Primal Approximation Spaces by $\kappa $-neighborhoods with Applications

Abstract

Real-world problems often involve imprecision and uncertainty, presenting challenges across various domains such as engineering, artificial intelligence, social sciences, and medical sciences. To bridge this gap, Pawlak introduced classical rough set models, focusing on upper and lower approximations based on equivalence relations. However, these relations restrict the applicability of rough sets in many contexts. This paper explores new approaches to extending rough set theory by introducing "primal approximation spaces". Primal is defined as novel structures designed to generalize rough set approximations beyond the traditional methods. This paper proposes a new technique for generating rough approximations by using κ-neighborhoods and primal which allows for creating diverse supra-topologies and enhancing the flexibility of rough set models. Furthermore, we here introduce bi-primal approximation spaces, a new form of approximation that can be examined through two distinct methods, as a result, revealing their unique characteristics and relationships. The research underlines the practical applications of these new methods by providing a detailed case study that demonstrates their effectiveness in solving decision-making problems. Moreover, this study also compares the primal-based methods with existing approaches based on ideals, illustrating their distinctive advantages and limitations. Overall, this work offers a remarkable advancement in rough set theory by expanding its theoretical framework and practical applicability through the introduction of primal and bi-primal approximation spaces.