Solving Traveling Salesman Problem Using Simulated Annealing Algorithm

Simulated annealing is an optimization algorithm inspired by the physical process of solid annealing, commonly used to solve combinatorial optimization problems like the Traveling Salesman Problem (TSP). The TSP requires finding the shortest possible route that visits each city exactly once and returns to the starting point. This represents a classic NP-hard problem where computational complexity increases exponentially with the number of cities.

The core concept of simulated annealing involves introducing random factors to escape local optima and gradually approach global optimal solutions. The algorithm implementation typically includes temperature initialization, new solution generation, acceptance/rejection criteria, and cooling schedules. For TSP implementation, the algorithm starts with an initial path, randomly swaps two cities' positions to generate new routes, and determines acceptance based on objective function changes (total path length). In code, this involves maintaining a current solution array and using Metropolis criterion probability calculations.

For a 20-city TSP scenario, the algorithm first generates a random initial path and calculates its total distance. During high-temperature phases, the code accepts worse solutions to explore broader solution spaces, while gradually favoring better solutions as temperature decreases. Convergence to a stable path is achieved through proper parameter tuning. Key implementation parameters include initial temperature setting (often based on initial solution quality), cooling rate (using geometric decay like T_new = α*T_old), and termination conditions (final temperature or iteration limits).

The algorithm's strength lies in its flexibility and global search capability for complex optimization problems. However, performance depends heavily on parameter configuration such as initial temperature, cooling strategy, and neighborhood generation methods, which require experimental tuning. For 20-city TSP instances, simulated annealing can find near-optimal solutions within reasonable computation time, making it suitable for practical route planning applications. Code implementation typically includes distance matrix calculation, solution perturbation functions (e.g., 2-opt swaps), and temperature-dependent acceptance probability functions.