Integrating Rough Sets and Multidimensional Fuzzy Sets for Approximation Techniques: A Novel Approach

This research introduces innovative rough approximation techniques for multidimensional fuzzy sets by integrating rough sets and multidimensional fuzzy sets. Departing from traditional methods that rely on predefined equivalence relations to delineate rough spaces, we propose a novel approach utilizing a function termed the ‘convenient function’. The values generated by this function facilitate the establishment of lower and upper estimations, enhancing the precision of rough approximations. The adept utilization of the parameter $\alpha $ , in conjunction with a multidimensional distance measure, allows for the attainment of desirable approximations within specified bandwidths. Additionally, we explore a more generalized version of $\alpha $ approximation, known as $\alpha $ - $\beta $ approximation, along with its associated properties, thereby expanding the applicability and flexibility of the proposed methodology. Finally, we present a comprehensive case study demonstrating the efficacy of the $\alpha $ approximation methodology in real-world decision-making processes, highlighting its practical utility and effectiveness.