Experimental Report on the Design and Application of Progressive Membership Functions

Traditional membership functions in fuzzy mathematics often have limitations in practical applications due to simplified assumptions about boundary characteristics. Particularly in open-boundary scenarios (e.g., variables with no fixed limits such as wealth or debt), they struggle to accurately depict the natural transition from extreme states to intermediate states. To address this, this paper proposes the design concept and implementation method of progressive membership functions. By distinguishing between scenarios with fixed boundaries and non-fixed boundaries, this function constructs a mathematical model that approximates an intermediate threshold from extreme values (either fixed values or infinity). Its core feature is a membership degree change rate that transitions from slow to fast, better aligning with real-world cognitive laws and physical properties. To verify its effectiveness, three experimental cases were designed: (1) Basic characteristics and parameter sensitivity experiments validated the function’s adjustability and continuity; (2) Experiments based on Weber-Fechner law confirmed consistency with human perception rules; (3) Financial risk assessment experiments (including small and micro enterprise scenarios) demonstrated its practical value. Results show that compared to traditional Sigmoid and triangular membership functions, progressive membership functions exhibit significant advantages in open-boundary handling, parameter flexibility, and noise robustness, providing a new supplement and perspective for membership function design in fuzzy mathematics.