Consistency for nonadditive measures: Analytical and algebraic methods

Abstract

We consider the problem of choosing an alternative in a set A = {A 1,A2, ⋯, Am} of alternatives, given a set D = {d1, d2, ⋯, dh} of decision makers and a set Ω = {O1,O2, ⋯, O n} of objectives. We assume that any decision maker dk assigns to any pair (alternative Ai, objective Oj) a number aijk that measures to what extent Ai satisfies Oj. We assume that Ω is a subset of a universal set U and, for every alternative Ai and decision maker dk, the function mik that associates a ijk to Oj is a fuzzy measure. We propose to aggregate the scores aijk by means of a t-conorm. λ a family Φλ of t-conorms such that every mik is a Φλ-decomposable measure. We consider also some algebraic and geometric representations of the Archimedean fuzzy unions and their additive generators in terms of the theory of hypergroups. By considering the Oj as events, we propose also to assign the scores aijk in such a way that for some. the assessment is consistent and to aggregate such evaluations with the correspondent t-conorm Φλ. Finally we generalize the previous procedure by considering fuzzy measures of type 2, having as a range a set of fuzzy numbers with the interval [0, 1] as support.