Revisiting transportation problems under Monge costs with applications to location problems

TL;DR

This paper explores transportation problems with Monge costs, providing new formulas for optimal solutions, and applies these insights to improve formulations and computational efficiency in facility location and median problems.

Contribution

It introduces compact formulas for dual solutions under Monge costs and applies them to enhance solution methods for location problems using Benders decomposition.

Findings

Formulas improve computational performance

New formulations achieve state-of-the-art results

Methods are robust across diverse instances

Abstract

We investigate the transportation problem under a Monge cost structure and derive compact formulas for optimal dual solutions based on the northwest-corner rule. As an application illustrating how these formulas yield structural insight while enhancing computational performance, we consider a broad class of facility location problems. In particular, the expressions are used within a Benders decomposition framework to derive novel formulations for the Discrete Ordered Median Problem with non-increasing weights. Numerical experiments validate that the resulting formulations achieve state-of-the-art performance and exhibit strong robustness across a wide range of instances.