An Approach to Interval Fuzzy Probabilities

Fuzzy sets and logic have been largely used for the treatment of the uncertainty, vagueness and ambiguity found in the modeling of real problems. However, there may exist also the case when there is uncertainty related to the membership functions to be used in the modeling of fuzzy sets and fuzzy numbers, as there are many ways to define the shape of this kind of number. Thus, one can use, for example, the theory of interval fuzzy sets to address this uncertainty, considering different modelings of fuzzy numbers into a single interval fuzzy number. In this paper, we use interval fuzzy numbers to represent probabilities that are difficult to be estimated and where the modeling of fuzzy numbers is not trivial. To elaborate the calculation of interval fuzzy probabilities, we introduce an approach based on the one used by Buckley and Eslami for fuzzy probabilities, where the probabilities respect an arithmetic restriction. We discuss several properties of the proposed approach.