Further results on concordance relations q

2005

The idea of concordance is central to many MCDM techniques. It leads to comparing alternatives by pairs on the basis of a comparison in terms of importance of the coalitions of attributes favoring each element of the pair. Such a way of comparing alternatives has a definite "ordinal" flavor. It is well-know that it may lead to relations that do not possess any remarkable transitivity properties. This paper shows how to use standard conjoint measurement techniques to characterize such relations. Their main distinctive feature is shown to lie in their very crude way to distinguish various levels of preference differences on each attribute.

European Journal of Operational Research, 2009

Outranking methods propose an original way to build a preference relation between alternatives evaluated on several attributes that has a definite ordinal flavor. Indeed, most of them appeal the concordance / non-discordance principle that leads to declaring that an alternative is "superior" to another, if the coalition of attributes supporting this proposition is "sufficiently important" (concordance condition) and if there is no attribute that "strongly rejects" it (non-discordance condition). Such a way of comparing alternatives is rather natural. However, it is well known that it may produce binary relations that do not possess any remarkable property of transitivity or completeness. This explains why the axiomatic foundations of outranking methods have not been much investigated, which is often seen as one of their important weaknesses. This paper uses conjoint measurement techniques to obtain an axiomatic characterization of preference relations that can be obtained on the basis of the concordance / non-discordance principle. It emphasizes their main distinctive feature, i.e., their very crude way to distinguish various levels of preference differences on each attribute. We focus on outranking methods, such as ELECTRE I, that produce a reflexive relation, interpreted as an "at least as good as" preference relation. The results in this paper may be seen as an attempt to give such outranking methods a sound axiomatic foundation based on conjoint measurement.

2004

This paper extends the preliminary results in Bouyssou and Pirlot (2002a). It is a much abridged version of Bouyssou and Pirlot (2003) to which the reader is referred for proofs. We thank Thierry Marchant for very helpful discussions.

Measurement, 2007

This paper discusses a relational modeling of measurement which is complementary to the standard representational point of view: by focusing on the experimental character of the measurand-related comparison between objects, this modeling emphasizes the role of the measuring systems as the devices which operatively perform such a comparison. The non-idealities of the operation are formalized in terms of non-transitivity of the substitutability relation between measured objects, due to the uncertainty on the measurand value remaining after the measurement. The metrological structure of traceability is shown to be an effective solution to cope with the problem of the general non-transitivity of measurement results. A preliminary theory is introduced as a possible formalization for the presented model.

2005

This paper presents a self-contained introduction to a general conjoint measurement framework for the analysis of nontransitive and/or incomplete binary relations on product sets. It is based on the use of several kinds of marginal traces on coordinates induced by the binary relation. This framework leads to defining three general families of models depending on the kind of trace that they use. Contrary to most conjoint measurement models, these models do not involve an addition operation. This allows for a simple axiomatic analysis at the cost of very weak uniqueness results.

Measurement, 2009

The aim of this paper is to give an algebraic description of an extensive measurement (with error), which consists of a series of direct comparisons of all copies of the measured object with all copies of the reference. The start point is the axiomatisation of measurement operations in contrast with the axiomatisation of properties of the order relation. We propose ''principles of measurement consistency" which describe an operation of ordering of comparisons results. This principle is more general than the assumption of transitivity or homothecity of precedence (or preference) relation. In the presented model, the final result of a measurement is described by four numbers which represent both the measurement value and the uncertainty with two components: systematic and non-systematic. The regular inexact measurement and the biased measurement are also considered as particular cases.

Psychometrika, 1979

Based on a simple nonparametric procedure for comparing two proximity matrices, a measure of concordance is introduced that is appropriate when K independent proximity matrices are available. In addition to the development of a general concept of concordance and specific techniques for its evaluation within and between the subsets of a partition of the K matrices, several methods are also suggested for comparing and/or for fitting a particular structure to the given data. Finally, brief indications are provided as to how the well-known notion of concordance for K rank orders can be included within the more general framework.

2013

Axioms for additive conjoint measurement and qualitative probability are given. Representation theorems and uniqueness theorems are proved for structures that satisfy these axioms. Both Archimedean and nonarchimedean cases are considered. Approximations of infinite structures by sequences of finite structures are also considered. At the present time, there is one set of techniques for proving representation theorems for finite measurement structures an another set for infinite structures. Techniques for finite structures were developed in Scott (1964) and basically consist of solving finite sets of inequalities; techniques for infinite structures in one way or another resemble those used in Holder (1901) an d consist of the construction of fundamental sequences. Although finite structures often admit good axiomatizations in the sense that necessary and sufficient conditions for their representations can be given, they do not admit good uniqueness results. Infinite structures, howeve...

Annals of Operations Research, 2015

Outranking relations such as produced by the Electre I or II or the Tactic methods are based on a concordance and non-discordance principle that leads to declaring that an alternative is "superior" to another, if the coalition of attributes supporting this proposition is "sufficiently important" (concordance condition) and if there is no attribute that "strongly rejects" it (non-discordance condition). Such a way of comparing alternatives is rather natural and does not require a detailed analysis of tradeoffs between the various attributes. However, it is well known that it may produce binary relations that do not possess any remarkable property of transitivity or completeness. The axiomatic foundations of outranking relations have recently received attention. Within a conjoint measurement framework, characterizations of reflexive concordance-discordance relations have been obtained. These relations encompass those generated by the Electre I and II methods, which are non-strict (reflexive) relations. A different characterization has been provided for strict (asymmetric) preference relations such as produced by Tactic. In this paper we briefly review the various kinds of axiomatizations of outranking relations proposed so far in the literature. Then we analyze the relationships between reflexive and asymmetric outranking relations in a conjoint measurement framework, consolidating our previous work. Co-duality plays an essential rôle in our analysis. It allows us to understand the correspondence between the previous characterizations. Making a step further, we provide a common axiomatic characterization for both types of relations. Applying the co-duality operator to concordance-discordance relations also yields a new and interesting type of preference relation that we call concordance relation with bonus. The axiomatic characterization of such relations results directly from co-duality arguments.