The probabilistic hesitant fuzzy preference relations (PHFPRs) provide the decision makers with an efficient means to express the preference information on pairwise comparisons over alternatives. In this paper, we propose an automatic consistency improving model for PHFPRs. First, we propose a novel normalization algorithm to normalize probabilistic hesitant fuzzy elements (PHFEs) by using probability splitting idea and develop novel operational laws. Then, we define the consistency index to compute the degree of deviation between the PHFPRs and their multiplicative consistent PHFPRs. We also develop a novel consistency threshold estimation method for obtaining the threshold of consistency index and then put forward an automatic consistency improving algorithm for repairing inconsistent PHFPRs. Moreover, two probabilistic hesitant fuzzy aggregation operators are put forward to aggregate preference values in acceptably multiplicative consistent PHFPRs for obtaining the ranking orders of alternatives. Finally, an illustrative example is given to show the implementation process of our proposed automatic consistency improving model and also we compare our proposed model with the existing studies.

中文翻译：

---

基于乘性一致性的概率犹豫模糊信息决策

概率犹豫模糊偏好关系 (PHFPR) 为决策者提供了一种有效的方式来表达对替代方案的成对比较的偏好信息。在本文中，我们提出了一种用于 PHFPR 的自动一致性改进模型。首先，我们提出了一种新的归一化算法，通过使用概率分裂思想来归一化概率犹豫模糊元素（PHFEs）并开发新的运算法则。然后，我们定义一致性指数来计算 PHFPR 与其乘法一致 PHFPR 之间的偏差程度。我们还开发了一种新的一致性阈值估计方法来获得一致性指标的阈值，然后提出了一种用于修复不一致 PHFPR 的自动一致性改进算法。而且，提出了两个概率犹豫模糊聚合算子，以在可接受的乘法一致 PHFPR 中聚合偏好值，以获得备选方案的排序顺序。最后，给出了一个说明性例子来展示我们提出的自动一致性改进模型的实现过程，并且我们将我们提出的模型与现有研究进行了比较。