In this article, we introduce two new rough set models based on the concept of double fuzzy relations. These models are called optimistic and pessimistic multi-granulation double fuzzy rough sets. We discuss their properties and explore the relationship between these new models and double fuzzy rough sets. Our study focuses on the lower and upper approximations of these models, which generalize the conventional rough set model. In addition, we suggest that the development of the multi-granulation double fuzzy rough set model is significant for the generalization of the rough set model.

In this paper, we introduce the notion of $\left(p,q\right)$-fuzzy local function and DF-ideal topological space. Also, we introduce the concepts DFU-ð-continuous and DFL-ð-continuous, almost ð-continuous, weakly ð-continuous and almost weakly ð-continuous multifunctions. Several properties and characterizations of the introduced multifunctions and types of continuity are established. Some examples are given to explain the correct implications between these notions.

To define the optimistic multiple-granulation (OMG) supra interior operator and the OMG-supra closure operator associated with a set A in a multiple-granulation approximation space, we used the concepts of the OMG-lower set OL (A) and the OMG-upper set OU (A). These operators create an approximate topological space known as optimistic multiple-granulation supratopological spaces. OMG-supra-open and OMG-supra-closed approximation sets are also defined, followed by OMG-supra-continuous approximation and OMG-irresolute approximation.

Based on the definition of double fuzzy relations, two types of new rough set models are constructed, which are multi-granulation double fuzzy rough sets called optimistic and pessimistic multi-granulation double fuzzy rough sets, and some of their properties were studied. In addition, we study the relationship between optimistic and pessimistic multi-granulation double fuzzy rough sets and double fuzzy rough sets.

To define the optimistic multiple-granulation(OMG) supra interior operator i n t R A and the OMG-supra closure operator c l R A associated with a set A in a multiple-granulation approximation space, we used the concepts of the OMG-lower set O L Σ m i = 1 R i ( A ) and the OMG-upper set O U Σ m i = 1 R i ( A ) . These operators establish an approximate topological space referred to as optimistic multiple-granulation supra topological spaces. Additionally, we introduce the concepts of OMG-supra-open and OMG-supra-closed approximation sets, as well as OMG-supra-continuous and OMG-irresolute approximations.