Abstract
Firefly algorithm (FA) is an efficient optimization technique, which has been widely used to solve various engineering problems. However, FA is sensitive to its control parameters. Recently, a memetic FA (called MFA) is proposed to improve the sensitivity of FA. To further enhance the performance of MFA, this paper proposes a new method to adaptively adjust the step factor. Experiments on several benchmark problems show that our approach is superior to the standard FA, MFA, and some other improved FAs.
Access provided by CONRICYT-eBooks. Download conference paper PDF
Similar content being viewed by others
Keywords
1 Introduction
Optimization problems exist in many engineering fields. With the growth of problem complexity, stronger optimization algorithms are required. In the past decades, several new optimization techniques have been proposed by the inspiration of swarm intelligence, such as artificial bee colony (ABC) [1,2,3,4], bat algorithm (BA) [5,6,7], firefly algorithm (FA) [8,9,10], cuckoo search (CS) [11, 12], fruit fly optimization (FFO) [13], and artificial plant optimization algorithm [14, 15]. FA is a new optimization technique originally invented by Prof. Yang [8]. In FA, every firefly moves toward new positions and find potential solutions by the attraction of other brighter fireflies. Some latest researches proved that FA is an efficient optimization tool.
However, the standard FA still has some drawbacks. For instance, FA is sensitive to the control parameters, and the convergence speed is slow. To tackle these issues, several improved strategies have been designed. Recently, a memetic FA (MFA) was proposed, in which two improved strategies are employed [16]. First, the step factor α is dynamically decreased based on an empirical model. Second, the attractiveness β is constrained into a predefined range [0.2, 1.0]. Moreover, the factor α is multiplied by the length of search interval. Gandomi et al. [17] presented a chaotic FA, in which different chaotic maps were used to adjust the step factor α and the light absorption coefficient γ. Besides Gandomi’s work, some researchers also combined FA with chaos to obtain a good performance. In [18], an adaptive selection method was used to choose the parameter α from a candidate set. In [19], Wang et al. investigated the relationships between convergence and the parameter α. Results show that α should tend to zero when FA converges to a solution. Based on this principle, Wang et al. designed two dynamic methods adjust α, in which α is gradually decreased based on different models as the generation increases [19, 20].
In this paper, we present a new adaptive firefly algorithm (AFA), which is an enhanced version of MFA. In AFA, we combine MFA with a new adaptive parameter strategy to dynamically adjust the step factor α. Thirteen famous test functions are used for performance verification. Results show that AFA is superior to the standard FA, MFA [16], chaotic FA (CFA) [17], and FA with random attraction (RaFA) [9].
2 Firefly Algorithm
For two fireflies X i and X j , their attractiveness β is defined as follows [21].
where β 0 is the attractiveness at r = 0, γ is the light absorption coefficient, and r ij is the distance between X i and X j . The distance r ij is computed as follows [21].
where D is the problem size.
When X j is brighter (better) than X i , X i will move towards X j because of the attraction. In the standard FA, this movement is defined as follows [21].
where x id and x jd are the dth dimensions of X i and X j , respectively, α ∈ [0, 1] is called step factor, and rand is a random value within [0, 1].
3 Proposed Approach
Recently, a memetic firefly algorithm (MFA) was designed to enhance the performance of FA [16]. The MFA made three improvements. Firstly, the step factor α is dynamically updated as follows.
where t indicates the generation index. We can find that the value of α decreases with the growth of t.
Secondly, the definitions of the attractiveness β is modified. The new β is calculated as follows.
where β min is the minimum value of the attractiveness β. The β is constrained into the range \( [\beta_{\hbox{min} } ,\beta_{0} ] \). In [16], β min and β 0 are equal to 0.2, and 1.0, respectively.
Thirdly, the step factor α is multiplied by the length of the search range by the suggestions of [21]. Then, the new movement equation is modified as follows.
where s d is the length of the search interval of the dth dimension.
Based on MFA, we propose an enhanced version by employing a new adaptive parameter strategy to adjust the step factor α. In our approach AFA, Eq. (4) is modified as follows.
where \( m > 0 \), \( FEs \) represents the number of fitness evaluations, \( MaxFEs \) indicates the maximum value for the \( FEs \), and t is the generation number. When \( 0 < m < 1 \), a small value is added to the weighed term of \( \alpha (t) \). It avoids that \( 1 - FEs/MaxFEs = 0 \). In our experiments, \( m \) is set to 0.5. In fact, Eq. (7) is a general version of our previous work [19].
The framework of AFA is given in Fig. 1. Compared to MFA, AFA only modifies the updating strategy of the step factor α. Therefore, AFA has the same complexity with MFA.
4 Experimental Study
4.1 Experimental Setup
In the experiment, thirteen benchmark functions are utilized for performance verification [22,23,24,25,26]. All test functions are minimization problems. Table 1 gives a brief description of these functions. More detailed descriptions of these functions can be found in [27,28,29,30].
In the comparison, AFA is compared with four FAs. The related FAs are presented as follows.
The parameters \( N \) and \( MaxFEs \) are equal to \( 20 \) and \( 5.0E + 05 \), respectively. In the standard FA, the parameters α, β 0, and γ are set to 0.2, 1.0, and γ = 1/Γ2, respectively. For MFA, RaFA, and AFA, the initial α, β min, β 0, and γ are set to 0.5, 0.2, 1.0, and 1.0, respectively. The parameter m used in AFA is set to 0.5. Besides AFA, RaFA is also an improved version of MFA.
4.2 Results
Table 2 presents the results of AFA, FA, MFA, CFA, and RaFA on thirteen test functions, where “Mean” is the mean best fitness value over on 30 runs. From the results, AFA outperforms the standard FA on 11 functions, while AFA achieves worse solutions on two functions f 3 and f 7. For function f 3, AFA is trapped into local optima and can hardly obtain reasonable solutions. For function f 7, the standard FA is a little better than AFA. Compared to MFA, the proposed adaptive parameter strategy helps AFA to achieve significant improvements, especially for f 1, f 2, and f 10–f 13. CFA outperforms AFA on three functions f 3, f 5 and f 8. For function f 6, AFA, CFA, RaFA, and MFA find the same solution. RaFA is better than AFA on 6 functions, while AFA achieves more accurate solutions than RaFA on 6 functions.
Figures 2 and 3 show the convergence graphs on some unimodal functions and multimodal functions, respectively. As seen, both RaFA and AFA converges faster than FA, MFA, and CFA.
To compare the optimization performance of the five FA variants on the whole test set, we calculate the mean rank values by the Friedman test. Table 3 gives the mean rank values of the five algorithms. The highest rank is marked in boldface. It is obvious that AFA obtains the highest rank. It demonstrates that AFA is the best one among AFA, FA, CFA, MFA, and RaFA.
5 Conclusions
In this paper, an adaptive firefly algorithm (AFA) is proposed. It is an enhanced version of MFA. In AFA, a new parameter method is designed to adaptively change the step factor. In the experiment, thirteen test functions are used for performance verification. Simulation results show that AFA is superior to the standard FA, MFA, CFA, and RaFA. The adaptive parameter strategy is a general version of our previous work. In this paper, an empirical value is used. More investigations will be conducted in the future work.
References
Sun, H., Wang, K., Zhao, J., Yu, X.: Artificial bee colony algorithm with improved special centre. Int. J. Comput. Sci. Math. 7(6), 548–553 (2016)
Yun, G.: A new multi-population-based artificial bee colony for numerical optimization. Int. J. Comput. Sci. Math. 7(6), 509–515 (2016)
Lv, L., Wu, L.Y., Zhao, J., Wang, H., Wu, R.X., Fan, T.H., Hu, M., Xie, Z.F.: Improved multi-strategy artificial bee colony algorithm. Int. J. Comput. Sci. Math. 7(5), 467–475 (2016)
Lu, Y., Li, R.X., Li, S.M.: Artificial bee colony with bidirectional search. Int. J. Comput. Sci. Math. 7(6), 586–593 (2016)
Cai, X., Gao, X.Z., Xue, Y.: Improved bat algorithm with optimal forage strategy and random disturbance strategy. Int. J. Bio-Inspired Comput. 8(4), 205–214 (2016)
Xue, F., Cai, Y., Cao, Y., Cui, Z., Li, F.: Optimal parameter settings for bat algorithm. Int. J. Bio-Inspired Comput. 7(2), 125–128 (2015)
Cai, X., Wang, L., Kang, Q., Wu, Q.: Bat algorithm with Gaussian walk. Int. J. Bio-Inspired Comput. 6(3), 166–174 (2014)
Yang, X.S.: Nature-Inspired Metaheuristic Algorithms. Luniver Press, Beckington (2008)
Wang, H., Wang, W.J., Sun, H., Rahnamayan, S.: Firefly algorithm with random attraction. Int. J. Bio-Inspired Comput. 8(1), 33–41 (2016)
Wang, H., Wang, W.J., Zhou, X.Y., Sun, H., Zhao, J., Yu, X., Cui, Z.: Firefly algorithm with neighborhood attraction. Inf. Sci. 382–383, 374–387 (2017)
Cui, Z., Sun, B., Wang, G., Xue, Y.: A novel oriented cuckoo search algorithm to improve DV-Hop performance for cyber-physical systems. J. Parallel Distrib. Comput. 103, 42–52 (2017)
Wang, G.G., Gandomi, A.H., Yang, X.S., Alavi, A.H.: A new hybrid method based on krill herd and cuckoo search for global optimization tasks. Int. J. Bio-Inspired Comput. 8(5), 286–299 (2016)
Zhang, Y.W., Wu, J.T., Guo, X., Li, G.N.: Optimising web service composition based on differential fruit fly optimisation algorithm. Int. J. Comput. Sci. Math. 7(1), 87–101 (2016)
Cui, Z., Fan, S., Zeng, J., Shi, Z.Z.: APOA with parabola model for directing orbits of chaotic systems. Int. J. Bio-Inspired Comput. 5(1), 67–72 (2013)
Cui, Z., Fan, S., Zeng, J., Shi, Z.Z.: Artificial plant optimisation algorithm with three-period photosynthesis. Int. J. Bio-Inspired Comput. 5(2), 133–139 (2013)
Fister Jr., I., Yang, X.S., Fister, I., Brest, J.: Memetic firefly algorithm for combinatorial optimization. arXiv preprint arXiv:1204.5165 (2012)
Gandomi, A.H., Yang, X.S., Talatahari, S., Alavi, A.H.: Firefly algorithm with chaos. Commun. Nonlinear Sci. Numer. Simul. 18(1), 89–98 (2013)
Roy, A.G., Rakshit, P., Konar, A., Bhattacharya, S., Kim, E., Nagar, A.K.: Adaptive firefly algorithm for nonholonomic motion planning of car-like system. In: IEEE Congress on Evolutionary Computation (CEC 2013), pp. 2162–2169. IEEE, Cancun (2013)
Wang, H., Zhou, X.Y., Sun, H., Yu, X., Zhao, J., Zhang, H., Cui, L.Z.: Firefly algorithm with adaptive control parameters. Soft. Comput. 1–12 (2016). doi:10.1007/s00500-016-2104-3
Wang, H., Cui, Z.H., Sun, H., Rahnamayan, S., Yang, X.S.: Randomly attracted firefly algorithm with neighborhood search and dynamic parameter adjustment mechanism. Soft. Comput. 1–15 (2016). doi:10.1007/s00500-016-2116-z
Yang, X.S.: Engineering Optimization: An Introduction with Metaheuristic Applications. Wiley, Hoboken (2010)
Wang, H., Wu, Z.J., Rahnamayan, S., Liu, Y., Ventresca, M.: Enhancing particle swarm optimization using generalized opposition-based learning. Inf. Sci. 181(20), 4699–4714 (2011)
Wang, H., Rahnamayan, S., Sun, H., Omran, M.G.H.: Gaussian bare-bones differential evolution. IEEE Trans. Cybern. 43(2), 634–647 (2013)
Guo, Z.L., Wang, S.W., Yue, X.Z., Yin, B.: Enhanced social emotional optimisation algorithm with elite multi-parent crossover. Int. J. Comput. Sci. Math. 7(6), 568–574 (2016)
Wang, H., Liu, Y., Li, C.H., Zeng, S.Y.: A hybrid particle swarm algorithm with Cauchy mutation. In: IEEE Swarm Intelligence Symposium (SIS 2007), pp. 356–360. IEEE, Honolulu (2007)
Yu, G.: An improved firefly algorithm based on probabilistic attraction. Int. J. Comput. Sci. Math. 7(6), 530–536 (2016)
Wang, H., Sun, H., Li, C.H., Rahnamayan, S., Pan, J.S.: Diversity enhanced particle swarm optimization with neighborhood search. Inf. Sci. 223, 119–135 (2013)
Wang, H., Wu, Z.J., Rahnamayan, S., Sun, H., Liu, Y., Pan, J.S.: Multi-strategy ensemble artificial bee colony algorithm. Inf. Sci. 279, 587–603 (2014)
Zhou, X.Y., Wu, Z.J., Wang, H., Rahnamayan, S.: Gaussian bare-bones artificial bee colony algorithm. Soft. Comput. 20(3), 907–924 (2016)
Zhou, X.Y., Wang, H., Wang, M.W., Wan, J.Y.: Enhancing the modified artificial bee colony algorithm with neighborhood search. Soft. Comput. 21(10), 2733–2743 (2017)
Acknowledgement
This work is supported by the Science and Technology Plan Project of Jiangxi Provincial Education Department (No. GJJ161115), the National Natural Science Foundation of China (No. 61663028), the Distinguished Young Talents Plan of Jaingxi Province (No. 20171BCB23075), the Natural Science Foundation of Jiangxi Province (No. 20171BAB202035), and the Open Research Fund of Jiangxi Province Key Laboratory of Water Information Cooperative Sensing and Intelligent Processing (No. 2016WICSIP015).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Wang, W., Wang, H., Zhao, J., Lv, L. (2017). A New Adaptive Firefly Algorithm for Solving Optimization Problems. In: Huang, DS., Bevilacqua, V., Premaratne, P., Gupta, P. (eds) Intelligent Computing Theories and Application. ICIC 2017. Lecture Notes in Computer Science(), vol 10361. Springer, Cham. https://doi.org/10.1007/978-3-319-63309-1_57
Download citation
DOI: https://doi.org/10.1007/978-3-319-63309-1_57
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-63308-4
Online ISBN: 978-3-319-63309-1
eBook Packages: Computer ScienceComputer Science (R0)