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The Quadratic Bin Packing Problem: Exact Formulations and Algorithm

Vítor Gomes Chagas, Alberto Locatelli, Flávio Keidi Miyazawa, Manuel Iori

Abstract

In this article, we introduce and study the Quadratic Bin Packing Problem (QBPP), which generalizes the classical bin packing problem by introducing a fixed cost for each used bin and a pairwise cost (or profit) incurred whenever two items are packed together. Beyond its theoretical relevance, the QBPP is of practical interest due to its numerous real-world applications, mainly related to cluster analysis. To address the QBPP, we propose three compact mixed-integer linear programming (MILP) formulations, along with a set-partitioning formulation. For each compact model, we present an enhanced version with a strengthened continuous relaxation, while, for the set-partitioning formulation, we develop a tailored Branch-and-Price algorithm. Computational experiments on benchmark instances demonstrated that, while the enhanced compact formulations can be effectively solved by a standard MILP solver for small-sized instances, the Branch-and-Price approach delivered superior performance overall, especially on larger and more challenging instances.

The Quadratic Bin Packing Problem: Exact Formulations and Algorithm

Abstract

In this article, we introduce and study the Quadratic Bin Packing Problem (QBPP), which generalizes the classical bin packing problem by introducing a fixed cost for each used bin and a pairwise cost (or profit) incurred whenever two items are packed together. Beyond its theoretical relevance, the QBPP is of practical interest due to its numerous real-world applications, mainly related to cluster analysis. To address the QBPP, we propose three compact mixed-integer linear programming (MILP) formulations, along with a set-partitioning formulation. For each compact model, we present an enhanced version with a strengthened continuous relaxation, while, for the set-partitioning formulation, we develop a tailored Branch-and-Price algorithm. Computational experiments on benchmark instances demonstrated that, while the enhanced compact formulations can be effectively solved by a standard MILP solver for small-sized instances, the Branch-and-Price approach delivered superior performance overall, especially on larger and more challenging instances.

Paper Structure

This paper contains 22 sections, 2 theorems, 18 equations, 5 figures, 4 tables.

Table of Contents

  1. Keywords
  2. Introduction
  3. Linear Formulations
  4. Fortet–Glover–Woolsey Formulation
  5. Extended Fortet–Glover–Woolsey Formulation
  6. Two-index Assignment Formulation
  7. Extended Two-index Assignment Formulation
  8. Representative Formulation
  9. Extended Representative Formulation
  10. Set-Partitioning Formulation
  11. Branch-and-Price Algorithm
  12. Column Generation
  13. Multiple Constructive Heuristic for the GQKP
  14. Exact Algorithm for the GQKP
  15. Bounding functions.
  16. ...and 7 more sections

Key Result

Proposition 1

Given a node $v = (I,O,U)$ of the B&B tree and a pattern $P$ such that $I \subseteq P$, the inequality $p(P) \leq p(I) + \sum_{i \in P \cap U} \overline{p}_i$ holds. $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure 1: Overall performance on small-sized instances for the compact formulations.
  • Figure 2: Computational results on small-sized instances for R and ER.
  • Figure 2: Number of small-sized instances solved to proven optimality by the implemented compact formulations across different values of $\sigma$.
  • Figure 3: Number of small- to medium-sized instances solved to proven optimality by the compact enhanced formulations and the two B&P implementations, for different values of $\sigma$.
  • Figure 4: Comparison of the two B&P implementations in terms of large-sized instances solved to proven optimality across $\sigma$ and $\mu$ values.

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Proposition 2
  • proof