A Catalog of Formulations for the Network Pricing Problem

TL;DR

This paper explores multiple reformulations of the network pricing problem, introduces new modeling techniques, and demonstrates their effectiveness through numerical comparisons, advancing solution approaches for bilevel optimization in network pricing.

Contribution

It establishes connections among various reformulations, introduces a new path enumeration scheme, and develops a hybrid framework to improve solving efficiency and robustness.

Findings

12 different reformulations generated and analyzed.

Path-based preprocessing improves computational performance.

Numerical results confirm the efficiency of the proposed methods.

Abstract

We study the network pricing problem where the leader maximizes their revenue by determining the optimal amounts of tolls to charge on a set of arcs, under the assumption that the followers will react rationally and choose the shortest paths to travel. Many distinct single-level reformulations to this bilevel optimization program have been proposed, however, their relationship has not been established. In this paper, we aim to build a connection between those reformulations and explore the combination of the path representation with various modeling options, allowing us to generate 12 different reformulations of the problem. Moreover, we propose a new path enumeration scheme, path-based preprocessing, and hybrid framework to further improve performance and robustness when solving the final model. We provide numerical results, comparing all the derived reformulations and confirming the efficiency of the novel dimensionality reduction procedures.