Rough approximation of a preference relation by dominance relations

doi:10.1016/S0377-2217(98)00127-1

European Journal of Operational Research

Volume 117, Issue 1, 16 August 1999, Pages 63-83

https://doi.org/10.1016/S0377-2217(98)00127-1 Get rights and content

Abstract

An original methodology for using rough sets to preference modeling in multi-criteria decision problems is presented. This methodology operates on a pairwise comparison table (PCT), including pairs of actions described by graded preference relations on particular criteria and by a comprehensive preference relation. It builds up a rough approximation of a preference relation by graded dominance relations. Decision rules derived from the rough approximation of a preference relation can be used to obtain a recommendation in multi-criteria choice and ranking problems. The methodology is illustrated by an example of multi-criteria programming of water supply systems.

Introduction

Solving a multi-criteria decision problem means to give the decision maker (DM) a recommendation (Roy, 1993) in terms of the best actions (choice), or of the assignment of the actions to pre-defined categories (classification or sorting), or of the ranking of actions from the best to the worst (ranking). None of these recommendations can be elaborated before the DM provides some preferential information suitable to the preference model assumed.

There are two major models used until now in multi-criteria decision analysis: functional and relational ones. The functional model has been extensively used within the framework of multi-attribute utility theory (Keeney and Raiffa, 1976). The relational model has its most widely known representation in the form of an outranking relation (Roy, 1991) and a fuzzy relation (Fodor and Roubens, 1994). These models require specific preferential information more or less explicitly related with their parameters. For example, in the deterministic case, the DM is often asked for pairwise comparisons of actions from which one can assess the substitution rates in the functional model or importance weights in the relational model (see Fishburn, 1967; Jacquet-Lagrèze and Siskos, 1982; Mousseau, 1993). This kind of preferential information seems to be close to the natural reasoning of the DM. He/she is typically more confident exercising his/her comparisons than explaining them. The transformation of this information into functional or relational models seems, however, less natural. According to Slovic (1975), people make decisions by searching for rules which provide good justification of their choices. So, after getting the preferential information in terms of exemplary comparisons, it would be natural to build the preference model in terms of “if..., then...” rules. Then, these rules can be applied to a set of potential actions in order to obtain specific preference relations. From the exploitation of these relations, a suitable recommendation can be obtained to support the DM in decision problem at hand.

The induction of rules from examples is a typical approach of artificial intelligence. It is concordant with the principle of posterior rationality by March (1988) and with aggregation–disaggregation logic by Jacquet-Lagrèze (1981). The rules represent the preferential attitude of the DM and enable his/her understanding of the reasons of his/her preference. The recognition of the rules by the DM justifies their use for decision support. So, the preference model in the form of rules derived from examples, fulfils both representation and recommendation tasks (see Roy, 1993).

This explains our interest in the rough sets theory (Pawlak, 1982, Pawlak, 1991), which proved to be a useful tool for analysis of vague description of decision situations (Pawlak and Slowinski, 1994). We remember that the rough set concept is founded on the assumption that with every object of the universe of discourse there is associated some information (data, knowledge). For example, if objects are potential projects, their technical and economic characteristics form information (description) about the projects. Objects characterized by the same information are indiscernible (similar) in view of available information about them. The indiscernibility relation generated in this way is the mathematical basis of the rough sets theory. Any set of indiscernible objects is called elementary set. Any subset of the universe can either be expressed precisely in terms of elementary sets or roughly only. In the latter case, this subset can be characterized by two ordinary sets, called lower and upper approximations. The lower approximation contains objects surely belonging to the subset considered; the upper approximation contains objects possibly belonging to the subset considered.

For algorithmic reasons, information about objects is represented in the form of an information table. The rows of the table are labelled by objects, whereas columns are labelled by attributes (or criteria) and entries of the table are attribute values (evaluations). An information table where the set of attributes is split into condition and decision attributes is called decision table.

An important advantage of the rough set approach is that it can deal with a set of inconsistent examples, i.e. objects indiscernible by condition attributes but discernible by decision attributes. Moreover, it provides useful information about the role of particular attributes and their subsets in the approximation of decision classes, and prepares the ground for generation of decision rules involving relevant attributes.

Until now, however, the use of rough sets has been restricted to the classification problems (Slowinski, 1993). This use is straightforward because the set of classification examples can be directly put in the information table analysed by the rough set approach. In the case of choice and ranking problems this straightforward use is not possible, because the information table in its original form does not allow the representation of preference binary relations between actions.

To handle binary relations within the rough set approach, Greco et al. (1995) proposed to operate on, so called, pairwise comparison table (PCT), i.e., with respect to a choice or ranking problem, a decision table whose objects are pairs of actions for which multi-criteria evaluations and a comprehensive preference relation are known.

The use of an indiscernibility relation on the PCT makes problems with interpretation of the approximations of the preference relation and of the decision rules derived from these approximations. Indiscernibility permits handling inconsistency, which arrives when two pairs of actions have preferences of the same strength on considered criteria, nevertheless, the comprehensive preference relations established for these pairs are not the same. However, when we deal with criteria, there may arrive also another type of inconsistency connected with the dominance principle: one pair of actions is characterised by some strength of preferences on a given set of criteria and another pair has all preferences of at least the same strength, however, for the first pair we have comprehensive preference and for the other – inverse comprehensive preference. This is why indiscernibility relation is not able to handle all kinds of inconsistencies connected with the use of criteria. For this reason, in this paper, we are proposing another way of defining the approximations and decision rules, which is based on the use of graded dominance relations.

The paper is structured as follows. In the next section we recall the concept of the PCT. In Section 3, we introduce the rough approximation of the preference relation by means of the graded dominance relations defined on PCT. Section 4is devoted to generation of decision rules observing the principle of non-dominance. In Section 5we investigate the exploitation of decision rules in the framework of a given multi-criteria decision problem. An illustrative example is given in Section 6, and Section 7groups conclusions.

Section snippets

The concept of the pairwise comparison table (PCT)

Let A be a finite set of actions (fictitious or not, feasible or not), considered in the multi-criteria decision problem at hand. The preference model is being built using a preferential information provided by the DM. This information concerns a set $B ⊆A$ of, so called, reference actions, with respect to which the DM is willing to express his/her attitude through pairwise comparisons. The pairwise comparisons are considered as exemplary. We are distinguishing two kinds of these decisions:

Rough approximation of a preference relation

Let $H_{P} =⋃_{q∈P} H_{q} ∀P⊆C$ . Given $x,y∈A, ∅≠P⊆C and h∈H_{p},$ we say that x positively dominates y by degree h with respect to the set of criteria P iff xP^fq_qy with $fq⩾h ∀q∈P$ . Analogously, $∀x,y∈A, P⊆C and h∈H_{P},$ x negatively dominates y by degree h with respect to the set of criteria P iff xP^fq_qy with $fq⩽h ∀q∈P$ . Thus, ∀h∈H_P every P⊆C generates two binary relations (eventually empty) on A which will be called P-positive-dominance of degree h, denoted by D^h_+P, and P-negative-dominance of degree h, denoted by D^h_−P,

Decision rules

We can derive a generalized description of the preferential information contained in a given PCT in terms of decision rules.

We will consider the following kinds of decision rules:

1.
D₊₊-decision rule, being a statement of the type: $xD^{h}_{+P} y ⇒ xSy;$
2.
D₊₋-decision rule, being a statement of the type: not $xD^{h}_{+P} y ⇒ xS^{c} y;$
3.
D₋₊-decision rule, being a statement of the type: not $xD^{h}_{−P} y ⇒ xSy;$
4.
D₋₋-decision rule, being a statement of the type: $xD^{h}_{−P} y ⇒ xS^{c} y.$

Speaking about decision rules we will simply understand all the

Application of decision rules and definition of a final recommendation

The decision rules derived from rough approximations of S and S^c are then applied to a non-empty set of actions M⊆A. The application of decision rules to any pair of actions (u,v)∈M×M means to state the presence (uSv) or the absence (uS^cv) of outranking using the following implications.

1.
if $xD^{h}_{+P} y ⇒ xSy$ is a D₊₊-decision rule and uD^h_+Pv, then we conclude that uSv,
2.
if not $xD^{h}_{+P} y ⇒ xS^{c} y$ is a D₊₋-decision rule and not uD^h_+Pv, then we conclude that uS^cv,
3.
if not $xD^{h}_{−P} y ⇒ xSy$ is a D₋₊-decision rule and not uD^h₋

Illustrative example

Let us consider a real example concerning the problem of programming water supply systems (WSSs) for use in the countryside, called regional WSSs.

According to the methodology proposed by Roy et al. (1992), the programming task is decomposed into two problems:

(a) setting up a priority order in which the water users should be connected to a new WSS, taking into account economic, agricultural and sociological consequences of the investment; and
(b) choosing the best technical variant of the

Conclusions

In this paper a rough sets methodology to the analysis of multi-criteria choice and ranking decision problems was proposed. It is based on the idea of approximating a preference relation represented in a PCT by graded dominance relations. This methodology supplies some very meaningful “if..., then...” decision rules, which synthesize the preferential information given by the DM and can be suitably applied to obtain a recommendation for the choice or ranking problem.

This new methodology has been

Acknowledgements

The research of the first two authors has been supported by grant No. 96.01658.CT10 from Italian National Council for Scientific Research (CNR); the research of the third author has been supported by grant No. 8 T11C 013 13 from State Committee for Scientific Research (Komitet Badan Naukowych) and from CRIT 2 Esprit Project no. 20288.

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Theory and MethodologyRough approximation of a preference relation by dominance relations

Abstract

Introduction

Section snippets

The concept of the pairwise comparison table (PCT)

Rough approximation of a preference relation

Decision rules

Application of decision rules and definition of a final recommendation

Illustrative example

Conclusions

Acknowledgements

Journal of Mathematical Psychology

European Journal of Operational Research

European Journal of Operational Research

Journal of Mathematical Psychology

Outranking relations: Do they have special properties?

Journal of Multi-Criteria Decision Analysis

A preference ranking organization method

Management Science

Multiple semiorders and multiple indifference graphs

SIAM Journal of Algebraic Discrete Methods

Threshold representation of multiple semiorders

SIAM Journal of Algebraic Discrete Methods

Methods for estimating additive utilities

Management Science

The Copeland choice function – An axiomatic characterization

Social Choice and Welfare

Theory and Methodology
Rough approximation of a preference relation by dominance relations