Decision matrix DM3 via PyFVs.

The diverse decision values may fail to capture an accurate perspective when multiple decision-makers are part of the process. To address this challenge, this work introduces the diamond Pythagorean fuzzy set (Dia‑PyFS), an advancement over both the Pythagorean fuzzy (PyFS) set and the interval-valued Pythagorean fuzzy set (IVPyFS). Due to the established extension of intuitionistic fuzzy sets into Pythagorean fuzzy sets, the Dia‑PyFS model, a broader form of the diamond intuitionistic fuzzy set model, demonstrates enhanced performance. We then introduce the Dia‑PyFS as an extension of PyFS. The Diamond Pythagorean fuzzy values that define the elements within Dia‑PyFS may share a common norm. For Dia‑PyFS, we define fundamental algebraic and arithmetic operations, including union, intersection, addition, multiplication, and scalar multiplication, and analyze their primary properties. Additionally, we propose some new Dia‑PyF weighted average and geometric aggregation operators as well as explore their unique properties. We also then propose several algebraic operations between Dia‑PyFVs using general triangular 𝓉-norms and 𝓉-conorms. To transform input values represented by Dia‑PyFs into a single output value, we also introduce specific weighted aggregation operators based on these algebraic methods. Additionally, the Dia‑PyFS framework builds upon the “combinative distance-based assess” (CODAS) methodology, which relies on both Euclidean and Hamming distances. To illustrate the applicability of this new approach, the feasibility and suitability of the Dia‑PyFS set approach for choosing the best options are demonstrated by the summary and comparative analysis of the produced reports.

当决策过程涉及多位决策者时，单一的多样化决策值往往无法准确反映整体视角。为解决这一难题，本研究提出菱形毕达哥拉斯模糊集（Diamond Pythagorean Fuzzy Set, Dia-PyFS），作为毕达哥拉斯模糊集（Pythagorean Fuzzy Set, PyFS）与区间值毕达哥拉斯模糊集（Interval-Valued Pythagorean Fuzzy Set, IVPyFS）的进阶拓展形式。鉴于直觉模糊集（Intuitionistic Fuzzy Set）已拓展至毕达哥拉斯模糊集范畴，作为菱形直觉模糊集模型的更广义形式，Dia-PyFS模型展现出更优异的性能。随后，本文将Dia-PyFS作为PyFS的拓展形式进行详细介绍。构成Dia-PyFS元素的菱形毕达哥拉斯模糊值可共享统一范数。针对Dia-PyFS，本文定义了基础代数与算术运算，包括并、交、加法、乘法与数乘，并分析了其核心性质。此外，本文提出了若干新型Dia-PyFS加权平均与几何聚合算子，并探究了其独有特性。本文还基于一般三角t模（triangular 𝓉-norms）与t余模（𝓉-conorms），构建了菱形毕达哥拉斯模糊值（Dia-PyFVs）间的多种代数运算规则。为将Dia-PyFS表征的输入值转化为单一输出值，本文还基于上述代数方法提出了特定的加权聚合算子。此外，Dia-PyFS框架依托组合距离评估法（Combinative Distance-based Assessment, CODAS）搭建，该方法同时融合欧氏距离与汉明距离两种度量方式。为验证该新方法的适用性，本文通过对生成报告的总结与对比分析，论证了Dia-PyFS集方法在最优方案选择场景中的可行性与适配性。