Randomized heuristic repair for large-scale multidimensional knapsack problem

TL;DR

This study tackles large-scale multidimensional knapsack problems (MKP) by enhancing the deterministic Chu & Beasley heuristic repair with a randomized, efficiency-group approach. It defines efficiency groups via rounding of dual-based efficiencies $e_j^{\text{dual}} = p_j / \sum_i \lambda_i w_{ij}$ and introduces two randomization operators (rg-swap and rg-shuffle) to explore item orderings without undermining the quality signal. Four variants, controlled by rounding level $d$ and operator type, are evaluated against the original CBGA on 270+ MKP instances, showing faster convergence and often smaller gaps to the best-known solutions, with occasional attainment of the best-known result. The work provides a practical mechanism to balance exploitation and exploration in large-scale MKP metaheuristics, highlighting interesting directions for parameter interaction analysis and integration with local search.

Abstract

The multidimensional knapsack problem (MKP) is an NP-hard combinatorial optimization problem whose solution is determining a subset of maximum total profit items that do not violate capacity constraints. Due to its hardness, large-scale MKP instances are usually a target for metaheuristics, a context in which effective feasibility maintenance strategies are crucial. In 1998, Chu and Beasley proposed an effective heuristic repair that is still relevant for recent metaheuristics. However, due to its deterministic nature, the diversity of solutions such heuristic provides is insufficient for long runs. As a result, the search for new solutions ceases after a while. This paper proposes an efficiency-based randomization strategy for the heuristic repair that increases the variability of the repaired solutions without deteriorating quality and improves the overall results.

Figures (1)

• Figure 1: Efficiency groups produced by rounding the original efficiencies to the first decimal case.