Solving the Traveling Salesman Problem (TSP) Using Genetic Algorithms

Using genetic algorithms to solve the Traveling Salesman Problem (TSP) represents a widely adopted optimization approach that mimics natural selection and genetic mechanisms to progressively refine path solutions. This method proves particularly effective for NP-hard problems like TSP, as it efficiently explores large search spaces to yield near-optimal solutions without exhaustive enumeration of all possible routes.

The core algorithmic workflow involves the following steps implemented in code: Population Initialization: Randomly generate an initial set of path solutions where each path represents an individual chromosome. In MATLAB implementation, this typically involves creating a matrix where each row encodes a city visitation sequence. Fitness Evaluation: Calculate fitness values for each individual, commonly using the reciprocal of total path distance - shorter paths yield higher fitness scores. The fitness function would compute Euclidean distances between consecutive cities in the permutation. Selection: Apply fitness-based selection mechanisms like roulette wheel selection or tournament selection to choose parents for reproduction. Code implementation often uses cumulative probability distributions for parent selection. Crossover (Recombination): Perform genetic exchange between selected parents using operators like Partially Mapped Crossover (PMX) or Order Crossover (OX) to generate offspring while preserving valid TSP tours. Mutation: Introduce random modifications (e.g., swapping two city positions) with a predefined mutation probability to maintain population diversity and prevent premature convergence. Iterative Optimization: Cycle through selection, crossover, and mutation operations until meeting termination criteria (maximum generations or fitness stabilization). The main loop typically includes elitism to preserve best solutions.

In the TSP1.m implementation, key parameters include population size, mutation rate, and maximum iterations. Parameter tuning balances convergence speed and solution quality. While genetic algorithms don't guarantee global optima, proper parameter configuration enables efficient generation of high-quality approximate solutions for large-scale TSP instances. The algorithm's performance can be enhanced through techniques like local search integration and adaptive parameter control.