Spherical Picture Fuzzy Sets with Application to Multicriteria Decision-Making (European Journal of Pure and Applied Mathematics, 2025)
Since the introduction of fuzzy sets by Zadeh in 1965 [1], a lot of new theories regarding imprecision and uncertainty have been introduced. Some of these theories are extensions of fuzzy set theory, other try to handle imprecision and uncertainty in different way. The extensions of ordinary fuzzy sets are classified into two broad categories: 1. Intuitionistic fuzzy sets [2] and their versions, 2. Neutrosophic sets [3] and their versions. The first group extensions can be defined by a membership degree and a non-membership degree, whereas the second class of extensions can be defined by a membership degree (truthiness), a non-membership degree (falsity), and a hesitancy degree (indeterminacy). Spherical and picture fuzzy sets fall into the same group because of the definition of membership functions. The squared sum of membership, nonmembership, and hesitancy degrees is equal to or less than 1.0 in spherical fuzzy sets whereas it is valid for the first degree sum in picture fuzzy sets. In this paper, we unify the concepts of picture fuzzy set and spherical fuzzy set into a broad class and name it as spherical picture fuzzy set (SP F S). In SP F Ss, every element of the universe is represented by a sphere. This unique geometrical representation is more adaptable and adequate for handling ambiguity in multi criteria decision-making. A new distance measure of spherical picture fuzzy sets is illustrated, and it is shown that it satisfies conditions of the distance measure. Besides investigating the structural properties of SP F S, set-theoretical operations along with some basic algebraic operations and aggregation operators are discussed. Oneof the most popular multi-criteria decision-making techniques, T OP SIS, is expanded to its SP F S form. To demonstrate the effectiveness and feasibility of the proposed SP F S-T OP SIS methodology for managing inherent vagueness in the given data, a numerical case study is analyzed wherein the methodology is applied to the pandemic hospital site selection problem.
Keywords: Spherical picture fuzzy set; Hesitancy degree; Multi-criteria; Decision making