Horizontal membership function (HMF) for fuzzy membership function type with two parameters

Membership function (MF), also called vertical membership function (VMF), is an essential object of study in fuzzy concepts which models fuzzy sets using different sorts of data or fuzzy hypotheses. Fuzzy membership functions are classified based on the number of fuzzy parameters. According to the definition of VMF, the degree of membership µ(x) can be ambiguous to the variable x. It is necessary to construct horizontal membership functions (HMF) to ensure that each function is unambiguous. HMF concept needs to be developed further to solve the problem containing uncertain variables. HMF are constructed using the definition of VMF. The paper presents the construction of HMF with two parameters like Gaussian, Sigmoid, Rectangular, S-Shaped, and Z-Shaped membership functions. Visualization of construction results are described to observe how similar the visualizations between the two-parameter HMF types. Some examples of fuzzy optimal control problems are given to be solved implementing two-parameter HMF types.