Improved firefly algorithm based on multi-strategy fusion

doi:10.11772/j.issn.1001-9081.2021101830

Journal of Computer Applications ›› 2022, Vol. 42 ›› Issue (12): 3847-3855.DOI: 10.11772/j.issn.1001-9081.2021101830

• Advanced computing • Previous Articles Next Articles

Improved firefly algorithm based on multi-strategy fusion

Xin YONG¹, Yuelin GAO¹^,²(), Yahua HE¹, Huimin WANG¹

^1.School of Computer Science and Engineering，North Minzu University，Yinchuan Ningxia 750021，China
^2.Ningxia Key Laboratory of Intelligent Information and Big Data Processing （North Minzu University），Yinchuan Ningxia 750021，China

Received:2021-10-27 Revised:2021-12-08 Accepted:2021-12-22 Online:2021-12-31 Published:2022-12-10
Contact: Yuelin GAO
About author:YONG Xin， born in 1998， M. S. candidate. Her research interests include intelligent optimization algorithm， intrusion detection.
HE Yahua， born in 1999， M. S. candidate. Her research interests include intelligent optimization algorithm， global optimization theory.
WANG Huimin， born in 1999， M. S. candidate. Her research interests include intelligent optimization algorithm， artificial intelligence.
Supported by:
National Natural Science Foundation of China(11961001);Project of First-class Discipline Construction Fund of Ningxia Higher Education(NXYLXK2017B09);Major Scientific Research Project of North Minzu University(ZDZX201901)

多策略融合的改进萤火虫算法

雍欣¹, 高岳林¹^,²(), 赫亚华¹, 王惠敏¹

^1.北方民族大学计算机科学与工程学院，银川 750021
^2.宁夏智能信息与大数据处理重点实验室（北方民族大学），银川 750021

通讯作者: 高岳林
作者简介:雍欣（1998—），女，宁夏中卫人，硕士研究生，主要研究方向：智能优化算法、入侵检测
赫亚华（1999—），女，安徽界首人，硕士研究生，主要研究方向：智能优化算法、全局优化理论
王惠敏（1999—），女，陕西延安人，硕士研究生，主要研究方向：智能优化算法、人工智能。
基金资助:
国家自然科学基金资助项目(11961001);宁夏高等教育一流学科建设基金资助项目(NXYLXK2017B09);北方民族大学重大科研专项(ZDZX201901)

Abstract

Abstract:

In order to solve the problems that the traditional Firefly Algorithm （FA） is easy to fall into local optimum and has low convergence speed， an improved FA based on multi-strategy fusion， named LEEFA （Levy flight-Elite participated crossover-Elite opposition-based learning Firefly Algorithm） was proposed after integrating Levy flight， elite participated crossover operator and elite opposition-based learning mechanism in the firefly optimization algorithm. Firstly， Levy flight was introduced based on the traditional FA， so that the global search ability of the algorithm was improved. Secondly， an elite participated crossover operator was proposed to improve the convergence speed and accuracy of the algorithm， as well as to enhance the diversity and quality of solutions in the iterative process. Finally， the elite opposition-based learning mechanism was combined to search for the optimal solution， which improved the ability of jumping out of local optimum and convergence performance of FA， and realized the rapid exploration of solution search space. In order to verify the effectiveness of the proposed algorithm， simulation experiments were carried out on the benchmark functions. The results show that compared with algorithms such as Particle Swarm Optimization （PSO） algorithm， traditional FA， Levy Flight Firefly Algorithm （LFFA）， Levy flight and Mutation operator based Firefly Algorithm （LMFA） and ADaptive logarithmic spiral-Levy Improved Firefly Algorithm （ADIFA）， the proposed algorithm performs better in both convergence speed and accuracy.

Key words: firefly optimization algorithm, intelligent optimization algorithm, Levy flight, elite participated crossover operator, elite opposition-based learning mechanism

摘要：

针对传统萤火虫算法（FA）中存在的易陷入局部最优及收敛速度慢等问题，把莱维飞行和精英参与的交叉算子及精英反向学习机制融入到萤火虫优化算法中，提出了一种多策略融合的改进萤火虫算法——LEEFA。首先，在传统萤火虫算法的基础上引入莱维飞行，从而提升算法的全局搜索能力；其次，提出精英参与的交叉算子以提升算法的收敛速度和精度，并增强算法迭代过程中解的多样性和质量；最后，结合精英反向学习机制进行最优解的搜索，从而提高FA跳出局部最优的能力和收敛性能，并实现对于解搜索空间的迅速勘探。为验证所提出的算法的有效性，在基准测试函数上进行了仿真实验，结果表明相较于粒子群优化（PSO）算法、传统FA、莱维飞行萤火虫算法（LFFA）、基于莱维飞行和变异算子的萤火虫算法（LMFA）和自适应对数螺旋-莱维飞行萤火虫优化算法（ADIFA）等算法，所提算法在收敛速度和精度上均表现得更为优异。

关键词: 萤火虫优化算法, 智能优化算法, 莱维飞行, 精英参与的交叉算子, 精英反向学习机制

CLC Number:

TP301.6

Xin YONG, Yuelin GAO, Yahua HE, Huimin WANG. Improved firefly algorithm based on multi-strategy fusion[J]. Journal of Computer Applications, 2022, 42(12): 3847-3855.

雍欣, 高岳林, 赫亚华, 王惠敏. 多策略融合的改进萤火虫算法[J]. 《计算机应用》唯一官方网站, 2022, 42(12): 3847-3855.

Figures/Tables 5

Tab.1 Benchmark functions

函数名称	描述	最优值	区间
Sphere	$f 1 (x) = ∑ i = 1 D x i 2$	$0$	$[- 10,10]$
Griewank	$f 2 (x) = x i 2 4 000 - ∏ i = 1 D c o s x i / i + 1$	$0$	$[- 100,100]$
Zakharov	$f 3 (x) = ∑ i = 1 D x i 2 + 12 ∑ i = 1 n i x i 2 + 12 ∑ i = 1 D i x i 4$	$0$	$[- 5,10]$
Ackley	$f 4 (x) = - 20 e - 0.2 D - 1 ∑ i = 1 D x i 2 - e D - 1 ∑ i = 1 D c o s (2 π x i) + 20 + e$	$0$	$[- 35,35]$
Schwedel 1.2	$f 5 (x) = ∑ i = 1 D ∑ j = 1 i x j 2$	$0$	$[- 100,100]$
Schwedel 2.22	$f 6 (x) = ∑ i = 1 D x i + ∏ i = 1 D x i$	$0$	$[- 100,100]$
Alpine	$f 7 (x) = ∑ i = 1 D x i s i n (x i) + 0.1 x i$	$0$	$[- 10,10]$
Csendes	$f 8 (x) = ∑ i = 1 D x i 6 2 + s i n 1 / x i$	$0$	$[- 1,1]$

Tab.1 Benchmark functions

函数名称	描述	最优值	区间
Sphere	$f 1 (x) = ∑ i = 1 D x i 2$	$0$	$[- 10,10]$
Griewank	$f 2 (x) = x i 2 4 000 - ∏ i = 1 D c o s x i / i + 1$	$0$	$[- 100,100]$
Zakharov	$f 3 (x) = ∑ i = 1 D x i 2 + 12 ∑ i = 1 n i x i 2 + 12 ∑ i = 1 D i x i 4$	$0$	$[- 5,10]$
Ackley	$f 4 (x) = - 20 e - 0.2 D - 1 ∑ i = 1 D x i 2 - e D - 1 ∑ i = 1 D c o s (2 π x i) + 20 + e$	$0$	$[- 35,35]$
Schwedel 1.2	$f 5 (x) = ∑ i = 1 D ∑ j = 1 i x j 2$	$0$	$[- 100,100]$
Schwedel 2.22	$f 6 (x) = ∑ i = 1 D x i + ∏ i = 1 D x i$	$0$	$[- 100,100]$
Alpine	$f 7 (x) = ∑ i = 1 D x i s i n (x i) + 0.1 x i$	$0$	$[- 10,10]$
Csendes	$f 8 (x) = ∑ i = 1 D x i 6 2 + s i n 1 / x i$	$0$	$[- 1,1]$

Tab.2 Parameter description

算法	参数设置
PSO	$c 1 = 1.8, c 2 = 1.8, w = 0.9, v m a x = 1, v m i n = - 1$
GA	$p m = 0.01, p c = 0.8$
ABC^［21］	$l l i m i t s = N ⋅ D$
FA^［5］	$α = 0.1, β 0 = 1, γ = 1$
LFFA^［15］	$α = 0.1, β 0 = 1, γ = 1$
LMFA^［6］	$α = 0.2, β 0 = 1, γ = 1 / d m a x 2, s d = 0.05 ⋅ (u b d - l b d), ε i, d ∈ (- 1,1) λ = 1.5, p m = 0.05, s g = 7$
ADIFA^［7］	$α = 0.1, β 0 = 1, γ = 1, u = 0.5$
LEEFA	$α = 0.1, β 0 = 1, γ = 0.002, e t h r e s h o l d = 0.1$

Tab.2 Parameter description

算法	参数设置
PSO	$c 1 = 1.8, c 2 = 1.8, w = 0.9, v m a x = 1, v m i n = - 1$
GA	$p m = 0.01, p c = 0.8$
ABC^［21］	$l l i m i t s = N ⋅ D$
FA^［5］	$α = 0.1, β 0 = 1, γ = 1$
LFFA^［15］	$α = 0.1, β 0 = 1, γ = 1$
LMFA^［6］	$α = 0.2, β 0 = 1, γ = 1 / d m a x 2, s d = 0.05 ⋅ (u b d - l b d), ε i, d ∈ (- 1,1) λ = 1.5, p m = 0.05, s g = 7$
ADIFA^［7］	$α = 0.1, β 0 = 1, γ = 1, u = 0.5$
LEEFA	$α = 0.1, β 0 = 1, γ = 0.002, e t h r e s h o l d = 0.1$

Fig.1 Threshold parameter selection

Tab. 3 Experimental results comparison of the proposed algorithm and other improved algorithms

函数	算法	最优值	最差值	平均值	标准差
$f 1$	PSO	0.601 6	0.953 3	0.782 1	0.0916 7
	GA	96.489 2	3.402 7E+02	1.939 2E+02	55.925 6
	ABC	2.560 9	22.832 6	11.915 4	5.198 9
	FA	443.110 3	616.453 3	20.357 1	33.456 9
	LFFA	275.053 0	362.690 8	307.758 9	20.599 7
	LMFA	2.644 5E-03	1.218 3E-02	7.386 9E-03	2.131 5E-03
	ADIFA	211.305 2	328.964 0	277.181 3	29.006 0
	LEEFA	7.702 5E-92	8.978 8E-81	2.9984E-82	1.6116E-81
$f 2$	PSO	0.026 8	0.686 6	0.127 8	0.181 8
	GA	3.235 2	8.896 7	5.986 0	1.331 6
	ABC	1.070 9	1.198 1	1.126 0	0.031 9
	FA	12.774 2	18.444 2	15.857 3	1.329 3
	LFFA	12.923 2	15.834 8	14.488 8	0.840 8
	LMFA	0.139 0	0.262 4	0.202 2	0.027 0
	ADIFA	11.459 9	15.227 3	13.846 0	0.889 0
	LEEFA	1.357 8E-12	9.078 5E-05	6.5387E-06	1.9040E-05
$f 3$	PSO	0.749 2	1.429 7	1.177 5	0.142 4
	GA	2.911 9E+02	8.349 8E+02	4.779 5E+02	1.069 4E+02
	ABC	1.678 3E+02	2.940 2E+02	2.608 7E+02	25.118 1
	FA	752.834 8	1.864 9E+08	5.319 0E+07	5.623 0E+07
	LFFA	222.202 6	333.352 1	294.423 9	29.251 7
	LMFA	0.173 8	0.283 3	0.236 0	0.031 3
	ADIFA	272.045 4	345.551 0	303.445 3	22.611 3
	LEEFA	1.8553E-08	1.0567E-06	1.3746E-07	3.1064E-06
$f 4$	PSO	1.259 3	1.659 3	1.516 7	0.091 8
	GA	20.317 6	20.981 7	20.591 2	0.156 5
	ABC	8.555 2	12.304 2	10.346 0	0.958 2
	FA	20.186 2	20.475 5	20.377 3	0.064 0
	LFFA	19.969 4	20.328 9	20.201 5	0.084 8
	LMFA	17.964 1	18.854 4	18.587 3	0.192 8
	ADIFA	19.841 7	20.324 3	20.162 9	0.117 3
	LEEFA	4.1599E-06	1.8025E-04	1.0109E-04	3.8139E-05
$f 5$	PSO	0.805 7	1.488 8	1.080 3	0.150 3
	GA	2.673 1E+02	9.398 7E+02	4.545 5E+02	1.424 6E+02
	ABC	2.027 9E+02	2.986 6E+02	2.496 5E+02	26.644 5
	FA	518.832 3	698.257 9	604.156 7	59.696 1
	LFFA	326.442 1	433.324 8	394.616 5	28.355 5
	LMFA	0.334 4	0.557 2	0.448 2	0.074 9
	ADIFA	332.946 8	431.034 6	390.475 8	31.199 1
	LEEFA	2.3705E-08	3.8520E-04	7.6522E-05	1.1365E-04
$f 6$	PSO	2.942 7	4.035 0	3.677 4	0.274 3
	GA	31.455 0	56.055 9	42.588 9	7.137 9
	ABC	5.717 2	9.794 1	7.922 9	0.959 5
	FA	163.703 4	1.117 0E+05	1.178 7E+04	3.235 1E+04
	LFFA	160.821 5	1.253 6E+05	2.286 7E+04	3.514 9E+04
	LMFA	1.539 3	1.833 3	1.667 2	0.073 4
	ADIFA	186.600 4	2.107 6E+04	7.895 6E+03	7 927.250 6
	LEEFA	6.8203E-05	9.6604E-03	2.5304E-03	2.6075E-03
$f 7$	PSO	0.533 0	0.880 5	0.720 2	0.082 5
	GA	19.826 7	39.879 9	28.431 4	5.386 3
	ABC	2.592 5	5.048 8	4.075 1	0.521 5
	FA	40.124 1	44.767 9	42.580 0	1.288 0
	LFFA	34.955 0	41.131 8	37.923 9	1.545 2
	LMFA	0.204 7	1.527 3	0.538 7	0.360 7
	ADIFA	30.667 3	40.947 6	38.454 3	2.853 3
	LEEFA	9.3186E-06	2.7447E-03	1.1964E-03	1.1168E-03
$f 8$	PSO	0.001 6	0.012 6	0.007 5	0.002 6
	GA	0.005 3	0.296 4	0.065 2	0.056 4
	ABC	4.106 3E-07	7.850 9E-05	1.474 2E-05	1.895 8E-05
	FA	1.025 7E-08	2.243 8E-08	1.652 1E-08	4.633 3E-09
	LFFA	1.397 9	1.832 9	1.531 6	0.178 5
	LMFA	9.088 4E-11	1.417 0E-10	1.192 9E-10	1.869 1E-11
	ADIFA	1.398 4	2.617 4	2.182 2	0.465 1
	LEEFA	1.1859E-35	8.0342E-28	2.1270E-28	3.4160E-28

Tab. 3 Experimental results comparison of the proposed algorithm and other improved algorithms

函数	算法	最优值	最差值	平均值	标准差
$f 1$	PSO	0.601 6	0.953 3	0.782 1	0.0916 7
	GA	96.489 2	3.402 7E+02	1.939 2E+02	55.925 6
	ABC	2.560 9	22.832 6	11.915 4	5.198 9
	FA	443.110 3	616.453 3	20.357 1	33.456 9
	LFFA	275.053 0	362.690 8	307.758 9	20.599 7
	LMFA	2.644 5E-03	1.218 3E-02	7.386 9E-03	2.131 5E-03
	ADIFA	211.305 2	328.964 0	277.181 3	29.006 0
	LEEFA	7.702 5E-92	8.978 8E-81	2.9984E-82	1.6116E-81
$f 2$	PSO	0.026 8	0.686 6	0.127 8	0.181 8
	GA	3.235 2	8.896 7	5.986 0	1.331 6
	ABC	1.070 9	1.198 1	1.126 0	0.031 9
	FA	12.774 2	18.444 2	15.857 3	1.329 3
	LFFA	12.923 2	15.834 8	14.488 8	0.840 8
	LMFA	0.139 0	0.262 4	0.202 2	0.027 0
	ADIFA	11.459 9	15.227 3	13.846 0	0.889 0
	LEEFA	1.357 8E-12	9.078 5E-05	6.5387E-06	1.9040E-05
$f 3$	PSO	0.749 2	1.429 7	1.177 5	0.142 4
	GA	2.911 9E+02	8.349 8E+02	4.779 5E+02	1.069 4E+02
	ABC	1.678 3E+02	2.940 2E+02	2.608 7E+02	25.118 1
	FA	752.834 8	1.864 9E+08	5.319 0E+07	5.623 0E+07
	LFFA	222.202 6	333.352 1	294.423 9	29.251 7
	LMFA	0.173 8	0.283 3	0.236 0	0.031 3
	ADIFA	272.045 4	345.551 0	303.445 3	22.611 3
	LEEFA	1.8553E-08	1.0567E-06	1.3746E-07	3.1064E-06
$f 4$	PSO	1.259 3	1.659 3	1.516 7	0.091 8
	GA	20.317 6	20.981 7	20.591 2	0.156 5
	ABC	8.555 2	12.304 2	10.346 0	0.958 2
	FA	20.186 2	20.475 5	20.377 3	0.064 0
	LFFA	19.969 4	20.328 9	20.201 5	0.084 8
	LMFA	17.964 1	18.854 4	18.587 3	0.192 8
	ADIFA	19.841 7	20.324 3	20.162 9	0.117 3
	LEEFA	4.1599E-06	1.8025E-04	1.0109E-04	3.8139E-05
$f 5$	PSO	0.805 7	1.488 8	1.080 3	0.150 3
	GA	2.673 1E+02	9.398 7E+02	4.545 5E+02	1.424 6E+02
	ABC	2.027 9E+02	2.986 6E+02	2.496 5E+02	26.644 5
	FA	518.832 3	698.257 9	604.156 7	59.696 1
	LFFA	326.442 1	433.324 8	394.616 5	28.355 5
	LMFA	0.334 4	0.557 2	0.448 2	0.074 9
	ADIFA	332.946 8	431.034 6	390.475 8	31.199 1
	LEEFA	2.3705E-08	3.8520E-04	7.6522E-05	1.1365E-04
$f 6$	PSO	2.942 7	4.035 0	3.677 4	0.274 3
	GA	31.455 0	56.055 9	42.588 9	7.137 9
	ABC	5.717 2	9.794 1	7.922 9	0.959 5
	FA	163.703 4	1.117 0E+05	1.178 7E+04	3.235 1E+04
	LFFA	160.821 5	1.253 6E+05	2.286 7E+04	3.514 9E+04
	LMFA	1.539 3	1.833 3	1.667 2	0.073 4
	ADIFA	186.600 4	2.107 6E+04	7.895 6E+03	7 927.250 6
	LEEFA	6.8203E-05	9.6604E-03	2.5304E-03	2.6075E-03
$f 7$	PSO	0.533 0	0.880 5	0.720 2	0.082 5
	GA	19.826 7	39.879 9	28.431 4	5.386 3
	ABC	2.592 5	5.048 8	4.075 1	0.521 5
	FA	40.124 1	44.767 9	42.580 0	1.288 0
	LFFA	34.955 0	41.131 8	37.923 9	1.545 2
	LMFA	0.204 7	1.527 3	0.538 7	0.360 7
	ADIFA	30.667 3	40.947 6	38.454 3	2.853 3
	LEEFA	9.3186E-06	2.7447E-03	1.1964E-03	1.1168E-03
$f 8$	PSO	0.001 6	0.012 6	0.007 5	0.002 6
	GA	0.005 3	0.296 4	0.065 2	0.056 4
	ABC	4.106 3E-07	7.850 9E-05	1.474 2E-05	1.895 8E-05
	FA	1.025 7E-08	2.243 8E-08	1.652 1E-08	4.633 3E-09
	LFFA	1.397 9	1.832 9	1.531 6	0.178 5
	LMFA	9.088 4E-11	1.417 0E-10	1.192 9E-10	1.869 1E-11
	ADIFA	1.398 4	2.617 4	2.182 2	0.465 1
	LEEFA	1.1859E-35	8.0342E-28	2.1270E-28	3.4160E-28

Fig. 2 Comparison of convergence curves on benchmark functions

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Improved firefly algorithm based on multi-strategy fusion

多策略融合的改进萤火虫算法

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