Piecewise Linear Cost Network Design

This chapter considers one of the most important extensions to “basic” network design models required to accurately model real-world applications: the fact that the capacity on the arcs does not come in an “all-or-nothing” fashion, but with a more complex cost. This often corresponds to the fact that providing capacity actually boils down to provisioning appropriate facilities on the arc. In some simple cases, such as when the facilities can be installed independently from one another or when they are all identical, the models of the previous chapters can be adapted via the simple trick of replicating the arcs. This leads to replicating the flow variables, which might be disadvantageous. Furthermore, in general, the shape of the cost-of-capacity function can be much more complex due to nontrivial interactions between installation of different facilities. However, every reasonable cost-of-capacity function can be approximated as a piecewise linear (possibly, non-convex) one, where the extent of the approximation can be tightly controlled at the cost of the number of breakpoints, and therefore are appropriate for a large variety of real-world applications. This chapter focuses on such models, considering both a very general case and some more restricted ones that serve to illustrate the main concepts. We show that the best models in terms of tightness of the continuous relaxation bound suffer from a significant increase in the number of variables, even more so than the already rather large ones corresponding to simpler cases. Because of this, we review the use of techniques that allow to efficiently solve very large formulations by dynamically generating smaller portions of them. These techniques are instrumental for the practical solution of network design problems with piecewise linear costs.