Extending the solid step fixed-charge transportation problem to consider two-stage networks and multi-item shipments

This paper develops a new mathematical model for a capacitated solid step fixed-charge transportation problem. The problem is formulated as a two-stage transportation network and considers the option of shipping multiple items from the plants to the distribution centers (DC) and afterwards from DCs to customers. In order to tackle such an NP-hard problem, we propose two meta-heuristic algorithms; namely, Simulated Annealing (SA) and Imperialist Competitive Algorithm (ICA). Contrary to the previous studies, new neighborhood strategies maintaining the feasibility of the problem are developed. Additionally, the Taguchi method is used to tune the parameters of the algorithms. In order to validate and evaluate the performances of the model and algorithms, the results of the proposed SA and ICA are compared. The computational results show that the proposed algorithms provide relatively good solutions in a reasonable amount of time. Furthermore, the related comparison reveals that the ICA generates superior solutions compared to the ones obtained by the SA algorithm.