Picture Fuzzy Rough Set and Rough Picture Fuzzy Set on Two Different Universes and Their Applications

Abstract

The major concern of this article is to propose the notion of picture fuzzy rough sets (PFRSs) over two different universes which depend on $\left(\delta ,\zeta ,\vartheta \right)$ -cut of picture fuzzy relation $\mathrm{ℛ}$ on two different universes (i.e., by combining picture fuzzy sets (PFSs) with rough sets (RSs)). Then, we discuss several interesting properties and related results on the PFRSs. Furthermore, we define some notions related to PFRSs such as (Type-I/Type-II) graded PFRSs, the degree $\alpha$ and $\beta$ with respect to ${\mathrm{ℛ}}_{\left[\left(\delta ,\zeta ,\vartheta \right)\right]}$ on PFRSs, and (Type-I/Type-II) generalized PFRSs based on the degree $\alpha$ and $\beta$ with respect to ${\mathrm{ℛ}}_{\left[\left(\delta ,\zeta ,\vartheta \right)\right]}$ and investigate the basic properties of above notions. Finally, an approach based on the rough picture fuzzy approximation operators on two different universes in decision-making problem is introduced, and we give an example to show the validity of this approach.