Abstract: Multi-attribute decision-making is a decision-making theory and method that uses multiple attributes to help decision makers choose alternatives, and it has been widely used in many fields. In addition to many attribute characteristics that can assist decision-making, the type of attribute value is also one of the important branches of multi-attribute decision-making research. With the higher complexity of decision-making environment, the research on multi-attribute decision-making with accurate values as attribute values cannot meet the current changeable decision-making environment, and then the research on multi-attribute decision-making with fuzzy numbers as attribute values has attracted extensive attention of many scholars. Fuzzy set theory has experienced the development from fuzzy set to intuitionistic fuzzy set and then to Pythagorean fuzzy set, and the explicable decision-making problems have also been improved in breadth and depth. The development of fuzzy sets more clearly depicts the uncertain nature of decision-making and ensures the authenticity of attribute information. Pythagorean triangular fuzzy number (PTFN) is an expanded data form. Although it can cope with the complexity and uncertainty of decision-making problems, the distribution characteristics of data are not considered in the process of aggregation. However, the decision-maker's preference for the density of data distribution will affect the final decision-making results in the face of realistic decision-making problems. Then, this is also the source of density operator. Density aggregation operator is the result of re-integration on the basis of classical integration operator, and it has also received some attention since it was put forward. It is a flexible aggregation method and can effectively enhance the accuracy of information aggregation process. Given this, this paper proposes a Pythagorean triangular fuzzy number density weighted operator (PTF-DM), which not only refines the fuzzy data types in multi-attribute decision-making, but also considers the decision-maker's preference for data distribution.
The research content of this paper is mainly aimed at the multi-attribute decision-making problem with Pythagorean triangular fuzzy number, and considering the density relationship of attribute information distribution, the Pythagorean triangular fuzzy number density weighted operator is proposed. Firstly, the Pythagorean triangular fuzzy numbers and related algorithms are introduced, and the effective clustering of Pythagorean triangular fuzzy numbers is realized by using the basic idea of fuzzy reference ideal method (FRIM) and score function. Then, the Pythagorean triangle fuzzy number density weighted operator and composition operator are proposed, and the programming model is established by entropy method to determine the final density weight. Finally, an example named investment choice of manufacturing enterprises is given to illustrate the application of Pythagorean triangular fuzzy number density operator. Based on the data type of Pythagorean triangular fuzzy number, this method can further expand the practical application range of density operator.
Pythagorean triangle fuzzy number density weighted operator is proposed, and in order to verify the effectiveness of Pythagorean triangular fuzzy number density weighted operator, four operators, PTF-DWA_WAA, PTF-DWGA_WAA, PTF-DWA_OWA and PTF-DWGA_OWA, are taken as this paper's examples for comparative analysis. Additionally, considering the preference of decision makers for the density of data distribution, the ranking results of comprehensive values of enterprises under the same preference and different operators, and the ranking changes of comprehensive values of enterprises under different preferences are obtained. From the results, it can be found that the Pythagorean triangular fuzzy number density weighting operator has good stability and can play a good role in assisting decision makers to make decisions. In addition, in order to further verify the effectiveness of Pythagorean triangular fuzzy number density weighted operators, this paper compares three Pythagorean triangular fuzzy number integration operators with Pythagorean triangular fuzzy number density weighted operators, and finds that Pythagorean triangular fuzzy number density weighted operators are the result of secondary aggregation, and the preference of decision makers for data distribution is the key reason that affects the aggregation results, which is also the reason why Pythagorean triangular fuzzy number density weighted operators and Pythagorean triangular fuzzy number integration operators have different results. Pythagoras fuzzy set and density information aggregation are the new development trend of information expression and information integration, which greatly expands the choice space of decision-making tools. In the future, the multi-attribute group decision-making problem under the three-dimensional information structure will be considered to realize the research on information expression and aggregation under a wide range of conditions.

Key words: multi-attribute decision making, density aggregation operators, Pythagorean triangular fuzzy numbers, clustering, fuzzy reference ideal method

摘要： 针对属性信息为毕达哥拉斯三角模糊数(PTFN)的多属性决策问题,考虑属性信息分布的疏密关系,提出毕达哥拉斯三角模糊数密度(PTF-DM)算子。首先介绍PTFN及其相关运算法则,并基于模糊参考理想法(FRIM)的基本思想以及得分函数,运用有序增量分割法以实现PTFN的有效聚类。然后,提出PTF-DM算子及相应的合成算子,并运用熵值法建立规划模型确定最终的密度权重。最后,通过一个算例对PTF-DM算子的应用做一简要说明,结果发现PTF-DM算子具有较好的稳定性,在辅助决策者做出决策方面可以发挥良好效用。该方法以PTFN数据类型为背景,呈现信息表达及信息集成理念的新发展趋势,并进一步拓展密度算子的实际应用范围与决策工具的选择空间。

关键词: 多属性决策, 密度集结算子, 毕达哥拉斯三角模糊数, 聚类, 模糊参考理想法