Hybrid Algorithm of Slime Mould Algorithm and Arithmetic Optimization Algorithm Based on Random Opposition-Based Learning

doi:10.3778/j.issn.1673-9418.2105016

Journal of Frontiers of Computer Science and Technology ›› 2022, Vol. 16 ›› Issue (5): 1182-1192.DOI: 10.3778/j.issn.1673-9418.2105016

• Theory and Algorithm • Previous Articles

Hybrid Algorithm of Slime Mould Algorithm and Arithmetic Optimization Algorithm Based on Random Opposition-Based Learning

JIA Heming¹^,⁺(), LIU Yuxiang², LIU Qingxin³, WANG Shuang¹, ZHENG Rong¹

1. Department of Information Engineering, Sanming University, Sanming, Fujian 365004, China
2. College of Physics and Information Engineering, Fuzhou University, Fuzhou 350108, China
3. School of Computer Science and Technology, Hainan University, Haikou 570228, China

Received:2021-05-07 Revised:2021-07-09 Online:2022-05-01 Published:2022-05-19
About author:JIA Heming, born in 1983, Ph.D., professor, M.S. supervisor. His research interests include swarm intelligence optimization algorithm, feature selection, etc.
LIU Yuxiang, born in 1999, M.S. candidate. Her research interest is swarm intelligence optimization algorithm.
LIU Qingxin, born in 1997, M.S. candidate. His research interest is swarm intelligence optimization algorithm.
WANG Shuang, born in 1992, Ph.D., associate professor. Her research interests include swarm intelligence optimization algorithm, feature selection, etc.
ZHENG Rong, born in 1992, Ph.D., associate professor. His research interests include swarm intelligence optimization algorithm, feature selection, etc.
Supported by:
General Project of Fujian Natural Science Foundation(2021J011128);Research Project on Education and Teaching Reform of Undergraduate Colleges and Universities in Fujian Province(FBJG20210338);Guiding Project of Science and Technology Plan of Sanming City(2020-G-61);Guiding Project of Science and Technology Plan of Sanming City(2021-S-8);Project of Funding Support for Introducing High-Level Talents Science and Research of Sanming University(20YG14);Open Research Fund of Key Laboratory of Agricultural Internet of Things in Fujian Province(ZD2101);Key Project of Education and Teaching Reform of Sanming University(J2010305);Higher Education Research Project of Sanming University(SHE2013)

融合随机反向学习的黏菌与算术混合优化算法

贾鹤鸣¹^,⁺(), 刘宇翔², 刘庆鑫³, 王爽¹, 郑荣¹

1.三明学院信息工程学院,福建三明 365004
2.福州大学物理与信息工程学院,福州 350108
3.海南大学计算机科学与技术学院,海口 570228

通讯作者: + E-mail: jiaheminglucky99@126.com
作者简介:贾鹤鸣（1983—）,男,辽宁辽阳人,博士,教授,硕士生导师,主要研究方向为群体智能优化算法、特征选择等。
刘宇翔（1999—）,女,福建宁德人,硕士研究生,主要研究方向为群体智能优化算法。
刘庆鑫（1997—）,男,福建福州人,硕士研究生,主要研究方向为群体智能优化算法。
王爽（1992—）,女,山东邹城人,博士,副教授,主要研究方向为群体智能优化算法、特征选择等。
郑荣（1992—）,男,江西上饶人,博士,副教授,主要研究方向为群体智能优化算法、特征选择等。
基金资助:
福建省自然科学基金面上项目(2021J011128);福建省本科高校教育教学改革研究项目(FBJG20210338);三明市科技计划引导性项目(2020-G-61);三明市科技计划引导性项目(2021-S-8);三明学院引进高层次人才科研启动经费支持项目(20YG14);福建省农业物联网应用重点实验室开放研究基金(ZD2101);三明学院教育教学改革重点项目(J2010305);三明学院高教研究课题(SHE2013)

Abstract

Abstract:

Slime mould algorithm (SMA) and arithmetic optimization algorithm (AOA) are new meta-heuristic optimization algorithms proposed recently. SMA has strong ability of global exploration, but the oscillation effect is weak in the late iteration. It is easy to fall into local optimum, and the contraction mechanism is not strong, which leads to slow convergence speed. AOA algorithm uses multiplication and division operator to update position, which has strong randomness and good ability to avoid premature convergence. To solve the above problems, this paper combines the two algorithms and uses random opposition-based learning strategy to improve the convergence speed, and proposes a hybrid algorithm of slime mould algorithm and arithmetic optimization algorithm based on random opposition-based learning (HSMAAOA) with superior performance and high efficiency. The improved algorithm retains the SMA’s exploration phase and the exploitation phase will be replaced by the multiplication and division operators, which improves the capacity of the algorithm and the ability to jump out of the local optimal solution. In addition, random opposition-based learning strategy is used to enhance the diversity of the improved algorithm population and improve the convergence speed. The experimental results show that the HSMAAOA algorithm has good robustness and optimization accuracy, and significantly improves the convergence speed. Finally, the applicability and effectiveness of HSMAAOA in engineering problems are verified through the design of welded beams and the design of pressure vessels.

Key words: slime mould algorithm (SMA), arithmetic optimization algorithm (AOA), hybrid optimization, random opposition-based learning

摘要：

黏菌优化算法（SMA）和算术优化算法（AOA）是最近提出的新型元启发式优化算法。SMA算法具有较强的全局探索能力,但迭代后期振荡作用较弱,易陷入局部最优,且收缩机制不强,导致收敛速度慢。AOA算法利用乘除算子进行位置更新,随机性强,具有较好的避免早熟收敛能力。针对上述问题,将两种算法结合并利用随机反向学习策略提高收敛速度,提出一种性能优越且高效的融合随机反向学习策略的黏菌与算术混合优化算法（HSMAAOA）。改进算法保留了SMA全局探索部分位置更新公式,局部开发阶段将乘除算子替换SMA收缩机制,提高算法随机性与跳出局部极值的能力。此外,通过随机反向学习策略增强改进算法种群多样性,提高收敛速度。实验结果表明,HSMAAOA算法具有良好的鲁棒性以及寻优精度,且明显提升了收敛速度。最后,通过焊接梁设计问题与压力容器设计问题,验证了HSMAAOA在工程问题上的适用性与有效性。

关键词: 黏菌优化算法（SMA）, 算术优化算法（AOA）, 混合优化, 随机反向学习

CLC Number:

TP301.6

JIA Heming, LIU Yuxiang, LIU Qingxin, WANG Shuang, ZHENG Rong. Hybrid Algorithm of Slime Mould Algorithm and Arithmetic Optimization Algorithm Based on Random Opposition-Based Learning[J]. Journal of Frontiers of Computer Science and Technology, 2022, 16(5): 1182-1192.

贾鹤鸣, 刘宇翔, 刘庆鑫, 王爽, 郑荣. 融合随机反向学习的黏菌与算术混合优化算法[J]. 计算机科学与探索, 2022, 16(5): 1182-1192.

Figures/Tables 11

Fig.1 Curve of math optimizer probability

Fig.2 Arbitrary solution and its randomly opposite solution

Fig.3 Flowchart of HSMAAOA

Table 1 Setting of each algorithm parameters

算法	参数设置
HSMAAOA	$z = 0.03, μ = 0.499, α = 5$
SMA^[7]	$z = 0.03$
AOA^[8]	$α = 5, M O P_M a x = 1$ $M O P_M i n = 0.2, u = 0.499$
HHO^[11]	$s = 0.01, β = 1.5$
WOA^[10]	$a 1 = [2,0], a 2 = [- 2,1], b = 1$
SSA^[6]	$c 1 ∈ [0,1], c 2 ∈ [0,1]$
GWO^[9]	$a = [2,0]$
PSO^[5]	$c 1 = 2, c 2 = 2, v m a x = 6, w m a x = 0.9, w m i n = 0.2$

Table 1 Setting of each algorithm parameters

算法	参数设置
HSMAAOA	$z = 0.03, μ = 0.499, α = 5$
SMA^[7]	$z = 0.03$
AOA^[8]	$α = 5, M O P_M a x = 1$ $M O P_M i n = 0.2, u = 0.499$
HHO^[11]	$s = 0.01, β = 1.5$
WOA^[10]	$a 1 = [2,0], a 2 = [- 2,1], b = 1$
SSA^[6]	$c 1 ∈ [0,1], c 2 ∈ [0,1]$
GWO^[9]	$a = [2,0]$
PSO^[5]	$c 1 = 2, c 2 = 2, v m a x = 6, w m a x = 0.9, w m i n = 0.2$

Table 2 Test results of benchmark functions of each algorithm

函数	统计值	HSMAAOA	SMA	AOA	HHO	WOA	SSA	GWO	PSO
$F 1$	Mean	0.000 0E+00	9.991 0E-300	2.267 7E-33	1.185 4E-92	9.266 7E-72	3.129 1E-07	1.187 7E-27	1.980 0E-04
$F 1$	Std	0.000 0E+00	0.000 0E+00	1.190 1E-32	6.492 4E-92	1.730 2E-70	3.748 8E-07	1.387 4E-27	2.120 0E-04
$F 2$	Mean	0.000 0E+00	9.182 6E-147	0.000 0E+00	6.632 9E-48	1.494 3E-50	1.682 9E+00	1.091 8E-16	3.872 2E-02
$F 2$	Std	0.000 0E+00	5.031 2E-146	0.000 0E+00	3.631 6E-47	2.117 2E-50	1.227 7E+00	9.630 3E-17	5.943 1E-02
$F 3$	Mean	0.000 0E+00	2.798 5E-297	5.757 0E-03	1.210 0E-76	3.985 8E+04	1.341 3E+03	8.971 1E-06	7.230 4E+01
$F 3$	Std	0.000 0E+00	0.000 0E+00	1.113 3E-02	6.509 0E-76	1.225 0E+04	7.894 9E+02	2.060 7E-05	3.582 5E+01
$F 4$	Mean	0.000 0E+00	3.524 6E-140	2.332 4E-02	1.189 5E-48	5.185 2E+01	1.091 3E+01	9.885 0E-07	1.104 5E+00
$F 4$	Std	0.000 0E+00	1.930 5E-139	2.079 8E-02	5.501 4E-48	2.724 2E+01	3.792 3E+00	1.773 4E-06	2.212 2E-01
$F 5$	Mean	6.072 2E+00	3.570 9E+00	2.846 0E+01	8.819 0E-03	2.790 7E+01	1.622 5E+02	2.703 7E+01	1.176 6E+02
$F 5$	Std	1.117 5E+00	7.108 0E+00	3.732 2E-01	1.556 8E-02	4.748 1E-01	3.005 4E+02	0.763 4E-01	1.232 7E+02
$F 6$	Mean	1.706 0E-03	5.938 0E-03	3.214 1E+00	1.210 0E-04	4.464 7E-01	2.003 7E-07	7.729 6E-01	2.440 0E-04
$F 6$	Std	1.822 0E-03	3.795 0E-03	2.827 6E-01	2.430 0E-04	2.368 7E-01	3.004 8E-07	4.798 8E-01	5.660 0E-04
$F 7$	Mean	1.690 0E-04	6.380 0E-04	1.069 5E-02	1.701 0E-04	2.199 0E-03	5.356 7E-01	2.492 0E-03	1.963 4E-01
$F 7$	Std	1.440 0E-04	5.570 0E-04	7.722 0E-03	1.619 0E-04	3.086 0E-03	2.087 6E-01	1.279 0E-03	6.862 6E-02
$F 8$	Mean	-1.256 9E+04	-1.256 8E+04	-5.230 1E+03	-1.256 1E+04	-1.054 6E+04	-7.279 2E+03	-6.265 2E+03	-5.205 8E+03
$F 8$	Std	6.327 6E-08	4.078 5E-01	4.355 3E+02	3.956 6E+01	1.533 7E+03	5.726 2E+02	5.826 1E+02	1.537 0E+03
$F 9$	Mean	0.000 0E+00	0.000 0E+00	0.000 0E+00	0.000 0E+00	1.894 8E-15	5.551 9E+01	3.143 2E+00	5.887 2E+01
$F 9$	Std	0.000 0E+00	0.000 0E+00	0.000 0E+00	0.000 0E+00	1.037 8E-14	1.842 7E+01	4.459 8E+00	1.694 3E+01
$F 10$	Mean	8.881 8E-16	8.881 8E-16	8.881 8E-16	8.881 8E-16	4.204 0E-15	2.566 7E+00	1.020 2E-13	2.117 9E-01
$F 10$	Std	0.000 0E+00	0.000 0E+00	0.000 0E+00	0.000 0E+00	2.788 6E-15	7.233 4E-01	1.423 3E-14	4.654 7E-01
$F 11$	Mean	0.000 0E+00	0.000 0E+00	1.913 7E-01	0.000 0E+00	4.506 0E-03	1.337 3E-02	4.113 0E-03	1.003 0E-02
$F 11$	Std	0.000 0E+00	0.000 0E+00	1.289 2E-01	0.000 0E+00	2.468 3E-02	1.528 0E-02	8.584 0E-03	1.077 0E-02
$F 12$	Mean	1.464 0E-03	4.222 0E-03	5.419 7E-01	1.613 0E-05	2.260 9E-02	7.438 1E+00	5.379 4E-02	2.073 8E-02
$F 12$	Std	3.349 0E-03	5.098 0E-03	5.557 3E-02	3.383 2E-05	1.409 3E-02	2.851 8E+00	2.642 0E-02	4.218 1E-02
$F 13$	Mean	2.090 0E-03	1.401 2E-02	2.794 4E+00	1.140 0E-04	4.965 9E-01	1.4463 E+01	6.559 5E-01	5.550 0E-03
$F 13$	Std	4.881 0E-03	4.327 4E-02	7.369 3E-02	2.030 0E-04	3.199 1E-01	1.218 1E+01	1.906 8E-01	8.988 0E-03
$F 14$	Mean	9.980 0E-01	9.980 0E-01	9.579 7E+00	1.426 1E+00	2.900 6E+00	1.262 3E+00	5.687 3E+00	3.364 6E+00
$F 14$	Std	6.356 4E-15	8.818 7E-13	4.035 4E+00	1.260 7E+00	2.819 8E+00	7.763 6E-01	4.848 0E+00	2.516 1E+00
$F 15$	Mean	5.000 0E-04	5.480 0E-04	6.705 0E-03	3.950 0E-04	9.870 0E-04	2.620 0E-03	6.384 0E-03	8.570 0E-04
$F 15$	Std	1.980 0E-04	2.570 0E-04	8.571 0E-03	2.350 0E-04	7.000 0E-04	5.331 0E-03	9.308 0E-03	1.350 0E-04
$F 16$	Mean	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00
$F 16$	Std	4.021 4E-11	1.355 3E-09	1.143 0E-07	2.946 1E-09	8.146 1E-10	1.913 2E-14	2.162 0E-08	6.387 7E-06
$F 17$	Mean	3.978 9E-01	3.978 9E-01	3.978 9E-01	3.978 9E-01	3.978 9E-01	3.978 9E-01	3.978 9E-01	3.978 9E-01
$F 17$	Std	1.543 2E-09	1.186 9E-08	5.057 6E-08	3.120 9E-06	1.323 5E-05	1.093 0E-14	9.500 1E-07	0.000 0E+00
$F 18$	Mean	3.000 0E+00	3.000 0E+00	7.500 2E+00	3.000 0E+00	3.000 0E+00	3.000 0E+00	3.000 0E+00	3.000 0E+00
$F 18$	Std	1.695 8E-15	8.406 3E-10	1.023 4E+01	1.125 0E-06	1.010 0E-06	1.657 5E-13	5.153 5E-05	1.573 3E-13
$F 19$	Mean	-3.862 8E+00	-3.862 8E+00	-3.850 6E+00	-3.861 5E+00	-3.859 4E+00	-3.862 8E+00	-3.861 6E+00	-3.862 8E+00
$F 19$	Std	1.388 2E-06	2.188 4E-07	4.072 0E-03	1.964 0E-03	3.309 0E-03	6.239 3E-12	2.707 7E-03	2.611 7E-15
$F 20$	Mean	-3.261 5E+00	-3.246 2E+00	-3.059 9E+00	-3.079 2E+00	-3.257 5E+00	-3.218 2E+00	-3.279 3E+00	-3.262 5E+00
$F 20$	Std	6.049 2E-02	5.867 7E-02	8.954 0E-02	1.201 4E-01	1.133 0E-01	5.395 5E-02	6.930 3E-02	6.046 3E-02
$F 21$	Mean	-1.015 3E+01	-1.015 3E+01	-4.174 2E+00	-5.208 5E+00	-8.493 6E+00	-7.315 8E+00	-9.133 5E+00	-7.016 9E+00
$F 21$	Std	2.840 0E-04	2.930 0E-04	1.663 5E+00	8.554 7E-01	2.511 8E+00	3.392 7E+00	2.070 3E+00	3.130 8E+00
$F 22$	Mean	-1.040 3E+01	-1.040 2E+01	-3.791 3E+00	-5.408 2E+00	-7.420 1E+00	-8.201 6E+00	-1.040 1E+01	-8.312 9E+00
$F 22$	Std	3.390 0E-04	3.610 0E-04	1.634 5E+00	1.239 2E+00	3.095 6E+00	3.241 6E+00	8.170 0E-04	3.066 2E+00
$F 23$	Mean	-1.053 6E+01	-1.053 8E+01	-3.887 1E+00	-5.299 5E+00	-7.734 3E+00	-8.959 6E+00	-9.993 9E+00	-9.185 4E+00
$F 23$	Std	3.630 0E-04	4.940 0E-04	1.612 4E+00	9.579 4E-01	3.304 0E+00	2.966 7E+00	2.058 4E+00	2.787 9E+00

Table 2 Test results of benchmark functions of each algorithm

函数	统计值	HSMAAOA	SMA	AOA	HHO	WOA	SSA	GWO	PSO
$F 1$	Mean	0.000 0E+00	9.991 0E-300	2.267 7E-33	1.185 4E-92	9.266 7E-72	3.129 1E-07	1.187 7E-27	1.980 0E-04
$F 1$	Std	0.000 0E+00	0.000 0E+00	1.190 1E-32	6.492 4E-92	1.730 2E-70	3.748 8E-07	1.387 4E-27	2.120 0E-04
$F 2$	Mean	0.000 0E+00	9.182 6E-147	0.000 0E+00	6.632 9E-48	1.494 3E-50	1.682 9E+00	1.091 8E-16	3.872 2E-02
$F 2$	Std	0.000 0E+00	5.031 2E-146	0.000 0E+00	3.631 6E-47	2.117 2E-50	1.227 7E+00	9.630 3E-17	5.943 1E-02
$F 3$	Mean	0.000 0E+00	2.798 5E-297	5.757 0E-03	1.210 0E-76	3.985 8E+04	1.341 3E+03	8.971 1E-06	7.230 4E+01
$F 3$	Std	0.000 0E+00	0.000 0E+00	1.113 3E-02	6.509 0E-76	1.225 0E+04	7.894 9E+02	2.060 7E-05	3.582 5E+01
$F 4$	Mean	0.000 0E+00	3.524 6E-140	2.332 4E-02	1.189 5E-48	5.185 2E+01	1.091 3E+01	9.885 0E-07	1.104 5E+00
$F 4$	Std	0.000 0E+00	1.930 5E-139	2.079 8E-02	5.501 4E-48	2.724 2E+01	3.792 3E+00	1.773 4E-06	2.212 2E-01
$F 5$	Mean	6.072 2E+00	3.570 9E+00	2.846 0E+01	8.819 0E-03	2.790 7E+01	1.622 5E+02	2.703 7E+01	1.176 6E+02
$F 5$	Std	1.117 5E+00	7.108 0E+00	3.732 2E-01	1.556 8E-02	4.748 1E-01	3.005 4E+02	0.763 4E-01	1.232 7E+02
$F 6$	Mean	1.706 0E-03	5.938 0E-03	3.214 1E+00	1.210 0E-04	4.464 7E-01	2.003 7E-07	7.729 6E-01	2.440 0E-04
$F 6$	Std	1.822 0E-03	3.795 0E-03	2.827 6E-01	2.430 0E-04	2.368 7E-01	3.004 8E-07	4.798 8E-01	5.660 0E-04
$F 7$	Mean	1.690 0E-04	6.380 0E-04	1.069 5E-02	1.701 0E-04	2.199 0E-03	5.356 7E-01	2.492 0E-03	1.963 4E-01
$F 7$	Std	1.440 0E-04	5.570 0E-04	7.722 0E-03	1.619 0E-04	3.086 0E-03	2.087 6E-01	1.279 0E-03	6.862 6E-02
$F 8$	Mean	-1.256 9E+04	-1.256 8E+04	-5.230 1E+03	-1.256 1E+04	-1.054 6E+04	-7.279 2E+03	-6.265 2E+03	-5.205 8E+03
$F 8$	Std	6.327 6E-08	4.078 5E-01	4.355 3E+02	3.956 6E+01	1.533 7E+03	5.726 2E+02	5.826 1E+02	1.537 0E+03
$F 9$	Mean	0.000 0E+00	0.000 0E+00	0.000 0E+00	0.000 0E+00	1.894 8E-15	5.551 9E+01	3.143 2E+00	5.887 2E+01
$F 9$	Std	0.000 0E+00	0.000 0E+00	0.000 0E+00	0.000 0E+00	1.037 8E-14	1.842 7E+01	4.459 8E+00	1.694 3E+01
$F 10$	Mean	8.881 8E-16	8.881 8E-16	8.881 8E-16	8.881 8E-16	4.204 0E-15	2.566 7E+00	1.020 2E-13	2.117 9E-01
$F 10$	Std	0.000 0E+00	0.000 0E+00	0.000 0E+00	0.000 0E+00	2.788 6E-15	7.233 4E-01	1.423 3E-14	4.654 7E-01
$F 11$	Mean	0.000 0E+00	0.000 0E+00	1.913 7E-01	0.000 0E+00	4.506 0E-03	1.337 3E-02	4.113 0E-03	1.003 0E-02
$F 11$	Std	0.000 0E+00	0.000 0E+00	1.289 2E-01	0.000 0E+00	2.468 3E-02	1.528 0E-02	8.584 0E-03	1.077 0E-02
$F 12$	Mean	1.464 0E-03	4.222 0E-03	5.419 7E-01	1.613 0E-05	2.260 9E-02	7.438 1E+00	5.379 4E-02	2.073 8E-02
$F 12$	Std	3.349 0E-03	5.098 0E-03	5.557 3E-02	3.383 2E-05	1.409 3E-02	2.851 8E+00	2.642 0E-02	4.218 1E-02
$F 13$	Mean	2.090 0E-03	1.401 2E-02	2.794 4E+00	1.140 0E-04	4.965 9E-01	1.4463 E+01	6.559 5E-01	5.550 0E-03
$F 13$	Std	4.881 0E-03	4.327 4E-02	7.369 3E-02	2.030 0E-04	3.199 1E-01	1.218 1E+01	1.906 8E-01	8.988 0E-03
$F 14$	Mean	9.980 0E-01	9.980 0E-01	9.579 7E+00	1.426 1E+00	2.900 6E+00	1.262 3E+00	5.687 3E+00	3.364 6E+00
$F 14$	Std	6.356 4E-15	8.818 7E-13	4.035 4E+00	1.260 7E+00	2.819 8E+00	7.763 6E-01	4.848 0E+00	2.516 1E+00
$F 15$	Mean	5.000 0E-04	5.480 0E-04	6.705 0E-03	3.950 0E-04	9.870 0E-04	2.620 0E-03	6.384 0E-03	8.570 0E-04
$F 15$	Std	1.980 0E-04	2.570 0E-04	8.571 0E-03	2.350 0E-04	7.000 0E-04	5.331 0E-03	9.308 0E-03	1.350 0E-04
$F 16$	Mean	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00	-1.031 6E+00
$F 16$	Std	4.021 4E-11	1.355 3E-09	1.143 0E-07	2.946 1E-09	8.146 1E-10	1.913 2E-14	2.162 0E-08	6.387 7E-06
$F 17$	Mean	3.978 9E-01	3.978 9E-01	3.978 9E-01	3.978 9E-01	3.978 9E-01	3.978 9E-01	3.978 9E-01	3.978 9E-01
$F 17$	Std	1.543 2E-09	1.186 9E-08	5.057 6E-08	3.120 9E-06	1.323 5E-05	1.093 0E-14	9.500 1E-07	0.000 0E+00
$F 18$	Mean	3.000 0E+00	3.000 0E+00	7.500 2E+00	3.000 0E+00	3.000 0E+00	3.000 0E+00	3.000 0E+00	3.000 0E+00
$F 18$	Std	1.695 8E-15	8.406 3E-10	1.023 4E+01	1.125 0E-06	1.010 0E-06	1.657 5E-13	5.153 5E-05	1.573 3E-13
$F 19$	Mean	-3.862 8E+00	-3.862 8E+00	-3.850 6E+00	-3.861 5E+00	-3.859 4E+00	-3.862 8E+00	-3.861 6E+00	-3.862 8E+00
$F 19$	Std	1.388 2E-06	2.188 4E-07	4.072 0E-03	1.964 0E-03	3.309 0E-03	6.239 3E-12	2.707 7E-03	2.611 7E-15
$F 20$	Mean	-3.261 5E+00	-3.246 2E+00	-3.059 9E+00	-3.079 2E+00	-3.257 5E+00	-3.218 2E+00	-3.279 3E+00	-3.262 5E+00
$F 20$	Std	6.049 2E-02	5.867 7E-02	8.954 0E-02	1.201 4E-01	1.133 0E-01	5.395 5E-02	6.930 3E-02	6.046 3E-02
$F 21$	Mean	-1.015 3E+01	-1.015 3E+01	-4.174 2E+00	-5.208 5E+00	-8.493 6E+00	-7.315 8E+00	-9.133 5E+00	-7.016 9E+00
$F 21$	Std	2.840 0E-04	2.930 0E-04	1.663 5E+00	8.554 7E-01	2.511 8E+00	3.392 7E+00	2.070 3E+00	3.130 8E+00
$F 22$	Mean	-1.040 3E+01	-1.040 2E+01	-3.791 3E+00	-5.408 2E+00	-7.420 1E+00	-8.201 6E+00	-1.040 1E+01	-8.312 9E+00
$F 22$	Std	3.390 0E-04	3.610 0E-04	1.634 5E+00	1.239 2E+00	3.095 6E+00	3.241 6E+00	8.170 0E-04	3.066 2E+00
$F 23$	Mean	-1.053 6E+01	-1.053 8E+01	-3.887 1E+00	-5.299 5E+00	-7.734 3E+00	-8.959 6E+00	-9.993 9E+00	-9.185 4E+00
$F 23$	Std	3.630 0E-04	4.940 0E-04	1.612 4E+00	9.579 4E-01	3.304 0E+00	2.966 7E+00	2.058 4E+00	2.787 9E+00

Fig.4 Convergence curves of various algorithms

Table 3 Wilcoxon rank sum test results of each algorithm

函数	SMA	AOA	HHO	WOA	SSA	GWO	PSO
$F 1$	N/A	6.866 2E-07	6.866 2E-07	6.866 2E-07	6.866 2E-07	6.866 2E-07	6.866 2E-07
$F 2$	2.172 8E-05	2.491 5E-06	2.491 5E-06	2.491 5E-06	2.491 5E-06	2.491 5E-06	2.491 5E-06
$F 3$	3.506 5E-03	6.866 2E-07	6.866 2E-07	6.866 2E-07	6.866 2E-07	6.866 2E-07	6.866 2E-07
$F 4$	1.260 4E-06	1.260 4E-06	1.260 4E-06	1.260 4E-06	1.260 4E-06	1.260 4E-06	1.260 4E-06
$F 5$	8.034 6E-03	1.329 5E-05	9.073 4E-06	1.605 3E-05	7.477 2E-06	3.356 8E-05	3.356 8E-05
$F 6$	3.400 9E-04	3.391 8E-06	4.806 3E-05	3.391 8E-06	3.391 8E-06	3.391 8E-06	4.020 0E-05
$F 7$	8.034 6E-03	1.099 2E-05	7.016 2E-03	5.452 1E-03	3.391 8E-06	3.391 8E-06	3.391 8E-06
$F 8$	3.195 1E-03	3.391 8E-06	4.807 3E-01	4.143 2E-06	3.391 8E-06	3.391 8E-06	3.391 8E-06
$F 9$	N/A	2.539 6E-06	N/A	N/A	6.866 2E-07	2.516 7E-06	6.866 2E-07
$F 10$	N/A	6.866 2E-07	N/A	2.298 8E-05	6.866 2E-07	6.667 7E-07	6.866 2E-07
$F 11$	N/A	6.866 2E-07	N/A	N/A	6.866 2E-07	1.643 9E-01	6.866 2E-07
$F 12$	6.783 0E-04	3.391 8E-06	3.391 8E-06	2.798 3E-05	3.391 8E-06	6.151 6E-06	5.452 1E-03
$F 13$	1.844 1E-05	3.391 8E-06	4.806 3E-05	3.391 8E-06	3.391 8E-06	3.391 8E-06	2.133 7E-01
$F 14$	3.195 1E-03	3.894 0E-04	3.391 8E-06	5.052 7E-06	2.706 1E-05	3.391 8E-06	3.593 5E-01
$F 15$	2.997 6E-03	7.400 2E-01	7.802 1E-04	2.627 5E-01	5.737 1E-05	1.985 1E-01	3.690 6E-03
$F 16$	7.940 3E-03	3.101 7E-02	1.844 1E-01	3.609 3E-04	5.737 1E-05	3.391 8E-06	6.866 2E-07
$F 17$	3.439 7E-02	6.709 0E-04	2.228 9E-04	1.099 2E-05	3.299 5E-06	7.477 2E-06	6.866 2E-07
$F 18$	2.253 1E-02	1.329 5E-05	3.609 3E-04	5.052 7E-06	1.617 8E-03	3.391 8E-06	2.458 8E-06
$F 19$	2.145 1E-03	6.709 0E-04	4.020 0E-05	3.391 8E-06	3.391 8E-06	3.391 8E-06	1.562 6E-06
$F 20$	1.844 1E-03	3.690 6E-03	1.050 0E-03	3.195 1E-01	4.197 0E-02	9.709 1E-02	6.785 7E-06
$F 21$	2.792 5E-02	2.822 6E-03	3.391 8E-06	3.391 8E-06	3.615 0E-01	1.605 3E-05	2.936 6E-02
$F 22$	8.034 6E-03	2.997 6E-01	3.391 8E-06	3.391 8E-06	6.709 0E-04	6.836 8E-05	1.178 6E-03
$F 23$	4.639 2E-02	1.8919E-04	3.391 8E-06	4.143 2E-06	5.452 1E-03	1.329 5E-05	5.790 5E-04

Table 3 Wilcoxon rank sum test results of each algorithm

函数	SMA	AOA	HHO	WOA	SSA	GWO	PSO
$F 1$	N/A	6.866 2E-07	6.866 2E-07	6.866 2E-07	6.866 2E-07	6.866 2E-07	6.866 2E-07
$F 2$	2.172 8E-05	2.491 5E-06	2.491 5E-06	2.491 5E-06	2.491 5E-06	2.491 5E-06	2.491 5E-06
$F 3$	3.506 5E-03	6.866 2E-07	6.866 2E-07	6.866 2E-07	6.866 2E-07	6.866 2E-07	6.866 2E-07
$F 4$	1.260 4E-06	1.260 4E-06	1.260 4E-06	1.260 4E-06	1.260 4E-06	1.260 4E-06	1.260 4E-06
$F 5$	8.034 6E-03	1.329 5E-05	9.073 4E-06	1.605 3E-05	7.477 2E-06	3.356 8E-05	3.356 8E-05
$F 6$	3.400 9E-04	3.391 8E-06	4.806 3E-05	3.391 8E-06	3.391 8E-06	3.391 8E-06	4.020 0E-05
$F 7$	8.034 6E-03	1.099 2E-05	7.016 2E-03	5.452 1E-03	3.391 8E-06	3.391 8E-06	3.391 8E-06
$F 8$	3.195 1E-03	3.391 8E-06	4.807 3E-01	4.143 2E-06	3.391 8E-06	3.391 8E-06	3.391 8E-06
$F 9$	N/A	2.539 6E-06	N/A	N/A	6.866 2E-07	2.516 7E-06	6.866 2E-07
$F 10$	N/A	6.866 2E-07	N/A	2.298 8E-05	6.866 2E-07	6.667 7E-07	6.866 2E-07
$F 11$	N/A	6.866 2E-07	N/A	N/A	6.866 2E-07	1.643 9E-01	6.866 2E-07
$F 12$	6.783 0E-04	3.391 8E-06	3.391 8E-06	2.798 3E-05	3.391 8E-06	6.151 6E-06	5.452 1E-03
$F 13$	1.844 1E-05	3.391 8E-06	4.806 3E-05	3.391 8E-06	3.391 8E-06	3.391 8E-06	2.133 7E-01
$F 14$	3.195 1E-03	3.894 0E-04	3.391 8E-06	5.052 7E-06	2.706 1E-05	3.391 8E-06	3.593 5E-01
$F 15$	2.997 6E-03	7.400 2E-01	7.802 1E-04	2.627 5E-01	5.737 1E-05	1.985 1E-01	3.690 6E-03
$F 16$	7.940 3E-03	3.101 7E-02	1.844 1E-01	3.609 3E-04	5.737 1E-05	3.391 8E-06	6.866 2E-07
$F 17$	3.439 7E-02	6.709 0E-04	2.228 9E-04	1.099 2E-05	3.299 5E-06	7.477 2E-06	6.866 2E-07
$F 18$	2.253 1E-02	1.329 5E-05	3.609 3E-04	5.052 7E-06	1.617 8E-03	3.391 8E-06	2.458 8E-06
$F 19$	2.145 1E-03	6.709 0E-04	4.020 0E-05	3.391 8E-06	3.391 8E-06	3.391 8E-06	1.562 6E-06
$F 20$	1.844 1E-03	3.690 6E-03	1.050 0E-03	3.195 1E-01	4.197 0E-02	9.709 1E-02	6.785 7E-06
$F 21$	2.792 5E-02	2.822 6E-03	3.391 8E-06	3.391 8E-06	3.615 0E-01	1.605 3E-05	2.936 6E-02
$F 22$	8.034 6E-03	2.997 6E-01	3.391 8E-06	3.391 8E-06	6.709 0E-04	6.836 8E-05	1.178 6E-03
$F 23$	4.639 2E-02	1.8919E-04	3.391 8E-06	4.143 2E-06	5.452 1E-03	1.329 5E-05	5.790 5E-04

Fig.5 Welded beam design problem

Table 4 Optimization results of each algorithm applied to welded beam design problem

算法	$h$	$l$	$t$	$b$	结果
HSMAAOA	0.202 6	3.319 7	9.034 5	0.205 8	1.699 9
SMA	0.194 3	3.470 5	9.033 4	0.205 9	1.707 9
AOA	0.199 1	4.248 9	10.000 0	0.216 3	2.084 7
HHO	0.233 0	2.983 0	8.464 9	0.234 5	1.800 4
WOA	0.125 1	6.705 3	7.167 8	0.327 0	2.450 7
SSA	0.133 6	5.935 9	7.244 6	0.325 1	2.376 1
GWO	0.177 7	3.840 2	9.034 7	0.205 9	1.730 8
PSO	0.262 1	4.335 6	6.314 7	0.421 3	2.675 9

Table 4 Optimization results of each algorithm applied to welded beam design problem

算法	$h$	$l$	$t$	$b$	结果
HSMAAOA	0.202 6	3.319 7	9.034 5	0.205 8	1.699 9
SMA	0.194 3	3.470 5	9.033 4	0.205 9	1.707 9
AOA	0.199 1	4.248 9	10.000 0	0.216 3	2.084 7
HHO	0.233 0	2.983 0	8.464 9	0.234 5	1.800 4
WOA	0.125 1	6.705 3	7.167 8	0.327 0	2.450 7
SSA	0.133 6	5.935 9	7.244 6	0.325 1	2.376 1
GWO	0.177 7	3.840 2	9.034 7	0.205 9	1.730 8
PSO	0.262 1	4.335 6	6.314 7	0.421 3	2.675 9

Fig.6 Pressure vessel design problem

Table 5 Optimization results of each algorithm applied to pressure vessel design problem

算法	$T s (x 1)$	$T h (x 2)$	$R (x 3)$	$L (x 4)$	结果
HSMAAOA	0.858 0	0.420 4	46.040 0	133.232 1	5 972.027 4
SMA^[7]	0.793 1	0.393 2	40.671 1	196.217 8	5 994.185 7
AOA^[8]	0.830 3	0.416 2	42.751 2	169.345 4	6 048.784 4
HHO^[11]	0.817 5	0.407 2	42.091 7	176.719 6	6 000.462 5
WOA^[10]	0.812 5	0.437 5	42.098 2	177.638 9	6 059.741 0
SSA^[6]	0.790 6	0.390 8	40.967 7	195.918 2	6 012.188 5
GWO^[9]	0.812 5	0.434 5	42.089 1	176.758 7	6 051.563 9
PSO^[5]	0.812 5	0.437 5	42.091 2	176.746 5	6 061.077 7

Table 5 Optimization results of each algorithm applied to pressure vessel design problem

算法	$T s (x 1)$	$T h (x 2)$	$R (x 3)$	$L (x 4)$	结果
HSMAAOA	0.858 0	0.420 4	46.040 0	133.232 1	5 972.027 4
SMA^[7]	0.793 1	0.393 2	40.671 1	196.217 8	5 994.185 7
AOA^[8]	0.830 3	0.416 2	42.751 2	169.345 4	6 048.784 4
HHO^[11]	0.817 5	0.407 2	42.091 7	176.719 6	6 000.462 5
WOA^[10]	0.812 5	0.437 5	42.098 2	177.638 9	6 059.741 0
SSA^[6]	0.790 6	0.390 8	40.967 7	195.918 2	6 012.188 5
GWO^[9]	0.812 5	0.434 5	42.089 1	176.758 7	6 051.563 9
PSO^[5]	0.812 5	0.437 5	42.091 2	176.746 5	6 061.077 7

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Hybrid Algorithm of Slime Mould Algorithm and Arithmetic Optimization Algorithm Based on Random Opposition-Based Learning

融合随机反向学习的黏菌与算术混合优化算法

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