Non additive measures and Choquet integral with uncertain coefficients

In Multi Attribute Decision Analysis, a finite number of alternatives has to be compared together with respect to a finite number of criteria. To this purpose, non additive measures and the Choquet integral were intensively used to compute the alternative score. Despite simpler methods, like the Weighted Average, approaches based on non additive measures permit to model interactions among criteria, and at the same time are methodologically well-founded. In the case of non monotonic measures, conflicting relationships among criteria can be considered. Anywise, up to now, only little effort was devoted to formalize and to solve problems where the measure values are uncertain. This paper suggests a way to overcome this lack. In particular, the Multi Attribute Decision problem will be translated into different optimization problems, depending on the way the uncertainty is represented and, more important, if the monotonicity constraints need to be included or not.