Advancing Disc-Based Pythagorean Fuzzy Sets with Distinct Radii

Abstract

Disc-Based Pythagorean Fuzzy Sets (D-PFSs) extend traditional fuzzy sets by incorporating the concept of distinct radii to model uncertainty more flexibly. Unlike classical fuzzy sets that use fixed radii, D-PFSs allow each element to have its own radius, which enhances the representation of varying degrees of uncertainty. In implementing D-PFSs, a systematic approach is crucial for effective uncertainty modeling. Each D-PFS element is defined by membership and non-membership degrees, and a distinct radius. The Pythagorean condition ensures that the sum of the squares of membership and non-membership degrees does not exceed the element's radius squared. Union and intersection operations involve combining degrees with specific formulas, where the union uses the minimum radius and the intersection uses the maximum radius of the involved sets. The complement operation swaps the membership and non-membership degrees while retaining the radius. Integrating D-PFSs into the TOPSIS method for multi-criteria decision-making involves replacing decision matrix elements with Pythagorean fuzzy numbers, normalizing these numbers, and calculating ideal solutions and distances using Euclidean measures. Alternatives are then ranked based on their proximity to ideal solutions. The results demonstrate that D-PFSs provide enhanced flexibility by accommodating distinct radii, allowing for a nuanced representation of uncertainty. Validation confirms adherence to the Pythagorean condition, and the varying radii effectively influence the degrees. The comparison of union and intersection operations further showcases D-PFSs' superior capability in managing complex uncertainties.