Roughness of linear Diophantine fuzzy sets by intuitionistic fuzzy relations over dual universes with decision-making applications

Rough sets (RSs) and fuzzy sets (FSs) are designed to tackle the uncertainty in the data. By taking into account the control or reference parameters, the linear Diophantine fuzzy set (LD-FS) is a novel approach to decision making (DM), broadens the previously dominant theories of the intuitionistic fuzzy set (IFS), Pythagorean fuzzy set (PyFS), and q-rung orthopair fuzzy set (q-ROFS), and allows for a more flexible representation of uncertain data. A promising avenue for RS theory is to investigate RSs within the context of LD-FS, where LD-FSs are approximated by an intuitionistic fuzzy relation (IFR). The major goal of this article is to create a novel method of roughness for LD-FSs employing an IFR over dual universes. The notions of lower and upper approximations of an LD-FS are established by using an IFR, and some axiomatic systems are carefully investigated in detail. Moreover, a link between LD-FRSs and linear Diophantine fuzzy topology (LDF-topology) has been established. Eventually, based on lower and upper approximations of an LD-FS, several similarity relations are investigated. Meanwhile, we apply the recommended model of LD-FRSs over dual universes for solving the DM problem. Furthermore, a real-life case study is given to demonstrate the practicality and feasibility of our designed approach. Finally, we conduct a detailed comparative analysis with certain existing methods to explore the effectiveness and superiority of the established technique.