New approaches in metaheuristics to solve the fixed charge transportation problem in a fuzzy environment

Fixed charge transportation problem (FCTP) is a primary and important problem which attracts researchers in the last decade. Recently, solution approaches typically metaheuristics are in focus. Therefore, metaheuristics have been developed to solve such a nondeterministic polynomial-time hard (NP-hard) problem. Since the real world is a complicated system and we could not formulate it as an exact problem, therefore it is necessary to describe an approximate and a fuzzy model. In this paper, both fixed costs and variable costs are considered as the fuzzy numbers. Three well-known algorithms that included a single point-based and two population-based metaheuristics are developed. Besides, a new population-based algorithm that has not been used in the previous works is developed: whale optimization algorithm (WOA). Contrary to previous works, this paper proposes new approaches in solution algorithms using both spanning tree-based Prufer number and priority-based representation. Also, Taguchi method is used to guarantee the proper performance of algorithms and calibration of parameters. In addition, several problems with different sizes are generated to assess the capability of the algorithms and commercial software according to the real-world case.