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Random-Key Metaheuristic and Linearization for the Quadratic Multiple Constraints Variable-Sized Bin Packing Problem

Natalia A. Santos, Marlon Jeske, Antonio A. Chaves

TL;DR

The paper tackles the challenging QMC-VSBPP by (i) linearizing the original quadratic Model to enable strong lower-bound computation with Gurobi, and (ii) developing RKO-ACO, a continuous-domain Ant Colony Optimization within the Random-Key Optimization framework, augmented with online Q-learning parameter control, caching, and local search. The linearized formulation provides substantially tighter lower bounds and four optimal small-instance solutions, while RKO-ACO consistently matches or improves best-known solutions across all benchmark sets, delivering new upper bounds for large-scale instances. The combined approach demonstrates that exact linearization and adaptive metaheuristics can synergistically advance complex packing problems with multiple capacities and interaction costs. The results establish new reference values for future QMC-VSBPP studies and highlight the efficacy of evolutionary and random-key methods in quadratic packing contexts.

Abstract

This paper addresses the Quadratic Multiple Constraints Variable-Sized Bin Packing Problem (QMC-VSBPP), a challenging combinatorial optimization problem that generalizes the classical bin packing by incorporating multiple capacity dimensions, heterogeneous bin types, and quadratic interaction costs between items. We propose two complementary methods that advance the current state-of-the-art. First, a linearized mathematical formulation is introduced to eliminate quadratic terms, enabling the use of exact solvers such as Gurobi to compute strong lower bounds - reported here for the first time for this problem. Second, we develop RKO-ACO, a continuous-domain Ant Colony Optimization algorithm within the Random-Key Optimization framework, enhanced with adaptive Q-learning parameter control and efficient local search. Extensive computational experiments on benchmark instances show that the proposed linearized model produces significantly tighter lower bounds than the original quadratic formulation, while RKO-ACO consistently matches or improves upon all best-known solutions in the literature, establishing new upper bounds for large-scale instances. These results provide new reference values for future studies and demonstrate the effectiveness of evolutionary and random-key metaheuristic approaches for solving complex quadratic packing problems. Source code and data available at https://github.com/nataliaalves03/RKO-ACO

Random-Key Metaheuristic and Linearization for the Quadratic Multiple Constraints Variable-Sized Bin Packing Problem

TL;DR

The paper tackles the challenging QMC-VSBPP by (i) linearizing the original quadratic Model to enable strong lower-bound computation with Gurobi, and (ii) developing RKO-ACO, a continuous-domain Ant Colony Optimization within the Random-Key Optimization framework, augmented with online Q-learning parameter control, caching, and local search. The linearized formulation provides substantially tighter lower bounds and four optimal small-instance solutions, while RKO-ACO consistently matches or improves best-known solutions across all benchmark sets, delivering new upper bounds for large-scale instances. The combined approach demonstrates that exact linearization and adaptive metaheuristics can synergistically advance complex packing problems with multiple capacities and interaction costs. The results establish new reference values for future QMC-VSBPP studies and highlight the efficacy of evolutionary and random-key methods in quadratic packing contexts.

Abstract

This paper addresses the Quadratic Multiple Constraints Variable-Sized Bin Packing Problem (QMC-VSBPP), a challenging combinatorial optimization problem that generalizes the classical bin packing by incorporating multiple capacity dimensions, heterogeneous bin types, and quadratic interaction costs between items. We propose two complementary methods that advance the current state-of-the-art. First, a linearized mathematical formulation is introduced to eliminate quadratic terms, enabling the use of exact solvers such as Gurobi to compute strong lower bounds - reported here for the first time for this problem. Second, we develop RKO-ACO, a continuous-domain Ant Colony Optimization algorithm within the Random-Key Optimization framework, enhanced with adaptive Q-learning parameter control and efficient local search. Extensive computational experiments on benchmark instances show that the proposed linearized model produces significantly tighter lower bounds than the original quadratic formulation, while RKO-ACO consistently matches or improves upon all best-known solutions in the literature, establishing new upper bounds for large-scale instances. These results provide new reference values for future studies and demonstrate the effectiveness of evolutionary and random-key metaheuristic approaches for solving complex quadratic packing problems. Source code and data available at https://github.com/nataliaalves03/RKO-ACO

Paper Structure

This paper contains 22 sections, 13 equations, 7 figures, 7 algorithms.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Problem Definition
  4. Model Linearization
  5. Random Key Optimization (RKO)
  6. RKO Framework
  7. ACO for Continuous Domains
  8. RKO-ACO
  9. Online Parameter Tuning
  10. Caching
  11. RKO-ACO for QMC-VSBPP
  12. Notations
  13. Generating Initial Solutions
  14. Decoding and Item to Bin Assignment
  15. Post-processing
  16. ...and 7 more sections

Figures (7)

  • Figure 1: RKO-ACO flowchart
  • Figure 2: Percentage gap from Gurobi results with original and linearized models
  • Figure 3: Relative gaps of RKO, VNS, and Gurobi to the best-known gap obtained by either VNS or Gurobi
  • Figure 4: Time required by RKO to reach the best solution reported by VNS
  • Figure 5: RKO's ARPD and BRPD values
  • ...and 2 more figures