Revisiting transportation problems under Monge costs with applications to the Discrete Ordered Median Problem

Abstract

We investigate the classical Hitchcock-Koopmans Transportation Problem under the assumption that the cost matrix satisfies the Monge property. In this setting, we 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 mixed-integer linear programming 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.

arXiv comment33 pages, 7 figures, 7 tables. Major revisions: expanded motivation (Section 1); refined section structure and theoretical proofs (Sections 2 & 3); significantly extended computational study with additional benchmark instances (Section 4); moved detailed numerical results to Mendeley Data. Models and main contributions remain unchanged