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Fast Neighborhood Search Heuristics for the Colored Bin Packing Problem

Renan F. F. da Silva, Yulle G. F. Borges, Rafael C. S. Schouery

TL;DR

This work tackles the Colored Bin Packing Problem (CBPP), a color-constrained generalization of the classic Bin Packing Problem. It introduces five fast initial heuristics, a Variable Neighborhood Search (VNS) framework, and a matheuristic (MH) that couples linear programming with VNS and GRASP to generate high-quality patterns and solutions. Empirical results show that MH often outperforms VNS and yields near-optimal solutions on large instances (up to $10^4$ items), frequently matching the offline optimal within a single extra bin on many tested cases. The findings highlight the strength of the pattern-based formulation, the efficacy of the Two-by-Two heuristic, and the value of combining LP relaxations with disciplined meta-heuristics for CBPP. Practically, the methods enable fast, high-quality packing for complex CBPP instances, suggesting promising directions for tighter bounds and broader CBPP extensions.

Abstract

The Colored Bin Packing Problem (CBPP) is a generalization of the Bin Packing Problem (BPP). The CBPP consists of packing a set of items, each with a weight and a color, in bins of limited capacity, minimizing the number of used bins and satisfying the constraint that two items of the same color cannot be packed side by side in the same bin. In this article, we proposed an adaptation of BPP heuristics and new heuristics for the CBPP. Moreover, we propose a set of fast neighborhood search algorithms for CBPP. These neighborhoods are applied in a meta-heuristic approach based on the Variable Neighborhood Search (VNS) and a matheuristic approach that combines linear programming with the meta-heuristics VNS and Greedy Randomized Adaptive Search (GRASP). The results indicate that our matheuristic is superior to VNS and that both approaches can find near-optimal solutions for a large number of instances, even for those with many items.

Fast Neighborhood Search Heuristics for the Colored Bin Packing Problem

TL;DR

This work tackles the Colored Bin Packing Problem (CBPP), a color-constrained generalization of the classic Bin Packing Problem. It introduces five fast initial heuristics, a Variable Neighborhood Search (VNS) framework, and a matheuristic (MH) that couples linear programming with VNS and GRASP to generate high-quality patterns and solutions. Empirical results show that MH often outperforms VNS and yields near-optimal solutions on large instances (up to items), frequently matching the offline optimal within a single extra bin on many tested cases. The findings highlight the strength of the pattern-based formulation, the efficacy of the Two-by-Two heuristic, and the value of combining LP relaxations with disciplined meta-heuristics for CBPP. Practically, the methods enable fast, high-quality packing for complex CBPP instances, suggesting promising directions for tighter bounds and broader CBPP extensions.

Abstract

The Colored Bin Packing Problem (CBPP) is a generalization of the Bin Packing Problem (BPP). The CBPP consists of packing a set of items, each with a weight and a color, in bins of limited capacity, minimizing the number of used bins and satisfying the constraint that two items of the same color cannot be packed side by side in the same bin. In this article, we proposed an adaptation of BPP heuristics and new heuristics for the CBPP. Moreover, we propose a set of fast neighborhood search algorithms for CBPP. These neighborhoods are applied in a meta-heuristic approach based on the Variable Neighborhood Search (VNS) and a matheuristic approach that combines linear programming with the meta-heuristics VNS and Greedy Randomized Adaptive Search (GRASP). The results indicate that our matheuristic is superior to VNS and that both approaches can find near-optimal solutions for a large number of instances, even for those with many items.
Paper Structure (31 sections, 2 theorems, 2 equations, 4 figures, 3 tables, 7 algorithms)

This paper contains 31 sections, 2 theorems, 2 equations, 4 figures, 3 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Our Contributions
  4. Colored Bin Packing Problem
  5. Notations
  6. Auxiliary Algorithm
  7. Initial Heuristics
  8. Best Fit Decreasing
  9. Good Ordering
  10. Minimum Bin Slack
  11. Hard BFD
  12. Two-by-Two
  13. Variable Neighborhood Search
  14. Objective function
  15. Neighborhoods
  16. ...and 16 more sections

Key Result

Lemma 1

Given a subset $B_r$ of items. Let $g$ be one of the most frequent colors of $B_r$, $B_r^g$ be the subset of $B_r$ of items with color $g$, and $B_r^o$ be the subset of $B_r$ of items with a color different from $g$. Then, there is a permutation $\pi$ of $B_r$ that satisfies color constraint if, and

Figures (4)

  • Figure 1: Comparison between Initial Heuristics in AI instances.
  • Figure 2: Comparison between Initial Heuristics in Random25to50 instances.
  • Figure 3: Comparison between Initial Heuristics in Random10to80 instances.
  • Figure 4: Nemenyi Test for the Initial Heuristics.

Theorems & Definitions (3)

  • Lemma 1
  • Lemma 2
  • proof