  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 $t$-test, 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.
