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

An Exact Algorithm for the Fixed Charge Transportation Problem Based on Matching Source and Sink Patterns

This paper describes an exact algorithm for the fixed charge transportation problem based on a new integer programming formulation that involves two sets of variables representing flow patterns from sources to sinks and from sinks to sources. The formulation states to select a pattern for each source and each sink and to match the corresponding flows. The linear relaxation of this new formulation is enforced by adding a pseudo-polynomial number of equations that are shown to contain, as special cases, different valid inequalities recently proposed for the problem. The resulting lower bound dominates the lower bounds proposed in the literature. Such a lower bound is embedded into an exact branch-and-cut-and-price algorithm. Computational results on benchmark instances show that the proposed algorithm is several times faster than the state-of-the-art exact methods and could solve all open instances. New harder instances with up to 120 sources and 120 sinks were solved to optimalit