A probabilistic dual hesitant fuzzy multi-attribute decision-making method based on entropy and cross-entropy

The probabilistic dual hesitant fuzzy set (PDHFS), as an extension of generalization of the dual hesitant fuzzy set, contains not only the hesitation values of membership degree (MD) and non-membership degree (NMD), but also considers the probabilities corresponding to MDs and NMDs, which are the degree of support and confidence of the decision makers in the evaluation value given by them. Distance measures, entropy measures, and cross-entropy measures are important tools in multi-attribute decision-making. In the PDHFS environment, distance and entropy measures are improved, and cross-entropy is proposed, and multi-attribute decision-making methods based on distance and entropy and cross-entropy are given. First, in order to effectively compare the distances between different PDHFSs, we improve the existing distance measures. Second, we review the existing formulations of probabilistic dual hesitant fuzzy entropy and find that they could not effectively distinguish the uncertainty of different PDHFSs due to ignoring the uncertainty caused by the differences between different MDs and between different NMDs, so we improve the existing entropy measure. Additionally, the formulas and properties of the cross-entropy of PDHFS and the axiomatic definition of the generalized cross-entropy of PDHFS are given. Finally, depending on the distance and entropy and cross-entropy built, we propose a new multi-attribute decision method to solve the multi-attribute decision problem with completely unknown attribute weights. We apply the proposed method to the protective decision-making for the release of radioactive substances, and the feasibility of the method is verified by comparative analysis.