Nowadays, stochastic method is generally utilized to cope with optimization problems [1]. Though there are many ways to classify them, a simple one is used to divide them PS-341 into two groups according to their nature: deterministic and stochastic. Deterministic algorithms can get the same solutions if the initial conditions are unchanged, because they always follow the rigorous move. However, regardless of the initial values, stochastic ones are based on certain stochastic distribution; therefore they generally generate various solutions. In fact, both of them can find satisfactory solutions after some generations. Recently, nature-inspired algorithms are well capable of solving numerical optimization problems more efficiently.

These metaheuristic approaches are developed to solve complicated problems, like permutation flow shop scheduling [2], reliability [3, 4], high-dimensional function optimization [5], and other engineering problems [6, 7]. In the 1950s, nature evolution was idealized as an optimization technology and this made a new type of approach, namely, genetic algorithms (GAs) [8]. After that, many other metaheuristic methods have appeared, like evolutionary strategy (ES) [9, 10], ant colony optimization (ACO) [11], probability-based incremental learning (PBIL) [12], big bang-big crunch algorithm [13�C16], harmony search (HS) [17�C19], charged system search (CSS) [20], artificial physics optimization [21], bat algorithm (BA) [22, 23], animal migration optimization (AMO) [24], krill herd (KH) [25�C27], differential evolution (DE) [28�C31], particle swarm optimization (PSO) [32�C35], stud GA (SGA) [36], cuckoo search (CS) [37, 38], artificial plant optimization algorithm (APOA) [39], biogeography-based optimization (BBO) [40], and FA method [41, 42].

As a global optimization method, FA [42] is firstly proposed by Yang in 2008, and it is originated from the fireflies swarm. Recent researches demonstrate that the FA is quite powerful and relatively efficient [43]. Furthermore, the performance of FA can be improved with feasible promising results [44]. In addition, nonconvex problems can be solved by FA [45]. A summarization of swarm intelligence containing FA is given by Parpinelli and Lopes [46].On the other hand, HS [17, 47] is a novel heuristic technique for optimization problems.

In engineering optimization, the engineers make GSK-3 an effort to find an optimum that can be decided by an objective function. While, in the music improvisation process, musicians search for most satisfactory harmony as decided by aesthetician. HS method originates in the similarity between them [1]. In most cases, FA can find the optimal solution with its exploitation. However, the search used in FA is based on randomness, so it cannot always get the global best values. On the one hand, in order to improve diversity of fireflies, an improvement of adding HS is made to the FA, which can be treated as a mutation operator.