Aggregating Fuzzy Sets Represented by Belief Structures

Abstract

Consonant belief structures provide a representation for fuzzy sets because their plausibility measures are possibility measures. However, in general the aggregation of these consonant belief structures are not consonant, not fuzzy sets. In this article, we attempt to overcome this problem. We first note that two belief structures are equivalent if their plausibility and belief functions are equal. This observation allows us to provide different equivalent representations for any belief structure. This allows us to induce for different consonant belief structures commensurate representations. We show that if we represent two consonant belief structures in a commensurate manner their aggregations are also consonant if we impose the additional requirement that the underlying probability distributions satisfy the condition of synonyminity, that is, they are completely correlated. The results of this work allow us to use belief structure representations to manipulate fuzzy subsets.