Hyperfuzzy and SuperHyperfuzzy Numbers: Generalizing Triangular, Trapezoidal, Pentagonal, Hexagonal, and Octagonal Fuzzy Numbers with Ranking Functions

Uncertainty modeling underpins decision-making in many domains. Over the years, numerous frameworks have been proposed to capture different facets of imprecision, including fuzzy sets [1, 2], rough sets [3, 4], hesitant fuzzy sets [5, 6], neutrosophic sets [7, 8], and plithogenic sets [9, 10]. More recently, hyperfuzzy sets and their recursive generalization, superhyperfuzzy sets, have been introduced to assign set-valued membership degrees at multiple hierarchical levels, thereby enriching the representation of uncertainty [11,12]. In parallel, a variety of fuzzy-number extensions—triangular, trapezoidal, pentagonal, hexagonal, and octagonal fuzzy numbers—together with ranking functions on fuzzy numbers, have been studied and applied in decisionmaking and related fields [13, 14]. In this paper, we develop hyperfuzzy and superhyperfuzzy analogues of these fuzzy-number constructs. Our aim is to provide a unified, hierarchically structured toolkit for modeling complex uncertainty in decision support and beyond.