Robust integrals

In decision analysis, and especially in multiple criteria decision analysis, several non additive integrals have been introduced in the last years. These include the Choquet integral, the Shilkret integral and the Sugeno integral, among others. In the context of multiple criteria decision analysis, these integrals are used to aggregate the evaluations of possible choice alternatives, with respect to several criteria, into a single overall evaluation. The use of mentioned integrals in the aggregation process requests the starting evaluations to be expressed in terms of exact evaluations. In this paper we present the robust Choquet, Shilkret and Sugeno integrals, computed with respect to an interval-capacity. These are quite natural generalizations of the Choquet, Shilkret and Sugeno integrals, useful to aggregate interval-evaluations of choice alternatives into a single overall evaluation. We show that, when the interval-evaluations collapse into exact evaluations, our definitions of robust integrals collapse into the previous definitions. We also provide an axiomatic characterization of the robust Choquet integral.