计算机科学与探索 ›› 2022, Vol. 16 ›› Issue (1): 88-105.DOI: 10.3778/j.issn.1673-9418.2107028
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。基金资助:
ZHANG Jiulong1,+(), WANG Xiaofeng1,2, LU Lei1, NIU Pengfei1
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.Supported by:
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。
中图分类号:
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.
算法 | 改进策略 | 优势 | 局限性 | 应用场景 |
---|---|---|---|---|
文献[ | 引入Lévy飞行策略[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 引入Lévy飞行策略[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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 |
算法 | 改进策略 | 优势 | 局限性 | 应用场景 |
---|---|---|---|---|
文献[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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 |
算法 | 改进策略 | 优势 | 局限性 | 应用场景 |
---|---|---|---|---|
文献[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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 |
算法 | 改进策略 | 优势 | 局限性 | 应用场景 |
---|---|---|---|---|
文献[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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 |
算法 | 简要分析 | 优势 | 局限性 | 应用场景 |
---|---|---|---|---|
文献[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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 |
算法 | 简要分析 | 优势 | 局限性 | 应用场景 |
---|---|---|---|---|
文献[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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[ | 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 |
Test function | Range | Dimension | |
---|---|---|---|
| [-100,100] | 30 | 0 |
| [-10,10] | 30 | 0 |
| [-100,100] | 30 | 0 |
| [-100,100] | 30 | 0 |
| [-30,30] | 30 | 0 |
| [-100,100] | 30 | 0 |
| [-1.28,1.28] | 30 | 0 |
表7 单峰测试函数
Table 7 Single peak test function
Test function | Range | Dimension | |
---|---|---|---|
| [-100,100] | 30 | 0 |
| [-10,10] | 30 | 0 |
| [-100,100] | 30 | 0 |
| [-100,100] | 30 | 0 |
| [-30,30] | 30 | 0 |
| [-100,100] | 30 | 0 |
| [-1.28,1.28] | 30 | 0 |
Test function | Range | Dimension | |
---|---|---|---|
| [-500,500] | 30 | -418.983 9n |
| [-5.12,5.12] | 30 | 0 |
| [-32,32] | 30 | 0 |
| [-600,600] | 30 | 0 |
| [-50,50] | 30 | 0 |
| [-50,50] | 30 | 0 |
表8 多峰测试函数
Table 8 Multimodal peak test function
Test function | Range | Dimension | |
---|---|---|---|
| [-500,500] | 30 | -418.983 9n |
| [-5.12,5.12] | 30 | 0 |
| [-32,32] | 30 | 0 |
| [-600,600] | 30 | 0 |
| [-50,50] | 30 | 0 |
| [-50,50] | 30 | 0 |
Test function | Range | Dimension | |
---|---|---|---|
| [-65.536,65.536] | 2 | 0.998 |
| [-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 |
| [1,3] | 3 | -3.86 |
| [0,1] | 6 | -3.32 |
| [0,10] | 4 | 10.153 2 |
| [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 | |
---|---|---|---|
| [-65.536,65.536] | 2 | 0.998 |
| [-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 |
| [1,3] | 3 | -3.86 |
| [0,1] | 6 | -3.32 |
| [0,10] | 4 | 10.153 2 |
| [0,10] | 4 | -10.536 3 |
算法 | 参数 | 取值 |
---|---|---|
BOA | | 0.01 |
| 0.8 | |
| 0.1 | |
| 0.025 | |
| [0,1] | |
MFO | | 1 |
| 50 | |
| [-1,1] | |
SCA | | 2 |
| | |
| [-2,2] | |
| [0,1] | |
GOA | | 1 |
| 0.000 04 | |
| 0.5 | |
| 2 | |
HHO | | 1.5 |
| [0,1] | |
| [-1,1] | |
| [0,2] | |
SSA | | 0.8 |
| 0.000 000 01 | |
| [0,1] | |
| [-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 | | 0.01 |
| 0.8 | |
| 0.1 | |
| 0.025 | |
| [0,1] | |
MFO | | 1 |
| 50 | |
| [-1,1] | |
SCA | | 2 |
| | |
| [-2,2] | |
| [0,1] | |
GOA | | 1 |
| 0.000 04 | |
| 0.5 | |
| 2 | |
HHO | | 1.5 |
| [0,1] | |
| [-1,1] | |
| [0,2] | |
SSA | | 0.8 |
| 0.000 000 01 | |
| [0,1] | |
| [-1,1] |
函数 | 统计值 | BOA | MFO | SCA | GOA | HHO | SSA |
---|---|---|---|---|---|---|---|
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 |
---|---|---|---|---|---|---|---|
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 |
函数 | BOA | MFO | SCA | GOA | HHO | SSA |
---|---|---|---|---|---|---|
| 0.452 1 | 0.289 1 | 0.265 6 | 132.030 7 | 0.802 6 | 0.787 5 |
| 0.443 2 | 0.304 2 | 0.273 4 | 129.928 6 | 0.479 2 | 0.883 9 |
| 2.514 1 | 1.295 8 | 1.294 3 | 136.860 9 | 2.918 8 | 1.861 5 |
| 0.501 6 | 0.318 2 | 0.309 9 | 148.181 3 | 0.623 9 | 0.893 7 |
| 0.777 1 | 0.393 2 | 0.364 1 | 146.655 2 | 0.910 4 | 0.947 4 |
| 0.427 1 | 0.278 7 | 0.238 5 | 144.043 2 | 0.603 6 | 0.800 4 |
| 0.738 0 | 0.435 4 | 0.420 8 | 148.610 4 | 0.899 5 | 1.030 7 |
| 0.725 4 | 0.335 9 | 0.301 6 | 127.057 3 | 0.720 8 | 0.947 4 |
| 0.773 9 | 0.354 2 | 0.340 6 | 142.702 6 | 0.807 8 | 0.916 2 |
| 0.737 5 | 0.405 7 | 0.404 2 | 147.189 1 | 0.908 3 | 1.016 1 |
| 0.703 7 | 0.378 1 | 0.352 1 | 144.982 3 | 0.899 4 | 0.963 5 |
| 1.519 3 | 0.785 9 | 0.750 5 | 149.763 5 | 1.827 1 | 1.362 5 |
| 1.464 1 | 0.778 7 | 0.736 9 | 149.042 7 | 1.733 9 | 1.317 7 |
| 3.414 1 | 1.626 3 | 1.623 4 | 11.192 7 | 4.121 9 | 1.705 2 |
| 0.571 9 | 0.221 3 | 0.202 1 | 18.654 2 | 0.714 6 | 0.370 3 |
| 0.467 7 | 0.159 9 | 0.150 5 | 9.525 4 | 0.575 5 | 0.270 8 |
| 0.445 3 | 0.151 6 | 0.156 8 | 9.895 8 | 0.603 1 | 0.279 2 |
| 0.914 1 | 0.301 6 | 0.285 4 | 19.445 3 | 0.913 0 | 0.433 3 |
| 0.963 0 | 0.309 9 | 0.295 3 | 28.769 3 | 0.951 6 | 0.482 8 |
| 1.061 5 | 0.401 0 | 0.388 5 | 19.716 1 | 1.152 1 | 0.557 8 |
| 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 |
---|---|---|---|---|---|---|
| 0.452 1 | 0.289 1 | 0.265 6 | 132.030 7 | 0.802 6 | 0.787 5 |
| 0.443 2 | 0.304 2 | 0.273 4 | 129.928 6 | 0.479 2 | 0.883 9 |
| 2.514 1 | 1.295 8 | 1.294 3 | 136.860 9 | 2.918 8 | 1.861 5 |
| 0.501 6 | 0.318 2 | 0.309 9 | 148.181 3 | 0.623 9 | 0.893 7 |
| 0.777 1 | 0.393 2 | 0.364 1 | 146.655 2 | 0.910 4 | 0.947 4 |
| 0.427 1 | 0.278 7 | 0.238 5 | 144.043 2 | 0.603 6 | 0.800 4 |
| 0.738 0 | 0.435 4 | 0.420 8 | 148.610 4 | 0.899 5 | 1.030 7 |
| 0.725 4 | 0.335 9 | 0.301 6 | 127.057 3 | 0.720 8 | 0.947 4 |
| 0.773 9 | 0.354 2 | 0.340 6 | 142.702 6 | 0.807 8 | 0.916 2 |
| 0.737 5 | 0.405 7 | 0.404 2 | 147.189 1 | 0.908 3 | 1.016 1 |
| 0.703 7 | 0.378 1 | 0.352 1 | 144.982 3 | 0.899 4 | 0.963 5 |
| 1.519 3 | 0.785 9 | 0.750 5 | 149.763 5 | 1.827 1 | 1.362 5 |
| 1.464 1 | 0.778 7 | 0.736 9 | 149.042 7 | 1.733 9 | 1.317 7 |
| 3.414 1 | 1.626 3 | 1.623 4 | 11.192 7 | 4.121 9 | 1.705 2 |
| 0.571 9 | 0.221 3 | 0.202 1 | 18.654 2 | 0.714 6 | 0.370 3 |
| 0.467 7 | 0.159 9 | 0.150 5 | 9.525 4 | 0.575 5 | 0.270 8 |
| 0.445 3 | 0.151 6 | 0.156 8 | 9.895 8 | 0.603 1 | 0.279 2 |
| 0.914 1 | 0.301 6 | 0.285 4 | 19.445 3 | 0.913 0 | 0.433 3 |
| 0.963 0 | 0.309 9 | 0.295 3 | 28.769 3 | 0.951 6 | 0.482 8 |
| 1.061 5 | 0.401 0 | 0.388 5 | 19.716 1 | 1.152 1 | 0.557 8 |
| 1.554 2 | 0.568 2 | 0.541 2 | 19.551 6 | 1.542 2 | 0.691 7 |
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