Some Information Measures for Interval-Valued Hesitant Fuzzy Sets in Multiple Criteria Decision-Making

In the fields of information science and artificial intelligence, dealing with uncertainty, fuzziness, and complexity has always been a hot and difficult research topic. Especially in modern society, with the continuous development of science and technology, people are facing more and more complex problems. Interval-valued hesitant fuzzy set (IVHFS) is an extended form of a fuzzy set. It can more flexibly express and handle uncertainty and fuzziness in decision-making processes. However, in the practical application of the IVHFS, its information measurement is crucial, which is directly related to the application value of the IVHFS in various fields. Therefore, studying the information measurement of the IVHFS has important theoretical significance and practical value for the fields of information science and artificial intelligence. In spite of significant advances, entropy and similarity as the well-known information measures for interval-valued hesitant fuzzy information have not yet been thoroughly researched. In this contribution, we investigate information measures in the IVHFS, including nonprobabilistic entropy, similarity, and cross-entropy. We first analyze the change law of hesitating uncertainty and fuzzy uncertainty in geometric space, and a nonprobabilistic entropy measurement method and its axiomatic definition for IVHFS are further developed. Then, a novel similarity measurement formula for IVHFS and its axiomatic requirements are proposed on the basis of the two nonfuzzy elements ($$ and $$). Furthermore, the novel similarity measure is used to construct the cross-entropy measure for IVHFS and its axiomatic requirements based on the association between the similarity and the cross-entropy. Lastly, a MAGDM method is proposed by using the developed three information measures, and the efficacy of the proposed method is demonstrated by a numerical example of emergency communication support capacity evaluation. Comparative analysis and computational cost analysis are implemented to demonstrate the superiority and validity of the proposed information measures.