Genetic Algorithm for the 0/1 Multidimensional Knapsack Problem

TL;DR

This paper introduces a genetic algorithm that efficiently solves the multidimensional 0/1 knapsack problem by leveraging Lagrangian multipliers and greedy crossover, outperforming previous algorithms in convergence speed.

Contribution

The paper presents a novel genetic algorithm incorporating Lagrangian multipliers and greedy crossover, significantly improving convergence speed over existing methods.

Findings

Achieves faster convergence to optimal or near-optimal solutions.

Successfully solves publicly available instances in very short computational times.

Outperforms the algorithm by Chu and Beasley in convergence speed.

Abstract

The 0/1 multidimensional knapsack problem is the 0/1 knapsack problem with m constraints which makes it difficult to solve using traditional methods like dynamic programming or branch and bound algorithms. We present a genetic algorithm for the multidimensional knapsack problem with Java and C++ code that is able to solve publicly available instances in a very short computational duration. Our algorithm uses iteratively computed Lagrangian multipliers as constraint weights to augment the greedy algorithm for the multidimensional knapsack problem and uses that information in a greedy crossover in a genetic algorithm. The algorithm uses several other hyperparameters which can be set in the code to control convergence. Our algorithm improves upon the algorithm by Chu and Beasley in that it converges to optimum or near optimum solutions much faster.