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

Mingxin Wang; Luping Liu; Mingxin Wang; Luping Liu

doi:10.3934/math.2026013

AIMS Mathematics

2026, Volume 11, Issue 1: 322-344. doi: 10.3934/math.2026013

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Research article

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

Mingxin Wang ,
Luping Liu ^,

College of Big Data and Intelligent Engineering, Guizhou University of Commerce, Guiyang 550014, China

Received: 23 August 2025 Revised: 10 November 2025 Accepted: 21 November 2025 Published: 05 January 2026
MSC : 03B52, 03E72

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.
Citation: Mingxin Wang, Luping Liu. Generalized probabilistic hesitant Pythagorean fuzzy aggregation operators and their application in teaching equipment procurement[J]. AIMS Mathematics, 2026, 11(1): 322-344. doi: 10.3934/math.2026013

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Abstract

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.

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