The p-Median Problem with Concave Costs

The p-median problem has been widely studied in the literature during the last decades especially its linear version. We propose a capacitated p-median problem with concave costs, in which the global cost incurred for each established facility is a concave function of the quantity q delivered by this facility. The unit distribution cost decreases with the increased quantity of output or demand generating economy of scales. This variant is in the class of non-linear location problem. We use DICOPT to solve this concave model. And then we transform this model into a linear programming problem and solve it using the commercial solver CPLEX. We also use the metaheuristic Variable Neighbourhood Search (VNS) to solve this problem. Computational results show that our linearization method helps to improve the calculations of the concave model. With VNS, we solve large size instances with up to 1500 facilities within a reasonable CPU time. Mathematical Non-linear Model and its linearization model In practice, distribution costs are not always dependent on the distance between sites and customers but on the total quantity delivered by the sites. For instance, if we outsource our transportation service to a third party logistics (3PL), the cost will be related to the quantity that we ask 3PL to deliver. That’s the linear part Bi · qi included in our objective function. And the more we delivered, the less we pay for the unit distribution cost. That’s the concave part Ci √ qi.