The Total Variation of Bounded Variation Functions to Evaluate and Rank Fuzzy Quantities

Abstract

In this paper, we present a different approach to introduce evaluation and ranking of fuzzy quantities. These general fuzzy sets are obtained by the union of several fuzzy sets. They are neither normal nor convex. The idea we have followed is to use the total variation and the bounded variation function definitions applied to the membership function of a fuzzy set to introduce its evaluation. This approach has produced that the well-known method of area compensation, introduced by Fortemps and Roubens only in a geometrical framework, is now presented in a general contest and useful for any fuzzy set. Moreover, this new representation formula provides an α-cut view. This aspect, absent in Fortemps and Roubens paper, offers an evaluation by a weighted average of alfa-cuts values, where the weights are connected with the number of subintervals that produce every α-cut. Following the same idea, we have introduced the ambiguity definition of a general fuzzy set. By this new definition of evaluation and the consequent ambiguity, we present a way to rank fuzzy quantities.