Fuzzy multi-objective nonlinear programming problems under various membership functions: a comparative analysis

Fuzzy sets have been applied to various decision-making problems when there is uncertainty in real-life problems. In decision-making problems, objective functions and constraints sometimes cannot be expressed linearly. In such cases, the problems discussed are expressed by nonlinear programming models. Fuzzy multi-objective programming models are problems containing multiple objective functions, where objective functions and/or constraints include fuzzy parameters. Membership functions are crucial to obtain optimal solution of fuzzy multi-objective programming model. In this study, a green supply chain network model with fuzzy parameters is proposed. Proposed model with nonlinear constraints is a fuzzy multi-objective nonlinear programming model that minimizes both transportation costs and emissions generated by two vehicle types during transportation. The model is used in Zimmermann's Min-Max approach by considering triangular, hyperbolic and exponential membership functions and optimal solutions are obtained. When optimal solutions are compared, it is seen that optimal solution obtained using the hyperbolic membership function is better than the optimal solutions obtained from triangular and exponential ones. Maximum common satisfaction level calculated using hyperbolic membership function for proposed model is λ=0.97. Sensitivity analysis is also carried out by taking into account distances between suppliers, manufacturers, distribution centers and customers, as well as customer demands.