《计算机应用》唯一官方网站 ›› 2025, Vol. 45 ›› Issue (7): 2304-2316.DOI: 10.11772/j.issn.1001-9081.2024060778
收稿日期:
2024-06-11
修回日期:
2024-09-10
接受日期:
2024-09-12
发布日期:
2025-07-10
出版日期:
2025-07-10
通讯作者:
邹德旋
作者简介:
杨志龙(1999—),男,河南三门峡人,硕士研究生,主要研究方向:智能优化算法基金资助:
Zhilong YANG, Dexuan ZOU(), Can LI, Yingying SHAO, Lejie MA
Received:
2024-06-11
Revised:
2024-09-10
Accepted:
2024-09-12
Online:
2025-07-10
Published:
2025-07-10
Contact:
Dexuan ZOU
About author:
YANG Zhilong, born in 1999, M. S. candidate. His research interests include intelligent optimization algorithm.Supported by:
摘要:
针对蜣螂优化(DBO)算法中存在的收敛速度慢、精度不高以及容易陷入局部最优的问题,提出一种融入限制反向学习与柯西-高斯变异的蜣螂优化算法(SI-DBO)。首先,用Circle映射初始化种群,以使种群的分布更加均匀和具有多样性,从而提升算法的收敛速度和寻优精度;其次,使用限制反向学习对蜣螂的位置进行更新,以提升蜣螂的搜索能力;最后,使用柯西-高斯变异策略帮助种群逃逸出局部最佳位置并寻找全局最佳位置。为了验证SI-DBO的性能,在测试函数上进行仿真实验并对实验结果进行Wilcoxon秩和检验,而且将该算法用于求解机器人夹持器问题。实验结果表明,与黑寡妇-蜣螂优化算法(BWDBO)和麻雀搜索算法(SSA)相比,SI-DBO在测试函数上均获得了较高的寻优精度和收敛速度,同时,SI-DBO在求解机器人夹持器问题时的效果优于粒子群优化(PSO)算法,验证了SI-DBO具有更好的寻优性能和工程实用性。
中图分类号:
杨志龙, 邹德旋, 李灿, 邵莹莹, 马乐杰. 融入限制反向学习与柯西-高斯变异的蜣螂优化算法[J]. 计算机应用, 2025, 45(7): 2304-2316.
Zhilong YANG, Dexuan ZOU, Can LI, Yingying SHAO, Lejie MA. Dung beetle optimizer algorithm with restricted reverse learning and Cauchy-Gauss variation[J]. Journal of Computer Applications, 2025, 45(7): 2304-2316.
运行次数 | DBO运行时间/s | SI-DBO运行时间/s |
---|---|---|
平均 | 1.01E-01 | 1.10E-01 |
1 | 9.96E-02 | 1.09E-01 |
2 | 1.00E-01 | 1.11E-01 |
3 | 9.91E-02 | 1.08E-01 |
4 | 1.00E-01 | 1.11E-01 |
5 | 1.06E-01 | 1.06E-01 |
6 | 1.01E-01 | 1.22E-01 |
7 | 1.01E-01 | 1.10E-01 |
8 | 1.06E-01 | 1.09E-01 |
9 | 1.04E-01 | 1.14E-01 |
10 | 1.03E-01 | 1.07E-01 |
11 | 9.58E-02 | 1.07E-01 |
12 | 9.91E-02 | 1.11E-01 |
13 | 9.97E-02 | 1.10E-01 |
14 | 1.00E-01 | 1.07E-01 |
15 | 9.78E-02 | 1.09E-01 |
16 | 1.01E-01 | 1.11E-01 |
17 | 1.06E-01 | 1.09E-01 |
18 | 9.91E-02 | 1.13E-01 |
19 | 1.03E-01 | 1.14E-01 |
20 | 9.69E-02 | 1.12E-01 |
表1 DBO算法与SI-DBO的运行时间
Tab. 1 Running time of DBO algorithm and SI-DBO
运行次数 | DBO运行时间/s | SI-DBO运行时间/s |
---|---|---|
平均 | 1.01E-01 | 1.10E-01 |
1 | 9.96E-02 | 1.09E-01 |
2 | 1.00E-01 | 1.11E-01 |
3 | 9.91E-02 | 1.08E-01 |
4 | 1.00E-01 | 1.11E-01 |
5 | 1.06E-01 | 1.06E-01 |
6 | 1.01E-01 | 1.22E-01 |
7 | 1.01E-01 | 1.10E-01 |
8 | 1.06E-01 | 1.09E-01 |
9 | 1.04E-01 | 1.14E-01 |
10 | 1.03E-01 | 1.07E-01 |
11 | 9.58E-02 | 1.07E-01 |
12 | 9.91E-02 | 1.11E-01 |
13 | 9.97E-02 | 1.10E-01 |
14 | 1.00E-01 | 1.07E-01 |
15 | 9.78E-02 | 1.09E-01 |
16 | 1.01E-01 | 1.11E-01 |
17 | 1.06E-01 | 1.09E-01 |
18 | 9.91E-02 | 1.13E-01 |
19 | 1.03E-01 | 1.14E-01 |
20 | 9.69E-02 | 1.12E-01 |
函数表达式 | 维度(dim) | 范围 | 最优值 |
---|---|---|---|
30或100 | 0 | ||
30或100 | 0 | ||
30或100 | 0 | ||
30或100 | 0 | ||
30或100 | 0 | ||
30或100 | 0 | ||
30或100 | 0 | ||
30或100 | |||
30或100 | 0 | ||
30或100 | 0 | ||
30或100 | 0 | ||
30或100 | 0 | ||
30或100 | 0 | ||
2 | 1 | ||
4 | 0.000 307 5 | ||
2 | -1.031 628 5 | ||
2 | 0.398 | ||
2 | 3 | ||
3 | -3.86 | ||
6 | -3.32 | ||
4 | -10.152 3 | ||
4 | -10.402 8 | ||
4 | -10.536 3 |
表2 CEC2005基准测试函数
Tab. 2 CEC2005 benchmark functions
函数表达式 | 维度(dim) | 范围 | 最优值 |
---|---|---|---|
30或100 | 0 | ||
30或100 | 0 | ||
30或100 | 0 | ||
30或100 | 0 | ||
30或100 | 0 | ||
30或100 | 0 | ||
30或100 | 0 | ||
30或100 | |||
30或100 | 0 | ||
30或100 | 0 | ||
30或100 | 0 | ||
30或100 | 0 | ||
30或100 | 0 | ||
2 | 1 | ||
4 | 0.000 307 5 | ||
2 | -1.031 628 5 | ||
2 | 0.398 | ||
2 | 3 | ||
3 | -3.86 | ||
6 | -3.32 | ||
4 | -10.152 3 | ||
4 | -10.402 8 | ||
4 | -10.536 3 |
算法 | 参数设置 |
---|---|
SI-DBO | |
MSADBO | |
TDBO | |
BWDBO | |
DBO | |
SSA | |
GWO | |
WOA | |
PSO | |
SO | |
GJO |
表3 不同算法参数设置
Tab. 3 Parameter settings for different algorithms
算法 | 参数设置 |
---|---|
SI-DBO | |
MSADBO | |
TDBO | |
BWDBO | |
DBO | |
SSA | |
GWO | |
WOA | |
PSO | |
SO | |
GJO |
函数 | 指标 | DBO | DBO1 | DBO2 | DBO3 | SI-DBO |
---|---|---|---|---|---|---|
平均值 | 1.74E-105 | 1.21E-101 | 4.13E-112 | 3.56E-131 | 0.00E+00 | |
标准差 | 9.33E-105 | 6.52E-101 | 2.22E-111 | 1.92E-130 | 0.00E+00 | |
平均值 | 4.45E-56 | 6.55E-64 | 1.40E-80 | 9.04E-141 | 0.00E+00 | |
标准差 | 2.40E-55 | 3.53E-63 | 7.53E-80 | 4.87E-140 | 0.00E+00 | |
平均值 | 4.16E-50 | 7.60E-63 | 1.43E-51 | 1.00E-74 | 8.76E-222 | |
标准差 | 2.17E-49 | 3.60E-62 | 7.70E-51 | 5.36E-73 | 0.00E+00 | |
平均值 | 1.56E-50 | 1.60E-32 | 2.02E-48 | 4.56E-71 | 4.81E-217 | |
标准差 | 8.40E-50 | 7.25E-32 | 8.80E-48 | 2.42E-70 | 0.00E+00 | |
平均值 | 2.97E+01 | 1.50E+01 | 2.59E+01 | 2.89E+01 | 1.36E+01 | |
标准差 | 2.11E+01 | 1.17E+01 | 2.40E-01 | 2.40E-01 | 1.36E+01 | |
平均值 | 7.91E-03 | 5.01E-04 | 7.37E-04 | 5.69E+00 | 6.75E-06 | |
标准差 | 3.85E-02 | 7.22E-04 | 1.35E-03 | 5.77E-01 | 1.52E-05 | |
平均值 | 1.04E-03 | 1.09E-03 | 6.70E-04 | 1.20E-03 | 5.58E-05 | |
标准差 | 9.45E-04 | 7.64E-04 | 6.15E-04 | 1.04E-03 | 6.15E-05 | |
平均值 | -8.80E+03 | -1.26E+04 | -8.50E+03 | -5.37E+03 | -1.26E+04 | |
标准差 | 1.63E+03 | 2.26E+00 | 1.62E+03 | 1.70E+03 | 8.12E+01 | |
平均值 | 2.32E-01 | 9.62E-01 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
标准差 | 1.25E+00 | 5.18E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
平均值 | 8.88E-16 | 8.88E-16 | 8.88E-16 | 8.88E-16 | 8.88E-16 | |
标准差 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
平均值 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
标准差 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
平均值 | 1.03E-04 | 4.74E-05 | 2.47E-05 | 9.15E-01 | 9.64E-07 | |
标准差 | 3.79E-04 | 1.82E-04 | 4.73E-05 | 2.52E-01 | 2.73E-06 | |
平均值 | 5.95E-01 | 8.98E-02 | 5.74E-03 | 2.77E+00 | 4.44E-03 | |
标准差 | 5.04E-01 | 3.26E-01 | 9.06E-03 | 3.58E-01 | 1.49E-02 |
表4 不同策略改进后的DBO算法的实验结果对比(F1~F13,30维)
Tab. 4 Comparison of experimental results of DBO algorithms improved by different strategies(F1-F13,30 dim)
函数 | 指标 | DBO | DBO1 | DBO2 | DBO3 | SI-DBO |
---|---|---|---|---|---|---|
平均值 | 1.74E-105 | 1.21E-101 | 4.13E-112 | 3.56E-131 | 0.00E+00 | |
标准差 | 9.33E-105 | 6.52E-101 | 2.22E-111 | 1.92E-130 | 0.00E+00 | |
平均值 | 4.45E-56 | 6.55E-64 | 1.40E-80 | 9.04E-141 | 0.00E+00 | |
标准差 | 2.40E-55 | 3.53E-63 | 7.53E-80 | 4.87E-140 | 0.00E+00 | |
平均值 | 4.16E-50 | 7.60E-63 | 1.43E-51 | 1.00E-74 | 8.76E-222 | |
标准差 | 2.17E-49 | 3.60E-62 | 7.70E-51 | 5.36E-73 | 0.00E+00 | |
平均值 | 1.56E-50 | 1.60E-32 | 2.02E-48 | 4.56E-71 | 4.81E-217 | |
标准差 | 8.40E-50 | 7.25E-32 | 8.80E-48 | 2.42E-70 | 0.00E+00 | |
平均值 | 2.97E+01 | 1.50E+01 | 2.59E+01 | 2.89E+01 | 1.36E+01 | |
标准差 | 2.11E+01 | 1.17E+01 | 2.40E-01 | 2.40E-01 | 1.36E+01 | |
平均值 | 7.91E-03 | 5.01E-04 | 7.37E-04 | 5.69E+00 | 6.75E-06 | |
标准差 | 3.85E-02 | 7.22E-04 | 1.35E-03 | 5.77E-01 | 1.52E-05 | |
平均值 | 1.04E-03 | 1.09E-03 | 6.70E-04 | 1.20E-03 | 5.58E-05 | |
标准差 | 9.45E-04 | 7.64E-04 | 6.15E-04 | 1.04E-03 | 6.15E-05 | |
平均值 | -8.80E+03 | -1.26E+04 | -8.50E+03 | -5.37E+03 | -1.26E+04 | |
标准差 | 1.63E+03 | 2.26E+00 | 1.62E+03 | 1.70E+03 | 8.12E+01 | |
平均值 | 2.32E-01 | 9.62E-01 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
标准差 | 1.25E+00 | 5.18E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
平均值 | 8.88E-16 | 8.88E-16 | 8.88E-16 | 8.88E-16 | 8.88E-16 | |
标准差 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
平均值 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
标准差 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
平均值 | 1.03E-04 | 4.74E-05 | 2.47E-05 | 9.15E-01 | 9.64E-07 | |
标准差 | 3.79E-04 | 1.82E-04 | 4.73E-05 | 2.52E-01 | 2.73E-06 | |
平均值 | 5.95E-01 | 8.98E-02 | 5.74E-03 | 2.77E+00 | 4.44E-03 | |
标准差 | 5.04E-01 | 3.26E-01 | 9.06E-03 | 3.58E-01 | 1.49E-02 |
函数 | 指标 | DBO | MSADBO | TDBO | BWDBO | SI-DBO |
---|---|---|---|---|---|---|
平均值 | 1.74E-105 | 0.00E+00 | 3.97E-172 | 0.00E+00 | 0.00E+00 | |
标准差 | 9.33E-105 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
平均值 | 4.45E-56 | 0.00E+00 | 1.03E-95 | 0.00E+00 | 0.00E+00 | |
标准差 | 2.40E-55 | 0.00E+00 | 3.89E-95 | 0.00E+00 | 0.00E+00 | |
平均值 | 4.16E-50 | 9.09E-172 | 1.89E-88 | 5.45E-182 | 8.76E-222 | |
标准差 | 2.17E-49 | 0.00E+00 | 1.02E-87 | 0.00E+00 | 0.00E+00 | |
平均值 | 1.56E-50 | 1.89E-173 | 3.95E-32 | 1.30E-201 | 4.81E-217 | |
标准差 | 8.40E-50 | 0.00E+00 | 2.02E-31 | 0.00E+00 | 0.00E+00 | |
平均值 | 2.97E+01 | 2.13E+01 | 2.34E+01 | 4.99E+00 | 1.36E+01 | |
标准差 | 2.11E+01 | 9.56E+00 | 2.17E+01 | 9.98E+00 | 1.36E+01 | |
平均值 | 7.91E-03 | 1.12E-03 | 1.33E+00 | 4.78E-07 | 6.75E-06 | |
标准差 | 3.85E-02 | 1.51E-03 | 8.40E-01 | 2.54E-06 | 1.52E-05 | |
平均值 | 1.04E-03 | 3.37E-04 | 1.45E-03 | 4.28E-04 | 5.58E-05 | |
标准差 | 9.45E-04 | 2.70E-04 | 1.00E-03 | 4.68E-04 | 6.15E-05 | |
平均值 | -8.80E+03 | -1.17E+04 | -1.26E+04 | -3.34E+04 | -1.26E+04 | |
标准差 | 1.63E+03 | 4.51E+03 | 5.18E+01 | 6.21E+03 | 8.12E+01 | |
平均值 | 2.32E-01 | 0.00E+00 | 4.04E+00 | 0.00E+00 | 0.00E+00 | |
标准差 | 1.25E+00 | 0.00E+00 | 1.03E+01 | 0.00E+00 | 0.00E+00 | |
平均值 | 8.88E-16 | 8.88E-16 | 8.88E-16 | 8.88E-16 | 8.88E-16 | |
标准差 | 0.00E+00 | 0.00E+00 | 9.86E-32 | 9.86E-32 | 0.00E+00 | |
平均值 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
标准差 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
平均值 | 1.03E-04 | 3.78E-05 | 4.57E-02 | 7.73E-06 | 9.64E-07 | |
标准差 | 3.79E-04 | 1.11E-04 | 4.82E-02 | 2.52E-05 | 2.73E-06 | |
平均值 | 5.95E-01 | 3.66E-01 | 9.30E-01 | 4.86E-02 | 4.44E-03 | |
标准差 | 5.04E-01 | 7.13E-01 | 8.60E-01 | 8.89E-02 | 1.49E-02 | |
平均值 | 1.06E+00 | 1.06E+00 | 1.30E+00 | 9.98E-01 | 9.98E-01 | |
标准差 | 3.56E-01 | 3.56E-01 | 6.29E-01 | 3.33E-16 | 0.00E+00 | |
平均值 | 7.85E-04 | 4.96E-04 | 4.97E-04 | 3.80E-04 | 3.11E-04 | |
标准差 | 4.15E-04 | 2.85E-04 | 2.60E-04 | 1.21E-02 | 3.40E-06 | |
平均值 | -1.03E+00 | -1.03E+00 | -1.03E+00 | -1.03E+00 | -1.03E+00 | |
标准差 | 0.00E+00 | 0.00E+00 | 1.34E-05 | 5.89E-16 | 0.00E+00 | |
平均值 | 3.98E-01 | 1.02E+00 | 3.98E-01 | 3.89E-01 | 3.98E-01 | |
标准差 | 0.00E+00 | 1.58E+00 | 4.10E-04 | 1.11E-16 | 1.18E-04 | |
平均值 | 5.70E+00 | 3.00E+00 | 3.00E+00 | 1.20E+01 | 3.00E+00 | |
标准差 | 8.10E+00 | 0.00E+00 | 2.95E-03 | 1.75E+01 | 1.29E-03 | |
平均值 | -3.86E+00 | -3.30E+00 | -3.86E+00 | -3.86E+00 | -3.86E+00 | |
标准差 | 3.16E-03 | 1.28E+00 | 2.98E-03 | 2.68E-03 | 1.11E-03 | |
平均值 | -3.23E+00 | -2.22E+00 | -3.12E+00 | -3.29E+00 | -3.30E+00 | |
标准差 | 1.07E-01 | 1.47E+00 | 1.54E-01 | 6.96E-02 | 4.10E-02 | |
平均值 | -6.03E+00 | -5.54E+00 | -9.10E+00 | -9.64E+00 | -1.01E+01 | |
标准差 | 2.15E+00 | 4.89E+00 | 1.57E+00 | 4.53E+00 | 3.79E-02 | |
平均值 | -7.84E+00 | -7.34E+00 | -8.79E+00 | -1.04E+01 | -1.04E+01 | |
标准差 | 3.02E+00 | 4.58E+00 | 1.91E+00 | 8.41E-04 | 8.01E-02 | |
平均值 | -9.01E+00 | -5.80E+00 | -8.62E+00 | -1.05E+01 | -1.05E+01 | |
标准差 | 2.57E+00 | 5.05E+00 | 2.22E+00 | 2.71E-12 | 9.65E-02 |
表5 本文算法与其他改进DBO算法的实验结果对比(F1~F13,30维;F14~F23,固定维度)
Tab. 5 Comparison of experimental results of proposed algorithm and other improved DBO algorithms (F1-F13,30 dim;F14-F23, fixed dimension)
函数 | 指标 | DBO | MSADBO | TDBO | BWDBO | SI-DBO |
---|---|---|---|---|---|---|
平均值 | 1.74E-105 | 0.00E+00 | 3.97E-172 | 0.00E+00 | 0.00E+00 | |
标准差 | 9.33E-105 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
平均值 | 4.45E-56 | 0.00E+00 | 1.03E-95 | 0.00E+00 | 0.00E+00 | |
标准差 | 2.40E-55 | 0.00E+00 | 3.89E-95 | 0.00E+00 | 0.00E+00 | |
平均值 | 4.16E-50 | 9.09E-172 | 1.89E-88 | 5.45E-182 | 8.76E-222 | |
标准差 | 2.17E-49 | 0.00E+00 | 1.02E-87 | 0.00E+00 | 0.00E+00 | |
平均值 | 1.56E-50 | 1.89E-173 | 3.95E-32 | 1.30E-201 | 4.81E-217 | |
标准差 | 8.40E-50 | 0.00E+00 | 2.02E-31 | 0.00E+00 | 0.00E+00 | |
平均值 | 2.97E+01 | 2.13E+01 | 2.34E+01 | 4.99E+00 | 1.36E+01 | |
标准差 | 2.11E+01 | 9.56E+00 | 2.17E+01 | 9.98E+00 | 1.36E+01 | |
平均值 | 7.91E-03 | 1.12E-03 | 1.33E+00 | 4.78E-07 | 6.75E-06 | |
标准差 | 3.85E-02 | 1.51E-03 | 8.40E-01 | 2.54E-06 | 1.52E-05 | |
平均值 | 1.04E-03 | 3.37E-04 | 1.45E-03 | 4.28E-04 | 5.58E-05 | |
标准差 | 9.45E-04 | 2.70E-04 | 1.00E-03 | 4.68E-04 | 6.15E-05 | |
平均值 | -8.80E+03 | -1.17E+04 | -1.26E+04 | -3.34E+04 | -1.26E+04 | |
标准差 | 1.63E+03 | 4.51E+03 | 5.18E+01 | 6.21E+03 | 8.12E+01 | |
平均值 | 2.32E-01 | 0.00E+00 | 4.04E+00 | 0.00E+00 | 0.00E+00 | |
标准差 | 1.25E+00 | 0.00E+00 | 1.03E+01 | 0.00E+00 | 0.00E+00 | |
平均值 | 8.88E-16 | 8.88E-16 | 8.88E-16 | 8.88E-16 | 8.88E-16 | |
标准差 | 0.00E+00 | 0.00E+00 | 9.86E-32 | 9.86E-32 | 0.00E+00 | |
平均值 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
标准差 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
平均值 | 1.03E-04 | 3.78E-05 | 4.57E-02 | 7.73E-06 | 9.64E-07 | |
标准差 | 3.79E-04 | 1.11E-04 | 4.82E-02 | 2.52E-05 | 2.73E-06 | |
平均值 | 5.95E-01 | 3.66E-01 | 9.30E-01 | 4.86E-02 | 4.44E-03 | |
标准差 | 5.04E-01 | 7.13E-01 | 8.60E-01 | 8.89E-02 | 1.49E-02 | |
平均值 | 1.06E+00 | 1.06E+00 | 1.30E+00 | 9.98E-01 | 9.98E-01 | |
标准差 | 3.56E-01 | 3.56E-01 | 6.29E-01 | 3.33E-16 | 0.00E+00 | |
平均值 | 7.85E-04 | 4.96E-04 | 4.97E-04 | 3.80E-04 | 3.11E-04 | |
标准差 | 4.15E-04 | 2.85E-04 | 2.60E-04 | 1.21E-02 | 3.40E-06 | |
平均值 | -1.03E+00 | -1.03E+00 | -1.03E+00 | -1.03E+00 | -1.03E+00 | |
标准差 | 0.00E+00 | 0.00E+00 | 1.34E-05 | 5.89E-16 | 0.00E+00 | |
平均值 | 3.98E-01 | 1.02E+00 | 3.98E-01 | 3.89E-01 | 3.98E-01 | |
标准差 | 0.00E+00 | 1.58E+00 | 4.10E-04 | 1.11E-16 | 1.18E-04 | |
平均值 | 5.70E+00 | 3.00E+00 | 3.00E+00 | 1.20E+01 | 3.00E+00 | |
标准差 | 8.10E+00 | 0.00E+00 | 2.95E-03 | 1.75E+01 | 1.29E-03 | |
平均值 | -3.86E+00 | -3.30E+00 | -3.86E+00 | -3.86E+00 | -3.86E+00 | |
标准差 | 3.16E-03 | 1.28E+00 | 2.98E-03 | 2.68E-03 | 1.11E-03 | |
平均值 | -3.23E+00 | -2.22E+00 | -3.12E+00 | -3.29E+00 | -3.30E+00 | |
标准差 | 1.07E-01 | 1.47E+00 | 1.54E-01 | 6.96E-02 | 4.10E-02 | |
平均值 | -6.03E+00 | -5.54E+00 | -9.10E+00 | -9.64E+00 | -1.01E+01 | |
标准差 | 2.15E+00 | 4.89E+00 | 1.57E+00 | 4.53E+00 | 3.79E-02 | |
平均值 | -7.84E+00 | -7.34E+00 | -8.79E+00 | -1.04E+01 | -1.04E+01 | |
标准差 | 3.02E+00 | 4.58E+00 | 1.91E+00 | 8.41E-04 | 8.01E-02 | |
平均值 | -9.01E+00 | -5.80E+00 | -8.62E+00 | -1.05E+01 | -1.05E+01 | |
标准差 | 2.57E+00 | 5.05E+00 | 2.22E+00 | 2.71E-12 | 9.65E-02 |
函数 | 指标 | SSA | GWO | WOA | PSO | SO | GJO | SI-DBO |
---|---|---|---|---|---|---|---|---|
平均值 | 2.65E-30 | 3.32E-67 | 8.94E-74 | 3.97E+02 | 1.56E-94 | 4.37E-54 | 0.00E+00 | |
标准差 | 1.38E-29 | 1.01E-66 | 4.00E-73 | 1.98E+02 | 3.71E-94 | 1.70E-53 | 0.00E+00 | |
平均值 | 2.86E-07 | 1.06E-06 | 4.53E+04 | 2.31E+03 | 2.60E-57 | 7.46E-17 | 0.00E+00 | |
标准差 | 1.54E-06 | 2.33E-06 | 1.54E+04 | 9.98E+02 | 8.82E-57 | 2.31E-16 | 0.00E+00 | |
平均值 | 1.97E-16 | 2.56E-42 | 2.34E-49 | 2.20E+01 | 3.14E-43 | 1.89E-32 | 8.76E-222 | |
标准差 | 8.89E-16 | 5.45E-42 | 1.24E-48 | 6.06E+01 | 6.11E-43 | 2.94E-32 | 0.00E+00 | |
平均值 | 1.96E-11 | 1.17E-21 | 5.07E+01 | 1.36E+01 | 3.79E-40 | 1.18E-15 | 4.81E-217 | |
标准差 | 9.86E-11 | 3.11E-21 | 2.91E+01 | 3.46E+00 | 1.38E-39 | 2.54E-15 | 0.00E+00 | |
平均值 | 6.13E-07 | 2.88E+01 | 2.80E+01 | 2.07E+04 | 2.03E+01 | 2.76E+01 | 1.36E+01 | |
标准差 | 1.44E-06 | 3.00E-02 | 4.00E-01 | 1.68E+04 | 1.21E+01 | 6.30E-01 | 1.36E+01 | |
平均值 | 3.91E-18 | 2.16E+00 | 4.30E-01 | 3.86E+02 | 8.10E-01 | 2.52E+00 | 6.75E-06 | |
标准差 | 1.30E-17 | 6.80E-01 | 2.00E-01 | 1.66E+02 | 7.60E-01 | 5.40E-01 | 1.52E-05 | |
平均值 | 2.97E-03 | 5.19E-04 | 3.89E-03 | 1.30E-01 | 2.89E-04 | 6.19E-04 | 5.59E-05 | |
标准差 | 4.20E-03 | 4.08E-04 | 4.64E-03 | 6.00E-02 | 2.01E-04 | 4.20E-04 | 6.15E-05 | |
平均值 | -7.98E+03 | -8.78E+03 | -9.75E+03 | -1.07E+04 | -1.25E+04 | -4.39E+03 | -1.25E+04 | |
标准差 | 7.64E+02 | 1.01E+03 | 1.56E+03 | 1.59E+03 | 1.09E+02 | 1.14E+03 | 8.72E+01 | |
平均值 | 0.00E+00 | 0.00E+00 | 5.68E-15 | 9.07E+01 | 8.05E+00 | 0.00E+00 | 0.00E+00 | |
标准差 | 0.00E+00 | 0.00E+00 | 2.25E-14 | 1.86E+01 | 1.23E+01 | 0.00E+00 | 0.00E+00 | |
平均值 | 4.94E-15 | 3.97E-15 | 3.85E-15 | 6.77E+00 | 3.07E-01 | 6.81E-15 | 8.88E-16 | |
标准差 | 1.25E-14 | 1.21E-15 | 2.44E-15 | 8.72E-01 | 7.89E-01 | 1.70E-15 | 0.00E+00 | |
平均值 | 0.00E+00 | 0.00E+00 | 1.49E-02 | 1.40E+00 | 3.18E-02 | 0.00E+00 | 0.00E+00 | |
标准差 | 0.00E+00 | 0.00E+00 | 6.44E-02 | 1.99E-01 | 7.51E-02 | 0.00E+00 | 0.00E+00 | |
平均值 | 2.14E-17 | 2.26E-01 | 2.71E-02 | 8.46E+00 | 3.20E-02 | 2.40E-01 | 9.64E-07 | |
标准差 | 7.03E-17 | 1.32E-01 | 2.32E-02 | 3.65E+00 | 4.57E-02 | 1.40E-01 | 2.73E-06 | |
平均值 | 2.94E-17 | 1.17E+00 | 3.73E-01 | 2.46E+01 | 2.78E-01 | 1.61E+00 | 4.44E-03 | |
标准差 | 7.37E-17 | 2.95E-01 | 1.97E-01 | 1.52E+01 | 4.26E-01 | 2.07E-01 | 1.49E-02 |
表6 本文算法与其他算法的实验结果对比(F1~F13,30维)
Tab. 6 Comparison of experimental results of proposed algorithm and other algorithms(F1-F13,30 dim)
函数 | 指标 | SSA | GWO | WOA | PSO | SO | GJO | SI-DBO |
---|---|---|---|---|---|---|---|---|
平均值 | 2.65E-30 | 3.32E-67 | 8.94E-74 | 3.97E+02 | 1.56E-94 | 4.37E-54 | 0.00E+00 | |
标准差 | 1.38E-29 | 1.01E-66 | 4.00E-73 | 1.98E+02 | 3.71E-94 | 1.70E-53 | 0.00E+00 | |
平均值 | 2.86E-07 | 1.06E-06 | 4.53E+04 | 2.31E+03 | 2.60E-57 | 7.46E-17 | 0.00E+00 | |
标准差 | 1.54E-06 | 2.33E-06 | 1.54E+04 | 9.98E+02 | 8.82E-57 | 2.31E-16 | 0.00E+00 | |
平均值 | 1.97E-16 | 2.56E-42 | 2.34E-49 | 2.20E+01 | 3.14E-43 | 1.89E-32 | 8.76E-222 | |
标准差 | 8.89E-16 | 5.45E-42 | 1.24E-48 | 6.06E+01 | 6.11E-43 | 2.94E-32 | 0.00E+00 | |
平均值 | 1.96E-11 | 1.17E-21 | 5.07E+01 | 1.36E+01 | 3.79E-40 | 1.18E-15 | 4.81E-217 | |
标准差 | 9.86E-11 | 3.11E-21 | 2.91E+01 | 3.46E+00 | 1.38E-39 | 2.54E-15 | 0.00E+00 | |
平均值 | 6.13E-07 | 2.88E+01 | 2.80E+01 | 2.07E+04 | 2.03E+01 | 2.76E+01 | 1.36E+01 | |
标准差 | 1.44E-06 | 3.00E-02 | 4.00E-01 | 1.68E+04 | 1.21E+01 | 6.30E-01 | 1.36E+01 | |
平均值 | 3.91E-18 | 2.16E+00 | 4.30E-01 | 3.86E+02 | 8.10E-01 | 2.52E+00 | 6.75E-06 | |
标准差 | 1.30E-17 | 6.80E-01 | 2.00E-01 | 1.66E+02 | 7.60E-01 | 5.40E-01 | 1.52E-05 | |
平均值 | 2.97E-03 | 5.19E-04 | 3.89E-03 | 1.30E-01 | 2.89E-04 | 6.19E-04 | 5.59E-05 | |
标准差 | 4.20E-03 | 4.08E-04 | 4.64E-03 | 6.00E-02 | 2.01E-04 | 4.20E-04 | 6.15E-05 | |
平均值 | -7.98E+03 | -8.78E+03 | -9.75E+03 | -1.07E+04 | -1.25E+04 | -4.39E+03 | -1.25E+04 | |
标准差 | 7.64E+02 | 1.01E+03 | 1.56E+03 | 1.59E+03 | 1.09E+02 | 1.14E+03 | 8.72E+01 | |
平均值 | 0.00E+00 | 0.00E+00 | 5.68E-15 | 9.07E+01 | 8.05E+00 | 0.00E+00 | 0.00E+00 | |
标准差 | 0.00E+00 | 0.00E+00 | 2.25E-14 | 1.86E+01 | 1.23E+01 | 0.00E+00 | 0.00E+00 | |
平均值 | 4.94E-15 | 3.97E-15 | 3.85E-15 | 6.77E+00 | 3.07E-01 | 6.81E-15 | 8.88E-16 | |
标准差 | 1.25E-14 | 1.21E-15 | 2.44E-15 | 8.72E-01 | 7.89E-01 | 1.70E-15 | 0.00E+00 | |
平均值 | 0.00E+00 | 0.00E+00 | 1.49E-02 | 1.40E+00 | 3.18E-02 | 0.00E+00 | 0.00E+00 | |
标准差 | 0.00E+00 | 0.00E+00 | 6.44E-02 | 1.99E-01 | 7.51E-02 | 0.00E+00 | 0.00E+00 | |
平均值 | 2.14E-17 | 2.26E-01 | 2.71E-02 | 8.46E+00 | 3.20E-02 | 2.40E-01 | 9.64E-07 | |
标准差 | 7.03E-17 | 1.32E-01 | 2.32E-02 | 3.65E+00 | 4.57E-02 | 1.40E-01 | 2.73E-06 | |
平均值 | 2.94E-17 | 1.17E+00 | 3.73E-01 | 2.46E+01 | 2.78E-01 | 1.61E+00 | 4.44E-03 | |
标准差 | 7.37E-17 | 2.95E-01 | 1.97E-01 | 1.52E+01 | 4.26E-01 | 2.07E-01 | 1.49E-02 |
函数 | 指标 | SSA | GWO | WOA | PSO | SO | GJO | SI-DBO |
---|---|---|---|---|---|---|---|---|
平均值 | 2.93E-22 | 1.07E-67 | 1.28E-68 | 2.23E+02 | 4.94E-83 | 8.50E-28 | 0.00E+00 | |
标准差 | 1.58E-21 | 4.27E-67 | 6.90E-68 | 6.70E+01 | 2.40E-82 | 1.06E-27 | 0.00E+00 | |
平均值 | 1.13E-15 | 3.77E-03 | 9.98E+05 | 3.46E+03 | 4.07E-38 | 1.64E+00 | 0.00E+00 | |
标准差 | 4.13E-15 | 1.82E-02 | 2.87E+05 | 1.35E+03 | 2.06E-37 | 4.82E+00 | 0.00E+00 | |
平均值 | 2.80E-12 | 1.35E-41 | 8.86E-51 | 1.33E+02 | 8.80E-36 | 9.07E-18 | 5.40E-224 | |
标准差 | 1.01E-11 | 3.29E-41 | 2.94E-50 | 2.26E+01 | 8.13E-36 | 5.67E-18 | 0.00E+00 | |
平均值 | 5.66E-13 | 7.40E-23 | 7.84E+01 | 6.81E+00 | 1.35E-37 | 6.79E+00 | 6.05E-219 | |
标准差 | 1.78E-12 | 1.48E-22 | 2.36E+01 | 8.62E-01 | 2.64E-37 | 8.68E+00 | 0.00E+00 | |
平均值 | 1.22E-04 | 9.83E+01 | 9.82E+01 | 3.42E+05 | 3.02E+01 | 9.83E+01 | 4.56E+01 | |
标准差 | 4.95E-04 | 9.31E-02 | 1.32E-01 | 2.58E+05 | 3.78E+01 | 5.29E-01 | 4.86E+01 | |
平均值 | 6.39E-08 | 4.79E+00 | 3.89E+00 | 2.18E+02 | 1.59E-01 | 1.64E+01 | 1.97E-05 | |
标准差 | 1.97E-07 | 1.53E+00 | 1.35E+00 | 9.12E+01 | 2.22E-01 | 8.50E-01 | 5.56E-05 | |
平均值 | 2.11E-03 | 5.15E-04 | 3.65E-03 | 2.04E+05 | 2.16E-04 | 1.30E-03 | 1.50E-04 | |
标准差 | 3.96E-03 | 4.32E-04 | 4.82E-03 | 1.80E+05 | 1.95E-04 | 6.70E-04 | 1.17E-04 | |
平均值 | -2.54E+04 | -3.10E+02 | -3.50E+04 | -5.17E+03 | -3.94E+02 | -9.20E+03 | -4.18E+04 | |
标准差 | 1.05E+03 | 1.62E+01 | 5.41E+03 | 1.00E+03 | 1.48E-01 | 3.35E+03 | 2.77E+02 | |
平均值 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 1.18E+03 | 1.82E+01 | 3.79E-15 | 0.00E+00 | |
标准差 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 1.57E+02 | 4.08E+01 | 2.04E-14 | 0.00E+00 | |
平均值 | 1.58E-12 | 4.20E-15 | 4.20E-15 | 7.40E+00 | 4.44E-15 | 5.26E-14 | 8.88E-16 | |
标准差 | 6.27E-12 | 8.86E-16 | 2.89E-15 | 8.80E-01 | 0.00E+00 | 9.43E-15 | 0.00E+00 | |
平均值 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 1.01E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
标准差 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 8.39E-02 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
平均值 | 2.39E-10 | 1.28E-02 | 3.53E-02 | 3.81E+00 | 2.00E-03 | 6.22E-01 | 7.21E-07 | |
标准差 | 6.44E-10 | 1.92E-02 | 1.75E-02 | 1.84E+00 | 3.22E-03 | 8.61E-02 | 1.86E-06 | |
平均值 | 2.87E+00 | 3.32E+00 | 2.22E+00 | 4.53E+01 | 1.03E-01 | 8.44E+00 | 2.70E-02 | |
标准差 | 1.04E+00 | 8.32E-01 | 7.72E-01 | 1.30E+01 | 1.28E-01 | 2.55E-01 | 8.90E-02 |
表7 本文算法与其他算法的实验结果对比(F1~F13,100维)
Tab. 7 Comparison of experimental results of proposed algorithm and other algorithms(F1-F13,100 dim)
函数 | 指标 | SSA | GWO | WOA | PSO | SO | GJO | SI-DBO |
---|---|---|---|---|---|---|---|---|
平均值 | 2.93E-22 | 1.07E-67 | 1.28E-68 | 2.23E+02 | 4.94E-83 | 8.50E-28 | 0.00E+00 | |
标准差 | 1.58E-21 | 4.27E-67 | 6.90E-68 | 6.70E+01 | 2.40E-82 | 1.06E-27 | 0.00E+00 | |
平均值 | 1.13E-15 | 3.77E-03 | 9.98E+05 | 3.46E+03 | 4.07E-38 | 1.64E+00 | 0.00E+00 | |
标准差 | 4.13E-15 | 1.82E-02 | 2.87E+05 | 1.35E+03 | 2.06E-37 | 4.82E+00 | 0.00E+00 | |
平均值 | 2.80E-12 | 1.35E-41 | 8.86E-51 | 1.33E+02 | 8.80E-36 | 9.07E-18 | 5.40E-224 | |
标准差 | 1.01E-11 | 3.29E-41 | 2.94E-50 | 2.26E+01 | 8.13E-36 | 5.67E-18 | 0.00E+00 | |
平均值 | 5.66E-13 | 7.40E-23 | 7.84E+01 | 6.81E+00 | 1.35E-37 | 6.79E+00 | 6.05E-219 | |
标准差 | 1.78E-12 | 1.48E-22 | 2.36E+01 | 8.62E-01 | 2.64E-37 | 8.68E+00 | 0.00E+00 | |
平均值 | 1.22E-04 | 9.83E+01 | 9.82E+01 | 3.42E+05 | 3.02E+01 | 9.83E+01 | 4.56E+01 | |
标准差 | 4.95E-04 | 9.31E-02 | 1.32E-01 | 2.58E+05 | 3.78E+01 | 5.29E-01 | 4.86E+01 | |
平均值 | 6.39E-08 | 4.79E+00 | 3.89E+00 | 2.18E+02 | 1.59E-01 | 1.64E+01 | 1.97E-05 | |
标准差 | 1.97E-07 | 1.53E+00 | 1.35E+00 | 9.12E+01 | 2.22E-01 | 8.50E-01 | 5.56E-05 | |
平均值 | 2.11E-03 | 5.15E-04 | 3.65E-03 | 2.04E+05 | 2.16E-04 | 1.30E-03 | 1.50E-04 | |
标准差 | 3.96E-03 | 4.32E-04 | 4.82E-03 | 1.80E+05 | 1.95E-04 | 6.70E-04 | 1.17E-04 | |
平均值 | -2.54E+04 | -3.10E+02 | -3.50E+04 | -5.17E+03 | -3.94E+02 | -9.20E+03 | -4.18E+04 | |
标准差 | 1.05E+03 | 1.62E+01 | 5.41E+03 | 1.00E+03 | 1.48E-01 | 3.35E+03 | 2.77E+02 | |
平均值 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 1.18E+03 | 1.82E+01 | 3.79E-15 | 0.00E+00 | |
标准差 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 1.57E+02 | 4.08E+01 | 2.04E-14 | 0.00E+00 | |
平均值 | 1.58E-12 | 4.20E-15 | 4.20E-15 | 7.40E+00 | 4.44E-15 | 5.26E-14 | 8.88E-16 | |
标准差 | 6.27E-12 | 8.86E-16 | 2.89E-15 | 8.80E-01 | 0.00E+00 | 9.43E-15 | 0.00E+00 | |
平均值 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 1.01E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
标准差 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 8.39E-02 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
平均值 | 2.39E-10 | 1.28E-02 | 3.53E-02 | 3.81E+00 | 2.00E-03 | 6.22E-01 | 7.21E-07 | |
标准差 | 6.44E-10 | 1.92E-02 | 1.75E-02 | 1.84E+00 | 3.22E-03 | 8.61E-02 | 1.86E-06 | |
平均值 | 2.87E+00 | 3.32E+00 | 2.22E+00 | 4.53E+01 | 1.03E-01 | 8.44E+00 | 2.70E-02 | |
标准差 | 1.04E+00 | 8.32E-01 | 7.72E-01 | 1.30E+01 | 1.28E-01 | 2.55E-01 | 8.90E-02 |
函数 | 指标 | SSA | GWO | WOA | PSO | SO | GJO | SI-DBO |
---|---|---|---|---|---|---|---|---|
平均值 | 3.55E+00 | 3.76E+00 | 3.36E+00 | 9.98E-01 | 1.28E+00 | 7.18E+00 | 9.98E-01 | |
标准差 | 3.43E+00 | 3.16E+00 | 3.01E+00 | 0.00E+00 | 1.07E+00 | 4.82E+00 | 0.00E+00 | |
平均值 | 3.22E-04 | 3.68E-03 | 7.86E-04 | 1.00E-03 | 5.49E-04 | 1.15E-03 | 3.11E-04 | |
标准差 | 4.95E-05 | 5.55E-03 | 5.06E-04 | 3.86E-04 | 2.91E-04 | 3.57E-03 | 3.40E-06 | |
平均值 | -1.03E+00 | -1.03E+00 | -1.03E+00 | -1.03E+00 | -1.03E+00 | -1.03E+00 | -1.03E+00 | |
标准差 | 0.00E+00 | 1.16E-02 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
平均值 | 3.98E-01 | 4.09E-01 | 3.98E-01 | 3.98E-01 | 3.98E-01 | 3.98E-01 | 3.98E-01 | |
标准差 | 0.00E+00 | 3.31E-02 | 9.20E-06 | 0.00E+00 | 0.00E+00 | 2.15E-04 | 1.18E-04 | |
平均值 | 7.50E+00 | 3.63E+00 | 4.80E+00 | 4.80E+00 | 4.80E+00 | 3.00E+00 | 3.00E+00 | |
标准差 | 1.01E+01 | 2.27E+00 | 6.73E+00 | 6.73E+00 | 6.73E+00 | 0.00E+00 | 1.29E-03 | |
平均值 | -3.86E+00 | -3.82E+00 | -3.85E+00 | -3.86E+00 | -3.86E+00 | -3.86E+00 | -3.86E+00 | |
标准差 | 3.11E-15 | 4.57E-02 | 2.57E-02 | 3.11E-15 | 3.11E-15 | 3.66E-03 | 1.11E-03 | |
平均值 | -3.27E+00 | -2.94E+00 | -3.23E+00 | -3.26E+00 | -3.32E+00 | -3.12E+00 | -3.29E+00 | |
标准差 | 5.89E-02 | 2.30E-01 | 1.13E-01 | 5.93E-02 | 2.13E-02 | 1.17E-01 | 4.10E-02 | |
平均值 | -7.94E+00 | -5.70E+00 | -9.46E+00 | -9.98E+00 | -1.01E+01 | -8.54E+00 | -1.01E+01 | |
标准差 | 2.53E+00 | 2.01E+00 | 1.73E+00 | 9.15E-01 | 2.56E-01 | 2.47E+00 | 3.79E-02 | |
平均值 | -6.68E+00 | -5.79E+00 | -8.48E+00 | -1.01E+01 | -1.03E+01 | -9.68E+00 | -1.04E+01 | |
标准差 | 2.44E+00 | 2.53E+00 | 2.53E+00 | 1.20E+00 | 1.31E-01 | 1.80E+00 | 8.01E-02 | |
平均值 | -7.11E+00 | -5.41E+00 | -6.17E+00 | -1.05E+01 | -1.03E+01 | -9.63E+00 | -1.05E+01 | |
标准差 | 2.61E+00 | 2.16E+00 | 3.01E+00 | 1.22E-03 | 4.65E-01 | 2.31E+00 | 9.68E-02 |
表8 本文算法与其他算法的实验结果对比(F14~F23,固定维度)
Tab. 8 Comparison of experimental results of proposed algorithm and other algorithms(F14-F23, fixed dimension)
函数 | 指标 | SSA | GWO | WOA | PSO | SO | GJO | SI-DBO |
---|---|---|---|---|---|---|---|---|
平均值 | 3.55E+00 | 3.76E+00 | 3.36E+00 | 9.98E-01 | 1.28E+00 | 7.18E+00 | 9.98E-01 | |
标准差 | 3.43E+00 | 3.16E+00 | 3.01E+00 | 0.00E+00 | 1.07E+00 | 4.82E+00 | 0.00E+00 | |
平均值 | 3.22E-04 | 3.68E-03 | 7.86E-04 | 1.00E-03 | 5.49E-04 | 1.15E-03 | 3.11E-04 | |
标准差 | 4.95E-05 | 5.55E-03 | 5.06E-04 | 3.86E-04 | 2.91E-04 | 3.57E-03 | 3.40E-06 | |
平均值 | -1.03E+00 | -1.03E+00 | -1.03E+00 | -1.03E+00 | -1.03E+00 | -1.03E+00 | -1.03E+00 | |
标准差 | 0.00E+00 | 1.16E-02 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
平均值 | 3.98E-01 | 4.09E-01 | 3.98E-01 | 3.98E-01 | 3.98E-01 | 3.98E-01 | 3.98E-01 | |
标准差 | 0.00E+00 | 3.31E-02 | 9.20E-06 | 0.00E+00 | 0.00E+00 | 2.15E-04 | 1.18E-04 | |
平均值 | 7.50E+00 | 3.63E+00 | 4.80E+00 | 4.80E+00 | 4.80E+00 | 3.00E+00 | 3.00E+00 | |
标准差 | 1.01E+01 | 2.27E+00 | 6.73E+00 | 6.73E+00 | 6.73E+00 | 0.00E+00 | 1.29E-03 | |
平均值 | -3.86E+00 | -3.82E+00 | -3.85E+00 | -3.86E+00 | -3.86E+00 | -3.86E+00 | -3.86E+00 | |
标准差 | 3.11E-15 | 4.57E-02 | 2.57E-02 | 3.11E-15 | 3.11E-15 | 3.66E-03 | 1.11E-03 | |
平均值 | -3.27E+00 | -2.94E+00 | -3.23E+00 | -3.26E+00 | -3.32E+00 | -3.12E+00 | -3.29E+00 | |
标准差 | 5.89E-02 | 2.30E-01 | 1.13E-01 | 5.93E-02 | 2.13E-02 | 1.17E-01 | 4.10E-02 | |
平均值 | -7.94E+00 | -5.70E+00 | -9.46E+00 | -9.98E+00 | -1.01E+01 | -8.54E+00 | -1.01E+01 | |
标准差 | 2.53E+00 | 2.01E+00 | 1.73E+00 | 9.15E-01 | 2.56E-01 | 2.47E+00 | 3.79E-02 | |
平均值 | -6.68E+00 | -5.79E+00 | -8.48E+00 | -1.01E+01 | -1.03E+01 | -9.68E+00 | -1.04E+01 | |
标准差 | 2.44E+00 | 2.53E+00 | 2.53E+00 | 1.20E+00 | 1.31E-01 | 1.80E+00 | 8.01E-02 | |
平均值 | -7.11E+00 | -5.41E+00 | -6.17E+00 | -1.05E+01 | -1.03E+01 | -9.63E+00 | -1.05E+01 | |
标准差 | 2.61E+00 | 2.16E+00 | 3.01E+00 | 1.22E-03 | 4.65E-01 | 2.31E+00 | 9.68E-02 |
函数 | 维度 | DBO | SSA | GWO | WOA | PSO | SO | GJO |
---|---|---|---|---|---|---|---|---|
30 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | |
100 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | |
30 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | |
100 | 3.02E-11 | 5.57E-10 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | |
30 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | |
100 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | |
30 | 3.02E-11 | 5.57E-10 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | |
100 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | |
30 | 3.02E-11 | 3.02E-11 | 3.47E-10 | 3.02E-11 | 3.70E-03 | 6.53E-08 | ||
100 | 1.61E-10 | 3.02E-11 | 6.70E-11 | 3.20E-11 | 3.32E-01 | 1.43E-08 | ||
30 | 4.62E-10 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.33E-11 | 3.02E-11 | |
100 | 3.02E-11 | 4.18E-09 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 6.07E-11 | 3.02E-11 | |
30 | 3.50E-09 | 2.60E-08 | 6.77E-05 | 7.12E-09 | 3.02E-11 | 3.00E-03 | 9.83E-08 | |
100 | 1.96E-10 | 2.83E-08 | 1.86E-06 | 8.99E-11 | 3.02E-11 | 1.22E-02 | 6.70E-11 | |
30 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.82E-09 | 3.02E-11 | 3.02E-11 | 3.02E-11 | |
100 | 3.02E-12 | 3.02E-12 | 3.02E-12 | 9.76E-10 | 3.02E-12 | 3.02E-12 | 3.02E-12 | |
30 | 1.21E-12 | 6.61E-05 | ||||||
100 | 1.21E-12 | 1.40E-03 | ||||||
30 | 5.50E-03 | 2.60E-11 | 1.08E-09 | 1.21E-12 | 1.20E-13 | 1.55E-13 | ||
100 | 6.59E-04 | 7.22E-12 | 4.16E-09 | 1.21E-12 | 1.69E-14 | 1.01E-12 | ||
30 | 1.21E-12 | 1.40E-03 | ||||||
100 | 1.21E-12 | NaN | ||||||
30 | 1.73E-07 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 7.39E-11 | 3.02E-11 | |
100 | 3.02E-11 | 3.47E-10 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 6.72E-10 | 3.02E-11 | |
30 | 1.46E-10 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 2.43E-05 | 3.02E-11 | |
100 | 3.02E-11 | 4.08E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 4.74E-06 | 3.02E-11 | |
2 | 1.70E-03 | 3.00E-11 | 1.60E-03 | 5.86E-06 | 2.73E-11 | 1.91E-02 | ||
4 | 1.07E-07 | 3.02E-11 | 3.02E-11 | 5.49E-11 | 3.02E-11 | 3.02E-11 | 3.32E-06 | |
2 | 1.25E-11 | 1.72E-12 | 1.46E-10 | 3.33E-11 | 1.45E-11 | 1.34E-11 | 5.07E-10 | |
2 | 1.72E-12 | 2.36E-12 | 6.91E-04 | 1.55E-09 | 3.69E-11 | 1.21E-12 | 4.44E-07 | |
2 | 2.74E-11 | 8.31E-09 | 5.57E-10 | 2.75E-11 | 8.25E-09 | 3.02E-11 | ||
3 | 3.60E-04 | 1.30E-11 | 8.99E-11 | 2.07E-02 | 1.14E-11 | 1.25E-11 | 2.06E-01 | |
6 | 2.67E-02 | 3.02E-11 | 8.00E-03 | |||||
4 | 7.60E-03 | 3.69E-11 | 6.10E-03 | 6.06E-08 | 4.00E-02 | |||
4 | 3.02E-11 | 4.90E-05 | ||||||
4 | 8.90E-04 | 4.98E-11 | 6.17E-05 |
表9 不同算法的Wilcoxon秩和检验的结果(p值)
Tab. 9 Wilcoxon rank-sum test results of different algorithms (p-value)
函数 | 维度 | DBO | SSA | GWO | WOA | PSO | SO | GJO |
---|---|---|---|---|---|---|---|---|
30 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | |
100 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | |
30 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | |
100 | 3.02E-11 | 5.57E-10 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | |
30 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | |
100 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | 1.21E-12 | |
30 | 3.02E-11 | 5.57E-10 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | |
100 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | |
30 | 3.02E-11 | 3.02E-11 | 3.47E-10 | 3.02E-11 | 3.70E-03 | 6.53E-08 | ||
100 | 1.61E-10 | 3.02E-11 | 6.70E-11 | 3.20E-11 | 3.32E-01 | 1.43E-08 | ||
30 | 4.62E-10 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.33E-11 | 3.02E-11 | |
100 | 3.02E-11 | 4.18E-09 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 6.07E-11 | 3.02E-11 | |
30 | 3.50E-09 | 2.60E-08 | 6.77E-05 | 7.12E-09 | 3.02E-11 | 3.00E-03 | 9.83E-08 | |
100 | 1.96E-10 | 2.83E-08 | 1.86E-06 | 8.99E-11 | 3.02E-11 | 1.22E-02 | 6.70E-11 | |
30 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.82E-09 | 3.02E-11 | 3.02E-11 | 3.02E-11 | |
100 | 3.02E-12 | 3.02E-12 | 3.02E-12 | 9.76E-10 | 3.02E-12 | 3.02E-12 | 3.02E-12 | |
30 | 1.21E-12 | 6.61E-05 | ||||||
100 | 1.21E-12 | 1.40E-03 | ||||||
30 | 5.50E-03 | 2.60E-11 | 1.08E-09 | 1.21E-12 | 1.20E-13 | 1.55E-13 | ||
100 | 6.59E-04 | 7.22E-12 | 4.16E-09 | 1.21E-12 | 1.69E-14 | 1.01E-12 | ||
30 | 1.21E-12 | 1.40E-03 | ||||||
100 | 1.21E-12 | NaN | ||||||
30 | 1.73E-07 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 7.39E-11 | 3.02E-11 | |
100 | 3.02E-11 | 3.47E-10 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 6.72E-10 | 3.02E-11 | |
30 | 1.46E-10 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 2.43E-05 | 3.02E-11 | |
100 | 3.02E-11 | 4.08E-11 | 3.02E-11 | 3.02E-11 | 3.02E-11 | 4.74E-06 | 3.02E-11 | |
2 | 1.70E-03 | 3.00E-11 | 1.60E-03 | 5.86E-06 | 2.73E-11 | 1.91E-02 | ||
4 | 1.07E-07 | 3.02E-11 | 3.02E-11 | 5.49E-11 | 3.02E-11 | 3.02E-11 | 3.32E-06 | |
2 | 1.25E-11 | 1.72E-12 | 1.46E-10 | 3.33E-11 | 1.45E-11 | 1.34E-11 | 5.07E-10 | |
2 | 1.72E-12 | 2.36E-12 | 6.91E-04 | 1.55E-09 | 3.69E-11 | 1.21E-12 | 4.44E-07 | |
2 | 2.74E-11 | 8.31E-09 | 5.57E-10 | 2.75E-11 | 8.25E-09 | 3.02E-11 | ||
3 | 3.60E-04 | 1.30E-11 | 8.99E-11 | 2.07E-02 | 1.14E-11 | 1.25E-11 | 2.06E-01 | |
6 | 2.67E-02 | 3.02E-11 | 8.00E-03 | |||||
4 | 7.60E-03 | 3.69E-11 | 6.10E-03 | 6.06E-08 | 4.00E-02 | |||
4 | 3.02E-11 | 4.90E-05 | ||||||
4 | 8.90E-04 | 4.98E-11 | 6.17E-05 |
编号 | 名称 | 维度 | 范围 | 最优值 |
---|---|---|---|---|
CEC01 | Shifted and Full Rotated Zakharov Function | 10 | 300 | |
CEC02 | Shifted and Full Rotated Rosenbrock’s Function | 10 | 400 | |
CEC03 | Shifted and Full Rotated Expanded Schaffer’s Function | 10 | 600 | |
CEC04 | Shifted and Full Rotated Non-Continuous Rastrigin’s Function | 10 | 800 | |
CEC05 | Shifted and Full Rotated Levy Function | 10 | 900 | |
CEC06 | Hybrid Function 1 (N=3) | 10 | 1 800 | |
CEC07 | Hybrid Function 2 (N=6) | 10 | 2 000 | |
CEC08 | Hybrid Function 3 (N=5) | 10 | 2 200 | |
CEC09 | Composition Function 1 (N=5) | 10 | 2 300 | |
CEC10 | Composition Function 2 (N=4) | 10 | 2 400 | |
CEC11 | Composition Function 3 (N=5) | 10 | 2 600 | |
CEC12 | Composition Function 4 (N=6) | 10 | 2 700 |
表10 CEC2022测试函数
Tab. 10 CEC2022 test functions
编号 | 名称 | 维度 | 范围 | 最优值 |
---|---|---|---|---|
CEC01 | Shifted and Full Rotated Zakharov Function | 10 | 300 | |
CEC02 | Shifted and Full Rotated Rosenbrock’s Function | 10 | 400 | |
CEC03 | Shifted and Full Rotated Expanded Schaffer’s Function | 10 | 600 | |
CEC04 | Shifted and Full Rotated Non-Continuous Rastrigin’s Function | 10 | 800 | |
CEC05 | Shifted and Full Rotated Levy Function | 10 | 900 | |
CEC06 | Hybrid Function 1 (N=3) | 10 | 1 800 | |
CEC07 | Hybrid Function 2 (N=6) | 10 | 2 000 | |
CEC08 | Hybrid Function 3 (N=5) | 10 | 2 200 | |
CEC09 | Composition Function 1 (N=5) | 10 | 2 300 | |
CEC10 | Composition Function 2 (N=4) | 10 | 2 400 | |
CEC11 | Composition Function 3 (N=5) | 10 | 2 600 | |
CEC12 | Composition Function 4 (N=6) | 10 | 2 700 |
函数 | 指标 | SSA | GWO | WOA | SO | SI-DBO |
---|---|---|---|---|---|---|
CEC01 | 平均值 | 1.44E+03 | 2.32E+09 | 1.59E+03 | 1.46E+09 | 1.03E+03 |
标准差 | 1.13E+02 | 5.99E+08 | 4.79E+02 | 2.38E-07 | 1.57E+02 | |
CEC02 | 平均值 | 4.15E+02 | 6.11E+03 | 4.84E+02 | 5.85E+03 | 4.42E+02 |
标准差 | 1.98E+01 | 1.55E+02 | 9.36E+01 | 1.67E-02 | 1.97E+01 | |
CEC03 | 平均值 | 6.32E+02 | 6.70E+02 | 6.40E+02 | 6.54E+02 | 6.19E+02 |
标准差 | 1.76E+01 | 5.83E+00 | 1.13E+01 | 2.16E+00 | 7.28E+00 | |
CEC04 | 平均值 | 8.33E+02 | 8.52E+02 | 8.43E+02 | 8.30E+02 | 8.38E+02 |
标准差 | 9.75E+00 | 1.04E+01 | 1.61E+01 | 1.75E+00 | 7.67E+00 | |
CEC05 | 平均值 | 1.43E+03 | 1.79E+03 | 1.46E+03 | 1.32E+03 | 1.01E+03 |
标准差 | 1.34E+02 | 1.36E+02 | 3.15E+02 | 2.33E+01 | 1.07E+02 | |
CEC06 | 平均值 | 4.74E+03 | 8.23E+09 | 5.35E+03 | 8.22E+09 | 5.66E+05 |
标准差 | 2.20E+03 | 1.47E+07 | 3.12E+03 | 7.22E+00 | 4.28E+05 | |
CEC07 | 平均值 | 2.09E+03 | 2.15E+03 | 2.08E+03 | 2.12E+03 | 2.05E+03 |
标准差 | 4.36E+01 | 1.19E+01 | 2.20E+02 | 2.99E+00 | 8.21E+00 | |
CEC08 | 平均值 | 2.25E+03 | 1.84E+04 | 2.24E+03 | 1.84E+04 | 2.23E+03 |
标准差 | 4.87E+01 | 1.57E+02 | 8.45E+00 | 4.48E+01 | 2.98E+00 | |
CEC09 | 平均值 | 2.53E+03 | 3.71E+03 | 2.63E+03 | 3.65E+03 | 2.56E+03 |
标准差 | 2.64E+01 | 3.59E+01 | 5.53E+01 | 1.75E+00 | 3.75E+01 | |
CEC10 | 平均值 | 2.62E+03 | 3.98E+03 | 2.59E+03 | 3.59E+03 | 2.58E+03 |
标准差 | 1.97E+02 | 2.34E+02 | 1.45E+02 | 1.69E+01 | 7.03E+01 | |
CEC11 | 平均值 | 2.85E+03 | 4.87E+03 | 2.97E+03 | 4.72E+03 | 2.78E+03 |
标准差 | 1.13E+03 | 5.50E+01 | 1.26E+02 | 1.97E+01 | 1.54E+01 | |
CEC12 | 平均值 | 2.87E+03 | 4.06E+03 | 2.90E+03 | 3.84E+03 | 2.87E+03 |
标准差 | 1.55E+01 | 1.30E+02 | 3.02E+01 | 3.72E+01 | 4.93E+00 |
表11 本文算法与其他算法在CEC2022测试函数上的实验结果对比(CEC01~CEC12,10维)
Tab. 11 Comparison of experimental results of proposed algorithm and other algorithms on CEC2022 test functions (CEC01-CEC12,10 dim)
函数 | 指标 | SSA | GWO | WOA | SO | SI-DBO |
---|---|---|---|---|---|---|
CEC01 | 平均值 | 1.44E+03 | 2.32E+09 | 1.59E+03 | 1.46E+09 | 1.03E+03 |
标准差 | 1.13E+02 | 5.99E+08 | 4.79E+02 | 2.38E-07 | 1.57E+02 | |
CEC02 | 平均值 | 4.15E+02 | 6.11E+03 | 4.84E+02 | 5.85E+03 | 4.42E+02 |
标准差 | 1.98E+01 | 1.55E+02 | 9.36E+01 | 1.67E-02 | 1.97E+01 | |
CEC03 | 平均值 | 6.32E+02 | 6.70E+02 | 6.40E+02 | 6.54E+02 | 6.19E+02 |
标准差 | 1.76E+01 | 5.83E+00 | 1.13E+01 | 2.16E+00 | 7.28E+00 | |
CEC04 | 平均值 | 8.33E+02 | 8.52E+02 | 8.43E+02 | 8.30E+02 | 8.38E+02 |
标准差 | 9.75E+00 | 1.04E+01 | 1.61E+01 | 1.75E+00 | 7.67E+00 | |
CEC05 | 平均值 | 1.43E+03 | 1.79E+03 | 1.46E+03 | 1.32E+03 | 1.01E+03 |
标准差 | 1.34E+02 | 1.36E+02 | 3.15E+02 | 2.33E+01 | 1.07E+02 | |
CEC06 | 平均值 | 4.74E+03 | 8.23E+09 | 5.35E+03 | 8.22E+09 | 5.66E+05 |
标准差 | 2.20E+03 | 1.47E+07 | 3.12E+03 | 7.22E+00 | 4.28E+05 | |
CEC07 | 平均值 | 2.09E+03 | 2.15E+03 | 2.08E+03 | 2.12E+03 | 2.05E+03 |
标准差 | 4.36E+01 | 1.19E+01 | 2.20E+02 | 2.99E+00 | 8.21E+00 | |
CEC08 | 平均值 | 2.25E+03 | 1.84E+04 | 2.24E+03 | 1.84E+04 | 2.23E+03 |
标准差 | 4.87E+01 | 1.57E+02 | 8.45E+00 | 4.48E+01 | 2.98E+00 | |
CEC09 | 平均值 | 2.53E+03 | 3.71E+03 | 2.63E+03 | 3.65E+03 | 2.56E+03 |
标准差 | 2.64E+01 | 3.59E+01 | 5.53E+01 | 1.75E+00 | 3.75E+01 | |
CEC10 | 平均值 | 2.62E+03 | 3.98E+03 | 2.59E+03 | 3.59E+03 | 2.58E+03 |
标准差 | 1.97E+02 | 2.34E+02 | 1.45E+02 | 1.69E+01 | 7.03E+01 | |
CEC11 | 平均值 | 2.85E+03 | 4.87E+03 | 2.97E+03 | 4.72E+03 | 2.78E+03 |
标准差 | 1.13E+03 | 5.50E+01 | 1.26E+02 | 1.97E+01 | 1.54E+01 | |
CEC12 | 平均值 | 2.87E+03 | 4.06E+03 | 2.90E+03 | 3.84E+03 | 2.87E+03 |
标准差 | 1.55E+01 | 1.30E+02 | 3.02E+01 | 3.72E+01 | 4.93E+00 |
算法 | 平均值 | 标准差 |
---|---|---|
SI-DBO | 3.12E-16 | 1.52E-16 |
SSA | 7.64E+00 | 1.34E+01 |
GWO | 2.67E+00 | 1.63E+00 |
PSO | 1.47E+00 | 1.47E+00 |
SO | 4.78E+00 | 7.11E-01 |
表12 不同算法求解机器人夹持器问题的结果对比
Tab. 12 Comparison of results of different algorithms for solving robot gripper problem
算法 | 平均值 | 标准差 |
---|---|---|
SI-DBO | 3.12E-16 | 1.52E-16 |
SSA | 7.64E+00 | 1.34E+01 |
GWO | 2.67E+00 | 1.63E+00 |
PSO | 1.47E+00 | 1.47E+00 |
SO | 4.78E+00 | 7.11E-01 |
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