Proportional Fuzzy Set Extensions and Their Operations

Following the initial introduction of fuzzy sets by Zadeh (1965), extensions of regular fuzzy sets have been documented in the literature. Among these expansions are type-2 fuzzy sets, intuitionistic fuzzy sets, image fuzzy sets, and neutrosophic sets. One issue with standard fuzzy set extensions is that they require experts to provide decimal values for an element's membership, non-membership, and indecision degrees, which might be difficult to ascertain. Assigning membership degrees for three or more digits will make this more challenging. Rather, it will be simpler to ascertain the degrees of membership, non-membership, and indecision using proportional relations between the degrees of parameters of a fuzzy set extension. We introduce the basic operations of proportional intuitionistic fuzzy sets, proportional Pythagorean fuzzy sets, proportional picture fuzzy sets, proportional spherical fuzzy sets, and proportional neutrosophic sets.