Multi-dimensional PF overlap functions and multi-granulation PF rough sets with their applications in three-way decision

Granular computing provides a multi-level knowledge representation framework for AI, while three-way decision (TWD) is an important paradigm for decision-making under uncertainty. This paper investigates multi-granulation Pythagorean fuzzy rough sets (MGPFRSs), TWD, and their applications to address several limitations of existing studies. First, existing multivariate Pythagorean fuzzy (PF) aggregation operators are typically restricted to fixed forms and lack sufficient generalization capability. Second, conventional MGPFRSs mainly rely on traditional AND/OR operators, limiting their flexibility in diverse decision scenarios. Third, existing TWD spaces and corresponding decision-making algorithms are not directly applicable to PF decision systems. To overcome these limitations, multi-dimensional PF overlap and grouping functions are introduced as generalized AND/OR operators. Based on these functions, two generalized MGPFRS models are constructed. Subsequently, PF (semi-)three-way decision spaces and several PF (semi-)decision evaluation functions are proposed, leading to the development of a single-granulation PF TWD algorithm. Furthermore, a novel MGPF-based TWD algorithm is developed by integrating generalized MGPFRSs, TOPSIS, and PF decision evaluation functions. The proposed algorithm can characterize 12 types of decision preferences. Finally, a mine-fan fault diagnosis case study, along with comparative and sensitivity analyses, demonstrates the effectiveness, flexibility, and practicality of the proposed methods.