Constrained type-2 fuzzy sets

Type-2 fuzzy sets extend the expressive capabilities of type-1 fuzzy sets in that, in addition to represent imprecise concepts, they are able to represent the imprecision in the membership function of fuzzy sets. Their use is particularly appropriate when modelling linguistic concepts such as words that mean slightly different things to different people. However, type-2 fuzzy sets do not place any constraints upon the continuity and other properties of their embedded sets. We argue that, for some concepts, additional properties constraining both the allowable footprint of uncertainty and the embedded sets of type-2 sets are desirable. We present a novel formulation of a constrained type-2 fuzzy set that achieves this, and then detail some of the properties of this novel form of type-2 fuzzy sets.