Extension of set functions to Interval Type-2 Fuzzy Sets: Applications to evidential reasoning

In this paper, we propose a method for extending set functions to Interval Type-2 Fuzzy Sets. We start from the Extension Principle for extending set functions to Type-1 Fuzzy Sets. Then we extend set functions to embedded Type-1 Fuzzy Sets of Interval Type-2 Fuzzy Sets. Consequently, we construct the Interval Type-2 Fuzzy Set result of the extension process (which resides in the co-domain of the extended set function) from its embedded Type-1 Fuzzy Sets. We show that the Extension Principle for set functions can be used to infer a single fuzzy probability measure from a linguistic belief structure, instead of lower and upper probabilities, and demonstrate how such an inference can be used to solve Advanced Computing with Words problems.