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université paris dauphine d.f.r. sciences des organisations

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THESE

Pour l’obtention du titre de DOCTEUR EN INFORMATIQUE

spécialité: aide à la décision

Les Structures Mathématiques et Logiques pour la Comparaison des Intervalles

Candidate: Meltem ÖZTÜRK

Directeur de thèse: Alexis TSOUKIAS

Directeur de recherche au CNRS Rapporteurs: Patrice PERNY

Professeur à l’Université Paris VI Marc PIRLOT

Professeur à la Faculté Polytechnique de Mons Suffrageants: Denis BOUYSSOU

Directeur de recherche au CNRS Fred ROBERTS

Professeur à l’Université de Rutgers Pierre MARQUIS

Professeur à l’Université d’Artois Philippe VINCKE

Professeur à l’Université Libre de Bruxelles Présentée et soutenue publiquement le 09 décembre 2005

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L’université n’entend donner aucune approbation ni improbation aux opinions émises dans les thèses: ces opinions doivent être considérées comme propres à leurs auteurs.

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Remerciements

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Contents

Introduction 9

I Preference Modelling with Intervals 13

1 Crisp Preference Structures 15

1.1 Preference modelling and decision aiding . . . 16

1.2 Basic notions . . . 25

1.3 Classical crisp orders . . . 28

1.4 Crisp orders with interval representation . . . 32

1.4.1 Crisp hP Ii orders with interval representation . . . 33

1.4.2 Extended crisp orders with interval representation . . . 46

1.5 Forbidden posets . . . 52

1.6 Conclusion . . . 60

2 n-Points Intervals 63 2.1 Basic notions . . . 65

2.2 hP≤ϕn , I≤ϕn i interval representation . . . 72

2.3 2-points intervals and 3-points intervals . . . 110

2.3.1 2-points intervals . . . 110

2.3.2 3-points intervals . . . 114

2.4 Coherence conditions . . . 123

2.5 Conclusion . . . 132

3 Fuzzy Preference Structures 135 3.1 Fuzzy set theory . . . 136

3.2 Fuzzy preference structures . . . 142

3.2.1 FuzzyhP Ii preference structures . . . 143

3.2.2 FuzzyhP QIi preference structures . . . 145

3.2.3 Comparison of intervals with valued hesitation . . . 153

3.3 Fuzzy orders with interval representation . . . 159

3.3.1 Classical fuzzy orders . . . 159

3.3.2 Fuzzy interval orders . . . 160 7

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3.3.3 Comparison of intervals with fuzzy Ferrers characteristic relation . . 164

3.4 Conclusion . . . 172

II Four Valued Logic and Its Continuous Extension 175

4 DDT Logic and Its Use for hP QIi Interval Orders 177 4.1 Bipolarity in value theory: historical discussion . . . 178

4.2 Four valued logic . . . 180

4.3 Known results concerning the use of DDT logic in preference modelling . . 188

4.3.1 PC preference structure . . . 188

4.3.2 (R+, R)- representation of a PQI interval order . . . 191

4.4 A monotone (R+, R)- representation for a PQI interval order . . . 193

4.4.1 Monotonicity . . . 193

4.4.2 Weak substitution . . . 198

4.4.3 Hybrid interval-representation . . . 201

4.5 Conclusion . . . 205

5 Fuzzy DDT Logic and Possibility Theory 207 5.1 Continuous extension of a four valued logic . . . 208

5.2 B(α) as a standard necessity mesure . . . 221

5.3 B(α) as a sub-normalised necessity measure . . . 223

5.4 Conclusion . . . 226

Conclusion 229

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Introduction

The comparison of objects plays an important role in several fields that call for the mod- elling of real situations. It is necessary to better understand and represent the problem studied, making it a crucial step for decision aid systems.

A decisional problem arises through the interaction between two factors: “the real sit- uation of the problem” and the “decision maker”. These two factors must be taken into consideration in the comparison of objects, which is done through the use of binary re- lations. The set of these relations is called a preference structure. This structure must reflect, on one hand, the preferential information given by the decision maker and, on the other hand, the real situation of the problem.

In practice, for a decision problem, there are two cases in which a preference structure may be needed (see [Vin01]):

- in the first case, calledcomparison problem, the objects that need to be compared are evaluated at the beginning of the decision process and we try to construct preference relations that will be coherent with the evaluations and the visions of the decision maker,

- in the second case, called numerical representation problem, the decision maker ex- presses his preferences over each couple of objects and we try to associate a numerical representation to each object coherent with the information given by the decision maker.

The two pillars of the decision problem - the real situation and the rational model of the decision maker - may lead to certain difficulties. These difficulties often require some kind of scientific intervention. The expert, in charge of such an intervention, tries to help the decision maker determine his “rational model”. In particular, we distinguish two sources of difficulties.

- The first difficulty concerns the uncertainties linked to the unknown, incomplete, and possibly even inconsistent information that is available.

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- The second difficulty is linked to the representation of the preferential information. In fact, the situation model must be done with respect to the decision maker’s rational model. This model is not independent of the context of the decisional problem, the experience, the vision, the conviction and the attitudes of the decision maker. All these aspects give rise to complex preferences.

Confronted with these difficulties, different preference structures propose diverse prefer- ences relations with different properties and numerical representations. The more classical ones, linear orders and weak orders, use two relations, the preference and the indifference.

These structures define the preference as an asymmetric and transitive relation, while the indifference is defined as a symmetric and transitive relation. However, the transitivity of the indifference is often criticized by researchers and is refuted through experiments.

These critiques gave rise to several preference structures defining the indifference relation as an intransitive one. Among the most well known are the semiorders, the interval orders, the split semiorders, the split interval orders and the tolerance orders. The majority of these structures uses interval representation. In fact, in practice, the intransitivity of the indifference relation is often related to the presence of discrimination thresholds and an interval representation is equivalent to a representation with thresholds.

The preference structures that use two preference relations suppose that, when com- paring two objects, a decision maker will either prefer one to the other (relation P) or be indifferent between the two (relation I). Nevertheless, the decision maker may have more complex preferences that require more complete structures that permit him to express a hesitation between a preference and an indifference. This need has given rise to other structures, such as orders with two thresholds, the pseudo orders etc., to emerge where such a hesitation is authorized through a third relation (relation Q). The introduction of this new relation is also done through thresholds and, in consequence, these structures that we call extended ones, can also be represented by intervals.

The use of intervals is also interesting for the comparison problem, since certain uncer- tain information can be represented by intervals.

The complexity of the situation, the uncertainty of the information and the vision of the decision maker can still render the construction of preference structures even more difficult. In certain types of situations, it may be preferable to have fuzzy preferences. The introduction of fuzzy relations can make the passage from one relation to another more

“gentle” and may further approach the decision maker’s vision.

All these remarks constitute the motivation of our thesis, where

- we are interested in preference structures with intransitive indifference,

- we consider the case of structures with two (P andI) or three (P, QandI) preference relations for which we distinguish two study cases: “crisp case” and “fuzzy case”, and

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11 - we use intervals to represent our preference relations.

The construction of preference structures requires a formal language. In this optic, we choose logic. The logics that we use are classic logic, fuzzy logic, four valued logic and the continuous extension of a four valued logic. The last two logics differ from the first two by the fact that they distinguish incomplete information from inconsistent information.

The objective of our thesis is to enrich the knowledge and results for preference struc- tures with interval representation.

Outline of the thesis:

In this thesis we differentiate two separate parts entitled :

• Preference modelling with intervals

• Four valued logic

The distinction of these two parts concerns the choice of the logic used: in the first part, we use classic logic and fuzzy logic, whereas, in the second part, we use a four valued logic and its continuous extension.

Part I:

In the first chapter, we introduce basic notions and present preference structures with interval representations. Certain new results concerning the existing link between two different characterization modes (the relational and the forbidden one) are also presented in this chapter. Chapter 2 is dedicated to a general framework of the study of interval comparison. The goal of this study is to get general results on preference structures having interval representation. We conclude this chapter with an analysis of intervals with or without one intermediate point for which pre-existing results appear as particular cases of our study. In chapter 3, we look at fuzzy preference structures. At first, we analyse the structures with three relations (P,Q, andI) and we propose a method to compare intervals.

Then we focus on fuzzy interval orders for which we give a new method of comparison of intervals that gives a fuzzy interval order.

Part II:

Chapter 4 is dedicated to the use of a four valued logic in preference modelling. We analyse in detail the comparison of intervals with three relations (P,Q, andI). In chapter 5, we look at the interpretation of the continuous extension of the four valued logic in the possibility theory.

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Part I

Preference Modelling with Intervals

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Chapter 1

Crisp Preference Structures

Following the well-known works of Luce ([Luc56]), a new line of research appeared in pref- erence modelling with the introduction of an intransitive indifference relation. The intran- sitivity of the indifference is often related to the presence of a discriminating threshold.

Studies showed that preference structures with intransitive indifferences, and in conse- quence those using thresholds, have, for the most part, an interval representation.

In this chapter, we study preference structures having an interval representation. More precisely, we present our problematic and introduce the basic notions and fundamental concepts for the preference modelling. We briefly present a variety of known results that have to do with our problematic. We have, however, added a few new results that we obtained in order to complete our study.

More precisely, we distinguish two cases in this chapter: the first case deals with struc- tures having two preference relations (P and I) and the second case deals with structures having three preference relations (P, Q, and I).

In the case of structures with two relations, we use the classification given by Fishburn ([Fis97]), where we have ten nonequivalent classes of preference structures (semiorders, interval orders, split semiorders, split interval orders, tolerance orders, bitolerance orders, unit bitolerance orders, bisemiorders, semitransitive orders and the subsemitransitive or- ders). We add three structures to this classification: the d-weak orders, the d-interval orders and the triangular orders. We propose interval representations for these last three structures.

In the case of structures with three relations, we look at four structures, the P QI interval orders, the P QI semiorders, the double threshold orders and the pseudo orders.

The definitions of all the structures are presented in the representational mode that defines a mapping to a class of preference structures through a function from the set of objects to the set of real numbers or the set of intervals. Apart from the representative mode, there exist two other definition modes. In the first one, called therelational mode, a preference structure is defined through the properties of its preference relations. The second mode, called theforbidden mode, defines a mapping to a class of preference structures by the absence of certain orders (the forbidden ones) among all the orders induced by the structure studied. Since the characterization of most structures is also known in the relational mode,

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we were able to present the properties of the majority of preference relations of the different structures. The characterization in the forbidden mode does not play an important role in our study for two reasons: first of all, we chose logic as a formal language, whereas the characterization in the forbidden mode is more oriented towards the use of graph theory, and second of all, the structures with three relations (P, Q and I) do not have an easy graph representation. In consequence, they often do not have a definition in the forbidden mode. Nevertheless, we have dedicated the last section of this chapter to a class of forbidden orders. Our motivation for this is the absence of a known characterization in the representational and the relational modes of semitransitive and subsemitransitive orders. The definitions of these call on one class of forbidden orders (m+n posets). It is for this reason that we analyse m +n posets in the last section of this chapter. Let us remark that this section does not give an overview of all the characterisations in the forbidden mode. Certain equivalences that we found between the forbidden posets and the properties of the preference relations allowed us to obtain a characterization in the relational mode of semitransitive and subsemitransitive orders and actually created a link between two modes of characterization (relational and forbidden). The new results that we present in this chapter can be found in [Özt].

This chapter is organised in the following way. In the first section, we present the moti- vations of our study. Section 1.2 is dedicated to the presentation of the basic notions. The fundamental concepts are introduced in section 1.3 where we also define certain classical preference structures. We consider the classes of total orders, preorders and partial orders in order to give a comprehensive overall vision of the relations that exist in these different structures. In the following two subsections (Subsections 1.4.1 and 1.4.2)), we look at preference structures that have an interval representation. We conclude with a section on a class of forbidden orders where we present new results concerning the relations between the relational and forbidden modes.

1.1 Preference modelling and decision aiding

The actions of comparing and ordering objects may be necessary in various fields. In order to establish an order between objects, it may be commonly necessary to do pairwise com- parisons and define a relation between every pair of objects. There can exist different types of relations (representing a preference, an indifference, an incomparability, an hesitation, etc.) with different properties (symmetry, reflexivity, transitivity, etc.). Such a diversity can have a determinant role for the characterization of the final order of alternatives. Pref- erence modelling is dedicated to the characterization of such orders and analyzes their properties, numerical representations, relations between different orders, etc.

As a result, preference modelling appears as an inevitable step in a variety of fields. Sci- entists build models in order to better understand and to better represent a given situation (see [BMP+00]). It is often necessary to compare objects in such models, basically in order to establish an order between them. Objects can be everything, from candidates to time in- tervals, from computer codes to medical patterns, from prospects (lotteries) to production

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1.1 Preference modelling and decision aiding 17 systems. This is the reason why preference modelling is used in a great variety of fields such as economy ([Arm39], [Arm48], [Arm50], [Deb59]), sociology, psychology ([CS73], [CS66], [KT79], [KST81], [Bro91]), political science ([BFM82], [Sen86]), artificial intelli- gence ([DW92]), computer science ([Sco82],[Tro92], [Fis99]), temporal logic (see [All83]) and the interval satisfiability problem ([GS93, PS97]) mathematical programming ([PS04], [PS02]), e-business, medicine and biology ([Ben62], [Car88], [Kar93], [Nag92], [JLM+03]), archaeology ([HKT71]), and obviously decision analysis.

In this study, we are going to focus on preference modelling for decision aiding purposes, although the results have a much wider validity (for more details see [ÖTV05]). For that reason we present first of all some basic notions of decision aiding

Decision Aiding

In every day life, we communally face decision problems. The approach that we choose to resolve them depends on several dimensions such as the context and difficulty of the problem, our experience, our knowledge etc. In general, we try to choose one or more actions among several ones or to order them by comparing them according to different points of view, called also criteria. When such a task is complicated, we may need the help of an expert (also called an analyst).

The concept of decision process is introduced for the first time by Simon ([Sim47]) and it mainly concerns the cognitive activities of an individual facing a question for which no automatic reply pattern is available. This definition gives place to the introduction of the concept of rationality which is expected to refer to the process more than to the final decision and is bounded in time, space and cognitive capacity of the decision maker.

After him many researchers are interested in decision problems and proposed different approaches. Due to such a diversity, even basic notions like “decision”, “alternative”, “crite- ria”, “decision maker”, etc. may have different interpretation. We undertake in this thesis, the notations and definitions given by Dias and Tsoukiàs ([DT03]).

We call an actor any person who has influence on the decision process. The decision maker is the one who is in charge of deciding on and confirming the final solution. The analyst is the person who aids the decision maker to structure the problem, to transform the processus of decision, to construct decision models.

Note to remove 1.1 on peut mettre ici et dans les cas de differents approaches les ref- erences cites par Ngo page 8. il faut bien separer les descriptists des prescriptists!

Dias and Tsoukiàs proposed in their paper [DT03] (see also [Tso03]) a classification of decision approaches. Their classification is general in the sense that it covers many of other classifications. They define four types of approaches: “normative,” “descriptive”,

“prescriptive” and “constructive” ones.

- In normative approaches, researchers develop models from norms established a priori which are postulated as necessary for rational behavior. Decision maker who deviates from

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these norms is considered as "not rational" and must be aided to learn to decide in rational way. More details can be found in [vM44, LR57, Rai70, Fis70b, Fis82, Wak89].

- In descriptive approaches, it is aimed to derive a rationality model by observing people. Such models pretend to be general in the sense that they may be applied similarly to all the analogous decision problems. For more details, see [MS76, All79, Mon83, vE86, Pou94, Sve96].

- In prescriptive approaches, analysts try to discover rationality models with the aid of some experiments. Moreover, these models do not pretend to be general, but to be suitable for contingent DM in particular case. Interested reader can find more details in [Tve77b, WC90, LM95, Van02, BS02].

- In constructive approaches the analyst, who is communally different from the de- cision maker, tries to help him to construct his own rationality model. The decision maker is not supposed to be objectively rational and even the situation, the set of ac- tions or preferences of decision maker are not supposed to be stable. The same prob- lem can have different models depending on the decision maker attitudes. Models aim to be suitable for the DM and his particular case. For more details reader might see [WBJ67, Sch88, Ros89, LBO96, Roy96, GP02].

Let us analyze now different steps of a decision aiding process: Tsoukiàs ([Tso03]) in- terprets such a process as a distributed cognition process with four stages :

i. A representation of the problem situation where the sets of participants, stakes and resources are defined.

ii. A problem formulation where the set of potential actions, points of view and the problem statement are constructed.

iii. An evaluation model where various elements are considered. Let’s define it as:

M=hA,{D,E},G,U,R}i where

A: Set of alternatives on which the model will apply;

D: Set of dimensions (attributes) under which elements ofA are observed, measured, described etc. (such a set can be structured, for instance through the definition of an hierarchy);

E: Set ofscales (set of numbers susceptible to code information relating to the objects) associated to each element of D;

G: Set of criteria (if any) under which each element of A is evaluated in order to take in account the client’s preferences;

U: Uncertainty structure;

R: Set of operators enabling to obtain synthetic information about the elements ofA or ofA×A, namely aggregation operators (of preferences, of measures, if uncertainties, etc.)

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1.1 Preference modelling and decision aiding 19 iv. A final recommandation

These four steps are equally important for decision aiding, especially, when we use constructive approach. However, in this document, we concentrate our efforts specially to the evaluation model by supposing that the two first stages are done by taking care of DM’s cognitive profile and different aspects of decision problem.

Our study focus on preference modelling under uncertainty. That is the reason for which two elements of the evaluation stage need a special attention: U andG. The relation between preference modelling andG may need some explication: we are interested in pref- erence modelling which aims to order alternatives according to a criterion. The properties that the final order will have depend on the nature of this criterion. The presence and the nature of thresholds, the evaluation of each alternative according to each criteria (ordinal evaluation or evaluation in form of number, interval, fuzzy number, etc.) have particular influence for the construction of such orders. In the rest of the chapter, we will present some well known and some recent orders. Their characterization, numerical representation and properties prove the importance that the nature of criteria has for preference modelling.

Preference Modelling for Decision Aiding

Preference modelling aims to construct preference relations on a set of alternatives which are evaluated with respect to a criterion. The representation of the decision maker’s prefer- ences over the setA of alternatives constitutes a crucial step in decision aiding. Depending on the context of the problem, the nature of information that we are able to handle and the expectations of the decision maker, different situations may appear. In this context, Vincke ([Vin01]) distinguishes two types of problems that he calls respectivelycomparison problem and numerical representation problem.

In the “comparison problems”, the alternatives are evaluated according to different points of view. These evaluations may be of different nature: symbols, linguistics expres- sions, numbers, intervals, geometrical figures, etc. Naturally, some additional information about the nature of the points of view, the scale type used and the meanings that the decision maker gives to the evaluations may be added. The aim is to define preference relations capable to represent these evaluations with the additional information.

Example 1.1 Suppose that we have to define preference relations between three alterna- tives a, b and c which are evaluated with respect to their performance such that:

- the performance of a: 10 - the performance of b: 12 - the performance of c: 14.

Let us imagine that the decision maker gives some additional information such that: he prefers one alternative to another one if the first one is greater than the second one and if the difference between their performance is greater than three; otherwise he is indifferent

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between them.

In the light of these remarks we can construct now our preference relations. Table 1.1 represents these relations. The notations cP a and aP−1c represent the sentence “ c is preferred to a” and the notation bIa represents the sentence “ b and a are indifferent”:

a b c

a I I P−1

b I I I

c P I I

Table 1.1: Preference relations between a,b and c

In the “numerical representation problem”, the decision maker expresses his preferences for each pair of alternatives. These preferences can be completed by some additional information like comparison of his preference1, presence of incomparability or thresholds, etc. The aim is then to assign a numerical representation to each alternative.

Example 1.2 Let us suppose that the decision maker gives us his preferences among three alternatives a, b and c such that:

- he is indifferent between a and b - he is indifferent between b and c but - he prefers c to a.

We remark that the indifference relation of our decision maker is not transitive: he is indifferent between a and b and between b and c but he prefers c to a.

We will see in next chapters that intervals are good candidates in order to represent such an intransitivity. Suppose that we say that we have a preference in the case of two disjoint intervals and that all the other cases are expressed with indifference. So in the light of this remark we can associate to a and c two disjoint intervals, while the intervals of a and of b (and also b and c) have a non empty intersection. We propose the following evaluations: a: [0,2], b: [1,4]and c: [3,5].

Comments on the Relations Used in Preference Modelling

Depending on the context of the decision problem and the attitudes of the decision maker, different relations can be used in order to represent preferences of the decision maker.

1In some cases, it can be fruitful to have information about the difference of preference between the alternatives. Leta,b,c,dbe four alternatives inAwithaP bandcP d. One can suggest that the preference of a overbis greater than that ofcoverd(it is denoted byabDcd). This kind of measurement is called a difference measurement and studied in general in the context of measurement theory: [Fis70b], [KLST71], [Rob79] and [Roy85].

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1.1 Preference modelling and decision aiding 21 The two examples that we present in the previous paragraphes (example 1.1 and 1.2) make use of the most basic and the most classic relations which are thepreference and the indifference, denoted respectively by P and I. In such case, two situations may happen:

- the decision maker prefers one alternative to another one (such a relation is denoted byP) or

- he is indifferent between them (such a relation is denoted by I).

In a large part of this thesis we make use of only these two relations (see sections 1.3 and 3.3, subsection 1.4.1 and chapter 2).

One can wish to give more freedom to the decision-maker and allow more detailed pref- erence models, introducing one or more intermediate relations between indifference and preference. Such relations might represent one or more zones of ambiguity and/or uncer- tainty where it is difficult to make a distinction between preference and indifference. One can distinguish two cases: one where only one such intermediate relation is introduced (usually called weak preference and denoted by Q), and another where several such inter- mediate relations are introduced. In this thesis, we are specially interested in the first case where only one intermediate relation is introduced (see subsection 1.4.2, section 3.2 and chapter 4). In such cases, we have three possibilities:

-the decision maker strictly prefers one alternative to the second one (such a relation is denoted by P) or

- he is indifferent between them (such a relation is denoted by I) or

- he hesitates between preferring one to another and being indifferent between them (such a relation is denoted by Q).

In the two previous cases, the decision-maker is supposed to be able to compare the alternatives. But certain situations, such as lack of information, uncertainty, ambiguity, multi-dimensional and conflicting preferences, can create uncertainty and/or incompara- bility between alternatives. In such cases, the decision maker may say that he can not compare two alternatives and these two alternatives can be considered as incomparable; such a incomparability relation is denoted by J.

Remark 1.1 Let us remark that if not otherwise mentioned, in the whole thesis, we inter- pret only cases where the decision maker is able to give a comparison between two objects (so that J =∅). Such a choice is coherent with the use of interval representation and will be clearer in the body of the thesis.

Another way to deal with the uncertainty is to use fuzzy preference relations. We dedicate the chapter 3 to fuzzy preference structures.

Comments on the Representations Used in Preference Modelling

As we already mentioned, we distinguish two decision situations: “comparison problems”

and “numerical representation problems”. In both of these cases, some evaluations must be

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associated to alternatives. In the first situation, these evaluations can be everything from symbols to linguistics expressions, from numbers to geometrical figures while in the second one, it can be preferred to have a numerical representation.

In the whole document, we make use ofintervals in order to represent alternatives. We justify this choice as follows:

The model that we develop must explore, on the one hand the situation of our problem with the most possible information and on the other hand the vision and the attitudes of the decision maker in view of this situation. These two pillars of the modelling reveal uncertainty related to two phenomena: imprecise, dubious, incomplete, even inconsistent character of the information and complex nature of the decision maker’s preferences like having intransitive preference or indifference relation. The first phenomenon may play an important role specially for the “comparison problems” while the second one becomes more important for the “numerical representation problems” because of the following reasons:

• Concerning the “comparison problem”, the uncertainty of the problem situation can be expressed by interval evaluation of alternatives. As it is known, intervals are suitable tools to represent uncertainty and are commonly used in various fields, such as computer science (with timing intervals), management (in inventory control for example), sociology, psychology (in classification problems), archaeology (when the exact date of an event is not known) and of course decision analysis. When it is difficult to give an exact evaluation of an alternative, using intervals may be useful in order to have more robust models. Intervals can be seen also as fuzzy numbers with uniform distribution. For that reason, their comparison can constitute a first step for the comparison of fuzzy numbers by using different relations2.

• Concerning the “numerical representation problem”, we try to obtain a representation of the decision maker’s preferences as conform as possible to the reality of the decision process of the decision maker. In order to reach this goal, the analyst tries to help the decision maker to build his own definition of rationality. The models that the analyst is supposed to propose to him might be flexible enough to present as much as possible his rationality, his wishes and his conviction. It is empirically shown that there exist many cases where the preference relations are not transitive and that such an intransitivity is generally related to the use of some thresholds. It is also known that a representation with threshold is equivalent to an interval one. For that reason, we propose to use intervals in order to represent intransitive relations. A more detailed explication on this subject can be found in section 1.4.

Another remark can be done concerning the weak preference relation. Its introduc- tion makes more flexible our model since it allows the decision maker to express an hesitation between the preference and the indifference. The numerical representa- tion of all the preference structures having the three relations P, Qand I (the strict preference, the weak preference and the indifference) make use of intervals.

2Let’s remark that all the existing interval and/or fuzzy number comparison use the relation "bigger than" and do not permit the introduction of weak preference)

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1.1 Preference modelling and decision aiding 23 As a conclusion using interval, we are able to represent uncertain, unknown situations, intransitive relations and to introduce the weak preference relation.

Comments on the characterisations of preference structures

We showed that different relations can be used in order to represent the decision maker preferences. Moreover, two relations referring to the same context may have different properties. For example one decision maker may interpret the indifference relation as a transitive one while another decision maker defines it as an intransitive relation. Such dif- ferences gave birth to different preference structures satisfying different properties. In the literature, three primary modes are used in order to characterize them: the representational mode, the relational mode and the forbidden mode.

• The representational mode characterizes preference structures by assigning a numer- ical representation to each of the preference relation.

Example 1.3 The binary relation R is a weak order if and only if∃ g: A7→R such that ∀x, y ∈A: xRy ⇐⇒g(x)≥g(y).

In our document, we specially make use of interval representations where intervals can be degenerate, as in the case of weak orders, or can have some special intermediate points.

• The relational mode characterizes preference structures by the properties of its pref- erence relations.

Example 1.4 The binary relation R is a weak order if and only if R is reflexive, complete and transitive.

• The forbidden mode characterizes preference structures by the absence among all induced orders, of every member of a family of minimal forbidden partially ordered set.

Example 1.5 The binary relation R is a weak order if and only if the partially ordered set (1 + 2) is minimally forbidden.

In our document, we define preference structures in the representational mode since we are specially interested in the use of intervals. Their characterization in the relational mode is commonly given when it is known. We make use of the forbidden mode when the characterization on the two first modes are not known. In other cases, this mode is not used since it is not intuitive enough for the characterization of fuzzy structures and structures with three preference relations, P, Qand I.

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Basic notations and the Language Used in Preference Modelling

We are ready now to present the basic notations that we will use in the whole document.

The list that we present here is not a complete one, it represents only notations which are common to every chapter.

In this thesis, we are interested in the comparison and the “numerical representation problems” of decision process. We consider the case of a finite set of alternatives which will be represented in the majority of time by intervals. Our results concern on the preference structures having two (the preference and the indifference) or three (the strict and the weak preferences and the indifference) relations which may be of crisp or fuzzy nature.

Each preference relation has different properties and there exist some relations between them. In order to represent all of these we use the following notation:

• Set of alternatives: We consider a finite set of alternatives that we denote by A.

Variables of the set A will be denoted by x, y, z, t, . . . while specific elements of the setA will be denoted by a, b, c, d, . . ..

• Binary relations: Capital letters like P, I, J, Q, R, . . . ,will be used in order to repre- sent our preference relations, which may be of crisp or fuzzy nature. In the crisp case we use the notation xRy or R(x, y) in order to say that the object x is in relation R with the object y. For the same purpose we use the notation r(x, y) in the fuzzy case.

Preference models are formal representations of comparisons of objects. As such they have to be established through the use of a formal and abstract language capturing both the structure of the world being described and the manipulations of it. In this thesis we make use of the formal logic as a language to model real life situation.

The logic is based on relational sentences which are interpreted in our case by binary relations: the sentence “the object a is preferred to the object b” is presented by the help of binary relationP and we get P(a, b).

Relations have properties, we need two quantificators "universal" (∀) and "existential"

and some connectives such as "¬,∨,∧,−→, . . .", in order to define them. For example, every elementxis equal to itself, so the relation "equal to" is saidreflexive. The reflexivity can be defined as: ∀x ∈ A, xEx. We present as an example the definition of transitivity which has more complicated presentation: a relationR istransitive if and only if ∀x, y, z ∈ A, ((xRy)∧(yRz))−→xRz.

Finally, we make use of some operators, such as "∩,∪,⊆, . . ." in order to present rela- tions which exist between different preference relations. For example the relation “at least as good as” (R) can be defined as “better than (P) or equal to (I)”: R =P ∪I.

In this thesis we make use of different logics. Classic logic is used in chapter 1 and chapter 2, fuzzy logic in chapter 3, a four valued logic in chapter 4 and finally the continuous extension of a four valued logic in chapter 5.

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1.2 Basic notions 25

1.2 Basic notions

The mathematical concept of binary relation is communally used as a formal representa- tion of preference relations defined on a finite set since such relations are results of the comparison of two elements. In this section we provide a survey of basic notions of binary relations. We present definitions of some operations and some properties that we need for the rest of the document.

The notion of binary relation appears for the first time in De Morgan’s study ([De 64]) and is defined as a set of ordered pairs in Peirce’s works ([Pei80, Pei81, Pei83]). Some of the first works dedicated to the study of preference relations can be found in [DM41] and in [SS58] (generally the concept of models of arbitrary relations is introduced in [Tar54, Tar55]).

Throughout this chapter, we adopt notations of Öztürk and Tsoukiàs ([ÖTV05]).

Definition 1.1 (Binary Relation) Let A be a finite set of elements (x, y, . . .), a binary relation R on the set A is a subset of the cartesian product A×A, that is, a set of ordered pairs (x, y) such that x and y are in A: R ⊆ A×A.

For an ordered pair (x, y) which belongs (resp. does not belong) to R, we indifferently use the notations:

xRy orR(x, y)(resp. not(xRy) or¬R(x, y)).

Let R and T be two binary relations on the same setA. Some set operations are:

Inclusion: R⊆T iff ∀x, y xRy =⇒xT y,

Union: ∀x, y, x(R∪T)y iff xRy or (inclusive) xT y, Intersection: ∀x, y, x(R∩T)y iff xRy and xT y,

Relative product: ∀x, y, x(R.T)y iff ∃z ∈A: xRz and zT y.

We denote by xRmy iff ∃zi, i∈ {1, . . . , m−1}:xRz1 and z1Rz2 and . . . zm−1Ry.

| {z }

m times

We respectively denote (Ra), (Rs), (R) the asymmetric, the symmetric and the com-¯ plementary part of the binary relationR:

xRay iff xRy and not(yRx), xRsy iff xRy and yRx,

xRy¯ iff not(xRy)and not(yRx).

The complement (Rc), the converse (R−1) and the dual (Rd) of R are respectively defined as follows:

xRcy iff not(xRy), xR−1y iff yRx, xRdy iff not(yRx).

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The relation R is called, with respect to a universe A,

reflexive, iff ∀x∈A, xRx,

irreflexive, iff ∀x∈A, xRcx,

symmetric, iff ∀x, y ∈A, xRy =⇒yRx,

antisymmetric, iff ∀x, y ∈A, (xRy∧yRx) =⇒x=y, asymmetric, iff ∀x, y ∈A, xRy =⇒yRcx,

complete, iff ∀x6=y∈A, xRy∨yRx, strongly complete, iff ∀x, y ∈A, xRy∨yRx,

transitive, iff ∀x, y, z∈A, (xRy∧yRz) =⇒xRz, negatively transitive, iff ∀x, y, z∈A, (xRcy∧yRcz) =⇒xRcz,

semitransitive, iff ∀x, y, z, w∈A, (xRy∧yRz) =⇒(xRw∨wRz), Ferrers relation, iff ∀x, y, z, w∈A, (xRy∧zRw) =⇒(xRw∨zRy).

Ferrers relations play an important role for the definitions of several preference struc- tures (especially interval orders -see section 1.4-) that we analyze in the rest of this docu- ment, this is the reason for which we present some properties of these relations.

Proposition 1.1 Let R be a binary Ferrers relation defined on a finite set A then Ra.Rs.Ra ⊂Ra

We continue by an immediate proposition related to the transitivity of a Ferrers relation.

Proposition 1.2 Let R be a binary relation defined on a finite set A. If R is a Ferrers relation then its asymmetric part is transitive.

Proof.

Let R be a Ferrers relation, then Ra.Rs.Ra ⊂ Ra (see proposition 1.1. In addition the identity relation Id is included in the symmetric part of R, so we have Ra.Id.Ra ⊂ Ra, which is equivalent to Ra.Ra ⊂ Ra. As a conclusion if R is a Ferrers relation then Ra is transitive.

A Ferrers relation has an equivalent definition in terms of the composition ofR and its dualRd. This proposition is well known proposition and presented in [Mon78] and [RV85].

We propose in the following a different proof.

Theorem 1.1 A binary relation R is a Ferrers relation if and only if R.Rd.R⊂R.

Proof.

∀x, y, z, w ∈A

((xRy∧zRw) =⇒(xRw∨zRy)) ⇐⇒ (¬(xRy∧zRw)∨(xRw∨zRy))

⇐⇒ ((not(xRy)∨not(zRw)∨xRw∨zRy)

⇐⇒ (not(xRy)∨not(yRdz)∨not(zRw)∨xRw)

⇐⇒ (xRy∧yRdz∧(zRw)) =⇒xRw)

⇐⇒ R.Rd.R⊂R.

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1.2 Basic notions 27

a b

c d

-

-

6 6

?

Figure 1.1: Graphical representation of R a b c d

a 0 1 0 0 b 0 0 1 1 c 1 0 0 1 d 0 1 0 0

Figure 1.2: Matrix representation ofR

The equivalence relation E associated with the relation Ris a reflexive, symmetric and transitive relation, defined by:

∀x∈A, xEy iff (xRz⇐⇒yRz)∧(zRx ⇐⇒zRy).

A binary relation R may be represented by a direct graph (A, R) where the nodes represent the elements of A, and the arcs, the relation R. Another way to represent a binary relation is to use a matrix MR; the elementMabR of the matrix (the intersection of the line associated toa and the column associated to b) is 1 if aRb and 0 if not(aRb).

Example 1.6 Let R be a binary relation defined on a set A, such that the set A and the relation R are defined as follows:

A={a, b, c, d} and R ={(a, b),(b, d),(b, c),(c, a),(c, d),(d, b)}

The graphical and matrix representations of R are given in figures 1.1 and 1.2.

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1.3 Classical crisp orders

LetAbe a finite set, the pairwise comparison of its elements may result in different binary relations having different properties. We say that such relations construct a preference structure if they satisfy some conditions:

Definition 1.2 (Preference Structure) [Roy85, RB93] A preference structure is a col- lection of binary relations hR1, ..., Rmi defined on the set A and such that:

∀x, y ∈A, ∃i∈ {1, m}, xRiy or yRix;

∀x, y ∈A, xRiy =⇒ ∀j 6=i, not(xRjy) and not(yRjx).

It means that for each couple x, y in A; at least one relation is satisfied and if one relation is satisfied, no other can be satisfied.

Alarge preference relation, called also outranking relation, denoted byR, is a collection of the preference relations. This relation is usually interpreted as “xRy if and only if x is at least as good as y”. A large number of classical orders can be characterized by this relation R (see theorems 1.2, 1.4, 1.7, 1.9); for this reason R is also called characteristic relation.

Convention 1.1 If not otherwise mentioned, when a collection of binary relationshR1, ..., Rmi forms a preference structure, the associated characteristic relation R will be denoted by R=hR1, ..., Rmi.

Let us notice that in the case of preference structures without incomparability R is the union of all the binary relations forming the preference structure: let hR1, ..., Rmi be a preference structure without incomparability, then its characteristic relation satisfies R=∪i∈{1,...,m}Ri.

The most traditional preference model considers that comparing two elements of a set A, either:

- one element is clearly preferred to the other, or - it is not the case and they are indifferent.

The subset of ordered pairs (a, b) belonging in A×A such that the statement “a is preferred to b” is true, is called preference relation and is denoted byP.

The subset of pairs (a, b) belonging to A×A such that the statement “a and b are indifferent” is true, is called indifference relation and is denoted byI (I being considered the complement of P ∪P−1 with respect toA×A).

A preference structure can be also defined by using some properties of its preference relations. For example, the pair(P, I)forms a preference structure if it is defined as in the following:

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1.3 Classical crisp orders 29 Definition 1.3 (hP, Ii Preference Structure) AhP, Iipreference structure on the set A is a pair of relations (P, I) on A such that:





P is asymmetric,

I is reflexive, symmetric, P ∪I is complete,

P and I are mutually exclusive (P ∩I =∅).

Another way to characterize a hP, Ii preference structure is to use the connections between P,I and the characteristic relation:

Proposition 1.3 LetR be a reflexive and complete binary relation on a finite set A, then a couple of relations (P, I) on A forms a hP, Ii preference structure iff

aP b iff aRdb, (1.1)

aIb iff aRb and bRa. (1.2)

The construction of orders is of a particular interest, especially in decision analysis since they allow an easy operational use of such preference structures.

As we already mentioned in the beginning of the chapter, we will present first of all the definition of each order on the representational mode which gives an intuitive idea of orders and provides an operational support for decision aiding.

In this section we will briefly represent the most elementary orders (linear order and weak order). All the definitions and proofs of theorems of this section can be found in [SS58, Fis85, KLST89, PV97].

Definition 1.4 (Linear Order) A reflexive relation R = hP, Ii on a finite set A, is a linear order if there exists a real-valued function g, defined on A, such that ∀x, y ∈A,

xP y ⇐⇒ g(x)> g(y),

x6=y =⇒ g(x)6=g(y). (1.3)

Figure 1.3 illustrates the graphical representation of a linear order where every object is represented by a node and the preference relation is represented by an arc.

x y

xP y

Figure 1.3: Linear Order

The connections between P, I and R, provide the following theorem:

Theorem 1.2 [PV97] Let R be a reflexive binary relation on the set A, the following assertions are equivalent:

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i. R is a linear order;

ii. ∃ g: A7→R satisfying for ∀x, y ∈A:

xRy ⇐⇒ g(x)> g(y), x6=y =⇒ g(x)6=g(y).

A linear order can be also characterized as:

Theorem 1.3 [PV97] Let R=hP, Ii be a binary relation on the set A, the following asser- tions are equivalent:

i. R is a linear order;

ii. R is reflexive, antisymmetric, complete and transitive;

iii. I ={(a, a),∀a ∈A}, P is transitive and P ∪I is complete.

In the literature, one can find different terms associated with this structure: total order, complete order, simple order or linear order.

With this relation, we have an indifference between any two objects only if they are identical. The linear order structure consists of an arrangement of objects from the best one to the worst one without any ex aequo. An order allowing ex aequo is the weak order.

Such an order is defined as follows:

Definition 1.5 (Weak Order) A reflexive relation R = hP, Ii on a finite set A, is a weak order if there exists a real-valued function g, defined on A, such that ∀x, y ∈A,

xP y ⇐⇒ g(x)> g(y),

xIy ⇐⇒ g(x) = g(y). (1.4)

The major difference between a weak order and a linear order comes from the definition of the indifference relation. Two distinct objects can not be indifferent in a linear order but they can be in an equivalence class of a weak order. Figure 1.4 illustrates graphical representation of a weak order where two different objects are considered indifferent.

x y

xP y

x y

xIy Figure 1.4: Weak Order

The connections between P, I and R, provide the following theorem:

Theorem 1.4 [PV97] Let R be a reflexive binary relation on the set A, the following assertions are equivalent:

i. R is a weak order;

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1.3 Classical crisp orders 31 ii. ∃ g: A7→R satisfying for ∀x, y ∈A: xRy ⇐⇒g(x)≥g(y).

A weak order can be also characterized as:

Theorem 1.5 [PV97] Let R=hP, Ii be a binary relation on the set A, the following asser- tions are equivalent:

i. R is a weak order;

ii. R is reflexive, complete and transitive;

iii. I is transitive, P is transitive and P ∪I is reflexive and complete;

iv. P is transitive,P.I ⊂P (or equivalentlyI.P ⊂P) andP∪I is reflexive and complete.

This structure is also called complete preorder or total preorder. In this structure, indifference is an equivalence relation. The associated order is indeed a linear order of the equivalence classes of A.

Numerical representations of preference structures are not unique. All monotonic strictly increasing transformations of the function g can be interpreted as equivalent nu- merical representations for linear and weak orders3.

Until now we present preference structures without incomparability but in the litera- ture, there exist also other preference structures introducing incomparability. They can be used in certain situations, such as lack of information, uncertainty, ambiguity, multi- dimensional and conflicting preferences. For such structures, called partial, a third sym- metric and irreflexive relationJ, called incomparability, is used. To have a partial structure hP, I, Ji, we add to the relational definitions of the preceding structures (linear order, weak order), the relation of incomparability (J 6=∅) which is irreflexive and symmetric; and we obtain respectively partial order, partial preorder (quasi-order) ([RV85]). As we already mentioned, we consider only preference structures without incomparability, however sec- tion 1.5 makes use of the concept of partial order, for this reason we give the definition of a partial order in the following:

Definition 1.6 (Partial order) A reflexive relation R =hP, I, Ji on a finite set A, is a partial order if there exists a real-valued function g, defined on A, such that ∀x, y ∈A,

aP b =⇒g(a)> g(b),

I ={(a, a),∀a∈A} (1.5) Let’s notice that for partial structures of preference, the functional representations admit the same formulas. But, although partial orders have numerical representation, such numerical representations are not necessarily partial orders.

A partial order can be also characterized as:

3the functiong defines an ordinal scale for both structures.

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Theorem 1.6 [PV97] Let R be a binary relation (R = P ∪I) on the set A, R being the characteristic relation of hP, I, Ji, the following definitions are equivalent:

i. R is a partial order;

ii. R is reflexive, antisymmetric and transitive;

iii.

P is asymmetric and transitive, J is irreflexive and symmetric, I ={(a, a),∀a∈A}.

A fundamental result ([DM41, Fis85]) shows that every partial order on a finite set can be obtained as an intersection of a finite number of linear orders.

The numerical representations of total orders and weak orders make use of functions, so that each element of the set A can be represented by a real number. Other orders having some more complex properties may need more complicated representations. In this perspective we continue our presentation by orders having interval representations.

1.4 Crisp orders with interval representation

In this section we present orders with intransitive indifference relation. We begin by orders covering only two relations: preference (P) and indifference (I) and we conclude by orders allowing an intermediate relation (Q).

Intransitivity of indifference and the appearance of intermediate hesitation relations are due to the use of thresholds. A threshold represents a quantity for which any difference smaller than this one is not significant for the preference relation. Thresholds can be constant or dependent on the value of the objects under comparison.

Example 1.7 Let a, b, c be elements of A with g(a) = 1000, g(b) = 1200, g(c) = 1400 and q= 300, a constant threshold related to the preference relation P, then

• there is not preference between a and b since |g(a)−g(b)|= 200<300, a and b can be seen as indifferent,

• there is not preference between b and c since |g(b)−g(c)|= 200< 300, b and c can be seen as indifferent,

• but there is preference between a andc since|g(a)−g(c)|= 400>300, cis preferred to a.

Graphical representations of the three objects a, b and c are illustrated in figure 1.5.

Numerical representations with thresholds are equivalent to numerical representations of intervals. It is sufficient to note that associating a value g(x) and a strictly positive value q(g(x))to each element x of A is equivalent to associating two values: f1(x) =g(x) (representing the left extreme of an interval) andf2(x) =g(x) +q(g(x))(representing the right extreme of the interval to each x; obviously: f2(x)> f1(x)always holds).

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1.4 Crisp orders with interval representation 33

a

b

c

1000 1300 1400 1700

1200 1500

Figure 1.5: Representation with thresholds

Example 1.8 Consider a, b, c defined as in example 1.7 with constant threshold q = 300 of relation P, then each element can be presented by an interval: a : [1000,1300], b : [1200,1500], c: [1400,1700] (see figure 1.6).

g(a) g(a) +q(g(a))

g(b) g(b) +q(g(b))

g(c) g(c) +q(g(c))

1000 1200 1300 1400 1500 1700

Figure 1.6: Representation with intervals

All the orders that will be presented in this section make use of thresholds, thus have interval representations. For extensions on the use of thresholds see [Fis97, Moo66, Han92].

We present first of all structures having two preference relations (P and I) and then in subsection 1.4.2 we extend our study in the case of structures having three relations (P, Q and I).

1.4.1 Crisp hP I i orders with interval representation

In the literature, we have a number of references on ordered sets accommodating thresholds in pairwise comparisons. It results that the semiorders which require uniform thresholds are the simplest ones and there exist other ordered sets generalizing semiorders. Fishburn, in [Fis97], distinguishes ten nonequivalent ordered sets having an interval representation.

These are semiorders, interval orders, split semiorders, split interval orders, tolerance or- ders, bitolerance orders, unit tolerance orders, bisemiorders, semitransitive orders and sub- semitransitive orders.

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In this section we present only structures being defined directly by an interval repre- sentation, such a case corresponds to the first eight structures of Fishburn’s list. Semitran- sitive orders and subsemitransitive orders, being defined only in the forbidden mode, will be presented in section 1.5.

Before analyzing one by one the eight structures, we present in figure 1.7 all proper inclusions between these ten nonequivalent classes. The definitions of these structures and their interrelations will be clearer in the body of this section. In the diagram, each box represents a class of preference structures having identical definitions. When there is an arrow from the box A to the box B, it means that the preference structures of the box B are properly included in the the class of preference structures of A. If there is not any arrow between two boxes, they are independent. The proofs of the illustrated equivalences and inclusions can be found in [Fis97].

Preference structures presented in the previous section consider the indifference relation as a transitive relation. The fact that such a property can be empirically falsifiable in some context, is shown in a number of studies. Undoubtedly the most famous example on this subject is that of [Luc56], with a cup of sweetened tea4. Before him, some authors have already suggested this phenomenon (see [Arm39, GR36, Fec60, Hal55] and [Poi05]). Fish- burn and Monjardet ([FM92]) present some historical comments on the subject. Relaxing the property of transitivity of indifference results in two well-known structures: semi-orders and interval orders. Like in the previous section all the definitions and proofs of theorems of these two structures can be found in [SS58, Fis85, KLST89, PV97].

Definition 1.7 (Semiorder) A reflexive relation R = hP, Ii on a finite set A, is a semiorder if there exists a real-valued function g, defined on A, and a nonnegative con- stant q such that ∀x, y ∈A,

xP y ⇐⇒ g(x)> g(y) +q,

xIy ⇐⇒ |g(x)−g(y)| ≤q. (1.6) The interval representation of semiorders makes use of uniform thresholds which forbids the inclusions between intervals. The preference relation P is associated to the case of two disjoint intervals and the cases of intersection and identity are presented by the indiffer- ence relation I. Figure 1.8 illustrates semiorders representation with intervals displaced vertically for visualization.

Since there is a connection between P, I and R, semiorders can be characterized also in term of their characteristic relation R as in the following.

Theorem 1.7 [PV97] Let R be a reflexive binary relation on the set A, the following assertions are equivalent:

4one can be indifferent between a cup of tea withnmilligrams of sugar and one withn+ 1milligrams of sugar, if one admits the transitivity of the indifference, after a certain step of transitivity, one will have the indifference between a cup of tea withnmilligram of sugar and that with n+N milligram of sugar with N large enough, even if there is a very great difference of taste between the two; which is contradictory with the concept of indifference

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1.4 Crisp orders with interval representation 35

- Bitolerance O.

- Trapezoid O.

- Bi-interval O.

- Split Interval O.

- Unit Bitolerance O.

- Proper Bitolerance O.

- Tolerance O.

- Parallelogram O.

- Unit tolerance O.

- Semitolerance O.

- Semitransitive O.

- Subsemitransitive O.

- Split Semiorder - Bisemiorder

- Equiparallelogram O. - Interval O.

- Bilinear O.

- Biweak Order - Semiorder

- Weak O.

- Linear O.

Figure 1.7: Inclusions between structures having interval representation

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y

x xP y

y x xIy

y x xIy

Figure 1.8: Semiorder i. R is a semiorder;

ii. ∃ g: A7→R and a constant q ≥0 satisfying ∀x, y ∈A:

xRy⇐⇒g(x)≥g(y)−q;

iii. ∃ g: A7→R and ∃q:R7→R satisfying ∀x, y ∈A:

xRy ⇐⇒ g(x)≥g(y)−q(g(y)),

(g(x)> g(y)) =⇒ (g(x) +q(g(x))≥g(y) +q(g(y))).

As we already mentioned, structures can be characterized also by the properties of each preference relation of the preference structure, such a characterization in the relational mode for semiorders is presented in the following theorem.

Theorem 1.8 [PV97] Let R=hP, Ii be a binary relation on the set A, the following asser- tions are equivalent:

i. R is a semiorder;

ii. R is reflexive, complete, Ferrers relation and semitransitive; 5 iii.

P.I.P ⊂P, P2∩I2 =∅,

P ∪ I is reflexive and complete;

iv.

P.I.P ⊂P,

P2I ⊂P (or equivalently IP2 ⊂P), P ∪ I is reflexive and complete.

Interval orders generalize the uniform threshold feature of semiorders in the following way:

Definition 1.8 (Interval order) A reflexive relation R = hP, Ii on a finite set A is an interval order if there exists a real-valued function g, defined on A, and a nonnegative function q such that ∀x, y ∈A,

xP y ⇐⇒ g(x)> g(y) +q(g(y)),

xIy ⇐⇒

g(x)≤g(y) +q(g(y)), g(y)≤g(x) +q(g(x)).

(1.7)

5A reflexive and Ferrers relation is always complete.

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1.4 Crisp orders with interval representation 37 The fact of having nonuniform thresholds implies the existence of inclusion cases for intervals. Such cases which are impossible for semiorders, are interpreted as indifference in the case of interval orders. The preference relation P and the indifference relationI of interval orders is graphically represented in figure 1.9.

y

x xP y

y x xIy

y x xIy

y x xIy

y x

xIy Figure 1.9: Interval Order

Interval orders can be characterized in the relational mode as in the following:

Theorem 1.9 [PV97] Let R=hP, Ii be a binary relation on the set A, the following asser- tions are equivalent:

i. R is an interval order;

ii. R is reflexive, complete and Ferrers relation;

iii.

P.I.P ⊂P,

P ∪ I is reflexive and complete.

A detailed study of this structure can be found in [PV97, Mon78, Fis85]. It is easy to see that this structure generalizes all the structures previously introduced.

One can further generalize the structure of interval order, by defining a threshold de- pending on both of the two elements. As a result, the asymmetric part appears without circuit: [AV93, AA93, Sub94, Abb95, Dia99, AM02].

Under this heading simple generalizations of semiorders and intervals orders that add an interior point to every semiorder or interval order can be considered. Such structures are called split semiorders and split interval orders and are studied in [MT76c, Tro92, FT99].

We present first of all the definition of split interval orders in the representational mode.

Definition 1.9 (Split interval order) A reflexive relation R=hP, Ii on a finite set A, is a split interval order if there exists a real-valued function g, defined on A, and nonnegative

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functions q and p such that ∀x, y ∈A,





































xP y ⇐⇒

g(x)> g(y) +q(g(y)),

g(x) +q(g(x))> g(y) +p(g(y)),

xIy ⇐⇒

















g(y) +q(g(y))≥g(x) and g(x) +q(g(x))≥g(y) or

g(y) +q(g(y))≥g(x) and g(x) +p(g(x))≥g(y) +q(g(y)) or

g(x) +q(g(x))≥g(y) and g(y) +p(g(y))≥g(x) +q(g(x)) or

g(y) +p(g(y))≥g(x)+ q(g(x)) and g(x) +p(g(x))≥g(y) +q(g(y)), 0 ≤q(g(x))≤ p(g(x)).

(1.8) Some instances of the preference and indifference relations of a split interval order are illustrated in figure 1.10. This example is proposed by Fishburn in its paper [Fis97].

d

c e

b a

g(x) g(x) +p(g(x))

g(y) g(y) +p(g(y))

xP y,aP bP cP e, dP cP e and I otherwise Figure 1.10: Split Interval Order

Like classical semiorders, split semiorders are defined as split interval orders plus a coherence condition.

Definition 1.10 (Split semiorder) A reflexive relation R =hP, Ii on a finite set A, is a split semiorder if it is a split interval order with ∀x∈A,

q(g(x)) =q, and p(g(x)) =p, q, p nonnegative constants.

(39)

1.4 Crisp orders with interval representation 39 Such structures relax the threshold requirement of semiorders and intervals orders for covering pairs.

Let us remark that the use of constant threshold for split semiorders reduces the number of the indifference cases as it can be seen in figure 1.11 which illustrates split semiorders by intervals.

y

x

xP y

y x

xIy

y x

xIy Figure 1.11: Split Semiorder

Other threshold relaxations, represented by intervals with special interior points are referred to tolerance orders. Detailed information on these structures can be found in [BT94, BI98, BT00, BJLM01]. We present in the following the definition of bitolerance orders in the relational mode. The indifference relation being defined as a disjunction of a number of conjunctions, has a very long representation. For this reason, in the following, we consider only the representation of the preference relation P; the indifference relation can be easily calculated by I(x, y) =¬P(x, y)∧ ¬P(y, x).

Definition 1.11 (Bitolerance order) A reflexive relation R = hP, Ii on a finite set A, is a bitolerance order if there exists a real-valued function g, defined on A, and nonnegative functions q, p and t such that ∀x, y ∈A,





xP y ⇐⇒

g(x)> g(y) +q(g(y)),

g(x) +p(g(x))> g(y) +t(g(y)), t(q(x))≥q(g(x)) and t(q(x))≥p(g(x)).

(1.9)

Figure 1.12 illustrates an instance of the preference relation P for a bitolerance order.

The quantity p(g(x)) is called theleft tolerance of x and q(g(x)) as the right tolerance of x. The name of this structure is referred to these two tolerances which are not always equal. Let us remark that there is not any ordering relation between q(g(x))and p(g(x)).

Like in the case of interval orders or split interval orders some subclasses of bitolerance orders where some particular conditions are required for tolerances are defined:

A proper bitolerance order is a bitolerance order with g(x)< g(y) iff g(x) +t(g(x))<

g(y) +t(g(y))for all x, y ∈A.

A unit bitolerance order is a bitolerance order with t(g(x)) = 1, for all x∈A.

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