Thesis of Nemery philippe ph_d Flowsort

Année Académique 2008-2009

Faculté des Sciences Appliquées

T HÈSE
soutenue à l’Université Libre de Bruxelles
En vue de l’obtention du grade académique de

DOCTEUR EN SCIENCES de L’INGENIEUR

On the use of multicriteria ranking methods in sorting problems

Philippe Nemery de Bellevaux

Directeur de thèse :

Prof. Philippe Vincke - Université Libre de Bruxelles

Encadrement de thèse :

Prof. Yves De Smet - Université Libre de Bruxelles

Jury de thèse :

Prof. Hugues Bersini - Université Libre de Bruxelles
Prof. Denis Bouyssou - Université Paris-Dauphine
Prof. Bertrand Mareschal - Université Libre de Bruxelles
Prof. Marc Pirlot - Faculté Polytechnique de Mons

"Suis-je plus ou moins ceci ou cela qu’une plante ou qu’un chimpanzé ?

La réponse est impossible et absurde. Car toute hiérarchie suppose une unidimensionnalité. Et
cela, c’est l’une de mes principales préoccupations: lequel de nous deux est supérieur à l’autre?
Eh bien, cela dépend en quoi. Dès qu’il y a une seule caractéristique, il y a une réponse. Mais
dès qu’il y a deux caractéristiques, il n’existe plus de réponse. Par conséquent, dire "je suis
plus complexe qu’un chimpanzé, parce que mon cerveau compte plus de neurones" est possible,
comme il serait possible de dire bien d’autres choses puisqu’il existe beaucoup d’autres critères
de performance."

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

Acknowledgment

The realization of this thesis is a long-term labour whose outcome is certainly due to the
contribution of several outstanding people. We would like to thank these persons ; not only for
their effective contribution to this work but also for the patience they have showed during these
four years of research.
After graduating, starting immediately by doing research was in our case, somewhat disconcerting. Finding a research direction is not always easy, especially when "so many" doors are open.
Nevertheless, we would like to thank sincerely and gratefully Professor Marie-Ange Remiche
and Professor Philippe Vincke for giving us the opportunity of doing research without imposing
us any subject nor direction. Thanks to them for the faith they have shown.
We wish to thank Professors Hugues Bersini, Denis Bouyssou, Yves De Smet, Bertrand
Mareschal and Marc Pirlot for accepting to be part of the jury. Moreover, having Professors
Hugues Bersini, Yves De Smet, Bertrand Mareschal, Marc Pirlot and Philippe Vincke during
our yearly accompaniment committees, certainly had a positive and fruitful contribution to the
achievement of this thesis.
We would like to thank Professor Yves De Smet and Professor Marie-Ange Remiche for being
present these four years and supporting our moods during the difﬁculties encountered when
researching. They have permit us to take some distance from our work and to keep two feet on
the ground.
We dedicate a special and warm regard to Professor Denis Bouyssou for his pertinent remarks
and words of encouragement, when we really needed them.
We learned a lot from Professor Betrand Mareschal, both scientiﬁcally and personally, while
working together on industrial multicriteria decision problems. Above all, he always has good
advices for choosing wine.
All the members of the Service des Mathématiques de la Gestion are unforgettable. We wish to
thank Olivier Cailloux, Aurélie Casier, Yves De Smet, Quantin Hayez, Claude Lamboray and
Karim Lidouh for their judicious remarks and for the interesting conversations we had all along
v

Acknowledgment

this work. Thank you all for having read (and corrected !) some parts of our work. Moreover,
Catherine Berard, Vinciane de Wilde, Rose-Marie Brynaert and Vanessa Palacios Perez have
been a very reliable and professional adminstrative staff. Their support and kindness during
these years were incredible.
We would like to thank all the members of the LAMSADE (Wassila, Sonia, Nicolas, Guilaume,
etc.), at the Université Paris-Dauphine for having welcomed and introduced us to la vie
parisienne. Special and warm thanks to Professor Vincent Mousseau, Professor Alexis Tsoukias
and Wassila Ouerdane. Moreover, our stay at Paris has been possible given the ﬁnancial supports
of the Cost IC0602 Action (STMS).
We are grateful to Iryna Yevseyeva, Claude, Michael, Pierre, Laurent, Slobodan, Nico and
Thomas for having read some parts of this work and for their judicious and pertinent remarks.
We can not forget the students from our faculty which permitted us to have the necessary
hindsight. Teaching courses, exercises and working with the students of our faculty, was a real
pleasure. Thanks to them for having accepted us like we are and for the human contacts we have
(had).
In particular, we are grateful to Laurent Huenaerts and Pierre Janssen for their kind collaboration
and their concrete contributions to some proofs of this work.
Thanks to all our friends, comrades and homemates for being what they are for us.
Last but not least, we would like to thank our family. Particularly, our parents and our little sister
for having supported us and our mood during these years and for accepting our decision to do
research. Was it so terrible ?
Besides, we are immensely thankful for Camille, who was present during these last years and
who gave us the necessary love, happiness and encouragements for achieving our goal.

Although this work will certainly not change the world nor the science, it has changed us, our
opinions and our insight towards research and towards multicriteria decision aid. That is, in our
point of view, the most important achievement.
So, might a reader hesitate to do research, we have only one advice: Just try it !

vi

Contents

Acknowledgment

v

Introduction

11

Resumé

15

Publications and Conferences

17

Notations

19

I

State of the A rt

21

1

Introduction to classiﬁcation problems

23

1.1

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

1.2

Classiﬁcation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.2.1

1.3
2

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

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

Introduction to Multicriteria Decision Aid

33

2.1

Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2

The actions, the criteria, . . . and the problems . . . . . . . . . . . . . . . . . . . 35
2.2.1

The set of actions A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2.2

The set of attributes F . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2.3

The set of criteria G . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.2.4

The different types of MCDA problems . . . . . . . . . . . . . . . . . . 37

2.3

The Pareto dominance relation . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4

Preference Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
ix

Contents

2.5

Consistent family of criteria and preferential independency . . . . . . . . . . . . 39

2.6

Pair-wise comparisons between actions based on outranking relations . . . . . . 41
2.6.1

The valued outranking degree S(a, b) . . . . . . . . . . . . . . . . . . . 41
2.6.1.1
2.6.1.2

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

2.6.1.3

Partial discordance degree d S (b, a) . . . . . . . . . . . . . . . 42
j

2.6.1.4
2.6.2

Partial concordance degree cS (b, a) . . . . . . . . . . . . . . . 41
j

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

Preference degree π(a, b) . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.6.2.1
2.6.2.2

3

Uni-criterion preference degree P j (a, b) . . . . . . . . . . . . 46
Global preference degree π(a, b) . . . . . . . . . . . . . . . . 47

Some multicriteria ranking methods

49

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2

Multi Attribute Utility Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.1

3.3

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

Outranking methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3.1

Electre III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.1.1
3.3.1.2

Qualiﬁcation of an action . . . . . . . . . . . . . . . . . . . . 58

3.3.1.3

Computation of the pre-orders O1 and O2 . . . . . . . . . . . . 58

3.3.1.4

Partial pre-order O . . . . . . . . . . . . . . . . . . . . . . . . 60

3.3.1.5

Model assumptions and some properties . . . . . . . . . . . . 64

3.3.1.6
3.3.2

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

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

Promethee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.3.2.1

Entering, leaving and net ﬂows . . . . . . . . . . . . . . . . . 65

3.3.2.2

The Gaia plane and the Walking Weigths . . . . . . . . . . . . 70

3.3.2.3

Model assumptions and some properties . . . . . . . . . . . . 72

3.3.2.4

Rank Reversal phenomenon . . . . . . . . . . . . . . . . . . . 74

3.3.2.5

Some extensions of the Promethee methodology . . . . . . . . 74

3.4
3.5
4

Other multicriteria ranking methods . . . . . . . . . . . . . . . . . . . . . . . . 75
How to choose a multicriteria ranking method ? . . . . . . . . . . . . . . . . . . 76

Some multicriteria sorting methods

81

4.1
4.2
x

Introduction to sorting problems . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Properties of sorting methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Contents

4.3

Sorting based on indifference indexes . . . . . . . . . . . . . . . . . . . . . . . 88
4.3.1

Indifference Index I (a, b)

. . . . . . . . . . . . . . . . . . . . . . . . . 88

4.3.1.1
4.3.1.2

Partial discordance index d Ij (a, b) . . . . . . . . . . . . . . . . 91

4.3.1.3
4.3.2

Partial indifference degree cIj (a, b) . . . . . . . . . . . . . . . 89
Global indifference index I (a, b) . . . . . . . . . . . . . . . . 92

PROAFTN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.3.2.1
4.3.2.2

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

4.3.2.3
4.4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Sorting based on similarity indexes . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.4.1

Similarity Index SI(a,b) . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.4.1.1
4.4.1.2

4.4.2

Partial Similarity Index SI j (a, b) . . . . . . . . . . . . . . . . 95
Global Similarity Index SI(a,b) . . . . . . . . . . . . . . . . . 96

TRINOMFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.4.2.1
4.4.2.2


4.4.2.3
4.5

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Sorting based on MAUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.5.1

UTADIS: UTilités Additives DIScriminantes . . . . . . . . . . . . . . . 98
4.5.1.1

4.5.2

MHDIS: Multi-group Hierarchical DIScrimination method . . . . . . . . 100
4.5.2.1

4.6

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Sorting based on outranking relations . . . . . . . . . . . . . . . . . . . . . . . 101
4.6.1

Electre-Tri with limiting proﬁles . . . . . . . . . . . . . . . . . . . . . . 101
4.6.1.1
4.6.1.2


4.6.1.3

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.6.1.4

Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.6.1.5
4.6.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

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

Trichotomic Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.6.2.1
4.6.2.2

4.6.3

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Comparison with Electre-Tri . . . . . . . . . . . . . . . . . . 114

nTomic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.6.3.1

4.6.4

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Filtering Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
xi

Contents

4.6.4.1
4.6.4.2
4.6.5

Filtering by strict preference . . . . . . . . . . . . . . . . . . 118
Filtering by indifference . . . . . . . . . . . . . . . . . . . . . 123

PairClass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.6.5.1
4.6.5.2

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

4.6.5.3
4.7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

II F lowS ort: a flow-based sorting method

133

5

137

Notation and conditions
5.1
5.2

6

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Flow-based assignment procedures
6.1

Limiting profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.1.1

6.2

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

Central profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.2.1

6.3

141

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

Influence of the preference parameters . . . . . . . . . . . . . . . . . . . . . . . 150
6.3.1
6.3.2

7

Limiting profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Central profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Analysis of some properties of F lowS ort

159

7.1

Coherence of the net-flow assignment rule . . . . . . . . . . . . . . . . . . . . . 159

7.2

Property of monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

7.3

Property of weak homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7.4

Properties of category conformity . . . . . . . . . . . . . . . . . . . . . . . . . 160

7.5

Relationship between Cφ− and Cφ+ . . . . . . . . . . . . . . . . . . . . . . . . . 162

7.6

Relationship between the assignments with limiting profiles and central profiles . 163

7.7

Property of weak stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.7.1
7.7.2

7.8

Negative flow assignment rules . . . . . . . . . . . . . . . . . . . . . . . 166
Positive flow assignment rules . . . . . . . . . . . . . . . . . . . . . . . 167

Strong stability conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.8.1

xii

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

Contents

8

Comparison between F lowS ort and some existing sorting methods
8.1

177

Comparison between F lowS ort and Electre-Tri . . . . . . . . . . . . . . . . . . 177
8.1.1
8.1.2

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

8.1.4

9

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

8.1.3
8.2

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

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

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

I nterval and F uzzy F lowS ort

191

9.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

9.2

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

9.3

Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

9.4

I nterval F lowS ort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
9.4.1
9.4.2

9.5

Limiting profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Central profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

F uzzy F lowS ort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
9.5.1

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

10 S oftware and applications

211

11 Conclusions

219

III

223

Outranking based sorting methods

12 Electre-Tri-Central

227

12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
12.2 Assignment rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
12.3 Properties of Electre-Tri-Central . . . . . . . . . . . . . . . . . . . . . . . . . . 232
12.4 Illustrative example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
12.5 Relationship between Electre-Tri-Central and PROAFTN . . . . . . . . . . . . . 236
12.6 Defining a category by several reference profiles. . . . . . . . . . . . . . . . . . 239
12.7 Comparison with ELECTRE-TRI-C . . . . . . . . . . . . . . . . . . . . . . . . 242
12.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
13 Partially ordered categories

245

13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
xiii

Contents

13.2 Assignment rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
13.3 Illustrative example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
13.4 Particular subproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
13.4.1 Completely non-ordered categories . . . . . . . . . . . . . . . . . . . . 255
13.4.2 Completely ordered categories . . . . . . . . . . . . . . . . . . . . . . . 257
14 Conclusion Part III

259

Conclusion

261

Bibliography

263

A Proof of Propositions 5.2.1 - 7.7.4

275

A.1 Proof of proposition 5.2.1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
A.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
xiv

Contents

D.2 Proof Proposition 12.3.1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
D.3 Link between PROAFTN and Electre Tri . . . . . . . . . . . . . . . . . . . . . . 301

xv

List of Figures

1.1

Representation of the classification model in classification problem taken from
[Doumpos and Zopounidis, 2002], p.7. . . . . . . . . . . . . . . . . . . . . . . . 25

1.2

Representation of a classification procedure fG which assigns each action ai to
none, one or several categories of the set C = {C1 , . . . ,CK }. . . . . . . . . . . . . 26

1.3

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

1.4

Illustration of the utilization of attributes. . . . . . . . . . . . . . . . . . . . . . 31

1.5

Illustration of the utilization of criteria. . . . . . . . . . . . . . . . . . . . . . . . 31

1.6

Comparison of an action a to 4 different reference profiles ri , (i = 1, 2, 3, 4). . . . 32

1.7

Comparison of an action a to 4 different reference profiles ri , (i = 1, 2, 3, 4). . . . 32

2.1

Representation of the partial concordance index cS (b, a). . . . . . . . . . . . . . 42
j

2.2

Representation of the partial discordance index d S (b, a). . . . . . . . . . . . . . 43
j

2.3

Outranking graph of A where a → b ⇔ aSb ; a ↔ b ⇔ aI b ; a b ⇔ aJ b and
© ⇔ cI c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.4

Preference function of type 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.1

Representation of the complete pre-order of the set A by using the MAUT theory. 52

3.2

Pair-wise linear marginal utility functions. . . . . . . . . . . . . . . . . . . . . . 53

3.3

Concave, linear and convex linear marginal utility functions. . . . . . . . . . . . 54

3.4

Outranking graph of A where a → b ⇔ aSb, a ↔ b ⇔ aI b; a b ⇔ aJ b and
© ⇔ cI c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.5

The partial pre-order O of A obtained with Electre III . . . . . . . . . . . . . . . 61

3.6

The partial pre-order O of A obtained by "reducing" the outranking graph. . . . . 62

3.7

Representation of the O1 ,O2 and O3 rankings as well as the concordance matrix
obtained with the Electre-III demo software [Lamsade, 2008] for the Example 3.2. 63

3.8

Representation of the O1 ,O2 and O3 rankings when suppressing action a4 from
A : illustration of the rank reversal phenomenon. [Lamsade, 2008]. . . . . . . . . 65

29

1

List of Figures

3.9

Chart representation of the flows. . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.10 Complete ranking of A on basis of the positive flows (φ+ ). . . . . . . . . . . . . 69
3.11 Complete ranking of A on basis of the negative flows (φ− ). . . . . . . . . . . . . 69
3.12 Complete Promethee I ranking of A . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.13 Complete Promethee II ranking of A . . . . . . . . . . . . . . . . . . . . . . . . 69
3.14 Gaia plane with δ = 72.8%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.15 Complete ranking of A on basis of the Gaia-plane. . . . . . . . . . . . . . . . . 71
3.16 Representations of the profiles of the uni-criterion net-flows of actions a3 and a5 .

71

3.17 Representations of the profiles of the uni-criterion net-flows of actions a3 and a2 .

72

3.18 Representations of the flows of the actions of set

A 4.

. . . . . . . . . . . . . . . 74

3.19 Representation of the ρ-relation. . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.1

Representation of the different classification problems on the basis of the different relations between the predefined groups. . . . . . . . . . . . . . . . . . . . . 82

4.2

Properties of sorting procedures according to the sorting problem where particular sorting procedures may verify classical properties (which is represented by
"→"). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.3

Representation of the partial indifference index cIj (a, b). . . . . . . . . . . . . . . 89

4.4

Representation of the partial indifference index cIj (a, b). . . . . . . . . . . . . . . 90

4.5

Representation of the partial discordance index d Ij (a, b). . . . . . . . . . . . . . . 91

4.6

Representation of the performances of the reference profiles r1 and r2 and the
˙
˙
actions a1 and a4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.7

Representation of some similarity functions for the computation of SI j (a, b). . . . 96

4.8

Representation of the UTADIS sorting model. . . . . . . . . . . . . . . . . . . . 98

4.9

Representation of the classification paradigm taken from [Doumpos and Zopounidis, 2002] p.83. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.10 Illustration of completely ordered categories defined by limiting profiles. . . . . . 102
4.11 Representation of preference relation between the limiting profiles: rk r j ⇔
r j ← rk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.12 Representation of reduced preference relation between the limiting profiles:
rk r j ⇔ r j ← . . . ← rk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.13 Reduced "optimistic

-graph": x

y ⇔ x → y . . . . . . . . . . . . . . . . . . . 105

4.14 Reduced "pessimistic S-graph": : xSy ⇔ x

y . . . . . . . . . . . . . . . . . . 105

4.15 Representation of the performances of the limiting profiles r1 , r2 , r3 , r4 . . . . . . 106
4.16 Illustration of the paradox of Condorcet where a → b means that a

b. . . . . . 109

4.17 Assignment of any point (x,y) of the plan with the " -optimistic" (right) and
"S-pessimistic" (S) rules when q = 0 and p = 0, and with w1 = w2 = 0.5. . . . . 111
2

List of Figures

4.18 Assignment of any point (x,y) of the plan when considering the “union” of the
optimistic and pessimistic result and where q = 0 and p = 0, and with w1 = w2 =
0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.19 Representation of the sets R2 , R3 for the definition of C1 ,C2 and C3

. . . . . . . 112

4.20 Assignment rules of Trichotomic Segmentation based on the decision tree. . . . . 113
4.21 Assignment rules of Trichotomic Segmentation based on the decision tree when
| R2 |=| R3 |= 1 and when s = t = s = t = λ. . . . . . . . . . . . . . . . . . . 115
4.22 Representation of the goodness D j (ai ) and badness d j (ai ) functions. . . . . . . . 116
4.23 Representation of the goodness and badness plan. . . . . . . . . . . . . . . . . . 117
4.24 Representation of the limiting profiles defining the ordered categories . . . . . . 119
4.25 Illustration of the paradox of Condorcet where a → b means that aPb. . . . . . . 122
4.26 Representation of the uni-criterion preference and indifference relation. . . . . . 124
4.27 Representation of the preference relation computed in the PairClass procedure. . 127
4.28 Illustration of the case where we define profiles as limiting one. . . . . . . . . . . 128
5.1

Representation of K completely ordered categories by limiting profiles . . . . . . 138

5.2

Representation of K completely ordered categories by central profiles . . . . . . 138

6.1

Representation of the complete ranking obtained by computing the positive
flows. This leads to the Cφ+ -assignment. . . . . . . . . . . . . . . . . . . . . . . 141

6.2

A flow and category representation with limiting profiles. . . . . . . . . . . . . . 143

6.3

Representation of the limiting profiles and the actions to be assigned. . . . . . . . 144

6.4

Flow-diagram for a1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.5

A flow and category representation with central profiles. . . . . . . . . . . . . . 149

6.6

Flow-diagram for a1

6.7

Assignment of any point (x,y) of the plan with the positive, negative and net
flows when q = 0 and p = 0, w1 = w2 = 0.5: identical assignments in the 3 cases. 153

6.8

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

6.9

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

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

6.10 Assignment of any point (x,y) of the plan with the negative flows when q = 0.05
and p = 0.075, w1 = w2 = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.11 Assignment of any point (x,y) of the plan with the positive, negative and net
flows when q = 0 and p = 0, w1 = w2 = 0.5. . . . . . . . . . . . . . . . . . . . 156
6.12 Assignment of any point (x,y) of the plan with the positive (left) and negative
(right) flows when q = 0 and p = 0, w1 = w2 = 0.5 (right) and by choosing the
worst category in case of equality. . . . . . . . . . . . . . . . . . . . . . . . . . 156
3

List of Figures

6.13 Assignment of any point (x,y) of the plan with the positive flows when q = 0.05
and p = 0.075, w1 = w2 = 0.5 (right) and by choosing the worst category in case
of equality (left). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.14 Assignment of any point (x,y) of the plan with the negative flows when q = 0.05
and p = 0.075, w1 = w2 = 0.5 (right) and by choosing the worst category in case
of equality (left). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.15 Assignment of any point (x,y) of the plan with the net flows when q = 0.05 and
p = 0.075, w1 = w2 = 0.5 (right) and by choosing the worst category in case of
equality (left). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
7.1
7.2

Illustration of the relationship between the assignments with limiting profiles and
centroids: case I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.3

Illustration of the relationship between the assignments with limiting profiles and
centroids: case II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

7.4

Representation of the relationship when defining reference profiles as either central (upper-figure) or limiting (lower-figure) profiles. . . . . . . . . . . . . . . . 165

7.5

Representation of the suppression (lower case) or addition (upper case) of a
’worse’ category when using the negative flows: weak stability. . . . . . . . . . 167

7.6

Representation of the suppression (lower case) or addition (upper case) of a ’better’ category when using the negative flows: stability. . . . . . . . . . . . . . . . 168

8.1

Comparison of F lowS ort and Electre-Tri : different scenarios. . . . . . . . . . . 178

8.2

Representation of the example of 8.1.2: assignments obtained with F lowS ort
which can not be obtained with Electre-Tri. . . . . . . . . . . . . . . . . . . . . 183

8.3

Illustration of an assignment with Electre-Tri: III . . . . . . . . . . . . . . . . . 184

8.4

Illustration of an assignment with Electre-Tri: I . . . . . . . . . . . . . . . . . . 185

8.5

Illustration of an assignment with Electre-Tri: II . . . . . . . . . . . . . . . . . . 185

8.6

Representation of the assignment rule of UTADIS . . . . . . . . . . . . . . . . . 186

8.7

Representation of the flows values of the limiting profiles . . . . . . . . . . . . . 187

8.8


8.9


9.1

Illustration of interval performances of the reference profiles and action a on one
criteria, where the stars and the bullet represent the mean values of the intervals. . 192

9.2

Illustration of the performances of limiting profiles defined by intervals. . . . . . 194

9.3

Illustration of the positive flow intervals of the reference profiles under the conditions 9.3.1-9.3.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

9.4
4

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

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

List of Figures

9.5

Illustration of the performances of reference profiles. . . . . . . . . . . . . . . . 200

9.6

Illustration of the interval flow-diagram for action a1 . . . . . . . . . . . . . . . . 202

9.7

Representation of a fuzzy interval x and its parameters xu , xl , α, β. . . . . . . . . . 204

9.8

Illustration of the fuzzy performances of the the actions of R1 on criterion 1 in
scenario 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

9.9

Illustration of the fuzzy performances of the actions of R1 on criterion 1 in scenario 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

10.1 Screen-shot of the software when encoding the preference parameters. . . . . . . 213
10.2 Screen-shot of the software when encoding the performances of the actions to be
sorted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
10.3 Screen-shot of the software representing the evaluations of action a1 with respect
to the reference profiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
10.4 Screen-shot of the software representing the positive and negative flow-plane for
action a1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
10.5 Screen-shot of the software representing all the actions assigned to C1 according
to net flow assignment rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
10.6 Screen-shot of the software representing the distribution of the actions into the
categories according to the positive and negative flow-plane for action a1 . . . . . 216
10.7 Screen-shot of the software representing the assignments of all the actions according to the different assignment rules. . . . . . . . . . . . . . . . . . . . . . . 217
11.1 Representation of the ρ-relation. . . . . . . . . . . . . . . . . . . . . . . . . . . 222
12.1 The reduced "optimistic S-graph": xSy ⇔ x → y . . . . . . . . . . . . . . . . . . 229
12.2 The reduced “pessimistic S-graph”: xSy ⇔ x

y . . . . . . . . . . . . . . . . 230

12.3 The reduced "optimistic S-graph" and "pessimistic S-graph"when a is indifferent
to more than one central profile. . . . . . . . . . . . . . . . . . . . . . . . . . . 233
˙
12.4 Example of categories defined by central profiles (R = {r1 , r2 , r3 }) and limiting
˙ ˙ ˙
profiles (R = {r1 , r2 , r3 , r4 }) and the actions a2 and a3 . . . . . . . . . . . . . . . 234
12.5 Representation of the performances of the central profiles r3 , r2 and r1 ; the
˙ ˙
˙
limiting profiles r4 , r3 , r2 , r1 and the actions a2 and a3 . . . . . . . . . . . . . . . 235
12.6 Situation I and II with ∀i = 1, 2 : pi = qi = 0, w1 = w2 . . . . . . . . . . . . . . 239
˙
˙
˙
˙
12.7 Case IV= Copt (a) = C2 and Cpess (a) = C2 . . . . . . . . . . . . . . . . . . . . . 240
˙
˙1
˙
˙1
12.8 Case XII: Copt (a) = C2 and Cpess (a) = C3 . . . . . . . . . . . . . . . . . . . . . 242
13.1 Representation of partially ordered categories. . . . . . . . . . . . . . . . . . . . 246
13.2 Example of partially ordered reference profiles in the optimistic and pessimistic
reduced "S-graph" where r1 = rI and r1 = rN . . . . . . . . . . . . . . . . . . . 247
˙1 ˙
˙1 ˙
5

List of Figures

13.3 Case I: Example of the optimistic and pessimistic reduced "S-graph" with a:
˙
˙1
˙
˙2
Copt (a) = C1 and Cpess (a) = C2 . . . . . . . . . . . . . . . . . . . . . . . . . . 248
13.4 Representation of the performances of the central profiles r1 , r2 , r2 and r3 . . . . 249
˙1 ˙1 ˙2
˙1
˙
˙1
˙
˙1
13.5 Case II: Copt (a) = C1 and Cpess (a) = C3 . . . . . . . . . . . . . . . . . . . . . . 252
˙
˙2 ˙1
˙
˙1
13.6 Case III: Copt (a) = C2 ∪ C2 and Cpess (a) = C3 . . . . . . . . . . . . . . . . . . . 252
˙
˙2
˙
˙2
13.7 Case IV= Copt (a) = C2 and Cpess (a) = C2 . . . . . . . . . . . . . . . . . . . . . 252
˙
˙2 ˙1
˙
˙2 ˙1
13.8 Case V: Copt (a) = C2 ∪ C2 and Cpess (a) = C2 ∪ C2 . . . . .
˙
˙2 ˙1
˙
˙1
13.9 Case VI: Copt (a) = C2 ∪ C2 and Cpess (a) = C2 . . . . . . . .
˙
˙1
˙
˙1
13.10 Case VII: Copt (a) = C3 and Cpess (a) = C3 . . . . . . . . .
˙
˙1
˙
˙1
13.11 Case VIII: Copt (a) = C1 and Cpess (a) = C3 . . . . . . . . .
13.12 Case
13.13 Case
13.14 Case
13.15 Case

˙
˙1
˙
˙1
IX: Copt (a) = C1 and Cpess (a) = C1 . . . .
˙
˙1
˙
˙1
X: Copt (a) = C3 and Cpess (a) = C3 . . . .
˙
˙1
˙
˙1
˙2
XI: Copt (a) = C1 and Cpess (a) = C2 and C2
˙
˙1
˙
˙1
XII: Copt (a) = C2 and Cpess (a) = C3 . . .

. . . . . . . . . . . 252
. . . . . . . . . . . 253
. . . . . . . . . . . 253
. . . . . . . . . . . 253

. . . . . . . . . . . . . . . . . 254
. . . . . . . . . . . . . . . . . 254
. . . . . . . . . . . . . . . . . 254
. . . . . . . . . . . . . . . . . 255

13.16 Representation of the ideal and nadir reference profile in case of nominal classification problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

6

List of Tables

2.1

Evaluation matrix with preferentially independent criteria. . . . . . . . . . . . . 40

2.2

The performance evaluation matrix . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3

The binary relations between the actions of A with λ = 0.6. . . . . . . . . . . . . 45

2.4


3.1

The performance evaluation matrix of A .

3.2


3.3

Evaluation matrix of the 9 candidates . . . . . . . . . . . . . . . . . . . . . . . . 62

3.4

Preference parameters of the Electre III method. . . . . . . . . . . . . . . . . . . 63

3.5

Evaluation of the performances of the actions of A . . . . . . . . . . . . . . . . . 67

3.6

Preference parameters of the Promethee method. . . . . . . . . . . . . . . . . . 67

3.7

Unicriterions net flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.8

Positive, negative, net flows and ranking of A . . . . . . . . . . . . . . . . . . . . 68

3.9

Stability intervals at different levels (in %). . . . . . . . . . . . . . . . . . . . . 72

. . . . . . . . . . . . . . . . . . . . . 60

3.10 Input-Output matrix: an extract of Table 7 in [Guitouni et al., 1999]. . . . . . . . 78
3.11 Input-Output matrix after application of the propagation rules: an extract of Table
8 in [Guitouni et al., 1999]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.1

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

4.2


4.3

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

4.4

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

4.5


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

4.9

Resume of the assignment results when using the Electre-Tri rules. . . . . . . . . 110

4.10 Assignment of the actions according to the different procedures: Copt ,Cpess and
CT S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.11 Pair-wise comparisons between the actions and the limiting profiles ri , ∀ j =
1, . . . , 4: valued preference relations. . . . . . . . . . . . . . . . . . . . . . . . . 121
4.12 Assignment of the actions according to the different procedures: Copt , Cpess and
CFP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.13 Resume of the assignment results when using the Electre-Tri rules. . . . . . . . . 123
4.14 Preference degrees between action a and the reference examples. . . . . . . . . . 129
4.15 Comparison of different sorting methods on the basis of their properties. The
following abbreviations are used: PO: partially ordered, CO: completely ordered,
CNO: completely not ordered ; CP: central profile, LP: limiting profile ; RelDeg.: Relation or Degree, Ind: Indifference, Sim.: Similarity, Out.: Outranking ;
IY: Yes, N: No ; W-S: Weak or Strong ; /: out of subject . . . . . . . . . . . . . . 132
6.1

The performances of the reference profiles. . . . . . . . . . . . . . . . . . . . . 145

6.2

The different thresholds and weights. . . . . . . . . . . . . . . . . . . . . . . . . 145

6.3

The performances of the actions to be sorted. . . . . . . . . . . . . . . . . . . . 145

6.4

The preference degrees between the reference profiles and the actions. . . . . . . 146

6.5

Computation of the different flow values. . . . . . . . . . . . . . . . . . . . . . 146

6.6

The assignments of the actions according to Electre-Tri and F lowS ort. . . . . . 147

6.7


6.8


6.9

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

6.10 The flow-values of the alternatives. . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.1
7.2


7.3

Computation of the different flow values when considering the set R . . . . . . . 172

7.4

Computation of the different flow values when considering the set R . . . . . . . 173

7.5

Number of instability occurrences in presence of preferred and strongly preferred
limiting profiles when working with the positive flows. . . . . . . . . . . . . . . 175

8.1

Comparison between F lowS ort and Electre-Tri in the case of limiting profiles. . 179

8.2

8


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

List of Tables

8.3

Comparison between F lowS ort and Electre-Tri in the case of limiting profiles
verifying Condition 6.1.1: analysis of the assignments. . . . . . . . . . . . . . . 181

8.4

Evaluation of the performances of the actions of A of example 8.1.2 . . . . . . . 182

8.5

Evaluation of the limiting profile rl . . . . . . . . . . . . . . . . . . . . . . . . . 182

8.6

The preference degrees between the rl and the actions. . . . . . . . . . . . . . . 183

9.1

Interval evaluations of the reference profiles on the different criteria. . . . . . . . 200

9.2


9.3

The preference matrix of the reference profiles. . . . . . . . . . . . . . . . . . . 201

9.4

The performances of the actions to be sorted. . . . . . . . . . . . . . . . . . . . 201

9.5

The preference degrees between the reference profiles and the actions. . . . . . . 201

9.6

Computation of the different flow values. . . . . . . . . . . . . . . . . . . . . . 202

9.7

Results of the assignments of the actions according to the different rules. . . . . . 203

9.8

Computation of the different flow values for am . . . . . . . . . . . . . . . . . . . 203
2

9.9

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

9.10 Computation of the different fuzzy flow values for R1 in scenario 2. . . . . . . . 208
9.13 Results of the assignments of the action a2 according to the different rules in
scenario 1, scenario 2 and when working directly with crisp evaluations. . . . . . 209
10.1 The limiting profiles of the 4 categories of different suppliers. . . . . . . . . . . . 212
10.2 The preference parameters associated to the 10 criteria of evaluation. . . . . . . . 212
˙
12.1 Evaluation of the performances of the central reference profiles of R . . . . . . . 234
12.2 Pair-wise comparisons between the actions and the central profiles r j , ∀ j = 1, 2, 3. 235
˙
12.3 Assignment of the actions according to the different procedures (with central and
limiting profiles). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
˙
12.4 Assignment results when defining C2 by two reference profiles r2 and r2 . . . . . . 241
˙1
˙2
13.1 Evaluation of the performances of the central reference profiles. . . . . . . . . . 249
13.2 Pair-wise comparisons between the actions and the reference profiles. . . . . . . 250
13.3 Classification result of the actions according to respectively the Optimistic and
Pessimistic version as well as the two PROAFTN assignment rules. . . . . . . . 250
B.1 The preference matrix of the reference profiles in case of scenario 1. . . . . . . . 287
B.2 The preference matrix of the reference profiles in case of scenario 2. . . . . . . . 287
B.3 The preference degrees between the reference profiles and the actions in scenario 1.288
9

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 corresponding 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, to
predefined classes. On the other hand, a multicriteria ranking method is a method which ranks
the actions from the best to the worst one while taking into account several (often conflicting)
criteria.
Sorting problems are known since the Antiquity. For instance, in the fourth century before
Christ, the ancient Greek philosopher Epicurus sorted the human desires into two classes: vain
desires (e.g. the desire of immortality) and natural desires (e.g. the desire of pleasure). This
sorting was supposed to help people in finding a peaceful mood.
Nowadays, sorting problems come up naturally in our daily life. A doctor for example will
diagnose a patient on the basis of his symptoms. Based on his examination, he will assign the
patient to a known pathology-class in order to prescribe the appropriate treatment. In enterprizes, projects are often sorted into priority-based categories. Recently, a study [Observador,
31th March 2008] showed that over 20 million Brazilians have moved from the lower social
categories ("D" and "E") to category "C", the first tier of the “middle class”, and are now active
consumers due to an increase in legal employment. Hurricanes or cyclones are sorted into one of
the 5 Saffir-Simpson categories based on their wind speed, superficial pressure and tide-hight.
Sorting aims thus to regroup actions with similar behaviors or characteristics for description,
organizational or predictive purposes. The possible caused damage of cyclones can be evaluated
using the Saffir-Sympson categories and the necessary protective measures can be taken. The
decision maker thus defines the classes relatively to the consequences that will be given to the
actions belonging to a same class.
In this work, we emphasize problems where the decision maker expresses a preference relation
among the classes. For instance, the human resources department might sort candidates in a
recruitment process into promising people or into unadapted persons. Obviously, the human
resources manager prefers the first category. Analogously, projects of the highest priority class
are considered to be of high potential for the company and will thus receive an immediate
financing and the necessary human support.
Furthermore, we will consider that the expressed preference relation among categories might be
partial.

11

Introduction

In addition, we will suppose that the decision maker is able to define the categories by means
of some norms or representative elements, called "reference profiles". The human resource
manager is likely to have a clear idea of the profile of a promising manager.
The actions to be sorted will be compared to these profiles in order to determine their assignment. This comparison can be based either on a similarity index or on preference relations. Let
us remark that the comparison of an action to the reference profiles is indeed independent from
the other actions. Clearly, the diagnosis of a patient for instance does not affect the examination
of another patient1 .

Similarly, ranking problems arise frequently in our daily life. A company may want to evaluate
its suppliers in order to develop special partnerships with the most promising ones. The father
(mother) of a family might be interested in the best energy-supplier for his (her) house. Besides,
no student can ignore the existence of university rankings. We have all been confronted once to
the delicate task of ordering actions from the best to the worst one. However, it might be difficult
to obtain a complete order. We are then is presence of a partial order.
The ranking problem is often somehow awkward and difficult since it usually (unfortunately)
involves conflicting points of view or criteria. Most of the time, no best candidate or supplier
exists. Methods of the multicriteria decision aid field, help a decision maker in this decision
process by proposing step stones and techniques to find a (compromise) solution. In ranking
methods, the actions are thus compared pairwise or by means of, for example, an aggregated
score. This constitutes a major difference with sorting methods since the actions to be sorted,
are not compared to each other.
Besides, nothing ensures the decision maker that the best ranked action is actually well suited
for his problem. For instance, all candidates, even the best candidate, may not be adapted for
a specific job. On the other hand, if an action is assigned to the best category, the decision
maker might be sure that the action answers his needs. Obviously, this depends on a good a
priori definition of the categories. This is another distinguishing feature between a ranking and
a sorting method.

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

1 We

12

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

Introduction

Besides developing new sorting methods, we are interested in the properties these ranking-based
sorting methods present. Are there some conditions to be imposed on, for example, the reference
profiles ? Moreover, what are the differences with some existing methods ? Is there a specific
reason or need to tackle sorting problems by ranking methods instead of sorting methods ? Is
there an advantage in defining the categories by means of criteria, even when there is no order on
the categories ? Can existing partially ranking methods be used in problems where the categories
are partially ordered ?
In this work, we try to give a first answer to these questions and to some emerging questions.

This thesis is divided into 3 parts. Part I (Chapter 1 - Chapter 4) contains a review of the
literature on multicriteria ranking and sorting methods. Our main contributions can be found in
Part II (Chapter 5 - Chapter 11) and Part III (Chapter 12 and Chapter 13).
In particular, Chapter 1 is devoted to a brief introduction to the classification problem and
the need of taking preference information into account. In Chapter 2, some aspects of the
multicriteria decision field are presented whereas Chapter 3 is devoted to the description of some
well known multicriteria ranking methods. In Chapter 4, we propose a deep analysis of some
multicriteria sorting methods which use reference profiles. The properties of these methods are
compared as well as their approach. This leads to Tab.4.15 where the methods are compared.
In Part II we use the Promethee ranking method for sorting problems where the categories are
completely ordered. This leads to the F lowS ort method which uses preference relations. In
Chapter 5, we precise the used notations and the conditions of the model. In Chapter 6 we define
the assignment rules. The properties of F lowS ort are analyzed in Chapter 7. The F lowS ort
method is compared theoretically and empirically to Electre-Tri and UTADIS in Chapter 8.
Furthermore, F lowS ort is extended in Chapter 9 to the case where the parameters of the model
are not precisely defined. Finally, we present in Chapter 10 the implementation of F lowS ort in
a user friendly software, developed by students. First conclusions on this approach are given in
Chapter 11.
In Part III we first present a slightly modified version of Electre-Tri for the cases where the
categories are defined by central profiles instead of limiting ones. Moreover, this permits us to
compare in Chapter 12 an outranking-based approach to similarity or indifference based sorting
methods. Finally, in Chapter 13 we propose a first investigation to treat problems where the
categories are partially ordered. The particular problems where the categories are completely
ordered and completely non-ordered are analyzed as well.
In Conclusion, we discuss the most interesting aspects which still deserve to be further investigated. Finally, the proofs of the propositions as well as the numerical data of some examples can
be found in the Appendix.

13

Resumé

Notre thèse est consacrée à l’étude des méthodes de rangements multicritères dans le cadre de
la problématique de tri.
Dans un problème de tri une personne, appelée décideur, désire assigner un objet, appelé action,
à des catégories prédéfinies. Des problèmes de tri surgissent régulièrement dans la vie de tous
les jours. Par exemple, un médecin ausculte son patient et sur base des symptômes observés, il
assigne son patient à une catégorie de pathologies. Ainsi, le médecin peut prescrire un traitement
approprié. Par ailleurs, on catégorise les cyclones tropicaux en fonction de leur vitesse, pression
superficielle et de la hauteur de marée. En fonction de la catégorie du cyclone, des dégâts
éventuels peuvent être prédits et des mesures de protection adéquates devront être prises.
Dans un problème de tri, un décideur regroupe ainsi les actions qu’il considère similaires, à des
fins descriptives, organisationnelles ou préventives. Nous supposerons en outre que le décideur
exprime une relation de préférence entre les classes préalablement définies.
D’autre part, les méthodes de rangement permettent de ranger les actions de la meilleure à la
moins bonne. Nul étudiant ne peut nier l’existence des " rankings " d’universités. Une société
ordonne les candidats à l’issu d’un entretien d’embauche. Une société désire par ailleurs établir
des partenariats avec les fournisseurs les plus performants. Nous sommes tous confrontés à cette
tâche délicate de ranger les actions de la meilleure à la moins bonne. Les méthodes d’aide à la
décision proposent des techniques permettant à un décideur d’obtenir un rangement d’actions.
L’objectif de cette thèse est d’étudier la possibilité de résoudre des problèmes de tri à l’aide de
méthodes de rangement. L’approche adoptée est de ranger une action particulière par rapport à
des normes ou profils définissant les catégories. L’assignation de l’action sera dès lors basée sur
sa position dans ce rangement particulier.
Quelles sont les hypothèses nécessaires pour un tel modèle ? Ces méthodes présentent-elles un
biais ou ont-elles d’autres avantages par rapport aux méthodes de tri existantes? Est-il préférable
de modéliser les catégories à l’aide de critères même si celles-ci ne présentent pas de relation de
préférence ? Dans cette thèse nous donnerons des premièrs éléments de réponse en développant
de nouvelles méthodes de tri basées sur des méthodes de rangement existantes.
15


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

Ph. Nemery, "A multicriteria sorting method for partially ordered categories", Proceedings of the doctoral work shop of EUROMOT 2008 - The Third European Conference
on Management of Technology, "Industry-University Collaborations in Techno Parks", Nice
Mareschal, B. and De Smet, Y. and Nemery, Ph.: "Rank reversal in the PROMETHEE
II method: some new results", to appear in the procedeeings of the IEEE International
Conference on Industrial Engineering and Engeneering Management
Nemery, Ph. and Lamboray, Cl. : "FlowSort : a flow-based sorting method with limiting and
central profiles", TOP (Official Journal of the Spanish Society of Statistics and Operations
Research), 16, 90-113, 2008
Nemery, Ph. and Lamboray, Cl. : "FlowSort : a sorting method based on flows" in Proceedings of the ORP3 Conference, Guimarães, Portugal, 2007, p. 45-60

Besides, two papers are currently under review (both in 4OR).
Two working papers have been published in proceedings of conferences, without peer-review:
Ph. Nemery: "An outranking-based sorting method for partially ordered categories",
DIMACS, Workshop and Meeting of the COST Action ICO602, Paris, Université Paris
Dauphine, 28-31 October 2008
Cailloux, O. and Lamboray, Cl. and Nemery, Ph.: "A taxonomy of clustering procedures" in
Proceedings of the 66th Meeting of the EWG on MCDA, Marrakech, Maroc, 2007

17


Most of the research results has also been presented in various conferences and seminars:
Nemery, Ph.: "On the use of outranking relations in all classification problems", Cost IC0602
International Doctoral School ; Algorithmic Decision Theory: MCDA, Data Mining and
Rough Sets ; Session 2008 : April 11-16, 2008, Troina, Italy
Nemery, Ph.: "FlowSort: a sorting method for group-decision making", Multiple Criteria
Sorting Workshop, February 19, 2008, Université Paris Dauphine, invited speaker
Nemery, Ph. and Janssen, P.: "A sorting method under uncertainty: extensions of FlowSort",
ORBEL 22, Brussels, Belgium, 2008
Nemery, Ph.: "Extensions of the FlowSort sorting method for group decision-making",
MCDM 2008 - 19th International Conference on Multiple Criteria Decision Making, Auckland, New-Zealand, 2008
Nemery, Ph.: "Resolving sorting problems with ranking methods" Cost IC0602 International
Doctoral School, Han-sur-Lesse, Belgium, 2007 (pdf)
Nemery, Ph. and Lamboray, Cl. : "FlowSort : a flow-based sorting method with limiting and
central profiles" ORP3, Guimarães, Portugal, 2007
Nemery, Ph. and Lamboray, Cl. : "FlowSort: a sorting method based on flows: some extensions" 22nd EUROPEAN CONFERENCE on Operational Research, Prague, Czech Republic, 2007
Casier, A and De Smet, Y and Mareschal, B and Nemery Ph.: "About the interpretation of
unicriterion net flows in the PROMETHEE method" 22nd EUROPEAN CONFERENCE on
Operational Research, Prague, Czech Republic, 2007
Nemery, Ph. and Lamboray, Cl. and Huenaerts, L: "FlowSort : a sorting method based on
flows" ORBEL 21, Luxembourg, January 2007
De Smet, Y. and Nemery Ph.: "The sorting problem based on disjunctive categories : a first
investigation", EURO 2006 Conference, Reykjavik, Iceland, July 2006

18

Notations

• A = {a1 , . . . , an }: a set of n actions
• F = { f1 , . . . , fq }: a set of q attributes
• G = {g1 , . . . , gq }: a set of q criteria
• Ω = {ω1 , . . . , ωq }: the set of weights associated to the q attributes or criteria
• C = {C1 , . . . ,CK }: a set of K classes or categories
• R = {r1 , . . . , rm }: a set of m reference proﬁles
• CX (ai ): the set of classes or categories to which ai is assigned, according to procedure X
• S(ai , a j ): the outranking degree of action ai over a j
• π(ai , a j ): the preference degree of action ai over a j
• Pk (ai , a j ): the uni-criterion preference degree of action ai over a j on criterion k
• cS (ai , a j ): the partial outranking concordance degree of ai over a j on criterion k
k
• cI (ai , a j ): the partial indifference concordance degree of ai over a j on criterion k
k
S
• dk (ai , a j ): the partial outranking discordance degree of ai over a j on criterion k
I
• dk (ai , a j ): the partial indifference discordance degree of ai over a j on criterion k

• | {.} |: the cardinality of the set {.}
• N: the set of natural numbers

19

1

Introduction to classification
problems

In this chapter we give a brief introduction to the classification paradigm and
differentiate several existing grouping problems (clustering problems, ordinal
and nominal classification problems). Some well-known "classical" classification methods are briefly described. The need for taking into account preference
information into the classification method is furthermore intuitively proposed.

1.1

General Introduction

Generally, we may define "to classify" by organizing data into groups which share common
characteristics. Grouping problems have been extensively studied in the literature and are
commonly encountered in various application fields such as health care, biology, finance,
marketing, agriculture, etc. [Richard et al., 2001], [Doumpos and Zopounidis, 2002]. Many
terms can be found such as problems of classification, segmentation, discriminant analysis,
filtering, clustering, etc. Nevertheless, two major families of problems are usually distinguished:
the supervised and the unsupervised grouping problems.
In unsupervised groupings problems, there is no a priori information available about the groups
(which are often called clusters in this context). The purpose is precisely to elicit a structure
in a given data set. Generally, the aim is, in this context, to obtain different clusters of objects
where objects of a same cluster are "similar" and objects of different clusters "dissimilar".
The similitude notion is often expressed in terms of object proximity, distance, similarity or
dissimilarity measures, etc.
One might for instance consider a marketing problem where the aim is to discover similar
customer behaviors in the retail industry which permits to detect different types of clients. In
biology for instance, scientists regroup species of organisms, according to (for example) shared
physical characteristics. This leads to a taxonomy of the species. In discriminant analysis, the
most famous example is the Fisher’s Iris data set analysis. On the basis of four features (length
and width of sepal and petal), three species (flowers) of Iris may be determined (iris setosa, iris
versicolor and iris virginica).
Among the most common unsupervised grouping or clustering procedures, one may cite the
23


K-means, hierarchical, finite mixture densities algorithms, etc. [Hartigan, 1975], [Oliver et al.,
1996],[Jain and R., 1998],[Jain et al., 1999],[Doreian et al., 2005a]. The reader will find a survey
of clustering procedures in [Hartigan, 1975],[Jain et al., 1999],[Cailloux et al., 2007],[Nemery,
2006].
On the other hand, the groups may be defined a priori. C. Zopounidis and M. Doumpos define
the (supervised) classification problem as follows: "Classification refers to the assignment of a
finite set of alternatives into predefined groups [Doumpos and Zopounidis, 2002]." The purpose
is thus not to discover or elicit the groups, but to label objects according to the definition of the
groups, called classes. We may think for instance about the medical diagnosis problem where
a new patient has to be assigned to a known pathology-class based on a set of symptoms. In
information science, documents are classified to one or more classes based on their contents. In
Australia, the Office of Film and Literature Classification is a government funded organization
which classifies all films that are released for public exhibition. There exists different classes
such as the E-class (films exempted of classification, e.g. documentaries), the G-class (general
films with a content which is very mild in impact), the PG-class (films for which parental guidance is recommended), the M-class (films recommended for mature audience), etc. [Wikipedia,
2008]. A new film will classified to one of these classes. In this work, we focus our interest
on supervised classification problems and we use abusively the term classification instead of
supervised classification.
The general idea behind classification is thus to predict the class membership of a set of new
objects on the basis of assignment rules. Most of the classification methods, proposed for the
development of classification models, exploit the knowledge that is provided through the a priori
definition of the groups. The model may be extracted from a set of classified examples, referred
to as the the training sample or reference set and noted R = {r1 , . . . , rm }. This set consists of a
collection of pairs: (an object, a class).
The class label will be denoted by y, taking its values in the discrete set C = {C1 , . . . ,CK } where
K is the number of classes. The objects, called actions or alternatives, are described by means
of independent variables, noted g1 , . . . , gq . The set of variables is denoted by G . Henceforth,
the independent variables will be referred to as criteria or attributes. The attributes, such as
properties or characteristics, define a nominal description of the actions (e.g. a color, a measure,
etc.) and allows to express (or measure) if two actions are similar. On the other hand, a criterion
defines an ordinal description enabling to specify if an action is preferred over another1 . Each
action of the training sample will be considered as a vector consisting of the performances of
the action on each variable: r j = [g1 (r j ), . . . , gq (r j )].
The goal of the classification model is to develop an application fG which maps any action,
defined by the vector of independent variables g, to the dependent variable, y, its classification laˆ
bel, where y ∈ C. Formally, we may represent the model as follows: G → C : g → y = fG (g) = C.
This is illustrated in Fig.1.1 taken from [Doumpos and Zopounidis, 2002]. Let us remark, that
an object may, according to some classification procedures, be assigned to none, one or several
1A

24

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

1.1. General Introduction

classes ( fG is thus not an application anymore).

Figure 1.1 — Representation of the classification model in classification problem taken from
[Doumpos and Zopounidis, 2002], p.7.

The development of such a model is done such that the a priori classification of the elements
ˆ
of the training sample (C), corresponds as much as possible to the estimated classification (C).
If the model performs "well" (i.e. if there is a high classification adequacy), the model can be
used for assigning a set of new objects, noted A = {a1 , . . . , an }, described by G to one or several
categories. This is illustrated in Fig. 1.2 where ai actually represents g(ai ), ∀i = 1, . . . , n.
In classification problems, classes are predefined, designed or conceived relatively to the treatment or the consequences that will be given to the actions belonging to a same group [Roy and
25


9

C

6
A

a1
•
a
•2
fG

aj
•
a
•i

.
z .
.
s

a
•3

a
•n
.
8. .

C1

E

Cj
.
.
.
CK

7

Figure 1.2 — Representation of a classification procedure fG which assigns each action ai to
none, one or several categories of the set C = {C1 , . . . ,CK }.

Bouyssou, 1993]. Actions assigned to a same group, will thus be investigated, treated, used, etc.
in a same manner. For instance, all the patients, assigned to a common known pathology-class,
may receive a similar treatment. All the documents of a same class may geographically be
regrouped in a library.
A decision maker confronted to a classification problem, needs to choose an assignment
procedure adapted to his problem. The choice of this procedure is obviously crucial and may
be influenced by several factors. Indeed the way of defining the classes, the properties that
the method should fulfill but also the meaning given to classes have to be considered before
opting for a particular method. In the literature we usually distinguish classification and sorting
problems. C. Zopounidis and M. Doumpos mentioned that "Classification refers to the case
where the groups are defined in a nominal way. On the contrary, sorting (a term which is used
by multicriteria decision aiding researchers) refers to the case where the groups are defined in
an ordinal way, starting from those including the most preferred actions to those including the
least preferred actions [Zopounidis and Doumpos, 2002a]."
In this work we consider a nominal classification problem as follows. The decision maker defines
the classes such that he considers the actions belonging to different classes as dissimilar or not
enough similar. Moreover, if he defines the classes by representative elements, he considers
them as not similar (different, dissimilar). This opinion may be based on attributes providing a
description of the classes or the representative elements.
On the other hand, several authors consider sorting problems when the classes (called categories
in this context) are defined in an ordinal way: the categories are completely ordered from the best
26

1.2. Classification Methods

to the worst [Doumpos and Zopounidis, 2004a]. As an example we may cite the different hotel
categories where a four-star hotel is considered to be better than a one-star hotel. Chemicals
may be sorted in different categories corresponding to different dangerousness. The revised
Annex II Regulations for the control of pollution by noxious liquid substances includes a new
four-category categorization system for noxious and liquid substances [Imo-Org, 2008]. In these
previous examples, the categories are completely ordered.
Let us remark, that in Chapter 4, we present a more exhaustive view of the different classification
problems.
This chapter is devoted to nominal classification methods. In Section 1.2 we describe briefly
some classical classification methods. Furthermore, in Section 1.3, we present intuitively the
needs for taking into account preference information in the classification model.

1.2

Classification Methods

Methodologies for addressing classification problems have been developed from a variety of
research areas, such as statistics and econometrics, artificial intelligence, operations research, etc.
Usually we distinguish two main families of classification methods: the parameter-based
and parameter-free techniques. In the former case, the classification problem is addressed by
statistical and econometric techniques using statistical assumptions on the data set. Among
others, one may cite the linear discriminant analysis and the quadratic discriminant analysis
(based on a priori probability distributions) [Fisher, 1939], the linear probability model, the
logit and probit analysis (based on the development of a non-linear function measuring the
group-membership probability) [Berkenson],[Bliss, 1934], etc. However, these techniques have
been severely criticized for their statistical assumptions [Altman et al., 1981].
On the other hand, in parameter-free techniques no statistical assumptions are made. The
methods will adjust themselves according to the characteristics of the data [Zopounidis and
Doumpos, 2002b]. One may cite among others the neural networks [Culloh and Pitts, 1943;
Zadeh and Nassery, 1999], machine learning [Goldberg, 1989], decision trees [Quinlan, 1986],
fuzzy set theories [Zadeh, 1965], rough sets [Pawlak, 1984a], k-nearest neighbors [Fix and
Hodges, 1951; Han and Kamber, 2001], etc.
The reader may find more information about these methods in [J.B.Mac Queen, 1966; Rulon
et al., 1967; Gower and Legendre, 1986; Wallace and Dowe, 1994; Batagelj and Ferligoj, 1998;
Lortie and Rizzo, 1999; Zopounidis and Doumpos, 2002b; Doumpos and Zopounidis, 2002].In
the next section, we will briefly describe one of these methods, namely the k-nearest neighbor.
Nevertheless, previous classification methods do not incorporate decision maker’s preferences.
As we will in Section 1.3, this can play a crucial role in the assignment results.
27


1.2.1

k-Nearest Neighbors

This method has been initially introduced by [Fix and Hodges, 1951] and its mathematical
properties have been given by [Hart, 1967]. Practical applications have been discussed by
[Fukugana and Hummels, 1987].
In this method, the assignment of a action ai is based on the proximity of ai to the actions of
the training set. The proximity is usually expressed by means of a distance, a (dis-) similarity
measure or a proximity measure. Generally, a dissimilarity measure has the following properties
[Doreian et al., 2005b] : ∀xi , x j ∈ A ∪ R : d (xi , x j ) → ℜ with
1. d (xi , xi ) = 0
2. d (xi , x j ) ≥ 0, non-negativity
3. d (xi , x j ) = d (x j , xi ), symmetry
When the following conditions are also satisfied, the dissimilarity measure is called a distance :
4. d (xi , x j ) = 0 ⇒ xi = x j
5. ∀xz : d (xi , x j ) ≤ d (xi , xz ) + d (xz , x j ), triangle inequality

For numerical data we can use the L p distance :
d (xi , x j ) = xi − x j

(1.1)

p

with

1/p

q

xi − x j

p

p

= ( ∑ |gk (xi ) − gk (x j )| )

(1.2)

k =1

where 1 ≤ p < ∞. The higher the values for p, the bigger the importance attached to the
differences. For p=2, we find the well-known Euclidean distance ; the Manhattan distance
corresponds when p=1 and when p = ∞ it is equal to the maximum of absolute difference in
coordinates.
On the other hand, different similarity indexes can be defined for numerical attributes such for
example the similarity, the cosine and the Dice coefficient as well as the distance exponent which
are respectively given by the following formulas :
s(xi , x j ) = 1/(1 + d (xi , x j ))

(1.3)

T
scos (xi , x j ) = xi x j / xi . x j

(1.4)

T
sDice = 2xi x j /( xi

2

+ x j 2)

(1.5)

α

(1.6)

sexp = exp(− xi − x j
28

)

1.2. Classification Methods

For categorical data, similarity measures can also be defined [Everitt, 1993; Dubes; Jain et al.,
1999]. Assuming binary attributes with values α, β = ±, let dαβ be a number of attributes having
outcomes α in xi and β in x j . We can then define respectively the Rand (Eq.1.7) and Jaccard
(Eq.1.8) indices :
d++ + d−−
R(xi , x j ) =
(1.7)
d++ + d+− + d−+ + d−−
J (xi , x j ) =

d++
d++ + d+− + d−+

(1.8)

where d++ corresponds thus to the number of attributes for which xi and x j have the same
response.
From the training set, constituted by a set of training actions and their labels, the subset of the
k (with k ∈ N) nearest training actions to ai (called the k neighbors) is extracted. The action ai
is assigned to the class which is the most represented among the k neighbors. Let us consider
Fig.1.3 where two classes, C1 and C2 , are defined by the training set R = {r1 , . . . , r6 } and where
{r1 , . . . , r3 } are representatives of C1 and {r4 , . . . , r6 } of C2 . Based on for instance the Euclidean
distance, if we fix k = 1, ai will be assigned to C2 (since d (ai , r5 ) < d (ai , r j ) with j = 5). On the
other hand, if k = 3, ai will be assigned to C1 .
g2
r6

r3
k=1

s ai

r1
r2

r4
r5

k=3

g1

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

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

29


A main advantage of this method lies indeed in its simplicity. Moreover, no assumption is needed
on the data and some interesting optimality features have been proven in [Bezdek, 1991]. Nevertheless, it suffers from the drawback of the need of a high memory-space [Belacel, 2000a].

1.3

Need of preference information

In this section we briefly motivate the need of taking preference information into account. This
section aims not to formally define some concepts but rather to give a first intuitive approach to
the reader.
As pointed out in previous section, classical classification methods used statistical assumptions
on the data, distances, similarity measures, etc. for assigning the actions to the categories. The
used measures of the model are most of the time symmetrical or do not consider preference
information.
For simplicity reasons we consider in this section "preference information" as information on
the basis of which a decision maker might express a preference of an action on another action.
For instance, when comparing two actions a and b on the basis of the price, a client might prefer
action a on b if the price-value of a is less than the price-value of b. The client aims to minimize
the price. On the other hand, a vendor might prefer b since he would like to maximize his profit.
Minimizing or maximizing the values of the characteristics of the actions permits to establish an
order on the set of characteristic values and thus to express a preference2 .
To illustrate intuitively the impact of taking into account this preference information let us
consider Fig.1.4 and 1.5. In the former case, we use so-called attributes ( f1 , f2 ) for describing the
actions whereas in the latter case, so-called criteria (g1 , g2 ) where we suppose that the features
of the objects have to be maximized. The main difference between attributes and criteria lies
in the fact that we associate preference information to the features describing the actions (e.g.
a decision maker’s preference orientation). Attributes and criteria will be precisely defined in
Chapter 2.
In the first case, we might use a similarity relation (or a distance) to compare the objects
a1 , a2 , a3 , a4 and b. In the first figure, when working with the attributes f1 and f2 , we can notice
that all the points of the circle (with b as midpoint of the circle) are at the same Euclidean
distance to point b. We can thus consider, that they are all similar or dissimilar.
Let us now consider that the decision maker considers that both criteria have to be maximized.
We can thus notice that a1 is, on both criteria, better than a2 , a3 , a4 , b. We will say that that a1 is
preferred to a2 since it is better than a2 on both criteria. This will be note as follows: a1 a2 .
We have thus moreover that a1 a j , with j = 2, 3, 4, and a1 b. Actions a2 , a3 and b are
analogously preferred to a4 . On the other hand, the decision maker might not be able to compare
2 In

30

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

1.3. Need of preference information

a2 to a3 since a2 is better than a3 on criterion g2 and a3 is better on criterion g1 . Actions a2 and
a3 are thus considered as incomparable (which is noted as follows: a3 J a2 ).
f2

g2
T
q a1

a2q

q a1

a2q

J
b
q

q

b

J
q

a4

qa

q

3

a4
f1

Figure 1.4 — Illustration of the utilization of attributes.

qa

3

E g1
Figure 1.5 — Illustration of the utilization of criteria.

We can thus remark that there exists three different "sub-zones" on the circle (with b as midpoint
of the circle) although all points are at the same distance: the points "preferred by" b, represented
in red (with for example a4 ), a zone of points which are incomparable to b (the brawn zone)
and points which are preferred to b, represented in green (with for example a1 ). We can thus
notice, that the fact of taking into account preference information, permits to refine or precise
the comparisons between the actions.
The aim of sorting procedures is exactly to take into account this granularity introduced by the
preference information. As illustration of this, let us consider the following basic classification
problem (represented in Fig.1.6) where four categories, noted Ci , (∀i = 1, . . . , 4), have been
defined by some typical representatives elements. These elements are called profiles and noted
ri , ∀i = 1, . . . , 4. If we use a similarity relation or a distance, we can conclude from the left figure
of Fig.1.6 that action a compares it-self in the same way to all the reference profiles. This can be
motivated by the fact that the Euclidean distance are the same: d (ri , a) = d (r j , a), ∀i, j = 1, . . . 4).
It might thus be difficult to assign action a to a category rather than to another.
On the contrary, if the decision maker considers that both criteria have to be maximized, he
obtains the following relations: r1 a, r2 J a, r3 J a and a r4 (Fig.1.6-right). In this context, the
decision maker might have his own reasons to assign a to a particular category considering the
preference relations between a and the profiles ri . He might adopt for:
• an optimistic approach by assigning a to category C1 since r4 , r3 , and r2 are not preferred
to action a.
• a compromise approach by assigning a to category C2 and C3 since a behaves as r2 and r3
do with respect to r1 and r4 .
• an pessimistic approach by assigning a to category C4 since a is not preferred to r2 , r3 and
r1
31


f2

g2
T
q r1

r2 q

q r1

r2 q

J
a

a
q

q

J
r4

qr

q

3

r4

qr

q

3

f1

E g1

Figure 1.6 — Comparison of an action a to 4 different reference profiles ri , (i = 1, 2, 3, 4).

• ...
The approach will obviously be chosen by the decision maker and depend on the classification
model.
Let us consider now, that the decision maker wants to minimize solely criterion g1 . We have
thus other preference relations between a and the reference profiles (Fig.1.7-right): r2 is now
preferred to a instead of incomparable, etc. The consequence of this, is that according the same
adopted approach, the assignments might be different.
g2
f2
T
q r1

r2 q

q r1

r2 q

J
a

a

q

q

J
r4

q

qr

3

r4
f1

q

qr

3

g1
'

Figure 1.7 — Comparison of an action a to 4 different reference profiles ri , (i = 1, 2, 3, 4).

We can thus conclude that the preference orientation plays a role in classification. This will
formally be defined in Chapter 4 by a property of preference-orientation dependency.
In this work, we will analyze classification procedures which take preference information into
account. In the next chapter, we define more formally the notions of preference, preference information, etc.

32

2

Introduction to Multicriteria
Decision Aid

The aim of this chapter is to give a short introduction to the Multicriteria Decision Aid (MCDA) field by presenting some important notions and definitions.
It is of course impossible to propose an exhaustive bibliography of what has
been done in almost thirty years, but it is not the purpose.
Firstly, we address the motivations of "aiding" actors in making decisions in
a complex world. This discipline offers, like suggested in the term-itself, an
"aid". It does not necessarily "solve" a problem.
As we will see, the philosophy of this young research field can be seen as one
of the logical continuations or extensions of the Operational Research (OR)
branch. Nevertheless, it will certainly not supplant the Operational Research.
The notions of actions, criteria, preference structures, etc. will be introduced
and discussed in this chapter. The different type of problems encountered in
MCDA will furthermore be briefly explained.

2.1

Motivations

Operational Research (OR) emerged just before the outbreak of World War II. General Pile,
Commander in Chief of the Anti-Aircraft Command in Great Britain, requested scientific
assistance for the coordination of the radar equipment at gun sites, which gave the slant rage and
bearing an attacking bomber with some newly approaches. Concretely, radars had to be placed
optimally to warn citizens of Great Britain of an eminent attack. Meanwhile, some other people
became involved in problems concerning the detection of ships and submarines by the use of
radar equipment in airplanes [Closkey and Trefethen, 1980; Pomerol and Barba-Romero, 1993].
Two years after the beginning of the war, Britain’s military services had acquired formally
established operational research groups.
The OR discipline has first been applied on military problems (hence its name) but after a
while it has known a huge expansion in the industrial world. The techniques and the developed
methods have been successfully applied on several different problems (problems of industrial
planning, transportation problems, scheduling problems, combinations problems, traveling
33


problems, etc.) and are still used in many different fields. Concrete examples of applications can
be for instance the determination of the shortest path between a source and destination, finding
the optimal sequence of set of operations to be performed on some goods, resolving some highly
constraints problems, etc.
The classical OR problems were generally tackled by the modeling of the problems with a
unique criterion function. The aim of the mono-criterion modeling is to obtain an optimization
problem (maximization or minimization) under several constraints, which optimal solution
represents the best choice.
Where in the beginning, only one criterion function had to be optimized (for example the price,
the distance and/or the cost), different aspects (like for example environmental, human, esthetics
or power criteria) need to be considered when making a decision. Most of the time, we have to
deal with conflicting criteria when facing a complex and global problem. For instance, when
engaging a new employee, a company will have to choose between several candidates. The
headhunter may be looking for an experienced candidate, with a high educational degree but
who is still young. If the headhunter chooses the person with the highest experience, he will
probably opt for an older candidate. Alternatively, a young candidate may present a lack of
experience. The headhunter will have thus to make a compromise: no best solution exists.
Moreover, when facing and comparing two alternatives or possible solutions, a decision maker
can express a preference, an indifference or an incomparability. The incomparability can be due
to the lack of information or to the fact that the solutions are too different.
One interesting feature of MCDA is pointing out these two aforementioned situations which is
difficult to bring to the light when using an unique criterion-function.
On the other hand, using an unique criterion-function or aggregating all the criteria to a
unique and artificial value (what is done with the multi-attribute utility function) leads to the
transitivity of preference and indifference. This can be severely criticized and can be refuted in
the following situation. Consider 401 cups of coffee noted C0 , C1 , ..., C400 . One assumes that
i
the cup Ci contains exactly 1 + 100 grams of sugar. In this context, any normal person is unable
to differentiate two successive cups. We have thus an indifference situation between C0 and C1 ,
C1 and C2 , C2 and C3 , ..., C399 and C400 . Nevertheless, like Luce explained in [Luce, 1956], it is
obvious that no one will consider that C0 and C400 are indifferent to him since there is now a
difference of 4 gram between both cups.
The MCDA field did not appear as the Messiah for solving all the complex problems of the
world. The MCDA is an aiding tool for a multicriteria paradigm but certainly not a decision
making one. There is thus a philosophical change in approaching the problem that can be
found in marketing as well. In the beginning, the product was the central point in a marketing
campaign. Nowadays, the client, and the associated services, are central. By the same way,
the decision maker is now the central actor: the optimization function has been placed like the
product, in the "back-yard".
As we will see, MCDA will first of all responsibilize the decision maker and make him aware of
the aspects, the aims and/or the consequences related to the decision that he will make. This is,

34

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

in our point of view, the first purpose of the MCDA. It is certainly not to give a final decision as
different methods can propose different results. Moreover, it is necessary to explain the decision
maker, the differences between the existing methods (hypothesis, advantages, disadvantages,...)
and discuss with him which one may be the most appropriate for his problem. These are some
of the roles of the analyst, who helps a decision maker, facing a problem.
Although the decision maker is usually not conscious of it, the process of making a decision
generally involves four phases. Moreover, the instant of decision can not (always) be identified.
Schärlig distinguishes during this process the phases of information, conception, choice and
retrospective analysis [Schärlig, 1985]. The order between these phases can of course be different
and is often characterized by passages from one phase to another, depending on the progression
of the consideration.
The phase of information corresponds to the horizon seeing. The candidates for a solution are
detected and the conceivable criteria are considered as well as. The phase of conception, on the
other hand, allows to define the set of choice (i.e. the candidates to be determined) and their
evaluations on the different chosen criteria. The decision making is done on the final set of
candidates, not necessarily corresponding to the initial one. The retrospective analysis is rarely
done by a formal study but is certainly presented in the decision maker’s mind.

2.2
2.2.1

The actions, the criteria, . . . and the problems
The set of actions A

When facing a decision problem, the first step may be to identify the different objects submitted
to the decision making process. These objects can be potential decisions, projects, feasible
solutions, items, units, alternatives, candidates, etc. and will be called the actions. The set of
actions will be noted in the rest of this work A .
A can be defined in extenso (an enumeration of all the actions is thus possible: A = {a1 , ..., an })
or by comprehension ( mathematical properties or characterizations) when the set is too big or
infinite.
As mentioned before, the decision process may be evolutive. This implicates that the actions are
not always defined once and for all. When the actions evolute, A is said to be evolutive. On the
other hand, A is called stable when it is defined a priori and will not change [Vincke, 1992].
Finally, A can be globalized, if each element excludes any other, or can be fragmented, if combinations of elements from A constitute possible issues [Vincke, 1992].

2.2.2

The set of attributes F

An attribute [Latin: attribuere ; attribut: ad- + tribuere: to allot [The Free Dictionary, 2008]] is
a function f , defined on A , taking its values in a set, noted V , ordered or not. It represents a
feature or a characteristic inherent in or ascribed to an action [The Free Dictionary, 2008].
35


As several attributes will be considered, we will note f j the j-th attributes and vij = f j (ai ) the
evaluation of the i-th action of A on this j-th attribute. The set of all the attributes will be noted
F = { f1 , ..., fq }.

2.2.3

The set of criteria G

The actions of a decision problem will be analyzed and evaluated according to the decision
maker’s (DM’s) point of view and preferences. A criterion [Greek: kriterion, from krites, judge,
from krinein: to separate, to judge [The Free Dictionary, 2008]] can be defined as "A standard,
a rule, or a test on which a judgment or decision can be based [The Free Dictionary, 2008]".
Vincke defines formally a criterion as follows [Vincke, 1992]:
Definition 2.1. A criterion is a function g, defined on A , taking its values in a totally ordered set
and representing the decision maker’s preferences according to some point of view.
g : A → V where V is a totally ordered set
If V is for instance the set of real-values, we suppose thus implicitly that the criterion has to be
maximized.
A more detailed and comprehensive definition can be found in [Roy and Bouyssou, 1993] where:
A function g with real values defined on A , is for a decision-maker a criterion-function or criterion ... if the decision maker recognizes the existence of an axis of significance on which two
possible actions ai and a j may be compared ... and he accepts to model this comparison as
follows:
g(ai ) ≥ g(a j ) ⇒ ai Sg a j ⇔ ai outranks a j on criterion g
⇔ ai is at least as good as a j on criterion g
where Sg defines a binary outranking relation restricted to the signification of criterion g.
As several criteria will be considered, we will note g j the j-th criterion and eij = g j (ai ) the
evaluation of the i-th action of A on this j-th criterion. An action ai will be represented
by the following vector : ai ≡ [ei , ..., ei ]. Moreover, the set of all the criteria will be noted
q
1
G = {g1 , ..., gq }. We will suppose, except of explicit counter-indication, that the criteria have to
be maximized. As we will see in Section 2.5, the set of criteria has to respect some conditions.
The complete characterization of the criterion (aspects, values, factors,...) is one of the most
difficult and crucial steps in a decision aiding process. Roy and Bouyssou has proposed a
methodology to construct G as a set of coherent criteria. This will be presented in Section 2.5
[Roy and Bouyssou, 1993] .
Let us remark that Vincke distinguishes, in [Vincke, 1992], several types of criteria such as
real-criterion, quasi-criterion, pseudo-criterion and interval-criterion depending on the induced
underlying preference structure.

36

2.3. The Pareto dominance relation

2.2.4

The different types of MCDA problems

When searching for an optimal solution in "traditional" problems, we model the situation such
that the set of considered actions is fixed once for all, such that every solution is exclusive
from the others and such that solutions can be ranked incontestably from the worst to the best.
However, the set of actions doesn’t necessarily fulfil this three characteristics. This is the reason
why it is sometimes preferred to analyze the problem differently.
Having the set of actions A and a set of criteria G , a decision maker may be facing different
type-problems [Roy and Bouyssou, 1993]:
• The choice problem : a subset of actions considered as the best according to the criterionset G , has to be chosen [the α − problem]
• The sorting problem : a partition of the set A must be done with respect to some preestablished norms [the β − problem]
• The ranking problem : a ranking of all the actions from the best to the worst must be
realized [the γ − problem]
• The description problem : a description, in an appropriate language, of the actions and
their consequences has to be given [the δ − problem]
Let us remark that other reference problems may be found in the literature (see for instance in
[Bana e Costa, 1990],[Henriet, 2000]). Furthermore, real problems often combine simultaneously several of these problems as we can cite for example the portfolio problem, the design
problem, choosing k among n actions [?], etc.
Moreover, same problems may lead to different elaborations of A , G and different problematics
[Vincke, 1992; Roy and Bouyssou, 1993].
Besides, as will explained later, the main idea of Part II of this work, is the using of an existing
ranking method for tackling a sorting problem.
To tackle these problems, we may use the Pareto dominance relation, define a preference
structure, compare the actions pair-wise, etc. This will be the subject of the next sections. We
will tackle the specific problems of ranking and sorting in the next chapters.

2.3

The Pareto dominance relation

Let us first, regardless of the sublying preference structure, the inter-criterion relations and the
type of MCDA problem, define the following notions. Let us remind that we suppose that the
criteria have to be maximized.
37

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