Abstract: Conventional approaches to Multi-Attribute Decision-Making (MADM) have largely relied on two-way decisions, which involve only acceptance or rejection. Three-way decision-making (3WD) reduces decision risks by introducing a deferred decision option. Given the increasing complexity and dynamism of decision environments, applying three-way decisions to multi-attribute problems has attracted growing research interest. A three-way multi-attribute decision model based on utility and similarity measures under Pythagorean fuzzy sets was proposed. The relative utility function was extended to the Pythagorean fuzzy context and integrated with a similarity measure of Pythagorean fuzzy numbers to establish a similarity-based relative utility function. The ideal positive and negative reference sequences were determined using the score functions. Grey relational coefficients were computed based on distance measures between Pythagorean fuzzy numbers. A method for calculating conditional probabilities was then developed using grey relational analysis. Based on this, a ranking strategy incorporating expected utility was introduced, leading to the construction of a three-way decision model based on the similarity-based utility under Pythagorean fuzzy information. The feasibility of the proposed method is validated through a site selection evaluation case for new domestic waste treatment plants. Experimental results demonstrate that the proposed method can precisely categorize alternatives and reduce decision risks. Compared to traditional MADM methods, it incorporates objective weighting with three-way decision classification, thereby overcoming the limitation of mere ranking. Relative to three-way MADM methods, the use of grey relational analysis to optimize conditional probabilities aligns more closely with the classification logic of three-way decisions.

Key words: Pythagorean fuzzy set, Multi-Attribute Decision-Making (MADM), three-way decisions, utility function, similarity measure, three-way multi-attribute decision-making

摘要： 传统的多属性决策（MADM）方法基于二支决策，即接受和拒绝，而三支决策（3WD）通过增加一个延迟决策来降低决策风险。在当前复杂多变的决策环境下，利用三支决策为多属性决策提供解决方案已经成为了一个热门研究。提出一种基于毕达哥拉斯模糊数相似度的效用三支多属性决策模型。首先将相对效用函数拓展到毕达哥拉斯模糊环境下，结合毕达哥拉斯模糊数的相似度度量，建立一个基于相似度的相对效用函数；其次利用记分函数得到理想优、劣母序列，结合毕达哥拉斯模糊数的距离度量计算灰色关联系数，进而提出一种基于灰色关联分析的条件概率计算方法；在此基础上，利用期望效用引入排序策略得到基于毕达哥拉斯模糊数相似度的效用三支多属性决策模型。通过开发新的生活垃圾处理厂中的选址评估问题，验证该方法的可行性。实验结果表明，该方法可以精准划分方案分类、降低决策风险的可行性，相较于传统MADM方法，以客观权重结合三支决策分类，突破了仅排序局限；相较于三支MADM方法，经灰色关联优化条件概率，更贴合三支决策的分类逻辑。

关键词: 毕达哥拉斯模糊集, 多属性决策, 三支决策, 效用函数, 相似度, 三支多属性决策