Introduction to q-Fractional Fuzzy Set

Many attempts have been made to generalize the concept of intuitionistic fuzzy sets (IFS) like Pythagorean (PFS), q-rung orthopair (q-OFS), and linear Diophantine (LDFS). However, these generalizations have many advantages and disadvantages. Among the disadvantages, the main concern with these sets is that they cannot capture the situation where both or at least one of the memberships and non-membership grades are equal to 1. Secondly, how to reduce the dependency between the membership and non-membership grades. Thus, any data in the form X = {< x₁; (1,0.9) > , < x₂; (0.3,1) > , < x₃; (1,1) >} is not handled by the IFS and other versions of IFS because 1 + 0.9 = 1.9 > 1, 0.3 + 1 = 1.3 > 1, and 1 + 1 = 2 > 1. We propose the new idea of the q-fractional fuzzy set ( \(qf_{r} s\) $q{f}_{r}s$ ), which can handle all such situations, using the q-intercept of the straight line and letting both membership and non-membership grades approach 100% without depending on each other. The q = 2 is the smallest value for which all the situations in the first quadrant are tackled, and the sum of membership and non-membership grades is near 1. For all other values of q > 2, the sum of membership and non-membership grades approaches 0, i.e., the larger the value of q, i.e., the intercepts, the sum of memberships and non-membership grades approaches 0. For q = 1, the first intercept is simply the intuitionistic fuzzy set. We provide the basic properties of the q-fractional fuzzy set using the extension principle of fuzzy sets and develop some aggregation operators. We also developed a new q-fractional fuzzy neural network and provided an example.