Mathematical foundations, structural embeddings, and complement operations in hyperbolic fuzzy set theory

Abstract

This paper presents a comprehensive study of Hyperbolic Fuzzy Sets (HyFS), a generalized fuzzy framework characterized by independent optimistic and pessimistic membership degrees constrained by a hyperbolic relation. We explore the embedding of q-Rung Orthopair Fuzzy Sets, (n, m)-Rung Orthopair Fuzzy Sets, and (n, m)-Power Root Fuzzy Sets into HyFS, thereby establishing its flexibility and generality. A comparative analysis based on computational complexity, runtime, and scalability demonstrates that HyFS achieves superior performance due to its simple, parameter-free structure. The study further investigates complement operations in HyFS, including classical and mixed-type complements using different fuzzy negation functions, and verifies theoretical properties such as De Morgan’s laws, distributivity, and difference operations. To validate practical utility, two real-world case studies are presented: cryptocurrency investment evaluation using HyFS-CoCoSo and sustainable transportation planning using HyFS-OCRA, both of which confirm the framework’s accuracy, stability, and scalability. Our results highlight the robustness, efficiency, and logical consistency of HyFS, positioning it as a promising tool for uncertainty modeling, data analytics, and real-time decision-making applications.