Strengthening Dual Bounds for Multicommodity Capacitated Network Design with Unsplittable Flow Constraints

TL;DR

This paper introduces new valid inequalities and solution techniques to strengthen dual bounds in the complex multicommodity network design problem with unsplittable flow constraints, improving solution quality for large-scale logistics networks.

Contribution

It develops novel classes of valid inequalities and a dynamic approach to enhance dual bounds in IP formulations of the MCND problem with unsplittable flow constraints.

Findings

Reduces IP gap by 26.5% and 22.5% on large instances.

Achieves over 85% gap reduction with new valid inequalities.

Demonstrates effectiveness on real-world e-commerce data.

Abstract

Multicommodity capacitated network design (MCND) models can be used to optimize the consolidation of shipments within e-commerce fulfillment networks. In practice, fulfillment networks require that shipments with the same origin and destination follow the same transfer path. This unsplittable flow requirement complicates the MCND problem, requiring integer programming (IP) formulations in which binary variables replace continuous flow variables. To enhance the solvability of this variant of the MCND problem for large-scale logistics networks, this work focuses on strengthening dual bounds. We investigate the polyhedra of arc-set relaxations, and we introduce two new classes of valid inequalities that can be implemented within solution approaches. We develop one approach that dynamically adds valid inequalities to the root node of a reformulation of the MCND IP with additional valid metric inequalities. We show the effectiveness of our ideas with a comprehensive computational study using path-based fulfillment instances, constructed from data provided by a large U.S.-based e-commerce company, and the well-known arc-based Canad instances. Experiments show that our best solution approach for a practical path-based model reduces the IP gap by an average of 26.5% and 22.5% for the two largest instance groups, compared to solving the reformulation alone, demonstrating its effectiveness in improving the dual bound. In addition, experiments using only the arc-based relaxation highlight the strength of our new valid inequalities relative to the linear programming relaxation (LPR), yielding an IP-gap reduction of more than 85%.