Defects of hesitant fuzzy set (HFS) manifest in actual decision‐making process, so adding probabilities to the values in HFS is necessary. The probabilistic HFS (PHFS) is a useful tool to describe the uncertainty of elements in HFS by introducing occurrence probabilities. However, some important issues in PHFS utilization remain to be addressed. In this study, an outranking method for multicriteria decision making (MCDM) with probabilistic hesitant information is presented. First, the binary relations between two probabilistic hesitant fuzzy elements (PHFEs) are defined on the basis of the elimination and choice translating reality method. Some outranking relations between the alternatives are then introduced. Second, we provide a Hausdorff distance between two PHFEs. The main characteristic of the proposed Hausdorff distance is that it does not require the same length and arrangement of the PHFEs. Third, a maximizing Hausdorff distance deviation method is developed to obtain the evaluation criteria weights under a probabilistic hesitant fuzzy environment. Finally, an illustrative example in conjunction with comparative analysis is used to demonstrate that the proposed method is feasible for practical MCDM problems.

中文翻译：

---

带有概率犹豫信息的多准则决策的排序方法

犹豫模糊集（HFS）的缺陷会在实际决策过程中体现出来，因此有必要在HFS中为值添加概率。概率HFS（PHFS）是通过引入出现概率来描述HFS中元素不确定性的有用工具。但是，PHFS使用中的一些重要问题仍有待解决。在这项研究中，提出了一种带有概率犹豫信息的多准则决策（MCDM）排序方法。首先，在消除和选择转换现实方法的基础上，定义了两个概率犹豫模糊元素（PHFE）之间的二元关系。然后介绍了替代方案之间的一些排位关系。其次，我们提供两个PHFE之间的Hausdorff距离。拟议的Hausdorff距离的主要特征是，它不需要相同长度和布置的PHFE。第三，发展了一种最大的Hausdorff距离偏差方法来获得概率犹豫模糊环境下的评估标准权重。最后，结合比较分析的说明性例子证明了该方法对于实际的MCDM问题是可行的。