A Novel Approach to Roughness of Linear Diophantine Fuzzy Sets by Fuzzy Relations and Its Application Toward Multi-Criteria Group Decision-Making

The notion of linear Diophantine fuzzy sets (LD-FSs) is a novel mechanism to combat uncertainties in decision analysis. Due to reference parameters associated with membership grade (MG) and non-membership grade (NMG), LD-FS is more efficient and reliable than ideas of the fuzzy set (FS), intuitionistic fuzzy set (IFS), Pythagorean fuzzy set (PyFS), and q-rung orthopair fuzzy set (q-ROFS). The main goal of this article is to present an innovative roughness strategy for LD-FSs using a fuzzy relation (FR) over dual universes, known as a linear Diophantine fuzzy rough set (LD-FRS). The lower and upper approximations of an LD-FS are formulated using fuzzy relation (FR) over dual universes, and several axiomatic systems are investigated. The suggested model of LD-FRS is more flexible to address fuzziness and roughness. Meanwhile, a relationship is made between LD-FRSs and linear Diophantine fuzzy topologies (LDF-topologies). It is shown that the collection of all lower approximations based on a reflexive FR leads to an LDF-topology. Moreover, several similarity relations among LD-FSs are also examined based on their lower and upper approximations. An application of multi-criteria group decision-making (MCGDM) is demonstrated by a supplier selection problem. Finally, a detailed comparative analysis with certain existing methods is given to verify the feasibility and superiority of the suggested model.