Discrete PSO Algorithm Routine for Traveling Salesman Problem

This routine implements a discrete Particle Swarm Optimization (PSO) algorithm to solve the Traveling Salesman Problem (TSP). The algorithm's core functionality involves dynamically generating path strings while maintaining path validity through specialized constraint-handling mechanisms. The implementation features velocity and position update operations adapted for discrete combinatorial optimization, where particle positions represent candidate tours and velocities guide transitions between solutions. Key algorithm components include: - A permutation-based representation where each particle encodes a complete tour - Discrete velocity operators that perform partial path swaps or inversions - Validity-preserving update mechanisms ensuring all generated paths form valid Hamiltonian cycles - Fitness evaluation using total tour distance calculation The discrete PSO approach effectively leverages swarm intelligence principles to explore the solution space, with built-in constraints that automatically maintain path legitimacy during the optimization process. This implementation demonstrates not only TSP solutions but also provides a framework adaptable to similar combinatorial optimization problems. The routine showcases how discrete PSO with dynamic path generation ensures both algorithmic effectiveness and practical feasibility through systematic solution exploration and constraint satisfaction.