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Nature-inspired metaheuristic algorithms, especially those based on swarm intelligence, have attracted much attention in the last ten years. Firefly algorithm appeared in about five years ago, its literature has expanded dramatically with diverse applications. In this paper, we will briefly review the fundamentals of firefly algorithm together with a selection of recent publications. Then, we discuss the optimality associated with balancing exploration and exploitation, which is essential for all metaheuristic algorithms. By comparing with intermittent search strategy, we conclude that metaheuristics such as firefly algorithm are better than the optimal intermittent search strategy. We also analyse algorithms and their implications for higher-dimensional optimization problems.

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Bat algorithm is a population metaheuristic proposed in 2010 which is based on the echolocation or bio-sonar characteristics of microbats. Since its first implementation, the bat algorithm has been used in a wide range of fields. In this paper, we present a discrete version of the bat algorithm to solve the well-known symmetric and asymmetric traveling salesman problems. In addition, we propose an improvement in the basic structure of the classic bat algorithm. To prove that our proposal is a promising approximation method, we have compared its performance in 37 instances with the results obtained by five different techniques: evolutionary simulated annealing, genetic algorithm, an island based distributed genetic algorithm, a discrete firefly algorithm and an imperialist competitive algorithm. In order to obtain fair and rigorous comparisons, we have conducted three different statistical tests along the paper: the Student's ttest, the Holm's test, and the Friedman test. We have also compared the convergence behaviour shown by our proposal with the ones shown by the evolutionary simulated annealing, and the discrete firefly algorithm. The experimentation carried out in this study has shown that the presented improved bat algorithm outperforms significantly all the other alternatives in most of the cases.