Treffer: Multicommodity network design with discrete node costs

Although there is an extensive literature dealing with network design, little attention has been devoted to networks with complicated node costs. Although node costs, depending linearly on the total passing flow, can be easily embedded into the more usual framework of networks with link costs, when the node costs are, for instance, a stepwise function of the facilities installed into the nodes, this is no longer possible. This feature seems to be crucial in modern telecommunications networks, but has also applications in other fields, where a limited set of technologies is available with discrete values of capacities and costs. In our specific application, we propose a mathematical programming model that explicitly accounts for node costs that are stepwise with nonlinear increments. Two families of valid inequalities are then introduced, one of which is an extension of those presented in a previous work by Stoer and Dahl for multifacility network models. As the separation problem for these inequalities is difficult, we develop a heuristic separation procedure. We devise a branch-and-cut method and test it on a set of real-world instances found in the network design literature. This new method proves to be efficient when compared to a commercial general purpose MIP algorithm.