Butterfly Optimization Algorithm for Chaotic Feedback Sharing and Group Synergy

doi:10.3778/j.issn.1673-9418.2012066

Journal of Frontiers of Computer Science and Technology ›› 2022, Vol. 16 ›› Issue (7): 1661-1672.DOI: 10.3778/j.issn.1673-9418.2012066

• Theory and Algorithm • Previous Articles Next Articles

Butterfly Optimization Algorithm for Chaotic Feedback Sharing and Group Synergy

LI Shouyu¹, HE Qing¹^,⁺(), DU Nisuo²

1. College of Big Data & Information Engineering, Guizhou University, Guiyang 550025, China
2. Guizhou Big Data Academy, Guizhou University, Guiyang 550025, China

Received:2020-11-30 Revised:2021-01-25 Online:2022-07-01 Published:2021-01-28
Supported by:
the Science and Technology Planning Project Major Special Project of Guizhou Province(黔科合重大专项字[2018]3002);the Science and Technology Planning Project Major Special Project of Guizhou Province(黔科合重大专项字[2016]3022);the Open Project of Guizhou Key Laboratory of Public Big Data(2017BDKFJJ004);the Youth Science and Technology Talent Growth Project of Guizhou Provincial Department of Education(黔科合KY 字[2016]124);the Guizhou University Cultivation Project(黔科合平台人才[2017]5788)

混沌反馈共享和群体协同效应的蝴蝶优化算法

李守玉¹, 何庆¹^,⁺(), 杜逆索²

1.贵州大学大数据与信息工程学院,贵阳 550025
2.贵州大学贵州省大数据产业发展应用研究院,贵阳 550025

作者简介:李守玉（1996—）,男,贵州安顺人,硕士研究生,主要研究方向为进化计算、深度学习。
LI Shouyu, born in 1996, M.S. candidate. His research interests include evolutionary computation and deep learning.
何庆（1982—）,男,贵州黔南州人,博士,副教授,主要研究方向为大数据应用、进化计算。
HE Qing, born in 1982, Ph.D., associate professor. His research interests include big data application and evolutionary computation.
杜逆索（1986—）,男,贵州六盘水人,博士研究生,讲师,主要研究方向为深度学习。
DU Nisuo, born in 1986, Ph.D. candidate, lecturer. His research interest is deep learning.
基金资助:
贵州省科技计划项目重大专项项目(黔科合重大专项字[2018]3002);贵州省科技计划项目重大专项项目(黔科合重大专项字[2016]3022);贵州省公共大数据重点实验室开放课题(2017BDKFJJ004);贵州省教育厅青年科技人才成长项目(黔科合KY 字[2016]124);贵州大学培育项目(黔科合平台人才[2017]5788)

Abstract

Abstract:

A butterfly optimization algorithm (BOA) based on chaotic feedback sharing and group synergy (CFSBOA) is proposed to solve the shortcomings of low precision and easy to fall into local optimum. Firstly, using Hénon chaos to initialize the population can make the population cover the search blind area as much as possible, increase the diversity of the population, and improve the ability of optimizing the algorithm. Secondly, using the ideas of positive and negative feedback mechanism in feedback control circuit, it builds butterfly feedback shared communication network, allowing individuals to receive information from multiple directions to help populations of positioning the location of the optimal solution and perform careful search, enhance the ability to escape from local optimum and accelerate the algorithm convergence speed. Finally, the collective synergistic effect mechanism is used to improve and balance the global and local search ability and enhance the global and local optimization ability of the algorithm. The performance of the improved butterfly optimization algorithm is verified by using different dimension benchmark test functions, statistical test, Wilcoxon test and multiple types of CEC2014 partial functions. Compared with the new improved butterfly algorithm and other swarm intelligence algorithms, the experimental results show that the proposed algorithm has obvious advantages.

Key words: butterfly optimization algorithm (BOA), chaotic mapping, feedback sharing, group synergy

摘要：

针对蝴蝶优化算法（BOA）寻优精度低和易陷入局部最优等缺点,提出了混沌反馈共享和群体协同效应的蝴蝶优化算法（CFSBOA）。首先,利用Hénon混沌初始化种群,能够使种群尽可能地覆盖搜索盲区,增加种群多样性,提高算法寻优性能;其次,利用反馈控制电路中正负反馈作用机制的思想,构建蝴蝶之间的反馈共享交流网络,使得蝴蝶个体能够接收来自多个方向的信息,帮助种群定位最优解的位置并执行精细搜索,增强算法逃离局部最优的能力和加快算法收敛的速度;最后,利用群体协同效应机制,提高和平衡全局与局部搜索的能力,增强算法的全局和局部的寻优能力。使用不同维度的基准测试函数、统计检验、Wilcoxon检验及多类型的CEC2014部分函数验证改进蝴蝶优化算法的性能,并与新改进的蝴蝶算法及其他群智能算法进行对比,实验结果表明该算法具有明显优势。

关键词: 蝴蝶优化算法（BOA）, 混沌映射, 反馈共享, 群体协同效应

CLC Number:

TP301.6

LI Shouyu, HE Qing, DU Nisuo. Butterfly Optimization Algorithm for Chaotic Feedback Sharing and Group Synergy[J]. Journal of Frontiers of Computer Science and Technology, 2022, 16(7): 1661-1672.

李守玉, 何庆, 杜逆索. 混沌反馈共享和群体协同效应的蝴蝶优化算法[J]. 计算机科学与探索, 2022, 16(7): 1661-1672.

Figures/Tables 10

Fig.1 Hénon chaotic mapping

Table 1 Benchmark functions

函数	维度	特征	范围	最佳值
$f 1$ Sphere	10	US	[-100,100]	0
$f 2$ Schwefel 2.22	50	UN	[-10,10]	0
$f 3$ Schwefel 1.2	50	UN	[-100,100]	0
$f 4$ Schwefel 2.21	50	US	[-100,100]	0
$f 5$ Matyas	2	UN	[-10,10]	0
$f 6$ Schaffer	2	MN	[-100,100]	0
$f 7$ Rastrigin	10	MS	[-5.12,5.12]	0
$f 8$ Ackley	50	MN	[-32,32]	0
$f 9$ Griewank	100	MN	[-600,600]	0
$f 10$ Foxholes	2	MN	[-65.56,65.56]	≈1
$f 11$ Apline	100	MN	[-10,10]	0
$f 12$ Salomon	200	MN	[-100,100]	0

Table 1 Benchmark functions

函数	维度	特征	范围	最佳值
$f 1$ Sphere	10	US	[-100,100]	0
$f 2$ Schwefel 2.22	50	UN	[-10,10]	0
$f 3$ Schwefel 1.2	50	UN	[-100,100]	0
$f 4$ Schwefel 2.21	50	US	[-100,100]	0
$f 5$ Matyas	2	UN	[-10,10]	0
$f 6$ Schaffer	2	MN	[-100,100]	0
$f 7$ Rastrigin	10	MS	[-5.12,5.12]	0
$f 8$ Ackley	50	MN	[-32,32]	0
$f 9$ Griewank	100	MN	[-600,600]	0
$f 10$ Foxholes	2	MN	[-65.56,65.56]	≈1
$f 11$ Apline	100	MN	[-10,10]	0
$f 12$ Salomon	200	MN	[-100,100]	0

Table 2 Main parameters

算法	主要参数
CFSBOA	P=0.8,c=0.01, $a = 0.1$ , $τ = 0.618033$ , $γ = 1.4$ , $φ = 0.3$
SBOA	P=0.8,c=0.01, $a = 0.1$ , $τ = 0.618033 …$
FBOA	P=0.8,c=0.01, $a = 0.1$
CBOA	P=0.8,c=0.01, $a = 0.1$ , $γ = 1.4$ , $φ = 0.3$
BOA	P=0.8,c=0.01, $a = 0.1$
MFO	—
SSA	m=2
GWO	$a m a x = 2$ , $a m i n = 0$

Table 2 Main parameters

算法	主要参数
CFSBOA	P=0.8,c=0.01, $a = 0.1$ , $τ = 0.618033$ , $γ = 1.4$ , $φ = 0.3$
SBOA	P=0.8,c=0.01, $a = 0.1$ , $τ = 0.618033 …$
FBOA	P=0.8,c=0.01, $a = 0.1$
CBOA	P=0.8,c=0.01, $a = 0.1$ , $γ = 1.4$ , $φ = 0.3$
BOA	P=0.8,c=0.01, $a = 0.1$
MFO	—
SSA	m=2
GWO	$a m a x = 2$ , $a m i n = 0$

Table 3 Results comparison of benchmark test functions

函数	算法	最优值	平均值	标准差	SR/%	T/s	函数	算法	最优值	平均值	标准差	SR/%	T/s
$f 1$	CFSBOA	0.00E+00	0.00E+00	0.00E+00	100	0.40	$f 7$	CFSBOA	0.00E+00	0.00E+00	0.00E+00	100	0.63
	SBOA	0.00E+00	0.00E+00	0.00E+00	100	0.25		SBOA	0.00E+00	0.00E+00	0.00E+00	100	0.46
	FBOA	6.51E-80	9.13E-79	9.00E-79	100	0.37		FBOA	0.00E+00	0.00E+00	0.00E+00	100	0.56
	CBOA	7.30E-13	1.09E-12	3.10E-13	100	0.23		CBOA	0.00E+00	5.68E-16	2.81E-15	100	0.40
	BOA	7.13E-12	1.01E-11	1.50E-12	100	0.22		BOA	2.84E-14	2.49E+01	2.11E+01	28	0.38
	MFO	3.91E-16	3.44E-13	5.72E-13	100	0.11		MFO	5.97E+00	2.47E+01	1.47E+01	0	0.16
	SSA	3.12E-10	9.43E-10	3.07E-10	100	0.14		SSA	2.98E+00	1.68E+01	8.34E+00	0	0.18
	GWO	7.01E-61	1.48E-56	7.76E-56	100	0.11		GWO	0.00E+00	4.25E-01	1.06E+00	88	0.16
$f 2$	CFSBOA	2.02E-225	2.53E-169	0.00E+00	100	0.47	$f 8$	CFSBOA	8.88E-16	8.88E-16	0.00E+00	100	0.82
	SBOA	4.51E-149	2.81E-113	1.99E-112	100	0.28		SBOA	8.88E-16	8.88E-16	0.00E+00	100	0.54
	FBOA	2.68E-180	6.61E+23	3.85E+24	70	0.43		FBOA	8.88E-16	8.88E-16	0.00E+00	100	0.78
	CBOA	3.00E-12	1.73E-01	6.91E-01	92	0.27		CBOA	7.55E-14	5.62E-12	6.87E-12	100	0.49
	BOA	4.38E-13	5.40E+24	3.64E+25	68	0.26		BOA	5.15E-09	5.94E-09	3.51E-10	100	0.47
	MFO	7.42E+00	6.99E+01	3.45E+01	0	0.23		MFO	1.29E+01	1.93E+01	1.20E+00	0	0.31
	SSA	4.02E+00	1.08E+01	7.95E+00	0	0.21		SSA	2.87E+00	4.64E+00	1.13E+00	0	0.29
	GWO	6.80E-13	2.80E-12	2.12E-12	100	0.25		GWO	1.14E-11	3.57E-11	2.08E-11	100	0.34
$f 3$	CFSBOA	0.00E+00	0.00E+00	0.00E+00	100	4.55	$f 9$	CFSBOA	0.00E+00	0.00E+00	0.00E+00	100	0.96
	SBOA	0.00E+00	0.00E+00	0.00E+00	100	2.31		SBOA	0.00E+00	0.00E+00	0.00E+00	100	0.62
	FBOA	2.94E-217	5.73E-198	0.00E+00	100	4.50		FBOA	0.00E+00	0.00E+00	0.00E+00	100	0.92
	CBOA	1.61E-16	1.38E-12	1.24E-12	100	2.29		CBOA	6.35E-13	1.36E-12	3.75E-13	100	0.57
	BOA	1.08E-11	1.32E-11	9.79E-13	100	2.31		BOA	1.98E-12	1.28E-11	4.12E-12	100	0.56
	MFO	3.27E+04	6.62E+04	2.13E+04	0	1.26		MFO	3.04E+02	5.42E+02	1.43E+02	0	0.49
	SSA	3.39E+03	9.13E+03	3.85E+03	0	1.23		SSA	7.20E+00	1.34E+01	3.20E+00	0	0.44
	GWO	3.05E-03	8.96E-01	5.13E+00	0	1.25		GWO	1.09E-13	1.94E-03	6.75E-03	88	0.52
$f 4$	CFSBOA	1.35E-193	2.08E-156	1.42E-155	100	0.56	$f 10$	CFSBOA	9.98E-01	9.98E-01	1.84E-10	100	3.53
	SBOA	1.71E-164	4.69E-153	1.66E-152	100	0.33		SBOA	9.98E-01	5.02E+00	4.21E+00	8	1.91
	FBOA	1.11E-28	2.75E-28	9.52E-29	100	0.54		FBOA	9.98E-01	1.41E+00	6.41E-01	30	3.49
	CBOA	6.68E-14	2.89E-11	8.86E-11	100	0.32		CBOA	9.98E-01	9.98E-01	2.11E-10	100	1.86
	BOA	5.51E-09	6.20E-09	3.67E-10	100	0.31		BOA	9.98E-01	1.31E+00	5.06E-01	16	1.80
	MFO	6.74E+01	8.32E+01	5.17E+00	0	0.25		MFO	9.98E-01	1.99E+00	1.49E+00	32	0.85
	SSA	1.22E+01	2.00E+01	3.49E+00	0	0.23		SSA	9.98E-01	1.22E+00	6.43E-01	88	0.88
	GWO	2.92E-05	3.51E-04	3.50E-04	10	0.26		GWO	9.98E-01	4.42E+00	4.25E+00	42	0.86
$f 5$	CFSBOA	0.00E+00	2.55E-297	0.00E+00	100	0.61	$f 11$	CFSBOA	2.17E-215	6.29E-165	0.00E+00	100	0.61
	SBOA	0.00E+00	0.00E+00	0.00E+00	100	0.41		SBOA	5.03E-165	1.76E-151	1.19E-150	100	0.45
	FBOA	3.41E-51	3.63E-49	8.70E-49	100	0.57		FBOA	1.41E-45	3.18E-44	3.64E-44	100	0.59
	CBOA	1.07E-18	6.74E-13	2.56E-13	100	0.37		CBOA	9.26E-12	3.79E-03	1.71E-02	76	0.44
	BOA	6.30E-17	9.28E-13	3.70E-13	100	0.37		BOA	3.24E-10	8.30E-10	3.61E-10	100	0.43
	MFO	1.34E-86	2.53E-22	1.79E-21	100	0.13		MFO	4.29E+01	6.64E+01	1.60E+01	0	0.39
	SSA	4.86E-18	1.29E-15	1.85E-15	100	0.16		SSA	1.93E+01	2.94E+01	5.58E+00	0	0.34
	GWO	5.97E-129	1.82E-102	9.01E-102	100	0.14		GWO	2.80E-08	3.39E-03	2.86E-03	6	0.43
$f 6$	CFSBOA	0.00E+00	0.00E+00	0.00E+00	100	0.66	$f 12$	CFSBOA	0.00E+00	0.00E+00	0.00E+00	100	38.72
	SBOA	0.00E+00	0.00E+00	0.00E+00	100	0.44		SBOA	1.97E-305	3.16E-258	0.00E+00	100	19.41
	FBOA	0.00E+00	1.53E-02	1.10E-02	4	0.58		FBOA	0.00E+00	6.38E-106	4.51E-105	100	38.54
	CBOA	1.44E-15	1.94E-04	1.37E-03	96	0.37		CBOA	3.99E-14	7.45E-02	4.45E-02	18	19.47
	BOA	1.65E-05	1.28E-02	7.97E-03	0	0.36		BOA	1.52E-16	3.98E-03	1.97E-02	94	19.44
	MFO	0.00E+00	9.49E-03	4.81E-03	10	0.14		MFO	0.00E+00	5.56E-30	9.42E-30	100	10.08
	SSA	2.60E-14	5.05E-03	4.90E-03	46	0.17		SSA	1.12E-16	2.35E-12	3.52E-12	100	10.00
	GWO	0.00E+00	5.64E-03	4.84E-03	42	0.14		GWO	7.21E-08	3.75E-04	7.36E-04	30	10.19

Table 3 Results comparison of benchmark test functions

函数	算法	最优值	平均值	标准差	SR/%	T/s	函数	算法	最优值	平均值	标准差	SR/%	T/s
$f 1$	CFSBOA	0.00E+00	0.00E+00	0.00E+00	100	0.40	$f 7$	CFSBOA	0.00E+00	0.00E+00	0.00E+00	100	0.63
	SBOA	0.00E+00	0.00E+00	0.00E+00	100	0.25		SBOA	0.00E+00	0.00E+00	0.00E+00	100	0.46
	FBOA	6.51E-80	9.13E-79	9.00E-79	100	0.37		FBOA	0.00E+00	0.00E+00	0.00E+00	100	0.56
	CBOA	7.30E-13	1.09E-12	3.10E-13	100	0.23		CBOA	0.00E+00	5.68E-16	2.81E-15	100	0.40
	BOA	7.13E-12	1.01E-11	1.50E-12	100	0.22		BOA	2.84E-14	2.49E+01	2.11E+01	28	0.38
	MFO	3.91E-16	3.44E-13	5.72E-13	100	0.11		MFO	5.97E+00	2.47E+01	1.47E+01	0	0.16
	SSA	3.12E-10	9.43E-10	3.07E-10	100	0.14		SSA	2.98E+00	1.68E+01	8.34E+00	0	0.18
	GWO	7.01E-61	1.48E-56	7.76E-56	100	0.11		GWO	0.00E+00	4.25E-01	1.06E+00	88	0.16
$f 2$	CFSBOA	2.02E-225	2.53E-169	0.00E+00	100	0.47	$f 8$	CFSBOA	8.88E-16	8.88E-16	0.00E+00	100	0.82
	SBOA	4.51E-149	2.81E-113	1.99E-112	100	0.28		SBOA	8.88E-16	8.88E-16	0.00E+00	100	0.54
	FBOA	2.68E-180	6.61E+23	3.85E+24	70	0.43		FBOA	8.88E-16	8.88E-16	0.00E+00	100	0.78
	CBOA	3.00E-12	1.73E-01	6.91E-01	92	0.27		CBOA	7.55E-14	5.62E-12	6.87E-12	100	0.49
	BOA	4.38E-13	5.40E+24	3.64E+25	68	0.26		BOA	5.15E-09	5.94E-09	3.51E-10	100	0.47
	MFO	7.42E+00	6.99E+01	3.45E+01	0	0.23		MFO	1.29E+01	1.93E+01	1.20E+00	0	0.31
	SSA	4.02E+00	1.08E+01	7.95E+00	0	0.21		SSA	2.87E+00	4.64E+00	1.13E+00	0	0.29
	GWO	6.80E-13	2.80E-12	2.12E-12	100	0.25		GWO	1.14E-11	3.57E-11	2.08E-11	100	0.34
$f 3$	CFSBOA	0.00E+00	0.00E+00	0.00E+00	100	4.55	$f 9$	CFSBOA	0.00E+00	0.00E+00	0.00E+00	100	0.96
	SBOA	0.00E+00	0.00E+00	0.00E+00	100	2.31		SBOA	0.00E+00	0.00E+00	0.00E+00	100	0.62
	FBOA	2.94E-217	5.73E-198	0.00E+00	100	4.50		FBOA	0.00E+00	0.00E+00	0.00E+00	100	0.92
	CBOA	1.61E-16	1.38E-12	1.24E-12	100	2.29		CBOA	6.35E-13	1.36E-12	3.75E-13	100	0.57
	BOA	1.08E-11	1.32E-11	9.79E-13	100	2.31		BOA	1.98E-12	1.28E-11	4.12E-12	100	0.56
	MFO	3.27E+04	6.62E+04	2.13E+04	0	1.26		MFO	3.04E+02	5.42E+02	1.43E+02	0	0.49
	SSA	3.39E+03	9.13E+03	3.85E+03	0	1.23		SSA	7.20E+00	1.34E+01	3.20E+00	0	0.44
	GWO	3.05E-03	8.96E-01	5.13E+00	0	1.25		GWO	1.09E-13	1.94E-03	6.75E-03	88	0.52
$f 4$	CFSBOA	1.35E-193	2.08E-156	1.42E-155	100	0.56	$f 10$	CFSBOA	9.98E-01	9.98E-01	1.84E-10	100	3.53
	SBOA	1.71E-164	4.69E-153	1.66E-152	100	0.33		SBOA	9.98E-01	5.02E+00	4.21E+00	8	1.91
	FBOA	1.11E-28	2.75E-28	9.52E-29	100	0.54		FBOA	9.98E-01	1.41E+00	6.41E-01	30	3.49
	CBOA	6.68E-14	2.89E-11	8.86E-11	100	0.32		CBOA	9.98E-01	9.98E-01	2.11E-10	100	1.86
	BOA	5.51E-09	6.20E-09	3.67E-10	100	0.31		BOA	9.98E-01	1.31E+00	5.06E-01	16	1.80
	MFO	6.74E+01	8.32E+01	5.17E+00	0	0.25		MFO	9.98E-01	1.99E+00	1.49E+00	32	0.85
	SSA	1.22E+01	2.00E+01	3.49E+00	0	0.23		SSA	9.98E-01	1.22E+00	6.43E-01	88	0.88
	GWO	2.92E-05	3.51E-04	3.50E-04	10	0.26		GWO	9.98E-01	4.42E+00	4.25E+00	42	0.86
$f 5$	CFSBOA	0.00E+00	2.55E-297	0.00E+00	100	0.61	$f 11$	CFSBOA	2.17E-215	6.29E-165	0.00E+00	100	0.61
	SBOA	0.00E+00	0.00E+00	0.00E+00	100	0.41		SBOA	5.03E-165	1.76E-151	1.19E-150	100	0.45
	FBOA	3.41E-51	3.63E-49	8.70E-49	100	0.57		FBOA	1.41E-45	3.18E-44	3.64E-44	100	0.59
	CBOA	1.07E-18	6.74E-13	2.56E-13	100	0.37		CBOA	9.26E-12	3.79E-03	1.71E-02	76	0.44
	BOA	6.30E-17	9.28E-13	3.70E-13	100	0.37		BOA	3.24E-10	8.30E-10	3.61E-10	100	0.43
	MFO	1.34E-86	2.53E-22	1.79E-21	100	0.13		MFO	4.29E+01	6.64E+01	1.60E+01	0	0.39
	SSA	4.86E-18	1.29E-15	1.85E-15	100	0.16		SSA	1.93E+01	2.94E+01	5.58E+00	0	0.34
	GWO	5.97E-129	1.82E-102	9.01E-102	100	0.14		GWO	2.80E-08	3.39E-03	2.86E-03	6	0.43
$f 6$	CFSBOA	0.00E+00	0.00E+00	0.00E+00	100	0.66	$f 12$	CFSBOA	0.00E+00	0.00E+00	0.00E+00	100	38.72
	SBOA	0.00E+00	0.00E+00	0.00E+00	100	0.44		SBOA	1.97E-305	3.16E-258	0.00E+00	100	19.41
	FBOA	0.00E+00	1.53E-02	1.10E-02	4	0.58		FBOA	0.00E+00	6.38E-106	4.51E-105	100	38.54
	CBOA	1.44E-15	1.94E-04	1.37E-03	96	0.37		CBOA	3.99E-14	7.45E-02	4.45E-02	18	19.47
	BOA	1.65E-05	1.28E-02	7.97E-03	0	0.36		BOA	1.52E-16	3.98E-03	1.97E-02	94	19.44
	MFO	0.00E+00	9.49E-03	4.81E-03	10	0.14		MFO	0.00E+00	5.56E-30	9.42E-30	100	10.08
	SSA	2.60E-14	5.05E-03	4.90E-03	46	0.17		SSA	1.12E-16	2.35E-12	3.52E-12	100	10.00
	GWO	0.00E+00	5.64E-03	4.84E-03	42	0.14		GWO	7.21E-08	3.75E-04	7.36E-04	30	10.19

Fig.2 Average convergence curve of benchmark test functions

Table 4 p value for Wilcoxon’s rank-sum test on benchmark functions

F	p-value win
F	BOA	CBOA	FBOA	SBOA	MFO	SSA	GWO
$f 1$	3.31E-20+	3.31E-20+	3.31E-20+	N/A=	3.31E-20+	3.31E-20+	3.31E-20+
$f 2$	7.07E-18+	7.07E-18+	2.78E-17+	7.07E-18+	7.07E-18+	7.07E-18+	7.07E-18+
$f 3$	3.31E-20+	3.31E-20+	3.31E-20+	N/A=	3.31E-20+	3.31E-20+	3.31E-20+
$f 4$	7.07E-18+	7.07E-18+	7.07E-18+	2.39E-13+	7.07E-18+	7.07E-18+	7.07E-18+
$f 5$	1.12E-18+	1.12E-18+	1.12E-18+	6.50E-05+	1.12E-18+	1.12E-18+	1.12E-18+
$f 6$	3.31E-20+	3.31E-20+	3.31E-20+	N/A=	7.55E-21+	3.28E-20+	5.25E-13+
$f 7$	3.31E-20+	1.59E-01-	N/A=	N/A=	3.31E-20+	3.31E-20+	1.82E-03+
$f 8$	3.31E-20+	3.31E-20+	N/A=	N/A=	3.31E-20+	3.31E-20+	3.31E-20+
$f 9$	3.31E-20+	3.31E-20+	N/A=	N/A=	3.31E-20+	3.31E-20+	3.31E-20+
$f 10$	7.07E-18+	5.74E-01-	7.07E-18+	7.07E-18+	2.38E-12+	3.59E-12+	1.27E-12+
$f 11$	7.07E-18+	7.07E-18+	7.07E-18+	7.07E-18+	7.07E-18+	7.07E-18+	7.07E-18+
$f 12$	3.31E-20+	3.31E-20+	N/A=	3.31E-20+	1.75E-06+	3.31E-20+	3.31E-20+
+/=/-	12/0/0	10/0/2	8/4/0	6/6/0	12/0/0	12/0/0	12/0/0

Table 4 p value for Wilcoxon’s rank-sum test on benchmark functions

F	p-value win
F	BOA	CBOA	FBOA	SBOA	MFO	SSA	GWO
$f 1$	3.31E-20+	3.31E-20+	3.31E-20+	N/A=	3.31E-20+	3.31E-20+	3.31E-20+
$f 2$	7.07E-18+	7.07E-18+	2.78E-17+	7.07E-18+	7.07E-18+	7.07E-18+	7.07E-18+
$f 3$	3.31E-20+	3.31E-20+	3.31E-20+	N/A=	3.31E-20+	3.31E-20+	3.31E-20+
$f 4$	7.07E-18+	7.07E-18+	7.07E-18+	2.39E-13+	7.07E-18+	7.07E-18+	7.07E-18+
$f 5$	1.12E-18+	1.12E-18+	1.12E-18+	6.50E-05+	1.12E-18+	1.12E-18+	1.12E-18+
$f 6$	3.31E-20+	3.31E-20+	3.31E-20+	N/A=	7.55E-21+	3.28E-20+	5.25E-13+
$f 7$	3.31E-20+	1.59E-01-	N/A=	N/A=	3.31E-20+	3.31E-20+	1.82E-03+
$f 8$	3.31E-20+	3.31E-20+	N/A=	N/A=	3.31E-20+	3.31E-20+	3.31E-20+
$f 9$	3.31E-20+	3.31E-20+	N/A=	N/A=	3.31E-20+	3.31E-20+	3.31E-20+
$f 10$	7.07E-18+	5.74E-01-	7.07E-18+	7.07E-18+	2.38E-12+	3.59E-12+	1.27E-12+
$f 11$	7.07E-18+	7.07E-18+	7.07E-18+	7.07E-18+	7.07E-18+	7.07E-18+	7.07E-18+
$f 12$	3.31E-20+	3.31E-20+	N/A=	3.31E-20+	1.75E-06+	3.31E-20+	3.31E-20+
+/=/-	12/0/0	10/0/2	8/4/0	6/6/0	12/0/0	12/0/0	12/0/0

Table 5 MAE ranking of each algorithm

算法	MAE	Rank
CFSBOA	1.66E-04	1
CBOA	1.04E-02	2
SBOA	2.98E-01	3
GWO	3.52E-01	4
SSA	8.46E+02	5
MFO	5.38E+03	6
FBOA	2.93E+22	7
BOA	5.38E+22	8

Table 6 CEC2014 function (part)

函数	特征	定义域	最佳值
CEC03	UF	[-100,100]	300
CEC05	MF	[-100,100]	500
CEC16	HF	[-100,100]	1 600
CEC25	CF	[-100,100]	2 500
CEC27	CF	[-100,100]	2 700
CEC30	CF	[-100,100]	3 000

Table 7 Comparison of CEC2014 test function results

函数	指标	CFSBOA	BOA	MFO	SSA	GWO	PSO
CEC03	Mean	8.951 0E+04	1.077 5E+10	9.546 7E+04	1.026 6E+05	5.368 9E+04	2.799 2E+04
CEC03	Std	4.765 1E+02	0.000 0E+00	5.072 7E+04	2.751 8E+04	1.387 3E+04	2.221 6E+04
CEC05	Mean	5.211 5E+02	5.217 6E+02	5.203 2E+02	5.200 6E+02	5.210 8E+02	5.210 0E+02
CEC05	Std	8.644 2E-03	3.468 9E-13	1.530 4E-01	9.554 9E-02	4.418 3E-02	7.922 0E-02
CEC16	Mean	1.613 4E+03	1.614 9E+03	1.612 8E+03	1.612 6E+03	1.612 2E+03	1.612 5E+03
CEC16	Std	2.494 5E-04	2.312 6E-13	5.504 8E-01	3.771 0E-01	5.385 8E-01	4.627 2E-01
CEC25	Mean	2.700 0E+03	5.125 0E+03	2.719 0E+03	2.718 6E+03	2.713 9E+03	2.714 9E+03
CEC25	Std	0.000 0E+00	0.000 0E+00	1.021 0E+01	4.834 9E+00	6.214 1E+00	3.990 2E+00
CEC27	Mean	2.900 0E+03	1.933 9E+04	3.627 7E+03	3.558 1E+03	3.464 5E+03	3.412 5E+03
CEC27	Std	0.000 0E+00	1.110 1E-11	1.995 7E+02	2.412 9E+02	1.317 1E+02	1.711 3E+02
CEC30	Mean	3.200 0E+03	3.236 5E+07	4.084 2E+04	6.971 4E+04	1.222 8E+05	1.806 6E+04
CEC30	Std	0.000 0E+00	1.515 6E-08	3.337 1E+04	3.733 2E+04	7.403 6E+04	8.312 1E+03

Table 8 Results comparison of mean and standard deviation of algorithms in references

算法	指标				$f 1$		$f 2$		$f 3$		$f 4$		$f 5$	$f 6$
CFSBOA		Mean			0.00E+00		3.40E-170		0.00E+00		1.13E-155		2.83E-296	0.00E+00
CFSBOA		Std			0.00E+00		0.00E+00		0.00E+00		6.18E-155		0.00E+00	0.00E+00
HPSOBOA^[23]		Mean			3.74E-104		2.63E-22		3.04E-71		3.61E-46		—	—
HPSOBOA^[23]		Std			2.05E-103		1.44E-21		1.67E-70		1.97E-45		—	—
LBOA^[23]		Mean			3.92E-12		1.39E-09		2.74E-12		2.30E-09		—	—
LBOA^[23]		Std			4.46E-12		2.08E-09		2.44E-12		2.36E-09		—	—
IBOA^[23]		Mean			1.61E-30		5.11E-19		6.15E-31		1.36E-19		—	—
IBOA^[23]		Std			3.90E-30		1.73E-18		1.16E-30		1.97E-19		—	—
CWBOA^[9]		Mean			0.00E+00		3.86E-134		—		3.29E-134		0.00E+00	0.00E+00
CWBOA^[9]		Std			0.00E+00		1.52E-133		—		1.80E-133		0.00E+00	0.00E+00
BOA-CE^[11]		Mean			1.26E-95		3.60E-47		2.44E-09		8.82E-07		—	—
BOA-CE^[11]		Std			9.78E-96		2.50E-47		1.83E-10		1.51E-07		—	—
CSO^[24]		Mean			3.50E-14		2.68E-08		7.17E-09		1.04E-02		—	—
CSO^[24]		Std			6.34E-14		2.61E-08		1.16E-08		7.96E-03		—	—
MPA^[25]		Mean			4.51E-23		3.01E-13		1.27E-04		3.26E-09		2.97E-51	3.84E-13
MPA^[25]		Std			3.56E-23		3.07E-13		2.19E-04		2.22E-09		1.57E-50	2.10E-12
MSIWOA^[26]		Mean			0.00E+00		7.17E-203		0.00E+00		4.59E-139		—	—
MSIWOA^[26]		Std			0.00E+00		1.64E-203		0.00E+00		2.51E-138		—	—
QOWOA-CS^[27]		Mean			1.01E-232		2.23E-125		2.61E-201		3.58E-103		—	—
QOWOA-CS^[27]		Std			0.00E+00		5.85E-125		0.00E+00		6.47E-103		—	—
算法	指标			$f 7$		$f 8$		$f 9$		$f 10$		$f 11$		$f 12$
CFSBOA			Mean	0.00E+00		8.88E-16		0.00E+00		9.98E-01		1.27E-166		0.00E+00
CFSBOA			Std	0.00E+00		0.00E+00		0.00E+00		1.20E-10		0.00E+00		0.00E+00
HPSOBOA^[23]			Mean	0.00E+00		8.96E-11		0.00E+00		—		2.54E-45		2.53E-08
HPSOBOA^[23]			Std	0.00E+00		4.73E-10		0.00E+00		—		1.39E-44		1.38E-07
LBOA^[23]			Mean	0.00E+00		2.34E-12		3.48E-13		—		6.32E-14		3.65E-02
LBOA^[23]			Std	0.00E+00		7.87E-12		8.78E-13		—		1.73E-13		4.88E-02
IBOA^[23]			Mean	0.00E+00		8.88E-16		0.00E+00		—		8.93E-20		2.25E-32
IBOA^[23]			Std	0.00E+00		0.00E+00		0.00E+00		—		1.19E-19		5.88E-32
CWBOA^[9]			Mean	0.00E+00		8.88E-16		0.00E+00		—		2.13E-137		—
CWBOA^[9]			Std	0.00E+00		0.00E+00		0.00E+00		—		8.15E-137		—
BOA-CE^[11]			Mean	1.01E+01		4.44E-15		0.00E+00		—		—		—
BOA-CE^[11]			Std	3.46E+01		0.00E+00		0.00E+00		—		—		—
CSO^[24]			Mean	2.40E+01		3.75E+00		3.56E-01		9.98E-01		—		—
CSO^[24]			Std	6.48E+00		1.68E+00		1.91E-01		3.39E-07		—		—
MPA^[25]			Mean	0.00E+00		1.59E-12		0.00E+00		9.98E-01		9.12E-14		5.58E-17
MPA^[25]			Std	0.00E+00		9.87E-13		0.00E+00		1.62E-16		8.01E-14		1.40E-16
MSIWOA^[26]			Mean	0.00E+00		0.00E+00		0.00E+00		—		—		—
MSIWOA^[26]			Std	0.00E+00		0.00E+00		0.00E+00		—		—		—
QOWOA-CS^[27]			Mean	0.00E+00		8.88E-16		0.00E+00		—		—		—
QOWOA-CS^[27]			Std	0.00E+00		0.00E+00		0.00E+00		—		—		—

Table 8 Results comparison of mean and standard deviation of algorithms in references

算法	指标				$f 1$		$f 2$		$f 3$		$f 4$		$f 5$	$f 6$
CFSBOA		Mean			0.00E+00		3.40E-170		0.00E+00		1.13E-155		2.83E-296	0.00E+00
CFSBOA		Std			0.00E+00		0.00E+00		0.00E+00		6.18E-155		0.00E+00	0.00E+00
HPSOBOA^[23]		Mean			3.74E-104		2.63E-22		3.04E-71		3.61E-46		—	—
HPSOBOA^[23]		Std			2.05E-103		1.44E-21		1.67E-70		1.97E-45		—	—
LBOA^[23]		Mean			3.92E-12		1.39E-09		2.74E-12		2.30E-09		—	—
LBOA^[23]		Std			4.46E-12		2.08E-09		2.44E-12		2.36E-09		—	—
IBOA^[23]		Mean			1.61E-30		5.11E-19		6.15E-31		1.36E-19		—	—
IBOA^[23]		Std			3.90E-30		1.73E-18		1.16E-30		1.97E-19		—	—
CWBOA^[9]		Mean			0.00E+00		3.86E-134		—		3.29E-134		0.00E+00	0.00E+00
CWBOA^[9]		Std			0.00E+00		1.52E-133		—		1.80E-133		0.00E+00	0.00E+00
BOA-CE^[11]		Mean			1.26E-95		3.60E-47		2.44E-09		8.82E-07		—	—
BOA-CE^[11]		Std			9.78E-96		2.50E-47		1.83E-10		1.51E-07		—	—
CSO^[24]		Mean			3.50E-14		2.68E-08		7.17E-09		1.04E-02		—	—
CSO^[24]		Std			6.34E-14		2.61E-08		1.16E-08		7.96E-03		—	—
MPA^[25]		Mean			4.51E-23		3.01E-13		1.27E-04		3.26E-09		2.97E-51	3.84E-13
MPA^[25]		Std			3.56E-23		3.07E-13		2.19E-04		2.22E-09		1.57E-50	2.10E-12
MSIWOA^[26]		Mean			0.00E+00		7.17E-203		0.00E+00		4.59E-139		—	—
MSIWOA^[26]		Std			0.00E+00		1.64E-203		0.00E+00		2.51E-138		—	—
QOWOA-CS^[27]		Mean			1.01E-232		2.23E-125		2.61E-201		3.58E-103		—	—
QOWOA-CS^[27]		Std			0.00E+00		5.85E-125		0.00E+00		6.47E-103		—	—
算法	指标			$f 7$		$f 8$		$f 9$		$f 10$		$f 11$		$f 12$
CFSBOA			Mean	0.00E+00		8.88E-16		0.00E+00		9.98E-01		1.27E-166		0.00E+00
CFSBOA			Std	0.00E+00		0.00E+00		0.00E+00		1.20E-10		0.00E+00		0.00E+00
HPSOBOA^[23]			Mean	0.00E+00		8.96E-11		0.00E+00		—		2.54E-45		2.53E-08
HPSOBOA^[23]			Std	0.00E+00		4.73E-10		0.00E+00		—		1.39E-44		1.38E-07
LBOA^[23]			Mean	0.00E+00		2.34E-12		3.48E-13		—		6.32E-14		3.65E-02
LBOA^[23]			Std	0.00E+00		7.87E-12		8.78E-13		—		1.73E-13		4.88E-02
IBOA^[23]			Mean	0.00E+00		8.88E-16		0.00E+00		—		8.93E-20		2.25E-32
IBOA^[23]			Std	0.00E+00		0.00E+00		0.00E+00		—		1.19E-19		5.88E-32
CWBOA^[9]			Mean	0.00E+00		8.88E-16		0.00E+00		—		2.13E-137		—
CWBOA^[9]			Std	0.00E+00		0.00E+00		0.00E+00		—		8.15E-137		—
BOA-CE^[11]			Mean	1.01E+01		4.44E-15		0.00E+00		—		—		—
BOA-CE^[11]			Std	3.46E+01		0.00E+00		0.00E+00		—		—		—
CSO^[24]			Mean	2.40E+01		3.75E+00		3.56E-01		9.98E-01		—		—
CSO^[24]			Std	6.48E+00		1.68E+00		1.91E-01		3.39E-07		—		—
MPA^[25]			Mean	0.00E+00		1.59E-12		0.00E+00		9.98E-01		9.12E-14		5.58E-17
MPA^[25]			Std	0.00E+00		9.87E-13		0.00E+00		1.62E-16		8.01E-14		1.40E-16
MSIWOA^[26]			Mean	0.00E+00		0.00E+00		0.00E+00		—		—		—
MSIWOA^[26]			Std	0.00E+00		0.00E+00		0.00E+00		—		—		—
QOWOA-CS^[27]			Mean	0.00E+00		8.88E-16		0.00E+00		—		—		—
QOWOA-CS^[27]			Std	0.00E+00		0.00E+00		0.00E+00		—		—		—

References 27

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Butterfly Optimization Algorithm for Chaotic Feedback Sharing and Group Synergy

混沌反馈共享和群体协同效应的蝴蝶优化算法

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