On Capacity Models for Network Design

TL;DR

This paper explores the relationships between different capacity modeling approaches in network design, providing a unified framework and methods to translate valid inequalities across models, enhancing formulation strength.

Contribution

It establishes a unifying approach linking directed, bidirected, and undirected capacity models, enabling translation of valid inequalities between them.

Findings

Projections of undirected and bidirected models onto capacity variables are identical.

Valid inequalities for undirected and bidirected models can be derived from directed models.

The relationships facilitate improved polyhedral analysis and formulation strengthening.

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.