PyFVs for Linguistic terms.

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，多样化的单一决策值往往无法准确反映整体视角。为解决这一挑战，本研究提出菱形毕达哥拉斯模糊集（diamond Pythagorean fuzzy set, Dia-PyFS），该模型是对毕达哥拉斯模糊集（Pythagorean fuzzy set, PyFS）与区间值毕达哥拉斯模糊集（interval-valued Pythagorean fuzzy set, IVPyFS）的拓展升级。鉴于直觉模糊集（intuitionistic fuzzy set）向毕达哥拉斯模糊集的已有拓展，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，我们将Dia-PyFS界定为PyFS的一类拓展形式。构成Dia-PyFS的元素所对应的菱形毕达哥拉斯模糊值可共享统一的范数。针对Dia-PyFS，我们定义了基础代数与算术运算，包括并、交、加法、乘法以及数乘，并分析了这些运算的核心性质。此外，我们提出了若干新型Dia-PyFS加权平均与几何聚合算子，并探究了其独有特性。我们还基于一般三角t模（triangular 𝓉-norms）与三角t余模（triangular 𝓉-conorms），提出了Dia-PyF值之间的多种代数运算。为将Dia-PyFS所表征的输入值转化为单一输出值，我们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框架依托于“组合距离评估法”（combinative distance-based assessment, CODAS），该方法同时依赖欧氏距离（Euclidean distance）与汉明距离（Hamming distance）。为阐明该新方法的适用性，我们通过对生成报告的总结与对比分析，验证了Dia-PyFS集方法在最优方案选择场景中的可行性与适配性。