A Lagrangian Relaxation for Capacitated Single Allocation P-hub Median Problem with Multiple Capacity Levels

In this paper we consider a capacitated single allocation p-hub median problem with multiple capacity levels (CSApHMPMC) in which the decisions are to determine the location of p hubs and their capacity levels, the single allocation of non-hub nodes to hubs in the logistics network. This problem is formulated as an integer programming model with the objective of minimizing the sum of total transportation cost and fixed cost of the selected p hubs with established capacity levels. A Lagrangian relaxation (LR) approach is proposed to solve the CSApHMPMC. The Lagrangian function that we formulated decomposed the original problem into smaller subproblems that can be solved easier. We only solve the CSApHMPMC using Gurobi optimizer for the small sized problems. The experimental results show that the proposed LR heuristic can be an effective solution method for the capacitated p hub median location problem with multiple capacity levels.