The hub location problem with market selection

Abstract We propose a hub location problem with market selection. The problem has a given number of markets with each market associated with predetermined demand and fixed revenue. The selected market demand has to be shipped from the origin to the destination via a hub-and-spoke system. The overall objective is to maximize the overall margins that are equal to the total revenue of the selected markets minus the total costs, including the leasing costs of hubs, transportation costs, and outsourcing costs. We propose a mixed integer programming model to formulate the problem with service and flow capacity constraints. For this problem, we propose a subgradient-based Lagrangian relaxation method that exploits the rich structure of the Lagrangian subproblems to obtain its objective values and bounds efficiently. Specifically, we treat the Lagrangian subproblems as knapsack problems and solve them by using a dynamic programming method. Furthermore, we propose a polynomial algorithm to quickly generate objective values of the original problem at each iteration of the Lagrangian relaxation. Computational results show that the proposed method can obtain optimal solutions for small-sized test instances and achieve much better solution qualities than the commercial software package (Cplex) for large-sized test instances. The proposed method also achieves better solutions than a Benders decomposition method for the tested instances.