This contribution reviews critically the existing entropy measures for probabilistic hesitant fuzzy sets (PHFSs), and demonstrates that these entropy measures fail to effectively distinguish a variety of different PHFSs in some cases. In the sequel, we develop a new axiomatic framework of entropy measures for probabilistic hesitant fuzzy elements (PHFEs) by considering two facets of uncertainty associated with PHFEs which are known as fuzziness and nonspecificity. Respect to each kind of uncertainty, a number of formulae are derived to permit flexible selection of PHFE entropy measures. Moreover, based on the proposed PHFE entropy measures, we introduce some entropy‐based distance measures which are used in the portion of comparative analysis. Eventually, the proposed PHFE entropy measures and PHFE entropy‐based distance measures are applied to decision making in the strategy initiatives where their reliability and effectiveness are verified.

中文翻译：

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多准则决策中概率犹豫模糊集的不确定性测度

该贡献批判性地审查了概率犹豫模糊集 (PHFS) 的现有熵度量，并表明这些熵度量在某些情况下无法有效区分各种不同的 PHFS。在续集中，我们通过考虑与 PHFE 相关的不确定性的两个方面（即模糊性和非特异性），为概率犹豫模糊元素 (PHFE) 开发了一个新的熵度量的公理化框架。针对各种不确定性，推导出了许多公式以允许灵活选择 PHFE 熵度量。此外，基于提出的 PHFE 熵度量，我们引入了一些基于熵的距离度量，用于比较分析部分。最终，