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A strong formulation for Multiple Allocation Hub Location based on supermodular inequalities

Elena Fernández, Nicolás Zerega

TL;DR

This paper tackles the MA-HLP by introducing reinforced 2-index formulations based on supermodular inequalities. It develops FZ-P and FZ-S, showing that their linear programming bounds match the tightest 4-index path formulations (HLP_{MA}) while using far fewer variables, greatly improving scalability. The authors provide theoretical connections between the two 2-index formulations and demonstrate, through computational experiments on CAB and AP datasets, that FZ-S outperforms existing CF-S and CF-P variants and solves up to 200-node instances within two hours. The work offers a practical, scalable approach to MA-HLP with potential extensions to longer routing paths and non-complete backbones, enhancing both theory and applications in hub location design.

Abstract

We introduce a new formulation for the multiple allocation hub location problem that exploits supermodular properties and uses 1- and 2-index variables only. We show that the new formulation produces the same Linear Programming bound as the tightest existing formulations for the studied problem, which use 4-index variables, outperforming existing supermodular formulations adapted to the considered problem. Computational results are presented with instances of up to 200 nodes optimally solved within a time limit of two hours.

A strong formulation for Multiple Allocation Hub Location based on supermodular inequalities

TL;DR

This paper tackles the MA-HLP by introducing reinforced 2-index formulations based on supermodular inequalities. It develops FZ-P and FZ-S, showing that their linear programming bounds match the tightest 4-index path formulations (HLP_{MA}) while using far fewer variables, greatly improving scalability. The authors provide theoretical connections between the two 2-index formulations and demonstrate, through computational experiments on CAB and AP datasets, that FZ-S outperforms existing CF-S and CF-P variants and solves up to 200-node instances within two hours. The work offers a practical, scalable approach to MA-HLP with potential extensions to longer routing paths and non-complete backbones, enhancing both theory and applications in hub location design.

Abstract

We introduce a new formulation for the multiple allocation hub location problem that exploits supermodular properties and uses 1- and 2-index variables only. We show that the new formulation produces the same Linear Programming bound as the tightest existing formulations for the studied problem, which use 4-index variables, outperforming existing supermodular formulations adapted to the considered problem. Computational results are presented with instances of up to 200 nodes optimally solved within a time limit of two hours.

Paper Structure

This paper contains 11 sections, 5 theorems, 15 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation and problem definition
  3. Problem definition
  4. Existing formulations for the $MA-HLP$
  5. Formulations with path variables
  6. A reinforcement of $CF-P$
  7. A reinforced supermodular formulation
  8. Relating $FZ-S$ and $FZ-P$
  9. Computational experiments
  10. Conclusions
  11. Acknowledgments

Key Result

Lemma 1

There is an optimal solution to $HLP_{MA}$, $({\mathbf{z}^*}, {\mathbf{X}}^*)$, such that, for all $r\in R$,

Figures (4)

  • Figure 1: Relationship between the relevant formulations and our contribution
  • Figure 2: Networks illustrating Remark \ref{['Remark1']}
  • Figure 3: LP bound comparison between $CF-S$ and $FZ-S$ on CAB dataset
  • Figure 4: LP bound comparison between $CF-S$ and $FZ-S$ on AP dataset

Theorems & Definitions (12)

  • Remark 1
  • Lemma 1
  • Remark 2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 2 more