Generalized probabilistic hesitant Pythagorean fuzzy aggregation operators and their application in teaching equipment procurement

Probabilistic hesitant Pythagorean fuzzy sets (PrHPyFSs) provide a robust framework for modeling decision-makers' preferences by simultaneously capturing probabilistic uncertainty, hesitation degrees, and the independent support/non-support relationships, thereby enabling a more accurate representation of real-world decision-making compared to traditional fuzzy sets. This study explores the aggregation of probabilistic hesitant Pythagorean fuzzy information in complex environments and its application to multi-criteria decision-making (MCDM). The research includes four main components: (1) developing an arithmetic operation system for probabilistic hesitant Pythagorean fuzzy elements (PrHPyFEs); (2) proposing four types of generalized aggregation operators for PrHPyFEs; (3) constructing a PrHPyFS-based MCDM framework using these operators, with effectiveness validated through a teaching equipment procurement case study; and (4) demonstrating the method's advantages via comparative analysis. The results confirm that the proposed solution effectively bridges the gap between theoretical foundations and practical decision-making applications.