Non-additive Robust Ordinal Regression: a multiple criteria decision model based on the Choquet integral

Within the multicriteria aggregation–disaggregation framework, ordinal regression aims at inducing the parameters of a decision model, for example those of a utility function, which have to represent some holistic preference comparisons of a Decision Maker (DM). Usually, among the many utility functions representing the DM’s preference information, only one utility function is selected. Since such a choice is arbitrary to some extent, recently robust ordinal regression has been proposed with the purpose of taking into account all the sets of parameters compatible with the DM’s preference information. Until now, robust ordinal regression has been implemented to additive utility functions under the assumption of criteria independence. In this paper we propose a non-additive robust ordinal regression on a set of alternatives A, whose utility is evaluated in terms of the Choquet integral which permits to represent the interaction among criteria, modelled by the fuzzy measures, parameterizing our approach. In our methodology, besides holistic pairwise preference comparisons of alternatives from a subset of reference alternatives A′⊆A, the DM is also requested to express the intensity of preference on pairs of alternatives from A′, and to supply pairwise comparisons on the importance of criteria, and the sign and intensity of interaction among pairs of criteria. The output of our approach defines a set of fuzzy measures (capacities) such that the corresponding Choquet integral is compatible with the DM’s preference information. Moreover, using linear programming, our decision model establishes two preference relations for any a,b∈A: the necessary weak preference relation, if for all compatible fuzzy measures the utility of a is not smaller than the utility of b, and the possible weak preference relation, if for at least one compatible fuzzy measure the utility of a is not smaller than the utility of b.

Within the multicriteria aggregation-disaggregation framework, ordinal regression aims at inducing the parameters of a decision model, for example those of a utility function, which have to represent some holistic preference comparisons of a Decision Maker (DM). Usually, among the many utility functions representing the DM's preference information, only one utility function is selected. Since such a choice is arbitrary to some extent, recently robust ordinal regression has been proposed with the purpose of taking into account all the sets of parameters compatible with the DM's preference information. Until now, robust ordinal regression has been implemented to additive utility functions under the assumption of criteria independence. In this paper we propose a non-additive robust ordinal regression on a set of alternatives A, whose utility is evaluated in terms of the Choquet integral which permits to represent the interaction among criteria, modeled by the fuzzy measures, parameterizing our approach. In our methodology, besides holistic pairwise preference comparisons of alternatives from a subset of reference alternatives A'subset of A, the DM is also requested to express the intensity of preference on pairs of alternatives from A' , and to supply pairwise comparisons on the importance of criteria, and the sign and intensity of interaction among pairs of criteria. The output of our approach defines a set of fuzzy measures (capacities) such that the corresponding Choquet integral is \textit{compatible} with the DM's preference information. Moreover, using linear programming, our decision model establishes two preference relations: for any $a,b \in A,$ the necessary weak preference relation, if for all \textit{compatible} fuzzy measures the utility of a is not smaller than the utility of b, and the possible weak preference relation, if for at least one compatible fuzzy measure the utility of a is not smaller than the utility of b.