Abstract: As a generalized form of hesitant fuzzy elements, the Probabilistic Hesitant Fuzzy Elements（PHFEs） can depict uncertainty more practically. In addition, because Frank’s operations have dexterity with parameters, it can reflect the subjective preferences of decision makers. Thus, combining the PHFEs and the Frank operations, the probabilistic hesitant fuzzy Frank operations are first defined, and the improved score function of PHFEs is presented. Then, a Probabilistic Hesitant Fuzzy Frank Weighted Average（PHFFWA） operator is investigated, and some propertied of the PHFFWA operator are studied. Based on which, a novel multi-attribute group decision making method is developed on the basis of PHFFWA operator. Finally, an example about selection of data products is provided to demonstrate that the effectiveness and feasibility of the proposed method.

Key words: multi-attribute group decision making, probabilistic hesitant fuzzy sets, Frank norms, information aggregation operator, data products

摘要： 作为犹豫模糊元的推广形式，概率犹豫模糊元能更好地刻画现实中的不确定性，此外由于Frank运算具有含参数的灵巧性，能反映决策者的主观偏好，因此将概率犹豫模糊元和Frank运算相结合，首先定义了概率犹豫模糊Frank运算法则，并提出改进的概率犹豫模糊元的得分函数；然后提出了概率犹豫模糊Frank加权平均算子，并讨论了算子的一些性质；在此基础上，构建了基于概率犹豫模糊Frank加权平均算子的多属性群决策方法；最后将其应用于数据产品的选择实例中，说明了该方法的有效性和可行性。

关键词: 多属性群决策, 概率犹豫模糊集, Frank范数, 信息集成算子, 数据产品