Generalized fuzzy sets with complexities and applications in decision-making problems

All the prevailing theories based on FS and their modifications, inconsistency, and uncertainties are involved in the form of truth grade TG whose value is also in the form of real numbers and certain user information may be lost and the decision-maker is affected by this. The principle of a complex fuzzy set (CFS) is a valuable procedure to manage inconsistent and awkward information genuine life troubles. CFS gives the TG against the value which is taken from the set of attributes in the form of a complex number whose real and unreal parts are limited to the unit interval. In this paper, we discussed some operations and formulas of set theory for complex fuzzy sets. We established the basic results of complex fuzzy sets using bounded sum, bounded product, bounded difference, simple difference, Cartesian product, algebraic product, and algebraic sums. We discussed particular examples of these operations and results. Moreover, a multicriteria decision-making (MCDM) technique is explored based on the elaborated complex fuzzy dominance matrix by using the complex fuzzy information. The application has been effectively demonstrated with numerical examples.