Optimistic multigranulation roughness of a fuzzy set based on soft binary relations over dual universes and its application

A fascinating extension of Pawlak rough set theory to handle uncertainty is multigranulation roughness, which has been researched by several researchers over dual universes. In light of this, we proposed a novel optimistic multigranulation roughness of a fuzzy set based on soft binary relations over dual universes and established two types of approximations of a fuzzy set with respect to forsets and aftersets of the finite number of soft binary relations in this article. We obtain two sets of fuzzy soft sets in this way, referred to as the lower approximation and upper approximation with respect to the aftersets and the foresets, respectively. Next, we look into some of the lower and higher approximations of the newly multigranulation rough set model's algebraic properties. Both the roughness and accuracy measurements were defined. In order to show our suggested model, we first develop a decision-making algorithm. Then, we give an example from a variety of applications.