Learning the Parameters of a Multiple Criteria Sorting Method
Multicriteria sorting methods aim at assigning alternatives to one of the predefined ordered categories. We consider a sorting method in which categories are defined by profiles separating consecutive categories. An alternative a is assigned to the lowest
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MATHRO, Facult´e Polytechnique, Universit´e de Mons 9, Rue de Houdain, Mons, Belgium [email protected] 2 Laboratoire G´enie Industriel, Ecole Centrale Paris Grande Voie des Vignes 92295 Chˆ atenay Malabry, France [email protected]
Abstract. Multicriteria sorting methods aim at assigning alternatives to one of the predefined ordered categories. We consider a sorting method in which categories are defined by profiles separating consecutive categories. An alternative a is assigned to the lowest category for which a is at least as good as the lower profile of this category, for a majority of weighted criteria. This method, that we call MR-Sort, corresponds to a simplified version of ELECTRE Tri. To elicit the values for the profiles and weights, we consider a learning procedure. This procedure relies on a set of known assignment examples to find parameters compatible with these assignments. This is done using mathematical programming techniques. The focus of this study is experimental. In order to test the mathematical formulation and the parameters learning method, we generate random samples of simulated alternatives. We perform experiments in view of answering the following questions: (a) assuming the learning set is generated using a MR-Sort model, is the learning method able to restore the original sorting model? (b) is the learning method able to do so even when the learning set contains errors? (c) is MR-Sort model able to represent a learning set generated with another sorting method, i.e. can the models be discriminated on an empirical basis? Keywords: Multicriteria Decision Aiding, Sorting, Preference Elicitation, Learning Methods.
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Introduction
In this paper we deal with multiple criteria sorting methods that assign each alternative to a category selected in a set of ordered categories. We consider assignment rules of the following type. Each category is associated with a “lower profile” and an alternative is assigned one of the categories above this profile as soon as the alternative is at least as good as the profile for a (weighted) majority of criteria. R.I. Brafman, F. Roberts, and A. Tsouki` as (Eds.): ADT 2011, LNAI 6992, pp. 219–233, 2011. Springer-Verlag Berlin Heidelberg 2011
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A. Leroy, V. Mousseau, and M. Pirlot
Such a procedure is a simplified version of ELECTRE Tri, an outranking sorting procedure in which the assignment of an alternative is determined using a more complex concordance non-discordance rule [16]. Several papers have recently been devoted to the elicitation by learning of the parameters of the ELECTRE Tri method. These learning procedures usually rely on a set of known assignment examples and use mathematical programming techniques to find parameters compatible with these assignments (see e.g. [13], [11], [14], [6]). Unfortunately, the number of parameters involved is rather high and the mathematical formulation of the constraints resulting from the assignment examples are nonlinear so that the proposed methods do not try in general to determine all parameters at the sam
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