Hybrid Membership Functions in Fuzzy Systems: A Survey of Structures, Applications, and Research Gaps

Abstract

The choice of membership function (MF) plays an important role in the ability of fuzzy systems to represent uncertainty. This survey reviews classical and hybrid membership functions, with emphasis on their mathematical structures, applications, strengths, limitations, and reported performance. The review covered publications from 1965 to 2026, with 35 sources retained for the review and theoretical background. The literature was identified through searches of Google Scholar, Scopus, ScienceDirect, SpringerLink, and IEEE Xplore, using terms related to membership functions, fuzzy membership functions, hybrid membership functions, combined membership functions, classical membership functions, and fuzzy systems. Ten classical membership functions and reported hybrid or mixed membership-function approaches were examined. The reviewed literature shows that hybridization can provide additional representational flexibility by combining useful characteristics of different membership functions. However, direct performance comparisons across studies remain difficult because of differences in datasets, fuzzy-system architectures, optimization procedures, applications, and evaluation measures. The review also indicates that much of the identified hybrid literature focuses on two-component or mixed membership-function approaches, while systematic evidence on higher-order hybridization remains limited. Higher-order hybrid membership functions therefore represent a research direction that requires further mathematical development and empirical benchmarking against well-tuned classical and binary-hybrid alternatives. Such evaluation should consider predictive performance, parameter complexity, computational cost, robustness, and interpretability