PSO Implementation for the Knapsack Problem

In combinatorial optimization, the Knapsack Problem (also known as the knapsack packing problem) represents a fundamental challenge: given a collection of items, each with a specific weight and value, the objective is to select a subset of items to place in a knapsack such that the total weight does not exceed a predetermined capacity while maximizing the total value of the included items. The problem's name originates from its practical interpretation—filling a fixed-size backpack with the most valuable combination of items. This optimization model finds applications in numerous real-world scenarios including resource allocation, item selection strategies, and planning problems. Code implementations often utilize dynamic programming for exact solutions or metaheuristic approaches like Particle Swarm Optimization (PSO) for near-optimal solutions in large-scale instances, where PSO particles represent potential item selections and fitness functions evaluate total value against weight constraints.