Interval type-2 fuzzy information measures and their applications to attribute decision-making approach.

Interval type-2 fuzzy set (IT2FS) offers great ability to depict high order information in reality while dealing with both extrinsic and intrinsic facets of uncertainty. In this paper, we try to develop a general framework of information measures and a flexible multiple attribute decision-making (MADM) approach in the interval type-2 fuzzy information environment. First, we propose the fuzzy factor, hesitant factor and interval factor to quantify the fuzziness, hesitancy and interval information of one IT2FS, respectively. An interval type-2 fuzzy cross-entropy has been initiated based on these three factors to measure the discrimination degree of uncertain information between two IT2FSs. Meanwhile, we exploit the axiomatic principles of interval type-2 fuzzy entropy and study the inherent relationship between cross-entropy and entropy measures. Then, some parameterized information measures are naturally investigated, and the decomposition formula suggests that the interval type-2 fuzzy entropy could be expressed as the weighted average of the fuzzy entropy, hesitant entropy and interval entropy. Finally, we construct two programming models based on the maximizing cross-entropy principle to determine attribute weights, and a novel MADM procedure is proposed and applied to a case study on the banks' liquidity risk evaluation.