferred or indifferent to b, we may notice that the Promethee II ranking correspondsto the well-known method of Borda [Bouyssou and Perny, 1992; Bouyssou, 1992] (see Section3.1), [Fishburn, 1973]. Moreover, if we consider that ∀a,b ∈ A : π(a,b) ∈ 0,1, the PrometheeII ranking method amounts to the Copeland ranking method [Bouyssou and Perny, 1992;Bouyssou, 1992].

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Furthermore, we may obtain the preference degrees π(a,b) by another way then proposed inprevious sections. In fact, these degrees may be directly expressed by the decision maker whencomparing a to b (and vice et versa). Moreover, when considering the preference degrees as anaggregation of different uni-criterion preference degrees, we may not use one of the 5 proposedpreference functions. A decision maker could propose other preference functions or give themdirectly 5. Nevertheless, regardless of the way of obtaining these preference degrees, we maystill compute the leaving, entering and net flows. These ranking methods are called rankingmethods based on leaving, entering and net flows [Bouyssou and Perny, 1992; Bouyssou, 1992].The Promethee I and Promethee II methods are thus particular ranking methods based on flows.

The ranking methods based on the net flows and on the difference of the entering and leavingflows make use of the "cardinal properties" of the valued preference degrees. In fact, if wetransform the preference degrees by a strict increasing transformation ℑ on the real line and suchthat ℑ(0) = 0 and ℑ(1) = 1, it may happen that the initial flow-ranking is not preserved. As aconsequence, it does not seem appropriate when the comparisons of the valuations (preferencedegrees) only have an ordinal meaning in term of credibility [Bouyssou and Perny, 1992;Bouyssou, 1992].Moreover, the Promethee methods may only be applied if the decision maker is able to expresshis preference between two actions, either on a certain criterion or on a ratio scale (and not onan ordinal scale) [Keyser and Peeters, 1996].

The ranking methods based on the net flows and on the difference of the entering and leavingflows are said to be neutral. Particularly, Promethee I and Promethee II do not discriminateactions in their ranking on the basis of their label or their given name [Bouyssou and Perny,1992; Bouyssou, 1992].

Moreover, Promethee I and Promethee II are strongly monotonic methods since the rankingsrespond "in the right direction" to a modification of the preference degrees. This propertyexcludes, in particular, the use of any threshold in the treatment of the valuations [Bouyssou andPerny, 1992; Bouyssou, 1992].

When two actions are compared similarly to any other action of the set A (in terms of preferencedegrees: π(a,x) = π(b,x); π(x,a) = π(x,b) and π(a,b) = π(b,a) ∀x ∈ A ), they will be con-sidered as globally indifferent. This is often called the non-discriminatory property of a rankingmethod. Nevertheless, it has as drawback that a situation of indifference or incomparabilitybetween two actions, may be threated similarly in the final ranking (see Example 3.1 wherethe Promethee I and II methods will lead to same pre-order: a2 and a3 are ex-aequo, althoughactions a2 and a3 are incomparable).

5The decision maker must be aware that some monotonicity constraints must nevertheless be fulfilled.

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Figure 3.18 — Representations of the flows of the actions of set A4.

3.3.2.4 Rank Reversal phenomenon

The Promethee II method is a complete ranking method based on pair-wise comparisons(preference degrees). All the pair-wise comparisons (which are not always consistent sincecycles may occur) are then aggregated into a global complete ranking. As a consequence, it mayhappen that the order in the final ranking does not correspond to these pair-wise comparisons[Mareschal et al., 2008]. It may suffer from pair-wise rank reversal (see Definition 3.7) since wemay have that π(a,b) > π(b,a) with φ(a) < φ(b).

Since Promethee II is not pair-wise rank reversal free, it may suffer from the rank reversal phe-nomenon (see Proposition 3.3.1) and thus Promethee I suffers also from this drawback. This canbe explained easily since the flow-score associated to each action a depends directly on the setA (see Eq.3.9,3.10 and 3.11, ). The order induced by the flows may vary when the initial set A isaltered. Some examples of pair-wise rank reversals and rank reversals are given in [Keyser andPeeters, 1996]. Nevertheless, the rank reversal phenomenon occur when the difference of flowsbetween two actions is "small". The interested reader may find more information on situationswhere there is no rank reversal [Mareschal et al., 2008].

Example 3.6. Let us consider Example 3.4. If we compute the net flows for all the actions ofA4 (i.e. A \ a4), we may remark that there is a rank reversal between actions a2 and a3 (seeFig.3.18).

3.3.2.5 Some extensions of the Promethee methodology

Some extensions of the Promethee methodology have been proposed in the literature. Amongstothers, let us cite a stochastic extension in case of uncertainty [Mareschal, 1986; D’Avignon andVincke, 1988] and an extension of a decision problem in presence of several decision makers

74

3.4. Other multicriteria ranking methods

("Group Decision Support System") in [Figueira et al., 2005].

3.4 Other multicriteria ranking methods

Let us briefly mention and describe some other multicriteria ranking methods:

ORESTE: The Oreste method is a multicriteria decision method which ranks the actions whenthe actions are ranked for the different criteria (with eventually indifference between theactions) and when the criteria themselves are ranked according to their importance (witheventually indifference between the criteria). The Oreste method proceeds in two phases:first a complete ranking of A is performed and then, after an indifference and incompara-bility analysis, a partial ranking of A is computed.The rank of an action ai is based on the associated ranks of ai on the different criteria g j.These ranks are based on the distance d j(ai) between an arbitrary origin and the projectionof ai, for every criterion, on a straight line. The interested reader may find more informationin [Roubens, 1981; Mercier et al., 1993]. Let us remark that this method suffers also fromrank reversal phenomenons.

Argus: The Argus method computes an outranking graph on the basis of qualitative and quan-titative data. "It begins with qualitative pairwise comparisons of the alternatives on the basisof each criterion and then imposes an order on the criteria. It calculates the number of crite-ria by which one alternative is better than another, and then compounds the ordinal rankingsof the alternatives and these criteria to produce a final ranking of the alternatives.6"

PRAGMA: Pragma (Preference Ranking Global Frequencies in Multicriterion Analysis) pro-vides the ranking frequencies of actions. "The method is based on the comparison of thepartial profiles of each alternative with reference to all the possible pairs of criteria consid-ered. The global frequencies are obtained as the weighted sum of the corresponding partialranking frequencies7[Matarazzo, 1988]." It permits to build complete and partial pre-orders.

Melchior: The Melchior method is a purely ordinal method which does not introduce quan-titative aspects in the the treatment of the data. Moreover, there is a relation, "is at leastimportant", between the criteria. Let us remark that Electre IV coincides with Melchior ina particular case [Vincke, 1992].

MACBETH: On the basis of pair-wise comparisons on each criterion, between actions, consis-tent value functions are inferred by linear programming [Bana e Costa and Vansnick, 1994].This permits to compute a complete pre-order.

Regime: "Regime is a qualitative multicriteria method based on ordinal information regardingboth the evaluation criteria and their respective weights. 8" An action ai is better than anaction a j if there exists a non-empty subset of criteria G on which ai is better than a j andif for any criteria on which ai is worse than a j there exists a criterion in G which is moreimportant.

6This paragraph is taken from [Moffett and Sarkar, 2006].7This paragraph is taken from the abstract of [Matarazzo, 1988].8This paragraph is taken from [Matarazzo, 1988].

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Qualiflex: It provides the probabilities that an action will be placed in any position in the rank-ings corresponding to each criterion. To each ranking, a concordance score is attributed.The aggregation phase of all the criteria takes further more into account the qualitative or-dering of the criteria to produce an overall concordance index for each possible ranking ofthe actions. The ranking with the highest global concordance index is chosen.

Topsis: The Topsis method computes a complete pre-order by ranking the actions in functionof their euclidean distance (in a n-dimensional space) to an optimum [Yoon and Hwang,1995].

Let us remark that Qualiflex [Paelinck, 1978], Oreste [Roubens, 1981], Argus [Keyser andPeters, 1994], Melchior [Leclercq, 1984], Pragma [Matarazzo, 1988] methods are characterizedby the fact that the relative importance of the criteria must not be "weighted". On the contrary,an importance relation on the criteria is needed.

Finally, let us mention AHP (Analytic Hierarchy Process) [Saaty, 1980] which have known ahuge success in the anglo-saxon community although it has been severely critiqued in the litera-ture [Schoner and Wedley, 1988; McCaffrey, 2005]. This method provides a complete pre-orderby successively pair-wise comparing the actions and the criteria what leads to value functionsfor the actions and the criteria.

3.5 How to choose a multicriteria ranking method ?

Considering the numerous existing ranking methods, a decision maker and an analyst, facing adecision problem, are confronted to the delicate choice of an appropriate decision tool. Theyhave at their disposal lots of methods and decision-analysis tools, but the selection and the useof a particular one, may be difficult (to justify). None of the methods is perfect nor applicablefor any problem. Each method has its own limitations, particularities, hypotheses, premises andperspectives. In this context, D. Bouyssou and B. Roy mentioned in [Roy and Bouyssou, 1993]:"... altough the great diversity of MCDA procedures may be seen as a strong point, it can also bea weakness. Up to now, there has been no possibility of deciding wether one method makes moresense than another in a specific problem situation. A systematic axiomatic analysis of decisionprocedures and algorithms is yet to be carried out."

In this context, A. Guitouni, J.M. Martel and Ph. Vincke propose in [Guitouni et al., 1999] aframework to choose an appropriate multicriteria procedure for a decision-making situation. Thesimilarities, differences and complementarities of certain methods are studied. For that purpose,preference modeling elements of the decision-making situation are represented to characterizethe input of a procedure i, noted as ιi. Moreover, the output of a multicriteria procedure,noted o j, can also be described. This leads to define a pair (ιi,o j) which is susceptible torepresent one or several decision-making situations. Based on propagation rules between theinputs on the one hand, and between the outputs on the other, the authors propose to associatemulticriteria procedures to the couples (ιi,o j) which lead to an input-output matrix. This matrixcontains empty cells, cells with only one procedure and cells with many procedures. This matrix

76

3.5. How to choose a multicriteria ranking method ?

reveals that for some particular problem, there does not exist yet a appropriate multicriteriaprocedure. Furthermore, the cells containing one procedure lead to a unique choice although thischoice may jeopardize the appropriateness of the method (since no comparison can be made).Moreover, the presence of several procedures for a specific couple, raises the question about theway of discriminating them. The reader will find more information in [Guitouni et al., 1999],but let us summarize some of their elements in the next paragraphs.

The MCDA-model of a decision making problem can be characterized by the following elements[Guitouni et al., 1999]:

• A = a1, . . . ,an representing the actions to be investigated9;

• G = g1, . . . ,gq representing the criteria or attributes10;

• E : the performance table where eij = g j(ai) represents the evaluation of the performance of

action ai with respect to criterion g j.

• P denoting the set of all preferences modeling elements: P = ,v j, p j,q j,P j,π,S, . . ..

The evaluations eij can be crisp quantities, discrete/continous random variables (e.g. with mass

or density probability functions), linguistic or fuzzy numbers (with membership functions).Moreover, the measurement scale of criterion g j can be a ratio, an interval or an ordinal scale.Since all combinations are possible, the evaluations contained in the multicriterion table E are,according to the authors, regrouped, without exhaustiveness, into 12 categories. Let us remarkthat nothing has been mentioned of the way of regrouping the inputs.

In the study of [Guitouni et al., 1999], the preference modeling elements of P where limited tothe following 10 elements: true-criterion, indifference thresholds, preference thresholds, vetothresholds, utility/value functions obtained on interval scales, utility/value functions obtainedon ratio scales, pair-wise comparisons, absence of information about the relative importance ofthe criteria, relative importance binary relation of the criteria, explicit vector Ω of the relativeimportance coefficients (weights) of the criteria.

The crosswise of the 12 categories of E-shapes with the elements of P , leads, according to theauthors, to the definition of 208 entering instance which have been categorized in 24 inputsιi, i = 1, . . . ,24. Some of the inputs are as follows: q structures of pre-orders, q structures ofsemi-orders and/or structures of pseudo-orders, q structures of semi-orders and/or structuresof pseudo-orders in addition of veto thresholds, q partial utility/value functions obtained frominterval scales, q structures of pre-orders in addition to a vector Ω of relative importancecoefficients of the criteria, etc. The reader may find all the proposed inputs in [Guitouni et al.,1999] p.8.

On the other hand, the output of a multicriteria procedure should be defined to characterize whatthe decision-maker expects from a decision-aid process. The 7 outputs, retained in the study

9In the study, this set is considered as finite and discrete.10In this section, we will not distinguish them .

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of [Guitouni et al., 1999] are the following: a global evaluation (e.g. a score) for each actionof A (o1), a complete ranking of the actions of A while considering an indifference threshold(semi-order) (o2), a complete pre-order of A (o3), a partial pre-order (o4), the choice of thebest action(s) (o5), the choice of a subset of actions within the best actions (o6), ordered sortingwhere the actions of A have to be assigned to ordered categories (o7)11.

On the basis of the inputs ιi and the outputs o j, most of the multicriteria procedures of theliterature have been assigned to a specific couple (ιi,o j) which lead to the input-output matrixof Table 7 in [Guitouni et al., 1999]. An extract of this table is given in Tab.3.10 where ι17and ι18 represents respectively q structures of pre-orders, in addition to a weight-vector Ω andq structures of semi-orders and/or structures of pseudo-orders, in addition to a weight-vectorΩ. This matrix characterizes more than 1400 decision-making situations. The multicriteriaprocedure is associated to a particular couple (ιi,o j) and is considered to be appropriate to tacklethe decision-making situation. We may thus define a multicriteria procedure as an function Ψfrom the inputs to the set of outputs: ψ ∈ Ψ : I→ O : ψ(ιi) = o j.

Table 3.10 — Input-Output matrix: an extract of Table 7 in [Guitouni et al., 1999].o3 o4 o5 o6

ι17 Regime, Qualiflex Electre II @ Electre Iι18 Promethee II Promethee I @ @

One may remark that some of the outputs can be deduced from other outputs (e.g. a global scoreleads to a complete ranking, a partial pre-order, etc.) and some inputs can be transformed intoother inputs (an numerical scale can be transformed to a, less richer, ordinal scale for example).For that purpose, "propagations relationships" have been defined between on the one handthe inputs, and between the outputs on the other hands. The propagation relations are used toenlighten relations between the inputs and outputs in the input-output matrix.

Let us define the following binary relation on the input:

Definition 3.9. Relation χ: where

(ιi, ι j) ∈ I× I : ιi χ ι j⇔ [ιi = ι j or ι j contains more information than ιi]

This relation can be interpreted as follows: "if a multicriteria procedure can handle an inputrepresenting a level of information, then it is able to handle an input whose information level isricher or more precise measurement level (by ignoring the measurement scale improvement)."Moreover, we have that the χ-relation is reflexive, transitive and antisymetric [Guitouni et al.,1999].As an example we may consider that q structures of semi-orders and/or structures of pseudo-orders (ι j) contains more information than q structures of pre-orders (ιi).

11Sorting and choice procedures should normally not be present in this section. Nevertheless, some elements ofthis analyze will be reconsidered when treating the sorting problem.

78

3.5. How to choose a multicriteria ranking method ?

Definition 3.10. Relation θ: where

(ιi, ι j) ∈ I× I : ιi θ ι j⇔ [ιi = ι j or ι j contains particular case of information of ιi]

This relation can be interpreted as follows: "if a multicriteria procedure can handle properly alevel of information, then it can handle any particular cases of this information level. Whereparticular information leading to this level take particular values. We are thus brought back to aless rich level of information." Moreover, we have that the θ-relation is reflexive, transitive andantisymetric [Guitouni et al., 1999].As an example we may remark that q structures of pre-orders (ι j) contains particular cases ofinformation of q structures of semi-orders and/or structures of pseudo-orders (ιi).

We may define a similar binary relation for the output:

Definition 3.11. Relation ρ: where

(oi,o j) ∈ O×O : oi ρ o j⇔ [oi = o j or o j can be ’naturally’ deduced from oi]

This relation can be interpreted as follows: "if a multicriteria procedure gives an output Oi, andwithout introducing any new information, it is possible to deduce naturally the output of O j fromthe output Oi".A complete ranking of the actions, with eventually ex aequo, as well as a complete ranking,while considering indifference thresholds, can be obtained from a global evaluation of theactions. This relation may be represented as in Fig.3.19 with e.g. (o1,o3), (o1,5 ).

uu

u

u

u

u

u 6 6I

-o7 o6

o4

o3 o5

o2o1

Figure 3.19 — Representation of the ρ-relation.

These relations may be used in the input-output matrix: a multicriteria procedure may be spreadto others couple (ιi,o j). The propagations are illustrated by an extract of Table 8 in [Guitouniet al., 1999] which is represented in Tab.3.11. This means that "even if a procedure was designedto a specific decision situation, it can be used else where; however with care" [Guitouni et al.,1999]. Furthermore, it is obvious from this table that no universal procedure exists. Moreover,one knows, now, which methods has to be compared and in which situation. The reader will findin Table 8 of [Guitouni et al., 1999], in which situation he may use a ranking method describedin this chapter.

79


Table 3.11 — Input-Output matrix after application of the propagation rules: an extract ofTable 8 in [Guitouni et al., 1999].

o3 o4 o5 o6

ι17

Regime, Qualiflex Electre II Bordaρ, Condorcetρ Electre IBordaχ, Condorcetχ Promethee Iθ, Electre IIIθ Regimeρ, Qualiflexρ Conjunctiveχ

Promethee IIθ, ... Oresteθ Promethee IIρ Disjunctiveχ

ι18

Promethee II Promethee I Promethee IIρ Electre Iχ

Bordaχ, Condorcetχ Oresteχ, Electre IIχ Bordaρ, Condorcetρ Conjunctiveχ

Regimeχ, Qualiflexχ Electre IIIθ Regimeρ, Qualiflexρ,... Disjunctiveχ

80

4 Some multicriteria sortingmethods

In this chapter we firstly explain the sorting paradigm. We clearly differentiatenominal classification problems from sorting problems. Moreover, we presentsome desirable properties that sorting procedures should verify.In the second section, we describe procedures based on indifference and simi-larity indexes, namely PROAFTN and TRINOMFC. Then, we introduce sort-ing methods based on the MAUT or outranking relations such as UTADIS,Electre-Tri, Trichotomic Segmentation, nTomic, filtering methods and Pair-Class. These methods are illustrated by means of examples and their proper-ties are analyzed. Some of the methods are described in specific situations inorder to compare them to each other. We finally bring to the light some aspectswhich, to the best of our knowledge, have not been considered in the literatureyet.

4.1 Introduction to sorting problems

The general classification problem has been introduced in Chapter 1. Likewise, we havedifferentiated the nominal classification problem from the ordinal one. In the latter case, weusually speak about sorting problems where the categories are completely ordered, starting fromthose including the most preferred actions to those including the least preferred actions.

Nevertheless, as we have seen in Chapter 2, a decision maker may express an indifference,an incomparability or a preference when comparing two actions. In this work, we considerthus sorting problems as classification problems where preferential information is added to thedefinition of the categories. This signifies that the actions, belonging to different categories,are for the decision maker incomparable, preferred or being preferred. Abusively, we may saythat there is a binary preference relation between the categories which induces an order on thecategories. If this preference relation is transitive, we have a partial order on the categories.Furthermore, this relation might be complete and transitive (i.e. the categories are completelyordered from the best to the worst) or empty (i.e. all the categories are incomparable orcompletely not-ordered). In Fig.4.1, we have represented different classification problems, based

81

Some multicriteria sorting methods

on the existing relations between the predefined classes.One may wonder that we differentiate nominal classification problems from sorting problemswith completely not-ordered (or incomparable) categories since in both cases there is noorder on the categories. The main difference lies in the way the decision maker considers thecategories. In the former case, the classes are considered as dissimilar whereas in the latter case,the categories are considered as incomparable. In the former case, we may describe the classesby means of attributes whereas in the latter case by criteria. We will see in this work that takinginto account preference information, may help a decision maker to clarify the assignments. This,even when the categories are completely not-ordered.

incomparable categories completely ordered categories

partially ordered categories

sorting problemsnominal classificationproblems

supervised classification

@@@@@@R

@@@@@@R

?

complete

transitive

preference relation

empty

(dis-)similarity relation

Figure 4.1 — Representation of the different classification problems on the basis of the dif-ferent relations between the predefined groups.

A sorting model may be defined analogously as a classification model. We will thus denoteby SG the sorting procedure which take into account a set of criteria G . Any action ai ∈ A ,defined by a vector g of performances for the set of criteria G , will be mapped to a label, noted

82

4.1. Introduction to sorting problems

CS (ai) ∈ C1. Formally, we may represent the model as follows: G → C : g→CS = SG (g). Letus remark that an action may, according to some procedures, actually be assigned to none, oneor several categories denoted by CS (ai). In such a case, we may define P(C) as the set of all thesubsets of C and SG as the following function: G →P(C) : g→CS = SG (g).

In previous paragraphs, we have differentiated the problems without considering the methods.Classification and sorting methods can also be distinguished on the basis of their properties.Methods may be for example neutral to the label of the actions and assign actions to none,one or several categories. The stability of the assignments when categories are split or fused,may be a feature requested by the decision maker. Moreover, the fact of taking into account thepreferential information in the assignment procedure can be crucial. This aspect is critical sinceit permits to point out that certain methods may be more appropriate or unappropriate for certainproblems. In this context, C. Zopounidis and M. Doumpos mentioned that: "using a similaritymeasure to perform the classification of the actions is of limited usefulness, simply becausethis would overlook the ordinal definition of the classes and the implications that it has for thepreferential system of a decision-maker [Doumpos and Zopounidis, 2004a]." This may be thecase, even when there is no order on the categories as will be illustrated in this work. For thatpurpose, a property, called the "preference-orientation dependency property", has been definedwhich may differentiate sorting methods from nominal classification methods (see Section 4.2).

Sorting methods may also be differentiated on the basis of the presence or absence of parametersin their model. Two types of sorting methods can be considered: parameter-based methodsand parameter-free methods (or non-parametric methods) [Yevseyeva, 2007]. In the formercase, it is possible to construct a model of the decision maker’s behavior on the basis of someparameters (e.g. preference parameters). In this case, these methods can be further subdividedon the basis of the used model (the use of indifference or similarity relations, outrankingrelations or utility functions) and the way of defining the parameters (directly by interactionwith the decision maker or indirectly by so-called elicitation procedures). Let us first mentionElectre-Tri [Yu, 1992b] since it is the most widely used multicriteria sorting method based onthe outranking relations approach [Zopounidis and Doumpos, 2002a]. Similar schemes are alsoemployed by other outranking based sorting methods such as the Trichotomic Segmentationmethod [Moscarola and Roy, 1977], the filtering methods [Perny, 1998], nTomic [Ostanelloand Massaglia, 1991], PairClass [Doumpos and Zopounidis, 2004a], TOMASO [Marichal et al.,2005], etc. An alternative form of these developed models is the utility-based functions with foramong others UTADIS [Jacquet-Lagrèze, 1995], MHDIS [Doumpos and Zopounidis, 2002],etc. Finally, similarity and indifference indexes are used in for example TRINOMFC [Léger andMartel, 2002] and PROAFTN [Belacel, 2000a].

On the other hand, in parameter-free methods the structure of the decision maker’s behavior isconsidered to be too complex in ordered to be defined in a formal model [Yevseyeva, 2007].Parameter-free methods contain for instance so-called decision rule or decision tree methodsbased on sorting examples [Doumpos and Zopounidis, 1998]. The decision rules speak thelanguage of the sorting examples given by the decision maker. They have thus a natural and easy

1For simplification, we will use CS (ai) in stead of CSG (g(ai)).

83


interpretation [Zopounidis and Doumpos, 2002a]. In this context, [Pawlak, 1984a,b; Pawlak andSlowinski, 1994] have proposed an axiomatized methodology for constructing decision rulepreference models from decision examples, based on the rough sets theory. Over the past decade,significant research has been conducted on the use of the rough set approach as a methodologyof preference modeling in multicriteria decision problems [Greco et al., 1998, 2000], even inpresence of inconsistent data.We refer the interested reader to a more exhaustive survey proposed by [Doumpos and Zopouni-dis, 2004a].

In this chapter we first present some desirable properties that sorting methods should verify(Section 4.2). Section 4.3 is devoted to sorting methods which use indifference indexes whereasin Section 4.4, methods based on similarity indexes. In Section 4.5 and 4.6 we address thesorting problem by means of utility based and outranking based sorting procedures. Thesemethods are illustrated by means of examples and their properties are analyzed. Some of themethods of Section 4.6 are described in specific situations in order to compare them to eachother and to point out their main differences.

Let us finally remark that we will not consider parameter-free methods in this work. This choicewill be clear when presenting our contributions to this field. Moreover, throughout this chapter,we suppose that all the parameters of the methods have been defined precisely.

4.2 Properties of sorting methods

Usually, we define different properties for a sorting method SG according to the type of sortingproblem. The most important desirable properties are presented in Fig.4.2 where "→" meansthat the specific sorting procedures for completely ordered or not-ordered categories, may verifythe "classical" properties of a sorting method.

Without being strictly formal, we may define the following properties:

Property 4.1. Property of neutrality [Belacel, 2000a; Yu, 1992a]The assignment of each action does not depend on its given label: if we give two different labelsai and bi to the same action (i.e. ai ≡ bi), we have:

CS (ai) = CS (bi)

Property 4.2. Property of uniqueness [Yu, 1992a]The sorting method assigns each action ai to exactly one category: | CS (ai) |= 1.

If SG assigns each action, to exactly one category, we have that SG defines an application fromA to C. Let us remark, that in certain conditions the decision maker may not desire this property.The decision maker may for instance be aware of a case of ambiguity.

84

4.2. Properties of sorting methods

Properties for sorting proceduresfor completely not-ordered categories:

- criteria dependency

Properties for sorting proceduresfor completely ordered categories:- monotonicity- stability-2- pairwise assignment consistency- conformity-l

Properties for sorting procedures:

- conformity-c- homogeneity- independency- uniqueness- universality- neutrality

- stability-1

>

Figure 4.2 — Properties of sorting procedures according to the sorting problem where partic-ular sorting procedures may verify classical properties (which is represented by "→").

Property 4.3. Property of independency [Belacel, 2000a; Yu, 1992a]The assignment of an action ai does not depend on the assignment of an action a j (with i 6= j):

C AS (ai) = C ai

S (ai) = C A\a jS (ai).

The way how the actions of A compared themselves, do certainly not influence their assignment.This is according to Bernard Roy a distinguishing feature with a choice or ranking procedure[Roy and Bouyssou, 1993]. This will be discussed more in detail in Chapter 5.

Property 4.4. Property of stability-1 [Yu, 1992a]The fusion of categories or the splitting of a category into several ones, may not alter the assign-ments of actions to the non-modified categories.

In other words, suppose that an action ai is initially assigned to a category (or a set of categories).If we fuse two categories (different from the one to which ai is initially assigned) or if we split acategory (different from the one to which ai is initially assigned) into several categories, ai willremain assigned to same initial category (initial set of categories).

In some sorting procedures, fictitious actions, noted r j, are used in order to define the categories.Henceforth, any fictitious action is referred to reference profile. These reference profilesconstitute the reference set, noted R = r1,r2, . . . ,rm. Each reference profile rk is evaluated onthe set of criteria which define completely rk.We distinguish central profiles and limiting profiles. Central profiles may be used wether thecategories are ordered or not. On the other hand, limiting profiles are used in the case where thecategories are completely ordered. The limiting profile define in these cases the boundary of

85


two consecutive categories: it plays the role of the upper limit and the lower limit of these twoconsecutive categories.The classification of the actions of A is, in these type of methods, processed in two phases. Thefirst consisting in comparing an action to be assigned to each reference profile on the basis of anoutranking relation, an outranking degree, an indifference relation, an indifference index, etc.The type of comparison depends among others on the fact that the categories are ordered or not.In the second phase, this comparison is exploited to decide upon the classification of the action[Doumpos and Zopounidis, 2002].

For sorting procedures which use central profiles or limiting profiles, we may have the followingproperties:

Property 4.5. Property of strong homogeneity [Yu, 1992a]If the outranking (preference) relations between an action ai and the reference profiles are thesame as the outranking relations between a j and the reference profiles, ai and a j are affected tothe same categories.

Property 4.6. Property of weak homogeneityIf the outranking (preference) degrees between an action ai and the reference profiles are thesame as the outranking degrees between a j and the reference profiles, ai and a j are affected tothe same categories.

If the categories are defined by a set of limiting profiles we may have the following conformityproperty:

Property 4.7. Property of conformity-l [Yu, 1992a]Each action, whose performances are in between two consecutive limiting profiles, is assignedto the category defined by the limiting profiles.

On the other hand, if the categories are defined by a set of central profiles we may have thefollowing conformity property:

Property 4.8. Property of conformity-cEach central profile defining a priori a category, is univocally assigned to its correspondingcategory.

Moreover, consider a sorting problem where the categories are completely ordered from the bestto the worst. We will denote by C1 the best category and CK the worst. Besides, CiDC j, with i≤j, denotes that category Ci is at least as good as category C j. In this context, we may define thefollowing properties.

86

4.2. Properties of sorting methods

Property 4.9. Property of monotonicity [Yu, 1992a]If an action ai dominates an action a j, action ai may not be assigned to a category worse thana j.

In particular, if the sorting procedure assigns each action to exactly one category we have:

ai W a j⇒ CS (ai)DCS (a j)

Property 4.10. Property of stability-2 [Yu, 1992a]The fusion of consecutive categories or the splitting of a category into several consecutive ones,may not alter the assignments of actions to the non-modified categories.

This property is analogous to Property 4.4 except that the fusion or the splitting is limited toconsecutive categories.

As mentioned previously, sorting methods where the categories are defined by profiles, exploita comparison between the actions and the reference profiles. When the categories are ordered,often a outranking based comparison is made. In this context, let us analyze the relation betweenthe assignment of two actions (to be assigned) and their pairwise comparison. For that purpose,let us define the following binary relation between two actions on the basis of their assignment(result):

Definition 4.1. ∀ai,a j ∈ A : ai PS a j⇔CS (ai)BCS (a j)

where CS (ai)BCS (a j) means that max[CS (ai)] < min[CS (a j)]. In other words, we have ai PS a j

if ai is assigned to a better category (categories) than a j.

Property 4.11. Property of pairwise assignment consistencyA sorting method SG is consistent with the pairwise comparisons between two actions, if ∀ai,a j ∈A , ai PS a j, implies that ai P a j.

This property analyzes if the preferential relational system used in the sorting procedure respects(or not) pairwise comparisons between two actions ai,a j, regardless the sorting problem. Inother words, a sorting procedure respects the pairwise comparisons, if, given CS (ai) better thanCS (a j), we necessarily have that ai is preferred to a j.As we will see, most of the sorting methods based on pairwise comparisons with referenceprofiles, do not respect this property (e.g. Electre-Tri, F lowSort, etc.) since the preferencerelations are often not transitive. On the other hand, the UTADIS sorting method respects thisproperty since it is based on an absolute score.

Finally, in a sorting problem, when the categories are completely not-ordered (i.e. the categoriesare considered as incomparable), we may analyze the use of the criteria. If a sorting methoduses a set of criteria, it seems natural that the results depend on the preferential informationassociated to the criteria. In other words, the assignments should sometimes change when thecriteria information change.

87


For that purpose, let us define the inverse function Inv(G ,g j). This functions transforms acriterion set G to G ′

, by transforming one criterion g j to g′j such that the induced preference

order on A , defined by the criterion g′j, is obtained by inverting the induced preference order

on A , defined by g j. In other words, we invert thus the preference orientation on criterion g′j: if

aSg j b we have that bSg′ja.

In case of a real-valued criterion, we may define the inverse function Inv(G ,g j) such thatInv(G ,g j) inverts the criterion g j by changing the sign of its criterion values:

Definition 4.2. ∀G ,∃g j ∈G : Inv(G ,g j) = (G \g j)∪g′j where g

′j(ai) =−g j(ai),∀ai ∈A

This can be justified by the fact that aSg j b ⇔ g j(a) ≥ g j(b) − q j and thus that−g j(b) ≥−g j(a)−q j, which leads to g

′j(b) ≥ g

′j(a)−q j and thus bSg′j

a.

Let us remark that we may define the inversion of a subset of criteria P(G) of G , notedInv(G ,P(G)), by replacing g j by P(G) in previous definitions such that if aSPb we have thatbSP

′a.

Property 4.12. Property of preference-orientation dependencyA sorting procedure is preference-orientation dependent, if there exists a set A , such that, invert-ing a subset of criteria P(G), leads to at least one change in the assignments. Formally:

∃A | ∃ai ∈ A ,P(G) ∈ G : CSG (ai) 6= CSG ′(ai) where G

′= Inv(G ,P(G))

When working with criteria in sorting problems, one may except that the preference orientationof the criteria play a role. If this property is not verified while using criteria, one may wonder onthe usefulness of the considered criteria.

Strictly speaking, this property should be analyzed when inverting one or several criteria at thesame time. The inverse function should for that purpose be defined for a subset of criteria of Ginstead of solely criterion g j.

4.3 Sorting based on indifference indexes

In this section we will briefly describe sorting methods based on indifference indexes. At first,we present an indifference index. used in the PROAFTN method (see Section 4.3.2).

4.3.1 Indifference Index I(a,b)

In this section, we present a way to measure to which extend two actions a,b are indifferent orsimilar. This indifference index has been proposed by Belacel [Belacel and Boulassel, 2000]. Itis based on the partial indifference and partial discordance indexes defined for each criterion orattribute. The global indifference index aggregates these partial indexes.

88

4.3. Sorting based on indifference indexes

4.3.1.1 Partial indifference degree cIj(a,b)

This partial indifference index measures the extent to which a and b are indifferent consideringthe criterion g j. It is a valued relation between a and b. Its value is obtained as follows [Belacel,2000b]:

cIj(a,b) =

1, if g j(b)− s−j (b) ≤ g j(a) ≤ g j(b)+ s+j (b)

g j(a)+d−j (b)−g j(b)+s−j (b)d−j (b) , if g j(b)− s−j (b)−d−j (b) < g j(a) < g j(b)− s−j (b)

g j(b)+s+j (b)−g j(a)+d+

j (b)d+

j (b) , if g j(b)+ s+j (b) < g j(a) < g j(b)+ s+

j (b)+ d+j (b)

0, otherwise.(4.1)

g j(b)+ d+j (b)

-0

6

1

g j(a)

g j(b)−d−j (b)

g j(b) g j(b)

cIj(a,b)

- -

g j(b)

s−j s+j

Figure 4.3 — Representation of the partial indifference index cIj(a,b).

where d+j (b), d−j (b) represent two discrimination thresholds and s+

j (b) and s−j (b) two indiffer-ence thresholds as illustrated in Fig. 4.3. The indifference thresholds have the following role:if the difference between a and b on criterion g j is smaller than this threshold, they are stillconsidered as indifferent on criterion g j (i.e. cI

j(a,b) = 1). If, on the other hand, the differenceis higher than d+

j (b) + s+j (b) (d−j (b) + s−j (b)), they are considered as totaly not indifferent on

criterion g j (i.e. cIj(a,b) = 0).

For commodity reasons, let us note s+j and s−j respectively q+

j and q−j . Moreover, s+j +d+

j = p+j

and s−j + d−j = p−j . This permits to interpret p+j and p−j as follows. If, the difference between

a and b on criterion g j is higher than p+j (p−j ), they are considered as totaly not indifferent

on criterion g j (i.e. cIj(a,b) = 0). With the previous notations, we have the following partial

indifference degree (represented in Fig.4.4) :

89


cIj(a,b) =

1, if g j(b)−q−j (b) ≤ g j(a) ≤ g j(b)+ q+j (b)

g j(a)+p−j (b)−g j(b)p−j (b)−q−j (b) , if g j(b)− p−j (b) < g j(a) < g j(b)−q−j (b)

g j(b)+p+j (b)−g j(a)

p+j (b)−q+

j (b) , if g j(b)+ q+j (b) < g j(a) < g j(b)+ p+

j (b)

0, otherwise.

(4.2)

g j(b)+ p+j (b)

-

6

1

g j(a)

g j(b)− p−j (b)

g j(b)−q−j g j(b)+ q+j

cIj(a,b)

- -

g j(b)

q−j q+j

0

Figure 4.4 — Representation of the partial indifference index cIj(a,b).

Let us remark that if we suppose that q j(b) = s+j , q j(a) = s−j (or q j(b) = q+

j and q j(a) = q−j ),p j(b) = d+

j +s+j and p j(a) = d−j −s−j (or p j(b) = p+

j and p j(a) = p−j ), we obtain the followingrelation between the partial indifference degree and the concordance index of the outrankingdegree (See Section 2.6.1.1):

cIj(a,b) = min[cS

j(a,b),cSj(b,a)] (4.3)

We can thus conclude that the only character measured by this symmetric index is the similarityor indifference degree. We loose all information about the preferences between these twoactions. Moreover, if we suppose that s+

j = s−j and that d+j = d−j , the preference direction of

the criterion g j does not play a role anymore in this degree. A criterion may in these cases beconsidered as an attribute.

The use of the indifference thresholds, permits to define an action b by the way of an inter-val: [g j(b),g j(b)] where g j(b) = g j(b)− s−j and g j(b) = g j(b)+ s+

j . In what follows, we willsuppose that b is defined by this interval. The partial indifference index can thus be written asfollows:

cIj(a,b) =

1, if g j(b) ≤ g j(a) ≤ g j(b)g j(a)+d−j (b)−g j(b)

d−j (b) , if g j(b)−d−j (b) < g j(a) < g j(b)g j(b)−g j(a)+d+

j (b)d+

j (b) , if g j(b) < g j(a) < g j(b)+ d+j (b)

0, otherwise.

(4.4)

90


This index is equal to 0 when a and b are considered as completely (not in-) different on criteriong j whereas it equals 1 when they are sufficiently indifferent.

4.3.1.2 Partial discordance index dIj(a,b)

This partial discordance index measures the extend to which a and b are not indifferent consider-ing the criterion g j. It defines a valued relation between a and b. Its value is obtained as follows[Belacel, 2000b]:

dIj(a,b) =

1, if g j(b)−υ−j (b) > g j(a) or g j(a) < g j(b)+ υ

+j (b)

g j(a)+d−j (b)−g j(b)d−j (b)−υ

−j (b) , if g j(b)−υ

−j (b) < g j(a) < g j(b)−d−j (b)

g j(b)−g j(a)+d+j (b)

d+j (b)−υ

+j (b) , if g j(b)+ d+

j (b) < g j(a) < g j(b)+ υ+j (b)

0, if g j(b)−d−j (b) < g j(a) < g j(b)+ d+j (b)

(4.5)

-

61

g j(a)g j(b)+ d+

j (b)

g j(b)−d−j (b)

dIj(a,b)

g j(b)−υ−j (b)

g j(b)+ υ+j (b)

0

Figure 4.5 — Representation of the partial discordance index dIj(a,b).

where υ−j (b) and υ

+j (b) represent two veto thresholds as illustrated in Fig.4.5. These verify the

following conditions: υ+j (b) > d+

j (b), υ−j (b) > d−j (b). For the same commodity reasons2, let

us write v+j = υ

+j − s+

j (v−j = υ−j − s−j ), such that the discordance is equal to 1 as soon as v+

j andv−j are exceeded. Moreover, if v−j = v+

j , the preference direction does not play role anymore: thediscordance index is symmetric. Furthermore, since the thresholds v−j and v+

j , may be interpretedin the same ways as the veto thresholds of the outranking relation, we have the following relation:

dIj(a,b) = min[dS

j (a,b),dSj (b,a)] (4.6)

2We may define the thresholds both in a "similarity" context as in "preference" context.

91


4.3.1.3 Global indifference index I(a,b)

On the basis of the partial discordance and indifference indexes, the global indifference indexaggregates the q criteria by taking into account the corresponding weights (denoted by w j,∀ j =1, . . . ,q). This measure is defined as follows:

I(a,b) =

[∑

j=1,..,qw j.cI

j(a,b)

]×

[∏V

[1−dI

j(a,b)1−CI(a,b)

]

](4.7)

where V represents respectively the set of criteria for which dIj(a,b) > CI(a,b) =

[∑ j=1,...,q w j×cIj(a,b))]. This index is small when there are no sufficient arguments to accept the

assertion that they are indifferent. It is composed of two parts: the left part represents the globalconcordance with the assertion that a and b are indifferent whereas the right part the discordance.

Let us remark that we have pointed out, in previous sections, that this index may in some casessymmetric. Moreover, under certain conditions (see Section 4.3.1.1), we have:

I(a,b) = min[S(a,b),S(b,a)] (4.8)

This indifference index is in this case symmetric and does not take "strict" preference informationinto account. An example of this indifference index will be given in Section 4.3.2, Example 4.1.

4.3.2 PROAFTN

4.3.2.1 Introduction

PROAFTN (PROcédure d’ Affectation Floue dans le cadre de la problématique de Tri Nominal)has been developed by Nabil Belacel in 1999 to address classification problems. It has been usedin many applications, especially in the medical sector [Belacel, 2000b; Belacel and Boulassel,2000, 2001].

The classes are defined in a nominal way, i.e. with no preference order on the classes and definedby central profiles to which an action of A , to be classified, is compared. We will suppose thatthe class Ch (∀h = 1, . . . ,K) is defined by one central profile, denoted by rh, although the methodpermits to use several profiles. According to the authors, the reference profiles of differentcategories have to be incomparable (in the sense of Section 2.6.1.4) between themselves.Nevertheless, if the thresholds (similarity and dissimilarity thresholds for instance) are notnecessarily defined in the sense of preference parameters, we suppose that the central profileshave to be non-indifferent. This is the unique condition imposed on the reference profiles suchthat the categories are defined in an acceptable manner.

To define a reference profile rh of class Ch, we will associate on each criterion g j (∀ j = 1, . . . ,q),an interval [g j(rh),g j(rh)].

92


The assignment rule is based on the following idea: "Any action which is considered asindifferent or sensibly equivalent to a reference profile, will be assigned to the correspondingclass "[Belacel, 2000b].

To evaluate the similarity between a reference profile rh and an action a of A , a indifferenceindex I (rh,a) (symmetric or not) is computed. This index results from the aggregation of thepartial indifference and discordance indexes and has been defined in Section 4.3.1.

4.3.2.2 Assignment rules

An action a ∈ A will be assigned to a class Ch according to one of these two different rules:

Assignment Rule 4.3.1. CPRO(a) = Ch⇔ I(a, rh) = maxI(a, rk), ∀k = 1, ...,K

Assignment Rule 4.3.2. CPRO(a) = Ch | I(a, rh) > λI;∀h = 1, ...,K

where λI > 0.5 represents a threshold to be fixed beforehand. The higher λI the "stricter" this rule.

The difference between these two assignment rules lies in the fact that the first rule will alwaysassign an action to a category. In presence of some ex-aequo between the indifference degrees,the assignment will nevertheless not be unique. On the other hand, according to the secondrule, it may happen that an action is assigned to none category. Furthermore, an action can beassigned to several categories without necessarily ex-aequo between the indifference degrees.

Let us remark that some authors have proposed an automatic method that helps to establish theparameters, used in this sorting procedure, from a given data set. The parameters are obtainedwith a variable neighborhood search metaheuristic [Belacel et al., 2007].

Example 4.1. As illustration, let us consider the following classification problem. This exampleis an adaptation of the example given by P. Perny in [Perny, 1998] (p.161) but will differ fromExample 4.2 since the classes are not ordered.The human resource department defines thetwo following classes: the engineers and the technical salespeople. No preference between theengineers and the technical salespeople may be expressed by the decision maker. These areconsidered as not similar or indifferent (incomparable). We will note the classes respectivelyC1 and C2.

The categories will be defined by central reference profiles which are evaluated according to the5 following criteria which have to be maximized:

1. g1: software knowledge

2. g2: programming experience

3. g3: commercial aptitude

93


Table 4.1 — Evaluation of the performances of the central reference profiles.

rn g1 g2 g3 g4 g5

r1 [9,11] [9,11] [14,16] [14,16] [11,13]r2 [14,16] [14,16] [9,11] [9,11] [11,13]


ai g1 g2 g3 g4 g5

a1 13 12 13 12 13a4 15 15 9 13 12a7 10 18 10 18 17a13 10 10 10 10 10.5

4. g4: potential mobility

5. g5: leadership attitude

The evaluations of the reference profiles on the different attributes are given in Tab. 4.1and represented in Fig.4.6. The parameters associated to the attributes are as follows:∀g j ∈ G : d+

j = d−j = 1,υ+j = υ

−j = 5 and w j = 0.2. We want to classify actions a1, a4, a7 and

a13 which evaluations are given in Tab.4.2.

The results of the assignments according to the 2 different assignment-rules are given in Tab.4.3.Moreover, we have chosen two different values for λI: 0.6 and 0.85. We may conclude that theassignments are not always unique and that sometimes an action is not assigned, as pointed outin section 4.3.2.

r2r1

-

g1

g2

g3

g4

g5

-

-

-

-

Figure 4.6 — Representation of the performances of the reference profiles r1 and r2 and theactions a1 and a4.

94

4.4. Sorting based on similarity indexes

Table 4.3 — Assignments of the actions.

ai I(ai,r1) I(ai,r2) CPROλI=0.6 CPRO

λI=0.85 CPROmax

a1 0.6 0.6 C1 and C2 /0 C1 and C2

a4 0 0.8 C2 /0 C2

a7 0 0 /0 /0 C1 and C2

a13 0.5 0.5 /0 /0 C1 and C2

4.3.2.3 Properties

The properties of PROAFTN are the following: neutrality, universality (Assignment Rule 4.3.1)or uniqueness (Assignment Rule 4.3.2), stability, independency, homogeneity (weak and strongaccording to the assignment rule) and conformity. The proofs can be found in [Belacel, 2000b].Let us remark that in certain cases (i.e. for certain values of the indifference, preference andveto thresholds), the partial indexes are symmetric and thus do not take into account the prefer-ence direction. In these cases, the method will not satisfy the preference-orientation dependencyproperty. This is due to the fact that the indifference degree reflects only the indifference ornon-indifference. As we will see, on the basis of the same preference information, we mightdistinguish the cases of non-indifference. This will further be analyzed in Section 12.5.

4.4 Sorting based on similarity indexes

In this section we will briefly describe sorting methods based on similarity indexes. At first, wepresent a similarity index used in the TRINOMFC method (see Section 4.4.1).

4.4.1 Similarity Index SI(a,b)

Léger and Martel have proposed a similarity index to measure to which extend two actions a,bare similar to each other. This similarity index will be used in the TRINOMFC sorting method(see Section 4.4.2). This index will be based on the set of attributes or criteria G3. The readerwill find more information in [Léger and Martel, 2002] and [Yevseyeva, 2007].

4.4.1.1 Partial Similarity Index SI j(a,b)

For each attribute or criterion g j, the similarity between two actions a,b will be evaluated ac-cording to a similarity function. The more the actions are considered as similar (i.e. the smaller| g j(a)− g j(b) |), the higher this degree. If a,b are not considered as similar, it will be closeto zero. The value of this index is nevertheless always in the interval [0,1] and is computed infunction of | g j(a)− g j(b) |. This measure is thus symmetric and does thus not depends on thepreference direction. Some4 examples of similarity functions are given in Fig.4.7 where s j(b)and d j(b) represent respectively the similarity and dissimilarity thresholds on g j. s j(b) indicates

3This will be discussed in Section 4.4.2.34Other similarity functions are proposed in [Léger and Martel, 2002].

95


6

- -

6

SI j(a,b)SI j(a,b)

s j(b) s j(b) d j(b)

11

| g j(a)−g j(b) | | g j(a)−g j(b) |0 0

Figure 4.7 — Representation of some similarity functions for the computation of SI j(a,b).

the largest difference between a and b on g j such that they remain similar for the decision maker.On the other hand, d j(b) indicates the smallest difference between a and b on g j such that theyare considered as dissimilar.

Let us remark that the similarity degrees SI j(a,b) may be obtained in a similar way when usingthe partial indifference degrees (see Section 4.3.1.1) if we suppose in the first case (left figureof Fig.4.7) that d+

j = d−j = 0 and s+j = s−j = s j(b). In the second case (right figure of Fig.4.7),

if s+j = s−j = s j(b) and if d+

j = d−j = d j(b)− s j(b), we obtain the same value as the partialindifference degree (Section 4.3.1.1).

4.4.1.2 Global Similarity Index SI(a,b)

The global similarity index is obtained by the following aggregation, where w j = w j(b) repre-sents the importance of attribute or criterion g j, associated to b:

SI(a,b) =q

∑j=1

w j×SI j(a,b) (4.9)

One of the main differences between the indifference index of PROAFTN and the global similar-ity, is that the latter one do not incorporate veto or discordances indexes. But, if no discordancesare taken into account and if the parameters respect the conditions given in previous section, themeasures will be equal. Both may be used for attributes or criteria (which preference directionwill not taken into account).

4.4.2 TRINOMFC


TRINOMFC (TRI NOMinal basé sur des Fonctions Critères) has been developed by Leger andMartel in 2002 to address classification problems [Léger and Martel, 2002]. The method is very

96

4.4. Sorting based on similarity indexes

similar 5 to the PROAFTN method.

The classes are defined in a nominal way, i.e. with no preference order on the classes andare defined by central profiles to which an action of A , to be classified, is compared. We willsuppose that the class Ch (∀h = 1, . . . ,K) is defined by one central profile which will be denotedby rh, although the method permits to use several profiles. Let us remark that no condition isimposed on the reference profiles.Each reference profile rh will be evaluated on a set of attributes or criteria6 G = g1, . . . ,gq.

The assignment rule is based on the following idea: an action which is considered as similar to areference profile, will be assigned to the corresponding class.

To evaluate the similarity between a reference profile rh and an action a of A , a similarity indexSI(rh,a) is computed. This index results from the aggregation of the partial similarity indexesand has been defined in Section 4.4.1. Let us remark that we have given in Section 4.4.1.2 theconditions for which the indifference of PROAFTN and similarity indexes of TRINOMFC arethe equal.


The assignment rules are analogous to the ones defined in the PROAFTN method. An actiona ∈ A will be assigned to a class Ch according to one of these two different rules:

Assignment Rule 4.4.1. CT RI(a) = Ch⇔ I(a, rh) = maxSI(a, rk), ∀k = 1, ...,K

Assignment Rule 4.4.2. CT RI(a) = Ch | SI(a, rh) > λI;∀h = 1, ...,K

where λI > 0.5 represents a threshold to be fixed at forehand. Again, the difference betweenthese two assignment rules lies in the fact that the first rule will always assign an action to acategory. In presence of some ex-aequo the assignment will nevertheless not be unique. On theother hand, it may happen that an action is assigned to none or to several categories according tothe second rule. The properties of these assignment rules are identical to the ones of PROAFTN.

4.4.2.3 Properties

The properties of TRINOMFC are similar to those of PROAFTN: neutrality, universality (As-signment Rule 4.4.1) or uniqueness (Assignment Rule 4.4.2), stability, independency, homogene-ity (weak and strong according to the assignment rule) and conformity.Moreover, given the definition of the partial similarity index (see Section 4.4.1.1), which is infunction of the absolute differences between the evaluations (| g j(a)− g j(b) |), the similarityindex does not take into account the preference direction on the criteria. As as consequence,

5This is a symmetrical assertion!6This will be discussed in Section 4.4.2.3

97


we might say that TRINOMFC does not fulfill the preference-orientation dependency property(Property 4.11). Criteria play thus an analogous role as attributes.

4.5 Sorting based on MAUT

In this section we will briefly describe sorting methods based on utility functions.

4.5.1 UTADIS: UTilités Additives DIScriminantes

The UTADIS sorting method permits to assign a set of actions to K completely ordered categoriesC1, . . . ,CK where C1 represents the best category and CK the worst one. The UTADIS method isbased on the Multi Attribute Utility method, UTA (see Section 3.2.1).It associates to each action ai to be assigned, a global score U(ai). This score is an aggregationof the q marginal utility functions associated to the q criteria. The global score of each action isthen compared to constant thresholds (called the utility cut-off points), denoted by δ1, . . . ,δK+1.These thresholds define the limits of the categories. If the conditions 3.2.1 and 3.2.2 are fulfilled,we have moreover that δ1 = 1 and δK+1 = 0.The global assignment scheme is given in Fig.4.8 and the assignment rules are as follows:

Assignment Rule 4.5.1.

CU (ai) = Ch, if δh > U(ai) ≥ δh+1

'

&

$

%

a1

ai

a j

a2

a3

an

. . .

•

•

•

••

•

A

s

U(g1(a j), . . . ,gq(a j))

-

6

~

*

j

*

U(.)

δ2

δK

?

6

6?

?

6

C1

CK

C j

δ j

1

0

Figure 4.8 — Representation of the UTADIS sorting model.

The UTADIS method requires thus a set of parameters such as the weights of the criteria, themarginal utility functions, the cut-off points, etc. Different methods have been proposed toelicit these parameters on the basis of preference disaggregation models [Beuthe and Scannella,

98

4.5. Sorting based on MAUT

1997; Zopounidis and Doumpos, 1996; Jacquet-Lagreze and Siskos, 1982; Doumpos andZopounidis, 1999; Jacquet-Lagrèze, 1995]. In these models, a decision maker gives a set ofassignment examples (called the reference set A ′

): ∀a′i ∈ A ′, whose evaluations on the criteria

are completely defined, the assignment is given by the decision maker, i.e. CU (a′i) = C j. On

the basis of these assignments, an assignment model and utility thresholds are proposed bymathematical programming in order to respect as much as possible the assignments given bythe decision maker (i.e. C ∼= C in Fig.4.9 which is taken from [Doumpos and Zopounidis, 2002]).

Figure 4.9 — Representation of the classification paradigm taken from [Doumpos and Zo-pounidis, 2002] p.83.

99


4.5.1.1 Properties

It is obvious that the UTADIS method presents the properties of neutrality, uniqueness, indepen-dency and stability. Given the condition of monotonicity on the marginal utility functions, theproperty of monotonicity is indeed verified.Moreover, the assignments are coherent with the preference relational system induced by theglobal utilities: if CU (ai)BCU (a j)⇒U(ai) > U(a j). In other words, if an action ai is assignedto a better category than a j, we are sure that ai is preferred to a j since its utility score is higher.The property of "pairwise assignment consistency" (Prop.4.11) is thus verified.Nevertheless, this method may suffer from the drawback of total compensation as presented inSection 3.2.1. Furthermore, situations of incomparability can not be modeled and the elicitationof the parameters may not always be easy.

4.5.2 MHDIS: Multi-group Hierarchical DIScrimination method

The MHDIS method can be considered as an alternative to the UTADIS method, since it proceedsas a sequential/hierarchical process [Doumpos and Zopounidis, 2002]. At each stage h of thehierarchical discrimination process, two choices are available for the classification of an actionai:

1. To decide that ai belongs into group Ch

2. To decide that ai belongs at most into the group Ch+1: to Ch+1, Ch+2, . . . or Cq.

This decision is based on the evaluation of the appropriateness of assigning action ai to Ch, notedAh(ai) and on the appropriateness of assigning ai to a worse category, noted A>h(ai). Thesemeasures are defined as follows:

∀ai ∈ A : Ah(ai) =q

∑j=1

ah j(g j(ai)) (4.10)

∀ai ∈ A : A>h(ai) =q

∑j=1

a>h j(g j(ai)) (4.11)

where Ah(ai) denotes the utility of classifying action ai into group Ch on the basis of its perfor-mances. Therefore, ah j(g j(ai)) denotes the corresponding marginal utility function regardingthe classification of action ai into Ch given its performances on criterion g j. Let us remarkthat these marginal utility function are defined in such a way that the monotonicity property issatisfied. The interested reader may find more information in [Doumpos and Zopounidis, 2002].Conceptually, Ah(ai) measures the similarity of the alternatives to the characteristics of groupCh. On the other hand, A>h denotes the utility to classify ai into a worse group than Ch.Let us remark that the marginal utility functions ah j have a completely different meaning asthe marginal utility functions u j in the UTA(DIS) model. Whereas, u j(g j(ai)) indicates theperformance of action ai on criterion g j, ah j serves as measure of the conditional similarity of ai

to the characteristics of group Ch.

100

4.6. Sorting based on outranking relations

On the basis of these measures, we may define the decision rule at stage h as follows:7


CM(ai) = Ch, if Ah(ai) ≥ A>h(ai)

CM(ai) = Ch+1∪ . . .∪Cq, if Ah(ai) < A>h(ai)

4.5.2.1 Properties

At the end of the K−1 stages, all the actions of A are assigned to exactly one category. MHDISverifies thus the property of uniqueness. As the UTADIS method, MHDIS enjoys the propertiesof neutrality, independency and monotonicity. When fusing or splitting two categories, it seemsappropriate to define new marginal utility functions a>h j(g j(ai)) and ah j(g j(ai)), which meansthus that the assignments may change. On the contrary of UTADIS, nothing can be said aboutthe pairwise comparison between the actions ai and a j on the basis of their assignment. Property4.11 does not apply for this method.

4.6 Sorting based on outranking relations

In this section we will describe briefly some sorting methods based on outranking relations.

4.6.1 Electre-Tri with limiting profiles


We will present the basics of Electre-Tri since it is one of the most used sorting methods [Tervoreet al., 2004]. For more information we refer the reader to [Yu, 1992a], [Roy and Bouyssou,1993]. Some real applications of this method may be found in [Yu, 1992b].

Electre-Tri is a multicriteria sorting method used for the assignment of a set of actions A intoK completely ordered categories C1, . . . ,CK where category Ch is better than (or preferred to)category Ch+1. Category C1 is thus the best category and CK the worst8. The categories aredefined by an upper and lower boundary or limiting profile. We will denote the upper and lowerlimiting profiles of category Ch respectively rh and rh+1. A limiting profile rh is thus the upperreference profile of category Ch and the lower reference profile of category Ch+1. This illustratedin Fig.4.10 where G = g1, . . . ,gq represents a coherent set of criteria (see Section 2.5).

7As pointed by the authors, during the model development the case Ah(ai) < A>h(ai) is considered as a misclas-sification. [Doumpos and Zopounidis, 2002]

8For the coherency of the notation in our work, we will adopt another notation as proposed by the authors in [Yu,1992a] and [Roy and Bouyssou, 1993]: we inverse the numbering of the categories. In fact, the best profile has therank 1 if we rank the reference profiles and will thus be noted r1. On the other hand, the worst one, will have the rankK + 1 and thus be noted rK+1.

101


-

-

-

-

g2

gq

gq−1

. . .CK . . .. . . C1C j

g1

r2rK r j+1 r jrK+1 r1

Figure 4.10 — Illustration of completely ordered categories defined by limiting profiles.

To assign an action a of the set A , the outranking relations S(a,rh) and S(rh,a) (∀h = 1, . . . ,K +1) will be built to compare the action a to the reference profiles rh according to the set of criteriaG . The outranking degree S(a,rh) ( S(rh,a) ) measures thus the strength of the assertion “a is atleast as good as rh” (and vice et versa). On the basis of these outranking degrees we may define4 different situations when comparing a and rh and defining a cutting-level λ:

1. aI rh : a and rh are indifferent ⇔ S(a,rh) ≥ λ and S(rh,a) ≥ λ

2. aJ rh : a and rh are incomparable ⇔ S(a,rh) < λ and S(rh,a) < λ

3. a rh : a is preferred to rh⇔ S(a,rh) ≥ λ and S(rh,a) < λ

4. rh a : rh is preferred to a⇔ S(a,rh) < λ and S(rh,a) ≥ λ

The actions of A will, on the basis of these four situations, be assigned to one of the predefinedcategories.

Since the limiting profiles define completely ordered categories (Fig.4.10), they fulfill the fol-lowing conditions: ∀h < l :

Condition 4.6.1. ∀ j ∈ G : g j(rl) ≤ g j(rh) and ∃ j ∈ G : g j(rl) < g j(rh) ("dominance relationbetween the profiles")

Condition 4.6.2. rl ≺ rh ("preference relation between profiles")

Most of the times, the extreme categories are supposed to be "closed", i.e. the performances ofthe actions are in between the performances of the best and worst limiting profiles. We maymoreover, strengthen this condition by imposing the following one:

Condition 4.6.3. ∀ai ∈ A : rK+1 ≺ ai ≺ r1

Let us remark that these conditions are independent of each other, in the sense that Condition4.6.1; Condition 4.6.2 and Condition 4.6.2; Condition 4.6.1.

102


To ensure that the categories are "large" enough 9, usually a conformity condition is imposed onthe data such that an action a may not be indifferent to more than one limiting profile ([Roy andBouyssou, 1993], p.394). As we will see, this condition may be violated and may have someimplications.

Previous conditions may be illustrated by the hand of a "-graph" as in Fig.4.11 where the arcsrepresent a preference relation between two actions (denoted by squares). Given the conditionson the reference profiles, we have transitive preference relations, which permit us to obtain a"reduced" graph presented in Fig.4.12.

r1r jr j+1rK+1 C j C1CK

)9 )9 ==

Figure 4.11 — Representation of preference relation between the limiting profiles: rk r j⇔ r j← rk

r1r jr j+1rK+1

C j C1CK

Figure 4.12 — Representation of reduced preference relation between the limiting profiles:rk r j⇔ r j← . . .← rk


When comparing the action to be sorted a to the reference profiles, three different situations mayappear10 [Roy and Bouyssou, 1993], p. 392: (with ∃ j ∈ 2, . . . ,K and k ∈ 0,1, . . . ,K +1− j):

1. r1 a,r2 a, . . . ,r j a,a r j+1,a r j+2, . . . ,a rK ,a rK+1 (I)

2. r1 a,r2 a, . . . ,r j−1 a,aI r j,aI r j+1, . . . ,aI r j+k,a r j+k+1, . . . ,a rK ,a rK+1 (II)

3. r1 a,r2 a, . . . ,r j−1 a,aJ r j,aJ r j+1, . . . ,aJ r j+k,a r j+k+1, . . . ,a rK ,a rK+1 (III)

9"For instance, consider the sequence of limiting profiles such that g j(ri) = i× ε,∀ j, i. If ε is very small, thelimiting profiles will not play a role in the assignment procedures [Perny, 1998]."

10We will consider that the conformity condition may be not fulfilled.

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In situation I, action a is, in the sense of the preference relation, “in between” two consecutivelimiting profiles. In situation II and III, action a is respectively indifferent and incomparable toone or several (i.e. k +1) consecutive limiting profiles. Let us remark, that if action a behaves ina similar way (i.e. indifferent or incomparable) to several limiting profiles, these profiles mustnecessarily be consecutive. In other words, there can not be a "hole". This monotone behavior, isdue to the conditions imposed on the limiting profiles and the way of computing the outrankingdegrees.

Two different approaches have been defined to address these three different situations: the opti-mistic and pessimistic assignment rules.

Optimistic assignment rule: An action a will be assigned to the category Ch, if the upperlimiting profile rh is the worst (lowest) profile which is preferred to a. Formally:

• Compare successively a and ri with i from K + 1 to 1.

• If rh is the first reference profile such that rh a, we then have that: Copt(a) = Ch

This assignment rule may easily be understood by analyzing the relative position of the actiona with respect to the reference profiles in the reduced "optimistic -graph" (see Fig.4.13)11. Ifwe consider the oriented path C1a from r1 to a where C1a ≡ r1→ r2→ . . .→ r j → a, a will beassigned to the category of which the upper limiting profile is the last reference profile in thispath. Formally:

If C1a @ : Copt(a) = C1Else : Copt(a) = C j

If C1a does not exists, this means that r1 a. Condition 4.6.3 it thus not fulfilled and category C1is thus "open". Action a may, in this case, be indifferent or preferred to r1. Nevertheless, sinceC1 is the best category, action a will be assigned to C1.Let us remark that the oriented path is defined in the inverse way as the assignment procedure(optimistic version) in order to obtain the same assignments.

Pessimistic version: An action a will be assigned to the category Ch if the lower limitingprofile rh+1 is the best (highest) profile which is outranked by a or with which a is at least asgood. Formally:

• Compare successively a and ri with i from 2 to K + 1.

• If rh+1 is the first reference profile such that aSrh+1, we have then that: Cpess(a) = Ch

This assignment rule may easily be visualized by analyzing the relative position of the action awith respect to the reference profiles in the reduced "pessimistic S-graph" (see Fig.4.14). If weconsider the oriented path CK+1a from rK+1 to a where CK+1a≡ rK+1 99K rK 99K . . . 99K r j→ a,

11This representation will be used later in this work

104


C jC1CK

+

Y

rK+1 r j r1r j+1

aj

Y

rK+1 r j r1r j+1

a

I :

II− III :

j

Figure 4.13 — Reduced "optimistic -graph": x y⇔ x→ y

a will be assigned to the category of which the lower limiting profile is the last reference profilein this path, i.e. C j. Analogously, let us remark that the oriented path is defined in the inverseway as the assignment procedure.

rK+1 r j−1 r1r j

C jCK C1

- - -- -

* q

rK+1 r j−1 r1r j

aj

ajrK+1 r j−1 r1r j

aj- -

- -

-

--

- -

*

?6

j-

I :

II :

III :

Figure 4.14 — Reduced "pessimistic S-graph": : xSy⇔ x L99 y

Let us remark that, given the assignment rules, it seems obvious that Electre-Tri can only beused when the categories are completely ordered. The assignment rules may be resumed asgiven in Tab.4.4 according to the three different situations.

An short example is given in Example 4.4.

Example 4.2. As illustration let us consider the following sorting problem adapted fromthe example given in [Perny, 1998] (p.159). The human resource department would like toevaluate the personnel of a computer company in order to propose them some salary evolutions.Nevertheless, the human resource department wants to avoid a direct comparison between

105


Table 4.4 — Resume of the assignment results when using the Electre-Tri rules.

rule Copt Cpes

relation Ssit-I C j C j

sit-II C j−1 C j−1

sit-III C j−1 C j+k

-

-

-

-

-

g1

g2

g3

g4

g5

r1r2r3r4

C1C2C3 15 200 10

rrr

rrr

a3

a2

r r

r r a3

Figure 4.15 — Representation of the performances of the limiting profiles r1, r2, r3, r4.

the personnel (i.e. they will not be pair-wise compared). For that purpose, they define threecompletely ordered categories: the non-proficient people, the average-performing people andthe promising people. Obviously, the category of the promising people is the best one and willreceive the biggest evolution.

The persons are evaluated on the basis of the 5 following criteria which have to be maximized:

1. g1: software knowledge

2. g2: programming experience

3. g3: commercial aptitude

4. g4: potential mobility

5. g5: leadership attitude

The parameters associated to the criteria are as follows: ∀g j ∈ G : q j = 1; p j = 2,v j = 4 andw j = 0.2. The λ-threshold is fixed at 0.6.

The categories are defined by the 4 limiting profiles: r1, r2, r3, r4 which performances are givenin Tab.4.6 and illustrated in Fig.12.5. The evaluations of the actions (the evaluated persons) ofA are given in Tab.4.5.

106



ai g1 g2 g3 g4 g5

a1 13 12 13 12 13a2 13 11 9 9 12a3 18 18 12 15 16a4 15 15 9 13 12a5 12 13 10 10 12a6 8 7 5 5 12a7 10 18 10 18 17a8 10 20 10 20 20a9 19 19 19 19 19a10 1 1 1 1 1a11 14 13 4 10 12

Table 4.6 — Evaluation of the performances of the limiting profiles of R .

rm g1 g2 g3 g4 g5

r4 0 0 0 0 0r3 10 10 10 10 10r2 15 15 15 14 15r1 20 20 20 20 20

The outranking degrees between the actions to be sorted and the limiting profiles are given inTab.12.2 and the corresponding assignments in Tab.12.3. The assignments of the actions a6 anda1 are immediate since their performances are in between two successive limiting profiles. Theassignments of actions a9 and a10 are also obvious since they are indifferent respectively to theworst and best limiting profile. Tab.12.2 gives the outranking relations between the actions andthe limiting profiles, which aid to understand the given assignments.We can thus conclude that we are in situation I for the actions a1,a3,a5,a6, in situation II forthe actions a7,a8,a11 and in situation III for the actions a2,a4,a9,a10.

4.6.1.3 Properties

Electre-Tri enjoys the following properties [Roy and Bouyssou, 1993],[Yu, 1992b] (see Section4.2):

• Every action a is assigned to one category according to one of the procedures. ("uniquenessproperty"). Let us stress, that both procedures may assign an action to different categories.

• The assignment of action a does not depend on the assignment of the other actions of A .("independency property")

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Table 4.7 — Pair-wise comparisons between the actions and the limiting profiles ri,∀ j =1, . . . ,4: outranking degrees and preference relations.

r4 r3 r2 r1 r4 r3 r2 r1

S(a1,rm) 1 1 0 0 ≺ ≺

S(rm,a1) 0 0 1 1S(a2,rm) 1 1 0 0

I ≺ ≺S(rm,a2) 0 0.6 1 1S(a3,rm) 1 1 0.8 0

I ≺S(rm,a3) 0 0 0.6 1S(a4,rm) 1 1 0 0

≺ ≺S(rm,a4) 0 0 1 1S(a5,rm) 1 1 0 0

≺ ≺S(rm,a5) 0 0.3 1 1S(a6,rm) 1 0 0 0

≺ ≺ ≺S(rm,a1) 0 0.8 1 1S(a7,rm) 1 1 0 0

J ≺S(rm,a7) 0 0 0 1S(a8,rm) 1 1 0 0

J ≺S(rm,a8) 0 0 0 1S(a9,rm) 1 1 1 1

IS(rm,a9) 0 0 0 1S(a10,rm) 1 0 0 0

I ≺ ≺ ≺S(rm,a10) 1 1 1 1S(a11,rm) 1 0 0 0

J ≺ ≺S(rm,a11) 0 0 1 1

Table 4.8 — Assignment of the actions according to the different procedures.

ai Copt Cpess

a1 C2 C2

a2 C2 C2

a3 C1 C1

a4 C2 C2

a5 C2 C2

a6 C3 C3

a7 C1 C2

a8 C1 C2

a9 C1 C1

a10 C3 C3

a11 C2 C3

108


• When two actions are compared similarly to the reference profiles (i.e. the outranking re-lations between the actions and the profiles are the same), they are affected to the samecategories. ("strong homogeneity property")

• If a′ dominates a, then a′ will be affected to category which is at least as good as the categoryto which a will be assigned. ("monotonicity property")

• The fusion or the separation of two neighboring categories do not affect the assignment ofthe actions to other categories. ("stability property")

• If the performances of an action a are "in between" the performances of two consecutivelimiting profiles, it will univocally be assigned to the category delimited by these profiles.("conformity property"). This property is further discussed in the Section 8.1.2 (see Fig.8.3):there might be some particular cases where this property is not verified.

• Since preference relations are not necessarily transitive (Paradox of Condorcet, see Section3.1), an action a j may be assigned to a worse category than ai although we have that a j ispreferred to ai. This is illustrated in Fig. 4.16 where two categories C1 and C2 are delimitedby one limiting profile r2. Since we have that ai r2 and r2 a j, the assignments are asfollows: Copt(ai) = C1 and Copt(a j) = C2. Nevertheless, it may happen that a j ai

12. Theproperty of pairwise assignment consistency is thus not always verified.

The proof of these properties may be found in [Roy and Bouyssou, 1993].

ss

s

R

6

ai

a j

r2

C2

C1

Figure 4.16 — Illustration of the paradox of Condorcet where a→ b means that a b.

4.6.1.4 Remarks

In Section 4.6.1.2 we have presented the optimistic and pessimistic assignment rules based onrespectively the - and S-relation. Nevertheless, a decision maker may want to use in both casethe -relation or the S-relations. Tab.4.9 presents the assignment results when modifying theused outranking relations.

We may thus remark, that the assignments do not change when using -relation instead ofthe S-relation (and vice versa) in the situations I and III. Nevertheless, the results change in

12This situation may be illustrated with the following actions evaluated on 3 true-criteria (to be maximized andwith same weights) and with λ = 0.6: ai ≡ [3,3,1], r2 ≡ [1,2,3] and a j ≡ [4,1,2].

109


situation II: the situation of indifference is thus treated differently according to the used relations.

If the optimistic and pessimistic rules are respectively defined on the basis of the and Srelations (columns 3 and 4 in Tab.4.9), we say that the categories are closed by their lower limit.Concretely, suppose that an action is indifferent to one limiting profile. Since the profile, is aboundary or a limit, we may have the choice between one of the two successive categories.Using, the and S relations, leads to assign the action to the better category. This means thus,that an action whose performances are in between two consecutive limiting profiles but thatis indifferent to the best one, might not be assigned to the category defined by these limitingprofiles. The conformity property is thus stricto sensu not verified (see Section 8.1.2 and Fig.8.3)On the other hand, if we use respectively the S and relations (columns 5 and 6 in Tab.4.9), thecategories are closed by their upper profile. This means concretely that if an action is indifferentto one (or several) limiting profile(s), it will be assigned to the lower category. The decisionmaker may thus choose between these two approaches on the basis of previous considerations.

When using only the or the S relation (e.g. columns 1 and 2 ; columns 7 and 8 in Tab.4.9),we may remark that an action a will, in this case, not univocally be assigned in situation II.This may be useful in the sense that the decision maker may, for each action separately, chooseone of the two categories (or the two categories). However, when using these rules (columns1,2,7,8 in Tab.4.9), we may remark that no difference is made between an indifference orincomparability situation. The use of the binary relations between the action and the profilespermits, nevertheless, to distinguish these cases.


rule Copt Cpes Copt Cpes Copt Cpes Copt Cpes

relation S S S Ssit-I C j C j C j C j C j C j C j C j

sit-II C j−1 C j+k C j−1 C j−1 C j+k C j+k C j+k C j−1

sit-III C j−1 C j+k C j−1 C j+k C j−1 C j+k C j−1 C j+k

In the rest of this work, we suppose that the optimistic and pessimistic assignment rules, userespectively the and S-relation.

4.6.1.5 Graphical illustration

To illustrate intuitively the assignment rules of Section 4.6.1.2, let us consider the followingexample. Consider a assignment problem with 3 ordered categories and 2 independent criteriawith the same weights. The limiting profiles are r4 = (0;0), r3 = (1/3;1/3), r2 = (2/3;2/3)and r1 = (1;1). These are represented by “” in Fig.4.17-4.18.

Let us consider that A = ai | g1(ai) ∈ [0,1] and g2(ai) ∈ [0,1]. In other words, we consider

110


that all the points from the cartesian product [0,1]× [0,1] have to be assigned with respect to the4 limiting profiles of R .

Let us suppose that p = q = 0. When applying the optimistic and pessimistic procedures tothis “map”, we obtain the assignments represented in Fig.4.18. Let us remark that there is nodifference between working with the or the S relations since p = q = 0.

-

6

g1

g21

2/3

1/3

1/3 12/3r4

r3

r2

r1

C1

C2

C3

-

6

g1

g21

2/3

1/3

1/3 12/3r4

r3

r2

r1

C1

C2

C3

Figure 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.

-

6

g1

g2

1

2/3

1/3

1/3 12/3

r4

r3

r2

r1

C1

C2

C3

C2∪C1

C2∪C1C3∪C2

C3∪C2∪C1

C3∪C2∪C1C3∪C2

Figure 4.18 — Assignment of any point (x,y) of the plan when considering the “union” of theoptimistic and pessimistic result and where q = 0 and p = 0, and with w1 = w2 = 0.5.

111


4.6.2 Trichotomic Segmentation

Trichotomic Segmentation is one of the pioneer’s work in the field of sorting. It has beenproposed by B. Roy and J. Moscarola [Moscarola and Roy, 1977; Roy, 1981]. This procedureenables to assign actions to one of three ordered and predefined categories, noted C1,C2 and C3.The assignments are based on the treatment the actions will receive later [Vincke, 1992]. Thesemantic definition of the categories may be explained as follows. The actions will be assignedto C1 if there are enough reasons to recommend them. If there are enough reasons to refuse them,they will be assigned to C3. On the other hand, actions will be assigned to C2 if the decision isnot clear. This unclear decision may be due to the lack of information (e.g. the performances ofthe actions are not well-known) or to the "average" behavior of the action.As an example, we may cite the segmentation of clients into three categories. The best categoryC1 corresponds to the promising clients (which perform well on a set of criteria) and the worstcategory C3 to bad-performing clients. On the other hand, category C2 may correspond to clientswhose performances are "average" performances, difficult to evaluate and thus not clear, or notcertain (e.g. investment factors which may lead to benefit or loose).

The categories are defined by a set of couples of reference profiles: R = r12,r1

3, . . . ,rk2,rk

3. Eachpair of reference profile (r j

2,r j3) may be seen as a pair of upper and lower limiting profiles. The

set R3 = r13, . . . ,rk

3 and set R2 = r12, . . . ,rk

2 demarcate respectively the categories C3 and C2and the categories C2 and C1

13. Fig. 4.19 represents the sets R2,R3, evaluated on the set G of qcriteria, for the definition of three ordered categories.

-

-

- g1

g j

gq

r12r1

3 r32r3

3 r23 r2

2

C1C2C3

Figure 4.19 — Representation of the sets R2,R3 for the definition of C1,C2 and C3

The actions of A will be pairwise compared to the reference profiles in order to assign them. Theassignment rules are either based on the Pareto-dominance relation (see Section 2.2) or on theoutranking degrees (see Section 2.6.1.4).

Assignment Rule 4.6.1. ∀ai ∈ A :

CT S(ai) = C1⇔∃ rk2 ∈R2 : ai W rk

2

CT S(ai) = C2⇔∃ rk2 ∈R2 and rk

3 ∈R2 : rk2 W ai W rk

3

CT S(ai) = C3⇔∃ rk3 ∈R3 : rk

3 W ai

13We choose this notation in order to adopt a coherent notation all along this work.

112


In other words, if action ai is preferred to at least one profile of the set R2, it will be assigned tothe best category. If, there exists at least one profile of the set R3 which is preferred to the actionai, it will be assigned to the lowest category C3. Finally, if there exists a couple (rk

1,rk2) which is

such that action ai has an average behavior (i.e. rk2 W ai W rk

3, it will be assigned to C2).

Since, many actions of A are in neither of the three situations of Assignment Rules 4.6.1, the au-thors recommend to use the outranking degrees and the preference relations as in Electre III. Theactions will be assigned on the basis of the decision tree given in Fig. 4.20 where the followingnotations are used14:

S(ai,r−2 ) = maxk

S(a,rk2) (4.12)

S(r+2 ,ai) = max

kS(rk

2,a)15 (4.13)

S(r+3 ,ai) = max

kS(rk

3,ai) (4.14)

S(ai,r−3 ) = maxk

S(ai,rk3)16 (4.15)

S(ai,r−3 ) < t′

yes no

S(r+3 ,ai) < t

S(ai,r−2 ) ≥ s

no

no

yes

yes

S(r+2 ,ai) < s

′

no yes

ai ∈C1ai ∈C2ai ∈C2

ai ∈C2ai ∈C3

Figure 4.20 — Assignment rules of Trichotomic Segmentation based on the decision tree.

and where s, t,s′, t′

are thresholds to be fixed in function of the considered application and inparticular, by taking into account the inconveniences due to an assignment error and those re-sulting from any assignment to category C2 [Vincke, 1992]. An example is given in Example 4.3.

Let us remark that this procedure is restricted to the case of three categories. Moreover, lotsof parameters need to be fixed (s, t,s

′, t′, and the indifference and preference thresholds, etc.).

14We have adopted another notation than in the original paper.15Let us remark that in the original paper, the authors impose the condition that r+

2 may not be equal to r−2 .16Idem as in previous footnote.

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Indeed, when k > 1, the role of the boundaries may be not very clear and no conditions havebeen clearly expressed between the reference profiles. This has as consequence that the relationbetween the profiles are not transitive (Condorcet paradox, see Section 3.1) even when k = 1[Henriet, 2000].

4.6.2.1 Properties

The reader may verify easily that the Trichotomic Segmentation enjoys the properties ofneutrality, uniqueness and independence. Moreover, since the outranking degrees are monotone,the procedure enjoys of the monotonicity property (Properties 4.1, 4.2, 4.3,4.9). Given the initialassignment rules (Assignment Rule 4.6.1), it is easy to verify that the conformity property isfulfilled. The homogeneity property is respected if we define "similar behavior in regards ofthe reference profiles" (see Property 4.6) on the basis of the outranking degrees. Furthermore,since this procedure only works for three ordered categories, we may not split categories. If onthe other hand, two categories are fused, by suppressing for instance the set R2 or R3, actionsassigned to C3 or C1 will remained assigned respectively to C3 and C1 considering that wework only we some parts of the decision tree. The suppression of a category do not alter theassignments to the non-modified categories.Furthermore, analogously to Electre-Tri, given the non-transitivity of preference and outrankingrelations, this method does not fulfill the pairwise assignment consistency.

4.6.2.2 Comparison with Electre-Tri

Let us analyze this procedure in a particular case. Actually, we will make some hypoth-esis in order to be in the same conditions as in Electre-Tri (see Section 4.6.1). This willpermit us to compare the methods easily. For that purpose, let us fix k = 1. This meansthus that we define the categories by the profiles r1

2 and r13, that we will note r2 and r3 to

adopt a coherent notation. We have thus that r2 is the upper and lower limiting profile ofrespectively C2 and C1 whereas r3 delimits in the same way C3 and C2. Moreover, supposethat s = t = s

′ = t′ = λ where λ represents the cutting-level as defined in Section 2.6.1.4. In

this case, we have obtain the decision tree given in Fig.4.21. An example is given in Example 4.3.

Let us remark that according to the authors, we may not obtain the proposed decisiontree since r−2 6= r+

2 and r−3 6= r+3 . Nevertheless, we may define r+

2 and r−3 , such that∀g j; i = 2,3 : g j(r+

i ) = g j(r−i ) ± ε with ε very small and such that they play the samerole.

From the decision tree of Fig.4.21, we have that if ai r2, CT S(ai) = C1. In other words, if ai

is preferred to the limiting profile between C2 and C1, it will be assigned to C1. Analogously,if ai is being preferred by the lower limiting profile of C2 (i.e. ai ≺ r3), we have CT S(ai) = C3.Moreover, if S(ai,r2) < λ and S(r3,ai) < λ, CT S(ai) = C2.

114


S(ai,r3) < λ

yes no

S(r3,ai) < λ

S(ai,r2) ≥ λ

no

no

yes

yes

S(r2,ai) < λ

no yes

ai ∈C1ai ∈C2ai ∈C2

ai ∈C2ai ∈C3

Figure 4.21 — Assignment rules of Trichotomic Segmentation based on the decision treewhen |R2 |=|R3 |= 1 and when s = t = s

′= t

′= λ.

Furthermore, if ai is indifferent or incomparable to at least one of the limiting profiles, it will beassigned to C2. This means thus that the segmentation method do not differentiate the followingcases: aiI r2, aiI r3, aiI r2∧aiI r3 ; aiJ r2, aiJ r3 or aiJ r2∧aiJ r3. Indifference and incomparabilityare thus not treated in a different way (such as in the optimistic approach of Electre-Tri). Thesesituations are, on the contrary, clearly differentiated in the Electre-Tri method when using boththe optimistic and the pessimistic approach. At the first sight, this may be seen as somewhatconfusing or ambiguous. Nevertheless, if category C2 is seen as an intermediate category forwhich the situation is considered as not clear or ambiguous, one can accept that the actionscorresponding to these situations, may be further analyzed and thus assigned to C2.Finally, let us remark that in this context, the homogeneity property is verified when workingwith the outranking degrees and the binary outranking relations (i.e. I ,J ,).

Example 4.3. As an illustration of the Trichotomic Segmentation method, we will reconsiderExample 4.2 (p.105). We have thus three ordered categories which are delimited by the twolimiting profiles r2 and r3. The performances of the actions to be sorted and the limiting profilesare given respectively in Tabs.4.5 and 4.6. The preference parameters are the same, which lead tothe outranking degrees given in Tab.12.2. On the basis of the decision tree given in Fig.4.21, weobtain the assignments presented in Tab.4.10. The reader may verify that in case of situation I, theassignments are identical. Nevertheless, in case of situations II and III (see Section 4.6.1.2), theassignments according to the Trichotomic Segmentation may correspond either to the optimisticor pessimistic assignments (e.g. actions a8 and a11). This illustrates the interpretation given tothe decision tree.

115


Table 4.10 — Assignment of the actions according to the different procedures: Copt ,Cpess andCT S.

ai Copt Cpess CT S

a1 C2 C2 C2

a2 C2 C2 C2

a3 C1 C1 C1

a4 C2 C2 C2

a5 C2 C2 C2

a6 C3 C3 C3

a7 C1 C2 C2

a8 C1 C2 C2

a9 C1 C1 C1

a10 C3 C3 C3

a11 C2 C3 C2

4.6.3 nTomic

A. Ostanello and M. Massaglia [Ostanello and Massaglia, 1991] have proposed an alternativeway to sort actions into 3 ordered categories (good, average and bad). The assignments will bebased on the "goodness" and "badness" measures of the actions with respect to the referenceprofile r2 and r3 which can be considered as limiting profiles. For each criterion, the performanceg j(ai) of an action ai to be assigned, will be compared to g j(r2) and g j(r3). g j(r2) correspondsto the threshold above which the action ai is considered as good on criterion j. On the otherhand, g j(r3) is the threshold below which ai is considered as bad. Obviously, g j(r2) andg j(r3) are not strict threshold. In order to take imprecision and uncertainty effects into account,indifference thresholds (noted q+

j ,q−j ) and discrimination thresholds (noted s+j ,s−j ) are used to

measure the goodness D j(ai) and the badness d j(ai) on criterion j. This is represented in Fig.4.22.

6D j(ai)d j(ai)

1

0 - g j(ai)g j(r2)g j(r2)−q−jg j(r2)− s−jg j(r3)+ s+

jg j(r3)+ q+jg j(r3)

Figure 4.22 — Representation of the goodness D j(ai) and badness d j(ai) functions.

Once for every criterion the goodness and the badness have been evaluated, several aggregation

116


techniques (compensatory and non-compensatory techniques, see [Henriet, 2000] p.45) permitto obtain a global goodness D(ai) and badness d(ai) measure. On the basis of a partition in thegoodness and badness plan, see Fig.4.23, the action ai is assigned.

-

6

β

β′

1

1D(ai)

d(ai)

/

uncertain

rather good

good very good

rather good

average

/

rather badvery bad

bad

Figure 4.23 — Representation of the goodness and badness plan.

4.6.3.1 Properties

Given the goodness and badness measures, the reader may verify that the properties of mono-tonicity, neutrality, independence, conformity (with limiting profiles) (Properties 4.1, 4.3,4.9,4.7) are verified. Categories may not be split, but the fusion of two consecutive ones does notalter the assignments of the actions to the non-modified categories. Moreover, depending on theuse of the goodness and badness plan, an action may be assigned to one or several categories.The reader will find more information in [Ostanello and Massaglia, 1991] and [Doumpos andZopounidis, 2002].Let us remark that depending of the partition, indifference and incomparability, may eventuallybe distinguished such as in Electre-Tri.

4.6.4 Filtering Methods

According to P. Perny, "filtering actions" means "comparing actions to reference points in orderto decide wether they belong to a given category or not." [Perny, 1998]" Two different situations

117


are distinguished: the categories may be defined either in an ordinal way or in a nominal way.According to the situation, a preference relation or an indifference relation will respectively beused to assign the actions. This section is based on [Perny, 1998].

4.6.4.1 Filtering by strict preference

This methodology is used when K categories, noted C1, . . . ,CK , are completely ordered fromthe best to the worst with C jBC j+1. The categories are defined by a set of reference profileswhich represent upper and lower limits of the categories. We denote the frontier or the set oflimiting profiles between the categories C j and C j+1 by the set R j ( j = 1, . . . ,K). Each set R j

is constituted by one profile (noted r j) or by several reference profiles (r1j ,r

2j , . . . ,r

lj) which are

mutually incomparable. A category C j is thus defined by a set of upper profiles R j−1 and aset of lower profiles R j. The categories are thus defined by the set R = R 1 ∪ . . .∪R K+1. Anexample of categories, defined by several profiles, is given in Fig.4.24.

The actions to be assigned, are compared to the reference profiles on the basis of a preferencerelation P. This relation is either binary or valued. The valued preference relation can beobtained by computing for example preference degrees π(a,b) (see Section 2.6.2) or on thebasis of outranking degrees (see Section 2.6.1.4) where ∀g j ∈ A : Pj(a,b) = 1− S j(b,a) andP(a,b) = 1−S(b,a) 17 [Perny, 1998]. This is illustrated in Fig.4.26.

Since the categories are completely ordered, the reference profiles verify the following condi-tions:

Condition 4.6.4. ∀rkh−1 ∈ R h−1,rl

h ∈ R h : rkh−1 W rl

h

This condition imposes that the reference profiles of better categories dominate profiles of worsecategories and particularly, that all the upper profiles of category dominate all the lower profiles.

We may strengthen previous Condition 4.6.4 by imposing the following condition of separability:

Condition 4.6.5. ∀rkh−1 ∈ R h−1,rl

h ∈ R h,∀g j ∈ G : g j(rkh−1)−g j(rl

h) > p j

where p j represents a preference threshold as in sections 2.6.1.4 or 2.6.2. This condition ensuresthat the categories are well-defined since P(rk

h−1,rlh) = 1,∀k, l,h = 2, . . . ,K + 1.

Moreover, all the reference profiles of a same frontier or set R h are incomparable:

Condition 4.6.6. ∀rkh,rl

h ∈ R h, rlh J rk

h

A representation of profiles respecting previous conditions, is given in Fig.4.24.

An action a will be assigned to Ch if and only if a is preferred to some elements of R h+1 (i.e. theprofiles of the lower limit of Ch) without being preferred to any element of R h (i.e. the profiles of

17This is valid if no discordances are considered.

118


-

-

- g1

g j

gq

Ch Ch−1

r1hr2

h

R hR h+1R h+2 R h−1

rh+2 rh−1

Ch+1

r3h

Figure 4.24 — Representation of the limiting profiles defining the ordered categories

the upper limit of Ch). Formally, if P denotes a binary preference relation between two actions,we have the following assignment rule:

Assignment Rule 4.6.2. Descending Assignment Rule:

CFP(a) = Ch⇔ (∃rkh+1 | aPrk

h+1) ∧ ( ∀rlh¬[aPrl

h] )

An alternative assignment rule, might be given by successively verifying wether at least oneelement of R K+1, R K , etc. is preferred to a:

Assignment Rule 4.6.3. Ascending Assignment Rule:

CFP(a) = Ch⇔ (∃rkh | rk

hPa) ∧ (∀rlh+1¬[rl

h+1Pa])

This is the dual assignment rule. This permits to assign an action a, eventually, to two differentconsecutive categories, especially in case of incomparability between action a and some limitingprofiles. Let us remark that the ascending assignment is always better than the descending result[Perny, 1998].

If on the other hand, the preference relation P is valued, we may18 define for each category Ch amembership function µh(a):

Definition 4.3.µh(a) = min

∀rkh+1∈R h+1;∀rl

h∈R h[P(a,rk

h+1),1−P(a,rlh)]

This membership function measures to which extend, a is preferred to the lower limits of thecategory and being preferred by the upper limits. Let us nevertheless remark that only the com-parison between a and the profiles is considered and not vice et versa, i.e. P(a,ri). Moreover,this measures takes into account two comparisons simultaneously (instead of one in Electre-Tri).As we will see, it may lead to a refinement (see Example 4.4).On the basis of these membership functions, we have the following assignment rules:


CFPµ(a) = Ch⇔ µh(a) = maxk=1,...,K

[µk(a)]

18Other definitions can be found in [Perny, 1998].

119


Assignment Rule 4.6.5.CFPµα (a) = Ch⇔ µh(a) ≥ α

where α is a parameter which needs to be fixed at forehand. The difference between thesetwo assignment rules is that in the first case, action a will be assigned to exactly one category(except in the case of ex aequo) whereas in the second case, a may be assigned to none orseveral categories. The conditions 4.6.4 and 4.6.5 ensure us moreover that an action a can notbe assigned both to a good and a bad category without being assigned to categories in between(see [Perny, 1998]). In this case, the multiple assignments can be considered as an hesitationbetween several consecutive categories.

Choosing α = 0.5 ensures us that an action a will be assigned at least to one category althoughit is a selective threshold (the proof is given in [Perny, 1998]).Let us remark that all the profiles of R h−1 will be assigned according to Assignment Rule 4.6.4to Ch:

∀rkh−1 ∈ R h−1 :

P(rkh−1,rl

h) = 1P(rk

h−1,rkh−1) = 0

⇒ µh(rk

h−1) = 1

This result is due to the chosen definition of µh (Definition 4.3) and can change by changing thedefinition given to µh.

Example 4.4. Let us reconsider Example 4.2 (p.105) where 3 ordered categories are defined by 4limiting profiles. The performances of the actions to be sorted and the limiting profiles are givenrespectively in Tabs.4.5 and 4.6 (p.107). Moreover, a representation is given in Fig.12.5 (p.235).The preference parameters are as follows: ∀ j = 1, . . . ,5 : q j = 1, p j = 2,v j = 5 and w j = 0.2.The valued preference relation P(x,y) is computed as follows:

P(x,y) = min[CP(x,y),1−DP(x,y)] (4.16)

where CP(x,y) and DP(x,y) represents respectively the global preference-concordance and dis-cordance indexes ([Perny, 1998]) obtained as follows:

CP(x,y) = 1−S(y,x) (4.17)

DP(x,y) = max[DPj (x,y)] with DP

j (x,y) = min(1,max[0,g j(y)−g j(x)− p j

v j− p j]) (4.18)

Let us remark that the global preference-concordance and discordance indexes or not equal tothe global outranking-concordance and discordance indexes of Section 2.6.1.4. The preferencerelation between the actions and the reference profiles are given in Tab.4.11 as well as the de-duced membership functions. The final assignments are given in Tab.4.12 with respect to theElectre-Tri assignments.The reader may remark from Tab.4.12 that the assignments obtained by filtering are not neces-sarily equal to those obtained with Electre-Tri (even when the optimistic and pessimistic assign-ments are the same). This may be explained by the fact that the membership function take onlyP(a,ri) into account, regardless of P(ri,a). Moreover, we may remark that action a10, although

120


Table 4.11 — Pair-wise comparisons between the actions and the limiting profiles ri,∀ j =1, . . . ,4: valued preference relations.

r4 r3 r2 r1 µ3 µ2 µ1

P(a1,rm) 1 0.33 0 0 0.67 0.33 0P(a2,rm) 1 0.33 0 0 0.67 0.33 0P(a3,rm) 1 0.67 0 0 0.33 0.67 0P(a4,rm) 1 1 0 0 0 1 0P(a5,rm) 1 0.33 0 0 0.67 0.33 0P(a6,rm) 1 0 0 0 1 0 0P(a7,rm) 1 0.6 0.6 0 0.4 0.4 0.6P(a8,rm) 1 0.6 0.6 0 0.4 0.4 0.6P(a9,rm) 1 1 1 0 0 0 1P(a10,rm) 0 0 0 0 0 0 0P(a11,rm) 1 0.6 0 0 0.4 0.6 0

Table 4.12 — Assignment of the actions according to the different procedures: Copt , Cpess andCFP.

ai Copt Cpess CFP

a1 C2 C2 C3

a2 C2 C2 C3

a3 C1 C1 C2

a4 C2 C2 C2

a5 C2 C2 C3

a6 C3 C3 C3

a7 C1 C2 C1

a8 C1 C2 C1

a9 C1 C1 C1

a10 C3 C3 C1,C2,C3

a11 C2 C3 C2

identical to the lower profile r4, can not be clearly assigned to C3 since all the membership func-tions are equal to 0. Nevertheless, when an action is indifferent to exactly one limiting profile(e.g. actions a2, a9, a10) the final assignment depends on the comparison between the actionand another limiting profile. It will thus not always be a systematic decision, on the contrary ofElectre-Tri (see the 3 different "assignment-cases", Section 4.6.1.2).

Properties

Given the monotonicity constraints on the preference relations, the reader may easily verify thatthe monotonicity property is verified. The property of neutrality is obliviously respected whereasan action can be assigned to none, exactly one or several categories, depending on the chosen

121


assignment rule. Moreover, the assignment of an action does not depend on the other actions.Given the definition of the assignment rule, the conformity property is indeed verified as well asthe assignment stability. Finally, since preference relations are not necessarily transitive (Paradoxof Condorcet, see Section 3.1), this filtering method suffers also from the same drawback asElectre-Tri: the property of pairwise assignment consistency is not verified.

ss

s

R

6

ai

a j

r2

C2

C1

Figure 4.25 — Illustration of the paradox of Condorcet where a→ b means that aPb.

Comparison with Electre-Tri

On the contrary of Electre-Tri, the limits between two categories may be defined by severalincomparable reference profiles. Since the outranking degrees permit to differentiate indif-ference and incomparability between two actions, it may be appropriate so specify severalcomplementary reference points within a single frontier [Perny, 1998].In Electre-Tri, an outranking relation is used instead of a preference relation. Nevertheless, therelation between these two is given in Section 4.6.4.1. Moreover, the condition of separability(Condition 4.6.5) in the filtering method is stronger than Condition 4.6.2 (i.e. condition ofpreference relation between the limiting profiles) of Electre-Tri.

In Electre-Tri, the binary relations (I ,J ,) between the limiting profiles and the actions (andvice et versa) are used for the assignment. On the other hand, in the filtering methods thepreference relations are valued and these values are explicitly used. This leads to differentassignments such as for action a10 in Example 4.4.

However, in order to compare the assignment philosophy between this two methods,let us suppose that the K ordered categories are delimited by exactly one limitingprofile (such as in Electre-Tri): ∀h : R h = rh. Moreover, Condition 4.6.5 leads to:∀h = 1, . . .K : S(rh,rh+1) = 1, S(rh+1,rh) = 0. This ensures us that Condition 4.6.2 ofElectre-Tri is respected and that we may use the set R for defining the K ordered categories inElectre-Tri.

Let us first define the binary preference relation of the filtering method as follows19:aPb⇔ a b⇔ S(a,b) ≥ λ and S(b,a) < λ where λ is the Electre cutting-level.

19We will use thus the preference relation of Electre-Tri in the filtering assignment rules.

122


As presented in Section 4.6.1.2, when comparing an action a to be sorted to the reference profiles,three different situations may appear: (with j ∈ 2, . . . ,K and k ∈ 1, . . . ,K + 1− j):

1. r1 a,r2 a, . . . ,r j a,a r j+1,a r j+2, . . . ,a rK ,a rK+1 (I)

2. r1 a,r2 a, . . . ,r j−1 a,aI r j,aI r j+1, . . . ,aI r j+k,a r j+k+1, . . . ,a rK ,a rK+1 (II)

3. r1 a,r2 a, . . . ,r j−1 a,aJ r j, . . . ,aJ r j+k−1,aJ r j+k,a r j+k+1, . . . ,a rK ,a rK+1 (III)

On the basis of these three cases, we have given the in Tab.4.13, the assignment results whenusing the Electre-Tri methodology and the preference filtering method. From this table, wemay remark that the ascending and descending assignment rules correspond respectively tothe optimistic and pessimistic assignment rules when working strictly with the -relation. It isnormal that the optimistic (pessimistic) corresponds to the ascending (descending) rule sincewe start in the optimistic procedure from the worst (best) category such as in the ascendingprocedure.Moreover, we may conclude that the filtering methods, based on the aforementioned preferencerelation do not distinguish the case of the indifference and incomparability situation (on thecontrary of the S-relations used in the Electre-Tri assignment rules).


rule Copt Cpes Copt Cpes Copt Cpes CAFP CD

FP

relation S S P Psit-I C j C j C j C j C j C j C j C j

sit-II C j−1 C j+k C j−1 C j−1 C j+k C j+k C j−1 C j+k

sit-III C j−1 C j+k C j−1 C j+k C j−1 C j+k C j−1 C j+k

When the preference relations are valued it seems not straightforward to obtain a relationshipbetween these two methods. This is confirmed by analyzing Tab.4.12 of Example 4.4: no rela-tionship may actually be stated.

4.6.4.2 Filtering by indifference

When the categories are completely not-ordered, we may use analogous assignment rulesbased on an indifference relation. Let us denote by C j, with j = 1, . . . ,K, the j− th category.Each category C j is characterized either by one reference point, noted r j, or by a set of notindifferent20 reference points, noted R j = r j

1, . . . , r jk.

An action a to be sorted will be compared to the reference profiles on the ba-sis of an indifference relation. This indifference relation may be valued and com-puted by taking the symmetric parts of the outranking relations on each criterion g j:I j(a,b) = min(S j(a,b),S j(b,a)) = min(1−Pj(b,a),1−Pj(a,b)). This is illustrated in Fig.4.26

20This prohibits redundancy.

123


where q j = q j(a) and p j = p j(a) representing respectively an indifference and preferencethreshold.

-

61 Pj(a,b) I j(a,b) = I j(b,a) Pj(b,a)

g j(a)− p j g j(a)+ p jg j(a)

g j(b)

g j(a)−q j g j(a)+ q j

Figure 4.26 — Representation of the uni-criterion preference and indifference relation.

Let us remark that this indifference relation is quite similar to the partial indifference degreepresented in Fig.4.3 and Section 4.3.1.1. In fact, if we define:

g j(a) =g j(a)+ g j(a)

2

q j(a) =g j(a)−g j(a)

2= s+

j = s−j

d+j = d−j = p j

we may obtain the same degrees.

Moreover, a discordance relation may be computed dIj(a,b) = min(dS

j (a,b),dSj (b,a)) (where

dSj (a,b) has been defined in Section 2.6.1.3).

These partial indifference and discordance relations can be aggregated to a global indifferencerelation by for example disjunctive operators, weighted sum, etc. An example is given in Eq.4.1921, where w j = w j(a) represents the relative importance of the criterion when compared to theother criteria that support the assignment to the category defined by a [Yevseyeva, 2007].

I(a,b) = [q

∑j=1

(I j(a,b)×w j]× [1−q

∏j=1

(1−dIj(a,b))

τ

q ] = CI(a,b)× (1−DI(a,b)) (4.19)

where DI(a,b) has been defined in Section 2.6.1.3 and τ is a technical parameter that is used formodifying the degree of synergy between the criteria [Yevseyeva, 2007].

On the basis of the indifference degrees between the action to be sorted and all the referenceprofiles of a category Ch, a membership function µh will be computed as follows:

21The reader will find more aggregation operators in [Perny, 1998]

124


Definition 4.4.

µh(a) = max∀rh

j∈R h[I(a, rh

j )]

This ensures us, that if an action a is indifferent to a reference profile, it will be considered asindifferent to all the profiles and thus assigned to that corresponding category. Let us remarkthat in some case, another definition may be given such that the membership degree is reinforcedwhen an action is indifferent to several reference profiles. We might define µh(a), when | R h |> 1,as follows:

Definition 4.5.

µh(a) = ∑∀rh

j∈R h

I(a, rhj )− ∏

∀rhj∈R h

I(a, rhj )

Finally, action a will be assigned to the category for which the membership degree is the highest.Formally, we have:


CFIµ(a) = Ch⇔ µh(a) = maxk=1,...,K

[µk(a)]

The reader may find an example of filtering by indifference in [Perny, 1998].

Comparison with PROAFTN

Let us remark that as mentioned in [Yevseyeva, 2007], the indifference-based filtering method israther analogous to PROAFTN (see Section 4.3.2) when the categories are defined in a nominalway. The thresholds (dissimilarity and preference) may have different names, but play equivalentroles. In PROAFTN, a central profile may be defined by intervals, whereas the imprecision of theevaluations are taken into account in the filtering method by the means of indifference thresholds.

Properties

The properties of the filtering methods are analogous to those of PROAFTN: neutrality, uni-versality and uniqueness (except in cases of ex-aequo), stability, independency, weak or stronghomogeneity (depending on the used assignment rule), stability and conformity.Let us remark that in certain cases (i.e. for certain values of the indifference, preference andveto thresholds), the partial indexes are symmetric and thus do not take into account the prefer-ence direction. In these cases, the method will not satisfy the preference-orientation dependencyproperty. This is due to the fact that the indifference degree reflects only the indifference ornon-indifference.

125


4.6.5 PairClass


Doumpos and Zopounidis have proposed a sorting method, called PairClass, for assigningactions to K completely ordered categories, C1B . . .B CK [Doumpos and Zopounidis, 2004a].PairClass is based on pairwise comparisons by means of preference degrees instead of outrank-ing degrees. The pairwise comparisons are an alternative to absolute comparisons among theactions, to be sorted, and some reference profiles (cut-off points).

Each category C j is defined by a set of representative examples noted R j = r1j , . . . , r

kj, to

which an action to be sorted is compared. The reference examples of a better category outrankthe examples of a worse category22, but no condition is imposed on the examples of a samecategory. The set of the reference examples is noted R = R 1∪ . . .∪ R K .

Each action a is compared to all the reference examples by computing the preference degreeswhich aggregate the preferences on q criteria (see Section 2.6.2): π(a,x) and π(x,a),∀x ∈ R . Onthat basis, a so-called category net-flow degree23 is defined for each action and for each category.

Let us remark, that the authors propose Linear Programming Techniques to infer preferenceparameters (such as indifference, preference thresholds, weights, etc.) from the examples givenby the decision maker. Moreover, the preference functions, used to obtain the preference degreesare more general than the one defined in Promethee (see Section 3.3.2).

4.6.5.2 Assignment Rules

Presence of two ordered categoriesIn order to give a more intuitive explanation of this procedure, let us consider the particular

case of two ordered categories defined by some reference profiles. This is represented in Fig.4.27.The category net-flow degree is defined as follows:

Definition 4.6. ∀a ∈ A : f1(a) = 1m2× f +

1 (a)− 1m1× f−1 (a)

where m2 =| R 2 |, m1 =| R 1 |, f +1 (a) and f−1 (a) are defined as follows:

f +1 (a) = ∑

x∈R 2

π(a,x) (4.20)

f−1 (a) = ∑x∈R 1

π(x,a) (4.21)

The term f +1 (a) represents the outranking character of a over the reference examples of R 2

whereas f−1 (a) the outranked character of a by all the reference actions of R 1. The factors

22This condition has been given but formally written in the original paper [Doumpos and Zopounidis, 2004a].23Let us remark that this name is not the name given by the authors. The authors have given the name "net flows"

but this may be confused with the "net flows" used in for example Promethee 3.3.2.

126


'

&

$

%

'

&

$

%

C1R 1C2 R 2

u

uuu

uuuu

/

kI

]M

r11

r21

r12

a

r31

rk2

Figure 4.27 — Representation of the preference relation computed in the PairClass procedure.

mr+1 and m1 are used to compute the "average" characters in Definition 4.6. On that basis, thefollowing assignment rule is given:

Assignment Rule 4.6.7. CPC(a) = C1, if f1(a) > b1

where b1 represents a so-called cut-off point. Action a will thus be assigned to C1 if f1(a) > b1,otherwise to C2. It means thus that the outranking character of a must be greater (with a certainthreshold) than the outranked character.

If we suppose that R 1 and R 2 are defined by exactly one reference profile each, we have thefollowing assignment rule:

Assignment Rule 4.6.8. ∀a ∈ A : f1(a) = π(a,r2)−π(r1,a) > b1⇔ a ∈ C1

This assignment rule will be discussed later in this work (Section 8.2).

Moreover, suppose that the categories are defined by 3 limiting profiles r1, r2, r3 such that R 1 =r1, r2 and R 2 = r2, r3 (see Fig.4.28). In order words, we consider that these profiles representlimiting profiles. If this might be strange to consider that r2 belongs either to C1 and C2, we maydefine r2 and r

′2 such that ∀g j; i = 2,3 : g j(r2) = g j(r

′2)± ε with ε very small and such that they

play the same role.In this case the assignment rule becomes:

Assignment Rule 4.6.9. ∀a ∈ A : f1(a) = π(a,r2)+π(a,r3)2 − π(r2,a)+π(r1,a)

2 > b1⇔ a ∈ C124

Assignment Rule 4.6.9 can be rewritten by taking the following information into account:π(r3,a) = 0 and π(a,r1) = 0,∀a ∈ A if ∀g j ∈ G : g j(r3) ≤ g j(a) ≤ g j(r1):

Assignment Rule 4.6.10. ∀a ∈ A : φR (a) = φ+R (a)−φ

−R (a) > b1⇔ a ∈ C1

24Since the reference profile play actually the role of limiting profile, we have noted them as limiting profiles, i.e.without " ˙ ".

127


C2 C1t tt r1r3 r2

Figure 4.28 — Illustration of the case where we define profiles as limiting one.

We may thus remark that in this assignment rule the net flows are compared to absolutethresholds. This will be discussed later in this work (Section 8.2).

Presence of K ordered categoriesWhen the actions, have to be assigned to one of K categories, we define the category net-flow

for each action as follows:

Definition 4.7. ∀C j,a ∈ A

f j(a) =1

m j+1× f +

j (a)− 1m1× f−j (a)

where m j+1 =| R j+1∪ . . .∪ R K |, m1 =| R 1∪ . . .∪ R j |, and

f +j (a) = ∑

x∈R j+1∪...∪R K

π(a,x) (4.22)

f−j (a) = ∑x∈R 1∪...∪R j

π(x,a) (4.23)

The term f +j (a) represents the outranking character of a over the reference examples of R j+1∪

. . .∪R K whereas f−j (a) the outranked character of a by all the reference actions of R 1∪ . . .∪R j.The factors m j+1 and m1 are used to compute the "average" characters. On the basis of thesevalues, the assignment rules are as follows:

Assignment Rule 4.6.11. Descending Procedure:if f1(a) > b1, a ∈ C1

else if f2(a) > b2, a ∈ C2

else if . . .

else fK−1(a) > bK−1, a ∈ CK−1

else a ∈ CK

where b1, . . . ,bK+1 represent cut-off points. As pointed out in the introduction of this procedure,the authors propose Linear Programming Techniques to infer preference parameters (such asindifference, preference thresholds, weights, cut-off points etc.) from the examples given by thedecision maker. This constitutes indeed one of the major advantages of the proposed approach.Nevertheless, the methodology do not take into account veto-concepts nor incomparability,which may be useful in certain situations. An example is given in Example 4.5.

128

4.7. Conclusions

Example 4.5. In order to illustrate the PairClass method, let us consider the following example.An action a must be assigned to one of three ordered categories. Each category is defined byexactly one reference example (noted ri, i = 1,2,3). The preference degrees between a and thereference examples are given in Tab.4.14. The category net flows are as follows: f1(a) = 0.5+0.6

2 −

Table 4.14 — Preference degrees between action a and the reference examples.

r1 r2 r3

π(ri,a) 0.4 0.2 0π(a, ri) 0.3 0.5 0.6

0.4 = 0.15 and f2(a) = 0.6− 0.4+0.22 = 0.3. Let us moreover suppose that b1 = 0.5 and b2 = 0.2.

By applying the Descending Procedure, a is assigned to C3.

Let us finally remark that no condition is mentioned such that the values f j(a) are not decreasing.If this would be the case, it should be possible to have f j(a) > b j and fh(a) < bh with h > j.This would lead to a different assignment while starting from the worst category (i.e. ascendingapproach). Some conditions (let us call them monotonicity conditions) may thus be expressedsuch that f j(a) > b j⇒ f j+h(a) > b j+h

25. This should be further analyzed.

4.6.5.3 Properties

The Descending property assigns an action to exactly one category. It verifies thus the propertyof uniqueness. Moreover, following properties are indeed fulfilled: neutrality, weak-homogeneity(in terms of preference degrees) and independency. Depending on the values of the cut-off points,the property of conformity-c is verified. However, it seems natural that the cut-off points are de-termined in that sense.Since preference relations are not necessarily transitive (Paradox of Condorcet, see Section 3.1),this procedure suffers also from the same drawback as Electre-Tri: the property of pairwise as-signment consistency is not verified.Finally, let us analyze the stability by fusing two consecutive categories C j and C j+1. Afterfusion, we have that fh(a) = f

′h(a),∀h 6= j and h 6= j + 1. If the monotonicity conditions are

verified, action a will after fusion be assigned to the same (corresponding) category. Neverthe-less, without satisfying the monotonicity conditions, it could be possible to have f

′j(a) > b j

and f′h(a) > bh and f

′h−1 < bh−1. This could lead to a different assignment. Hence, the property

stability might be thus further analyzed by looking at some monotonicity conditions.

4.7 Conclusions

This chapter has been devoted to a survey of well-known existing sorting methods issued fromthe multicriteria decision aid field. At first, some desirable properties have been defined withouttaking into account the particularities of the methods. In a second step, we have analyzed more

25According to our opinion, it is not clear why this should be verified without imposing some "monotonicity"conditions.

129


deeply certain particular methods. Some concrete examples have been given in order to explainclearly the procedures. The results of these investigations are summarized in Tab.4.15, p.132.

All the methods fulfill the properties of neutrality, uniqueness (depending on the used assign-ment rule) and independency. Most of the methods are stable whereas on the basis of Tab.4.15,we may conclude that solely the UTADIS method (among the studied methods), respects theproperty of pair-wise assignment consistency. Furthermore, some methods respect the stronghomogeneity property whereas other methods the weak one.

We have pointed out that in some cases, the preference-orientation dependency property is notverified by indifference or similarity based methods. In other words, the preference orientationof a criteria does not always play a role in these methods. This happens for instance in thePROAFTN, TRINOMFC and filtering methods (see Tab.4.15, p.132) although these methodscould be used in the case where there is an order on the categories.

All the methods we presented, suppose that a set of parameters is (directly or indirectly) defineda priori by the decision maker. However, in order to compare these methods, we have supposedthat the set of parameters needed for each method is similar. We have for instance comparedPROAFTN, TRINOMFC and the indifference-based filtering methods. In particular situations,we can conclude that these methods lead to the same assignments.

On the other hand, in outranking based methods, we have for instance imposed that the referenceprofiles used in the filtering methods respect the same conditions as the one fulfilled by thereference profile of Electre-Tri. We proceed analogously with the Trichotomic Segmentationmethod. We compared the sorting methods to Electre-Tri since Electre-Tri is, according to[Zopounidis and Doumpos, 2002a], the most widely used multicriteria sorting method basedon the outranking relations approach. This permitted us to compare directly the assignmentsobtained by the different methods. As illustration Tab.4.13 and Fig.4.21 summarize the com-parison of Electre-Tri and the filtering methods on one hand, and Electre-Tri and TrichotomicSegmentation on the other. We can conclude that the main difference between these methodslies in the way a situation of incomparability or indifference, between an action to be assignedand a reference profile, is treated (see Tab.4.15, p. 132).

To the best of our knowledge, no such comparison between these methods has been done before.It constitutes a contribution to the decision aid field, since it allows us to better understand themethods and to distinguish their approach. Moreover, it enlightens a decision maker who needsto choose a method among all the possible ones. A (more) systematic comparison between (all)the methods might be a further research direction.

As presented in this chapter, specific sorting methods have been proposed for specific category-definitions (i.e. either by means of central profiles or limiting profiles). Nevertheless, these twoapproaches have not been reconciled yet. In the next part, we will address the sorting problemwith a more general approach (namely a ranking based approach) which permits to address

130

4.7. Conclusions

several situations. This will be the object of Part II where the Promethee ranking methodologywill be applied in a sorting context. Moreover, we will only consider ranking methods based onpairwise comparisons since it permits to handle situations of pairwise incomparability.

In Chapter 12, we will moreover compare outranking based methods to similarity based sortingmethods. Therefore, we will first slightly adapt the Electre-Tri method to the case of categoriesdefined by central profiles. This will permit us to compare directly PROAFTN to (the modifiedversion of) Electre-Tri and to analyze the appropriateness of using similarity or indifferencebased methods when the categories are ordered.

Finally, let us remark that the proposed outranking based sorting procedures are used in thecase where the categories are completely ordered. No method has been proposed yet in orderto address partially ordered categories. One might think that indifference or similarity basedmethods are appropriate in such case. As we will see, developing outranking based methods forproblems where the categories are partially ordered might enlighten the assignment results. Thiswill be addressed in Chapter 13.

131


Tabl

e4.

15—

Com

pari

son

ofdi

ffer

ents

ortin

gm

etho

dson

the

basi

sof

thei

rpr

oper

ties.

The

follo

win

gab

brev

iatio

nsar

eus

ed:P

O:p

artia

llyor

dere

d,C

O:c

ompl

etel

yor

dere

d,C

NO

:com

-pl

etel

yno

tord

ered

;CP:

cent

ralp

rofil

e,L

P:lim

iting

profi

le;R

el-D

eg.:

Rel

atio

nor

Deg

ree,

Ind:

Indi

ffer

ence

,Sim

.:Si

mila

rity

,Out

.:O

utra

nkin

g;I

Y:Y

es,N

:No

;W-S

:Wea

kor

Stro

ng;

/:ou

tofs

ubje

ct

XXXXXXXXXXXX

Met

hod

Prop

erty

Orderon

categ

ories

Catego

ryde

finition

Assign

mentr

elatio

n

IandJdif

feren

tiatio

n

Homog

eneit

y Confor

mity Stabilit

yM

onoto

nicity Pair

-Wise

Assign

mentC

onsis

tency

Prefere

nceOrie

ntatio

nDepen

denc

y

PRO

AFT

NC

NO

Mul

tiC

PIn

d.R

el.-D

eg.

YW

-SY

Y/

/N

TR

INO

MFC

POM

ulti

CP

Sim

.Deg

.Y

WY

Y/

/N

I-Fi

lteri

ngM

etho

dsPO

Mul

tiC

PIn

d.R

el.-D

eg.

YW

-SY

Y/

/N

UTA

DIS

CO

Thr

esho

lds

Util

ities

.N

W/

YY

YY

MH

DIS

CO

Thr

esho

lds

Util

ities

NW

/Y

YY

Y

Ele

ctre

-Tri

CO

Mon

oL

PO

ut.R

el.

YS

NY

YN

/Tr

icht

omic

Segm

enta

tion

CO

Mul

tiL

PO

ut.R

el.-D

eg.

NW

-SN

/Y

N/

nTom

icC

OT

hres

hold

sO

ut.D

eg.

NW

N/

YN

/P-

Filte

ring

Met

hods

CO

Mul

tiL

PPr

ef.R

el.-D

eg.

NW

-SN

YY

N/

Pair

Cla

ssC

OM

ulti

CP

Pref

.Deg

.N

WY

NY

N/

132

Part

F lowSort: a flow-based sortingmethod

133

The F lowSort method is the outcome of the following main idea: when completely orderedcategories are defined by central or limiting profiles, we may address the sorting problematic bymeans of a complete ranking method. The ranking method is applied on the data-set consistingof one action to be assigned and the reference profiles. The category, to which the action will beassigned, will then be deduced from its relative position with respect to the references profilesin the obtained ranking (e.g. Fig.6.1). The assignment will thus depend on a global comparisonwith all the profiles simultaneously. The consequences of this will be addressed in Section8. Moreover, using a complete ranking method (without ex-aequo) implies that an action isassigned to a unique category.

One advantage of such an approach is that a decision maker who is familiar with a particularranking method can easily understand the corresponding sorting method. We illustrate thisconcept by the well-known Promethee method ([Brans and Vincke, 1985]). Let us remarkthat an extension of Promethee, called Promethee Tri, has been proposed to address nominalclassification problems [Figueira et al., 2004]. A critical analysis of this method is given by[Cailloux, 2006]. However, F lowSort and Promethee Tri differ significantly.

Moreover, a category may either be delimited by two boundaries or limiting profiles (like forinstance in Electre-Tri, Section 4.6.1), or it can be represented by central profiles (such as inthe model proposed by Doumpos and Zopounidis, [Doumpos and Zopounidis, 2004a]). Nosorting method has, to the best of our knowledge, been proposed yet which handles both typesof profiles. The proposed method is thus somehow more flexible.

Throughout this part, we suppose that all the parameters (weights and thresholds of the criteria,evaluations of the reference profiles on the different criteria, etc.) have been determined directlyby the decision maker or by an elicitation process. In other words, the parameters are fixed.Except for Chapter 9, we consider that the parameters and that the evaluation of the actions tobe sorted are known precisely.

This part is organized as follows. First, we shall introduce some notation and definitions andclearly state the assumptions on which the model is built (Chapter 5). Chapter 6 is devoted topresenting the assignments rules, their analysis, some examples and an application. Chapter7 presents some properties of the method. Chapter 8 presents a empirical and theoreticalcomparison with Electre-Tri and UTADIS. In Chapter 9 we tackle the sorting problematic whenthe data is imprecise. In this context, we propose an interval and fuzzy extension of F lowSort,called respectively I nterval and F uzzy F lowSort. Finally, some general conclusions are givenin Chapter 11. The proofs of the propositions can be found in the Appendix A.

This first part is based on the following works: [Nemery and Lamboray, 2007], [Nemery andLamboray, 2008], [Janssen and Nemery, 2007] and [Huenaerts and Nemery, 2007].

Prerequisites for this part: Ranking and Sorting Problems (See Section 2.2.4 and chapters 3 and4), Promethee(see Section 3.3.2).

135

5 Notation and conditions

In this chapter we give the notations used in the F lowSort method. Moreover,we explain the conditions that the reference profiles should respect as well asthe assumptions on which the model is built. Finally, the main proposition onwhich the assignment rules are based, is given.

5.1 Notation

We will adopt the same notation as in previous chapters. A denotes the set of the actions to besorted: A = a1, ...,an. Each action ai is evaluated on q criteria g j ( j = 1, ...,q) and we suppose,without any loss of generality, that these q criteria have to be maximized.

The categories to which the actions will be assigned, are denoted by C1,C2, . . . ,CK . Furthermorewe suppose that the categories are completely ordered as follows: C1 B . . . B Cl B . . . B CK

where Ch BCl , with h < l, designates that category Ch is preferred to category Cl .

Each category is defined either by two limiting profiles or one central profiles. First we shallconsider the case where the categories are defined by limiting profiles. Consequently, a categoryCh is defined by a upper and lower profile, noted as rh and rh+1 respectively. At the same time,rh is thus the lower profile of Ch−1 and rh+1 is the upper profile of Ch+1. The set of limitingprofiles is denoted as R = r1, ...,rK+1. Let us remark that the limiting profiles r1 and rK+1are not necessary to define respectively categories C1 and CK . The decision maker may want toconsider these two categories as “semi-open”. In the rest of this part, we will suppose that r1 andrK+1 have been defined (for a preprocessing purpose for instance).

Besides, we suppose that the performances of all the actions in A are between the worst rK+1and best r1 limiting profiles. We have thus formally that ∀ai ∈ A , ∀g j ∈ G : g j(r1) ≥ g j(ai) ≥g j(rK+1). This is not a hard constraint since the decision maker is free to define them as hewants. Fig.5.1 give an illustration of five limiting profiles defining 4 categories.

On the other hand, instead of using two limiting profiles for the definition of each category, thedecision maker may choose to define these by one central profile (often called the centroid of

137

Notation and conditions

-

-

-

-

g2

gq

gq−1

. . .CK CK−1 . . .. . . C1C j

g1

rK−1rK r j r j−1rK+1 r1

Figure 5.1 — Representation of K completely ordered categories by limiting profiles

-

-

-

-

CK

rK

g2

gq

gq−1

. . .C1C j

g1

. . .

r1r j

. . .

Figure 5.2 — Representation of K completely ordered categories by central profiles

the category). In such a situation, for K completely ordered categories, there are K centroidsdenoted by R = r1, · · · , rK, where r j is the centroid of category C j. The subscript “.” will beused to indicate that a category is defined by a central reference profile. Fig.5.2 illustrates anexample where centroids are used for defining 4 categories.

We shall further note the reference profiles by R ∗ = r∗1,r∗2, ... when no distinction has to bemade between a set of limiting profiles and a set of centroids.

Finally, let us define for any action ai of A to be assigned the following set: R ∗i = R ∗∪ai,∀ai ∈A .

5.2 Conditions

Since the reference profiles define ordered categories, we shall assume that two consecutivereference profiles dominate each other (see Section 2.2). This is formulated by the followingcondition:

Condition 5.2.1. ∀r∗h,r∗l ∈R ∗ such that h < l: gk(r∗l )≤ gk(r∗h) , ∀gk ∈G and ∃gx ∈G | gx(r∗l ) <

gx(r∗h).

Moreover, we suppose that the actions of A will be compared to the reference profiles by meansof preference degrees π(x,y) (∀x,y of R ∗i ), which evaluate the preference strength of action x overan action y according to the decision maker’s insight. The preference degrees can for instance be

138

5.2. Conditions

obtained as in the Promethee methodology [Brans and Vincke, 1985] as introduced in Section3.3.2. Furthermore, we suppose that ∀x,y ∈ R ∗i the following conditions hold:

Condition 5.2.2. 0≤ π(x,y) ≤ 1.

Condition 5.2.3. π(x,y)+ π(y,x) ≤ 1.

Condition 5.2.4. π(x,x) = 0.

Condition 5.2.5. ∀x,y,x′,y′ ∈ R ∗i , If ∀ j ∈ 1, . . . ,q : g j(x)−g j(y) ≤ g j(x′)−g j(y′),then π(x,y) ≤ π(x′,y′).

Since the reference profiles define ordered categories, we will impose that a reference profile ofa lower (better) category is “preferred”, according to the decision maker, to the reference profilesof a higher (worse) category. Formally, we have that:

Condition 5.2.6. ∀r∗h,r∗l ∈ R ∗ such that h < l: π(r∗h,r∗l ) > 0 and π(r∗l ,r∗h) = 0.

Practically, Condition 5.2.6 will be verified if Condition 5.2.1 is fulfilled and if there exists atleast one criterion g j such that g j(r∗h)− g j(r∗l ) > q j g j is modeled with for example a linearpreference function and q j representing the indifference threshold.

On the basis of these preference degrees, the positive (leaving), negative (incoming) and netflows of each action x of R ∗i are computed as follows [Brans and Vincke, 1985]:

φ+R ∗i

(x) =1

|R ∗i |−1 ∑y∈R ∗i

π(x,y) (5.1)

φ−R ∗i

(x) =1

|R ∗i |−1 ∑y∈R ∗i

π(y,x) (5.2)

φR ∗i (x) = φ+R ∗i

(x)−φ−R ∗i

(x) (5.3)

Let us stress that these flows are computed for all the actions of the set R i and not for the set Asuch as in a ranking problem. This means thus that, after having pair-wise compared an actionai to the reference profiles, global scores (flows) are calculated. These flows lead to completerankings. These rankings will be computed each time an action ai has to be assigned.

If the reference profiles verify condition 5.2.1 to 5.2.6, we have that the order of the flows of thereference profiles is (always) the same as the order of the reference profiles regardless the actionto be sorted. In other words, there will not be a rank reversal phenomenon (see Section 3.3.2.4)between the reference profiles. This is formalized in the next proposition:

Proposition 5.2.1. Under conditions 5.2.1 to 5.2.6, we have that the order of the flows of thereference profiles is invariant with respect to the action ai to assign: ∀ai ∈ A:

∀h ∈ 1, ...,K ∀h ∈ 1, ...,K−1φ

+R i

(rh) > φ+R i

(rh+1)φ−R i

(rh) < φ−R i

(rh+1)φR i

(rh) > φR i(rh+1)

φ

+R i

(rh) > φ+R i

(rh+1)φ−R i

(rh) < φ−R i

(rh+1)φR i

(rh) > φR i(rh+1)

139

Notation and conditions

In other words, although the actual flow values of the reference profiles directly depend on theaction ai, their order always respects the order of the categories. This allows us to delimit acategory Ch by the flow values of rh and rh+1 in the case that the categories are defined by anupper and lower limit. Alternatively, a category, defined by a centroid, is represented by theflows of that centroid with respect to the other centroids.

This proposition will be the basis of the F lowSort assignment rules.

140

6 Flow-based assignmentprocedures

Based on Proposition 5.2.1, the assignment rules will be given in the case oflimiting and central profiles. The assignment rules are illustrated by means ofsome basic examples. Moreover, a graphical representation as well as someproperties are proposed. Finally, the influence of the preference parameters ofthe model are represented by graphical illustrations.

6.1 Limiting profiles

We assume throughout this section that a set of limiting profiles R has been defined and thatpreference degrees between the actions of R i and the flows have been computed.

Two different assignment rules, based on the previous considerations (i.e. Proposition 5.2.1), aredefined as follows:


Cφ+(ai) = Ch, if φ+R i

(rh) ≥ φ+R i

(ai) > φ+R i

(rh+1)


Cφ−(ai) = Ch, if φ−R i

(rh) < φ−R i

(ai) ≤ φ−R i

(rh+1)

In other words, we evaluate the “preferred to” (Assignment Rule 6.1.1) and the “being-preferredby” character (Assignment Rule 6.1.2) of an action ai with respect to the reference profiles by

-

10

r1. . . . . .rK+1Fai

φ+Ri

(.)r jr j+1

Figure 6.1 — Representation of the complete ranking obtained by computing the positiveflows. This leads to the Cφ+ -assignment.

141

Flow-based assignment procedures

means of respectively the positive and the negative flows. In the first rule, the action is thenassigned to the category Ch if the flow φ+(ai) is contained in the interval defined by the positiveflows of the reference profiles of category Ch (see Fig.6.1). In the second rule, the action isassigned to the category Ch if the flow φ−(ai) is contained in the interval defined by the negativeflows of the reference profiles of category Ch.

As in the Promethee I methodology, two aspects are considered. On one hand, a ranking iscomputed based on incoming flows, which in our model leads to the assignment Cφ+ . On theother hand, a ranking is computed based on leaving flows, which in our model leads to theassignment Cφ− .

Moreover, the actions are assigned according to each rule to exactly one category. The propertyof uniqueness (Property 4.2) is thus fulfilled.

We can thus obtain two different assignments: Cφ+(ai) and Cφ−(ai), where either Cφ+(ai) D 1

Cφ−(ai), or Cφ−(ai)DCφ+(ai). Let us denote by Cb(ai) the best category and by Cw(ai) the worstcategory obtained with these two assignment rules for ai:

Cb(ai) = min[Cφ+(ai),Cφ−(ai)] (6.1)

Cw(ai) = max[Cφ+(ai),Cφ−(ai)] (6.2)

Furthermore, if a decision maker would impose the assignment to strictly one category, we coulddefine a similar assignment rule using the net flows:


Cφ(ai) = Ch, if φR i(rh) ≥ φR i(ai) > φR i(rh+1)

In fact, this net flow assignment rule is analogous to the Promethee II ranking. This appearsreasonable since the assignment obtained with the net flow rule is consistent with the twoassignments obtained with the positive and negative flow rules: the net flow assignment isalways in between these two (see Property 7.1.1).

To give an easily understandable visualization of the assignments, we can represent the φ−R i

andφ

+R i

flows of all the actions of R i in a [φ−,φ+]-flow space. In this space, given a category Ch, thepoints (φ−R i

(rh),φ+R i

(rh)) and (φ−R i(rh+1),φ+

R i(rh+1)) naturally define a rectangle (see Fig.6.2).

The flows of the action ai to be assigned also defines a point (φ−R i(ai),φ+

R i(ai)) in this space.

If this point lies in the rectangle of category Ch (e.g. action 2 in Fig.6.2), then ai is obviouslyassigned to Ch (with both the positive and the negative flow rules). However, the assignmentsobtained on the basis of the negative and the positive flow may also differ, as it is the case ofactions 1 and 3 in Fig.6.2. These actions are then assigned to a set of consecutive categories.

1C D C means that either C B C or C = C.

142

6.1. Limiting profiles

Let us moreover remark that this flow-diagram is not “fixed” for a particular problem (i.e. set oflimiting profiles). On the contrary, since the flows directly depend on the action ai to be assigned(cf. Table 6.5), this flow-representation varies with every action of A .

6

-

rrr

r

Ch−1

Ch

Ch+1

φ−

φ+

φ−Ri

(rh−1) φ−Ri

(rh) φ−Ri

(rh+1) φ−Ri

(rh+2)

φ+Ri

(rh+2)

φ+Ri

(rh+1)

φ+Ri

(rh)

φ+Ri

(rh−1)

F3 2F

F1

Figure 6.2 — A flow and category representation with limiting profiles.

The assignment of an action depends thus on the comparison with all the reference profilessimultaneously, and not, as for instance in Electre-Tri (see Section 4.6.1), on successive pairwisecomparisons. Although, this more “global” approach can appear unconventional, it is a directconsequence of using a ranking method in a sorting context. This has as consequence, thatthe categories are not defined in an absolute manner. On the contrary, they are defined in aordinal and relative way. In Section 8.1.2, we study the impact of such a complete-ranking orglobal-comparison approach with respect to an absolute assignment procedure.Nevertheless, let us stress that each action to be sorted is compared individually to the referenceprofiles. The actions are thus not ranked with respect to each other but ranked, individually, withrespect to the norms or profiles. The Property of independency (Property 4.3) is indeed verified.

6.1.1 Strongly preferred limiting profiles

So far, we simply supposed that rh is preferred to rh+1: π(rh,rl) > 0 with h < l. We canstrengthen this condition by accepting that the upper profile r j of a category C j is “stronglypreferred” to the lower profile r j+1. This may be formalized as follows:

Condition 6.1.1. ∀rh,rl ∈ R such that h < l: π(rh,rl) = 1

This condition on the reference profiles might at first sight be considered as too strong. In fact, it

143


just ensures that the dominance between reference profiles is a strong preference according to thedecision maker. Let us remind that in Electre-Tri a similar condition is imposed on the referenceprofiles (page 29 in [Yu, 1992b]).

In case the reference profiles verify Condition 6.1.1, we have moreover the following property:

Proposition 6.1.1. ∀ai, if Cφ−(ai) = Cl and Cφ+(ai) = Ch, then l−h≤ 0.

One consequence of this proposition is that not every category combination (Cφ−(ai),Cφ+(ai))is possible. In fact, under the assumption that Condition 6.1.1 is verified, category Cφ−(ai) isalways as least as good as category Cφ+(ai).

Moreover, as we will see in Section 7.7, this condition of strong preference play a crucial role inthe stability of the sorting procedure.

Example 6.1. In order to illustrate the assignment rules in the case with limiting profiles, let usconsider the following example. Suppose that we have 4 categories defined by 5 limiting profiles.Each profile is evaluated on the basis of 5 quantitative criteria which have to be maximized. Thecorresponding performances are given in Table 6.1 and illustrated in Fig.6.3. To compute thepreference degree we have used the Promethee methodology. For each criterion we have fixedthe indifference and preference threshold equal to 0 2. The profiles verify thus Condition 6.1.1.Moreover, all criteria have the same weights. The parameters are resumed in Table 6.2.

-

-

-

-

s s

sss

ss

s ssss

sss

ss

s

ss

s -s s s s g5(.)

g4(.)

g3(.)

g2(.)

g1(.)

a3

23

9

19

5

15

a2 a1

r5 r3 r4 r2 r1

63

45

71

17

54

75

75

50

25

25

Figure 6.3 — Representation of the limiting profiles and the actions to be assigned.

Suppose now that we want to sort a following set of actions A = a1,a2,a3 whose performanceson the different criteria are as given in Table 6.3 and illustrated in Fig.6.3. The preferencesdegree of a1, a2 and a3 with respect to the reference profiles are given below (see Table 6.4). This

2One may remark that we actually use the Borda ranking method for sorting the actions given these particularthresholds (Section 3.1 and 3.3.2.3). Moreover, we have thus that ∀h < l : π(rh,rl) = 1 and π(rl ,rh) = 0).

144


Table 6.1 — The performances of the reference profiles.

R g1 g2 g3 g4 g5

r1 100 100 100 100 100

r2 75 75 75 75 75

r3 50 50 50 50 50

r4 25 25 25 25 25

r5 0 0 0 0 0

Table 6.2 — The different thresholds and weights.

g1 g2 g3 g4 g5

qk 0 0 0 0 0

pk 0 0 0 0 0

wk 1 1 1 1 1

allows us to compute the flows and determine their assignment (see Table 6.5). For instance,φ

+R 1

(r2) = ∑ j π(r2,r j)+π(r2,a1)5 = 1+1+1+0.6

5 . The flow-diagram for a1 is given in Fig.6.4.

Table 6.3 — The performances of the actions to be sorted.

A g1 g2 g3 g4 g5

a1 25 25 50 75 75

a2 54 17 71 45 63

a3 15 5 19 9 23

We can conclude that action a1 should be assigned to respectively category 3 or 2 consideringthe positive flows and the negative flows. The assignment of action a1 is graphically representedin Fig.6.4. On the other hand, action a2 behaves analogous as reference profile r3 with respectto the other profiles (traduced by the flow equality). It should thus be assigned consideringthe positive, negative and net flows to categories 2 and 3. Finally, action a3 is unambiguouslyassigned to category 4.

145


Table 6.4 — The preference degrees between the reference profiles and the actions.

r1 r2 r3 r4 r5

π(a1,r j) 0 0 0.4 0.6 1

π(r j,a1) 1 0.6 0.4 0 0

π(a2,r j) 0 0 0.6 0.8 1

π(r j,a2) 1 1 0.4 0.2 0

π(a3,r j) 0 0 0 0 1

π(r j,a3) 1 1 1 1 0

Table 6.5 — Computation of the different flow values.

r1 r2 r3 r4 r5 ai C(ai)

φ+R 1

1 0.72 0.48 0.2 0 0.4 3

R 1 φ−R3

0 0.2 0.48 0.72 1 0.4 2

φR 11 0.52 0 -0.52 -1 0 2,3

φ+R 2

1 0.8 0.48 0.24 0 0.48 2,3

R 2 φ−R1

0 0.2 0.52 0.76 1 0.52 2,3

φR 21 0.6 -0.04 -0.48 -1 -0.04 2,3

φ+R 3

1 0.8 0.6 0.4 0 0.2 4

R 3 φ−R2

0 0.2 0.4 0.6 1 0.8 4

φR 31 0.6 0.2 -0.2 -1 -0.6 4

146


-

6

φ−R 1

φ+R 1

C1

C2

C3

C4

F

10.720.2 0.48

0.24

0.48

0.7

1

a1

Figure 6.4 — Flow-diagram for a1.

Let us note that, considering the same parameters, Electre-Tri assigns action a1 according tothe optimistic and pessimistic procedures to category 2 with a λ-threshold equal to 0.5. Withλ = 0.85, the optimistic and the pessimistic procedures give respectively category 2 and 3. More-over, Electre-Tri assigns unambiguously action a2 to category 4. Finally, if λ > 0.6 the optimisticand the pessimistic procedures assign a3 respectively to category 2 and 3, and if λ≤ 0.6 unam-biguously to category 3. All the results are summarized in Table 6.6.One may remark that the results of the assignments are quite similar. Action a3 will be assignedto the same category according to both methods. Depending on the values of λ, actions a1 and a2

may be assigned to the same categories. In any case, the difference in assignment is very smallbut may differ depending on the value of λ.

Table 6.6 — The assignments of the actions according to Electre-Tri and F lowSort.

ai λ Cpess Copt Cφ+ Cφ− Cφ

[0.5 ; 0.6] 2 2

a1 ]0.6 ; 0.8] 3 2 2,3 2,3 2,3

]0.8 ; 1] 4 2

a2 [0.5;1] 4 4 4 4 4

a3 [0.5;0.6] 3 3 2 3 3

]0.6;1] 3 2

147


6.2 Central profiles

We assume throughout this section that a set of central profiles R has been defined and thatpreference degrees between the actions of R i and the flows have been computed.

When assigning an action to a category, we compare its flows with the centroids’ flows and assignan action to the category whose centroid has similar (or the most similar) flows. Formally, theassignment rules based on either the positive or the negative flow are defined as follows:


Cφ+(ai) = Ch, if: | φ+R i

(rh)−φ+R i

(ai) |= min∀ j | φ+

R i(r j)−φ

+R i

(ai) |


Cφ−(ai) = Ch, if: | φ−R i(rh)−φ

−R i

(ai) |= min∀ j | φ−R i

(r j)−φ−R i

(ai) |

We may interpret the previous assignment rules as assigning action ai to the nearest centralprofile considering the positive and negative flows.

The philosophy of the assignment rule can be interpret as in the case of limiting profiles: anaction is assigned, according to the first rule, to the category whose centroid has the same (or themost similar) “preferred character” over the remaining reference actions. On the other hand, thesecond rule is based on the ’being preferred by-’ character of that action.Previous assignment rules may be written as follows (∀ h 6= K and h 6= 1) :


Cφ+(ai) = Ch, if:φ

+R i

(rh)+ φ+R i

(rh+1)

2< φ

+R i

(ai) ≤φ

+R i

(rh)+ φ+R i

(rh−1)

2.

Assignment Rule 6.2.4. ∀ h 6= K and h 6= 1

Cφ−(ai) = Ch, if:φ−R i

(rh)+ φ−R i

(rh+1)

2≥ φ

−R i

(ai) >φ−R i

(rh)+ φ−R i

(rh−1)

2.

The assignment rules are quite similar to the assignment rules defined for limiting profiles.Actually, we have a virtual limiting profile defined by two consecutive central profiles.

Based on the two different assignment rules, two possible categories Cφ−(ai) and Cφ+(ai) can beobtained.As in the case of limiting profiles, let us denote by Cb the best and by Cw the worst ofthese two categories:

Cb(ai) = min[Cφ+(ai),Cφ−(ai)] (6.3)

Cw(ai) = max[Cφ+(ai),Cφ−(ai)] (6.4)

148

6.2. Central profiles

If an action has to be assigned to one unique category, we can naturally define an assignmentrule based on the net flows:


Cφ(ai) = Ch, if: | φR i(rh)−φR i

(ai) |= min∀ j | φR i

(r j)−φR i(ai) |

Previous assignment rule may be rewritten as follows (∀ h 6= K and h 6= 1):


Cφ(ai) = Ch, if:φR i

(rh)+ φR i(rh+1)

2< φ

+R i

(ai) ≤φR i

(rh)+ φR i(rh−1)

2.

The different positive and negative flows of the centroids and the action to be sorted can againbe represented in the [φ−,φ+] flow-space (Figure 6.5). Let us notice that the “rectangles”representing the ordered categories are now obtained in a slightly different way. The upper-leftpoint of the rectangle representing category Ch is the average between the flows values of rh andrh−1, whereas the lower-right point is the average between the flows values of rh and rh+1. If thepoint (φ−R i

(ai),φ+R i

(ai)) lies in the rectangle of category Ch, then ai is obviously assigned to Ch

(both with the positive and negative flow rules). Let us again stress that this diagram depends onthe action to be assigned.

6

-

s

Ch−1

Ch

Ch+1

φ−

φ+

φ−Ri

(rh−1) φ−Ri

(rh) φ−Ri

(rh+1)

φ+Ri

(rh+1)

φ+Ri

(rh)

φ+Ri

(rh−1) •

•F3 2F

F1

Figure 6.5 — A flow and category representation with central profiles.

6.2.1 Strongly preferred central profiles

As in the case of limiting profiles, we could accept that two successive centroids are “stronglypreferred”. Formally, they may respect the following condition:

149


Condition 6.2.1. ∀rh, rl ∈ R such that h < l: π(rh, rl) = 1

If the centroids satisfy Condition 6.2.1, we have a similar property as Property 7.5.1 concerningthe possible category-combination: ∀ai, if Cφ−(ai) = Cl and Cφ+(ai) = Ch, then l−h≤ 0.

Example 6.2. Let us reconsider Example 6.1 where the 4 categories are now defined by centralprofiles. The performances of the central profiles are given in Table 6.7. We assign the sameactions as in previous section (6.1) but the preference thresholds and the weights are different(see Table 6.8).


R g1 g2 g3 g4 g5

r1 80 90 90 75 82

r2 56 62 58 61 52

r3 40 35 35 30 33

r4 10 5 10 10 15


g1 g2 g3 g4 g5

qk 4 6 0 2 3

pk 8 13 7 5 12

wk 1 1 1 1 1

The preferences degrees of a1, a2 and a3 with respect to the reference profiles are given inTable 6.9. This allows us to compute the flows which determine their assignment (see Table 6.10).

Hence, a1 is assigned to C2 in all three assignment rules. Action a2 is unambiguously assignedto C4 and action a3 to C2.

6.3 Influence of the preference parameters

A crucial issue in multicriteria decision procedures is the determination of the preferenceparameters of the model. The preference parameters can either be given directly by the decisionmaker or elicited by means of a learning set. As stressed in the introduction of this work, theelicitation through examples of the parameters can be considered as a problem it-self. Several

150

6.3. Influence of the preference parameters

Table 6.9 — The preference degrees between the reference profiles and the alternatives.

r4 r3 r2 r1

π(a1, r j) 0 0.38 0.8 0.97

π(r j,a1) 1 0.4 0.2 0

π(a2, r j) 0 0 0 0.36

π(r j,a2) 1 1 0.95 0

π(a3, r j) 0 0.4 0.6 1

π(r j,a3) 0.69 0.6 0.31 0

Table 6.10 — The flow-values of the alternatives.

r1 r2 r3 r4 ai C(ai)

φ+R 1

1 0.6 0.3 0 0.54 2

R 1 φ−R 1

0 0.34 0.7 0.99 0.4 2

φR 11.0 0.25 -0.4 -0.99 0.14 2

φ+R 2

1 0.75 0.49 0 0.09 4

R 2 φ−R 2

0 0.25 0.5 0.84 0.74 4

φR 21 0.5 -0.01 -0.84 -0.65 4

φ+R 3

0.92 0.65 0.33 0 0.5 2

R 3 φ−R 3

0 0.35 0.65 1 0.40 2

φR 30.92 0.3 -0.32 -1 0.1 2

researchers have been active in this field for previously mentioned methods like UTADIS,Electre-Tri. We refer the reader for more information to the following works [Dias et al., 2002;The and Mousseau, 2002], etc.

Recently, some methods have been proposed for the determination of the preference parametersof the Promethee method [Frikha et al., 2007, 2004],[Vilain, 2007],[Huenaerts and Nemery,2007]. These methods may be adapted to the sorting context. In this work, we do not havefocussed on the determination of the preference parameters of the different proposed methodsbut rather on the consequences and issues of the methods. Nevertheless, to be a direct applicablemethod, it should certainly be worth to pursue this research direction.

151


6

φ+

-

r1

r2a1

r3

r4

0.70.34

0.8

0.6

0.3

φ−

Ft

t

t

tFigure 6.6 — Flow-diagram for a1

However, to illustrate the impact of the indifference and preference thresholds on the assign-ments, let us consider the following examples which may be either an aid for their determinationand for a intuitive understanding of the assignment rules. We will distinguish the case where weare working with limiting or central profiles.

6.3.1 Limiting profiles

Consider an assignment problem with 3 ordered categories and 2 criteria with the same weights.Suppose that the limiting profiles are r1 = (0;0), r2 = (1/3;1/3), r3 = (2/3;2/3) andr4 = (1;1). These are represented by “” in Fig.6.7-6.8.

Let us consider that A = ai | g1(ai) ∈ [0,1] and g2(ai) ∈ [0,1]. In other words, we considerthat all the points from the cartesian product [0,1]× [0,1] have to be assigned with respect to the4 limiting profiles of R .

Case 1: p = q = 0.Let us first suppose that p = q = 0. When applying the F lowSort procedures (according tothe positive, negative and net flows) to this “map”, we obtain the assignments represented inFig.6.7. The Cφ+- and Cφ−-assignments are in this case identical. Moreover, the Cφ-assignmentsequal them obviously (Proposition 7.1).

Case 2: q = 0.05 and p = 0.075.Suppose now that q = 0.05 and p = 0.075 and that the preference functions are the linearfunctions. For the same set A and R we obtain respectively for the positive, the negative and the

152


1

2/3

1/3

1/3 12/3-

6

g1

g2

r4

r3

r2

r1

C1

C2

C3

Figure 6.7 — Assignment of any point (x,y) of the plan with the positive, negative and netflows when q = 0 and p = 0, w1 = w2 = 0.5: identical assignments in the 3 cases.

net flows the Fig.6.9, 6.10 and 6.8.

The purpose of the figures of this chapter, is to give the reader a first intuitive idea of theassignment philosophy. The reader can, moreover, make a first intuitive comparison with theassignments of UTADIS (see Section 8.2) and Electre-Tri (see Section 8.1, see Fig.4.17) in thesame situations.

6.3.2 Central profiles

Consider a assignment problem with 3 ordered categories defined by central profiles. Supposethat there are 2 independent criteria with the same weights and that the reference profilesare as follows: r1 = (0.75;0.75), r2 = (0.5;0.5) and r3 = (0.25;0.25) (represented by "?" inFig.6.12-6.13).

Let us consider that A = ai | g1(ai) ∈ [0,1] and g2(ai) ∈ [0,1]. In other words, we considerthat all the points from the cartesian product [0,1]× [0,1] have to be assigned with respect to the3 central profiles.

Case 1: p = q = 0.When applying the F lowSort procedures (according to the positive, negative and net flows) tothis "map", we obtain the assignments represented in Fig.6.11. The Cφ+- and Cφ−-assignments

153


1

2/3

1/3

1/3 12/3-

6

g1

g2

r4

r1

C1

C2

C3

r3

r2-

?6p

q

q

q

-

-

Figure 6.8 — Assignment of any point (x,y) of the plan with the net flows when q = 0.05 andp = 0.075, w1 = w2 = 0.5.

1

2/3

1/3

1/3 12/3-

6

g1

g2

r4

r2

r1

C1

C2

C3

-

6?p

r3

q

Figure 6.9 — Assignment of any point (x,y) of the plan with the positive flows when q = 0.05and p = 0.075, w1 = w2 = 0.5.

154


1

2/3

1/3

1/3 12/3-

6

g1

g2

r4

r1

C1

C2

C3

r3

r2

?6

--

-

q

q

pq

Figure 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.

are, in this case, identical. Moreover, the Cφ-assignments equal them obviously (Proposition7.1). Let us remark that in the case of two different possible assignments (due to the equalitycase in the "≤" of the assignment rules), the decision maker may want to choose the worstcategory as represented in Fig.6.12.

Case 2: q = 0.05 and p = 0.075.Suppose now that q = 0.05 and p = 0.075 and that the preference functions are the linearfunctions. For the same set A and R , we obtain with respectively the positive, negative and netflows the (right) the assignments illustrated in Fig.6.13,6.14 and 6.15. When the decision makerchoose the worst category in case of several possibilities, we obtain the left sub-figures of theprevious one.

155


-

6

g1

g2

F

F

F

r3

r2

r1

0.25 0.5 0.75

0.25

0.5

0.75

C2

C2

C2

C2C3

C1

C1 ∪C2

C1 ∪C2

C1 ∪C2

C2 ∪C3

C2 ∪C3

C2 ∪C3

Figure 6.11 — Assignment of any point (x,y) of the plan with the positive, negative and netflows when q = 0 and p = 0, w1 = w2 = 0.5.

-

6

g1

g2

F

F

F

r3

r2

r1

0.25 0.5 0.75

0.25

0.5

0.75

C1

C2

C3

F

F

F

r3

r2

r1

-

6

g1

g2

?

?

?

0.25 0.5 0.75

0.25

0.5

0.75

C1

C3

C2

Figure 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 categoryin case of equality.

156


-

6

g1

g2

F

F

F

r3

r2

r1

0.25 0.5 0.75

0.25

0.5

0.75

C1

C3

C2

-

6

g1

g2

F

F

F

r3

r2

r1

0.25 0.5 0.75

0.25

0.5

0.75

C1

C3

C2C3 ∪C2

C3 ∪C2

C2 ∪C1

C2 ∪C1

C3 ∪C2

C2 ∪C1

Figure 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 ofequality (left).

-

6

g1

g2

F

F

F

r3

r2

r1

0.25 0.5 0.75

0.25

0.5

0.75

C1

C3

C2

-

6

g1

g2

F

F

F

r3

r2

r1

0.25 0.5 0.75

0.25

0.5

0.75

C1

C3

C2C2 ∪C1

C2 ∪C1

C3 ∪C2

C2 ∪C1

C3 ∪C2

C3 ∪C2

Figure 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 ofequality (left).

157


-

6

g1

g2

F

F

F

r1

r2

r3

0.25 0.5 0.75

0.25

0.5

0.75

C1

C3

C2

-

6

g1

g2

F

F

F

r3

r2

r1

0.25 0.5 0.75

0.25

0.5

0.75

C1

C3

C2

C2 ∪C1

C3 ∪C2

Figure 6.15 — Assignment of any point (x,y) of the plan with the net flows when q = 0.05and p = 0.075, w1 = w2 = 0.5 (right) and by choosing the worst category in case of equality(left).

158

7 Analysis of some properties ofF lowSort

In this chapter we analyze the properties (defined in Section 4.2) that the as-signment rules 6.1.1-6.2.6 verify. Most of the properties are valid when usingeither limiting or central profiles. In the contrary case, it will be mentionedexplicitly. The proof are given in Appendix A.

7.1 Coherence of the net-flow assignment rule

The assignment procedure based on the net flows complies with the one based on the positiveand negative flows since the assignment result is always in between the positive and the negativeassignment result. More formally, we have:

Proposition 7.1.1. ∀ai ∈ A ; ∀R ,∀R :

Cb(ai)DCφ(ai)DCw(ai) & Cb(ai)D Cφ(ai)D Cw(ai)

7.2 Property of monotonicity

If an action ai dominates another action a j, then ai can not be assigned to a higher (worse)category than action a j. This is stated in the following proposition:

Proposition 7.2.1. ∀ai,a j ∈ A ; ∀R ,∀R ,

gk(a j) ≤ gk(ai) , ∀gk ∈ G and ∃gx ∈ G | gx(a j) < gx(ai)

.⇓

Cφ+(ai)DCφ+(a j) & Cφ−(ai)DCφ−(a j) & Cφ(ai)DCφ(a j)

159

Analysis of some properties of F lowSort

Cφ+(ai)D Cφ+(a j) & Cφ−(ai)D Cφ−(a j) & Cφ(ai)D Cφ(a j)

7.3 Property of weak homogeneity

If two action ai and a j compare themselves (in term of preference degrees) in the same way tothe reference profiles, they are assigned to the same categories. Formally:

Proposition 7.3.1. ∀ai;a j ∈ A ; ∀R ,

∀rh ∈ R : π(rh,ai) = π(rh,a j) & π(ai,rh) = π(a j,rh)

⇓

Cφ+(ai) = Cφ+(a j) & Cφ−(ai) = Cφ−(a j); & Cφ(ai) = Cφ(a j)

Proposition 7.3.2. ∀ai;a j ∈ A ; ∀R ,

∀rh ∈ R : π(rh,ai) = π(rh,a j) & π(ai, rh) = π(a j, rh)

⇓

Cφ+(ai) = Cφ+(a j) & Cφ−(ai) = Cφ−(a j) & Cφ(ai) = Cφ(a j)

7.4 Properties of category conformity

A crucial issue in multicriteria decision aid is the ability to explain the assignments of theactions in the different situations. This is addressed by the following proposition.

When working with limiting profiles, if the performances of an action ai (on all the criteria) arein between two successive limiting profiles, it will obviously be assigned to that correspondingcategory. The proof is immediate when using Proposition 5.2.1. Formally:

Proposition 7.4.1. ∀ai ∈ A

∀g j ∈ G : g j(rh+1) ≤ g j(ai) ≤ g j(rh)

⇓

Cφ+(ai) = Cφ−(ai) = Cφ(ai) = Ch

160

7.4. Properties of category conformity

When working with central profiles, we have the following conformity property: each centralprofile rh, is assigned strictly to its corresponding category Ch.

Proposition 7.4.2. ∀rh ∈ R ,

Cφ+(rh) = Cφ−(rh) = Cφ(rh) = Ch

The proof is immediate when using Proposition 5.2.1.

Moreover, let us define the indifference between ai and a reference profile rh when the preferencedegrees π(ai, rh) and π(rh,ai) equal zero (or if they are "very small"). When an action ai isindifferent to exactly one centroid rh, it seems natural that this action should be assigned to thecorresponding category Ch.

Moreover, if an action ai is incomparable to exactly one centroid rh (as action a2 in Fig.7.1),traduced by similar preference degrees between ai and rh (i.e. π(ai, rh) ≈ π(rh,ai)), the actionmay be assigned to category Ch.This might be, at first sight, somehow confusing since ai is assigned to category Ch although itis not comparable with rh. The underlying idea is that, although they are “locally” incomparable(i.e. pair-wise not comparable), they are “globally” indifferent (i.e. they behave in terms of being“preferred to” and “preferred by”-character in the same way with respect to the other profiles).

Let us remark that another approach will be adopted when using outranking relations forassigning actions with respect to central profile (cf. Electre-Tri-Central, Chapter 12).

All these previous considerations, may be formally written as follows:

Proposition 7.4.3. ∀0≤ γ < 0.5

if ∃1! rh ∈ R | π(ai, rh) ≤ γ & π(rh,ai) ≤ γ

⇓

Ch D Cφ+(ai)D Ch+1

Ch−1 D Cφ−(ai)D Ch

Cφ(ai) = Ch

Furthermore, when the performances of an action are between the performances of two centralprofiles (such as a1 between rh and rh+1; and a3 between rh and rh−1 in Fig.7.1) it will beassigned to the two “nearer” categories according to negative and positive flows.

161


This can be illustrated as follows. If we consider the example represented in Fig.7.1 where∀i = 1,2 : pi = qi = 0; wi = 0.5, we have that π(rh,a2) = π(a2, rh) = 0.5, π(rh−1,a3) = 1 =π(a3, rh) and π(rh,a1) = 1 = π(a1, rh+1). This leads to the announced assignments: we havethat: Cφ+(a3) = h; Cφ−(a3) = h−1 and Cφ+(a1) = h + 1, Cφ+(a3) = h.

6

- g1

g2

rh−1

rh

rh+1

F

F

F

ca3ca2

ca1

Figure 7.1 — Illustration of particular situation where π(ai, r2) ≤ γ and π(r2,ai) ≤ γ withγ = 0.5

7.5 Relationship between Cφ− and Cφ+

When the reference profiles verify Condition 6.1.1 or 6.2.1 (i.e. strong preference), we havethat not every category combination (Cφ−(ai),Cφ+(ai)) is possible when using the positive andnegative flows. Under this assumption, category Cφ−(ai) is always as least as good as categoryCφ+(ai). Formally:

Proposition 7.5.1. ∀ai ∈ A ,R , R verifying Condition 6.1.1 and Condition 6.2.1:

Cφ−(ai) = Cl & Cφ+(ai) = Ch

Cφ−(ai) = Cl & Cφ+(ai) = Ch

⇓

l− h≤ 0

l−h≤ 0.

To have an idea about the possible category-combination some empirical tests have been made[Janssen and Nemery, 2007]. For that purpose, a set of strongly preferred limiting profiles andactions, have been randomly generated: 100 actions have been assigned for each of the 15000

162

7.6. Relationship between the assignments with limiting profiles and central profiles

generated reference sets of 4 limiting profiles. The actions were evaluated on the basis of 3linear criteria.It appears that the difference between the assignments is never higher than 3 (and thus that anaction will not be assigned to more than 3 different categories)1. Moreover, the case where thedifference is equal to 2 is very small (about 2,73%) and is mainly characterized by the presenceof equalities between the flows of the action and a limiting profile. There was just a proportion of0,068% where the difference of categories was equal to 2 without the presence of flow equalities.

7.6 Relationship between the assignments with limiting profiles andcentral profiles

In this section we compare the assignment of an action when the same set of ordered categoriesC1 .C2 .. . ..CK is either defined by limiting profiles or by centroids. For this purpose, we supposethat the centroid of a category is in between the limiting profiles of that same category: ∀h ∈1, . . . ,K, ∀ j ∈ 1, . . . ,q:

g j(rh) ≥ g j(rh) ≥ g j(rh+1).

Let us note that rh is not necessarily the “midpoint”or the “average” of rh and rh+1. This isillustrated in Fig.7.2

-

-

-

-

s ssss

ss

s s ssss

sss

ss

s

s

gq(.)

g j(.)

gi(.)

g1(.)r5 r4 r3 r2 r1

C4 C3 C2 C1

r4 r3 r2 r1

a1

Figure 7.2 — Illustration of the relationship between the assignments with limiting profilesand centroids: case I.

We shall furthermore assume that R verifies Condition 6.1.1 and R verifies Condition 6.2.1.This leads respectively to the assignments Cφ+(ai) and Cφ+(ai) and to the assignments Cφ−(ai)and Cφ−(ai). In such situation, there exists a relationship between the categories to which anaction will be assigned according to the limiting profiles or the centroids:

Proposition 7.6.1. ∀ai ∈A ;∀R , R verifying Condition 6.1.1 and Condition 6.2.1 and such that:

∀h ∈ 1, . . . ,K;∀ j ∈ 1, . . . ,q : g j(rh) ≥ g j(rh) ≥ g j(rh+1)

Cφ+(ai) = Ch & Cφ+(ai) = Ch

1 Let us remark, that some effort has been spent to proof this formally, but in vain...[Huenaerts and Nemery, 2007]

163


-

-

-

-

s ssss

ss

s s ssss

sss

ss

s

s

gq(.)

g j(.)

gi(.)

g1(.)r5 r4 r3 r2 r1

C4 C3 C2 C1

r4 r3 r2 r1

a2

Figure 7.3 — Illustration of the relationship between the assignments with limiting profilesand centroids: case II.

Cφ−(ai) = Cl & Cφ−(ai) = Cl

⇓

|h− h| ≤ 1 and |l− l| ≤ 1

In other words, the difference in rank between the category to which ai is assigned when workingwith strongly preferred limiting profiles or centroids (respecting the previous conditions) is atmost one. Let us remark that this proposition does not necessarily hold when conditions 6.1.1and 6.2.1 are not satisfied. This proposition is illustrated in Fig.7.2 and 7.3 where the bold linesrepresent limiting profiles and the dotted lines centroids. In Fig. 7.2, we can remark that actiona1 (thin line) is very close to r3. It is obvious, that Cφ(a1) = C3 whereas Cφ(a1) = C2. Hence theassignment differs by one category. In Fig. 7.3, we have that Cφ(a2) = Cφ(a2) = C4.

Previous considerations, may also be simply illustrated as follows. If we consider the assign-ment rules with limiting or central profiles, we remark that they are analogous in the sense thatthey have the same structure: only the boundaries change from one rule to another (see e.g.assignment rules 6.1.1 and 6.2.3). In the case of central profiles, we take as boundaries the meanof the positive (negative or net) flows of two consecutive central profiles.So, let us consider that the same reference profiles, play either the role of the limiting profile andthe central profile. We remark that there is a shift or translation of “half a category” between thesame categories defined either by limiting or central profiles. This is illustrated in Fig.7.4 wherer1, . . . ,rK+1 will play the role of limiting profiles (lower sub-figure) or central profiles (uppersub-figure). The “|” represents a profile and the (square) the limits of the categories in thesetwo cases. In the lower sub-figure the squares represent thus the limiting profiles. In the uppersub-figure, they represent the mean of consecutive central profiles.

164

7.7. Property of weak stability

| | | | | -

10

. . . . . .

φ+Ri

(ai)

φ+Ri

(r1)φ+Ri

(rK) φ+Ri

(r j)

CK C j

φ+Ri

(r j+1)

C1

-

10

. . . . . .

φ+Ri

(ai)

C1CK C j−1 C j

φ+Ri

(r1)φ+Ri

(rK) φ+Ri

(r j)φ+Ri

(r j+1)

| | | |

Figure 7.4 — Representation of the relationship when defining reference profiles as eithercentral (upper-figure) or limiting (lower-figure) profiles.

7.7 Property of weak stability

The property of stability is usually defined in the context of sorting as the fact that the assignmentof an action is not altered when one or several categories (different from the category to whichthe action is assigned) are suppressed or added.The suppression of categories may result from the fusion of two or several consecutive categoriesto just one. On the other hand, the addition of a category occurs when one category is split intotwo or several ones. This may for example happen when the decision maker wants to refinethe assignments of a category: he may subdivide the category in two categories by adding onelimiting profile. We have thus a new category.Formally:

• suppose that the initial set of reference profiles is R and that an action ai is initially assignedto CR

h

• split or fuse one or several categories Ci−m, . . . ,Ci, . . .Ci+l to Ci−m′ , . . . ,Ci′ , . . . ,Ci+l′ with h 6∈[i−m, i+ l] and h 6∈ [i−m

′, i+ l

′ ] and with l,m, l′,m′ ∈N and such that i−m > 0,i−m

′> 0

and i + l < K,i + l′< K ; we will denote by R ′

the new reference set

• a sorting method enjoys the property of stability if ai is still assigned to the same corre-

sponding category CR ′

h′≡CR

h , after the splitting or fusion.

This property usually results from the absolute definition given to the categories or from theassignment rules based on a “local” comparison.

Since the categories have a relative definition in F lowSort, we can expect that is this property isnot respected. Nevertheless, the F lowSort assignment rules respect what we will call the weakstability. We will discuss only the case while working with limiting profiles, since the results aresimular when working with central profiles.

165


Throughout this section, we will suppose that action ai is initially assigned to Ch and that thereference profiles verify Condition 6.1.1.

7.7.1 Negative flow assignment rules

Let us first use the negative flows assignment rules. We evaluate thus the “preferred by-” char-acter of the action ai with respect to the reference profiles (see Section 6.1). As a consequence,if we add one category which is worse than Ch (by adding a new worse limiting profile forexample), the action ai may change from Ch to C

′

h′with h

′ ∈ [h,h + 1]. It may only “decrease”to the next worse category or remain in the same.This can be explained by the fact that ai may be “preferred by” this new profile but not thelimiting profiles of the categories which are better than Ch. The global “preferred by”-character(i.e. thus the negative flows) of ai may increase while not the negative flows of the betterreference profiles rk with k ≤ h. This may explain the possible move-downwards. However, theflow increase of ai, and thus the assignment, is limited given Condition 6.1.1.

On the other hand, if we add one category which is better than Ch, the action ai can change fromCh to to C

′

h′with h

′ ∈ [h−1,h]. It may thus only “increase” to the next better category or remainin the same.Since the new profile is (strongly) preferred to the profiles of a higher rank than Ch (and maybenot so strongly preferred to ai), the action may be placed in the better category Ch−1. Thenegative flows of the profiles rk (with k ≥ h) will increase by one while the negative flow of ai

may not follow the same increase. Its position may thus change.

If, instead of splitting a category, we suppress one category, the same reasoning may be appliedto explain a possible difference in the assignments of ai. These properties are formally writtenin the following propositions and illustrated in Fig.7.5 and Fig.7.6.

In these figures, we have represented 3 different situations. The middle situation is the initialone. In the upper case, we have added one category (by the addition of a new profile) whereasin the lower case, we have deleted a category (by suppression of a profile). We have illustratedthe fact that there can be a difference in assignment when adding or suppressing a category.However, this difference is maximum one and not both, up and down-wards.

Formally, we have that: In case of splitting of categories:

Proposition 7.7.1. ∀ai; R = r1,r2, . . . ,rh, . . . ,rK+1 verifying Condition 6.1.1 andCR

φ−(ai) = Ch

If R ′ = R \rm with m < h and CR ′

φ− (ai) = Ch′ =⇒ h′ ∈ [h−1,h]

If R ′ = R \rm with h < m and CR ′

φ− (ai) = Ch′ =⇒ h′ ∈ [h,h + 1]

166

7.7. Property of weak stability

C′h+1

C′h+3

C′h+2

Ch+1Ch+2Ch

Ch−1?ai

ChCh−1

?ai

Ch

?ai

Ch+1

?ai

Ch−1

?ai

6 O

W

Ch−2

Ch−2

Ch−2

fusion

splitting

Figure 7.5 — Representation of the suppression (lower case) or addition (upper case) of a’worse’ category when using the negative flows: weak stability.

In case of fusion of categories:


φ−(ai) = Ch

If R ′ = R ∪r′m with π(r′m,rh) = 1 and CR ′

φ− (ai) = Ch′ =⇒ h′ ∈ [h,h + 1]

If R ′ = R ∪r′m with π(rh,r′k) = 1 and CR ′

φ− (ai) = Ch′ =⇒ h′ ∈ [h−1,h]

7.7.2 Positive flow assignment rules

If we now consider the case of the positive flows. We evaluate thus the “preferred”-character ofthe action ai with respect to the reference profiles. As a consequence, if we add one categorywhich is worse than Ch (by adding a new profile rm for example), the action ai may changefrom Ch to C

′

h′with h

′ ∈ [h,h + 1]. This can be explained by the fact that the initial profilesrk (with k ≤ h) are stronger preferred to this new profile rm than ai is preferred to rk (i.e.π(rk,rm) ≥ π(ai,rm)). This means thus that the increase in terms of positive flows of thesereference profiles can only be higher than the increase of the positive flows of ai. This explain

167


Ch

?ai

Ch

?ai

Ch

?ai

Ch−2Ch+2

Ch+2

Ch+2

Ch−1

Ch−1

?ai

Ch+1

Ch+1

?

Ch+1

C′h−1

C′

h−3

C′h−2

Ufusion

O splitting

Figure 7.6 — Representation of the suppression (lower case) or addition (upper case) of a’better’ category when using the negative flows: stability.

the possible move downwards.

On the other hand, if we add one category which is better than Ch, the action ai can change fromCh to C

′

h′with h

′ ∈ [h−1,h]. Since the new profile is (strongly) preferred to the profiles rk (withk ≥ h) and maybe not so strongly preferred to ai, the action may (or not) be assigned in thebetter category Ch−1.

If, instead of splitting a category, we suppress one, the same reasoning maybe applied to explaina possible difference in the assignments of ai. These properties are illustrated in Fig. 7.6 andformally written in the following propositions.

In case of splitting of categories:


φ+(ai) = Ch

If R ′ = R \rm with m < h and CR ′

φ+ (ai) = C′

h′=⇒ h

′ ∈ [h−1,h]

If R ′ = R \rm with h < m and CR ′

φ+ (ai) = C′

h′=⇒ h

′ ∈ [h,h + 1]

168

7.8. Strong stability conditions

In case of fusion of categories:


φ+(ai) = Ch

If R ′ = R ∪r′m with π(r′m,rh) = 1 and CR ′

φ+ (ai) = C′

h′=⇒ h

′ ∈ [h,h + 1]

If R ′ = R ∪r′m with π(rh,r′k) = 1 and CR ′

φ+ (ai) = C′

h′=⇒h

′ ∈ [h−1,h]

This “weak” stability property is a direct consequence of the relative definition given to thecategories. Actually, it can be explained very simply when we consider the sorting procedurewithin a ranking context. An action ai is assigned to a category according to its relative positionin the ranking (of the reference profiles and ai). On the other hand, the complete ranking methodis based on pair-wise comparisons which may thus suffer from rank reversal effects (see Section3.3.2.4; [Mareschal et al., 2008]). The rank reversal effect in rankings, is traduced, in a sortingcontext, by a non-stability property (when adding or suppressing a reference profile of theactions to be ranked). Nevertheless, imposing Condition 6.1.1 limits, in terms of category-move,this non-stability effect.

Let us remind the reader that previous considerations hold when working with central profilesand if conditions 6.1.1 are verified. Nevertheless, we were neither able to find a counter-exampleof previous propositions when conditions 6.1.1 are not verified, nor able to proof the proposi-tions rigourously without imposing conditions 6.1.1. Intuitively, we would say that previouspropositions are still valid. This can be motivated by the following considerations: suppose thatthere is a difference of assignment higher than one (e.g. two). This means thus that the actionhas swapped with a profile which had, initially, two ranks better or worse. Since the instabilityis due to rank reversal, this means that the action and the profile have swapped and thus thatthere is moreover a rank reversal between reference profiles which is impossible consideringProposition 5.2.1. Let us nevertheless remark that the non-increasing or non-decreasing of anassignment, can be proven without imposing conditions 6.1.1 (see next section).

7.8 Strong stability conditions

Despite previous results, it is possible to “predict” the (non-) stability of the assignment of anaction ai when splitting or fusing two categories. This permits us to increase the confidence of theassignment results. In other words, we can analyze a kind of sensitivity of the sorting assignment.

a. Let us suppose that ai is assigned to category Ch defined by the limiting profiles rh and rh+1.According to the positive flows (Assignment Rule 6.1.1), we know thus that:

φ+R i

(rh) ≥ φ+R i

(ai) > φ+R i

(rh+1) (7.1)

169


Suppose that we fuse two categories by suppressing the limiting profile rl with l < h. We notethus R ′ = R \rl and R ′

i = R i \rl. The action ai might be assigned to the categories Ch orCh+1, according to this new set R ′

(see Proposition 7.7.4). If ai is assigned to category Ch+1, wehave thus necessarily that:

φ+R ′i

(rh+1) ≥ φ+R ′i

(ai) > φ+R ′i

(rh+2) (7.2)

To be sure that action ai will not change from category (after fusing) and thus that there will notbe a rank reversal between ai and rh+1, we need that:

φ+R ′i

(ai) > φ+R ′i

(rh+1)

⇔

φ+R i

(ai)−π(ai,rl)| R i | −1

> φ+R i

(rh+1)−π(rh+1,rl)| R i | −1

⇔ 2

φ+R i

(ai)−φ+R i

(rh+1) >π(ai,rl)K + 1

− π(rh+1,rl)K + 1

(7.3)

In other words, Inequality 7.3, may be interpreted as a sufficient condition that ensures usstability in case of fusion of the categories Cl−1 and Cl . This condition expresses the fact thatai might not be to close from rh+1 in terms of positive flows. Let us remark, that the higher thenumber of categories, the smaller the difference needed between the positive flows of ai and rh+1

for ensuring the stability. Moreover, we may notice that this condition is necessarily fulfilledwhen suppressing a reference profile rl with l > h + 1 (a worse profile) since π(rh+1,rl) = 1which makes left member negative in Inequality 7.3. This illustrates the fact that not all theinstabilities are possible (cf. previous section).

Let us remark, that one might think that it is possible to generalize Inequality 7.3 such that it’sindependent of the suppressed profile rl (i.e. better or worse). If we write Inequality 7.3 for anyfusion of any two consecutive categories, we obtain: ∀l ∈ 1,2, . . . ,h−1,h + 2, . . . ,K + 1

φ+R i

(ai)−φ+R i

(rh+1) >1

K + 1(7.4)

Nevertheless, under previous sorting conditions, it seems difficult (or impossible)3 that the dif-ference in terms of positive flows can be higher than 1

K+1 . This renders the last inequality useless.

Similarly, for the suppression of a limiting profile rl with l > h+1 we have the following condi-tion of stability:

φ+R i

(rh)−φ+R i

(ai) >π(rh,rl)K + 1

− π(ai,rl)K + 1

(7.5)

2When working with limiting profiles we have that:| R i | −1 =| R |=K+13We were not able to find a counter-example to this assertion.

170


As a conclusion, when working with the positive flows, if the inequalities 7.3 and 7.5 are satisfied,the action ai will be assigned to the same category after the fusion of two consecutive categories.This is formally expressed in the following proposition:

Proposition 7.8.1. ∀R ;∀ai ∈ A ;∀rh 6= rl 6= rh+1 ∈ RThe assignment of an action ai to Ch, according to the positive flows, will not be affected by thefusion of two consecutive categories if the following conditions are satisfied:

φ+R i

(rh)−φ+R i

(ai) >π(rh,rl)K + 1

− π(ai,rl)K + 1

φ+R i

(ai)−φ+R i

(rh+1) >π(ai,rl)K + 1

− π(rh+1,rl)K + 1

In case of splitting of two categories (by the addition of a new limiting profile rl) and if Cφ+(ai) =Ch, we may obtain similar conditions ensuring Cφ+(ai) = C

′

h′≡Ch. Formally:

Proposition 7.8.2. ∀R ,∀ai ∈ A ;∀rh 6= rl 6= rh+1 ∈ R :The assignment of an action ai to Ch, according to the positive flows, will not be affected by thesplitting of a category into two consecutive categories, if the following conditions are satisfied:

φ+R i

(ai)−φ+R i

(rh+1) >π(rh+1,rl)

K + 1− π(ai,rl)

K + 1

φ+R i

(rh)−φ+R i

(ai) >π(ai,rl)K + 1

− π(rh,rl)K + 1

Analogous stability conditions might be found when working with the negative and with the netflows or when working with central reference profiles.

Let us finally remark, that previous propositions have been generalized in a ranking contextwhen dealing with the rank reversal phenomenon. The results can be found in [Mareschal et al.,2008].

Example 7.1. The reader may verify that all the actions of the example given in Section 6.1verify the stability conditions. The fusion of any category does not affect the assignments of theactions.

Example 7.2. In order to illustrate the stability properties, let us consider the followingexample. Suppose that we have initially 4 categories defined by 5 limiting profiles. Eachprofile is evaluated on the basis of 4 quantitative criteria which have to be maximized. Thecorresponding performances are given in Table 7.1. To compute the preference degree we haveused the Promethee methodology. All the preference parameters are given in Table 7.2. Let us

171



R g1 g2 g3 g4

r1 10 10 10 10

r2 6 8 7 8

r3 5 5 4 4

r4 3 4 3 2

r5 0 0 0 0


g1 g2 g3 g4

qk 2 2 1 2

pk 3 4 3 3

wk 1 1 1 1

remark that the reference profiles do not verify Condition 6.1.1.

Suppose now that we want to sort an action a1 whose performances are given by the followingperformance vector: [3,3,1,1]. The flows of the action and its assignment are given in Table 7.3.

Table 7.3 — Computation of the different flow values when considering the set R .

r1 r2 r3 r4 r5 ai C(ai)

φ+R1

0.883 0.633 0.258 0.133 0 0.142 3

φ−R1

0 0.083 0.292 0.442 0.842 0.392 3

φR10.88 0.55 -0.03 -0.31 -0.84 -0.25 3

If we suppose that the reference set become R ′ = R \r2 we obtain the flows given in Tab.7.4(with the initial category-numbering). We may thus notice that the assignment of the action isunstable when fusing the initial category C1 and C2 into C

′1. The being preferred character of

action a1 is not altered by the suppression of r2 since a1 was dominated by r2. The assignmentaccording to the positive flows remains unchanged. On the other hand, the suppression of profile

172


r2 provokes a rank reversal between r4 and a1. Profile r4 gains now from the point of view of thebeing preferred character since it is no longer being (strongly) preferred by r2 whereas a1 gainsless (which explains the rank reverse): r4 is now better than a1 on the basis of the negative flows.This is moreover an illustration of the simultaneous comparison.

Analogously, if we consider the initial set R ′, and that we split category C

′1 into two consecutive

categories C1 and C2, the assignment of action a1 may change.

Table 7.4 — Computation of the different flow values when considering the set R ′.

r1 r3 r4 r5 ai C(ai)

φ+R ′1

1 0.323 0.167 0 0.177 3

φ−R ′1

0 0.25 0.302 0.802 0.312 4

φR ′1

1 0.07 -0.14 -0.80 -0.14 4

One may notice that the instability does not occur in Example 7.2 where the limiting profiles arestrongly preferred (i.e. verifying Condition 6.1.1). In the next section, we will analyze the impactof Condition 6.1.1 on the stability of the assignments.

7.8.1 Influence of Condition 6.1.1 on the stability.

When the reference profiles are strongly preferred, we know that not every category combinationis possible (cf. Proposition 6.1.1). Moreover, since the preferences between the profiles,expressed by the decision maker, are strong, we might think that the categories are clearlyseparated (by these profiles). A consequence of a good separation or delimitation of thecategories, should be that the assignment of the actions are (more) stable.

For this purpose, we have generated randomly a set of reference profiles either verifyingCondition 6.1.1 or not. First, we compute a (K + 1)× q-dimensional performance matrix ofthe K + 1 reference profiles evaluated on the q criteria. This performance matrix is obtainedby sorting each column of a random matrix generated with a uniform distribution for eachelement. To obtain reference profiles verifying Condition 6.1.1, the preference threshold foreach criterion is chosen as the smallest difference between any two consecutive evaluationson that criterion. Alternatively, in order to generate non strongly preferred reference profiles,the preference threshold is chosen as the largest difference between two consecutive evalu-ations. In both situations, the indifference threshold is sampled from a uniform distributionbetween 0 and the chosen preference threshold. Finally, q non-normalized weights are gener-ated uniformly on [0,1] (and then normalized), as well as the q evaluations of the action to assign.

173


For each reference set R , we have generated randomly a set of actions A (with | A |= 100)and assigned the actions with respect to R . For any assignment of ai, we have counted thetotal number of times the stability conditions of Section 7.8 were violated. These violationsare synonym of assignment instability due to the fusion of a possible pair of two consecutivecategories (different from the one to which the action ai was assigned to) or the splitting ofa category. Since the negative and net flows assignment rules are similar to the positive flowassignment rules, we have only focussed on these last one.

For the simulations we have fixed successively the number of categories K at 2, 4, 6, 8, 10and the number of criteria at 3, 6, 9. For each case we have generated 100 different referencedata sets. The results are given in Table 7.5. We have given the total number of instabilityoccurrences (when fusing any pair of consecutive categories) as well as the number of actionswhich assignment has changed (expressed in %). We may moreover precise that the averagenumber of instability occurrences per reference set (when there is instability) is about 2 and 1.5when respectively preference and strong preference.

From Table 7.5 we may remark that the number of occurrences of instability are clearly lowerwhen working with strongly preferred limiting profiles (i.e. when respecting Condition 6.1.1).Particularly, when the number of categories is relatively small, we can notice that this number iseven very low, which may lead to the conclusion that the assignments are quite stable.The percentage of actions whose assignment change, is very low when in presence of a smallnumber of categories (7.75% and 0.12% respectively). Nevertheless, when the number ofcategories rises (for example K=8), we may notice that Condition 6.1.1 does not guarantee"much" more stability than with preferred limiting profiles. The stability % falls even down to75%.The fact that the Condition 6.1.1, does not play a role any more when K is high, can be explainedby fact that the flow differences become smaller (due to a higher K). This has as consequencethat the left member of the Inequalities 7.3 and 7.5, decrement which render the assignmentmore unstable or less stable.

Table 7.5 has as purpose to give an idea about the possible instabilities. However, it may be moreappropriate to verify if there might be instabilities for the assigned actions in a particular sortingproblem.

174


Table 7.5 — Number of instability occurrences in presence of preferred and strongly preferredlimiting profiles when working with the positive flows.

Preferred limiting Strongly preferred

profiles limiting profiles

K q occurrences % occurrences %

3 840 7,75 21 0,12

2 6 666 8,72 0 0,01

9 741 7,05 0 0

3 3464 25,68 288 2,62

4 6 2986 26,06 331 3,16

9 2710 23,59 263 2,81

3 5272 32,66 1858 16,66

6 6 4541 20,89 2367 26,94

9 3695 24,41 2753 27,75

3 7061 31,86 4298 25,7

8 6 6178 30,56 5429 36

9 5840 27,80 5817 40

175

8 Comparison between F lowSortand some existing sortingmethods

In order to analyze the assignments given by F lowSort, we compare its as-signments results to those obtained by Electre-Tri and the UTADIS method.Some basic examples are given in order to analyze the differences. In this con-text, an empirical comparison with Electre-Tri is realized by means of simula-tions. Furthermore, the global or simultaneous approach of F lowSort is con-fronted to Electre-Tri. Finally, the relative definition given to the categories inthe F lowSort method is emphasized with respect to the UTADIS method.

8.1 Comparison between F lowSort and Electre-Tri

In order to analyze the assignments given by F lowSort, we compare its results to those ofElectre-Tri presented in Section 4.6.1. As in the F lowSort method, Electre-Tri also uses lim-iting profiles and proposes two different assignment rules, an optimistic and pessimistic one. Atfirst, an empirical comparison will be done by means of simulations. The simulations will showus to what extent the assignments given by F lowSort are similar (or not) to the ones obtainedwith Electre-Tri. A more theoretical approach will furthermore be adopted to answer the questionif both methods are (always) similar (or not).

8.1.1 Empirical comparison with Electre-Tri

At first we will analyze the differences between Electre-Tri on the basis of the assignmentresults. For this purpose, we have done some simulations such as in Section 7.8.1. We havegenerated randomly actions, preference parameters, reference profiles (strongly and not stronglypreferred), weights, etc. The actions have been assigned according to the F lowSort method.These results have been compared to the one obtained with Electre-Tri while using the sameweights and thresholds, although they do not have necessarily the same meaning. Moreover,

177

Comparison between F lowSort and some existing sorting methods

the λ-threshold of the outranking relation in Electre-tri is fixed at 0.85 and no veto has been used.

When comparing the assignments of Electre-Tri optimistic and pessimistic (respectively notedas Copt and Cpess) with Cb and Cw, we distinguish 5 different situations (see also Fig. 8.1):

1. Cb .Copt DCw DCpess or Copt DCb DCpess .Cw

2. Cb .Copt DCpess .Cw

3. Cb DCw .Copt DCpess or Copt DCpess .Cb DCw

4. Copt .Cb DCw .Cpess

5. Copt = Cb DCw = Cpess

- - - -

- -

- -

? ?

? ?

? ?

? ?

-?? ? ?

? ?

? ? ? ?6 6

6 6

6 6 6 6 6 6

6 6

6 6

I:

II:

III:

IV:

V:

VI:

Cpess Copt Cpess

Cpess

Cpess Cpess

Cpess

Cpess

Cpess Cpess

Copt

Copt

Copt Copt

Copt

Copt

Copt Copt

Cw Cb

Cw Cw

Cw

Cb

Cw

Cb

Cb

CbCw

Cφ Cφ

Cb

Figure 8.1 — Comparison of F lowSort and Electre-Tri : different scenarios.

Finally, another point of interest is to count the cases where the unique assignment Cφ is incon-sistent with the assignments of Electre-Tri:

6. Cφ .Copt DCpess or Copt DCpess .Cφ

For the simulations we have fixed successively the number of categories K at 2, 4, 6, 8, 10 andthe number of criteria at 3, 6, 9. For each case we have generated 1000 different reference datasets. The results are given in Table 8.1, 8.2 and 8.3.

First we analyze Table 8.1. The assignments given by F lowSort can be seen as completelyconsistent with Electre-Tri in situations 4 and 5. These two situations occur in a large majorityof cases. In general, these frequencies rise when the number of categories increases. More-over, we can also notice that situation 6 does not happen very often, which means that the

178

8.1. Comparison between F lowSort and Electre-Tri

Table 8.1 — Comparison between F lowSort and Electre-Tri in the case of limiting profiles.

Preferred limiting Strongly preferred

profiles (%) limiting profiles (%)

K q Sit-1 Sit-2 Sit-3 Sit-4 Sit-5 Sit-6 Sit-1 Sit-2 Sit-3 Sit-4 Sit-5 Sit-6

3 5 0 0,4 29,7 64,9 2,2 5 0,1 1,8 28,6 64,5 4

2 6 2,8 0 0 49,6 47,6 1,1 2,8 0 0,4 44,8 52 1,1

9 1,7 0 0 62,3 36 0,5 1,4 0 0 50 48,6 0,4

3 6,9 0 1,4 55,1 36,6 3,8 0,6 0 0,6 71,7 27,1 0,9

4 6 1,7 0 0,2 83,2 14,9 0,9 0,3 0 0 90,1 9,6 0

9 0,3 0 0 90 9,7 0 0 0 0 94,3 5,7 0

3 5,6 0,1 1,3 67,4 25,6 3,7 0,2 0 0,4 82,9 16,5 0,6

6 6 1 0 0,1 91,4 7,5 0,5 0 0 0 96,8 3,2 0

9 0,5 0 0 96,8 2,7 0 0 0 0 99,1 0,9 0

3 5,4 0,2 1 74 19,4 2,2 0,2 0 0,7 87,6 11,5 0,7

8 6 0,8 0 0 95,3 3,9 0,1 0 0 0 99 1 0

9 0,1 0 0 99 0,9 0 0 0 0 100 0 0

3 3,7 0 1,8 82,3 12,2 3,3 0,4 0 0,6 89,2 9,8 0,7

10 6 0,9 0 0,3 96,3 2,5 0,5 0,1 0 0 99,4 0,5 0

9 0,1 0 0 99,4 0,5 0 0 0 0 99,8 0,2 0

net-flow assignment procedure usually gives results compatible with Electre-Tri. Finally, whenimposing Condition 6.1.1 (strongly preferred limiting profiles), then situations 4 and 5 happeneven more often and situation 6 even less often. This can be explained by the fact that Condi-tion 6.1.1 ensures that classes are "well-defined", which may lead to less ambiguous assignments.

Concerning Table 8.2 and 8.3, when the number of classes is increasing we can notice that theassignment-interval is becoming larger. Nevertheless, we can conclude that the assignment-interval is significantly smaller in F lowSort than in Electre-Tri. This can be explained by thefact, that in a large majority the incoming-flow assignment coincides with the leaving-flowassignment. Finally, the last two columns in Table 8.2 and 8.3, which describe symmetricsituations, are almost equal as the data is randomly generated.

179


Table 8.2 — Comparison between F lowSort and Electre-Tri in the case of limiting profiles:analysis of the assignments.

Preferred limiting

profiles

K q mean of mean of Cφ+ = Cφ− |Cφ+ −Cφ|< |Cφ+ −Cφ|>

Copt −Cpess |Cφ+ −Cφ− | (%) |Cφ−Cφ− |(%) |Cφ−Cφ− |(%)

3 0,366 0,119 88,1 5,2 6,7

2 6 0,608 0,14 86 5,9 8,1

9 0,735 0,129 87,1 4,7 8,2

3 1,097 0,283 72,5 15,9 11,6

4 6 1,714 0,234 77,4 11,1 11,5

9 1,86 0,23 77,3 11,2 11,5

3 1,81 0,312 71,3 14 14,7

6 6 2,719 0,288 72,8 13,9 13,3

9 2,948 0,242 76,4 11,8 11,8

3 2,582 0,352 67,7 16,5 15,8

8 6 3,896 0,304 71,4 15,1 13,5

9 4,289 0,294 71 14,4 14,6

3 3,422 0,356 67,6 17,5 14,9

10 6 4,89 0,319 69,2 16,2 14,6

9 5,365 0,275 73,1 13,4 13,5

180


Table 8.3 — Comparison between F lowSort and Electre-Tri in the case of limiting profilesverifying Condition 6.1.1: analysis of the assignments.

Strongly preferred

limiting profiles

K q mean of mean of Cφ+ = Cφ− |Cφ+ −Cφ|< |Cφ+ −Cφ|>

Copt −Cpess |Cφ+ −Cφ− | (%) |Cφ−Cφ− |(%) |Cφ−Cφ− |(%)

3 0,373 0,146 85,4 8,3 6,3

2 6 0,628 0,208 79,2 10,2 10,6

9 0,741 0,255 74,5 13,8 11,7

3 1,338 0,124 87,6 6 6,4

4 6 1,93 0,153 84,7 7,7 7,6

9 2,13 0,191 80,9 9,1 10

3 2,195 0,119 88,1 6 5,9

6 6 3,159 0,162 83,8 7,3 8,9

9 3,448 0,149 85,1 8,3 6,6

3 3,061 0,108 89,2 5,4 5,4

8 6 4,348 0,127 87,3 6,3 6,4

9 4,779 0,124 87,6 6,2 6,2

3 3,906 0,1 90 5,2 4,8

10 6 5,516 0,095 90,5 4,8 4,7

9 5,937 0,098 90,2 6,1 3,7

181


It is well-known that Electre-Tri can be used with vetoes. This aspect has not been consideredin our simulations as vetoes cannot easily be taken into account in the F lowSort methodology.Nevertheless, working with vetoes in Electre-Tri will in general further increase the differenceCopt −Cpess and so strengthen our previous conclusions.

8.1.2 Intuitive comparison with Electre-Tri

In previous section, we have shown to which extend Electre-Tri and F lowSort give analogousresults. Nevertheless, this does not enable us to statue if both method are the "same". Indeed,both methods use analogous parameters but their meaning (and thus their values) may bedifferent. In order to analyze if the two models are different, instead of fixing the set ofparameters, we may formulate the problem as follows: "is it possible that F lowSort, with acertain set of parameters, leads to the assignment of an action which is impossible to obtainwith Electre-Tri, for any set of parameters (and vice-versa) ?"If the answer is positive, we can state that both methods are different. Previous ideas arebased on the work of D. Bouyssou and T. Marchant [Bouyssou and Marchant, 2007a] whereElectre-Tri is axiomatically compared to UTADIS.

Let us suppose that we have a sorting problem with two categories: C1 and C2. Having the sameinitial problem (same number of categories and same set A = a1,a2,a3,a4), we may want tohave the same assignment results for all the actions of A . The actions are evaluated according totwo criteria g1 and g2 which have to be maximized. The evaluations of the actions are given inTab.8.4 and represented in Fig.8.2.

Table 8.4 — Evaluation of the performances of the actions of A of example 8.1.2

g j a1 a2 a3 a4

g1 5 10 10 15g2 2 2 1 1

Table 8.5 — Evaluation of the limiting profile rl .

g1 g2

rl 1.72 9

Let us consider that the parameters of F lowSort are such that p1 = 0.8 and p2 = 4; q1 = q2 =0 and w1 = w2 = 0.5. We will moreover suppose that the preference functions are linear. Toseparate the two categories, we define one limiting profile rl which performances are given inTab.8.5.

On the basis of the preference parameters, we have that: π(rl ,a1) > π(a1,rl), π(rl ,a3) >

182


6

- g1

g2

u

uu u

a1

a2a3

a4

C1

C2

1 2

5

10

15

Figure 8.2 — Representation of the example of 8.1.2: assignments obtained with F lowSortwhich can not be obtained with Electre-Tri.

π(a3,rl), π(a2,rl) > π(rl ,a2), π(a4,rl) > π(rl ,a4) (see Tab.8.6).

According to these preference degrees, F lowSort will assign actions a2 and a4 to C1 and actionsa1 and a3 to C2 (when using the positive, negative and net flows 1).

Table 8.6 — The preference degrees between the rl and the actions.

a1 a2 a3 a4

π(ai,rl) 0.175 0.3 0.125 0.5

π(rl ,ai) 0.5 0 0.45 0.45

In [Bouyssou and Marchant, 2008]2, D. Bouyssou and T. Marchant have attempted to determinepreference parameters for the Electre-Tri model in order to assign the actions of A similarly(i.e. as previously done by F lowSort). When using the Electre-Tri model, the authors obtain thefollowing conclusions:

g2(a1) = g2(a2)Cpess(a1) = C2 and Cpess(a2) = C1

=⇒

g1(a1) < g1(rl)−q1 (1)g1(a2) ≥ g1(rl)−q1 (2)

g2(a3) = g2(a4)Cpess(a3) = C2 and Cpess(a4) = C1

=⇒

g1(a3) < g1(rl)−q1 (3)g1(a4) ≥ g1(rl)−q1 (4)

1Let us remark that in this special case where there are only two categories defined by one limiting profile, wehave identical assignments.

2In this work, the authors analyzed the differences between UTADIS and Electre-Tri

183


We may thus remark that equations (2) and (3) lead to a contradiction with the initial conditionthat g1(a2) = g1(a3) (see Tab.8.4). This proofs thus that it is impossible to determine a set ofpreference parameters (especially a limiting profile rl) in the Electre-Tri model in order to obtainthe same assignments as in the F lowSort model.

Let us besides consider the following example where the performances of an action a are inbetween two consecutive limiting profiles r j and r j+1, and such that action a is indifferent toboth of them. This is represented in Fig.8.5 where the actions are evaluated on two criteria to bemaximized.According to the optimistic assignment rule of Electre-Tri, a will be assigned to C j−1 (see Section4.6.1.2), although the performances of a are in between those of r j and r j+1. Furthermore, thepessimistic assignment rule assigns a to the same category C j−1 (see Section 4.6.1.3).

r j r j−1 r j−2r j+1r j+2 g1

g2

-

-

C j C j−1rra

Figure 8.3 — Illustration of an assignment with Electre-Tri: III

According to Proposition 5.2.1 and the strongly monotonic character of Promethee (see Section3.3.2.3), the order between r j, r j+1 and a (given the conditions on their performances) willalways be as follows: r j, a and r j+1. This leads to the assignment of a to category C j

3.

We may thus conclude, that the F lowSort and the Electre-Tri model are different given thetwo previous examples. A further step, which will not be investigated rigourously in this work,is to study "how" they are different. A first intuitive attempt to differentiation will be given innext section. The interested reader might find more information on this topic in [Bouyssou andMarchant, 2007a], [Bouyssou and Marchant, 2007b].

8.1.3 Impact of a simultaneous comparison

The assignment of an action in F lowSort depends on the comparison with all the referenceprofiles simultaneously, and not, as for instance in ELECTRE-Tri, on successive pairwisecomparisons. Although, this more “global” approach can appear unconventional, it is a directconsequence of using a ranking method in a sorting context. Let us analyze more deeply theimpact of a complete-ranking or global-comparison approach.

Suppose that an action a is incomparable to exactly one limiting profile r j and such that theperformances of a are re in between r j+1 and r j−1. This situation is represented in Fig.8.5.

3Even in case of ex-aequo according to the definition of the assignment rules.

184


Action a will, according to the pessimistic and optimistic assignment rules, be assignedrespectively to C j and C j−1 (situation III in Section 4.6.1.2). The same result may be obtainedwith the F lowSort method when working with the positive and negative flows4.

r j r j−1 r j−2r j+1r j+2 g1

g2

-

-rra

C j C j−1

Figure 8.4 — Illustration of an assignment with Electre-Tri: I

Suppose now, that some new limiting profiles are added to the set of limiting profiles:R ′ = R ∪r′j,r

′j+1,r

′j+1,r

′j+2,r

′j+4,r

′j+5,r

′j+6. The performances of the added limiting profiles

are in between r j+1 and r j−1 (see Fig.8.5) and such that r′j+7 = r j+1 and r

′j+3 = r j.

r j−1 r j−2g1

g2

-

-rrr

′j

r′j+1

r′j+2

r′j+3

r′j+4

r′j+5

r′j+6r

′j+7 a

C j−1C′j+6

Figure 8.5 — Illustration of an assignment with Electre-Tri: II

Action a will this time be assigned respectively to C′j+6 and C

′j−1 with C

′j+6 = C j and

C′j−1 = C j−1. In other words, action a will be assigned to the same categories as previously,

although a is incomparable to new profiles.This can be motivated by the fact that the categories C j and C j−1 have indeed received a new(relative) definition and label. Nevertheless, with respect to the best and the worst category (i.e.in terms of "ranks"), action a is assigned to the same category.

Given the simultaneous comparisons, F lowSort will take into account that action a is incompa-rable to these added profiles. Action a may be assigned in between the categories C

′j+6 and C

′j−1

5. The assignment of an action a is thus based on the ordinal character of its performances withrespect to the profiles. Nevertheless, taken into account the ordinal character of the performancesof a may lead to rank reversal phenomenons, and thus to instability (see Section 7.7).

To our opinion, this constitutes the main difference with Electre-Tri. Besides, we consider

4Let us remark that, strictly speaking, no pair-wise incomparability has been defined in Promethee. Anyway, wemay define an incomparability relation analogous as the one defined by outranking relations (see Section 2.6.1.4).

5This may be verified in previous example by considering ∀i = 1,2 : pi = qi = 0 and computing the positive,negative and net-flows.

185


that Electre-Tri can not, strictly speaking, be considered as a ranking procedure since it seemsdifficult to rank all the actions of A with using solely the Electre-Tri procedure (i.e. withoutadding reference profiles). Nevertheless, [Rolland, 2008] proposes to extend this sortingprocedure in order to rank actions by adding reference points to the set A .

8.1.4 Conclusions

Throughout this section we have first shown that the results of Electre-Tri and F lowSort arequite similar on the basis of empirical comparisons. Nevertheless, some small examples haveproven that the models differ in some particular situations. As previously mentioned, furtherresearch should be done in order to distinguish them more deeply. In the next section, a briefcomparison between F lowSort and UTADIS will be given.

8.2 Comparison with the UTADIS model

The aim of this section is to compare the F lowSort method with UTADIS, presented in Sec-tion 4.5.1. For that purpose, let us consider that the set of actions A have to be assigned to Kcompletely ordered categories: C1BC2B . . .BCK . Fig.8.6 remind briefly the sorting procedureUTADIS (see Section 4.5.1): the global utility score of action ai, U(ai), is compared to constantthresholds δ2, . . . ,δK (with ∀ j : δ j > δ j+1) defining the categories. This utility score is computedon the basis of the marginal utility functions:

U(ai) =q

∑k=1

Uk(gk(ai))×wk (8.1)

The UTADIS assignment rule is defined as follows:


CUTA(ai) = C j⇔ δ j+1 < U(ai) ≤ δ j

-Fai U(.)

CK

δ1δ2δ j+1δKδK+1 δ j

C j C1

Figure 8.6 — Representation of the assignment rule of UTADIS

On the other hand, in F lowSort, when working with the limiting profiles of the set R , theassignment rule based on the net-flows is defined as follows (see Section 6, A.R. 6.1.3):


Cφ(ai) = C j⇔ φR i(r j+1) < φR i(ai) ≤ φR i(r j)

186

8.2. Comparison with the UTADIS model

where

φR i(ai) =q

∑k=1

wk×φk(ai) (8.2)

with φk(ai) representing the uni-criterion net flow of action ai. Considering the formulas 8.1and 8.2 on the one hand, and the assignment rules on the other hand, one may wonder about thedifferences between the procedures. In this section, we will not consider the differences betweenthe aggregations phases of these methods (i.e. the differences between the formulas 8.1 and 8.2),since this has been done in Chapter 3. We will only focus on the aspects peculiar to the sortingproblematic.

Let us first remark that the thresholds δ2, . . . ,δK in UTADIS are absolute and constant.UTADIS does not need profiles. However, these thresholds may correspond to certain limitingprofiles whose global utility are equal to these threshold. We may define a set R such that∀ j : U(r j) = δ j. An action a j will thus be assigned to C j if U(r j+1) < U(ai) ≤U(r j): in otherwords, ai is assigned to C j if it is better than profile r j+1 and worse than r j (in terms of utilities).

On the other hand, in the F lowSort method, the thresholds of the categories are not constant norabsolute. If we suppose that the limiting profiles fulfill Condition 6.1.1 (i.e. if they are stronglypreferred), the net-flows of the limiting profiles with respect to each other (without consideringany action to be assigned) take the values given in Fig. 8.7.

- φR (.)

φR (r1)φR (r2)φR (r j+1) φR (r j)φR (rK)φR (rK+1)

1K−2 j−2K

K−2 j+2K−1

Figure 8.7 — Representation of the flows values of the limiting profiles

Since 0≤ π(a,b) ≤ 1 (see Section 2.6.2), we have that ∀ai ∈ A ,∀r j ∈ R :

φR i(r j) = φR (r j)+π(r j,ai)−π(ai,r j)

K + 1

⇓

φR i(r j) ∈ [φR (r j)−1

K + 1,φR (r j)+

1K + 1

]

Let us note previous interval as follows: φR i(r j) = [φR i(r j),φR i(r j)].

The variable limits of the categories, are thus always in these intervals associated to the limitingprofiles. We have thus, on the contrary of the UTADIS model, that the limits of the categoriesvary, in a determined interval (see Fig.8.8). Let us motivate the use of this variable thresholds inthe next paragraphs.

187


- φRi(.)

φRi(r1)φRi(r2)φRi(r j+1) φRi(r j)φRi(rK)φRi(rK+1)

---CK C j C1-

ς j


In Fig.8.8, the intervals φR i(r j) are represented by the square boxes6. The constant valuesφR (r j), which are always in the middle of these boxes7 are indicated by small vertical verticesand the category limits (φR i(r j)) by the "thicker" vertical vertices. In this figure, ς j representsthe difference between the constant value φR i(r j) and φR (r j):

| ς j |=| φR (r j)−φR i(r j) |=|π(r j,ai)−π(ai,r j)

2|

- φRi(.)

φRi(r1)φRi(r2)φRi(r j+1) φRi(r j)φRi(rK)φRi(rK+1)

- U(.)CK C j C1

δ1δ2δ j+1δKδK+1 δ j

-ς j


If π(r j,ai) > π(ai,r j), we have that this value ς j is positive and thus that the category-limitis "higher" (than the constant threshold φR (r j)). Fig.8.9 represents the difference between thecategory limits of UTADIS and F lowSort.

We may moreover remark that the assignment rules are different to a successive pair-wisecomparison approach. We have that π(r j,ai) > π(ai,r j) is not sufficient in order to assign ai

to C j. This can be motivated by the fact that the difference might be small and that globalinformation might be considered.In order to be assigned to the category C j−1, if π(r j,ai) > π(ai,r j), action a j needs to have aglobal behavior better than the global behavior of profile r j. Analogous conclusions may bemade about the assignment of ai to C j if π(r j,ai) < π(ai,r j). We can thus remark that the globalbehavior is taken into account in the assignment rule and not only the pair-wise comparisonswith the profiles.

Let us finally that this constitutes the main difference between PairClass (A.R.4.6.9, see Section4.6.5.2) and F lowSort. In particular conditions given 4.6.5.2, we have proven that PairClassuses net flows as well. Nevertheless, the net flow is, in PairClass, compared to a fixed threshold

6The intervals are smaller for the limiting profiles r1 and rK+1 since ∀ai ∈ A : π(ai,r1) = 0 and π(rK+1,ai) = 0.7Excepted for r1 and rK+1 given footnote 6.

188

8.2. Comparison with the UTADIS model

although this might be not recommended when using net flows (see the model assumptions ofPromethee, Section 3.3.2.3).

189

9 I nterval and F uzzy F lowSort

In this chapter we consider the case of a sorting problem where the perfor-mances of the actions to be assigned and/or reference profiles are imprecise.Imprecision on the decision maker’s parameters of the model are considered aswell. We propose, on the basis of an interval and fuzzy version of Promethee,an extension of F lowSort to the case where the data is defined by intervals orfuzzy numbers. We analyze furthermore the properties and the implications ofthe use of intervals or fuzzy numbers on the assignments.

9.1 Introduction

In this chapter we will present a way of treating different sources of imprecision in a sortingproblem. As identified by Roy and Bouyssou [Roy and Bouyssou, 1993], imprecision maybe due for example to the fact that the decision maker is unable to clearly determine theevaluations of the actions or the parameters of the model. It may also be due to the fact thatthese values may evolve over time and space. The consequence might be that the evalua-tion of the performances, the parameters (e.g. the weights of the criteria) or the preferencescan not be well-determined by crisp values [Gelderman et al., 2000],[Teno and Mareschal, 1998].

In this context, we propose, based on the previous works of [Gelderman et al., 2000],[Araz andOzkarahan, 2007a],[Nemery and Lamboray, 2008] and [Janssen and Nemery, 2007], an intuitiveextension of the F lowSort method1 to the case where the values of the performances of theactions, weights or preference’s degrees are defined by means of mere intervals rather than crispvalues. Instead of reducing these intervals to a single value (e.g. the mean value of the inter-val), we will keep this information all along the method in order to avoid any loss of information.

This is for example illustrated in Fig.9.1 where the mean performances of the reference profilesand an action a are represented respectively by stars and a bullet. Suppose, we consider only this

1This method will in this chapter be referred as C risp F lowSort.

191

I nterval and F uzzy F lowSort

criterion to assign a. On the basis of the mean-values it will univocally be assigned to C j althoughthere might be some cases where the performances of a are higher than the performance of r j.Working with intervals permit to take theses cases into account.

-. . .. . .

g(.)C j

g(r1)g(rK+1) g(r j)g(r j+1)

C1g(r j)g(r j+1)

g(a)g(a)

? ? ? ? ?•

Figure 9.1 — Illustration of interval performances of the reference profiles and action a onone criteria, where the stars and the bullet represent the mean values of the intervals.

We will thus compute the flow intervals for an action to be sorted. The relative position of theseflow intervals, with respect to the flow intervals of the reference profiles, will determine thecategory (categories) to which the action will be assigned. This approach will be called I ntervalF lowSort.

Let us notice that several works have been proposed to determine the parameters of a sortingmodel by assignment examples given by the decision maker [Ngo and Mousseau, 2002],[Hue-naerts and Nemery, 2007], etc. Furthermore, we could imagine that the given examples imposesome constraints on these values, without fixing them precisely, which are then traduced interms of intervals.

In a second step, we will suppose that the decision maker is able to define a membership func-tion associated to the intervals. We will model the evaluations by using fuzzy numbers whosemembership function grade the inner interval. Since the fuzzy flows contain more information,we will compute these flows. This allows us a better differentiation when comparing an actiona to the reference profiles. Based on the fuzzy flows, assignment rules are thus proposed. Thiswill be called F uzzy F lowSort.

This chapter is organized as follows. First we introduce some notations and definitions andclearly state the assumptions on which the model is built (Section 9.2 and 9.3 ). In Section 9.4we present the different assignment rules while working with mere intervals. In Section 9.5 wepresent the assignment rules when working with fuzzy numbers. The different assignment rulesare illustrated by some concrete examples in Section 9.1 and 9.5. Let us remark that the proofsof the propositions and the numerical details of the examples are given in Appendix B.

192

9.2. Notation

9.2 Notation

The actions to be sorted and the reference profiles may be defined by interval performances onthe different criteria instead of crisp values due to imprecision. We will denote by g j(x) andg j(x) respectively the lower and upper bound of the evaluation of x on criterion j (∀x ∈ R ∗i ).Moreover, the interval [g j(x),g j(x)] will be denoted by g j(x). Let us remark that the decisionmaker has no particular idea about an eventual distribution on this interval.

Since our approach is to keep the information about the evaluations all along, instead of reducingit, we will use basic arithmetic operations with intervals. These are defined as follows [Neumaier,1990]:

• x + y = [x,x]+ [y,y] = [x + y,x + y]

• x− y = [x,x]− [y,y] = [x− y,x− y]

• x∗ y = [x,x] ∗ [y,y] = [min(xy,xy,xy,xy),max(xy,xy,xy,xy)]

We will abusively note in this chapter x≤ y when actually x≤ y and x = y when x = x = y = y.

We suppose that a uni-criterion preference degree Pj(x,y),∀ j ∈ 1, . . . ,q can be computedfor all the actions x,y of R ∗i . This uni-criterion preference degree (which may be an interval)evaluates the preference strength on criterion j of action x over an action y according tothe preferences of the decision maker. It can for instance be obtained as in the Prometheemethodology [Brans and Vincke, 1985], [Teno and Mareschal, 1998], [Gelderman et al., 2000].Let us furthermore note the normalized weights associated to each criterion j as follows w j.

The global interval preference degree can thus be computed as follows:

π(x,y) =q

∑j=1

w j ∗ Pj(x,y) =q

∑j=1

[w j,w j] ∗ [Pj(x,y),Pj(x,y)]. (9.1)

In the particular case where the weights have crisp values, we obtain:

π(x,y) =q

∑j=1

w j ∗ Pj(x,y) = [q

∑j=1

w j ∗Pj(x,y),q

∑j=1

w j ∗Pj(x,y)] = [π(x,y),π(x,y)]. (9.2)

9.3 Conditions

Since the reference profiles define ordered categories, we shall assume that two consecutivereference profiles dominate each other. This is formulated by the following condition:

193


Condition 9.3.1. ∀r∗h,r∗l ∈ R ∗ such that h < l: g j(r∗h) ≥ g j(r∗l ) ⇔ g j(r∗h) ≥ g j(r∗l ), ∀ j ∈1, . . . ,q.

An illustration is given in Fig.9.2.

-

-ssss

s

ss

ss

s

ss

sss

C1C4 C3 C2

r1r5

g5

g4

g3

g2

g1

r3r3r4 r2rm3

60 75 95

35

30

55

40

20

40

22

16

55

30

18

9

75 955

12

18 0

120

0

120

0

Figure 9.2 — Illustration of the performances of limiting profiles defined by intervals.

Furthermore, we suppose that ∀x,y ∈ R ∗i the following conditions hold:

Condition 9.3.2. 0≤ π(x,y) ≤ 1 2.

Condition 9.3.3. π(x,y)+ π(y,x) ≤ 1.

Condition 9.3.4. π(x,x) = 0.

Condition 9.3.5. ∀x′,y′ ∈R ∗i , If ∀ j ∈ 1, . . . ,q : g j(x)− g j(y)≤ g j(x′)− g j(y′), then π(x,y)≤π(x′,y′).

Since the reference profiles define ordered categories, we will impose that a reference profile ofa lower (better) category is "preferred", according to the decision maker, to the reference profilesof a higher (worse) category. Formally, we have that:

Condition 9.3.6. ∀r∗h,r∗l ∈ R ∗ such that h < l: π(r∗h,r∗l ) > 0 and π(r∗l ,r∗h) = 0

On the basis of these interval preference degrees, the positive (leaving), negative (incoming) andnet flows intervals of each action x of R ∗i are computed as follows [Brans and Vincke, 1985]:

φ+R ∗i

(x) =1

|R ∗i |−1 ∑y∈R ∗i

π(x,y) (9.3)

φ−R ∗i

(x) =1

|R ∗i |−1 ∑y∈R ∗i

π(y,x) (9.4)

2We will in the rest of this work write a when a = [a,a]

194

9.4. I nterval F lowSort

φR ∗i(x) = φ

+R ∗i

(x)− φ−R ∗i

(x) (9.5)

Furthermore, under previous conditions, we are sure that the order of the flow intervals of thereference profiles is the same as the order of the reference profiles. This is formalized in the nextproposition:

Proposition 9.3.1. If conditions 9.3.1-9.3.6 are verified, we have that the order of the flows ofthe reference profiles is invariant with respect to the action ai to assign: ∀ai ∈ A :

∀h ∈ 1, ...,K + 1 ∀h ∈ 1, ...,Kφ

+R i

(rh) > φ+R i

(rh+1)

φ−R i

(rh) < φ−R i

(rh+1)φR i

(rh) > φR i(rh+1)

φ

+R c

i(rc

h) > φ+R c

i(rc

h+1)

φ−R c

i(rc

h) < φ−R c

i(rc

h+1)φR c

i(rc

h) > φR ci(rc

h+1)

-. . .. . .

φ+Ri

(.)C j

φ+Ri

(r1)φ+Ri

(rK+1) φ+Ri

(r j)φ+Ri

(r j+1)

C1φ+Ri

(r j)φ+Ri

(r j+1)

Figure 9.3 — Illustration of the positive flow intervals of the reference profiles under theconditions 9.3.1-9.3.6.

Imposing analogous conditions (Conditions 9.3.1-9.4.1) on the reference profiles (as in the C rispF lowSort model [Nemery and Lamboray, 2008]), preserves the main proposition (Proposition5.2.1) on which the assignment procedure will be based. The consequence of this property, is thatwe may equally define a category Ch by the interval flow values of rh and rh+1 and a category Ch

by the flow intervals of rh.

9.4 I nterval F lowSort

9.4.1 Limiting profiles

As in the C risp F lowSort model, the assignment rules will be based on the "preferred to" and"being-preferred by" character of the action ai with respect to the reference profiles. More pre-cisely, the relative position of the positive and negative flow intervals of the action to be sortedwith respect to the intervals of the reference profiles will determine the assignment. When work-ing with the positive flows we define thus the following assignment rules:


Cφ+(ai) = [Cφ+(ai),Cφ+(ai)]⇐⇒

Cφ+(ai) = Cl ⇔ φ+R i

(rl) ≥ φ+R i

(ai) > φ+R i

(rl+1)

Cφ+(ai) = Ch⇔ φ+R i

(rh) ≥ φ+R i

(ai) > φ+R i

(rh+1)with h≤ l.

195


Since the parameters (e.g. weights) and the performances of the actions are defined by intervals,the flows may thus be intervals. These assignment rules are a direct extension of the assignmentrules when working with crisp values.The assignment of an action may thus not always be unique but can consist in a set of consecutivecategories.

Since we are working with intervals, 4 different scenarios may occur when comparing therelative positions of the intervals. These scenarios are illustrated in the case of the positive flowsin Fig. 9.4. where the "black" rectangles represent the positive flow intervals of action ai.

. . .

φ+Ri

(r j+1)

. . .

C jφ+Ri

(r j+1)

. . .

C jφ+Ri

(r j+1)

C j

. . .

C j−1

C j−1

C j−1φ

+Ri

(r j−1)

φ+Ri

(r j−1)

φ+Ri

(r j−1)φ

+Ri

(.)

φ+Ri

(.)

φ+Ri

(.)-

-

φ+Ri

(r j+1)

φ+Ri

(r j+1)

. . .

. . .φ

+Ri

(r j−1)

φ+Ri

(r j−1)

φ+Ri

(r j−1)

φ+Ri

(ai)

φ+Ri

(ai)

φ+Ri

(ai)

. . .

C jφ+Ri

(r j+1) C j−1φ

+Ri

(r j−1)

φ+Ri

(.)-

φ+Ri

(r j+1). . .

φ+Ri

(r j−1)φ+Ri

(ai)

-

Figure 9.4 — Illustration of the assignment rules when working with the positive flow inter-vals.

Similarly, when working with the negative flows, we define thus the assignments rules as follows:


Cφ−(ai) = [Cφ−(ai),Cφ−(ai)]⇐⇒

Cφ−(ai) = l⇔ φ−R i

(rl) < φ−R i

(ai) ≤ φ−R i

(rl+1)

Cφ−(ai) = h⇔ φ−R i

(rh) < φ−R i

(ai) ≤ φ−R i

(rh+1)with h≤ l.

We may obtain different assignments, having these two different assignment rules: Cφ+(ai) andCφ−(ai). Let us denote furthermore by Cb(ai) the best category and Cw(ai) the worst defined asfollows:

196


Cb(ai) = min[Cφ−(ai);Cφ+(ai)] (9.6)

Cw(ai) = max[Cφ−(ai);Cφ+(ai)] (9.7)

Taking both perspectives, permit us to assign ai to the following set of categories:Cb∪Cb+1∪ . . .∪Cw−1∪Cw.

If the decision maker should impose the assignment to an eventually smaller set of consecutivecategories, we may use the net flow intervals with the following assignment rule:


Cφ(ai) = [Cφ(ai),Cφ(ai)]⇐⇒

Cφ(ai) = Cl ⇔ φR i

(rl) ≥ φR i(ai) > φR i

(rl+1)

Cφ(ai) = Ch⇔ φR i(rh) ≥ φR i

(ai) > φR i(rh+1)

with h≤ l.

As in the C risp F lowSort model, this assignment rule appears to be reasonable since thecategories obtained with the net flows are consistent with the one obtained with the positive andnegative flows. In other words, we have the following analogous property:

Proposition 9.4.1. ∀ai ∈ A :Cφ(ai) ⊆ [Cb(ai),Cw(ai)]

The previous assignment rules are completely coherent with the one defined in the C rispF lowSort model since the assignment rules are the same when all the parameters andthe actions of R i are precisely defined (i.e. they have crisp values for all the evaluations∀k ∈ 1, . . . ,q).

Let us define the following notations:

Definition 9.1. R i = r1, . . . , rK+1, ai

Definition 9.2. xmi | ∀k ∈ 1, . . . ,q : g j(xm

i ) ∈ g j(xi)⇔ g j(xi) ≤ g j(xmi ) ≤ g j(xi)

Definition 9.3. R mi = rm

1 , . . . ,rmK+1,am

i

If starting for any action xi ∈ R i we take a crisp value in these intervals for all criteria and thenapply the C risp F lowSort procedure we know that this last assignment will always be in theinterval assignment. Formally, we have the following proposition:

197


Proposition 9.4.2. ∀xi ∈ R i,∀xmi ∈ R m

i

Cφ+(ami ) ∈ Cφ+(ai) &Cφ−(am

i ) ∈ Cφ−(ai) & Cφ(ami ) ∈ Cφ(ai)

The more an action is poorly defined (i.e. the larger the intervals of its performances is), the lessprecise is the assignment (i.e. the larger the set of categories to which it will be assigned).Particularly, suppose that the only information we have is that the performancesof an action ai is, on all criteria between, the best and worst limiting profile (i.e.∀k ∈ 1, . . . ,q : gk(rK+1) ≤ gk(ai) ≤ gk(r1)), then it will be assigned to all the cate-gories (i.e. Cφ(ai) = [C1,CK ]).

Let us remark that the interval version of Promethee [Teno and Mareschal, 1998] may sufferfrom the fact that the flow intervals are large. Consequently, the relative position of the referenceprofiles may be not clear. This is not the case under the conditions 9.3.1-9.3.6. These conditionsensures us that Proposition 9.3.1 is fulfilled. This will be illustrated in section 9.1.

Let us now emphasize some properties of the assignment rules defined above. The previousassignment rules verify the monotonicity property such that if an action ai dominates anotheraction a j, then ai can not be assigned to a higher (worse) category than action a j. Formally:

Proposition 9.4.3. ∀ai,a j ∈ A :

∀k ∈ 1, . . . ,q : gk(ai) ≥ gk(a j)

⇓

Cφ+(ai)DCφ+(a j) and Cφ+(ai)DCφ+(a j)3

Previous property holds also for the assignment rules based on the negative and net flows.

Although we are dealing with imprecision, we may impose that the upper profile r j of a categoryC j is "strongly preferred" to the lower profile r j+1. This can be written as follows:

Condition 9.4.1. ∀rh,rl ∈ R such that h < l: π(rh,rl) = 1

This condition may be interpret as in the C risp F lowSort model. It leads identically to the factthat not every category combination, according to the positive and negative flows, is feasible.Analogously, we have that the "best" (worst) categories determined with the negative flows arealways as least as good as the "best" (worst) one determined with the positive flows. Formally:

3C D C means that either C B C or C = C.

198


Proposition 9.4.4. ∀ai ∈ A :

Cφ−(ai) = Cl and Cφ+(ai) = Ch⇒ l−h≤ 0

Cφ−(ai) = Cl and Cφ+(ai) = Ch⇒ l−h≤ 0

An illustrative example is given in Example 9.1.

Example 9.1. In this section we will apply previous considerations on an illustrative example.The example and the data are taken from [Araz and Ozkarahan, 2007a] (Araz, p 123). We firstbriefly introduce the context. The reader may find more useful information in the original text.

A company wants to evaluate its suppliers to determine specific partnerships. For this purpose,it evaluates the suppliers on the basis of 5 criteria which are then compared to some norms(limiting profiles). These norms define 4 ordered categories. The meaning of the categories areas follows:

1. C1: suppliers for strategic partnerships (the best category)

2. C2: promising suppliers

3. C3: suppliers for competitive partnerships

4. C4: suppliers to be pruned (worst category)

The five criteria of evaluation are the following:

1. g1: delivery performance (to be maximized)

2. g2: processing time (in days): time needed to develop product structural design (to be min-imized)

3. g3: design revision time (in days): time needed to perform project revisions (to be mini-mized)

4. g4: prototyping time (in days): time needed to construct prototypes (to be minimized)

5. g5: cost reduction performance (to be maximized)

In this context, it is not unrealistic to assume that the performances of the limiting profiles andof the suppliers are not completely well-defined. For instance, delivery performances may ingeneral determined with precision. We will treat this imprecision on the performances by consid-ering interval performances on the different criteria (see Tab. 9.1). Moreover, we suppose thatthe decision maker of the company uses linear preferences functions for the different criteria. Theweights and the different preference parameters have nevertheless crisp values (see Tab. 9.2).

199


The norms to which the suppliers are compared are represented in Fig. 9.5 and given in Tab.9.1. Let us remark that gm

j (.) simply refers to a crisp value in the interval.

-

-ssss

s

ss

ss

s

ss

sss

C1C4 C3 C2

r1r5

g5

g4

g3

g2

g1

r3r3r4 r2rm3

60 75 95

35

30

55

40

20

40

22

16

55

30

18

9

75 955

12

18 0

120

0

120

0

Figure 9.5 — Illustration of the performances of reference profiles.

g1 g2 g3 g4 g5

ri g1 gm1 g1 g2 gm

2 g2 g3 gm3 g3 g4 gm

4 g4 g5 gm5 g5

r5 40 40 40 55 55 55 30 30 30 20 20 20 35 35 35r4 55 60 65 38 40 42 21 22 23 15 16 17 50 55 60r3 70 75 80 28 30 32 17 18 19 8 9 10 70 75 80r2 90 95 100 16 18 20 11 12 13 4 5 6 90 95 100r1 120 120 120 0 0 0 0 0 0 0 0 0 120 120 120

Table 9.1 — Interval evaluations of the reference profiles on the different criteria.

g1 g2 g3 g4 g5

qk 2 1 0 0 0pk 10 8 5 3 15wk 0.2 0.23 0.17 0.15 0.25


At first, the preference degrees between the reference profiles, according to the given parameters,are computed in order to verify if the conditions of the model are fulfilled. These preferencedegrees are given in Tab. 9.3.

Then the suppliers are compared to the limiting profiles by means of preference degrees. Thesepairwise preferences degrees are given in Tab. 9.5.

On the basis of these preference degrees, we compute the positive, negative and net flows of thereference profiles and the actions are given in Tab. 9.6. This leads to the results resumed in Tab.

200


r5 r4 r3 r2 r1

π(r5,r j) 0 0 0 0 0π(r4,r j) 1 0 0 0 0π(r3,r j) 1 [0.624;1] 0 0 0π(r2,r j) 1 1 1 [0.8327;1] 0π(r1,r j) 1 1 1 1 0

Table 9.3 — The preference matrix of the reference profiles.

A g1 gm1 g1 g2 gm

2 g2 g3 gm3 g3 g4 gm

4 g4 g5 gm5 g5

a1 50 60 70 43 45 55 20 22 30 18 20 28 45 60 60a2 57 75 66 18 30 24 15 21 18 8 12 10 65 85 75a3 46 47 48 46 47 48 25 26 27 18 18 18 42 43 44a4 77 77 77 49 49 49 13 13 13 12 12 12 94 94 94

Table 9.4 — The performances of the actions to be sorted.

r1 r2 r3 r4 r5

π(a1,r j) 0 0 0 [0;0.4867] [0.3667;0.95]π(r j,a1) 1 1 [0.587;1] [0.05;1] [0;0.15]π(a2,r j) 0 [0;0.0329] [0; 0.791] [0.4633;1] 1π(r j,a2) 1 [0.4513;1] [0;0.7689] [0;0.15] 0π(a3,r j) 0 0 0 0 1π(r j,a3) 1 1 1 [0.4416;1] 0π(a4,r j) 0 0 [0;0.025] [0.20;0.3] 1π(r j,a4) 1 1 [0.78;0.925] [0.55;0.6167] 0

Table 9.5 — The preference degrees between the reference profiles and the actions.

9.7: we compare the relative position of the flows of an action with respect to the flows of theprofiles.

Moreover, we may represent a flow-diagram as illustrated in Fig. 9.6. This plan may be obtainedsuch as in the C risp F lowSort model while replacing the crisp flows (positive and negative) bytheir intervals instead of crisp values.

Finally, we have compared the interval flow values with the one obtained by taking a crispvalue4 (∀k : gm

k (.)) in the interval performance of each reference profile and action am2 (for all

the criteria). These crisp values are given in Tab. 9.1 and Tab. 9.4.

4not necessarily the mean of the interval

201


r1 r2 r3 r4 r5 ai

φ+R1

1 [0.7665;0.8] [0.4409;0.6] [0.21;0.4] [0;0.03] [0.0733;0.2837]

R1 φ−R1

0 0.2 [0.3665;0.4] [0.8733;0.99] 1 [0.5261;0.83]

φR11 [0.5665;0.6] [0.0409;0.2335] [-0.4837;-0.1248] [-0.99;-0.843] [-0.7567;-0.244]

φ+R2

1 [0.6568;0.8] [0.3448;0.5538] [0.2;0.23] 0 [0.2927;0.5648]

R2 φ−R2

0 [0.2;0.2066] [0.3665;0.5582] [6175;0.8] 1 [0.2903;0.5838]

φR21 [0.4502;0.6] [-0.2334;0.1872] [-0.6;-0.3875] -1 [-0.2911;-0.2745]

φ+R3

1 [0.7665;0.8] [0.5248;0.6] [0.2883;0.4] 0 0.2

R3 φ−R3

0 0.2 [0.3665;0.4] [0.5248;0.6] 1 [0.6883;0.8]

φR31 [0.5665;0.6] [0.1248;0.2335] [-0.3117;-0.125] -1 [-0.6;-0.4883]

φ+R4

1 [0.6065;0.66] [0.3248;0.405] 0.2 0 [0.667;0.708]

R4 φ−R4

0 [0.31;0.33] [0.523;0.585] [0.72;0.8] 1 [0.24;0.26]

φR41 [0.283;0.35] [-0.26;-0.118] [-0.6;-0.52] -1 [0.40;0.468]

Table 9.6 — Computation of the different flow values.

-

6

φ+R1

φ+R1

(r1)

φ+R1

(r2)φ

+R1

(r2)

φ+R1

(r3)

φ+R1

(r3)

φ+R1

(a1)

φ+R1

(r4)

φ+R1

(r5)

φ−R1

(a1)

φ−R1

(r1) φ−R1

(r2) φ−R1

(r3) φ−R1

(r5)φ−R1

(r4) φ−R1

(r4)

C1

C2

φ−R1

1

1

φ+R1

(r4)

?

?

C3

C2∪C3

C1∪C2

C3∪C4

C4

C1∪C2∪C3∪C4

Figure 9.6 — Illustration of the interval flow-diagram for action a1.

202

9.5. F uzzy F lowSort

ai Cφ+ Cφ− Cφ

a1 [3,4] [3,4] [3,4]a2 [2,3] [2,3] [2,3]a3 4 3 [3,4]a4 1 1 1

Table 9.7 — Results of the assignments of the actions according to the different rules.

This leads to crisp values for the flows and to obtain thus a unique assignment according to eachrule. The results are given in Tab.9.8. We can conclude that with the interval assignments, thecategories to which a2 should be assigned are C2∪C3 whereas am

2 should be assigned differentlyaccording to the considered flows (categories C2 and C3). This attests thus from the eventual lossof information when working with crisp values instead of interval values.

rm1 rm

2 rm3 rm

4 rm5 am

i C(am2 )

φ+R m

2m0.97 0.728 0.429 0.728 0.97 0.392 3

R m2m φ

−R m

2m0 0.184 0.425 0.721 0.963 0.403 2

φR m2m

0.97 0.544 0.004 -0.544 -0.963 -0.011 3

Table 9.8 — Computation of the different flow values for am2 .

9.4.2 Central profiles

As mentioned in Section 9.1, the decision maker may also define the categories by central pro-files. The assignment rules as well as the the propositions of the C risp F lowSort may analo-gously be formulated and proven by adopting the same formalism.

9.5 F uzzy F lowSort

In previous sections we have implicitly supposed that all values of the intervals are possible withthe same distribution. However, it can be assumed that the extreme values for example are lesspossible. Moreover, the decision maker could be more confident with some subintervals. Forthat purpose, fuzzy sets and fuzzy numbers seems to be appropriate to deal with this "richer"information given by the decision maker.

Once the extreme possibilities presented by the hand of the flow intervals and the intervalassignment results, we may use defuzzification techniques to exploit the obtained distributionassociated to the fuzzy flows.

At first we present briefly fuzzy numbers. The reader will find more information on fuzzy

203


numbers in [Rommelfanger, 1996]. The flow computation with these fuzzy intervals will thenbe proposed and finally we will show how they may be used to refine the final assignments.

9.5.1 Fuzzy numbers

We propose here a fuzzy extension of the I nterval F lowSort model. Actually, it consists toassociate a membership function to the interval defining for example, the performances of theactions, the weights interval, etc. To each interval we shall associate a membership functionµI(x) representing the grade of membership of x in the interval I. This illustrated in Fig.9.7.This value is comprised between 0 and 1 and increases with the grade of membership. We willdenote by x the fuzzy interval. In most of the cases we may define the membership functionsby triangular and trapezoidal functions [Araz and Ozkarahan, 2007a], [Rommelfanger, 1996].In this context we note the fuzzy interval as follows: x = (xu,xl ,α,β) where xu,xl ,α and β aredefined as in Fig.9.7.

-

6µI

1

xxl- α β

x xxu

-

Figure 9.7 — Representation of a fuzzy interval x and its parameters xu,xl ,α,β.

The algebraic operations with fuzzy intervals are defined in [Dubois and Prade, 1980], [Rom-melfanger, 1996] as follows:

• X⊕ Y =(xl ,xu,α,β)⊕ (yl ,yu,γ,δ)=(xl + yl ,xu + yu,α + γ,β + δ)

• X Y =(xl ,xu,α,β) (yl ,yu,γ,δ) = (xl− yu,xu− yl ,α + δ,β + γ)

• X⊗Y =(xl ,xu,α,β)⊗(yl ,yu,γ,δ)≈ (xl ∗yl ,xu∗yu,xl ∗γ+yl ∗α−α∗γ,xu∗δ+yu∗β+β∗δ)

Moreover, we will note the fuzzy uni-criterion preference degree of an action x on action y asfollows: Pj(x,y),∀ j ∈ 1, . . . ,q. This fuzzy uni-criterion preference degree can for examplebe obtained on the basis of the difference between the fuzzy performances on the considered

204


criterion as given in [Gelderman et al., 2000], p.54. Let us remark that the fuzzy uni-criterionpreference degree do not necessarily have the same type as the fuzzy difference. It might happenthat a fuzzy difference is triangular or rectangular and the uni-criterion preference degree istrapezoidal [Gelderman et al., 2000].

The global fuzzy preference degree can thus be obtained as follows:

Π(x,y) =q

∑j=1

w j⊗ Pj(x,y). (9.8)

On the basis of these fuzzy preference degrees, the positive (leaving), negative (incoming) andnet fuzzy flows of each action x of R ∗i are computed as follows:

Φ+R ∗i

(x) =1

|R ∗i |−1 ∑y∈R ∗i

Π(x,y) (9.9)

Φ−R ∗i (x) =1

|R ∗i |−1 ∑y∈R ∗i

Π(y,x) (9.10)

ΦR ∗i(x) = Φ+

R ∗i(x)− Φ−R ∗i (x) (9.11)

Thus, we obtain fuzzy flow intervals for the reference profiles and the action to be sorted. Let usstress that the supports of these fuzzy numbers are the same intervals as the one that we obtainwith I nterval F lowSort.

Let us remind that the obtained fuzzy flows may not always have the same membership functionas the membership function defined on the (performance) intervals. I nterval F lowSort cantherefore not be seen as a particular case of fuzzy numbers where the associated membershipis a constant value for all intervals. Nevertheless, the properties of previous sections (such asmonotonicity, category combination, net flow assignment consistency, etc.) are still valid in thiscontext when considering the support of the fuzzy numbers.

Since the decision maker gives more information (by associating a membership function to theintervals), the fuzzy flows contain also more information than the mere flow intervals. To assignan action ai to a category on the basis of this fuzzy flows several approaches can be proposed. Asmentioned in [Teno and Mareschal, 1998], we may use the relative and geometrical propertiesof these fuzzy flows to finally rank the alternatives. From the relative position of the action ai

with respect to the reference profiles, we can assign an action to a category.

205


Another approach may be a defuzzification of the fuzzy flows which reduce these fuzzy flows tocrisp numbers taking into account the initial membership given by the decision maker. For thatpurpose we may use for example the "Centre of Area" defuzzification method [Rommelfanger,1996] which is defined as follows:

xDe f =∫

x∗µ(x)dx∫µ(x)dx

(9.12)

When supposing a trapezoidal membership function we have:

xDe f =x2

u− x2l + α∗ xl + β∗ xu +(1/3) ∗ (β2−α2)

α + β + 2∗ xu−2∗ xl. (9.13)

On the other hand, when supposing a triangular membership function we have then:

xDe f = xl +β−α

3. (9.14)

We obtain thus for instance the crisp positive flows ((Φ+R ∗i

(x))De f ) from the fuzzy positive flows

(Φ+R ∗i

(x) = ((φ+R ∗i

(x))l , (φ+R ∗i

(x))u,α,β)) by defuzzification.

Finally, these crisp flow values (calculated for the reference profiles and the action ai) can beused as in the C risp F lowSort assignment rules. This leads thus to a unique category (accordingto the different flows) but while having the initial interval assignments. It permits to refine theresults.

Let us remark that the defuzzification operator should be chosen in such a way that thedefuzzified numbers of the flow values of the reference profiles verify Proposition 5.2.1 in theC risp F lowSort. Since the intervals of the flows verify Proposition 9.3.1 and since, moreover,the defuzzified number is in the interval when using for instance the "Centre of Area" method,Proposition 5.2.1 will be verified.

Example 9.2. To illustrate previous considerations, let us reconsider the example given inExample 9.1. We will suppose here that the reference profiles and the actions to be sorted aredefined by fuzzy numbers on the same intervals. We suppose moreover that to each interval atriangular membership function is associated.

At first, we will consider the membership functions chosen as follows (see Fig.9.8):∀x ∈ R i, ∀ j ∈ 1, . . . ,q :

g j(x) = (g j(xm),g j(xm),α,β)

where α = g j(xm)− g j(x) and β = g j(x)− g j(xm) with g j(x) = [g j(x),g j(x)]. We denote this

206


situation by scenario 1.

6

1

µ

40s60srm

3

r3r375

s95 120

- g1

r1r2r3r4r5

a1

-

Figure 9.8 — Illustration of the fuzzy performances of the the actions of R1 on criterion 1 inscenario 1.

To illustrate the impact of the membership functions on the final assignment, let us considerthe case where the membership is defined differently on the same support (scenario 2). We willassociate to all the interval performances of the reference profiles the following membership:∀ri ∈ R , ∀ j ∈ 1, . . . ,q :

g j(ri) = (g j(ri),g j(ri),0,β)

where β = g j(ri)−g j(ri). This is illustrated in Fig.9.9.

On the other hand, the membership function associated to the interval performances of theaction ai to be assigned will be as follows: ∀ j ∈ 1, . . . ,q,∀ai ∈ A :

g j(ai) = (g j(ai),g j(ai),α,0)

where α = g j(ai)−g j(ai).

In other words, in scenario 1, the decision maker, believes that the mean performances aremore likely to happen (either for the reference profiles and the actions). On the other hand, inscenario 2, the lowest performances for the reference profiles are more probable and the highestperformances for the actions to be assigned.

Furthermore, we consider the same preference parameters and the same weights (defined bycrisp values, see Tab. 9.2). The fuzzy preference degrees between the reference profiles are givenin Tab.B.1 (Appendix B). The pairwise preference degrees between the reference profiles and theactions a1 and a2 are given in Tab.B.2, B.3 and B.4 (see Appendix B).

The fuzzy flows of the actions are given in Tab.B.5 and B.6 for action a1 in the two differentscenarios and in Tab.B.7 and B.8 for action a2. Let us remark that we have written the fuzzy

207


6

1

µ

40s60s

r3r375

s95 120

- g1

r1r5

a2

-

r2r3r4

Figure 9.9 — Illustration of the fuzzy performances of the actions of R1 on criterion 1 inscenario 2.

numbers abusively in the following aggregated form since we are working with triangular fuzzynumbers: x = (xt ,xt ,α,β) = x≡ (xt ,α,β).

We have given in Tab.9.9 - 9.12 solely the corresponding defuzzified flow values.

(Φ+R1

)De f (Φ−R1)De f (ΦR1)

De f

r1 1 0 1r2 0.7888 0.2 0.589r2 0.542 0.389 0.154r4 0.288 0.609 -0.321r5 0 0.944 -0.934a1 0.181 0.679 -0.497

Table 9.9 — Computation of the different fuzzy flow values for R1 in scenario 1.

(Φ+R1


De f

r1 1 0 1r2 0.7888 0.2 0.589r2 0.517 0.389 0.128r4 0.273 0.636 -0.362r5 0.01 0.9514 -0.941a1 0.214 0.637 -0.424


The assignment of action a1 will thus unambiguously be category C4 according to the differentflows in both scenarios. This means thus that for this two different membership functions, theassignment does not change and "strengthens" the assignment.

On the contrary, when assigning a2, by taking into account the membership function, a2 willunivocally be assigned to C3 in scenario 1 whereas in scenario 2 to C2. Let us remark that when

208


(Φ+R2


De f

r1 1 0 1r2 0.748 0.2022 0.5457r2 0.542 0.4525 -0.0136r4 0.288 0.728 -0.518r5 0 1 -1a2 0.421 0.679 -0.0141


(Φ+R2


De f

r1 1 0 1r2 0.7157 .2044 0.5113r2 0.424 0.4943 -0.0704r4 0.21 0.7369 -0.5269r5 0 1 -1a2 0.4741 0.3881 0.086


working with the "mean" crisp evaluations (gmi ) of the actions and reference profiles (Section

9.1), we obtained that a2 was assigned to C2 or C3. The results are summarized in Tab.9.13.

Scen-1 Scen-2 gm(.)Cφ+ 3 2 3Cφ− 3 2 2Cφ 3 2 3

Table 9.13 — Results of the assignments of the action a2 according to the different rules inscenario 1, scenario 2 and when working directly with crisp evaluations.

We can thus notice that the complement of information given by the decision maker may precisethe assignments (see Tab. 9.13). The use of fuzzy numbers, and particularly the membershipfunction, may thus be a useful way of taking into account the imprecise decision maker’s opinion.

Furthermore, it can be interesting to compare the assignment rules when different scenarios(i.e. different membership functions) are considered. Nevertheless, some additional effort mustbe made to determine these functions or to propose one to the decision maker.

209

10 Software and applications

In the context of this work, we have supervised the implementation of the F lowSort method ina user friendly program. Dieungang Wekop Marlyse, Perrault Laurent and Otschudi OmangaOlivier have developed in the JAVA language, a software for the sorting of actions into com-pletely ordered categories. This software is downloadable on the web-site given in [DieungangWekop et al., 2007].

As illustration of the use and the features of this software, let us consider the followingreal-based application taken from [Araz and Ozkarahan, 2007b].A manufacture company, in the field of electronic industry, would like to develop strategicpartnerships with a set of (hopefully) innovative suppliers. "Integration of the right suppliersin concurrent engineering teams is an important factor for success" ([Araz and Ozkarahan,2007b],p.143). For this purpose, the company would like to evaluate its suppliers (and someemergent ones) in order to distinguish them.

The purpose is obviously not to rank the suppliers. A ranking of the suppliers may not be adaptedsince for example the worst supplier may be completely in line with the need of the company.On the other hand, the best supplier may not be adapted at all for the company. In this situation,new suppliers need to be find.For these reasons the company wants to compare the suppliers according to some norms or ref-erence profiles which will be representative of their strategy and needs. Therefore, the companydefines 4 completely ordered categories by means of limiting profiles:

1. C1: suppliers for strategic partnerships

2. C2: promising suppliers that must be supported via supplier development programs

3. C3: suppliers for competitive partnerships: they have to be considered for competitive part-nerships for some products

4. C4: suppliers to be pruned: they should no longer be considered for the partnership in anylevel

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Software and applications

The criteria chosen for evaluating the suppliers are the following [Araz and Ozkarahan, 2007b]:

1. g1: Support in Product Structural Design

2. g2: Support in Process Design and Engineering

3. g3: Design Revision time

4. g4: Prototyping time

5. g5: Level of Technology

6. g6: Quality Performance

7. g7: Financial Strength

8. g8: Cost Reduction Performance

9. g9: Delivery Performance

10. g10: Ease of communication

All criteria, except of g3 and g4, have to be maximized and the preference parameters are givenin Tab.10. The values of the parameters are determined by the interaction with the concurrentdesign team. The 4 categories are defined by means of 5 limiting profiles which performancesare given in Tab.10.

Table 10.1 — The limiting profiles of the 4 categories of different suppliers.

ri g1 g2 g3 g4 g5 g6 g7 g8 g9 g10

r5 0 0 40 40 0 0 0 0 0 0r4 65 70 25 25 65 60 70 70 70 65r3 80 80 18 15 75 80 80 80 85 80r2 90 90 8 7 90 90 95 90 95 90r1 100 100 0 0 100 100 100 100 100 100

Table 10.2 — The preference parameters associated to the 10 criteria of evaluation.

g1 g2 g3 g4 g5 g6 g7 g8 g9 g10

qi 0 0 0 0 5 5 5 0 5 5pi 10 10 7 8 10 10 10 10 10 10wi 0.15 0.1 0.1 0.1 0.08 0.15 0.05 0.12 0.1 0.05

212

When using the software, the decision maker needs to introduce the exact number of referenceprofiles (by adding new profiles, see the "menu-bar" in Fig.10.1) and may moreover choose thepreference function needed to define a criterion (see Fig.10.1). He has the choice between the6 preference functions defined in Promethee (see Section 2.6.2) by using the "pop-up" menu.After encoding the values of the different thresholds (if needed), the weights of the criteria,minimizing or maximizing the criteria, the program check if the reference profiles verifyconditions 5.2.1-5.2.6.

Figure 10.1 — Screen-shot of the software when encoding the preference parameters.

One may remark that the decision maker may easily understand the signification of the thresh-olds given the small illustrations associated to each preference function (see Fig.10.1).

The performances of the suppliers to be sorted are given in Fig.10.2 and Appendix C, Tab.C. Letus remark that the decision maker may enter the data in an "Excel"-format or in a specific binarydata file of the application (".fs").

The program permits indeed to visualize the performances of each action individually, withrespect to the profiles on the defined criteria. In Fig.10.3 we may remark that the decision makerhas asked to represent the performances of action a1. Let us remark that the criteria are denotedby c j in the program.

213


Figure 10.2 — Screen-shot of the software when encoding the performances of the actions tobe sorted.

Figure 10.3 — Screen-shot of the software representing the evaluations of action a1 withrespect to the reference profiles.

When all the data has been given and verified by the program, the actions are assigned accordingto the positive, negative and net flows. The results are given in a table as presented in Fig.10.7,217.The different flow-values with respect to the reference profiles are given in Appendix C.

The assignment of each action may be easily visualized in the positive and negative flow-plane.

214

This permits to obtain an idea about its "being preferred to" and "being preferred by character".An example for action a1 is given in Fig.10.4. The assignment of a1 is unambiguously categoryC2. According to the positive flows, supplier a1 is just in between r2 and r3 whereas accordingto the negative flows, he behaves more as r2.

Figure 10.4 — Screen-shot of the software representing the positive and negative flow-planefor action a1.

Furthermore, it is possible to render visible in the "criteria-plan" all the actions which are as-signed to the same category according to a specified assignment rule. In Fig.10.5 we have repre-sented all the actions which are assigned to C1 (according to the net-flow rule) with respect to thereference profile. This permits the decision maker to analyze if the actions, assigned to a samecategory, are rather similar or not, if there are some disparities, etc. The decision maker may thusremark that the performances of action a15 (green action) are for each criteria in between r1 andr2 (it will thus be assigned to C1). On the contrary, the actions a11 (red action) and a18 (yellowaction) present some strong points and some weak points with respect to r2. Besides, we mayremark that these two actions have quite similar performances.

The distribution of the actions according to the positive and negative assignment rules are rep-resented in Fig.10.6. It is thus easy to notice that action a16 is the only action which is assignedunivocally to the category of the strategic partners, C1, according to both optimistic and pes-simistic rules. Actions a12 and a19 may be assigned to C1 or C2. On the other hand, solely a10

and a20 belong to the suppliers which have to be pruned. Beyond, most of the suppliers belong

215


Figure 10.5 — Screen-shot of the software representing all the actions assigned to C1 accord-ing to net flow assignment rule.

Figure 10.6 — Screen-shot of the software representing the distribution of the actions intothe categories according to the positive and negative flow-plane for action a1.

to category C2 or C3.

The actual program allows only to work with precise data. Nevertheless, a new part, permittingthe definition of the performances and parameters by means of mere intervals has been devel-oped and will be incorporated in this software. Moreover, the software enables to make somereports about the sorting problem. Beyond, the data and the results are compatible with clas-

216

Figure 10.7 — Screen-shot of the software representing the assignments of all the actionsaccording to the different assignment rules.

sical file format (such as Excel), which permit to import data and retrieve them in other programs.

Some more effort should be spent to aid the decision maker to enter and determine his preferencesparameters from for example a given data set. Linear programming techniques may for thatpurpose be added.

217

11 Conclusions

In this part, we have developed a new multicriteria sorting method, the F lowSort method,for problems where the categories are completely ordered. F lowSort is based (directly) onan existing ranking approach, the Promethee method. The categories are defined by profiles,limiting or central ones, which are thus naturally ranked from the best to worst.To assign an action to a category, the F lowSort method computes its ranking-position withrespect to solely the reference profiles. This permits to ensure the independency between theassignments of the actions.To the best of our knowledge, it is one of the first sorting methods enabling the decision makerto handle both types of profiles. A relationship has been moreover given between these twoapproaches.

Since Promethee may suffer from rank reversal phenomenons, some conditions are imposedon the reference profiles such that their order (when comparing an action to sort) is alwaysconsistent with the order of the categories they define. Actually, such conditions allow the useof any ranking method for sorting problems.

The fact that Promethee is a complete ranking method involves that the assignment of an actionis based on a simultaneous or global comparison to the reference profile (on the contrary ofElectre-Tri for example). This has as consequence that the property of strong stability is notfulfilled since the adding or the suppression of a reference profile may alter the ranking (andthus the assignment). F lowSort does thus not respect the strong stability property. However, itrespect the property of weak stability.Besides, some properties of F lowSort have been put forward in order to analyze this approachin a sorting context. F lowSort respects the properties of uniqueness, neutrality, monotonicity,weak homogeneity, weak stability and conformity.

We presented some empirical and theoretical comparisons with Electre-Tri. Although per-forming empirical comparisons between two multicriteria approaches can always be criticized,we concluded that, in our context, the assignments between both approaches are consistent.

219

Conclusions

However, it seems that the difference between the categories obtained with the positive and thenegative flows is smaller than the difference between the categories obtained with the optimisticand pessimistic Electre-Tri rules.

Nevertheless, we have presented some examples for which the assignment with Electre-Triand UTADIS may differ. This points out that the methods are not the same. In order todistinguish more fundamentally F lowSort from other existing sorting methods, some furthertheoretical investigation needs to be done. More particularly, a future direction of research is toaxiomatically characterize F lowSort.

In the last part, an extension of the F lowSort method has been proposed to deal with impreci-sion. Actually, the evaluation of the performances of the reference profiles (limiting or central)and the actions to be sorted, the weights and the preference degrees may not always be knownprecisely (due to for example measures, etc.).To tackle this problem, we have at first defined these values by mere intervals. This leads to theI nterval F lowSort model where the leaving, entering and net interval flows are calculated usingsimple interval arithmetic. The assignment rules are based on the flow intervals, which involvesthat the property of uniqueness may not be verified. Nevertheless, we have moreover proven thatthe properties of the C risp F lowSort version hold when working with intervals.

In a second phase, we have supposed that a membership function may be associated to theinterval. We have used the fuzzy numbers and fuzzy logic to deal with this information. Thisleads to the F uzzy F lowSort model. We have illustrated through examples that this addedinformation may be used to clarify the assignment of the actions while taking into account thedecision maker’s opinion. This last approach should be confirmed through the application to realsorting problems and may be compared to a probabilistic approach (not presented in this work).

It seems natural to extend the F lowSort model to define the categories by several profiles foreach category. F lowSort needs to be adapted to handle this kind of situations.Beyond, some further effort should be spent on facilitating the determination of the parametersof the model, such as for instance the reference profiles, the weights, the thresholds, themembership functions. Assignment examples given by the decision maker may be for instanceused. For this purpose, one may adapt previous works on the determination of these parametersfor Promethee or for Electre-Tri. Deducing the performances of limiting profiles from those ofcentral profiles (and vice-versa) such that the assignment are the same, may be an interestingpoint to analyze.

∵

220

The proposed approach may be furthermore applied with other ranking methods. One may usefor instance Electre III or the Rough-Set-based ranking method to sort actions with respectto reference profiles. However, conditions have to be imposed on these profiles such that thesorting properties (e.g. independency, monotonicity, etc.) are satisfied. This permits moreover tostudy the considered ranking method from a new insight. A future research could be the studyof the comparison of Electre III (applied on an action to be sorted an the reference profiles) andElectre-Tri under the conditions of Electre-Tri.

In addition, adopting this ranking-based sorting approach permits to tackle new problems.In Chapter 3, Section 3.5, we have presented the input-output matrix after application ofpropagation rules. Multicriteria procedures were associated to a particular couple (ιi,o j).Furthermore, [Guitouni et al., 1999] proposed propagations in order to "fill" the input-outputmatrix representing possible decision-making situations. This was based on the idea that someinputs may be transformed into other inputs and outputs may be deduced from other outputs.Based on previous chapter we may complete this input-output matrix.

Let us first define some new outputs: (o7) will represent a sorting problem where the actionsof A have to be assigned to completely ordered categories defined by reference profiles, (o8)will represent a sorting problem where the actions of A have to be assigned to partially orderedcategories defined by reference profiles and (o9) will represent a sorting problem where theactions of A have to be assigned to completely non-ordered categories defined by referenceprofiles.

At first, one may notice that o7 and o9 are particular outputs of o8. This leads thus theoreticallyto the ρ−propagations couple (o8,o9) and (o8,o7) (see Fig.11.1).

As we have seen, Promethee has been used, under specific conditions, in order to address (o7).This permits us to define a possible ρ−propagation couple (o3,o7) as illustrated in Fig.11.1.Nevertheless, conditions must be imposed such that the order of the reference profiles is notaltered. In addition, partial ranking methods could be used to sort actions to completely orderedcategories: we have thus (o4,o7).

Let us remark remark that UTADIS can be considered an example of the ρ−propagation coupleof (o1,o7) since the output o1 represents a global evaluation (e.g. a score) for each action. Thereference profiles can be represented by constant thresholds (i.e. scores) and the actions are thussorted with respect to these thresholds.

Furtheremore, one might define the ρ−propagation couple (o4,o8) since partial ranking methodsmay be used in sorting problems where the categories are partially ordered (and defined bycentral profiles). However, although Promethee I and Electre III are partial ranking methods,

221

Conclusions

uu

u

u

u

u 6 6I

-o7 o3 o5

o2o1

iy

o4

u

o6

o8u

o9u

6

..

..

..

..

......

.. ??

?

Figure 11.1 — Representation of the ρ-relation.

they do not distinguish situations of indifference and incomparability. A direct propagationseems thus not straightforward. Some further conditions should be analyzed (and imposed) inorder to define this type of propagation.

As we see, some further research is thus needed in order to define possible propagations couplesto complete the input-output matrix.

In stead of analyzing an existing partially ranking method which could be used in sortingproblem, where the categories are partially ordered (i.e. the ρ−propagtion couple (o4,o8)), wehave extended the Electre-Tri method to this case (i.e. o8). This can be motivated by the fact, thatin the particular case where the categories are completely ordered we can compare the methodseasily. Moreover, distinguishing incomparability and indifference, is a crucial issue when thecategories are completely ordered. This excludes thus the use of Electre III and Promethee I.

Given this problem, the first step will be to extend Electre-Tri to the case where the categories aredefined by central profiles. A second step, will then to extend this method to partially ordered cat-egories. Moreover, we will analyze the possible ρ−propagtion couple (o8,o9) and verify (o8,o7).This will be the aim of the next part.

222

Part

Outranking based sorting methods

223

In this part, we propose a sorting method that allows us to treat assignment problems in whichthe categories, defined by central profiles, are partially ordered. The proposed approach totackle this type of problem, is similar as the one of F lowSortbut where we use a partial rankingmethod (based on outranking graphs) instead of a complete ranking method.As we will see, not all partial ranking methods are adpated to tackle this problems.

At first, we consider the particular subproblem of completely ordered categories and propose aslightly modified version of Electre-Tri (Chapter 12). The method, called Electre-Tri-Central,permits to define the categories by means of central profiles. Outranking degrees between theactions to be assigned and the reference profiles are computed. On that basis, we propose anoptimistic and pessimistic approach to assign an action. The assignment rules may be visualizedby means of outranking graphs.Furthermore, although PROAFTN, presented in Chapter 4.3.2, is a sorting method for com-pletely not-ordered categories, we analyze the use of PROAFTN in the particular case wherethe categories are completely ordered. In this context, we study the similarities and differencesbetween Electre-Tri-Central and PROAFTN. A relationship between these two methods isproposed.

In Chapter 13, we propose an extension of the assignment rules of Electre-Tri-Central forassigning actions to partially ordered categories. Some illustrative examples are given andtreated. The assignment rules, based on outranking graphs, are furthermore analyzed in theparticular subproblems where the categories are completely ordered and completely not-ordered.This part ends with some conclusions and further directions of research.

225

12 Electre-Tri-Central

In this chapter, we propose a slightly modified version of the Electre-Trimethod for the case where the completely ordered categories are defined a pri-ori by central profiles, instead of limiting profiles. This will be called Electre-Tri-Central.Besides, we point out that similarity based methods may not be suitable for thistype of problems although some relations exist with the proposed outranking-based approach.

12.1 Introduction

In this chapter we suppose that the categories are completely ordered and suppose that eachcategory is defined by one central profile instead of two limiting profiles (as in Electre-Tri,Section 4.6.1, that will be called in this section Electre-Tri-Limit).Let us remark that [Dias et al., 2008] have proposed an alternative work to tackle the sameproblem. We distinguish both approaches in Section 12.7.

We will note Ch a category defined by a unique central profile rh. Since the categories arecompletely ordered, we will impose analogous conditions as in Electre-Tri-Limit (see Section4.6.1.1): ∀h < l :

1. ∀ j ∈ G : g j(rl) ≤ g j(rh) and ∃ j ∈ G : g j(rl) < g j(rh) ("dominance relation between theprofiles")

2. rl ≺ rh1 ("preference relation between the profiles")

Moreover, we will impose an analogous conformity condition on the data: an action a ∈ A maynot be indifferent to more than one central profile. This condition can be seen as restrictive sincethe model should be developed regardless of the actions to be sorted. Nevertheless, we will seein which situation this condition plays a role and how it can be avoided.

1a b⇔ aSb and b¬Sa and where we note aSb⇔ S(a,b) ≥ λ

227

Electre-Tri-Central

12.2 Assignment rules

The actions to be sorted will be pair-wise compared to the central profiles of the categories bycomputing the outranking degrees.

In an ideal case, an action a will be assigned to a category Ch, if a and rh are considered asindifferent. In this case, we can notice that rh is the least good (worst) profile which outranksa or which is at least as good as a. Alternatively, rh is the best profile with which a is at least.These two considerations will be the basis of the optimistic and pessimistic assignment rules.

When comparing an action a to the central profiles, several situation may appear as in Electre-Tri-Limit [Roy and Bouyssou, 1993] (with k ∈N, 1≤ k ≤ j):

1. r1 a, r2 a, . . . , r j a,a r j+1,a r j+2, . . . ,a rK (I)

2. r1 a, r2 a, . . . , r j−1 a,aI r j,a r j+1, . . . ,a rK (II)

3. r1 a, r2 a, . . . , r j+1 a,aJ r j, . . . ,aJ r j+k−1,aJ r j+k,a r j+k+1, . . . ,a rK (III)

4. a r1, . . . ,a r j, . . . ,a rK (IV)

5. aJ r1, . . . ,aJ r j−1,a r j, . . . ,a rK (V)

6. aI r1,a r2, . . . ,a rK (VI)

7. r1 a, r2 a, . . . , r j a, rK a (VII)

8. r1 a, r2 a, . . . , r j−1 a, r jJ a, rKJ a (VIII)

9. r1 a, r2 a, . . . , r2 a, rKI a (IX)

When working with limiting profiles, we obtain only three different situations since theperformances of the actions to be sorted are “in between” the performances of the best and worstlimiting profiles (see [Roy and Bouyssou, 1993]). Situations (IV) to (IX) are thus not presentwhen working with limiting profiles.Nevertheless, we may obtain the same 3 situations when working with central profiles byimposing that the performances of the actions are in between two virtual ideal and nadirreference profiles.

Analogously as Electre-Tri-Limit, we will define the optimistic and pessimistic assignment ruleswhich will tackle these situations differently.

Assignment Rule 12.2.1. Optimistic version:An action a will be assigned to the category Ch, if rh is the "worst" central profile which is atleast as good as a. Formally:

228

12.2. Assignment rules

• Compare successively a and ri with i from K to 1 by computing S(ri,a).

• If rh is the first central profile such that rhSa, we then have that: Copt(a) = Ch

Let us first remark that this assignment rule, differs from the Electre-Tri-Limit optimisticassignment rule since the assignment rule is based on the outranking relation S instead of thepreference relation .This rule may be easily visualized by the of a so-called “optimistic S-graph” as given in Fig.12.1.In this graph, we denote the central profiles and an action a to be assigned, by circles and theoutranking relations between them by arcs. Moreover, given the conditions on the referenceprofiles, we may reduce this graph by representing only the transitive outranking relations(analogously proposed in Section 4.6.1.2).

If we consider the oriented path C1a from r1 to a: C1a ≡ r1 → . . .→ r j → a, we may write theassignment rule as follows:

Assignment Rule 12.2.2. Optimistic Path-version:If C1a @ : Copt(a) = C1

Else : Copt(a) = C j

Let us finally remark, that we have chosen to define the “optimistic path” starting from the bestprofile and not from the worst one (as in the optimistic version, A.R.12.2.1) in order to obtainthe same assignments when working with the outranking paths and the optimistic version.

i

i

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I :

II :

III :

˙rK r j r1

a

r1˙rK r j

a

˙rK r j

a

r1

r j+1

r j+1

r j+1

Figure 12.1 — The reduced "optimistic S-graph": xSy⇔ x→ y

In situation II of Fig.12.1, we have that action a is indifferent to reference profile r j which isthus the worst profile which is at least as good as a. Action a will thus be assigned according tothis rule to C j. Moreover, if C1a does not exists (in case IV and V), it means that r1¬Sa 2 andthus that a will be assigned to the best category: Copt(a) = C1.

2¬ stands for the logic negation operator.

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Electre-Tri-Central

Assignment Rule 12.2.3. Pessimistic version:An action a is assigned to the category Ch, if rh is the “best” central profile which is at least asgood as a. Formally:

• Compare successively a and ri with i from 1 to K by computing S(a, ri).

• If rh is the first central profile such that aSrh, we then have that: Cpess(a) = Ch

Let us first remark that the assignment rule differs slightly from the pessimistic one of Electre-Tri-Limit (see Section 4.6.1.2). Moreover, in this case, the reduced “pessimistic S-graph” (seeFig.12.2) may be used by defining the path CKa from rK to a: CKa ≡ rK 99K . . . 99K r j 99K a:

Assignment Rule 12.2.4. Pessimistic Path-version:If CKa @ : Cpess(a) = CK

Else: Cpess(a) = C j

i

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III :

˙rK r j r1

a

r1˙rK r j

a

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a

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r j+1

r j+1

Figure 12.2 — The reduced “pessimistic S-graph”: xSy⇔ x L99 y

In situation II in Fig.12.2, we have that action a is indifferent to reference profile r j. Action a isthus, at best, at least as good rh. Action a will thus be assigned according to this rule to C j. Letus remark that if CKa does not exists (in case VII and VIII), it means that a¬SrK and thus that awill be assigned to the worst category: Cpess(a) = CK .

In order to analyze the features of these assignment rules let us look at the assignment results inthe situations defined by [Roy and Bouyssou, 1993] (with k ∈N, 0≤ k ≤ j)3:

1. r1 a, r2 a, . . . , r j a,a r j+1,a r j+2, . . . ,a rK (I)We have Copt = C j and Cpess = C j+1

2. r1 a, r2 a, . . . , r j−1 a,aI r j,a r j+1, . . . ,a rK (II)We have Copt = C j and Cpess = C j

3When imposing the conformity condition, there are no assignment differences between A.R.12.2.1 andA.R.12.2.2 and between A.R.12.2.3 and A.R.12.2.4

230


3. r1 a, r2 a, . . . , r j−1 a,aJ r j, . . . ,aJ r j+k−1,aJ r j+k,a r j+k+1, . . . ,a rK (III)We have Copt = C j−1 and Cpess = C j+k+1

4. a r1, . . . ,a r j, . . . ,a rK (IV)We have Copt = C1 and Cpess = C1

5. aJ r1, . . . ,aJ r j−1,a r j, . . . ,a rK (V)We have Copt = C1 and Cpess = C j

6. aI r1,a r2, . . . ,a rK (VI)We have Copt = C1 and Cpess = C1

7. r1 a, r2 a, . . . , r j a, rK a (VII)We have Copt = CK and Cpess = CK

8. r1 a, r2 a, . . . , r j−1 a, r jJ a, rKJ a (VIII)We have Copt = C j−1 and Cpess = CK

9. r1 a, r2 a, . . . , r2 a, rKI a (IX)We have Copt = CK and Cpess = CK

The assignments are rather straightforward in situations I, IV, VII and II, VI, IX. Let us considerthe situation III with k = 1. The number k represents the number of central profiles which areincomparable with a. If k = 1, action a is incomparable to r j and being preferred by r1, . . . , r j−1

and preferred to r j+1, . . . , rK . Since it is incomparable r j, it will be assigned to the nearest betterand worse category according respectively to the optimistic and pessimistic assignment rule. Itwill thus be assigned to C j−1 and C j+1. This can be motivated by the fact that since it is notindifferent to r j, we may exclude category C j from the possible categories and thus r j from theset of reference profiles. If we eliminate r j, we are then in a situation I. Action a will thus beassigned to the nearest better and worse category. Situation V and VIII are analogous as situationIII.

We may remark that the the situations of indifference and incomparability are threaten dif-ferently. This constitutes the main difference between F lowSort (when working with centralprofiles) and Electre-Tri-Central. Electre-Tri-Central adopts a local approach and uses theuses the partial pre-order between the action a and the reference profiles. In F lowSort, if anaction is incomparable to exactly one reference profile (see Proposition 7.4), it will univocallybe assigned to that category (on the contrary of Electre-Tri-Central) since F lowSort uses acomplete pre-order for the assignment of a. This other point of view, may be motivated bythe fact that, although r j and a are not similar, they behave similarly with respect to the otherreference profiles (i.e. the "being preferred by" and "preferred to" character).

The decision maker may thus make a choice between different sorting methods on the basisof the assignments in such situation. The choice is thus analogous when choosing between a

231

Electre-Tri-Central

complete and partial ranking method. Let us remark that another difference between Electre-Tri-Central and F lowSort concerns the stability property as given in the next paragraph.

Previous assignment rules are rather intuitive and are completely analogous to those defined inElectre-Tri when working with limiting profiles. Moreover, the the outranking graphs enable adecision maker to understand them easily.

12.3 Properties of Electre-Tri-Central

1. General PropertiesThe assignment procedures being analogous to those of Electre-Tri-Limit, we have thus the sameproperties as proposed by [Yu, 1992a]) when working with central profiles (see Section 4.2 forthe definition of the properties):

• Every action a is assigned to one category according to one of the procedures. ("uniquenessproperty").

• The assignment of action a does not depend on the assignment of the other actions of A("independency property").

• When two actions are compared similarly to the reference profiles, they are affected to thesame categories ("strong homogeneity property").

• If a′ dominates a, then a′ will be affected to category which is at least as good as the categoryto which a will be assigned ("monotonicity property").

• The fusion or the separation of two neighboring categories do not affect the assignmentof the actions to other categories ("stability property"). This constitutes another differencewith F lowSort (see Proposition 7.7).

• Each central profile is univocally assigned to its corresponding categories with both assign-ment rules. ("conformity property").

The proof of these properties are analogous as those of Electre-Tri-Limit. Moreover, similarlyas in the case of Electre-Tri-Limit ([Yu, 1992a]), we have a relationship between the optimisticand pessimistic assignment results when working with central profiles and imposing the dataconformity condition. Formally, we have that:

Proposition 12.3.1.∀a ∈ A : Copt(a) ≥ Cpess(a)

232

12.3. Properties of Electre-Tri-Central

The proof is given in D.2. Let us remark that this property does not hold if the data conformitycondition is not satisfied. When a is indifferent to several successive central profiles, we may havethat Cpess(a) ≥ Copt(a). However, the assignment rules based on the optimistic and pessimisticoutranking graphs remain completely coherent. This is thus an advantage since it helps to avoidconfusion without imposing a restrictive and difficult applicable condition. On the basis of Fig.12.3, we may conclude that action a will be assigned respectively to C j and C j+1 whereas wewould have obtained Copt(a) = C j+1 and Cpess(a) = C j .

j

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*

jY

Figure 12.3 — The reduced "optimistic S-graph" and "pessimistic S-graph"when a is indif-ferent to more than one central profile.

2. Relationship between Electre-Tri-Limit and Electre-Tri-CentralIn order to analyze the difference between the Electre-Tri-Limit and Electre-Tri-Central, let usconsider the following situation. Consider that we note a set of central profiles R and a set oflimiting profiles R for defining the same categories. Suppose moreover, that the central profilesof R are defined as follows: ∀h = 1, . . . ,K

∀ j = 1, . . . ,q : g j(rh+1) ≤ g j(rh) ≤ g j(rh)

We suppose thus that the performances of rh are comprised between the performances of theneighboring limiting profiles rh+1 and rh on all criteria. Let us remark that rh is not necessarilythe mean or the median of two consecutive limiting profiles (see Fig.12.4). In such situation wehave that:

Proposition 12.3.2.|Copt(a)−Copt(a)| ≤ 1

|Cpess(a)−Cpess(a)| ≤ 1

We have thus that the difference in assignment is maximum one when defining categories eitherby limiting or central profiles. Previous proposition is illustrated in Fig.12.4 where we haverepresented the categories by limiting and central profiles which satisfy the aforementioned con-ditions. This proposition is thus completely analogous to Proposition 7.6 given for F lowSort.An illustrative example of this proposition is given in Example 12.1 and the proof in D.1.

233

Electre-Tri-Central

-

-

-

-

-

g1

g2

g3

g4

g5

r1r2r3r4 C1C2C3

15 200 10

rrr

rr r ra3

a2

r r

rr rrrr

rr rr r

rr rr r

r3 r2 r1

Figure 12.4 — Example of categories defined by central profiles (R = r1, r2, r3) and lim-iting profiles (R = r1,r2,r3,r4) and the actions a2 and a3.

Table 12.1 — Evaluation of the performances of the central reference profiles of R .

rm g1 g2 g3 g4 g5

r3 8 8 8 8 10r2 13 12 13 12 12r1 18 17 18 17 17

A further research interest might be the determination of the limiting profiles given the centralprofiles (and vice et versa) such that the actions are assigned to the same categories. This maygive some additional information to the decision maker.

12.4 Illustrative example

Example 12.1. As illustration let us reconsider Example 4.2 but where the three completelyordered categories are defined by central profiles instead of limiting profiles. The evaluationsof the reference profiles are given in Tab.12.1 and represented in Fig.12.5. We will consider thesame set of criteria with their corresponding parameters as well as the same set of actions Agiven in Tab.4.5, p.107.

The outranking degrees between the central profiles and the actions are given in Tab.12.2 aswell as the corresponding cases. This enables us to obtain the different assignments which aregiven in Tab.12.3. We have moreover given the assignments when using the PROAFTN ruleswhere in the first case we have fixed λI = 0.6 and such that the parameters verify Conditions12.5.1.

In Tab.12.3 we may for example consider action a2 and a3 where there is a difference inassignment like presented in Proposition 12.3.2 and illustrated in Fig.12.4. On the other hand,we observe that the other actions are assigned to the same category when working either with

234

12.4. Illustrative example

-

-

-

-

-

g1

g2

g3

g4

g5

C1C2C3

15 200 10

rrr

rr r ra3

a2

r r

rr rrrr

rr rr r

rr rr r

r3 r2 r1

Figure 12.5 — Representation of the performances of the central profiles r3, r2 and r1 ; thelimiting profiles r4, r3, r2, r1 and the actions a2 and a3.

Table 12.2 — Pair-wise comparisons between the actions and the central profiles r j,∀ j =1,2,3.

r3 r2 r1 case

S(a1, rm) 1 1 0II

S(rm,a1) 0 1 1S(a2, rm) 1 0.5 0

IIIS(rm,a2) 0 1 1S(a3, rm) 1 1 0

IS(rm,a3) 0 0 1S(a4, rm) 1 0.8 0

IIS(rm,a4) 0 0.6 1S(a5, rm) 1 0.6 0

IIS(rm,a5) 0 1 1S(a6, rm) 0.6 0 0

V IS(rm,a1) 0.8 1 1S(a7, rm) 1 0.6 0

IS(rm,a7) 0 0 1S(a8, rm) 1 0.6 0

V IIIS(rm,a8) 0 0 0.4S(a9, rm) 1 1 1

IVS(rm,a9) 0 0 0S(a10, rm) 0 0 0

V IIS(rm,a10) 1 1 1S(a11, rm) 0.8 0 0

IIS(rm,a11) 0 1 1

235

Electre-Tri-Central

Table 12.3 — Assignment of the actions according to the different procedures (with centraland limiting profiles).

ai Copt Cpess Copt Cpess CPROAλI=0.6 CPROA

max

a1 C2 C2 C2 C2 C2 C2

a2 C2 C3 C2 C2 /0 C2

a3 C1 C2 C1 C1 /0 C1,C2,C3

a4 C2 C2 C2 C2 C2 C2

a5 C2 C2 C2 C2 C2 C2

a6 C3 C3 C3 C3 C3 C3

a7 C1 C2 C1 C2 C2 C2

a8 C1 C2 C1 C2 /0 C1,C2,C3

a9 C1 C1 C1 C1 /0 C1,C2,C3

a10 C3 C3 C3 C3 /0 C1,C2,C3

a11 C2 C3 C2 C1 /0 C1,C2,C3

central or limiting profiles.

12.5 Relationship between Electre-Tri-Central and PROAFTN

Let us analyze the similarities and differences between similarity (indifference)-based andoutranking-based sorting methods when the categories are completely ordered. For that purpose,we will study the particular methods PROAFTN4 (see sections 4.3.1 and 4.3.2) and Electre-Tri-Central since they both use central reference profiles for defining categories. Moreover,they are based ares quite analogous concepts, namely similarity (indifference). The purpose ofthis section is to study to which extend a similarity (indifference)-based classification method,particularly PROAFTN, can be used in presence of completely ordered categories.

Let us first point out that the family of parameters of both methods are quite similar:

• A category Ch is defined in PROAFTN by a the profile rh and the following parameters:

– a profile is defined on each attribute/criterion by means of an interval which extremevalues are g j(rh) and g j(rh)

– ∀ j : p+j (rh) and p−j (rh): discrimination thresholds are used to compute the partial

concordance indexes (see 4.4.1)

• A category Ch is defined in Electre Tri-Central by the reference profile rh and the followingparameters:

4Let us remark that we have mentioned in Section 4.4.2.3 that PROAFTN, TRINOMFC and Indifference-filteringmethods are in certain circumstances equivalent.

236

12.5. Relationship between Electre-Tri-Central and PROAFTN

– ∀ j : g j(rh) represents the crisp values of the performances of the central profile rh oncriterion g j

– ∀ j : p j(rh) and q j(rh): the preference and indifference thresholds

Furthermore, both methods uses weights w j, ∀ j = 1, . . . ,q in the aggregation phase. Supposemoreover, that for any reference profiles, rh, ∀h = 1, . . . ,K, we define the following relationsbetween previous parameters:

Condition 12.5.1. Conditions on the parameters of PROAFTN and Electre-Tri-Central:

• ∀ j = 1, . . . ,q :g j(rh)+g j(rh)

2 = g j(rh)

• ∀ j = 1, . . . ,q :g j(rh)−g j(rh)

2 = q j(rh)

• ∀ j = 1, . . . ,q : p+j (rh) = p−j (rh) = p j(rh)

• λ = λI

• Let us moreover suppose that the criteria have the same weights wi in the aggregation phase,although they may have a different meaning.

Under previous hypothesis, if we consider that λ = λI , we have the following relations (p.73 in[Belacel, 2000b]): ∀h = 1, . . . ,K

cIj(a, rh) = cS

j(a, rh)∩ cSj(rh,a) = min(cS

j(a, rh),cSj(rh,a)) (12.1)

CI(a, rh) = [S(a, rh)∩S(rh,a)] = min(S(a, rh),S(rh,a)) (12.2)

When using the second assignment rule of PROAFTN, these relations lead us to the followingproposition: ∀a ∈ A

Proposition 12.5.1. If the parameters of Electre-Tri-Central and PROAFTN are such thatEq.12.2 is verified (i.e. verifying Conditions 12.5.1), we have ∀h 6= 1 and h 6= K:

PROAFTN assigns action a to the unique category Ch

m

Electre-Tri-Central optimistic and pessimistic affects the action a to the same category Ch.

This ensures us that when both optimistic and pessimistic assignments results are equal, we arein the case of similarity (indifference) between the action and the reference profile. This resultis thus completely in line with the issues pursued when defining ordered categories by centralprofiles while keeping the notion of preference relations. The proof is given in Appendix D.3.This is illustrated in Example 12.4.

237

Electre-Tri-Central

The assignment rules are thus such that if an action a is considered as similar to a central profilerh (with h 6= K, h 6= 1) with PROAFTN, a is indifferent to rh and vice et versa. Moreover,since the relationship between PROAFTN and the similarity based method TRINOMFC hasbeen presented in Section 4.4.2, similarity between two actions is obviously closely linked toindifference between two actions (when defining appropriate parameters).

We exclude the cases h 6= K, h 6= 1 since the best and the worst categories are defined as"open" in Electre-Tri-Central, on the contrary of PROAFTN. To overcome this, the reader mayintroduce some virtual reference profiles (r0 and rK+1) and the proposition is then valid for any"real" category.

Nevertheless, similarity or indifference-based approaches (e.g. TRINOMFC, PORAFTN) maydiffer in some simple situations from outranking-based approaches. Actually, in absence ofsimilarity or indifference between the action a and a reference profile r j, the assignment resultmay differ significantly. This is due to the fact that non-similarity or dissimilarity, which inoutranking-based approaches englobe preference, non-preference or incomparability, is treateddifferently.

As illustration, let us consider Fig.13.2 where the completely ordered categories are defined bycentral profiles evaluated on two criteria (which have to be maximized). The preference andindifference thresholds are equal to 0 (see Fig.13.2).

1. Situation I (Fig.13.2-left): rh+1 a, a rh−1 and aJ rh:CPROAFT N(a) = Ωa = C1, . . . ,CK or /0

Copt(a) = Ch−1 ; Cpess(a) = Ch+1

In this situation, the similarity index I(a, rh) is 0 (case of incomparability). According tothe first rule of PROAFTN, a will be assigned to all the categories (given the ex-aequo). Onthe other hand, according to the second rule of PROAFTN, a is assigned to no category.Let us remark that Electre-Tri-Central treats thus the incomparability case differently (seeSection 12.2).

2. Situation II (Fig.13.2-right): rh+1 a, a rh :CPROAFT N(a) = Ωa = C1, . . . ,CK or /0

Copt(a) = Ch−1 ; Cpess(a) = Ch

In this situation (case of preference), the similarity index between a and all the referenceprofiles is again equal to 0. Action a is assigned according to the rules of PROAFTN to noneor all of the categories. Let us remark that Electre-Tri-Central treats this case differently.

By these simple examples we may conclude that similarity (indifference)-based classification

238

12.6. Defining a category by several reference profiles.

6

- g1

g2

rh−1

rh+1

rh

?

?

? ia

6

- g1

g2

rh−1

rh+1

rh

?

?

?

ia

Figure 12.6 — Situation I and II with ∀i = 1,2 : pi = qi = 0,w1 = w2

methods may not be completely well-adapted for problems in which the categories are com-pletely ordered. In situation of non-indifference or non-similarity, there is thus obviously a lossof information when working with PROAFTN (this can be verified in Example 12.4).This is obviuously not surprising since it is due to the fact that the preference information isnot taken into account (anymore) in the used index (non preference-orientation dependencyproperty, see Section 4.3.2.3).

12.6 Defining a category by several reference profiles.

Suppose that each category is defined by a set of reference profiles which are all incomparable.Let us note rk

h the k-th reference profile of category Ch and where the categories are completelyordered: ChBCl ,∀h < l. Moreover, the profile respect the following conditions:

Condition 12.6.1. ∀m;∀k, l : rlmJ rk

m: all the reference profiles of a category Cm are incomparable.

Condition 12.6.2. ∀m < n;∀k, l;∀g j ∈ G : g j(rlm) ≥ g j(rk

n) and ∃ j ∈ G : g j(rlm) > g j(rk

n): theprofiles of categories of a better rank dominate all the profiles of a lower rank.

Condition 12.6.3. ∀m < n;∀k, l : rln ≺ rk

m: the profiles of categories of a better rank are preferredto all the profiles of lower rank.

Let us first remark that we will suppose that the best category and the worst category are definedby exactly one profile. If it is not the case, we may define the virtual ideal and nadir categoriesby the virtual ideal and nadir reference profiles as follows: ∀g j ∈ G :

g j(rI) = max∀m,k

[g j(rkm)] (12.3)

239

Electre-Tri-Central

g j(rN) = min∀m,k

[g j(rkm)] (12.4)

Moreover, we will note the best category C1 (r1) and the worst category CK (rK) respectively CI

(rI)and CN (rN). This is represented in Fig.12.7 where category C2 is defined by two profiles (r12

and r22) and C1 and C3 by one profile each.

r22 r1

2

rI

rN

ha R

/w

C2

C3

C1

/- r2

2 r12

rI

rN

7 I

ha C2

C3

C1

-

o

Figure 12.7 — Case IV= Copt(a) = C2 and Cpess(a) = C2

An action a of A will be pair-wise compared to all the reference profiles by computing theoutranking degrees S(a, rk

m) and S(rkm,a), ∀m,k. These outranking relations will be represented

in the the reduced optimistic and pessimistic "S-graph" and then exploited by the followingassignment rules.

Optimistic version: On the basis of the reduced optimistic "S-graph", we may define the op-timistic oriented path CIa as follows: CIa ≡ rI → . . .→ rk

m→ a. Based on CIa, we may define thefollowing optimistic assignment rule :

Assignment Rule 12.6.1. Optimistic version in case of completely ordered categories definedby several profiles:

If CIa @ : Copt(a) = CI

Else Copt(a) = Cm

Pessimistic version: On the basis of the reduced pessimistic "S-graph", we may define thepessimistic oriented path CIa as follows: CNa ≡ rN → . . .→ rk

m → a. Based on CNa, we maydefine the following pessimistic assignment rule:

Assignment Rule 12.6.2. Pessimistic version in case of completely ordered categories definedby several profiles::

If CNa @ : Copt(a) = CN

Else Cpess(a) = Cm

We may remark that if an action is indifferent to one or several profiles, it will, according to bothrules, be assigned to the corresponding category (categories) although it may be incomparable to

240

12.6. Defining a category by several reference profiles.

some other profiles of the same category. This can be justified as follows: as a measure has beentaken for a certain profile, we will apply the same measure for that action (by assigning it to thecategory of that profile) since they are considered as indifferent. Moreover, we do not considera global (or average) behavior of an action to be assigned, with respect to the central profiles ofone category5.

Example 12.2. As an example, let us reconsider previous example, Example 13.1 p.248, butwhere we are in presence of 3 completely ordered categories C1 = CN , C2 and C3 = CI . Thecategories C1

2 and C22 have been fused to one category, C2. The categories C1 and C3 will be

defined respectively by the profiles r11 and r1

3 whereas C2 will be defined by two incomparablecentral profiles r1

2 and r22 (see Fig.12.7).

We suppose that the preference parameters, performances of the reference profile and the actionsto be sorted are the same as in Example 13.1 (see Tab.13.1, Tab.4.5). We obtain the assignmentsgiven in Tab.12.4 on the basis of the outranking degrees (see Tab.13.2) and the outrankinggraphs (Fig.13.3-13.14).

Table 12.4 — Assignment results when defining C2 by two reference profiles r12 and r2

2 .

ai Copt Cpess

a1 C2 C2

a2 C2 C3

a3 C1 C2

a4 C2 C2

a5 C2 C2

a6 C3 C3

a7 C1 C3

a8 C1 C3

a9 C1 C1

a10 C3 C3

a11 C2 C3

Actions a4, a1, a6 are indifferent to one of the central profiles, their assignment is thus immediate.Action a9 (a10), is preferred to the best profile (being preferred by the worst profile) and willthus be assigned to best (worst) category. Action a7 is incomparable to all the profiles of C2 andis for that reason assigned respectively to the best and worst category. Moreover, action a8 isincomparable to all the profiles of C1 and C2, which leads to the given assignment.On the other hand, action a2 (a3) is preferred to the worst profile, but being preferred by (being

5This is not the case when using for instance a K-nearest neighbor procedure, since this procedure uses a votingprocedure (based on a global behavior).

241

Electre-Tri-Central

preferred to) all the profiles of category C2. Action a2 (a3) is thus assigned respectively tocategory C2 (C1) and C3 (C2).

Since action a11 is being preferred by to r22 and incomparable to r1

2, and since its being preferredto the worst profile, it is assigned respectively to C2 and C3.

One may remark that the results are similar to the case where C2 is defined by one centralprofile r2 (see Example 12.1, p. 234, Electre-Tri-Central) ; except for the actions a7 and a8. Thedifference lies in the fact that a7 and a8 are preferred to the central profile r2 and incomparableto r2

2 and r12.

r22

w

r12

rI

rN

haC2

C3

C1

/

/

Rr2

2 r12

rI

rN

7 I

o haC2

C3

C1

7

Figure 12.8 — Case XII: Copt(a) = C12 and Cpess(a) = C1

3

12.7 Comparison with ELECTRE-TRI-C

Simultaneously to Electre-Tri-Central, a sorting method, called Electre-Tri-C [Dias et al., 2008],has been proposed, as an alternative to Electre-Tri-Limit, when the categories are defined bycentral profiles. Electre-Tri-C uses outranking degrees as well. These outranking degrees areused in a descending and an ascending assignment rule by computing the following terms:ξ+h (a,λ) = S(a, rh)−λ and ξ

−h (a,λ) = S(rh,a)−λ (∀h = 1, . . . ,K).

Although both approaches use outranking degrees, both methods are significantly different[Dias et al., 2008] as will be illustrated in the next paragraphs.

The main differences between the two methods reside in the way of how the outranking degreesare used in the assignment rules. In Electre-Tri-Central, the outranking degrees are computedfor the determination of the I ,J or -relation between the action to be sorted and the profiles.On the basis of these, several assignment-situations have been proposed (see Section 12.2).In Electre-Tri-C, the differences between λ and the outranking degrees (between the referenceprofiles and an action) is computed: ξ

+h (a,λ), ξ

−h (a,λ),∀h = 1, . . . ,K. These values are com-

242

12.8. Conclusions

pared to each other for the assignment. The I ,J or -relation are thus not (explicitly) used forthe assignment.This has as consequence that it is not possible to define similar situations when comparingan action to the reference profiles as done for Electre-Tri-Limit and Electre-Tri-Central (seesections 4.6.1.2 and 12.2).

Moreover, the results may differ as can be illustrated by means of the example given p.25in [Dias et al., 2008]. Action a5 which is indifferent to r4 (or b1 in the original notation), isassigned to C3 and C4 (C2 and C1 in the original notation) whereas using Electre-Tri-Centralleads unambigously to the same category: Copt = Cpess = C4.However, this permits to assign univocally an action to a category although this action and thecorresponding profile are incomparable, which is not possible with Electre-Tri-Central6. This isa direct consequence of using the outranking values instead of outranking relations.

Another consequence of the assignment rules is that the proposed homogeneity property (p.11in [Dias et al., 2008]) is the weak homogeneity property. Electre-Tri-Central respects the stronghomogeneity property. The consequence of this might be illustrated by the following situationwhere two actions ai and a j compare themselves with respect to the reference actions in anidentical manner, in terms of I , J , relations, but are assigned differently. (e.g. a7 and a8 ; a3

and a4 p.25 in [Dias et al., 2008]) since the outranking degrees are different.

Finally, let us remark that there exists no general relation between the descending and theascending assignment as in Proposition 12.3.1.

12.8 Conclusions

In this chapter, we proposed the sorting method Electre-Tri-Central, inspired by the Electre-Trimethodology, for sorting problems where the categories are completely ordered and defined bycentral profiles. The assignment rules are similar to those of Electre-Tri-Limit which permit aneasy understanding. Moreover, the outranking graphs give a graphical representation of theserules and permit to define each category by several central profiles.

First, we showed that the assignments results are linked to those of Electre-Tri-Limit and theproperties of these sorting methods are analogous.

We have then compared the method to the similarity-based classification method PROAFTN. Wehave proven that in case of indifference or similarity between an action and a reference profiles,

6But it is possible with F lowSort, see Section 7.4.

243

Electre-Tri-Central

both methods give the same results. Nevertheless, as illustrated in some basic examples, usingoutranking relations permits to refine the assignments in case of non-similarity. PROAFTNseems not to be well-adapted for tackling problems where the categories are completely ordered.

Furthermore, we have clearly distinguished Electre-Tri-Central from the Electre-Tri-C.

A natural extension of this Electre-Tri-Central model is the adaptation of the assignment ruleswhen the categories are partially ordered. This will be proposed in Chapter 13.

Some effort should be spent on facilitating the determination of the parameters of the modelsuch as for instance the central profiles or the thresholds. Existing elicitation methods shouldtherefore be analyzed and if necessary, adapted. Finally, the determination of central profiles bymeans of limiting profiles (and vice-versa) may help the decision maker to apprehend the use ofcentral profiles.

244

13 Partially ordered categories

In this chapter we propose assignment rules for sorting problems where thecategories may be partially ordered. The procedures are based on the outrank-ing graphs representing the outranking relations between an action to be sortedand the reference profiles.These new assignment rules may also be applied in the particular sub-problemswhere the categories are completely ordered or completely not-ordered (in-comparable categories). This is illustrated by some concrete examples.

13.1 Introduction

In Section 4.1 we have differentiated sorting problems from classification problems. We havemoreover pointed out that neither Electre-Tri is suited for classification problems nor PROAFTNfor sorting problems (see Section 12.5 ). Nevertheless, these two problems may be consideredas sub-problems of a more general problem where there is a partial order on the categories (orclasses).In this chapter we define assignment rules (which are an extension of the ones defined inElectre-Tri-Central) for this more general problem. As we will see, these new assignment rulesmay also be applied in the particular sub-problems where the categories are completely orderedor completely not-ordered (incomparable categories).

Let us first give some examples where there might be a partial order on the categories.Suppose that the human resource department of a computer enterprize wants to evaluate theperformances of the employees or their evolution in order to attribute promotions or to reveal(predict) possible "managers". Therefore, the human resources defines four different categories:the less good people, the engineers, the technical salespeople and the managers. Obviously, theclass of the less good people is preferred by all the other categories. On the contrary, the classof managers may be seen as the "best" category. On the other hand, no preference between theengineers and the technical salespeople may be expressed: these are considered as incomparable.We are thus in presence of partially ordered categories. Let us remark, that the purpose is not

245

Partially ordered categories

to compare the employees between them (i.e. to rank them respectively) but to compare themindependently with respect but to some norms.

Moreover, suppose that a school admits new students on the basis of an application file whichtakes into account aspects such as their motivation to enter the school and their educationalbackground. The students may be sorted in order to admit their application or to give somerecommendation. At first we may define a category of students which are considered asinadmissible since their previous school results or background skills are not good enough. Somestudents might have a strong background but whose choice (motivation) to enter the school isnot sure. On the other hand, we may have students whose choice is unambiguous but whosebackground seems to be too poor. Finally, there might be a category of students whose schoolresults are good enough and whose choice to enter the school is unambiguous. Obviously, theschool direction might express a preference relation such that the first category of studentsis the least preferred one and the last category the most preferred. Moreover, no preferencemay be expressed between the two "medium" classes but differentiating the students might beinteresting to propose incentives to the indecisive ones or complementary education to the oneswith an "average" background.

Given a partial preference structure on the categories, we define these by central profiles sinceits seems difficult (or impossible) to define them by limiting profiles. We note rk

m the centralprofile of the kth category of rank m, noted as Ck

m, and where Ckn is preferred to Cl

m, ∀k, l, ifm < n. This is illustrated in Fig. 13.1.

CN

. . . . . .

CI

C1m

rank 1

rank m

rank N

...

...

Clm

. . .

. . .> 6

k6

Figure 13.1 — Representation of partially ordered categories.

We suppose that there is only one category of the best rank, CI , and one of the worst rank, CN ,which reference profiles are noted rI and rN . If this is not the case, we may always define theideal and nadir (virtual) categories, CI and CN , defined by the (virtual) reference profiles rI andrN defined in Eq.12.3 and Eq.12.4.

246


r13

r11

r22

/

w

r12

R

/r1

3

r11

r22 r1

2

7

I

o

Figure 13.2 — Example of partially ordered reference profiles in the optimistic and pes-simistic reduced "S-graph" where r1

1 = rI and r11 = rN

We moreover suppose that no incomparability relation exists between categories of differ-ent ranks. This ensures us that the ranks of the categories, starting from the best category tothe worst category, are the same as the ranks starting from the worst category to the best category.

Given this partial order on the categories, we may impose some conditions on the referenceprofiles:

Condition 13.1.1. ∀m;∀k, l : rlmJ rk

m: all the reference profiles of categories of the same rank areincomparable.

Condition 13.1.2. ∀m < n;∀k, l;∀g j ∈ G : g j(rlm) ≥ g j(rk

n) and ∃ j ∈ G : g j(rlm) > g j(rk

n): theprofiles of categories of a better rank dominate all the profiles of a lower rank on each criterion.

Condition 13.1.3. ∀m < n;∀k, l : rln ≺ rk

m: the profiles of categories of a better rank are preferredto all the profiles of a lower rank.

These outranking relations between the reference profiles may be represented by the reducedoptimistic and pessimistic "S-graph" as for instance in Fig.13.2.

13.2 Assignment rules

An action a of A will be pair-wise compared to the reference profiles by computing the out-ranking degrees S(a, rk

m) and S(rkm,a), ∀m,k. These outranking relations will be represented in

the reduced optimistic and pessimistic "S-graph", with respect to the reference profiles, as forexample in Fig.13.3. These graphs will be exploited to determine the assignment of an action.

Pessimistic version: On the basis of the reduced optimistic "S-graph", we may define theoptimistic oriented path CIa as follows: CIa ≡ rI → . . .→ rk

m→ a. Based on CIa, we may definethe following optimistic assignment rule such that:

247


r22

w

r12

R

/

ha rI

rN

r22 r1

2

rI

rN

ha7

7 I

7o

Figure 13.3 — Case I: Example of the optimistic and pessimistic reduced "S-graph" with a:Copt(a) = C1

1 and Cpess(a) = C22

Assignment Rule 13.2.1. Optimistic version in case of partially ordered categories:If CIa @ : Copt(a) = CI

Else Copt(a) = Ckm

From Fig.13.3 we may deduce for example that CIa ≡ rI → a and thus that Copt(a) = CI = C13 .

Pessimistic version: On the basis of the reduced pessimistic "S-graph", we may define thepessimistic oriented path CIa as follows: CNa ≡ rN → . . .→ rk

m → a. Based on CNa, we maydefine the following pessimistic assignment rule:

Assignment Rule 13.2.2. Optimistic version in case of partially ordered categories:If CNa @ : Copt(a) = CN

Else Cpess(a) = Ckm

From Fig.13.3 we may deduce that CNa ≡ rN → r22→ a and that Cpess(a) = C2

2 .

Let us remark that it might happen that more than one optimistic and pessimistic paths exist.In this case, an action a may be assigned, according to a procedure, to several categories. Theprocedures do not respect the property of uniqueness. This will be illustrated in Example 13.1.Besides, the properties of neutrality, independency, strong homogeneity and stability are verified.Nevertheless, as in Electre-Tri, the pairwise assignment consistency may not be fulfilled.

13.3 Illustrative example

Example 13.1. As an example, let us reconsider the examples 4.1 and 12.4. We suppose acombination of both problems. In this case, suppose that the human resource department definesfour different categories: the less good people, the engineers, the technical salespeople and themanagers. Obviously, the class of the less good people is preferred by all the other categories.On the contrary, the class of managers may be seen as the "best" category. No preference

248


Table 13.1 — Evaluation of the performances of the central reference profiles.

rmn g1 g2 g3 g4 g5

r13 8 8 8 8 10

r12 10 10 15 15 12

r22 15 15 10 10 12

r11 18 17 18 17 17

-

-

-

-

-

g1

g2

g3

g4

g5r1

1

r22

r13

r12

8

8

8

810 12

10

10

10

10 15

15

15

15 18

17

18

17

17

Figure 13.4 — Representation of the performances of the central profiles r11 , r1

2 , r22 and r1

3 .

between the engineers and the technical salespeople may be expressed: these are considered asincomparable. We are thus in presence of partially ordered categories: C1

3 = CI; C12 and C2

2 andfinally C1

1 = CN .

The categories will be defined by central reference profiles which are evaluated according to thesame five criteria (which have to be maximized). The performances of the reference profiles aregiven in Tab.13.1 and represented in Fig.13.4, p.249. The parameters associated to the criteriaare the same: ∀g j ∈ G : q j = 1; p j = 2,v j = 4 and w j = 0.2. The λ-threshold is fixed at 0.6. Theevaluations of the actions are given in Tab.4.5, p. 107.

The actions to be sorted are pair-wise compared with the reference profiles by computing theoutranking degrees which are given in Tab.13.2. On the basis of the outranking degrees andthe outranking graphs, the optimistic and pessimistic assignment rules sort the actions into thecorresponding categories. The results are given in Tab.13.3.

Fig.13.3 - Fig.13.15 represent several possible situations when comparing a to the partiallyordered reference profiles.In case X, Fig.13.13 (case VII, Fig.13.10), we have that a is indifferent to exactly one profile r2

2(rN). It will thus be assigned, according to both rules, to C2

2 (CN = C3).In case V, Fig.13.7, a is indifferent to two profiles r2

1 and r22 and will thus be assigned to both

categories C12 and C2

2 . One may notice that there exist two different optimistic and pessimisticpaths, which lead to the assignment to two categories according to each rule.

249


Table 13.2 — Pair-wise comparisons between the actions and the reference profiles.

r13 r1

2 r22 r1

1 Partial Electre-III ranking

S(a1, rmn ) 1 0.6 0.6 0

r11 r2

2, r12,a1 r1

3S(rmn ,a1) 0 0.6 0.6 1

S(a2, rmn ) 1 0 0.5 0

r11 r2

2 r12 a2 r1

3S(rmn ,a2) 0 0.8 1 1

S(a3, rmn ) 1 0.8 1 0

r11 a3 r1

2 r22 r1

3S(rmn ,a3) 0 0 0 1

S(a4, rmn ) 1 0 1 0

r11 a4 r1

2 r22 r1

3S(rmn ,a4) 0 0 0.8 1

S(a5, rmn ) 1 0 0.6 0

r11 r2

2 r12 a5 r1

3S(rmn ,a5) 0 0.6 1 1

S(a6, rmn ) 0.6 0 0 0

r11 r1

2, r22 r1

3 a6S(rmn ,a6) 0.8 1 1 1

S(a7, rmn ) 1 0 0 0

r11 r2

2, r12,a7 r1

3S(rmn ,a7) 0 0 0 1

S(a8, rmn ) 1 0 0 0

r11 a8 r2

2, r12 r1

3S(rmn ,a8) 0 0 0 0.231

S(a9, rmn ) 1 1 1 1

a9 r11 r1

2, r22 r1

3S(rmn ,a9) 0 0 0 0

S(a10, rmn ) 0 0 0 0

r11 r1

2, r22 r1

3 a10S(rmn ,a10) 1 1 1 1

S(a11, rmn ) 0.8 0 0 0

r11 r2

2 r12 a11 r1

3S(rmn ,a11) 0 0.5 1 1

Table 13.3 — Classification result of the actions according to respectively the Optimistic andPessimistic version as well as the two PROAFTN assignment rules.

ai Copt Cpess case CPRO−I CPRO−II

a1 C22 , C1

2 C22 , C2

2 V C22 , C1

2 C22 , C1

2a2 C2

2 ,C12 C1

3 III /0 C22

a3 C11 C2

2 ,C12 XI /0 C

a4 C22 C2

2 IV C22 C2

2a5 C1

2 C22 XI /0 C2

2a6 C1

3 C13 VII C1

3 C13

a7 C11 C1

3 II /0 C

a8 C11 C1

3 VIII /0 C

a9 C11 C1

1 IX /0 C

a10 C13 C1

3 X /0 C

a11 C22 C1

3 XII /0 C

250


In case VIII, Fig.13.11, a is outranking rN , but it is incomparable to all the other referenceprofiles. For this reason, it will be assigned to all the categories.In situations IX, Fig.13.12 (X, Fig. 13.13), a is preferred to the best profile (being preferred bythe worst profile) and will thus be assigned to best (worst) category.In Fig.13.15, we have that action a is incomparable to r2

2 but outranked by r12 and outranking

rN . It will thus be assigned, according to the pessimistic and optimistic assignment rule, to CN

and C12 .

In the cases III, V and XI more than one optimistic and pessimistic paths exist. This simplymeans that action a may be assigned, according to a procedure, to several categories.

On the basis of Eq.4.8, we may use the second assignment rule of PROAFTN and fix λI = 0.6.We have that CPRO(a1) = C2

2 ∪ C12 , CPRO(a4) = C2

2 , CPRO(a6) = C13 . These last assignments

are similar to the one obtained by using the outranking graphs. Nevertheless, the actionsa2,a3,a5,a7,a8,a9,a10,a11 will be assigned to none category (on the contrary when workingwith the outranking graphs, see Tab.13.3).

If we use the first rule of PROAFTN, the actions a3,a5,a7,a9,a10,a11 will be assigned to allthe categories (which is clearly not the case when working with the outranking graphs, seeTab.13.3). Action a2 will in this case be assigned univocally to C2

2 whereas according to theoutranking graphs to C2

2 ,C12 and C1

3 . The remaining actions (i.e. a1, a4, a6) are assigned tothe same categories. From these assignments, we might conclude that there might be a loss ofinformation when using indifference-based sorting methods. These results are given in Tab.13.3where C = C1

1 , . . . ,C13.

Let us finally analyze the results if we would assign the actions on the basis of their rank in thepartial ranking given by Electre III, which is given in Tab.13.2. Only the actions a6,a9 and a10

would lead to the same assignment.Moreover, although a4 is indifferent to r2

2, it is ranked between r11 and r1

2 which should lead to acompletely different assignment. The same can be stated for action a5 indifferent to r2

2 and r12.

In the partial rankings, a1 and a7 are ranked similarly, although the outranking relationsbetween the actions and the reference profiles are different (and leads, according to the out-ranking graphs, to different assignments). As pointed out in the Example 3.1 of Chapter 3.3.1,using Electre III might not always be appropriated since the indifference and incomparabilityrelations are not differentiated.

251


r22 r1

2

rI

rN

w

R

/

/ ha?

?

r22 r1

2

rI

rN

ha7

o

6I

6

Figure 13.5 — Case II: Copt(a) = C11 and Cpess(a) = C1

3

r22 r1

2

rI

rN

ha?

/ R

9q

r22 r1

2

rI

rN

ha7

Y 1

I

6

Figure 13.6 — Case III: Copt(a) = C22 ∪C1


r22 r1

2

rI

rN

ha/-

R

/w

r22 r1

2

rI

rN

7 I

hao

-

Figure 13.7 — Case IV= Copt(a) = C22 and Cpess(a) = C2

2

r22 r1

2

rI

rN

ha/ w

w /

- - r22 r1

2

rI

rN

I

ha7

o

- -

Figure 13.8 — Case V: Copt(a) = C22 ∪C1

2 and Cpess(a) = C22 ∪C1

2

252


r22

w

r12

/

rI

rN

ha/ R- - r2

2 r12

rI

rN

ha7

o

I

7

-

Figure 13.9 — Case VI: Copt(a) = C22 ∪C1


r22 r1

2

rI

rN ha-

/

w

R

/

r22 r1

2

rI

rN

7

o

I

ha-

Figure 13.10 — Case VII: Copt(a) = C13 and Cpess(a) = C1

3

r22

w

r12

R

/

rI

rN

/

ha

r22 r1

2

rI

rN

7

o

I

ha

Figure 13.11 — Case VIII: Copt(a) = C11 and Cpess(a) = C1

3

253


r22

w

r12

R

/

rI

rN

/

ha?

r22 r1

2

rI

rN

7

o

I

6

ha

Figure 13.12 — Case IX: Copt(a) = C11 and Cpess(a) = C1

1

r12

rI

harN

r22

?

w

R

/

/

h

r22 r1

2

rI

rN

7

o

I

a

6

Figure 13.13 — Case X: Copt(a) = C13 and Cpess(a) = C1

3

r22

w

r12

/rN

rI

ha s

?

r22 r1

2

rN

rI

ha6

1 i

o 7

Figure 13.14 — Case XI: Copt(a) = C11 and Cpess(a) = C1

2 and C22

254

13.4. Particular subproblems

r22

/

/

R

w

r12

rI

rN

har2

2 r12

rI

rN

7 I

hao

7

Figure 13.15 — Case XII: Copt(a) = C12 and Cpess(a) = C1

3

13.4 Particular subproblems

Let us analyze the previous assignment rules in the two particular subproblems of completelyordered and completely not-ordered or incomparable classes.

13.4.1 Completely non-ordered categories

Let us suppose that we are in presence of completely non-ordered categories which are definedby one central profile. When there is no order on the classes, we may define the virtual idealand nadir reference profiles in order to apply the assignment rules 13.2.1 and 13.2.2 for these"partially" ordered categories (see Fig.13.16).

Actions, considered as indifferent to a reference profile are assigned to that correspondingcategory. In this case, we obtain similar assignments as when using a method based on asimilarity or indifference index such as PROAFTN for instance (cf. Eq.4.8).

Nevertheless, an action which is not indifferent or similar to any reference profile are neverthe-less assigned. This may not be the case for the PROAFTN method for example, as pointed outin Section 4.3.2 and in Example 13.1. Dissimilarity may due to different reasons. Two dissimilaractions may be incomparable or the one action preferred to the other one. The differencebetween this two situations may be easily detected by means of the outranking graphs. Therelative position of an action with respect to the reference profiles, will be exploited in the partialranking to refine the result when there is no similarity or indifference.

In Example 13.1, let us consider that rI and rN are virtual reference profiles which do notrepresent a particular category. They are thus introduced for the assignment procedure.Consider moreover the simplified example represented in Fig.13.16 where two incomparablecategories are defined by two central profiles (when working with 2 criteria which have to be

255


maximized). The circles define conceptually all the actions considered as indifferent or similarto the corresponding central profiles. Similarity based methods do not distinguish the cases ofthe actions b1,b2,b3,b4. These actions are considered as not similar and will thus be assigned tonone category (or to all categories) when using PROAFTN.

-

6

g1

g2

?r2

?r1

C1

C2

?rI

?rN

b1

b2

b4

b3

Figure 13.16 — Representation of the ideal and nadir reference profile in case of nominalclassification problems.

In case I of the example of Section 13.1 (Fig.13.3), action a will be assigned to class C22 and CI .

The same results may be obtained for b3 in Fig.13.16. This should be interpreted as follows:action a (b3) actually belongs to the “upper” (better) part of category C2

2 since it is incomparableto the central profile of C1

2 (C1) and preferred to the central profile of C22 (C2). On the contrary,

in situation XII, action a (b2) should belong to the “lower” (worse) part of category C12 (C1). It

is assigned to C12 (C1) and CN since it is incomparable to r2

2 (r1) and being preferred by r12 (r2).

We can thus remark that the use of the outranking degrees may be useful in situations wherethere is no strict similarity or indifference. It permits thus to refine the assignment whereasdissimilarity includes different situations which may be distinguished on the basis of outrankingrelations. This is due to the fact that the criteria dependency property (Property 4.12) is verifiedsince the preferences are taken into account.

Let us remark, that the actions b1,b2,b3,b4 may point out the existence (or the need for) of newcategories.

A further research direction is the design of assignment rules when each category is defined byseveral profiles in the case that the categories are completely not-ordered.

256

13.4. Particular subproblems

13.4.2 Completely ordered categories

When the categories are completely ordered and defined by one central profile, the assignmentrules are (by construction of the rules) completely analogous to the graphical assignmentrules defined in Electre-Tri-Central. The assignment results will thus be similar to those ofElectre-Tri-Central 1.

1As mentioned in Section 12.3, this is the case when the data-conformity condition is respected

257

14 Conclusion Part III

In this part, we developed a new multicriteria sorting method inspired by the Electre-Trimethodology for assignment problems where the categories are partially ordered and defined bycentral profiles. Our approach is based on the use of a partial ranking method, using outrankingrelations. The assignment rules are directly based on the graphical representation of theseoutranking relations.In the particular case where the categories are completely ordered, we showed that the resultsare similar to the results obtained with Electre-Tri-Central and thus closely linked to Electre-Tri-Limit. This method can thus be used ("propagated") in these particular subproblems.On the other hand, when there is no order on the classes, we obtain the same results as thesimilarity or indifference based method PROAFTN, when the assignment is unambiguous(case of indifference or similarity between an action and a reference profile). Nevertheless, weillustrated that the proposed method permits moreover to precisely point out the issues involvedwhen the assignments are not straightforward. The graphical representations contribute evenmore to an easy comprehension of the assignment results.

As illustrated in Example 13.1, not all partial ranking methods may be used to handle thistype of problems. In particular, since Promethee and Electre III do not differentiate the case ofindifference from the case of incomparability, it seems delicate to use these methods in problemswhere the categories are partially ordered. This has been the basis of developing assignmentrules based on solely outranking graphs.

Some effort should be spent on facilitating the determination of the parameters of the modelsuch as for instance the central profiles or the thresholds. As already mentioned, some previousmethods may be adapted to this kind of problems.

More generally, a natural extension of this model should be the adaptation of the assignmentrules when the partially ordered categories are defined by several central profiles.

259

Conclusion

At the beginning of this thesis we asked ourselves the questions whether it makes sense to usea (complete or partial) ranking method in a sorting problem. What conditions are necessary inorder to respect desirable sorting properties ? Does this approach present some advantages ?

Following these motivations, we first focused on gaining a better understanding of the existingsorting methods. We focalized our study on multicriteria sorting methods using referenceprofiles for the definition of the categories. On the basis of their model and the properties theypresent, we have compared some of them in order to understand their approach. This has ledto point out differences and similarities between the existing approaches. Moreover, it haspermitted to establish that existing classification methods might not be appropriate for sortingproblems where the categories are partially ordered.

Based on this, we tackled first the sorting problem where the categories where completelyordered. To answer our main concern, we proposed a sorting method, the F lowSort method,based on the complete ranking method Promethee. In F lowSort we compare an action to besorted to solely the reference profiles. These comparisons lead to a complete ranking of thereference profiles and the action. The assignment of the action is then based on its relativeposition in the ranking.Since Promethee might suffer from rank reversals phenomena, we have imposed the necessaryconditions in order to avoid rank reversals between the profiles. This ensures us that thereference profiles define completely ordered categories. Furthermore, we analyzed the propertiesof F lowSort and compared the assignments to the one obtained with Electre-Tri and UTADIS.Although the results are coherent, the approaches are different. The main difference lies in thefact that a complete ranking method aggregates pairwise comparisons to a global comparison.This implies that the assignment of an action to be sorted, will be based on a global orsimultaneous comparison with respect to the reference profiles.We concluded this part, by asking ourselves wether this approach can be propagated to newsorting problems.

In the last part of this work, we were mainly concerned by considering a more general problem

261

Conclusion

where the categories might be partially ordered. In this context, we have illustrated that thepartial ranking methods Electre III and Promethee I are not appropriate since they do notdistinguish the case of pairwise indifference and incomparability between actions. This haspushed us to develop a sorting method based on the relative position of an action with respectto the profiles in an outranking graph. We have also shown that this method can be used("propagated") in the particular subproblems where the categories are completely ordered orincomparable. This has been motivated by the fact that our approach give, in the particularsubproblems, identical results to the results obtained with specific sorting methods. Moreover,when there is no order on the classes, we illustrated that such a method allows in some cases toprecisely refine the assignments and to point out the issues involved.

All along this thesis, we have given some concrete examples in order to illustrate our proposalsand the differences between the approaches.

However, there are still many open questions and problems which deserve further attention. Wedid not concentrate our effort on developing all the aspects of the sorting problem. The prefer-ence parameters and profiles were supposed to be determined (precisely or not). Nevertheless,in real applications, the decision maker might have some problems with their determination. Inthis context, it seems necessary to develop elicitation models (using for instance assignment ex-amples) in order to confirm this approach. In this context, some previous works could be adapted.

Some research should deal with the study of other ranking methods in the typical sorting context.One might for instance analyze Electre III under the "Electre-Tri conditions". Their comparisonshould certainly bring light to some new facets of Electre III and help in its understanding. In thesame context, one may used the Rough-Set based ranking method in case the decision maker isable to define reference profiles. This should further more help to differentiate Electre-Tri fromthe Rough Sets.

Finally, the input and output of multicriteria methods can be further investigated (by studyingfor instance the choice problem by means of ranking methods or vice-versa) and eventuallypropagated in order to tackle new decision making situations.

262

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273

A Proof of Propositions 5.2.1 -7.7.4

A.1 Proof of proposition 5.2.1:

In this proof we will denote by R the set of limiting or central profiles and rh the limiting orcentral profile of category Ch or Ch. In other words, we will not distinguish these two differentcases.

Conditions 5.2.1 requires that the reference profiles dominate each other:

g j(rh) ≥ g j(rh+1) ∀ j ∈ 1, . . . ,q,∀h ∈ 1, . . . , |R |−1.

Combining this with Condition 5.2.5, we obtain that:

∀r j ∈ R , π(rh,r j) ≥ π(rh+1,r j).

Condition 5.2.6 requires that π(rh,rh+1) > 0, whereas Condition 5.2.4 tells us thatπ(rh+1,rh+1) = 0. That is why we can conlude that:

π(rh,rh+1) > π(rh+1,rh+1).

These last two observations tell us that:

φ+R (rh) = ∑

r j∈Rπ(rh,r j) > ∑

r j∈Rπ(rh+1,r j) = φ

+R (rh+1).

This completes the first part of the proof.

In a similar way, the fact that the reference profiles dominate each other

g j(rh) ≥ g j(rh+1) ∀ j ∈ 1, . . . ,q,∀h ∈ 1, . . . , |R |−1,

combined with Condition 5.2.5 implies that:

∀ai ∈ A , π(rh,ai) ≥ π(rh+1,ai).

Since:

275

Proof of Propositions 5.2.1 - 7.7.4

• φ+R i

(rh) = φ+R (rh)+ π(rh,ai)

• φ+R i

(rh+1) = φ+R (rh+1)+ π(rh+1,ai)

• φ+R (rh) > φ

+R (rh+1) (see first part of the proof),

we can conclude that:φ

+R i

(rh) > φ+R i

(rh+1).

This completes the second part of the proof. The proofs for the incoming and net flows aresimilar.



We suppose that Cφ−(ai) = Ck and Cφ+(ai) = Cl which, by constuction of the assignment rules,implies that:

φ−R i

(rk) ≤ φ−R i

(ai) < φ−R i

(rk+1) (A.1)

φ+R i

(rl) ≥ φ+R i

(ai) > φ+R i

(rl+1) (A.2)

By substracting A.1 from A.2, we obtain:

φ+R i

(rl)−φ−R i

(rk) ≥ φ+R i

(ai)−φ−R i

(ai) > φ+R i

(rl+1)−φ−R i

(rk+1) (A.3)

If k=l: φ+R i

(rk)−φ−R i

(rk)≥ φ+R i

(ai)−φ−R i

(ai) > φ+R i

(rk+1)−φ−R i

(rk+1) In this case, the net flowrule assigns alternative ai to class Ck:

Ck = Cl = Cφ(ai).

If k < l: From proposition 5.2.1 we know that:

φ+R i

(rk) > φ+R i

(rl) (A.4)

φ−R i

(rk+1) > φ−R i

(rl+1) (A.5)

Combining B.6 with A.4 and A.5, we obtain that:

φ+R i

(rk)−φ−R i

(rk) ≥ φ+R i

(ai)−φ−R i

(ai) > φ+R i

(rl+1)−φ−R i

(rl+1).

In terms of flows, this is equivalent to:

φR i(rk) ≥ φR i(ai) > φR i(rl+1).

276

A.3. Proof of proposition 7.2.1:

Consequently, the net flow assignement must lie between class Ck and class Cl:

Ck DCφ(ai)DCl .

If k > l: Similar to k < l.



Let us suppose that a j is assigned to category Ch using the outgoing flows. In such a case, byconstruction of the assignment rule, we must have that:

φ+R j

(a j) > φ+R j

(rh+1). (A.6)

Since∀k ∈ 1, . . . ,q : gk(ai) ≥ gk(a j),

Condition 5.2.5 ensures us that

π(rh+1,a j) ≥ π(rh+1,ai).

That is why we have that:

(|R |−1) ·φ+R (rh+1)+ π(rh+1,a j) ≥ (|R |−1) ·φ+

R (rh+1)+ π(rh+1,ai).

This implies that:

φ+R j

(rh+1) ≥ φ+R i

(rh+1). (A.7)

Similarly, since∀k ∈ 1, . . . ,q : gk(ai) ≥ gk(a j),


∀r ∈ R , π(ai,r) ≥ π(a j,r)

Since Condition 5.2.4 we have that π(ai,ai) = 0 and that π(a j,a j) = 0, we can conclude that:

∑r∈R i

π(ai,r) ≥ ∑r∈R j

π(a j,r).

In terms of flows, this means that:

φ+R i

(ai) ≥ φ+R j

(a j). (A.8)

277


From A.6, A.7, A.8, we can conclude that:

φ+R i

(ai) ≥ φ+R j

(a j) > φ+R j

(rh+1) ≥ φ+R i

(rh+1).

Hence, we have that:Cφ+(ai)DCh = Cφ+(a j).

This proves the proposition considering the outgoing flow assignment. The proof for theincoming flow assignment is analogous.


The proof is immediate since we have for the preference degrees ∀rh ∈ R : π(ai,rh) = π(a j,rh)and π(rh,ai) = π(rh,a j). The flows of ai and a j, with respect to the reference profiles, will thusbe the same φ

+R i

(ai) = φ+R j

(a j), φ−R i

(ai) = φ−R j

(a j) and φR i(ai) = φR j(a j) as well as the flows

taken by the profiles with respect to the two actions: ∀rh ∈ R : φ+R i

(rh) = φ+R j

(rh); φ−R i

(rh) =φ−R j

(rh) and φR i(rh) = φR j(rh).


The proof is similar to previous proof (A.4).


The proof is immediate when using Proposition 5.2.1 since the order between the actions will bepreserved. This ensures us that the action ai will have a rank between rh and rh+1.


The proof is immediate since “rh” will have the same flows as rh. The assignment of rh will thusunivocally be to Ch.


∀ R = r1, . . . , rK and with R verifying Condition 6.1.1.If alternative ai is indifferent or incomparable to just one centroid rh (i.e. π(ai, rh) ≤

278


λ and π(rh,ai) ≤ λ, with λ≤ 0.5), we have that :

• Cφ+(ai) ∈ [h,h + 1]

• Cφ−(ai) ∈ [h−1,h]

• Cφ(ai) = h

In this demonstration we will note R as R and r j as r j.

Proof :If alternative ai is indifferent or incomparable to just one centroid rh we should have thatπ(ai,h)≤ λ and π(rh,ai)≤ λ, with λ≤ 0.5, and moreover that ∀k | φk(rh−1)≥ φk(ai)≥ φk(rh+1).

From the hypothesis on the reference profiles we have :

r1 . . . rh−2 rh−1 rh rh+1 rh+2 . . . rK

π(ai,r j) 0 . . . 0 0 < λ > λ 1 . . . 1π(r j,ai) 1 . . . 1 > λ < λ 0 0 . . . 0

We have thus :

φ+R i

(ai) = (K− (h + 1))+ π(ai,h−1)+ π(ai,h)+ π(ai,h + 1) (A.9)

⇒ K−h−1 + λ < φ+R i

(ai) < K−h + λ (A.10)

φ−R i

(ai) = h−2 + π(h−1,ai)+ π(h,ai)+ π(h + 1,ai) (A.11)

⇒ h−2 + λ < φ−R i

(ai) < h−1 + λ (A.12)

φR i(ai) = K−2 ·h+1+[(π(ai,h−1)+π(ai,h)+π(ai,h+1))−(π(h−1,ai)+π(h,ai)+π(h+1,ai))](A.13)

⇒ [K−2 ·h] < φR i(ai) < [K−2 ·h + 2] (A.14)

1. Cφ+(ai) ∈ [h,h + 1] :φ

+R i

(rh−1)+ φ+R i

(rh) = 2 · [K−h]+ 1 + π(rh−1,ai)+ π(rh,ai)φ

+R i

(rh+1)+ φ+R i

(rh+2) = 2 · [K−h]−3 + π(rh+1,ai)+ π(rh+2,ai))

φ+R i

(rh−1)+ φ+R i

(rh)2

> [K−h]+1 + λ

2(A.15)

279


K−h− 32

>φ

+R i

(rh+1)+ φ+R i

(rh+2)2

(A.16)

Cφ+(ai) ∈ [h,h + 1]⇔

φ

+R i

(rh−1)+φ+R i

(rh)2 ≥ φ

+R i

(ai)

φ+R i

(ai) >φ

+R i

(rh+1)+φ+R i

(rh+2)2

⇔A.15&A.10⇔ [K−h]+ 1+λ

2 ≥ K−h + λ

A.16&A.10⇔ K−h−1 + λ > K−h− 32

⇔−1 < λ≤ 1⇒Cφ+(ai) ∈ [h,h + 1]

2.Cφ−(ai) ∈ [h−1,h] :

φ−R i

(rh−1)+ φ−R i

(rh−2) = 2 ·h−5 + π(ai,rh−2)+ π(ai,rh−1)φ−R i

(rh+1)+ φ−R i

(rh) = 2 ·h−1 + π(ai,rh+1)+ π(ai,rh)⇒

φ−R i

(rh−1)+ φ−R i

(rh−2)2

< h−2− 12

(A.17)

h +−1 + λ

2<

φ−R i

(rh+1)+ φ−R i

(rh)2

(A.18)

Cφ−(ai) ∈ [h−1,h]⇔

φ−R i

(rh−1)+φ−R i

(rh−2)2 ≤ φ

−R i

(ai)

φ−R i

(ai) <φ−R i

(rh+1)+φ−R i

(rh)2

⇔A.17&A.12⇔ h−2− 1

2 + λ≤ h−2 + λ

A.18&A.12⇔ h−1 + λ < h + −1+λ

2

⇔−12

< λ <12⇒Cφ−(ai) ∈ [h−1,h]

3. Cφ(ai) = h :

φR i(rh−1)+φR i(rh) = [K−h+1+π(rh−1,ai)−h+2−π(ai,rh−1)]+[K−h−h+1+π(rh,ai)−π(ai,rh)](A.19)

= [2 ·K−4 ·h + 4 +(π(rh−1,ai)−π(ai,rh−1)+ π(rh,ai)−π(ai,rh))] (A.20)

φR i(rh)+φR i(rh+1) = [K−h−h+1+π(rh,ai)−π(ai,rh)]+[K−h−1+π(rh+1,ai)−h−1+1−π(ai,rh)](A.21)

= [2 ·K−4 ·h +(π(rh,ai)−π(ai,rh)+ π(rh+1,ai)−π(ai,rh+1)] (A.22)

⇒

280


[K−2 ·h + 2] <φR i(rh−1)+ φR i(rh)

2(A.23)

φR i(rh)+ φR i(rh+1)2

< [K−2 ·h] (A.24)

A.23,A.24,A.14=⇒φR i(rh)+ φR i(rh+1)

2< [K−2 ·h] < φR i(ai) < [K−2·h+2] <

φR i(rh−1)+ φR i(rh)2

(A.25)⇒Cφ(ai) = h


We will only prove the proposition when the categories are determined by limiting profiles. Thecase where the categories are defined by central profiles may be demonstrated in a analogousway.

Since the reference profiles respect Condition 6.1.1 we have the following non-normalized flowvalues :

• (|R i|−1) ·φ+R i

(rh) = K + 1−h + π(rh,ai)

• (|R i|−1) ·φ−R i(rl) = l−1 + π(ai,rl)

Since we suppose that Cφ+(ai) = Ch and that Cφ−(ai) = Cl , the definition of the assignment rulesimply that:

1. (|R i|−1) ·φ+(rh+1) = K + 1− (h + 1)+ π(rh+1,ai) < (|R i|−1) ·φ+R i

(ai)

2. (|R i|−1) ·φ−(rl) = l−1 + π(ai,rl) ≤ (|R i|−1) ·φ−R i(ai)

Adding these two inequalities together, we obtain:

K + 1− (h + 1)+ π(rh+1,ai)+ l−1 + π(ai,rl) < (|R i|−1) · (φ+R i

(ai)+ φ−R i

(ai)).

Since(|R i|−1) ·φ+

R i(ai) = ∑

r∈R i

π(ai,r),

and(|R i|−1) ·φ−R i

(ai) = ∑r∈R i

π(r,ai),

this can be rewritten as follows:

K + l−h + 1 + π(rh+1,ai)+ π(ai,rl) < ∑r∈R i

(π(ai,r)+ π(r,ai)). (A.26)

281


Moreover, Condition 5.2.3 tells us that

∀ai ∈ A ,∀r ∈ R , π(ai,r)+ π(r,ai) ≤ 1.

Since Condition 5.2.4 requires that π(ai,ai)+ π(ai,ai) = 0, we can conclude that:

∀ai ∈ A , ∑r∈R i

(π(ai,r)+ π(r,ai))≤ (|R i|−1) = K.

Combining this observation with B.21, we obtain that:

K + l−h−1 + π(rh+1,ai)+ π(ai,rl) < K.

Since π(rh+1,ai) ≥ 0 and π(ai,rl) ≥ 0, we have that:

K + l−h−1+ < K.

In other words:

l−h < 1.

Since l and h are integers, this implies that l−h≤ 0.


We recall that R = r1, . . . ,rK+1 denotes the set of limiting profiles, whereas R =r1, . . . , rK+1 denotes the set of central profies. The proof of Proposition 7.6.1 is based on thefollowing lemma:

Lemme 1.

(|R i|−1) ·φ+R i

(ai) ≥ (|Ri|−1) ·φ+R i

(ai) ≥ (|R i|−1) ·φ+R i

(ai)−π(ai,rK+1)

Proof of Lemma 1:We assume that:

∀h ∈ 1, . . . ,K,∀ j ∈ 1, . . . ,q, g j(rh) ≥ g j(rh) ≥ g j(rh+1).

Condition 5.2.6 requests that:

∀ai ∈ A ,∀h ∈ 1, . . . ,K, π(ai,rh+1) ≥ π(ai, rh) ≥ π(ai,rh).

This implies that:K

∑h=1

π(ai,rh+1) ≥K

∑h=1

π(ai, rh) ≥K

∑h=1

π(ai,rh).

282


In terms of flows, this can be rewritten as follows:

(|R i|−1) ·φ+R i

(ai)−π(ai,r1) ≥ (|Ri|−1) ·φ+Ri

(ai) ≥ (|R i|−1) ·φ+R i

(ai)−π(ai,rK+1).

Since we assumed that π(ai,r1) = 0, we must have that:

(|R i|−1) ·φ+R i

(ai) ≥ (|Ri|−1) ·φ+Ri

(ai) ≥ (|R i|−1) ·φ+R i

(ai)−π(ai,rK+1)

This completes the proof of the lemma.

Proof of 7.6.1: We recall that the assignment rule in the limit case is such that

Cφ+(ai) = Ch if φ+R i

(rh+1) < φ+R i

(ai) ≤ φ+R i

(rh).

The assignment rule in the centroid case is such that

Cφ+(ai) = Ch if

φ+Ri

(rh)+ φ+Ri

(rh+1)

2< φ

+Ri

(ai) ≤φ

+Ri

(rh)+ φ+Ri

(rh−1)

2.


LF: ∀h ∈ 1, . . . ,K + 1, (|R i|−1) ·φ+R i

(rh) = K + 1−h + π(rh,ai).

CF: ∀h ∈ 1, . . . ,K, (|Ri|−1) ·φ+Ri

(rh) = K−h + π(rh,ai)

We are going to show that 1) h≥ h−1 and that 2) h≤ h+1, which will complete the proof. Wewant to show that h≥ h−1. If h≤ 2, then it is trivial that h≥ h−1. Let us from now on supposethat h≥ 3. We assume that:

∀h ∈ 3, . . . ,K,∀ j ∈ 1, . . . ,q, g j(rh−2) ≥ g j(rh−1) ≥ g j(rh).


∀ai ∈ A ,∀h ∈ 3, . . . ,K, π(rh,ai) ≥ π(rh−1,ai) ≥ π(rh−2,ai). (A.27)

Given LF, A.27, and the fact that π(rh−1,ai) ≤ 1, we obtain that:

(|R i|−1) ·φ+R i

(rh) = K + 1−h + π(rh,ai) ≤ K +32−h +

π(rh−1,ai)+ π(rh−2,ai)2

. (A.28)

283


Using CF, the reader can check that

K +32−h +

π(rh−1,ai)+ π(rh−2,ai)2

=(|Ri|−1) ·φ+

Ri(rh−1)+ (|Ri|−1) ·φ+

Ri(rh−2)

2. (A.29)

Combining A.28 and A.29, we obtain that:

(|R i|−1) ·φ+R i

(rh) ≤(|Ri|−1) ·φ+

Ri(rh−1)+ (|Ri|−1) ·φ+

Ri(rh−2)

2(A.30)

Given the assignment rule in the limit case, we know that

φ+R i

(ai) ≤ φ+R i

(rh),

which implies that(|R i|−1) ·φ+

R i(ai) ≤ (|R i|−1) ·φ+

R i(rh).

Given Lemma 1, we know that:

(|Ri|−1) ·φ+Ri

(ai) ≤ (|R i|−1) ·φ+R i

(ai).

Combining the two last observations with A.29, we finally obtain that:

φ+Ri

(ai) ≤φ

+Ri

(rh−1)+ φ+Ri

(rh−2)

2

This means that Ch−1 D Cφ+(ai), or, in other words, that h≥ h−1. This concludes the first partof the proof.

We want to show that h ≤ h + 1. If h ≥ K− 1, then it is trivial that h ≤ h + 1. Let us from nowon suppose that h≤ K−2. We assume that:

∀h ∈ 1, . . . ,K−2,∀ j ∈ 1, . . . ,q, g j(rh) ≥ g j(rh+1) ≥ g j(rh+2).


∀ai ∈ A ,∀h ∈ 1, . . . ,K−2, π(rh,ai) ≥ π(rh+1,ai) ≥ π(rh+2,ai). (A.31)

Combining LF with A.31, we obtain that:

(|R i|−1) ·φ+R i

(rh+1)−π(ai,rK+1) ≥ K−h− 32

+π(rh+1,ai)+ π(rh+2,ai)

2(A.32)

Using CF, the reader can check that:

K−h− 32

+π(rh+1,ai)+ π(rh+2,ai)

2=

(|Ri|−1) ·φ+Ri

(rh+1)+ (|Ri|−1)φ+Ri

(rh+2)

2. (A.33)

284

A.11. Proof of propositions 7.7.1-7.7.4:

Combining A.32 and A.33, we obtain that:

(|R i|−1) ·φ+R i

(rh+1)−π(ai,rK+1) ≥(|Ri|−1) ·φ+

Ri(rh+1)+ (|Ri|−1)φ

+Ri

(rh+2)

2(A.34)

Given the assignement rule in the limit case, we know that

(|R i|−1) ·φ+R i

(ai) > (|R i|−1) ·φ+R i

(rh+1).

Given Lemma 1, we know that:

(|Ri|−1) ·φ+R i

(ai) ≥ (|R i|−1) ·φ+R i

(ai)−π(ai,rK+1).

Combining these two observations with A.34, we obtain that:

φ+Ri

(ai) ≥φ

+R i

(rh+1)+ φ+R i

(rh+2)2

(A.35)

This means that Cφ+(ai) D Ch+1, or, in other words, that h ≤ h + 1. This concludes the secondpart of the proof.

The proof for the incoming flows is similar.

A.11 Proof of propositions 7.7.1-7.7.4:

We will only prove the proposition when the categories are determined by limiting profiles. Thecase where the categories are defined by central profiles may be demonstrated in a analogousway.

We suppose that CRφ+(ai) = Ch with initially the following set of limiting profiles: R =

r1,r2, . . . ,rh,rh+1, . . . ,rK+1. We have thus that:

φ+R i

(rh+1) ≤ φ+R i

(ai) < φ+R i

(rh) (A.36)

Since the reference profiles respect Condition 6.1.1 we have the following non-normalized flowvalues:

(|R i|−1) ·φ+R i

(rh) = K + 1−h + π(rh,ai) (A.37)

(|R i|−1) ·φ+R i

(rh+1) = K−h + π(rh+1,ai) (A.38)

1. Let us first suppose that we add a reference profile r′l to R and such that π(r

′l ,rh) = 1 and

π(rh,r′l) = 0. In other words, we add a "better" category than the one to which ai is assigned. We

have thus with this new of set of reference profiles R ′ = r1,r2, . . . ,r′l , . . . ,rh,rh+1, . . . ,rK+1:

(|R i|−1) ·φ+R ′i

(rh) = K + 1−h + π(rh,ai) = φ+R i

(rh) (A.39)

285


(|R i|−1) ·φ+R ′i

(rh+1) = K−h + π(rh+1,ai) = φ+R i

(rh+1) (A.40)

But in this new set R ′, we have that φ

+R ′i

(ai) = φ+R i

(ai)+π(ai,r′l) with 0≤ π(ai,r

′l)≤ 1 which en-

sures us with A.36, A.38 and A.40 that φ+R ′i

(ai)≥ φ+R ′i

(rh+1) and thus that CR ′

φ+ (ai)DCR ′

h+1 = CRh .

Moreover, we have that from (A.39) φ+R ′i

(rh−1) ≥ φ+R ′i

(rh) + 1 and thus we have that

φ+R ′i

(ai) ≤ φ+R ′i

(rh−1). This leads to the fact that CR ′

φ+ (ai)ECR ′

h = CRh−1.

We have thus CR ′

φ+ (ai) ∈ [h−1,h].

2. Let us now suppose that we add a reference profile r′l to R and such that π(r

′l ,rh) = 0 and

π(rh,r′l) = 1. In other words, we add a "worse" category than the one to which ai is assigned.

We have thus with this new of set of reference profiles R ′ = r1,r2, . . . ,rh,rh+1, . . . ,r′l , . . . ,rK+1:

R ′ = r1,r2, . . . ,r′l , . . . ,rh,rh+1, . . . ,rK+1:

(|R i|−1) ·φ+R ′i

(rh) = K + 2−h + π(rh,ai) = φ+R i

(rh)+ 1 (A.41)

(|R i|−1) ·φ+R ′i

(rh+1) = K + 1−h + π(rh+1,ai) = φ+R i

(rh+1)+ 1 (A.42)

But in this new set R ′, we have that φ

+R ′i

(ai) = φ+R i

(ai)+ π(ai,r′l) with 0 ≤ π(ai,r

′l) ≤ 1 which

ensures us with A.36, A.38 and A.42 that φ+R ′i

(ai)≤ φ+R ′i

(rh) and thus that CR ′

φ+ (ai)DCR ′

h = CRh .

Moreover, we have from A.41 that φ+R ′i

(rh+2) ≤ φ+R ′i

(rh+1) − 1 and thus we have that

φ+R ′i

(ai) ≥ φ+R ′i

(rh+2). This leads to the fact that CR ′

φ+ (ai)ECR ′

h+1 = CRh+1.

We have thus CR ′

φ+ (ai) ∈ [h,h + 1].

The proofs for the other cases are similar.

286

B Interval and Fuzzy FlowSort:proofs

B.1 Numerical example

r5 r4 r3 r2 r1

Π(r5,r j) (0,0,0) (0,0,0) (0,0,0) (0,0,0) (0,0,0)Π(r4,r j) (1,0,0) (0,0,0) (0,0,0) (0,0,0) (0,0,0)Π(r3,r j) (1,0,0) (0.967,0.343,0.033) (0,0,0) (0,0,0) (0,0,0)Π(r2,r j) (1,0,0) (1,0,0) (1,0,0) (1,0.167,0) (0,0,0)Π(r1,r j) (1,0,0) (1,0,0) (1,0,0) (1,0,0) (0,0,0)

Table B.1 — The preference matrix of the reference profiles in case of scenario 1.

r5 r4 r3 r2 r1

Π(r5,r j) (0,0,0) (0,0,0) (0,0,0) (0,0,0) (0,0,0)Π(r4,r j) (1,0,0) (0,0,0) (0,0,0) (0,0,0) (0,0,0)Π(r3,r j) (1,0,0) (0.966,0.308,0) (0,0,0) (0,0,0) (0,0,0)Π(r2,r j) (1,0,0) (1,0,0) (1,0,0) (1,0.083,0) (0,0,0)Π(r1,r j) (1,0,0) (1,0,0) (1,0,0) (1,0,0) (0,0,0)

Table B.2 — The preference matrix of the reference profiles in case of scenario 2.

287

Interval and Fuzzy FlowSort: proofs

r 1r 2

r 3r 4

r 5Π

(a1,

r j)

(0,0

,0)

(0,0

,0)

(0.0

833,

0.08

33,0

.385

3)(0

.85,

0.48

3,0.

1)(1

,0,0

)Π

(rj,

a 1)

(1,0

,0)

(1,0

,0)

(0.9

66,0

.385

3,0.

0.03

4)(0

.281

4,0.

2314

,0.7

186)

(0,0

,0.1

5)Π

(a2,

r j)

(0,0

,0)

(0,0

,0.0

329)

(0.1

643,

0.16

43,0

.626

7)(0

.866

,0.4

027,

0.13

4)(1

,0,0

)Π

(rj,

a 2)

(1,0

,0)

(0.9

343,

0.48

3,0.

0657

)(0

.225

,0.2

25,0

.543

9)(0

,0,0

.15)

(0,0

,0)

Tabl

eB.3

—T

hepr

efer

ence

degr

eesb

etw

een

the

refe

renc

epr

ofile

sand

the

actio

nsin

scen

ario

1.

r 1r 2

r 3r 4

r 5Π

(a1,

r j)

(0,0

,0)

(0,0

,0)

(0,0

,0)

(0.4

687,

0.46

87,0

)(1

,0,0

)Π

(rj,

a 1)

(1,0

,0)

(1,0

,0)

(0.5

867,

0,0.

4193

)(0

.966

,0.3

42,0

.034

)(0

,0,0

)Π

(a2,

r j)

(0,0

,0)

(0.0

329,

0.03

29,0

)(0

.791

,0.7

91,0

)(1

,0.5

367,

0)(1

,1,0

)Π

(rj,

a 2)

(1,0

,0)

(0.4

513,

0,0.

5487

)(0

,0,0

.768

9)(0

.966

,0.3

42,0

.034

)(0

,0,0

)

Tabl

eB.4

—T

hepr

efer

ence

degr

eesb

etw

een

the

refe

renc

epr

ofile

sand

the

actio

nsin

scen

ario

2.

288

B.1. Numerical example

Φ+ R 1

(Φ+ R 1

)Def

Φ− R 1

(Φ− R 1

)Def

ΦR 1

(ΦR 1

)Def

r 1(1

,0,0

)1

(0,0

,0)

0(1

,0,0

)1

r 2(0

.8,0

.033

5,0)

0.78

88(0

.2,0

,0)

0.2

(0.6

,0.0

36,0

)0.

589

r 2(0

.586

4,0.

146,

0.01

4)0.

542

(0.4

,0.0

34,0

)0.

389

(0.1

86,0

.146

,0.0

47)

0.15

4r 4

(0.2

56,0

.046

,0.1

44)

0.28

8(0

.61,

0.08

5,0.

084)

0.60

9(-

0.36

9,0.

13,0

.229

)-0

.321

r 5(0

,0,0

)0

(0.9

7,0.

097,

0.02

)0.

944

(-0.

97,0

.02,

0.12

7)-0

.934

a 1(0

.187

,0.1

13,0

.097

)0.

181

(0.6

5,0.

123,

0.21

)0.

679

(-0.

463,

0.32

3,0.

22)

-0.4

97

Tabl

eB

.5—

Com

puta

tion

ofth

edi

ffer

entf

uzzy

flow

valu

esfo

rR1

insc

enar

io1.

Φ+ R 1

(Φ+ R 1

)Def

Φ− R 1

(Φ− R 1

)Def

ΦR 1

(ΦR 1

)Def

r 1(1

,0,0

)1

(0,0

,0)

0(1

,0,0

)1

r 2(0

.8,0

.033

5,0)

0.78

88(0

.2,0

,0)

0.2

(0.6

,0.0

36,0

)0.

589

r 2(0

.51,

0.07

,0.0

91)

0.51

7(0

.4,0

.034

,0)

0.38

9(0

.109

,0.0

68,0

.124

)0.

128

r 4(0

.21,

0,0.

19)

0.27

3(0

.68,

0.16

2,0.

007)

0.63

6(-

0.47

7,0.

007,

0.35

2)-0

.362

r 5(0

,0,0

.03)

0.01

(0.9

9,0.

117,

0)0.

9514

(-0.

99,0

,0.1

47)

-0.9

41a 1

(0.2

84,0

.21,

0)0.

214

(0.5

27,0

,0.3

34)

0.63

7(-

0.24

2,0.

544,

0)-0

.424

Tabl

eB

.6—

Com

puta

tion

ofth

edi

ffer

entf

uzzy

flow

valu

esfo

rR1

insc

enar

io2.

289


Φ+ R 2

(Φ+ R 2

)Def

Φ− R 2

(Φ− R 2

)Def

ΦR 2

(ΦR 2

)Def

r 1(1

,0,0

)1

(0,0

,0)

0(1

,0,0

)1

r 2(0

.787

,0.1

3,0.

013)

0.74

8(0

.2,0

,0.0

07)

0.20

22(0

.587

,0.1

37,0

.013

1)0.

5457

r 2(0

.438

,0.1

13,0

.116

)0.

542

(0.4

33,0

.066

3,0.

1253

)0.

4525

(0.0

53,0

.238

8,0.

1819

)-0

.013

6r 4

(0.2

,0.0

03,0

.21)

0.28

8(0

.767

,0.1

49,0

.033

6)0.

728

(-0.

5664

,0.0

336,

0.17

89)

-0.5

18r 5

(0,0

,0)

0(1

,0,0

)1

(-1,

0,0)

-1a 2

(0.4

06,0

.113

,0.1

59)

0.42

1(0

.432

,0.1

42,0

.152

)0.

679

(-0.

0258

,0.2

653,

0.30

03)

-0.0

141

Tabl

eB

.7—

Com

puta

tion

ofth

edi

ffer

entf

uzzy

flow

valu

esfo

rR2

insc

enar

io1.

Φ+ R 2

(Φ+ R 2

)Def

Φ− R 2

(Φ− R 2

)Def

ΦR 2

(ΦR 2

)Def

r 1(1

,0,0

)1

(0,0

,0)

0(1

,0,0

)1

r 2(0

.690

3,0.

0335

,0.1

097)

0.71

57(0

.206

6,0.

0066

,0)

0.20

44(0

.483

7,0.

036,

0.11

63)

0.51

13r 2

(0.3

932,

0.06

84,0

.160

6)0.

424

(0.5

582,

0.19

17,0

)0.

4943

(-0.

165,

0.06

8,0.

3522

)-0

.070

4r 4

(0.2

,0,0

.03)

0.21

(0.7

932,

0.17

57,0

.007

)0.

7369

(-0.

5932

,0.0

07,0

.205

2)-0

.526

9r 5

(0,0

,0)

0(1

,0,0

)1

(-1,

0,0)

-1a 2

(0.5

648,

0.27

4,0)

0.47

41(0

.290

3,0,

0.29

35)

0.38

81(0

.274

5,0.

5656

,0)

0.08

6

Tabl

eB

.8—

Com

puta

tion

ofth

edi

ffer

entf

uzzy

flow

valu

esfo

rR2

insc

enar

io2.

290

B.2. Proof proposition 9.3.1:

B.2 Proof proposition 9.3.1:

Condition 9.3.1 requires that the reference profiles dominate each other:∀ j ∈ 1, . . . ,q,∀h ∈ 1, . . . , |R |−1 :

g j(rh) ≥ g j(rh+1)⇔ g j(rh) ≥ g j(rh+1)

We have thus that ∀rl ∈ 1, . . . , |R |−1 ::

g j(rh)−g j(rl) ≥ g j(rh+1)−g j(rl)

This leads to ∀w j > 0:w j ∗ P j(rh,rl) ≥ w j ∗ P j(rh+1,rl)

Combining this with Condition 9.3.5, we obtain that:

∀rl ∈ R , π(rh,rl) ≥ π(rh+1,rl)⇔ π(rh,rl) ≥ π(rh+1,rl)

Condition 9.3.6 requires that π(rh,rh+1) > 0, whereas Condition 9.3.4 tells us thatπ(rh+1,rh+1) = 0. That is why we can conclude that:

π(rh,rh+1) > π(rh+1,rh+1).

These last two observations tell us that:

φ+R (rh) = ∑

r j∈Rπ(rh,r j) > ∑

r j∈Rπ(rh+1,r j) = φ

+R (rh+1).

This completes the first part of the proof.

In a similar way, the fact that the reference profiles dominate each other

g j(rh) ≥ g j(rh+1) ∀ j ∈ 1, . . . ,q,∀h ∈ 1, . . . , |R |−1,

combined with Condition 9.3.5 implies that:

∀ai ∈ A , π(rh,ai) ≥ π(rh+1,ai).

Since:

• φ+R i

(rh) = φ+R (rh)+ π(rh,ai)

• φ+R i

(rh+1) = φ+R (rh+1)+ π(rh+1,ai)

• φ+R (rh) > φ

+R (rh+1) (see first part of the proof),

we can conclude that:

φ+R i

(rh) > φ+R i

(rh+1)⇔ φ+R i

(rh) > φ+R i

(rh+1)

This completes the second part of the proof. The proofs for the incoming and net flows aresimilar.

291



We will suppose that the action ai has been assigned according to the positive and negative flowsto the following categories: Cφ+(ai) = [Ch,Cl ] and Cφ−(ai) = [C j,Ci]. This means thus that:

φ+R i

(rl) ≥ φ+R i

(ai) > φ+R i

(rl+1) (B.1)

φ+R i

(rh) ≥ φ+R i

(ai) > φ+R i

(rh+1) (B.2)

φ−R i

(ri+1) > φ−R i

(ai) ≥ φ−Ri

(ri) (B.3)

φ−R i

(r j+1) > φ−R i

(ai) ≥ φ−R i

(r j) (B.4)

We have thus moreover that h≤ l and j ≤ i. By subtracting B.3 from B.1 and B.2 from B.4 , weobtain respectively B.5 and B.6 :

φ+R i

(rl+1)−φ−R i

(ri) < φ+R i

(ai)−φ−R i

(ai) ≤ φ+R i

(rl)−φ−R i

(ri+1) (B.5)

φ+R i

(rh+1)−φ−R i

(r j) ≤ φ+R i

(ai)−φ−R i

(ai) < φ+R i

(rh)−φ−R i

(r j+1) (B.6)

If l<j:We have thus that h≤ l < j ≤ i and with proposition 9.3.1 we know that:

φ+R i

(rl) ≥ φ+R i

(rl+1) > φ+R i

(ri) (B.7)

And thus that:φ

+R i

(ri)−φ−Ri

(ri) < φ+R i

(rl+1)−φ−R i

(ri) (B.8)

Combining B.8 with the left inequality of B.5, this leads to:

φ+R i

(ri)−φ−R i

(ri) < φ+R i

(ai)−φ−R i

(ai)⇔ φR i(ri) < φR i(ai) (B.9)

Consequently, from B.9 the upper net flow assignment must lie between Ci and Cl .

292

B.4. Proof proposition 9.4.2

Analogously, and from proposition 9.3.1 we know that:

φ−R i

(r j+1) > φ−Ri

(r j) > φ−R i

(rh) (B.10)

And thus that:φ

+R i

(rh)−φ−Ri

(rh) > φ+R i

(rh)−φ−Ri

(r j+1) (B.11)

Combining B.11 with the right inequality of B.6, this leads to:

φR i(ai) < φR i(rh) (B.12)

Consequently, from B.12 the upper net flow assignment must lie between Ch and C j and we havethus finally that Cφ+(ai) = [Ch,C j].

The other cases are similar.

B.4 Proof proposition 9.4.2

Let us note Cφ+(ai) = Ch and Cφ+(ai) = Cl . We have thus to prove that words that

φ+Ri

(rml+1) < φ

+Ri

(ami ) < φ

+Ri

(rmh ) (B.13)

According to the assignment rules, we have that:

Cφ+(ai) = [Cφ+(ai),Cφ+(ai)]⇐⇒

Cφ+(ai) = Cl ⇔ φ+Ri

(rl) ≥ φ+Ri

(ai) > φ+Ri

(rl+1)

Cφ+(ai) = Ch⇔ φ+Ri

(rh) ≥ φ+Ri

(ai) > φ+Ri

(rh+1)with h≤ l.

This implies with B.13 that we need to prove that φ+Ri

(ami ) ∈ [φ+

Ri(ai),φ+

Ri(ai)],

φ+Ri

(rl+1) > φ+Ri

(rml+1) and that φ

+Ri

(rmh ) > φ

+Ri

(rh).

Since ∀k ∈ 1, . . . ,q, ∀xi ∈ Ri, g j(xi) ≤ gmj (xm

i ) ≤ g j(xi) and since the preference functionsmonotone are, we necessarily have that:

Pj(ami ,rm

j ) ≤ Pmj (am

i ,rmj ) ≤ Pj(am

i ,rmj ) (B.14)

We have thus the same result as B.14 for the preference degrees and the flows, which proveimplies that φ

+Ri

(ai) ∈ [φ+Ri

(ai),φ+Ri

(ai)]. We obtain the same for the positive flows of rh and rl

which proves the proposition.

The proof is similar when working with the negative and net flows.

293



Let us suppose that a j is assigned to category [Ch,Cl ] using the outgoing flows. In such a case,

by construction of the assignment rule, we must have that: C+φ

= [Ch,Cl ]⇔

φ+Ri

(rl) ≥ φ+Ri

(ai) > φ+Ri

(rl+1) (B.15)

φ+Ri

(rh) ≥ φ+Ri

(ai) > φ+Ri

(rh+1) (B.16)

Since∀k ∈ 1, . . . ,q : gk(ai) ≥ gk(a j)


π(rh+1,a j) ≥ π(rh+1,ai)

andπ(rl+1,a j) ≥ π(rl+1,ai).

That is why we have that:

(|R |−1) ·φ+R (rh+1)+ π(rh+1,a j) ≥ (|R |−1) ·φ+

R (rh+1)+ π(rh+1,ai)

and(|R |−1) ·φ+

R (rl+1)+ π(rl+1,a j) ≥ (|R |−1) ·φ+R (rl+1)+ π(rl+1,ai).

This implies that:

φ+R j

(rh+1) ≥ φ+Ri

(rh+1) (B.17)

φ+R j

(rl+1) ≥ φ+Ri

(rl+1). (B.18)

Similarly, since∀k ∈ 1, . . . ,q : gk(ai) ≥ gk(a j),


∀r ∈ R , π(ai,r) ≥ π(a j,r)

and∀r ∈ R , π(ai,r) ≥ π(a j,r).

294

B.6. Proof proposition 9.4.4:

Since Condition 9.3.4 we have that π(ai,ai) = 0 and that π(a j,a j) = 0, we can conclude that:

∑r∈Ri


π(a j,r)

and

∑r∈Ri


π(a j,r).

In terms of flows, this means that:

φ+Ri

(ai) ≥ φ+R j

(a j). (B.19)

φ+Ri

(ai) ≥ φ+R j

(a j). (B.20)

From B.15, B.17, B.19, we can conclude that:

φ+Ri

(ai) ≥ φ+R j

(a j) > φ+R j

(rl+1).

Hence, we have that:Cφ+(ai)DCφ+(a j) = Cl .

Finally, from B.16, B.18, B.20, we can conclude that:

φ+Ri

(ai) ≥ φ+R j

(a j) > φ+R j

(rh+1).

Hence, we have that:Cφ+(ai)DCφ+(a j) = Ch.

This proves the proposition considering the outgoing flow assignment. The proof for the incom-ing flow assignment is analogous.



• (|Ri|−1) ·φ+Ri

(rh) = K + 1−h + π(rh,ai)

295


• (|Ri|−1) ·φ−Ri(rl) = l−1 + π(ai,rl)

Since we suppose that Cφ+(ai) = Ch and that Cφ−(ai) = C j, the definition of the assignment rulesimply that:

1. (|Ri|−1) ·φ+(rh+1) = K + 1− (h + 1)+ π(rh+1,ai) < (|Ri|−1) ·φ+Ri

(ai)

2. (|Ri|−1) ·φ−(r j) = j−1 + π(ai,rl) ≤ (|Ri|−1) ·φ−Ri(ai)

Adding these two inequalities together, we obtain:

K + 1− (h + 1)+ π(rh+1,ai)+ l−1 + π(ai,rl) < (|Ri|−1) · (φ+Ri

(ai)+ φ−Ri

(ai)).

Since(|Ri|−1) ·φ+

Ri(ai) = ∑

r∈Ri

π(ai,r),

and(|Ri|−1) ·φ−Ri

(ai) = ∑r∈Ri

π(r,ai),

this can be rewritten as follows:

K + l−h−1 + π(rh+1,ai)+ π(ai,rl) < ∑r∈Ri

(π(ai,r)+ π(r,ai)). (B.21)

Moreover, Condition 9.3.3 tells us that

∀ai ∈ A ,∀r ∈ R , π(ai,r)+ π(r,ai) ≤ 1.

Since Condition 9.3.4 requires that π(ai,ai)+ π(ai,ai) = 0, we can conclude that:

∀ai ∈ A , ∑r∈Ri

(π(ai,r)+ π(r,ai))≤ (|Ri|−1) = K.

Combining this observation with B.21, we obtain that:

K + l−h−1 + π(rh+1,ai)+ π(ai,rl) < K.

Since π(rh+1,ai) ≥ 0 and π(ai,rl) ≥ 0, we have that:

K + l−h−1 < K.

In other words:

l−h < 1.

Since l and h are integers, this implies that l−h≤ 0.

296

C Data of the application ofchapter 10

Table C.1 — The performances of the suppliers to be sorted.

ai g1 g2 g3 g4 g5 g6 g7 g8 g9 g10

a1 84 83 12 7 85 85 80 85 95 90a2 72 78 7 5 70 70 80 75 95 90a3 70 82 7 7 80 85 89 65 90 95a4 70 68 20 25 75 70 60 90 70 90a5 70 95 15 5 95 50 95 95 80 95a6 90 85 30 32 85 60 70 77 80 85a7 80 75 15 7 80 95 70 84 90 80a8 86 90 10 5 85 85 92 75 99 90a9 92 85 30 26 90 60 92 75 90 90a10 70 65 25 28 60 60 75 70 60 60a11 75 85 30 32 65 50 90 80 89 60a12 92 90 8 5 90 90 85 92 99 90a13 72 75 27 10 80 70 80 70 89 80a14 55 60 28 32 70 85 60 65 70 60a15 95 90 8 5 90 90 85 85 98 90a16 95 95 8 7 95 95 95 92 95 90a17 70 75 24 12 85 80 84 70 86 80a18 80 70 10 7 85 60 80 60 95 90a19 95 90 7 7 95 85 85 95 97 95a20 60 70 30 30 60 60 80 70 60 80a21 90 90 15 5 80 90 80 75 99 90a22 70 60 30 15 60 50 60 75 70 65

297

Data of the application of chapter 10

Table C.2 — Flows of the suppliers with respect to the reference profiles and the correspond-ing assignments.

ai φ+R (r2) φ

+R (r3) φ

+R (r4) φ

+R (ai) φ

−R (r2) φ

−R (r3) φ

−R (r4) φ

−R (ai) Cφ+ Cφ− Cφ Copt Cpess

a1 0.8 0.33 0 0.5 0 0.5 1 0.13 2 2 2 2 2a2 0.88 0.42 0 0.43 0.01 0.46 0.96 0.3 2 2 2 2 3a3 0.81 0.42 0.02 0.4 0 0.45 0.94 0.24 3 2 2 2 3a4 0.95 0.57 0.05 0.22 0 0.38 0.83 0.57 3 3 3 2 3a5 0.82 0.42 0.05 0.5 0.04 0.53 0.93 0.29 2 2 2 2 3a6 0.9 0.5 0.05 0.26 0 0.42 0.82 0.44 3 3 3 2 4a7 0.88 0.39 0 0.38 0 0.39 0.94 0.27 3 2 2 2 2a8 0.73 0.35 0 0.56 0.01 0.57 0.98 0.01 2 2 2 2 2a9 0.82 0.46 0.03 0.36 0.01 0.48 0.88 0.31 3 2 2 2 3a10 1 0.62 0.06 0.04 0 0.33 0.68 0.69 4 3 3 4 4a11 0.92 0.5 0.09 0.25 0 0.39 0.85 0.51 3 3 3 3 4a12 0.71 0.65 0.33 0 0.03 0.62 1 0.05 2 2 2 1 1a13 0.96 0.5 0.01 0.26 0 0.35 0.9 0.46 3 3 3 3 3a14 0.95 0.6 0.18 0.05 0 0.33 0.71 0.73 4 4 4 4 4a15 0.73 0.33 0 0.63 0.03 0.6 1 0.06 2 2 2 1 1a16 0.67 0.33 0 0.71 0.05 0.67 1 0.05 1 1 1 1 1a17 0.96 0.46 0 0.27 0 0.37 0.9 0.41 3 3 3 3 3a18 0.88 0.45 0.04 0.35 0 0.46 0.89 0.36 3 2 2 2 3a19 0.71 0.33 0 0.62 0.05 0.58 1 0.05 2 1 1 1 1a20 1 0.61 0.09 0.03 0 0.33 0.73 0.7 4 3 3 4 4a21 0.8 0.35 0 0.53 0.01 0.54 0.98 0.15 2 2 2 2 2a22 1 0.62 0.14 0.07 0 0.33 0.74 0.76 4 4 4 3 4

298

D Proofs of Part III

D.1 Proof Proposition 12.3.2:

We have the following conditions on the profiles: ∀h = 1, . . . ,K; ∀ j = 1, . . . ,q : g j(rh−1) ≤g j(rh) ≤ g j(rh)

1. Let us proof that Copt(a) = h⇒ Copt(a) = h ∈ [h−1,h + 1]

Copt(a) = h⇔

rh a ⇔ et

S(rh,a) ≥ λ (1)S(a,rh) < λ (2)

rh−1¬ a ⇔ ou

S(rh−1,a) < λ (3a)S(a,rh−1) ≥ λ (3b)

- Consider the first case: (1)&(2)&(3a):(1)⇒ S(rh+1,a) ≥ λ ⇒ Copt(a) ≤ h + 1

(3a)⇒ S(rh−1,a) < λ ⇒ Copt(a) > h−1

- Consider the second case: (1)&(2)&(3b):(1)⇒ S(rh+1,a) ≥ λ ⇒ Copt(a) ≤ h + 1

Moreover, we have either S(rh−1,a) < λ orS(rh−1,a) ≥ λ:

• If: S(rh−1,a) < λ⇒ Copt(a) > h−1

• If: S(rh−1,a) ≥ λ⇒ S(rh−2,a) < λ otherwise action a is indifferent to more than one suc-cessive limting profile which is in contradiction with the conformity hypothesis.⇒ S(rh−2,a) < λ

⇒ Copt(a) ≥ h−1

299

Proofs of Part III

We have thus that Copt(a) = h ∈ [h−1,h + 1]

2. Let us proof that Copt(a) = h⇒Copt(a) ∈ [h−1, h + 1]

Copt(a) = h⇔

rhSa (1) : S(rh,a) ≥ λ

rh−1¬Sa (2) : S(rh−1,a) < λ

But, we have moreover that:

• Either: aSrh and thus that a¬Srh+1 (3a) according to the hypothesis

• or: a¬Srh (3b)

- Consider the first case: (1)&(2)&(3a):(1)⇒ rh+1Sa,(3a)⇒ a¬Srh+1

⇒ rh+1 a⇒Copt(a) ≤ h + 1

- Consider the second case: (1)&(2)&(3b):

(2)⇒ ou

rh−1¬Sa⇒ Copt(a) ≥ h

rh−1Sa⇒ ou

aSrh−1 ⇒Copt(a) > h−1a¬Srh−1 ⇒Copt(a) = h−1

We have thus that Copt(a) = h ∈ [h−1, h + 1].This permits us to have finally that |Copt(a)−Copt(a)| ≤ 1.

The proof for the pessimistic version is analogous.

1. Let us proof that Cpess(a) = h⇒ Cpess(a) = h ∈ [h−1,h + 1]

Cpess(a) = h⇔

S(a,rh−1) ≥ λ (1)S(a,rh) < λ (2)

(1)⇒ S(a, rh−1) ≥ λ⇒ Cpess(a) ≥ h−1.(2)⇒ S(a, rh+1) < λ⇒ Cpess(a) ≤ h + 1.

2. Let us proof that Cpess(a)⇒Cpess(a) = h ∈ [h−1, h + 1]

Cpess(a)⇔

S(a, rh) ≥ λ (1)S(a, rh+1) < λ (2)

300

D.2. Proof Proposition 12.3.1:

(1)⇒ S(a,rh−1) ≥ λ⇒Cpess(a) ≥ h.(2)⇒ S(a,rh+1) < λ⇒Cpess(a) ≤ h + 1.

D.2 Proof Proposition 12.3.1:

Suppose that Copt(a) = h et Cpess(a) = l with h<l.

Copt(a) = h⇒ S(rh,a) ≥ λ

S(rh+1,a) ≥ λ

S(rh+i,a) ≥ λ with i = 1, . . . ,K−hparticularly: S(rl ,a) ≥ λ

Cpess(a) = l⇒ S(a, rl) ≥ λ

S(a, rl−1) ≥ λ

S(a, rl− j) ≥ λ with j = 1, . . . , lparticularly: S(a, rh) ≥ λ

⇒

aIrh

. . . contradictory with the

. . . con f ormity hypothesisaIrl

D.3 Link between PROAFTN and Electre Tri

1. Let us first proof that if Copt(a) = Cpess(a) = Ch, with h 6= 1 and h 6= K, we have thatCPRO(a) = Ch.

According to the pessimistic and optimistic assignment rules we have that:

• Copt(a) = Ch:- aSrh ⇔ S(a, rh) ≥ λ⇒ S(a, rh−1) ≥ λ

- a¬Srh+1⇔ S(a, rh+1) < λ

• Cpess(a) = Ch:- rhSa ⇔ S(rh,a) ≥ λ⇒ S(rh−1,a) < λ

- rh−1¬Sa⇔ S(rh−1,a) < λ

We have thus that:

• ...

• min [S(a, rh−1); S(rh−1,a)] < λ

• min [S(a, rh); S(rh,a)] ≥ λ

301

Proofs of Part III

• min [S(a, rh+1); S(rh+1,a)] < λ

• ...

This ensures us that CPRO(a) = Ch.

2. Let us proof that CPRO(a) = Ch ⇒ Copt(a) = Cpess(a) = Ch,with h 6= 1 and h 6= K.

CPRO(a) = Ch⇔

min[S(a, rh); S(rh,a)] ≥ λ (1)min∀ j 6=h[S(a, r j); S(r j,a)] < λ (2)

We have from (1) that:

∀ j ≥ h : S(r j,a) ≥ λ and thus with (2)⇒ S(a, r j) < λ

∀ j ≤ h : S(a, r j) ≥ λ and thus with (2)⇒ S(r j,a) < λ

This ensures us that Copt(a) = Cpess(a) = Ch.

302

thesis of nemery philippe ph_d flowsort

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