若干新型智能优化算法对比分析研究

doi:10.3778/j.issn.1673-9418.2107028

计算机科学与探索 ›› 2022, Vol. 16 ›› Issue (1): 88-105.DOI: 10.3778/j.issn.1673-9418.2107028

若干新型智能优化算法对比分析研究

张九龙¹^,⁺(), 王晓峰¹^,², 芦磊¹, 牛鹏飞¹

1.北方民族大学计算机科学与工程学院,银川 750021
2.北方民族大学图像图形智能处理国家民委重点实验室,银川 750021

收稿日期:2021-06-10 修回日期:2021-08-18 出版日期:2022-01-01 发布日期:2021-08-26
通讯作者: + E-mail: 1137073437@qq.com
作者简介:张九龙（1997—）,男,山东济宁人,硕士研究生,主要研究方向为算法分析与设计。
王晓峰（1980—）,男,甘肃会宁人,博士,副教授,硕士生导师,CCF会员,主要研究方向为算法分析与设计。
芦磊（1995—）,男,山东聊城人,硕士研究生,主要研究方向为算法分析与设计。
牛鹏飞（1997—）,男,宁夏银川人,硕士研究生,主要研究方向为算法分析与设计。
基金资助:
国家自然科学基金(62062001);宁夏自然科学基金(2020AAC03214);北方民族大学重大专项(ZDZX201901)

Analysis and Research of Several New Intelligent Optimization Algorithms

ZHANG Jiulong¹^,⁺(), WANG Xiaofeng¹^,², LU Lei¹, NIU Pengfei¹

1. School of Computer Science and Engineering, North Minzu University, Yinchuan 750021, China
2. Key Laboratory for Intelligent Processing of Computer Images and Graphics of National Ethnic Affairs Commission of the PRC, North Minzu University, Yinchuan 750021, China

Received:2021-06-10 Revised:2021-08-18 Online:2022-01-01 Published:2021-08-26
About author:ZHANG Jiulong, born in 1997, M.S. candidate. His research interests include algorithm analysis and design.
WANG Xiaofeng, born in 1980, Ph.D., associate professor, M.S. supervisor, member of CCF. His research interests include algorithm analysis and design.
LU Lei, born in 1995, M.S. candidate. His res-earch interests include algorithm analysis and design.
NIU Pengfei, born in 1997, M.S. candidate. His research interests include algorithm analysis and design.
Supported by:
National Natural Science Foundation of China(62062001);Natural Science Foundation of Ningxia(2020AAC03214);Major Scientific Research Projects of North Minzu University(ZDZX201901)

摘要/Abstract

摘要：

智能优化算法（IOA）指的是一类以自然界的生物生存进化过程或物理现象为算法原理,用于解决最优化问题的算法,较为知名的智能优化算法有遗传算法、粒子群算法、模拟退火算法等。智能优化算法属于启发式方法,广泛应用在解决最优化问题上,传统的群智能算法为解决一些实际问题提供了新思路。随着科学技术的进步和应用场景的改变,传统的智能优化算法在收敛速度、求解精度等方面已无法满足日益复杂的优化问题,因此不断有新的更高效的智能优化算法被提出。选取了近几年国内外提出的几种新型智能优化算法：蝴蝶优化算法（BOA）、飞蛾扑火算法（MFO）、正弦余弦优化算法（SCA）、蝗虫优化算法（GOA）、哈里斯鹰优化算法（HHO）、麻雀搜索算法（SSA）。阐述了各算法的基本原理、算法步骤、相关的改进策略及存在的优缺点。为客观对比各算法性能,进一步通过3种类型共21个测试函数及6个指标评价各算法性能,最后归纳总结各算法的特点并对智能优化算法的发展前景进行展望。

关键词: 智能优化算法（IOA）, 蝴蝶优化算法（BOA）, 飞蛾扑火算法（MFO）, 正弦余弦优化算法（SCA）, 蝗虫优化算法（GOA）, 哈里斯鹰优化算法（HHO）, 麻雀搜索算法（SSA）

Abstract:

Intelligent optimization algorithms (IOA) refer to a kind of algorithm that is used to solve optimization problems by imitating the survival and evolution process of natural creatures or physical phenomena as the algorithm principle. The well-known intelligent optimization algorithms include genetic algorithm, particle swarm optimization, simulated annealing algorithm, etc. Intelligent optimization algorithm is a heuristic method, which is widely used in solving optimization problems and provides some new ideas for solving some practical problems. With the advan-cement of science and technology and the increase in the complexity of application scenarios, traditional intelligent optimization algorithms can no longer satisfy optimization problems in terms of solving effects and accuracy. Therefore, new and more efficient intelligent optimization algorithms are constantly being proposed. Several new intelligent optimization algorithms have been proposed at home and abroad in recent years, such as butterfly optimization algorithm (BOA), moth-flame optimization (MFO), sine cosine algorithm (SCA), grasshopper optimization algorithm (GOA), Harris hawks optimization (HHO) and sparrow search algorithm (SSA). This paper describes the basic principle, algorithm steps, corresponding improvement strategies with advantages and disadvantages of each algorithm. To objectively compare the performance of each algorithm, this paper further evaluates the performance of each algorithm through 21 test functions of 3 types and 6 indicators. Finally, this paper summarizes the characteristics of the algorithm and prospects the development direction of intelligent optimization algorithm.

Key words: intelligent optimization algorithms (IOA), butterfly optimization algorithm (BOA), moth-flame optimi-zation (MFO), sine cosine algorithm (SCA), grasshopper optimization algorithm (GOA), Harris hawks optimization (HHO), sparrow search algorithm (SSA)

中图分类号:

TP18

张九龙, 王晓峰, 芦磊, 牛鹏飞. 若干新型智能优化算法对比分析研究[J]. 计算机科学与探索, 2022, 16(1): 88-105.

ZHANG Jiulong, WANG Xiaofeng, LU Lei, NIU Pengfei. Analysis and Research of Several New Intelligent Optimization Algorithms[J]. Journal of Frontiers of Computer Science and Technology, 2022, 16(1): 88-105.

图/表 13

表1 蝴蝶优化算法改进分析

Table 1 Improvement analysis of BOA

算法	改进策略	优势	局限性	应用场景
文献[27]	引入Lévy飞行策略^[16,28]	增加了种群多样性	跳出局部最优的能力不强	函数优化
文献[29]	结合2-opt算子及模拟退火策略	增强寻优能力和算法鲁棒性	数据集规模不能太大	旅行商问题
文献[30]	结合单纯形法,将正弦余弦优化算法作为算子	改善种群多样性,加快算法跳出局部最优	增加了算法时间复杂度	函数优化
文献[31]	引入柯西分布函数,加入自适应权重因子和动态切换概率	增强全局搜索能力,提升算法鲁棒性	对高维多极值问题易陷入局部最优且不易跳出	函数优化
文献[32]	引入具有密集开发阶段的全局优化方案	提高收敛速度,减少计算量	复杂的基准测试函数下不能保证找到最优值	设计优化问题
文献[33]	加入反向学习策略和柯西变异	加快了收敛速度,提高了算法的寻优精度	对多目标优化问题提升不大	电联供微网优化调度

表2 飞蛾扑火算法改进分析

Table 2 Improvement analysis of MFO

算法	改进策略	优势	局限性	应用场景
文献[40]	引入交叉扰动策略	提升了求解精度和鲁棒性	算法时间成本增加	旅行商问题
文献[41]	与Lévy飞行策略结合	增强了局部搜索能力	对复杂函数的收敛率和求解精度仍有不足	函数优化
文献[42]	优化火焰更新机制和飞行路径	提升了算法稳健性和鲁棒性	时间复杂度增加	最优潮流计算
文献[43]	引入外部储存机制,自适应网格和筛选机制	提升了算法精度和收敛性	只能解决低维度的多目标问题	电力系统无功优化
文献[44]	融合粒子群优化算法	增强算法收敛性,降低时间消耗	求解精度一般	网络入侵检测
文献[45]	改进火焰更新过程	提升收敛速度和收敛精度	时间复杂度增加	雷达信号集设计
文献[46]	引入Lévy飞行策略结合萤火虫算法	提升收敛速度和精度	求解效果依赖初始阈值	红外图像分割

表3 正弦余弦优化算法改进分析

Table 3 Improvement analysis of SCA

算法	改进策略	优势	局限性	应用场景
文献[53]	设计出转换参数抛物线函数递减和指数递减形式	优化精度更高,收敛速度更快	算法时间复杂度增加	优化协同过滤推荐算法
文献[54]	加入惯性权重和反向学习策略	提升求解精度和收敛速度	低维优化问题提升效果不明显	高维优化问题
文献[55]	与二进制粒子群优化算法相结合	提升算法求解精度	求解效果与使用的数据集有关	特征选择
文献[56]	非线性调整控制参数,加入精英混沌策略和反向学习机制	提升算法收敛能力和稳定性	时间复杂度略微增加	函数优化问题
文献[57]	加入非线性转换参数和随机差分编译策略	增强跳出局部最优能力,提升收敛速度	时间复杂度增加	感知器网络参数优化
文献[58]	加入自学习机制,融入交叉开发技术	提升算法收敛速度和鲁棒性	时间复杂度增加	图像分割
文献[59]	加入双曲正弦调节因子和动态余弦波权重系数,引入拉普拉斯和高斯分布的变异策略	提升收敛精度和收敛速度	测试函数类型单一,反映不全面	传感器节点部署优化

表4 蝗虫优化算法改进分析

Table 4 Improvement analysis of GOA

算法	改进策略	优势	局限性	应用场景
文献[64]	加入外部存档和目标选择机制	提高算法收敛性	目标数量不能大于4个	多目标优化问题
文献[65]	改进惯性权重,融合模拟退火算法	提升求解精度和收敛速度	易陷入局部最优	函数优化问题
文献[66]	加入Lévy飞行机制和随机跳出策略	提高搜索精度	时间复杂度增加	任务调度问题
文献[67]	加入强化Lévy飞行机制	提高求解精度及鲁棒性	解决高维问题优势不明显	函数优化问题
文献[68]	加入量子旋转门操作	提升全局搜索能力和求解精度	需要额外的可变存储空间	作业车间调度
文献[69]	改进算法表示方法	提高精度和局部搜索能力	仿真实验数据集单一	三维航迹规划
文献[70]	引入百分点算子进行二值化	提高求解精度和算法鲁棒性	增加了算法时间复杂度	多维背包问题

表5 哈里斯鹰优化算法改进分析

Table 5 Improvement analysis of HHO

算法	简要分析	优势	局限性	应用场景
文献[77]	融入长时记忆概念,参考过去个体位置信息	增加种群多样性	增加了算法时间复杂度和空间复杂度	发电系统中的最优潮流问题
文献[78]	引入基于信息交换和共享的改进策略,加入非线性逃逸能量因子	提升了算法求解精度和鲁棒性	运行时间增加	函数优化和工程结构优化问题
文献[79]	与模拟退火算法结合	增强局部搜索能力和收敛速度	运行时间增加	特征选择问题
文献[80]	融入精英反向学习策略,与黄金正弦算法结合	缩小了搜索空间,增强了算法局部开发能力	时间复杂度增加	函数优化问题
文献[81]	利用最大似然估计改进算法适应度函数	提升求解精度和收敛速度	时间复杂度增加	时差定位问题
文献[82]	引入精英等级策略,加入Tent混沌映射和高斯随机游走策略	提升算法鲁棒性和收敛精度	前期收敛速度慢,固定维数问题上表现不佳	函数优化问题

表6 麻雀搜索算法改进分析

Table 6 Improvement analysis of SSA

算法	简要分析	优势	局限性	应用场景
文献[88]	加入柯西变异和反向学习策略	加快收敛速度,提升精度	测试函数少,反映不全面	函数优化
文献[89]	加入自适应学习因子	加快收敛速度	时间复杂度增加	电堆模型参数辨识
文献[90]	加入logistic映射、自适应参数、变异算子等	提升求解精度	时间成本增加	随机配置网络
文献[91]	加入Tent混沌序列,加入高斯变异	提升求解精度和算法稳定性,加快收敛速度	时间复杂度增加	图像分割
文献[92]	加入立方映射和反向学习策略,混合正弦余弦算法和高斯变异	增加了种群多样性,增强算法鲁棒性	前期收敛速度慢	无人机路径规划
文献[93]	引入多项式变异因子,设计一种新型拥堵距离计算策略	提高算法收敛性和解的质量	算法所需空间成本增加	多目标优化问题

表7 单峰测试函数

Table 7 Single peak test function

Test function	Range	Dimension
$F 1 = ∑ i = 1 n x i 2$	[-100,100]	30
$F 2 (x) = ∑ i = 1 n \| x i \| + ∏ i = 1 n \| x i \|$	[-10,10]	30
$F 3 (x) = ∑ i = 1 n ∑ j = 1 i x j 2$	[-100,100]	30
$F 4 (x) = max {\| x i \|, 1 ≤ i ≤ n}$	[-100,100]	30
$F 5 (x) = ∑ i = 1 n - 1 [100 (x i + 1 - x i 2) 2 + (x i - 1) 2]$	[-30,30]	30
$F 6 (x) = ∑ i = 1 n ([x i + 0.5]) 2$	[-100,100]	30
$F 7 (x) = ∑ i = 1 n i x i 4 + random [0,1)$	[-1.28,1.28]	30

表7 单峰测试函数

Table 7 Single peak test function

Test function	Range	Dimension
$F 1 = ∑ i = 1 n x i 2$	[-100,100]	30
$F 2 (x) = ∑ i = 1 n \| x i \| + ∏ i = 1 n \| x i \|$	[-10,10]	30
$F 3 (x) = ∑ i = 1 n ∑ j = 1 i x j 2$	[-100,100]	30
$F 4 (x) = max {\| x i \|, 1 ≤ i ≤ n}$	[-100,100]	30
$F 5 (x) = ∑ i = 1 n - 1 [100 (x i + 1 - x i 2) 2 + (x i - 1) 2]$	[-30,30]	30
$F 6 (x) = ∑ i = 1 n ([x i + 0.5]) 2$	[-100,100]	30
$F 7 (x) = ∑ i = 1 n i x i 4 + random [0,1)$	[-1.28,1.28]	30

表8 多峰测试函数

Table 8 Multimodal peak test function

Test function	Range	Dimension	$f min$
$F 8 (x) = ∑ i = 1 n - x i sin \| x i \|$	[-500,500]	30	-418.983 9n
$F 9 (x) = ∑ i = 1 n [x i 2 - 10 cos (2 π x i) + 10]$	[-5.12,5.12]	30	0
$F 10 (x) = - 20 exp - 0.2 1 n ∑ i = 1 n x i 2 - exp 1 n ∑ i = 1 n cos (2 π x i) + 20 + e$	[-32,32]	30	0
$F 11 (x) = 14000 ∑ i = 1 n x i 2 - ∏ i = 1 n cos x i i + 1$	[-600,600]	30	0
$F 12 (x) = π n {10 sin (π y 1) + ∑ i = 1 n - 1 (y i - 1) 2 [1 + 10 si n 2 (π y i + 1)] + (y n - 1) 2} + ∑ i = 1 n u (x i, 10,100,4) y i = 1 + x i + 1 4, u (x i, a, k, m) = k (x i - a) m, x i > a 0, - a < x i < a k (- x i - a) m, x i < - a$	[-50,50]	30	0
$F 13 (x) = 0.1 {si n 2 (3 π x 1) + ∑ i = 1 n (x i - 1) 2 [1 + si n 2 (3 π x i + 1)] + (x n - 1) 2 [1 + si n 2 (2 π x n)]} + ∑ i = 1 n u (x i, 5,100,4)$	[-50,50]	30	0

表8 多峰测试函数

Table 8 Multimodal peak test function

Test function	Range	Dimension	$f min$
$F 8 (x) = ∑ i = 1 n - x i sin \| x i \|$	[-500,500]	30	-418.983 9n
$F 9 (x) = ∑ i = 1 n [x i 2 - 10 cos (2 π x i) + 10]$	[-5.12,5.12]	30	0
$F 10 (x) = - 20 exp - 0.2 1 n ∑ i = 1 n x i 2 - exp 1 n ∑ i = 1 n cos (2 π x i) + 20 + e$	[-32,32]	30	0
$F 11 (x) = 14000 ∑ i = 1 n x i 2 - ∏ i = 1 n cos x i i + 1$	[-600,600]	30	0
$F 12 (x) = π n {10 sin (π y 1) + ∑ i = 1 n - 1 (y i - 1) 2 [1 + 10 si n 2 (π y i + 1)] + (y n - 1) 2} + ∑ i = 1 n u (x i, 10,100,4) y i = 1 + x i + 1 4, u (x i, a, k, m) = k (x i - a) m, x i > a 0, - a < x i < a k (- x i - a) m, x i < - a$	[-50,50]	30	0
$F 13 (x) = 0.1 {si n 2 (3 π x 1) + ∑ i = 1 n (x i - 1) 2 [1 + si n 2 (3 π x i + 1)] + (x n - 1) 2 [1 + si n 2 (2 π x n)]} + ∑ i = 1 n u (x i, 5,100,4)$	[-50,50]	30	0

表9 固定维度测试函数

Table 9 Fixed dimension test function

Test function	Range	Dimension	$f min$
$F 14 (x) = 1500 + ∑ j = 1 25 1 j + ∑ i = 1 2 (x i - a ij) 6 - 1$	[-65.536,65.536]	2	0.998
$F 15 (x) = ∑ i = 1 11 a i - x i (b i 2 + b i x 2) b i 2 + b i x 3 + x 4 2$	[-5,5]	4	0.000 307 5
$F_{16}(x)=4 x_{1}^{2}-2.1 x_{1}^{4}+\frac{1}{3} x_{1}^{6}+x_{1} x_{2}-4 x_{2}^{2}+4 x_{2}^{4}$	[-5,5]	2	1.031 63
$F_{17}(x)=[1+(x_{1}+x_{2}+1)^{2}(19-14 x_{1}+3 x_{1}^{2}-14x_{2}+6x_{1} x_{2}+3x_{2}^{2}]\times[30+(2 x_{1}-3 x_{2})^{2} \times(18-32 x_{1}+12 x_{1}^{2}+48 x_{2}-36 x_{1} x_{2}+27 x_{2}^{2})]$	[-2,2]	2	3
$F 18 (x) = - ∑ i = 1 4 c i exp - ∑ j = 1 3 a ij (x j - p ij) 2$	[1,3]	3	-3.86
$F 19 (x) = - ∑ i = 1 4 c i exp - ∑ j = 1 6 a ij (x j - p ij) 2$	[0,1]	6	-3.32
$F 20 (x) = - ∑ i = 1 5 [(X - a i) (X - a i) T + c i] - 1$	[0,10]	4	10.153 2
$F 21 (x) = - ∑ i = 1 10 [(X - a i) (X - a i) T + c i] - 1$	[0,10]	4	-10.536 3

表9 固定维度测试函数

Table 9 Fixed dimension test function

Test function	Range	Dimension	$f min$
$F 14 (x) = 1500 + ∑ j = 1 25 1 j + ∑ i = 1 2 (x i - a ij) 6 - 1$	[-65.536,65.536]	2	0.998
$F 15 (x) = ∑ i = 1 11 a i - x i (b i 2 + b i x 2) b i 2 + b i x 3 + x 4 2$	[-5,5]	4	0.000 307 5
$F_{16}(x)=4 x_{1}^{2}-2.1 x_{1}^{4}+\frac{1}{3} x_{1}^{6}+x_{1} x_{2}-4 x_{2}^{2}+4 x_{2}^{4}$	[-5,5]	2	1.031 63
$F_{17}(x)=[1+(x_{1}+x_{2}+1)^{2}(19-14 x_{1}+3 x_{1}^{2}-14x_{2}+6x_{1} x_{2}+3x_{2}^{2}]\times[30+(2 x_{1}-3 x_{2})^{2} \times(18-32 x_{1}+12 x_{1}^{2}+48 x_{2}-36 x_{1} x_{2}+27 x_{2}^{2})]$	[-2,2]	2	3
$F 18 (x) = - ∑ i = 1 4 c i exp - ∑ j = 1 3 a ij (x j - p ij) 2$	[1,3]	3	-3.86
$F 19 (x) = - ∑ i = 1 4 c i exp - ∑ j = 1 6 a ij (x j - p ij) 2$	[0,1]	6	-3.32
$F 20 (x) = - ∑ i = 1 5 [(X - a i) (X - a i) T + c i] - 1$	[0,10]	4	10.153 2
$F 21 (x) = - ∑ i = 1 10 [(X - a i) (X - a i) T + c i] - 1$	[0,10]	4	-10.536 3

表10 算法控制参数的初始值

Table 10 Initial value of control parameters

算法	参数	取值
BOA	$c$	0.01
	$P$	0.8
	$a$	0.1
	$b$	0.025
	$r$	[0,1]
MFO	$b$	1
	$N$	50
	$θ$	[-1,1]
SCA	$a$	2
	$r 2$	$[0,2 π]$
	$r 3$	[-2,2]
	$r 4$	[0,1]
GOA	$c max$	1
	$c min$	0.000 04
	$f$	0.5
	$δ$	2
HHO	$β$	1.5
	$r 1, r 2, r 3, r 4, r 5, l, m$	[0,1]
	$E 0$	[-1,1]
	$J$	[0,2]
SSA	$ST$	0.8
	$ε$	0.000 000 01
	$α$ , $β$	[0,1]
	$K$	[-1,1]

表10 算法控制参数的初始值

Table 10 Initial value of control parameters

算法	参数	取值
BOA	$c$	0.01
	$P$	0.8
	$a$	0.1
	$b$	0.025
	$r$	[0,1]
MFO	$b$	1
	$N$	50
	$θ$	[-1,1]
SCA	$a$	2
	$r 2$	$[0,2 π]$
	$r 3$	[-2,2]
	$r 4$	[0,1]
GOA	$c max$	1
	$c min$	0.000 04
	$f$	0.5
	$δ$	2
HHO	$β$	1.5
	$r 1, r 2, r 3, r 4, r 5, l, m$	[0,1]
	$E 0$	[-1,1]
	$J$	[0,2]
SSA	$ST$	0.8
	$ε$	0.000 000 01
	$α$ , $β$	[0,1]
	$K$	[-1,1]

表11 测试函数优化结果

Table 11 Test function optimization results

函数	统计值	BOA	MFO	SCA	GOA	HHO	SSA
$F 1$	mean	7.759 4E-10	1.333 3E+03	2.313 2E-02	7.386 4E+00	3.465 1E-190	5.249 2E-98
	std	2.079 6E-10	3.457 5E+03	6.346 4E-02	6.276 7E+00	0	2.875 1E-97
	best	3.381 8E-10	9.355 6E-06	2.732 5E-08	2.001 5E-01	5.246 5E-221	0
	worst	1.272 9E-09	1.000 5E+04	2.819 8E-01	2.531 7E+01	9.714 3E-189	1.574 8E-96
	rank	3	6	4	5	1	2
$F 2$	mean	7.198 1E-14	3.733 3E+01	1.891 5E-05	8.750 7E+00	6.735 6E-97	5.627 8E-46
	std	2.526 1E-13	2.391 6E+01	4.960 4E-05	1.315 8E+01	2.774 7E-96	3.081 7E-45
	best	4.662 3E-21	2.286 3E-04	8.203 3E-08	6.844 4E-01	7.970 9E-109	0
	worst	1.265 8E-12	1.016 4E+02	2.694 7E-04	7.516 0E+01	1.465 9E-95	1.687 9E-44
	rank	3	6	4	5	1	2
$F 3$	mean	5.925 3E-10	1.584 7E+04	5.455 9E+03	2.004 8E+03	2.930 2E-143	1.176 7E-16
	std	1.881 7E-10	1.268 7E+04	4.653 3E+03	8.752 8E+02	1.604 9E-142	5.729 0E-16
	best	3.113 4E-10	7.117 8E+02	1.062 4E+03	8.899 8E+02	7.769 4E-179	5.962 8E-207
	worst	1.044 4E-09	5.167 3E+04	1.751 9E+04	4.040 2E+03	8.790 3E-142	3.134 9E-15
	rank	3	6	5	4	1	2
$F 4$	mean	4.524 8E-07	6.997 0E+01	1.956 4E+01	1.079 4E+01	1.464 5E-93	1.048 2E-09
	std	7.906 9E-08	7.791 4E+00	1.201 5E+01	3.554 5E+00	4.262 1E-93	5.614 1E-09
	best	3.200 2E-07	5.384 4E+01	2.625 1E+00	4.023 7E+00	1.835 6E-108	2.972 2E-51
	worst	5.837 7E-07	8.465 4E+01	5.475 9E+01	1.789 4E+01	1.929 7E-92	3.077 1E-08
	rank	3	6	5	4	1	2
$F 5$	mean	2.889 2E+01	1.565 9E+04	5.024 9E+02	2.063 7E+03	4.307 4E-03	1.974 4E-04
	std	2.424 6E-02	3.384 9E+04	1.533 8E+03	3.997 6E+03	5.047 3E-03	4.981 4E-04
	best	2.884 7E+01	2.364 1E+01	2.888 2E+01	9.269 7E+01	4.239 2E-05	5.738 1E-16
	worst	2.893 3E+01	9.007 5E+04	8.305 5E+03	1.934 9E+04	2.002 9E-02	2.485 2E-03
	rank	3	6	4	5	2	1
$F 6$	mean	5.176 1E+00	1.010 0E+03	4.915 6E+00	7.836 3E+00	3.852 1E-05	5.726 9E-06
	std	6.071 1E-01	3.081 9E+03	1.565 0E+00	6.705 1E+00	5.333 5E-05	9.466 9E-06
	best	2.929 3E+00	1.648 2E-05	3.683 6E+00	4.169 2E-01	2.216 3E-07	2.490 8E-09
	worst	6.040 3E+00	1.010 0E+04	1.290 8E+01	2.367 1E+01	2.620 7E-04	3.900 1E-05
	rank	4	6	3	5	2	1
$F 7$	mean	9.065 1E-04	3.030 7E+00	2.688 9E-02	1.758 6E-03	6.458 4E-05	2.128 9E-04
	std	3.455 4E-04	8.367 4E+00	2.637 0E-02	7.618 8E-03	5.537 4E-05	1.398 8E-04
	best	2.514 8E-04	4.833 0E-02	8.848 7E-04	6.585 6E-03	5.479 3E-06	1.269 1E-05
	worst	1.998 4E-03	4.582 4E+01	1.299 1E-01	4.015 2E-02	2.573 9E-04	5.248 6E-04
	rank	3	6	5	4	1	2
$F 8$	mean	-3.863 8E+03	-8.713 3E+03	-3.969 2E+03	-7.356 0E+03	-1.256 2E+04	-1.105 5E+04
	std	3.920 7E+02	8.097 8E+02	-3.278 0E+02	7.761 4E+02	4.012 4E+01	2.219 8E+03
	best	-3.570 2E+03	-1.029 79E+04	-4.718 6E+03	-8.943 8E+03	-1.256 9E+04	-1.256 9E+04
	worst	-4.719 9E+03	-7.003 5E+03	-3.426 1E+03	-5.498 1E+03	-1.234 9E+04	-5.396 0E+03
	rank	6	3	5	4	1	2
$F 9$	mean	0	1.581 4E+02	9.808 9E+00	1.004 3E+02	0	0
	std	0	3.304 3E+01	1.418 8E+01	2.736 2E+01	0	0
	best	0	8.165 8E+01	9.412 1E-06	5.028 1E+01	0	0
	worst	0	2.173 3E+02	4.901 9E+01	1.688 7E+02	0	0
	rank	1	4	2	3	1	1
$F 10$	mean	3.152 5E-07	1.343 6E+01	1.182 9E+01	4.224 4E+00	8.881 8E-16	1.598 7E-15
	std	4.968 1E-08	8.206 9E+00	9.721 5E+00	7.980 2E-01	0	2.704 1E-15
	best	2.181 5E-07	2.154 5E-03	1.702 3E-05	2.677 7E+00	8.881 8E-16	8.881 8E-16
	worst	4.044 2E-07	1.996 2E+01	2.034 6E+01	6.244 5E+00	8.881 8E-16	1.509 9E-14
	rank	3	6	5	4	1	2
$F 11$	mean	8.209 5E-11	2.708 6E+01	2.540 9E-01	7.240 9E-01	0	0
	std	2.502 6E-10	5.383 2E+01	2.668 4E-01	2.135 0E-01	0	0
	best	6.783 5E-14	4.076 9E-05	5.486 2E-05	3.241 5E-01	0	0
	worst	1.247 5E-09	1.809 6E+02	7.689 9E-01	1.025 7E+00	0	0
	rank	4	5	2	3	1	1
$F 12$	mean	5.538 0E-01	8.533 3E+06	6.151 6E+00	7.463 8E+00	2.766 1E-06	1.146 5E-07
	std	1.526 3E-01	4.673 9E+07	1.735 2E+01	3.624 9E+00	3.580 5E-06	3.651 4E-07
	best	2.846 3E-01	7.291 3E-06	5.366 3E-01	3.207 2E+00	1.436 4E-10	8.462 4E-12
	worst	9.085 4E-01	2.560 0E+08	8.264 9E+01	1.674 9E+01	1.355 8E-05	2.003 8E-06
	rank	3	6	4	5	2	1
$F 13$	mean	2.859 4E+00	1.366 9E+07	9.143 6E+02	2.399 4E+01	1.731 9E-05	1.372 5E-06
	std	2.053 0E-01	7.486 9E+07	4.790 7E+03	1.388 3E+01	1.854 1E-05	2.134 8E-06
	best	2.387 3E+00	2.381 7E-04	2.103 1E+00	2.000 6E+00	4.585 1E-08	9.155 3E-12
	worst	2.992 3E+00	4.100 6E+08	2.626 1E+04	4.794 1E+01	6.039 1E-05	9.505 8E-06
	rank	3	6	5	4	2	1
$F 14$	mean	1.068 1E+00	2.084 9E+00	1.527 2E+00	9.981 0E-01	1.097 4E+00	3.831 5E+00
	std	2.516 0E-01	1.798 7E+00	8.923 6E-01	3.987 4E-16	3.033 1E-01	4.410 4E+00
	best	9.982 1E-01	9.980 1E+00	9.980 2E-01	9.980 2E-01	9.980 2E-01	9.980 2E-01
	worst	1.993 6E+00	7.874 2E+00	2.982 1E+00	9.981 0E-01	1.992 0E+00	1.267 1E+01
	rank	2	5	4	1	3	6
$F 15$	mean	4.343 1E-04	1.099 0E-03	9.958 1E-04	5.987 1E-03	3.265 1E-04	3.125 9E-04
	std	7.742 8E-05	3.880 2E-04	3.718 9E-04	8.663 5E-03	2.525 4E-05	1.666 3E-05
	best	3.254 5E-04	5.202 8E-04	3.410 8E-04	3.079 6E-04	3.079 6E-04	3.074 9E-04
	worst	6.232 6E-04	1.655 4E-03	1.528 3E-03	2.036 3E-02	4.145 9E-04	3.979 6E-04
	rank	4	5	3	6	2	1
$F 16$	mean	-4.501 2E+18	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00
	std	2.411 3E+19	6.775 2E-26	2.687 9E-07	1.749 3E-14	5.826 4E-11	4.701 2E-16
	best	1.782 7E+04	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00
	worst	-1.321 4E+20	-1.031 6E+00	-1.031 5E+00	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00
	rank	6	2	5	3	4	1
$F 17$	mean	3.139 6E+00	3	3	1.111 1E+01	3	1.020 4E+01
	std	2.394 1E-01	1.643 1E-15	4.120 5E-09	2.471 5E+01	7.791 6E-09	1.214 4E+01
	best	3.000 4E+00	3	3	3	3	3
	worst	4.024 7E+00	3	3	8.445 1E+01	3	3.001 4E+01
	rank	4	1	3	6	2	5
$F 18$	mean	-3.811 1E+00	-3.862 8E+00	-3.855 0E+00	-3.746 5E+00	-3.861 2E+00	-3.785 5E+00
	std	1.961 4E-01	2.710 1E-15	2.102 8E-03	2.449 4E-01	2.393 5E-03	2.358 7E-01
	best	-3.105 9E+00	-3.862 8E+00	-3.861 6E+00	-3.862 8E+00	-3.862 8E+00	-3.862 8E+00
	worst	-4.049 2E+00	-3.862 8E+00	-3.852 5E+00	-3.089 7E+00	-3.855 7E+00	-3.089 8E+00
	rank	4	1	3	6	2	5
$F 19$	mean	-2.587 6E+00	-3.237 8E+00	-2.918 1E+00	-3.277 4E+00	-3.150 8E+00	-3.278 2E+00
	std	3.874 3E-01	6.863 1E-02	2.280 6E-01	5.974 1E-02	1.010 8E-01	5.860 7E-02
	best	-1.951 4E+00	-3.322 0E+00	-3.250 7E+00	-3.322 3E+00	-3.317 1E+00	-3.322 0E+00
	worst	-3.144 7E+00	-3.137 6E+00	-2.213 8E+00	-3.194 5E+00	-2.956 9E+00	-3.198 8E+00
	rank	6	3	5	2	4	1
$F 20$	mean	-4.650 5E+00	-5.887 1E+00	-2.272 5E+00	-4.799 4E+00	-5.218 3E+00	-8.453 9E+00
	std	2.617 2E-01	3.226 0E+00	1.898 6E+00	2.680 1E+00	9.000 3E-01	2.444 3E+00
	best	-4.956 3E+00	-1.015 3E+01	-4.993 9E+00	-1.015 3E+01	-9.983 7E+00	-1.015 3E+01
	worst	-3.833 4E+00	-2.630 5E+00	4.972 6E-01	-2.630 5E+00	-5.049 4E+00	-5.055 2E+00
	rank	5	2	6	4	3	1
$F 21$	mean	-4.658 8E+00	-7.513 2E+00	-4.329 1E+00	-4.159 9E+00	-5.302 1E+00	-9.094 3E+00
	std	2.256 3E-01	3.567 2E+00	1.951 4E+00	2.979 9E+00	9.556 6E-01	2.432 4E+00
	best	-4.934 2E+00	-1.053 6E+01	-8.491 8E+00	-1.053 6E+01	-1.036 2E+01	-1.053 6E+01
	worst	-4.029 4E+00	-2.427 3E+00	-9.453 6E-01	-2.421 7E+00	-5.125 9E+00	-5.128 5E+00
	rank	4	2	5	6	3	1
Mean rank		3.67	4.43	4.14	4.24	1.90	1.95

表11 测试函数优化结果

Table 11 Test function optimization results

函数	统计值	BOA	MFO	SCA	GOA	HHO	SSA
$F 1$	mean	7.759 4E-10	1.333 3E+03	2.313 2E-02	7.386 4E+00	3.465 1E-190	5.249 2E-98
	std	2.079 6E-10	3.457 5E+03	6.346 4E-02	6.276 7E+00	0	2.875 1E-97
	best	3.381 8E-10	9.355 6E-06	2.732 5E-08	2.001 5E-01	5.246 5E-221	0
	worst	1.272 9E-09	1.000 5E+04	2.819 8E-01	2.531 7E+01	9.714 3E-189	1.574 8E-96
	rank	3	6	4	5	1	2
$F 2$	mean	7.198 1E-14	3.733 3E+01	1.891 5E-05	8.750 7E+00	6.735 6E-97	5.627 8E-46
	std	2.526 1E-13	2.391 6E+01	4.960 4E-05	1.315 8E+01	2.774 7E-96	3.081 7E-45
	best	4.662 3E-21	2.286 3E-04	8.203 3E-08	6.844 4E-01	7.970 9E-109	0
	worst	1.265 8E-12	1.016 4E+02	2.694 7E-04	7.516 0E+01	1.465 9E-95	1.687 9E-44
	rank	3	6	4	5	1	2
$F 3$	mean	5.925 3E-10	1.584 7E+04	5.455 9E+03	2.004 8E+03	2.930 2E-143	1.176 7E-16
	std	1.881 7E-10	1.268 7E+04	4.653 3E+03	8.752 8E+02	1.604 9E-142	5.729 0E-16
	best	3.113 4E-10	7.117 8E+02	1.062 4E+03	8.899 8E+02	7.769 4E-179	5.962 8E-207
	worst	1.044 4E-09	5.167 3E+04	1.751 9E+04	4.040 2E+03	8.790 3E-142	3.134 9E-15
	rank	3	6	5	4	1	2
$F 4$	mean	4.524 8E-07	6.997 0E+01	1.956 4E+01	1.079 4E+01	1.464 5E-93	1.048 2E-09
	std	7.906 9E-08	7.791 4E+00	1.201 5E+01	3.554 5E+00	4.262 1E-93	5.614 1E-09
	best	3.200 2E-07	5.384 4E+01	2.625 1E+00	4.023 7E+00	1.835 6E-108	2.972 2E-51
	worst	5.837 7E-07	8.465 4E+01	5.475 9E+01	1.789 4E+01	1.929 7E-92	3.077 1E-08
	rank	3	6	5	4	1	2
$F 5$	mean	2.889 2E+01	1.565 9E+04	5.024 9E+02	2.063 7E+03	4.307 4E-03	1.974 4E-04
	std	2.424 6E-02	3.384 9E+04	1.533 8E+03	3.997 6E+03	5.047 3E-03	4.981 4E-04
	best	2.884 7E+01	2.364 1E+01	2.888 2E+01	9.269 7E+01	4.239 2E-05	5.738 1E-16
	worst	2.893 3E+01	9.007 5E+04	8.305 5E+03	1.934 9E+04	2.002 9E-02	2.485 2E-03
	rank	3	6	4	5	2	1
$F 6$	mean	5.176 1E+00	1.010 0E+03	4.915 6E+00	7.836 3E+00	3.852 1E-05	5.726 9E-06
	std	6.071 1E-01	3.081 9E+03	1.565 0E+00	6.705 1E+00	5.333 5E-05	9.466 9E-06
	best	2.929 3E+00	1.648 2E-05	3.683 6E+00	4.169 2E-01	2.216 3E-07	2.490 8E-09
	worst	6.040 3E+00	1.010 0E+04	1.290 8E+01	2.367 1E+01	2.620 7E-04	3.900 1E-05
	rank	4	6	3	5	2	1
$F 7$	mean	9.065 1E-04	3.030 7E+00	2.688 9E-02	1.758 6E-03	6.458 4E-05	2.128 9E-04
	std	3.455 4E-04	8.367 4E+00	2.637 0E-02	7.618 8E-03	5.537 4E-05	1.398 8E-04
	best	2.514 8E-04	4.833 0E-02	8.848 7E-04	6.585 6E-03	5.479 3E-06	1.269 1E-05
	worst	1.998 4E-03	4.582 4E+01	1.299 1E-01	4.015 2E-02	2.573 9E-04	5.248 6E-04
	rank	3	6	5	4	1	2
$F 8$	mean	-3.863 8E+03	-8.713 3E+03	-3.969 2E+03	-7.356 0E+03	-1.256 2E+04	-1.105 5E+04
	std	3.920 7E+02	8.097 8E+02	-3.278 0E+02	7.761 4E+02	4.012 4E+01	2.219 8E+03
	best	-3.570 2E+03	-1.029 79E+04	-4.718 6E+03	-8.943 8E+03	-1.256 9E+04	-1.256 9E+04
	worst	-4.719 9E+03	-7.003 5E+03	-3.426 1E+03	-5.498 1E+03	-1.234 9E+04	-5.396 0E+03
	rank	6	3	5	4	1	2
$F 9$	mean	0	1.581 4E+02	9.808 9E+00	1.004 3E+02	0	0
	std	0	3.304 3E+01	1.418 8E+01	2.736 2E+01	0	0
	best	0	8.165 8E+01	9.412 1E-06	5.028 1E+01	0	0
	worst	0	2.173 3E+02	4.901 9E+01	1.688 7E+02	0	0
	rank	1	4	2	3	1	1
$F 10$	mean	3.152 5E-07	1.343 6E+01	1.182 9E+01	4.224 4E+00	8.881 8E-16	1.598 7E-15
	std	4.968 1E-08	8.206 9E+00	9.721 5E+00	7.980 2E-01	0	2.704 1E-15
	best	2.181 5E-07	2.154 5E-03	1.702 3E-05	2.677 7E+00	8.881 8E-16	8.881 8E-16
	worst	4.044 2E-07	1.996 2E+01	2.034 6E+01	6.244 5E+00	8.881 8E-16	1.509 9E-14
	rank	3	6	5	4	1	2
$F 11$	mean	8.209 5E-11	2.708 6E+01	2.540 9E-01	7.240 9E-01	0	0
	std	2.502 6E-10	5.383 2E+01	2.668 4E-01	2.135 0E-01	0	0
	best	6.783 5E-14	4.076 9E-05	5.486 2E-05	3.241 5E-01	0	0
	worst	1.247 5E-09	1.809 6E+02	7.689 9E-01	1.025 7E+00	0	0
	rank	4	5	2	3	1	1
$F 12$	mean	5.538 0E-01	8.533 3E+06	6.151 6E+00	7.463 8E+00	2.766 1E-06	1.146 5E-07
	std	1.526 3E-01	4.673 9E+07	1.735 2E+01	3.624 9E+00	3.580 5E-06	3.651 4E-07
	best	2.846 3E-01	7.291 3E-06	5.366 3E-01	3.207 2E+00	1.436 4E-10	8.462 4E-12
	worst	9.085 4E-01	2.560 0E+08	8.264 9E+01	1.674 9E+01	1.355 8E-05	2.003 8E-06
	rank	3	6	4	5	2	1
$F 13$	mean	2.859 4E+00	1.366 9E+07	9.143 6E+02	2.399 4E+01	1.731 9E-05	1.372 5E-06
	std	2.053 0E-01	7.486 9E+07	4.790 7E+03	1.388 3E+01	1.854 1E-05	2.134 8E-06
	best	2.387 3E+00	2.381 7E-04	2.103 1E+00	2.000 6E+00	4.585 1E-08	9.155 3E-12
	worst	2.992 3E+00	4.100 6E+08	2.626 1E+04	4.794 1E+01	6.039 1E-05	9.505 8E-06
	rank	3	6	5	4	2	1
$F 14$	mean	1.068 1E+00	2.084 9E+00	1.527 2E+00	9.981 0E-01	1.097 4E+00	3.831 5E+00
	std	2.516 0E-01	1.798 7E+00	8.923 6E-01	3.987 4E-16	3.033 1E-01	4.410 4E+00
	best	9.982 1E-01	9.980 1E+00	9.980 2E-01	9.980 2E-01	9.980 2E-01	9.980 2E-01
	worst	1.993 6E+00	7.874 2E+00	2.982 1E+00	9.981 0E-01	1.992 0E+00	1.267 1E+01
	rank	2	5	4	1	3	6
$F 15$	mean	4.343 1E-04	1.099 0E-03	9.958 1E-04	5.987 1E-03	3.265 1E-04	3.125 9E-04
	std	7.742 8E-05	3.880 2E-04	3.718 9E-04	8.663 5E-03	2.525 4E-05	1.666 3E-05
	best	3.254 5E-04	5.202 8E-04	3.410 8E-04	3.079 6E-04	3.079 6E-04	3.074 9E-04
	worst	6.232 6E-04	1.655 4E-03	1.528 3E-03	2.036 3E-02	4.145 9E-04	3.979 6E-04
	rank	4	5	3	6	2	1
$F 16$	mean	-4.501 2E+18	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00
	std	2.411 3E+19	6.775 2E-26	2.687 9E-07	1.749 3E-14	5.826 4E-11	4.701 2E-16
	best	1.782 7E+04	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00
	worst	-1.321 4E+20	-1.031 6E+00	-1.031 5E+00	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00
	rank	6	2	5	3	4	1
$F 17$	mean	3.139 6E+00	3	3	1.111 1E+01	3	1.020 4E+01
	std	2.394 1E-01	1.643 1E-15	4.120 5E-09	2.471 5E+01	7.791 6E-09	1.214 4E+01
	best	3.000 4E+00	3	3	3	3	3
	worst	4.024 7E+00	3	3	8.445 1E+01	3	3.001 4E+01
	rank	4	1	3	6	2	5
$F 18$	mean	-3.811 1E+00	-3.862 8E+00	-3.855 0E+00	-3.746 5E+00	-3.861 2E+00	-3.785 5E+00
	std	1.961 4E-01	2.710 1E-15	2.102 8E-03	2.449 4E-01	2.393 5E-03	2.358 7E-01
	best	-3.105 9E+00	-3.862 8E+00	-3.861 6E+00	-3.862 8E+00	-3.862 8E+00	-3.862 8E+00
	worst	-4.049 2E+00	-3.862 8E+00	-3.852 5E+00	-3.089 7E+00	-3.855 7E+00	-3.089 8E+00
	rank	4	1	3	6	2	5
$F 19$	mean	-2.587 6E+00	-3.237 8E+00	-2.918 1E+00	-3.277 4E+00	-3.150 8E+00	-3.278 2E+00
	std	3.874 3E-01	6.863 1E-02	2.280 6E-01	5.974 1E-02	1.010 8E-01	5.860 7E-02
	best	-1.951 4E+00	-3.322 0E+00	-3.250 7E+00	-3.322 3E+00	-3.317 1E+00	-3.322 0E+00
	worst	-3.144 7E+00	-3.137 6E+00	-2.213 8E+00	-3.194 5E+00	-2.956 9E+00	-3.198 8E+00
	rank	6	3	5	2	4	1
$F 20$	mean	-4.650 5E+00	-5.887 1E+00	-2.272 5E+00	-4.799 4E+00	-5.218 3E+00	-8.453 9E+00
	std	2.617 2E-01	3.226 0E+00	1.898 6E+00	2.680 1E+00	9.000 3E-01	2.444 3E+00
	best	-4.956 3E+00	-1.015 3E+01	-4.993 9E+00	-1.015 3E+01	-9.983 7E+00	-1.015 3E+01
	worst	-3.833 4E+00	-2.630 5E+00	4.972 6E-01	-2.630 5E+00	-5.049 4E+00	-5.055 2E+00
	rank	5	2	6	4	3	1
$F 21$	mean	-4.658 8E+00	-7.513 2E+00	-4.329 1E+00	-4.159 9E+00	-5.302 1E+00	-9.094 3E+00
	std	2.256 3E-01	3.567 2E+00	1.951 4E+00	2.979 9E+00	9.556 6E-01	2.432 4E+00
	best	-4.934 2E+00	-1.053 6E+01	-8.491 8E+00	-1.053 6E+01	-1.036 2E+01	-1.053 6E+01
	worst	-4.029 4E+00	-2.427 3E+00	-9.453 6E-01	-2.421 7E+00	-5.125 9E+00	-5.128 5E+00
	rank	4	2	5	6	3	1
Mean rank		3.67	4.43	4.14	4.24	1.90	1.95

图1 各算法在测试函数上收敛曲线对比

Fig.1 Comparison of convergence curves of each algorithm on test functions

表12 各算法的执行时间

Table 12 Execution time of each algorithm s

函数	BOA	MFO	SCA	GOA	HHO	SSA
$F 1$	0.452 1	0.289 1	0.265 6	132.030 7	0.802 6	0.787 5
$F 2$	0.443 2	0.304 2	0.273 4	129.928 6	0.479 2	0.883 9
$F 3$	2.514 1	1.295 8	1.294 3	136.860 9	2.918 8	1.861 5
$F 4$	0.501 6	0.318 2	0.309 9	148.181 3	0.623 9	0.893 7
$F 5$	0.777 1	0.393 2	0.364 1	146.655 2	0.910 4	0.947 4
$F 6$	0.427 1	0.278 7	0.238 5	144.043 2	0.603 6	0.800 4
$F 7$	0.738 0	0.435 4	0.420 8	148.610 4	0.899 5	1.030 7
$F 8$	0.725 4	0.335 9	0.301 6	127.057 3	0.720 8	0.947 4
$F 9$	0.773 9	0.354 2	0.340 6	142.702 6	0.807 8	0.916 2
$F 10$	0.737 5	0.405 7	0.404 2	147.189 1	0.908 3	1.016 1
$F 11$	0.703 7	0.378 1	0.352 1	144.982 3	0.899 4	0.963 5
$F 12$	1.519 3	0.785 9	0.750 5	149.763 5	1.827 1	1.362 5
$F 13$	1.464 1	0.778 7	0.736 9	149.042 7	1.733 9	1.317 7
$F 14$	3.414 1	1.626 3	1.623 4	11.192 7	4.121 9	1.705 2
$F 15$	0.571 9	0.221 3	0.202 1	18.654 2	0.714 6	0.370 3
$F 16$	0.467 7	0.159 9	0.150 5	9.525 4	0.575 5	0.270 8
$F 17$	0.445 3	0.151 6	0.156 8	9.895 8	0.603 1	0.279 2
$F 18$	0.914 1	0.301 6	0.285 4	19.445 3	0.913 0	0.433 3
$F 19$	0.963 0	0.309 9	0.295 3	28.769 3	0.951 6	0.482 8
$F 20$	1.061 5	0.401 0	0.388 5	19.716 1	1.152 1	0.557 8
$F 21$	1.554 2	0.568 2	0.541 2	19.551 6	1.542 2	0.691 7

表12 各算法的执行时间

Table 12 Execution time of each algorithm s

函数	BOA	MFO	SCA	GOA	HHO	SSA
$F 1$	0.452 1	0.289 1	0.265 6	132.030 7	0.802 6	0.787 5
$F 2$	0.443 2	0.304 2	0.273 4	129.928 6	0.479 2	0.883 9
$F 3$	2.514 1	1.295 8	1.294 3	136.860 9	2.918 8	1.861 5
$F 4$	0.501 6	0.318 2	0.309 9	148.181 3	0.623 9	0.893 7
$F 5$	0.777 1	0.393 2	0.364 1	146.655 2	0.910 4	0.947 4
$F 6$	0.427 1	0.278 7	0.238 5	144.043 2	0.603 6	0.800 4
$F 7$	0.738 0	0.435 4	0.420 8	148.610 4	0.899 5	1.030 7
$F 8$	0.725 4	0.335 9	0.301 6	127.057 3	0.720 8	0.947 4
$F 9$	0.773 9	0.354 2	0.340 6	142.702 6	0.807 8	0.916 2
$F 10$	0.737 5	0.405 7	0.404 2	147.189 1	0.908 3	1.016 1
$F 11$	0.703 7	0.378 1	0.352 1	144.982 3	0.899 4	0.963 5
$F 12$	1.519 3	0.785 9	0.750 5	149.763 5	1.827 1	1.362 5
$F 13$	1.464 1	0.778 7	0.736 9	149.042 7	1.733 9	1.317 7
$F 14$	3.414 1	1.626 3	1.623 4	11.192 7	4.121 9	1.705 2
$F 15$	0.571 9	0.221 3	0.202 1	18.654 2	0.714 6	0.370 3
$F 16$	0.467 7	0.159 9	0.150 5	9.525 4	0.575 5	0.270 8
$F 17$	0.445 3	0.151 6	0.156 8	9.895 8	0.603 1	0.279 2
$F 18$	0.914 1	0.301 6	0.285 4	19.445 3	0.913 0	0.433 3
$F 19$	0.963 0	0.309 9	0.295 3	28.769 3	0.951 6	0.482 8
$F 20$	1.061 5	0.401 0	0.388 5	19.716 1	1.152 1	0.557 8
$F 21$	1.554 2	0.568 2	0.541 2	19.551 6	1.542 2	0.691 7

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若干新型智能优化算法对比分析研究

Analysis and Research of Several New Intelligent Optimization Algorithms

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