Lagrangian relaxation heuristics for deterministic transshipment problems

Transshipment problems (TPs) have been widely applied in academic and practical areas. The deterministic TPs, which include four characteristics: commodities, modes (transportation methods), generalized network, and time periods, are NP-hard problems. Therefore, this research analyzes the structure of the multi-commodity multi-mode generalized networks over a multi-time-period problem and develops three Lagrangian-relaxation heuristic algorithms to minimize total costs for delivering commodities. The proposed mathematical model incorporates an upper bound on traffic flow for each segment of the distribution network. This is the so-called knapsack constraint. Further, two application networks are tested for medium and large-scale problems. As a result, simulation results show that the three proposed algorithms can provide more feasible solutions with attractive levels of near-optimality within reasonable time compared to the branch-and-bound method.