Grey Pythagorean Fuzzy Sets for Decision-Making

As a generalized extension of fuzzy sets, Pythagorean fuzzy sets (PFSs) provide a sophisticated framework for modeling subjective uncertainty inherent in human cognition. Real-world decision-making systems frequently exhibit concurrent uncertainties characterized by incompleteness and fuzziness. This work synthesizes grey system theory with PFSs to establish grey Pythagorean fuzzy sets (GPFSs), a novel uncertainty formalism capable of simultaneously representing vague and incomplete information. GPFSs achieve a unified characterization of both subjective and objective uncertainty. Initially, we express PFS membership and non-membership degrees as generalized grey numbers, thereby defining the degree of greyness for PFSs. Subsequently, fundamental set-theoretic operations for GPFSs are formalized—including whitening, subset relations, complementation, union, intersection, merging, and meet—along with a distance measure between arbitrary GPFSs. Finally, we develop an extended TOPSIS methodology for multi-criteria decision-making (MCDM) under grey Pythagorean fuzzy uncertainty. The proposed MCDM framework is empirically validated through a failure mode and effects analysis case study.