A variable step size firefly algorithm for numerical optimization

doi:10.1016/j.amc.2015.04.065

Applied Mathematics and Computation

Volume 263, 15 July 2015, Pages 214-220

https://doi.org/10.1016/j.amc.2015.04.065 Get rights and content

Abstract

Firefly algorithm is a novel nature-inspired optimization algorithm, which has been demonstrated to perform well on various numerical optimization problems. However, in standard firefly algorithm, it adopted the fixed step size throughout all iterations. This will result in the algorithm easily getting trapped in the local optima and causing low precision. In order to remedy this defect, we use a variable strategy for step size setting. The results show that the proposed algorithm enhances the performance of the standard firefly algorithm.

Introduction

Firefly algorithm (FA) is one of the novel swarm intelligence methods for optimization problems introduced by Xin-She Yang [1]. It is inspired by social behavior of fireflies and is a kind of nature-inspired algorithm that can be applied for solving the hardest optimization problems. Owing to FA is relatively easy to implement without requiring complex evolutionary operations [2], it has been widely utilized as an optimization tool in various applications such as optimization of the quality of continuously cast steel slabs [3], promoting products online [4], optimal power flow with emission controlled [5], dynamic multidimensional knapsack problems [6], linear antenna array failure correction [7], multi-objective economic emission dispatch solution [8], optimized gray-scale image watermarking [9], and so on.

Though FA has shown good performance in solving various optimization problems, it has a tendency to premature converge to local optima. To improve this default of FA, several variants of FA have been proposed. Gandomi et al. [10] introduced chaos into FA so as to increase its global search mobility for robust global optimization. Coelho and Bora [11] propose a novel multi-objective variant which uses the beta probability distribution in the tuning of control parameters and it is useful to maintain the diversity of solutions. Hassanzadeh and Kanan proposed a fuzzy-based FA to increase the exploration and improve the global search of the FA [12]. In [13], an accelerated FA was presented, which imposed some improvements on the searching procedure by both reduction of randomness and scaling the random term in fireflies’ motion.

In fact, in the standard FA, firefly movement is based on the fixed step size. This is obviously inappropriate and may impact on the balance between the global and local search. Therefore, it is desirable to adaptively and dynamically adjust the appropriate value of step at different generation. In this study, we adopt a variable step size strategy in search stages. At early stage, we adopt large step size, with the search progressing, the step size decreases nonlinearly from iteration to iteration. By using this strategy to adjust the step of FA can achieving a good balance between exploration and exploitation.

The rest of the paper is structured as follows. Review of FA is summarized in Section 2. Section 3 describes the proposed method. In Section 4, the testing of the proposed method through a set of 16 benchmark functions is carried out and the simulation results are compared. Section 5 summarized the main of this study.

Section snippets

Firefly algorithm schema

In natural world, firefly uses the flashing as a signal to attract other fireflies. FA imitates the social behaviors of fireflies. There are three idealized assumptions as following: (1) Each firefly will be attracted to other fireflies regardless of their sex because they are unisexual; (2) Fireflies attract each other, in proportionally to their brightness. The lesser bright flashing firefly will move towards the brighter one, the more the distance the less attractiveness. If there is no

Analysis of standard firefly algorithm

In population-based optimization methods, proper control of global exploration and local exploitation is crucial in finding the optimum solution efficiently. Therefore, it is desirable to encourage the individuals to wander through the entire search space, without clustering around local optima, during the early stages of the optimization. During the latter stages, it is very important to enhance convergence toward the global optima, to find the optimum solution efficiently. [14].

In standard

Experimental verifications

The proposed VSSFA and standard FA are tested on sixteen benchmark functions which be given as Table 1. All the problems used here are minimization problems. The simulations are run with 2GB-RAM, WIN-XP OS and MATLAB 2010b software. To avoid stochastic discrepancy, we adopted 100 independent runs for each of the optimization methods involving 100 different initial trial solutions. The number of fireflies was 30, the dimensions of $f_{11} - f_{16}$ was 20 and the maximum iteration number was 1000.

Conclusion

This paper presents a variable step firefly algorithm called VSSFA to solve numerical optimization problems. The proposed algorithm employed a dynamic strategy to adjust the step α in search phases. This strategy is incorporated into the dynamic changing step to balance the ability between exploration and exploitation. Experiments on 16 standard benchmark functions show that the proposed method has some significant improvements.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (grant no. 71131002), and the Universities Natural Science Foundation of Anhui Province (grant no. KJ2011A268). The authors of the paper express great acknowledgment for these supports.

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View full text

A variable step size firefly algorithm for numerical optimization

Abstract

Introduction

Section snippets

Firefly algorithm schema

Analysis of standard firefly algorithm

Experimental verifications

Conclusion

Acknowledgments

Nature-Inspired Metaheuristic Algorithms

Firefly algorithm, stochastic test functions and design optimisation

Int. J. Bio-Inspir. Commun.

Optimization of the quality of continuously cast steel slabs using the firefly algorithm

Mater. Tehnol.

Promoting products online using firefly algorithm

Optimal power flow with emission controlled using firefly algorithm

An improved firefly algorithm for solving dynamic multidimensional knapsack problems

Expert Syst. Appl.

A linear antenna array failure correction with null steering using firefly algorithm

Defence Sci. J.

Multi-objective economic emission dispatch solution using hybrid FFA (firefly algorithm) and considering wind power penetration

Energy