A Step-by-Step Mathematical Derivation of Ant Colony Optimization Algorithm for solving 01 Knapsack Problem

Abstract

This article provides a meticulous step-by-step mathematical derivation of the Ant Colony Optimization (ACO) algorithm designed for solving the 0/1 Knapsack Problem. Each facet of the algorithm, from initialization to pheromone updates, solution construction, and the delicate balance between exploration and exploitation, is dissected. This comprehensive understanding of ACO's underlying mathematical framework offers a clear roadmap for researchers and practitioners, guiding them in harnessing ACO's power to optimize combinatorial challenges. The 0/1 Knapsack Problem, a significant challenge in computer science due to its NP-hard nature, involves strategically loading objects into a knapsack with limited capacity. Despite its computational complexity, this problem finds applications in operations research and finance. The article addresses the urgency for swift solutions by exploring heuristic algorithms inspired by ant behavior, providing a valuable guide for efficiently solving the 0/1 Knapsack Problem.