Abstract
A q-rung orthopair fuzzy set is one of the effective generalizations of fuzzy set for dealing with uncertainties in information. Under this environment, in this study, we define a new type of extensions of fuzzy sets called n,m-rung orthopair fuzzy sets and investigate their relationship with Fermatean fuzzy sets, Pythagorean fuzzy sets and intuitionistic fuzzy sets. The n,m-rung orthopair fuzzy sets can supply with more doubtful circumstances than Fermatean fuzzy sets, Pythagorean fuzzy sets and intuitionistic fuzzy sets because of their larger range of depicting the membership grades. There is a symmetry between the values of this membership function and non-membership function. Here, any power function scales are utilized to widen the scope of the decision-making problems. In addition, the novel notion of an n,m-rung orthopair fuzzy set through double universes is more flexible when debating the symmetry between two or more objects that are better than the diffusing concept of an n-rung orthopair fuzzy set, as well as m-rung orthopair fuzzy set. The main advantage of n,m-rung orthopair fuzzy sets is that it can describe more uncertainties than Fermatean fuzzy sets, which can be applie in many decision-making problems. Then, we discover the essential set of operations for the n,m-rung orthopair fuzzy sets along with their several properties. Finally, we introduce a new operator, namely, n,m-rung orthopair fuzzy weighted power average (n,m-ROFWPA) over n,m-rung orthopair fuzzy sets and apply this operator to the MADM problems for evaluation of alternatives with n,m-rung orthopair fuzzy information.