Type-2 fuzzy cross-entropy and entropy measures and their applications

Type-2 fuzzy sets (T2FSs) have received increasing attention due to its great ability to handle imprecise and ambiguous information in reality. The purpose of this paper is to develop a general framework of type-2 fuzzy cross-entropy and entropy measures, which provides a fresh new look into the uncertainty of T2FSs. We first point out that uncertainty of a T2FS is consist of fuzziness and hesitancy, which could be described by the fuzzy factor and hesitant factor, respectively. A novel type-2 fuzzy cross-entropy has been initiated based on these two factors to measure the discrimination of uncertain information between two T2FSs. Meanwhile, we refine the axiomatic principles of type-2 fuzzy entropy and study the inherent relationship between type-2 cross-entropy and entropy. Moreover, some parameterized type-2 fuzzy cross-entropy and entropy measures are also investigated, and decomposition formula suggests that type-2 fuzzy entropy could be expressed as the weighted average of the fuzzy entropy and hesitant entropy. Finally, we apply the proposed uncertainty measures to the clustering pattern of T2FSs and develop a new multiple attribute decision-making (MADM) approach.