Another approach to linear Diophantine fuzzy rough sets on two universes and its application towards decision-making problems

Theories of the rough set (RS) and the fuzzy set (FS) are constructed to accommodate the uncertainty in the data analysis. Linear Diophantine FS (LD-FS) as a novel approach to decision-making (DM), broadening the predominating theories of intuitionistic FS (IFS), Pythagorean FS (PFS), q-rung orthopair FS (q-ROFS) deals with uncertain and vague information by considering the control or reference parameters. Exploring RSs in the framework of LD-FS is a propitious direction in RS theory, where LD-FSs are approximated by Linear Diophantine fuzzy relation (LD-FR). The primary aim of this article is to develop a new linear Diophantine fuzzy RS (LDF-RS) model based on an LD-FR over dual universes. The notions of lower and upper approximations of an LD-FS are introduced by using an LD-FR, and several fundamental structural properties are explored. Moreover, a connection between LDF-RSs and linear Diophantine fuzzy topology (LDF-topology) is established. In addition, some similarity relations among LD-FSs based on their lower and upper approximations are studied. Finally, a DM approach is crafted for the ranking of alternatives using the notions of LDF-RS. Moreover, a numerical example is designed and compared with some existing techniques.