This paper proposes an approach to linguistic three-way decision making problem with double hierarchy linguistic term evaluation information. Double hierarchy linguistic term set consists of the first hierarchy and second hierarchy linguistic term set, which can describe uncertainty and fuzziness more flexibly. First, the Hamacher operational rules, score function and distance measure of double hierarchy linguistic elements are defined. Next, we construct the double hierarchy linguistic decision-theoretic rough set model. And the conditional probability is calculated based on the double hierarchy linguistic term environment with grey relational analysis, which makes the process of decisions more rational. Then the loss functions are aggregated by the double hierarchy linguistic Hamacher weighted averaging operator, which takes into account different decision attitudes of decision makers. And the results of decision are deduced by the minimum-loss principle. Finally, a case study about the selection of cooperation companies during the COVID-19 is used to demonstrate the practicability of our proposed method.

中文翻译：

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双层次语言环境下基于一些Hamacher聚合算子的三向决策

本文提出了一种利用双层次语言术语评估信息解决语言三向决策问题的方法。双层次语言术语集由第一层次和第二层次语言术语集组成，可以更灵活地描述不确定性和模糊性。首先，定义了双层次语言元素的Hamacher运算规则、评分函数和距离测度。接下来，我们构建了双层次语言决策理论粗糙集模型。并且基于双层次语言术语环境，通过灰色关联分析计算条件概率，使得决策过程更加合理。然后通过双层次语言 Hamacher 加权平均算子聚合损失函数，这考虑到了决策者的不同决策态度。并根据最小损失原则推导出决策结果。最后，关于在 COVID-19 期间选择合作公司的案例研究用于证明我们提出的方法的实用性。