Learning the Parameters of a Non Compensatory Sorting Model

We consider a multicriteria sorting procedure based on a majority rule, called MR-Sort. This procedure allows to sort each object of a set, evaluated on multiple criteria, in a category selected among a set of pre-defined and ordered categories. With MR-Sort, the ordered categories are separated by profiles which are vectors of performances on the different attributes. Using the MR-Sort rule, an object is assigned to a category if it is at least as good as the category lower profile and not better than the category upper profile. To determine whether an object is as good as a profile, the weights of the criteria on which the object performances are better than the profile performances are summed up and compared to a threshold. If the sum of weights is at least equal to the threshold, then the object is considered at least as good as the profile. In view of increasing the expressiveness of the model, we substitute additive weights by a capacity to represent the power of coalitions of criteria. This corresponds to the Non-Compensatory Sorting model characterized by Bouyssou and Marchant. In the paper we describe a mixed integer program and a heuristic algorithm that enable to learn the parameters of this model from assignment examples.