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Revisiting transportation problems under Monge costs with applications to location problems

Stefan Nickel, Justo Puerto, Simon Ramoser, Alberto Torrejon

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.

Revisiting transportation problems under Monge costs with applications to location problems

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.
Paper Structure (15 sections, 11 theorems, 25 equations, 3 figures, 1 table, 4 algorithms)

This paper contains 15 sections, 11 theorems, 25 equations, 3 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Literature review
  3. The Transportation Problem
  4. The Discrete Ordered Median Problem
  5. Contributions
  6. Transportation problems under Monge costs
  7. Application to the Discrete Ordered Median Problem
  8. Theoretical notes
  9. Practical notes
  10. Numerical experiments
  11. Instances
  12. Reference models
  13. Discussion
  14. Conclusion and further research
  15. Tables

Key Result

Proposition 2.2

For the cells $\mathcal{A} = \{a^1, \dots, a^T\}$ traversed by Algorithm NW_corner_rule, it holds

Figures (3)

  • Figure 1: Number of instances solved to optimality vs. average runtime per instance size $n$.
  • Figure 2: Cumulative distribution of optimality gaps at termination.
  • Figure 3: Average callback times and numbers of added lazy constraints across all tested instances.

Theorems & Definitions (22)

  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3: Correctness of Algorithm \ref{['dual_algo_forward']}
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Corollary 2.7
  • ...and 12 more