A note on capacity models for network design

Abstract

In network design problems capacity constraints are modeled in three different ways depending on the application and the underlying technology for installing capacity: directed, bidirected, and undirected. In the literature, polyhedral investigations for strengthening mixed-integer formulations are done separately for each model. In this paper, we examine the relationship between these models to provide a unifying approach and show that one can indeed translate valid inequalities from one to the others. In particular, we show that the projections of the undirected and bidirected models onto the capacity variables are the same. We demonstrate that valid inequalities previously given for the undirected and bidirected models can be derived as a consequence of the relationship between these models and the directed model.

Data-Driven Transit Network Design at Scale

“…where, for any γ > 0, 1 /2γ r∈R k∈K x k,r i,j 2 is a ridge regularization term that improves Problem (1)'s practical tractability (c.f. Bertsimas et al 2021), by augmenting the hard constraint k∈K x k,r i,j ≤ u i,j on each edge's capacity with a soft penalty in the objective (see also Atamtürk and Günlük (2018) for a discussion of capacity constraints in network design problems). Correspondingly, Problem (1) is an upper approximation of the optimal objective value without regularization.…”