In the standard firefly algorithm, each firefly has the same step settings and its values decrease from iteration to iteration. Therefore, it may fall into the local optimum. Furthermore, the decreasing of step is restrained by the maximum of iteration, which has an influence on the convergence speed and precision. In order to avoid falling into the local optimum and reduce the impact of the maximum of iteration, a selfadaptive step firefly algorithm is proposed in the paper. Its core idea is setting the step of each firefly varying with the iteration, according to each firefly’s historical information and current situation. Experiments are made to show the performance of our approach compared with the standard FA, based on sixteen standard testing benchmark functions. The results reveal that our method can prevent the premature convergence and improve the convergence speed and accurateness.
Firefly algorithm (FA) is inspired by biochemical and social aspects of real fireflies [
Despite these advantages, the FA is also a metaheuristic algorithm; the standard FA can easily get trapped in the local optima when solving complex multimodal problems. These weaknesses have restricted wider applications of the FA. Therefore, avoiding the local optima and accelerating convergence speed have become the two most important and appealing goals in the FA research. To overcome these disadvantages, many researchers have proposed a variety of modifications to the original FA [
Compared with other evolutionary algorithms, such as Genetic Algorithm and Simulated Annealing, standard FA has the following problem: it is not rational that each firefly uses the same step or the linear step just depends on maximum iteration not related to experience of fireflies, which may impact on the balance between the global and local search. Based on the above problem, a selfadaptive step firefly algorithm (SASFA) is proposed in the paper, which considers the historical information and current situation of each firefly.
The rest of this paper is organized as follows. Section
Firefly algorithm is based on the idealized behavior of the flashing feature of fireflies. The following three rules are idealized for the basic formulation of FA:
In the FA, there are two critical issues: the formulation of the attractiveness and the variation of light intensity. We can always suppose that a firefly’s attractiveness is determined by its light intensity or brightness, which in turn is associated with the encoded objective function [
The third term is the randomization with the step
Begin
Objective function
Generate initial population of
Formulate light intensity
While (
Define absorption coefficient
for
for
if (
move firefly
end if
Vary attractiveness with distance
Evaluate new solutions and update light intensity
end for
end for
Rank the fireflies and find the current best
end while
Postprocessing the results and visualization
end
In standard FA, the third term of (
Finally, we can see from (
When we make a decision, two important messages are usually integrated, which is known by Boyd and Richerson, on the exploration of human decisionmaking process. One is the experience of themselves and their neighbors. The other is the current situation. Inspired by this idea, we use it to set the step of each firefly to guide its search in the search space [
To solve the problems mentioned in Section
The implementation procedure of our proposed selfadaptive step firefly algorithm (SASFA) can be described as follows.
Generate the initial population of fireflies,
Compute intensity for each firefly member,
Update the step of each firefly. The step is calculated by (
Move each firefly
Update the solution set.
Terminate if a termination criterion is fulfilled; otherwise go to Step
The proposed SASFA and standard FA are tested on sixteen benchmark functions which are given in Table
Benchmark functions.
Functions  Formulations  Limits 

















































Table
Comparison of standard FA and SASFA.
Functions  Optimization method  Best solution  Worst solution  Medium of solutions  Standard deviation 


Standard FA 




SASFA 






Standard FA 




SASFA 






Standard FA 




SASFA 






Standard FA 




SASFA 






Standard FA 




SASFA 






Standard FA 




SASFA 






Standard FA 




SASFA 






Standard FA 




SASFA 






Standard FA 




SASFA 






Standard FA 




SASFA 






Standard FA 




SASFA 






Standard FA 




SASFA 






Standard FA 




SASFA 






Standard FA 




SASFA 






Standard FA 




SASFA 






Standard FA 




SASFA 




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In the paper, we have proposed a selfadaptive step firefly algorithm which considers the current situation and historical information of each firefly. Simulation results demonstrated the potential of the proposed algorithm. Considering more iteration’s information of the algorithm could be an exciting direction in future.
This research is financially supported by the National Natural Science Foundation of China (NSFC) for Professor Shanlin Yang (no. 71071045) and Professor Shoubao Su (no. 61075049) and the Universities natural science foundation of anhui province (nos. KJ2011A268 and KJ2012Z429). The authors of the paper express great acknowledgment for these supports.