It is well known that the traveling salesman problem (TSP) is one of the most studied NP-complete problems, and evolutionary technique such as simulated annealing has mostly been used to solve various NP-complete problems. In this paper, a two-stage simulated annealing (two-stage SA) algorithm is proposed to solve the TSP, and the two-stage SA algorithm is made up of two stages. In the first stage, a simple simulated annealing (simple SA) algorithm is proposed to obtain some appropriate solutions or closed tours. In the second stage, an effective simulated annealing (effective SA) algorithm is proposed to obtain solutions with good quality based on the solutions or closed tours obtained by the simple SA algorithm. To assess the effectiveness of the two-stage SA algorithm, simulations were carried out on 23 benchmark TSP instances. The simulation results show that the two-stage SA algorithm can obtain solutions with better quality than four recent algorithms such as ant colony optimization algorithm, self-organizing maps algorithm, particle swarm optimization algorithm and constructive optimizer neural network algorithm.

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Local search techniques, such as simulated annealing and greedy search, have been used to solve various NP-hard problems, and it is well known that the graph coloring problem (GCP) is one of the most studied NP-hard optimization problems. In this paper, an adaptive local search algorithm is proposed to solve the GCP. The proposed algorithm is based on the standard simulated annealing technique, and makes the best of greedy search technique and search local strategy to speed up the convergence rate. Aim at the greedy search parameters, the run time for given benchmark instance, annealing temperature and its cool coefficient, adaptive control of the parameters and multistage simulated annealing are designed in order to improve quality-time trade-off of the proposed local search algorithm. As a result, our algorithm achieves a reasonable trade-off between computation time, solution quality, and complexity of implementation, and simulation results indicate that the proposed algorithm performs well on a set of 119 benchmark instances.

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