Continuous Function Valued q-Rung Orthopair Fuzzy Sets and an Extended TOPSIS

Fuzzy sets, which have a crucial role in the decision making theory, model uncertainty by means of membership and non-membership functions. q-rung orthopair fuzzy sets, which are the natural extension of fuzzy, intuitionistic fuzzy and Pythagorean fuzzy sets, are quite successful in modeling data thanks to their larger domains. However, in a q-rung orthopair fuzzy set the membership and non-membership degrees of an element to a set are given just by a pair of certain numbers from the closed interval [0, 1] that causes a strict modelling. Various types of interval valued fuzzy sets, multi fuzzy sets or circular fuzzy sets change these strict modelling with a sensitive one. In this paper, we introduce a new fuzzy set notion via continuous functions that take values on a closed interval to provide a more sensitive tool in decision making theory. In this new fuzzy set notion, the membership and non-membership degrees of an element to a fuzzy set are represented by continuous functions instead of numbers. Actually, we study not only with points, but also with functions by taking into account the sufficiently large and continuous neighborhoods of the points. Thus more sensitive and realistic models are made by relieving the precision of the fuzzy data or linguistic argument. The data carried to function space environment is processed with the function theoretic tools via aggregation functions, distance measures or score functions. This fact distinguishes the new fuzzy set notion from the other continuous extensions in the literature such as interval valued or circular structures. Moreover, we provide an extended Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) in this new fuzzy environment and apply it to a multi criteria group decision making problem from the literature. Finally, we provide a comparison analysis and a complexity analysis. We also visualise the time complexity of the proposed extended TOPSIS for different numbers of decision makers.