Extensions of fuzzy aggregation

Abstract

We discuss the equivalence between aggregation of fuzzy sets and integration with respect to a special class of non-additive set functions. Both fuzzy integral and Choquet integral are considered. First, we study aggregation of a finite family of fuzzy sets and then we extend our results to aggregation of an infinite family. The concepts of comonotonic maxitivity and additivity play a central role. We argue that, for the purpose of aggregating fuzzy sets, comonotonic maxitivity is a more desirable requirement than comonotonic additivity. In the absence of any such requirement we explore a wider class of aggregation procedures. Another important subject that we study is a comparison between two aggregation functions, and we obtain a simple characterization in both the finite and infinite case. Our results are related to concepts that were studied in the three areas where non-additive set-functions and integrals were found particularly useful: fuzzy sets theory, mathematical economics, and mathematical statistics.