The Construction of Non-Convex Fuzzy Sets

Although non-convex fuzzy set (FS) has high potential in modeling data and controlling, it is seldom used and discussed because of the lack of linguistic explanation. A method named "parametric qualitative fuzzy set (PQ FS) plus choice strategy" is put forward in this paper for the construction of non-convex FS, in which PQ FS is defined as a collection of convex FSs with specific structure and choice strategy is an approach to choose one convex FS in PQ FS for each situation. Some theorems are shown to demonstrate that the way "PQ FS plus choice strategy" can construct a non-convex FS by some proper choice strategy. Furthermore, the algorithm of constructing given non-convex FSs is also proposed by homotopy. Based on the construction way, the linguistic explanation of non-convex FS is obtained. The result shows that Non-convex FSs do not violate the routine of representing linguistic variables by convex FSs, for linguistic viable is represented by convex FS at each situation and non-convex FS is just the trajectory of those chosen convex FSs under all the situations.