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多子群的共生非均匀高斯变异樽海鞘群算法

陈忠云 张达敏 辛梓芸

陈忠云, 张达敏, 辛梓芸. 多子群的共生非均匀高斯变异樽海鞘群算法. 自动化学报, 2022, 48(5): 1307−1317 doi: 10.16383/j.aas.c190684
引用本文: 陈忠云, 张达敏, 辛梓芸. 多子群的共生非均匀高斯变异樽海鞘群算法. 自动化学报, 2022, 48(5): 1307−1317 doi: 10.16383/j.aas.c190684
Chen Zhong-Yun, Zhang Da-Min, Xin Zi-Yun. Multi-subpopulation based symbiosis and non-uniform Gaussian mutation salp swarm algorithm. Acta Automatica Sinica, 2022, 48(5): 1307−1317 doi: 10.16383/j.aas.c190684
Citation: Chen Zhong-Yun, Zhang Da-Min, Xin Zi-Yun. Multi-subpopulation based symbiosis and non-uniform Gaussian mutation salp swarm algorithm. Acta Automatica Sinica, 2022, 48(5): 1307−1317 doi: 10.16383/j.aas.c190684

多子群的共生非均匀高斯变异樽海鞘群算法

doi: 10.16383/j.aas.c190684
基金项目: 贵州省自然科学基金([2017]1047)资助
详细信息
    作者简介:

    陈忠云:贵州大学大数据与信息工程学院硕士研究生. 主要研究方向为智能优化算法和认知无线网络. E-mail: chenzhongyun315@hotmail.com

    张达敏:贵州大学大数据与信息工程学院教授. 主要研究方向为智能优化算法和认知无线网络. 本文通信作者.E-mail: dmzhang@gzu.edu.cn

    辛梓芸:贵州大学大数据与信息工程学院硕士研究生. 主要研究方向为智能优化算法和认知无线网络.E-mail: muz_e@sina.cn

Multi-subpopulation Based Symbiosis and Non-uniform Gaussian Mutation Salp Swarm Algorithm

Funds: Supported by Natural Science Foundation of Guizhou Province ([2017]1047)
More Information
    Author Bio:

    CHEN Zhong-Yun  Master student at the School of Big Date and Information Engineering, Guizhou University. His research interest covers intelligent optimization algorithm and cognitive wireless network

    ZHANG Da-Min Professor at the School of Big Date and Information Engineering, Guizhou University. His research interest covers intelligent optimization algorithm and cognitive wireless network. Corresponding author of this paper

    XIN Zi-Yun Master student at the School of Big Date and Information Engineering, Guizhou University. Her research interest covers intelligent optimization algorithm and cognitive wireless networkreless network

  • 摘要: 针对樽海鞘群算法求解精度不高和收敛速度慢等缺点, 提出一种多子群的共生非均匀高斯变异樽海鞘群算法. 根据不同适应度值将樽海鞘链群分为三个子种群, 各个子种群分别进行领导者位置更新、追随者共生策略和链尾者非均匀高斯变异等操作. 使用统计分析、收敛速度分析、Wilcoxon检验、经典基准函数和CEC 2014函数的标准差来评估改进樽海鞘群算法的效率. 结果表明, 改进算法具有更好的寻优精度和收敛速度. 尤其在求解高维和多峰测试函数上, 改进算法拥有更好性能.
  • 图  1  c1变化曲线

    Fig.  1  c1 change curve

    图  2  基准函数平均收敛曲线

    Fig.  2  Function average convergence curves

    表  1  参数m对SSA的影响

    Table  1  Influence of parameter m on SSA

    m最佳值平均值标准差平均收敛代数
    1.02.98 × 10−81.40 × 10−32.51 × 10−3883
    1.54.67 × 10−92.68 × 10−34.34 × 10−3937
    2.01.20 × 10−143.41 × 10−34.64 × 10−3975
    2.5000847
    3.09.72 × 10−39.72 × 10−32.08 × 10−17719
    3.59.72 × 10−39.72 × 10−31.99 × 10−17621
    下载: 导出CSV

    表  2  基准函数

    Table  2  Benchmark function

    函数维度特征定义域最佳值
    f1 Sphere10US[−100, 100]0
    f2 Schwefel 1.250UN[−100, 100]0
    f3 Schwefel 2.2150US[−100, 100]0
    f4 Quartic100US[−1.28, 1.28]0
    f5 Rosenbrock100UN[−30, 30]0
    f6 Step200US[−100, 100]0
    f7 Schaffer2MN[−100, 100]0
    f8 Foxholes2MN[−65.56, 65.56]≈1
    f9 Kowalik4MN[−5, 5]3.075×10−4
    f10 Rastrigin10MS[−5.12, 5.12]0
    f11 Ackley50MN[−32, 32]0
    f12 Griewank100MN[−600, 600]0
    f13 Penalized1100MN[−50, 50]0
    f14 Penalized2200MN[−50, 50]0
    下载: 导出CSV

    表  3  参数设置

    Table  3  Parameter settings

    算法主要参数
    MSNSSAm = 2.5, R = 1 或 2
    SSSAm = 2.0, R = 1 或 2
    NSSAm = 2.0
    BOAp = 0.8, c = 0.01, a = 0.1
    MFO
    SCAa = 2
    PSOc1 = c2 = 1.5
    下载: 导出CSV

    表  4  基准函数结果对比

    Table  4  Comparison of benchmark function results

    函数算法最佳值平均值标准差SR (%)T (s)
    f1 MSNSSA 1.06 × 10−34 5.01 × 10−30 1.30 × 10−29 100 0.80
    SSSA 1.53 × 10−18 1.18 × 10−11 3.54 × 10−11 100 0.73
    NSSA 3.46 × 10−24 7.94 × 10−15 2.36 × 10−14 100 0.56
    SSA 2.52 × 10−10 6.55 × 10−10 2.35 × 10−10 100 0.28
    BOA 1.04 × 10−14 1.50 × 10−14 1.83 × 10−15 100 0.44
    MFO 1.06 × 10−32 5.64 × 10−29 2.35 × 10−28 100 0.26
    SCA 7.48 × 10−32 2.18 × 10−24 1.02 × 10−23 100 0.23
    PSO 4.39 × 10−5 2.66 × 10−4 2.10 × 10−4 0 0.25
    f2 MSNSSA 3.31 × 10−30 2.84 × 10−27 2.32 × 10−27 100 3.62
    SSSA 4.95 × 10−14 1.75 × 10−9 3.26 × 10−9 100 4.79
    NSSA 1.91 × 10−21 1.00 × 10−10 4.20 × 10−10 100 3.32
    SSA 1.29 × 103 4.81 × 103 2.25 × 103 0 2.98
    BOA 1.59 × 10−14 1.85 × 10−14 1.00 × 10−15 100 5.67
    MFO 2.21 × 104 5.52 × 104 2.49 × 104 0 3.09
    SCA 1.08 × 104 3.41 × 104 1.46 × 104 0 3.05
    PSO 7.10 × 101 1.68 × 102 6.46 × 101 0 2.89
    f3 MSNSSA 1.15 × 10−17 2.71 × 10−16 1.65 × 10−16 100 1.03
    SSSA 9.96 × 10−9 3.87 × 10−6 5.10 × 10−6 82 0.99
    NSSA 3.74 × 10−13 1.55 × 10−8 3.20 × 10−8 100 0.78
    SSA 1.11 × 101 1.82 × 101 3.61 × 100 0 0.47
    BOA 1.04 × 10−11 1.21 × 10−11 8.36 × 10−13 100 0.65
    MFO 6.77 × 101 8.19 × 101 4.86 × 100 0 0.60
    SCA 3.65 × 101 5.85 × 101 8.18 × 100 0 0.55
    PSO 4.06 × 100 5.90 × 100 1.11 × 100 0 0.40
    f4 MSNSSA 1.66 × 10−6 1.16 × 10−5 5.52 × 10−6 38 1.78
    SSSA 2.58 × 10−5 1.66 × 10−4 1.38 × 10−4 0 2.03
    NSSA 2.09 × 10−5 2.84 × 10−4 2.90 × 10−4 0 1.53
    SSA 8.23 × 10−1 1.36 × 100 3.27 × 10−1 0 1.18
    BOA 1.36 × 10−4 6.89 × 10−4 3.33 × 10−4 0 2.13
    MFO 2.37 × 101 1.78 × 102 1.20 × 102 0 1.52
    SCA 3.37 × 100 5.93 × 101 4.20 × 101 0 1.43
    PSO 2.10 × 100 4.48 × 100 2.24 × 100 0 1.01
    f5 MSNSSA 0 3.99 × 10−28 3.80 × 10−28 100 1.33
    SSSA 9.77 × 101 9.78 × 101 8.27 × 10−2 0 1.30
    NSSA 4.44 × 10−27 2.48 × 10−12 4.33 × 10−12 100 1.05
    SSA 6.60 × 102 2.20 × 103 2.71 × 103 0 0.68
    BOA 9.89 × 101 9.89 × 101 2.86 × 10−2 0 1.23
    MFO 3.03 × 106 7.18 × 107 5.39 × 107 0 1.02
    SCA 2.19 × 107 6.79 × 107 3.34 × 107 0 0.93
    PSO 6.47 × 102 1.12 × 103 3.10 × 102 0 0.52
    f6 MSNSSA 0 4.56 × 10−29 3.92 × 10−29 100 1.58
    SSSA 3.39 × 100 4.93 × 100 6.31 × 10−1 0 1.48
    NSSA 2.85 × 10−27 2.27 × 10−13 7.95 × 10−13 100 1.30
    SSA 2.16 × 103 3.34 × 103 6.74 × 102 0 0.87
    BOA 4.58 × 101 4.83 × 101 8.80 × 10−1 0 1.06
    MFO 1.27 × 105 1.82 × 105 2.02 × 104 0 1.61
    SCA 5.91 × 103 2.88 × 104 1.53 × 104 0 1.44
    下载: 导出CSV

    4  基准函数结果对比 (续表)

    4  Comparison of benchmark function results (continued table)

    函数算法最佳值平均值标准差SR (%)T (s)
    f6 PSO 4.43 × 101 7.07 × 101 1.11 × 101 0 0.63
    f7 MSNSSA 0 0 0 100 0.82
    SSSA 0 8.61 × 10−13 1.54 × 10−12 100 0.80
    NSSA 0 1.60 × 10−14 2.62 × 10−14 100 0.59
    SSA 3.66 × 10−14 5.25 × 10−3 4.89 × 10−3 46 0.33
    BOA 1.68 × 10−14 1.04 × 10−2 4.35 × 10−3 2 0.93
    MFO 0 8.16 × 10−3 3.60 × 10−3 16 0.27
    SCA 0 4.92 × 10−7 3.48 × 10−6 98 0.25
    PSO 6.15 × 10−9 3.30 × 10−3 4.65 × 10−3 62 0.30
    f8 MSNSSA 9.98 × 10−1 9.98 × 10−1 6.34 × 10−17 100 2.68
    SSSA 9.98 × 10−1 9.98 × 10−1 2.21 × 10−16 100 3.51
    NSSA 9.98 × 10−1 9.98 × 10−1 1.21 × 10−16 100 2.40
    SSA 9.98 × 10−1 1.22 × 100 6.11 × 10−1 86 2.11
    BOA 9.98 × 10−1 1.16 × 100 3.69 × 10−1 26 4.63
    MFO 9.98 × 10−1 2.28 × 100 2.05 × 100 52 2.05
    SCA 9.98 × 10−1 1.51 × 100 8.79 × 10−1 34 2.03
    PSO 9.98 × 10−1 1.38 × 100 8.44 × 10−1 66 2.08
    f9 MSNSSA 3.07 × 10−4 3.08 × 10−4 1.47 × 10−7 8 0.96
    SSSA 3.07 × 10−4 4.29 × 10−4 1.56 × 10−4 0 0.98
    NSSA 3.08 × 10−4 4.48 × 10−4 2.29 × 10−4 0 0.71
    SSA 4.83 × 10−4 3.22 × 10−3 6.40 × 10−3 0 0.45
    BOA 3.12 × 10−4 3.66 × 10−4 5.91 × 10−5 0 1.17
    MFO 3.20 × 10−4 9.25 × 10−4 3.45 × 10−4 0 0.39
    SCA 3.27 × 10−4 8.72 × 10−4 3.79 × 10−4 0 0.37
    PSO 3.22 × 10−4 1.15 × 10−3 2.80 × 10−3 0 0.41
    f10 MSNSSA 0 0 0 100 0.95
    SSSA 0 9.59 × 10−11 1.35 × 10−10 100 0.95
    NSSA 0 6.62 × 10−14 1.25 × 10−13 100 0.71
    SSA 1.99 × 100 5.33 × 100 1.27 × 100 0 0.43
    BOA 0 2.93 × 101 1.81 × 101 20 1.10
    MFO 5.97 × 100 2.32 × 101 1.22 × 101 0 0.40
    SCA 0 6.31 × 10−1 3.16 × 100 90 0.38
    PSO 3.01 × 100 1.01 × 101 4.25 × 100 0 0.38
    f11 MSNSSA 4.44 × 10−15 2.01 × 10−14 1.77 × 10−14 100 1.15
    SSSA 1.14 × 10−8 1.46 × 10−6 2.17 × 10−6 100 1.16
    NSSA 4.44 × 10−15 4.74 × 10−8 7.31 × 10−8 100 0.89
    SSA 1.56 × 100 1.97 × 100 1.58 × 10−1 0 0.59
    BOA 1.06 × 10−11 1.22 × 10−11 5.97 × 10−13 100 1.25
    MFO 1.09 × 101 1.91 × 101 1.81 × 100 0 0.72
    SCA 3.55 × 10−2 1.70 × 101 7.24 × 100 0 0.69
    PSO 4.48 × 100 6.78 × 100 1.03 × 100 0 0.50
    f12 MSNSSA 0 0 0 100 1.50
    SSSA 1.19 × 10−14 4.83 × 10−10 1.18 × 10−9 100 1.57
    NSSA 0 1.93 × 10−13 4.31 × 10−13 100 1.22
    SSA 2.11 × 10−1 3.28 × 10−1 3.39 × 10−2 0 0.86
    BOA 3.11 × 10−15 1.34 × 10−14 6.68 × 10−15 100 1.51
    下载: 导出CSV

    4  基准函数结果对比 (续表)

    4  Comparison of benchmark function results (continued table)

    函数算法最佳值平均值标准差SR (%)T (s)
    f12MFO3.82 × 1012.80 × 1021.19 × 10201.19
    SCA9.68 × 1005.36 × 1014.38 × 10101.12
    PSO8.91 × 1011.17 × 1021.36 × 10100.79
    f13MSNSSA4.78 × 10−334.97 × 10−299.06 × 10−291004.34
    SSSA3.01 × 10−25.40 × 10−21.13 × 10−205.63
    NSSA4.83 × 10−283.29 × 10−165.56 × 10−161004.00
    SSA3.04 × 1014.57 × 1019.46 × 10003.61
    BOA9.80 × 10−11.11 × 1005.44 × 10−206.74
    MFO7.21 × 1081.31 × 1092.86 × 10804.38
    SCA3.71 × 1081.01 × 1093.02 × 10804.24
    PSO3.75 × 1005.75 × 1001.01 × 10003.30
    f14MSNSSA1.35 × 10−323.61 × 10−277.51 × 10−271002.88
    SSSA5.36 × 1009.72 × 1008.22 × 10−103.61
    NSSA2.50 × 10−261.18 × 10−144.32 × 10−141002.56
    SSA1.34 × 1021.76 × 1022.38 × 10102.22
    BOA9.98 × 1009.99 × 1004.50 × 10−304.21
    MFO8.43 × 1062.38 × 1081.98 × 10802.51
    SCA8.61 × 1073.31 × 1081.46 × 10802.49
    PSO1.07 × 1021.46 × 1022.28 × 10101.99
    下载: 导出CSV

    表  5  基准函数Wilcoxon 秩和检验的p

    Table  5  p-value for Wilcoxon's rank-sum test on benchmark function

    函数SSSANSSASSABOAMFOSCAPSO
    f17.05 × 10−18 +7.05 × 10−18 +7.05 × 10−18 +7.05 × 10−18 +3.92 × 10−5 +1.16 × 10−13 +7.05 × 10−18 +
    f27.07 × 10−18 +7.07 × 10−18 +7.07 × 10−18 +7.07 × 10−18 +7.07 × 10−18 +7.07 × 10−18 +7.07 × 10−18 +
    f37.07 × 10−18 +7.07 × 10−18 +7.07 × 10−18 +7.07 × 10−18 +7.07 × 10−18 +7.07 × 10−18 +7.07 × 10−18 +
    f47.07 × 10−18 +7.07 × 10−18 +7.07 × 10−18 +7.07 × 10−18 +7.07 × 10−18 +7.07 × 10−18 +7.07 × 10−18 +
    f54.26 × 10−18 +4.26 × 10−18 +4.26 × 10−18 +4.26 × 10−18 +4.26 × 10−18 +4.26 × 10−18 +4.26 × 10−18 +
    f66.79 × 10−18 +6.79 × 10−18 +6.79 × 10−18 +6.79 × 10−18 +6.79 × 10−18 +6.79 × 10−18 +6.79 × 10−18 +
    f71.26 × 10−19 +5.96 × 10−18 +3.23 × 10−20 +3.31 × 10−20 +2.61 × 10−17 +8.22 × 10−23.31 × 10−20 +
    f83.51 × 10−18 +4.12 × 10−19 +1.25 × 10−20 +1.23 × 10−19 +1.86 × 10−6 +1.23 × 10−19 +1.23 × 10−19 +
    f94.28 × 10−11 +9.53 × 10−17 +7.07 × 10−18 +7.07 × 10−18 +7.06 × 10−18 +7.07 × 10−18 +7.07 × 10−18 +
    f104.67 × 10−19 +1.14 × 10−12 +3.31 × 10−20 +1.69 × 10−18 +3.30 × 10−20 +1.82 × 10−3 +3.31 × 10−20 +
    f115.90 × 10−18 +6.96 × 10−17 +5.90 × 10−18 +5.90 × 10−18 +5.90 × 10−18 +5.90 × 10−18 +5.90 × 10−18 +
    f123.31 × 10−20 +1.84 × 10−10 +3.31 × 10−20 +3.29 × 10−20 +3.31 × 10−20 +3.31 × 10−20 +3.31 × 10−20 +
    f137.04 × 10−18 +7.48 × 10−18 +7.04 × 10−18 +7.04 × 10−18 +7.04 × 10−18 +7.04 × 10−18 +7.04 × 10−18 +
    f147.05 × 10−18 +7.05 × 10−18 +7.05 × 10−18 +7.05 × 10−18 +7.05 × 10−18 +7.05 × 10−18 +7.05 × 10−18 +
    + / = / −12 / 0 / 012 / 0 / 012 / 0 / 012 / 0 / 012 / 0 / 011 / 0 / 112 / 0 / 0
    下载: 导出CSV

    表  6  MAE算法排名

    Table  6  MAE algorithm ranking

    算法MAE排名
    MSNSSA7.12641 × 10−21
    NSSA7.12655 × 10−22
    SSSA7.67841 × 1003
    BOA1.11882 × 1014
    PSO6.98046 × 1015
    SSA3.06178 × 1026
    SCA3.42235 × 1077
    MFO5.23029 × 1078
    下载: 导出CSV

    表  7  CEC 2014基准函数

    Table  7  CEC 2014 benchmark function

    函数维度特征定义域最佳值
    CEC0330UN[−100, 100]300
    CEC0530MN[−100, 100]500
    CEC1830HF[−100, 100]1800
    CEC2330CF[−100, 100]2300
    CEC2430CF[−100, 100]2400
    CEC2530CF[−100, 100]2500
    下载: 导出CSV

    表  8  CEC 2014优化结果对比

    Table  8  Comparison of optimization results of CEC 2014

    函数指标MSNSSASSABOAMFOSCAPSO
    CEC03 平均值 3.48173 × 104 7.13409 × 104 7.71258 × 104 1.05168 × 105 5.95030 × 104 4.93113 × 104
    标准差 3.98187 × 103 1.97053 × 104 7.95590 × 103 4.35140 × 104 1.31496 × 104 7.30314 × 103
    CEC05 平均值 5.20018 × 102 5.20177 × 102 5.21049 × 102 5.20275 × 102 5.21035 × 102 5.20990 × 102
    标准差 6.58164 × 10−3 1.35252 × 10−1 6.03943 × 10−2 1.73345 × 10−1 4.83807 × 10−2 9.88705 × 10−2
    CEC18 平均值 2.78893 × 103 1.21041 × 104 4.31418 × 109 2.18893 × 107 3.16304 × 108 5.78922 × 103
    标准差 7.12803 × 102 9.17792 × 103 2.03175 × 109 8.30950 × 107 1.92087 × 108 3.37197 × 103
    CEC23 平均值 2.50000 × 103 2.63108 × 103 2.50000 × 103 2.67493 × 103 2.71333 × 103 2.61612 × 103
    标准差 0 7.36331 × 100 0 4.24126 × 101 2.39342 × 101 1.24951 × 100
    CEC24 平均值 2.70000 × 103 2.71750 × 103 2.70000 × 103 2.71882 × 103 2.73876 × 103 2.72053 × 103
    标准差 0 5.83456 × 100 0 8.00616 × 100 7.90888 × 100 6.35266 × 100
    CEC25 平均值 2.60000 × 103 2.64087 × 103 2.60000 × 103 2.68247 × 103 2.61048 × 103 2.63564 × 103
    标准差 0 7.27821 × 100 0 3.48329 × 101 1.81045 × 101 1.02263 × 101
    下载: 导出CSV

    表  9  与参考文献中算法均值的对比

    Table  9  Comparison of the mean with algorithm in references

    算法f1f2f3f4f5f6f7
    MSNSSA 7.35 × 10−36 9.69 × 10−32 1.47×10−21 3.53 × 10−6 0 0 0
    MFOA-SQP[18] 0 5.62 × 10−11 5.96 × 10−6 5.71 × 10−3 2.88 × 101 2.53 × 10−4 0
    CSO[19] 0 1.79 × 10−9 1.63 × 10−5 6.15 × 10−4 1.65 × 102 6.06 × 10−3 0
    HCPSO[20] 8.71 × 10−28 3.39 × 10−3 1.38 × 10−2 2.57 × 10−4 3.14×10−5 5.76 × 10−3 3.68 × 10−10
    DMS-PSO[21] 4.29 × 10−12 4.54 × 10−6 2.06 × 101 1.07 × 10−2 2.77 × 101 5.68 × 10−2 7.31 × 10−4
    PSO-SMS[11] 3.55 × 10−20 9.82 × 10−8 1.53 × 10−5 2.09 × 10−2 2.59 × 101 3.54 × 10−4 7.19 × 10−3
    CASSA[22] 9.35 × 10−147 2.84×10−52 8.66 × 10−6 1.88 × 10−5 2.77 × 101 9.81 × 10−2 0
    CESSA[23] 2.50 × 10−23 4.22 × 10−3 1.73 × 10−15 5.90 × 10−5 2.86 × 101 7.51 × 10−2 0
    EHO[5] 9.63 × 10−7 4.71 × 10−4 6.93 × 10−1 1.25 × 10−5 2.85 × 101 6.55 × 100 6.30 × 10−5
    EWA[6] 7.25 × 101 2.15 × 100 2.48 × 10−5 1.10 × 10−1 5.14 × 103 2.74 × 103 9.93 × 10−3
    MBO[7] 8.53 × 10−3 4.17 × 10−4 2.66 × 10−1 4.46 × 10−1 2.05 × 10−7 1.42 × 100 1.52 × 10−2
    MABC[24] 6.02 × 10−4 6.53 × 10−6 9.55 × 100 1.07 × 10−2 2.25 × 10−8 5.98 × 100 2.91 × 10−1
    MIWO[25] 3.17 × 10−5 5.41 × 10−10 9.34 × 10−13 8.41 × 10−3 5.28 × 10−1 7.68 × 10−2 4.50 × 10−1
    MPEA[26] 2.70 × 10−11 1.52 × 10−20 1.04 × 10−2 2.35 × 10−1 6.74 × 10−12 3.26 × 10−5 8.74 × 10−3
    算法 f8 f9 f10 f11 f12 f13 f14
    MSNSSA 9.98 × 10−1 3.08 × 10−4 0 8.88 × 10−16 0 1.39 × 10−34 3.27 × 10−30
    MFOA-SQP[18] 9.98 × 10−1 1.06 × 10−3 0 3.55 × 10−15 0 3.71 × 10−6 7.53 × 10−10
    CSO[19] 9.98 × 10−1 6.03 × 10−4 1.12 × 10−7 1.24 × 10−12 0 1.64 × 10−7 2.24 × 10−1
    HCPSO[20] 9.98 × 10−1 1.40 × 10−2 2.49 × 10−5 2.26 × 10−4 8.67 × 10−5 2.69 × 10−13 4.18 × 10−3
    DMS-PSO[21] 2.13 × 100 5.68 × 10−1 3.88 × 101 1.88 × 100 2.24 × 10−2 2.87 × 10−3 6.88 × 10−1
    PSO-SMS[11] 9.98 × 10−1 2.09 × 10−2 1.53 × 101 2.99 × 100 7.23 × 10−2 1.12 × 10−5 1.76 × 10−8
    CASSA[22] 9.98 × 10−1 4.81 × 10−3 0 8.88 × 10−16 0 2.33 × 10−20 1.68 × 10−2
    CESSA[23] 9.98 × 10−1 2.59 × 10−3 1.48 × 101 1.06 × 10−2 2.88 × 10−1 5.68 × 10−18 2.62 × 101
    EHO[5] 1.67 × 100 1.27 × 10−1 1.21 × 10−6 2.39 × 10−4 1.89 × 10−6 1.35 × 10−1 6.86 × 101
    EWA[6] 1.50 × 100 1.76 × 10−3 3.10 × 101 3.05 × 100 1.53 × 100 5.77×10−1 1.35 × 10−3
    MBO[7] 9.98 × 10−1 1.78 × 10−1 5.86 × 10−1 1.13 × 10−1 8.05 × 10−1 7.20 × 10−15 7.12 × 10−1
    MABC[24] 1.41 × 100 4.32 × 10−4 4.15 × 10−2 2.05 × 10−1 5.63 × 10−2 8.05 × 10−9 1.18 × 10−2
    MIWO[25] 1.89 × 100 1.01 × 10−2 4.62 × 10−1 1.86 × 10−1 3.29 × 10−2 1.98 × 10−1 1.35 × 10−3
    MPEA[26] 9.98 × 10−1 1.46 × 10−3 6.38 × 10−6 2.84 × 10−1 4.38 × 10−7 2.01×10−6 2.27 × 10−5
    下载: 导出CSV

    表  10  与参考文献中算法标准差的对比

    Table  10  Comparison of the standard deviation with algorithms in reference

    算法f1f2f3f4f5f6f7
    MSNSSA 1.04 × 10−35 4.52 × 10−32 4.75×10−21 1.75 × 10−6 0 0 0
    MFOA-SQP[18] 0 2.20 × 10−11 2.03 × 10−6 4.12 × 10−3 5.10 × 10−2 2.17 × 10−4 0
    CSO[19] 0 1.04 × 10−9 4.73 × 10−6 3.12 × 10−2 7.39 × 102 4.75 × 10−3 0
    HCPSO[20] 3.55 × 10−28 2.04 × 10−3 5.80 × 10−3 1.88 × 10−5 1.07 × 10−4 3.48 × 10−3 5.97 × 10−10
    DMS-PSO[21] 3.00 × 10−11 2.23 × 10−5 7.48 × 100 1.03 × 10−3 2.69 × 100 4.87 × 10−2 3.81 × 10−1
    PSO-SMS[11] 4.61 × 10−20 1.47 × 10−7 4.65 × 10−6 2.50 × 10−3 2.19 × 100 2.21 × 10−4 1.57 × 10−4
    CASSA[22] 2.32 × 10−147 2.27×10−50 4.15 × 10−6 1.21 × 10−5 1.16 × 10−1 4.08 × 10−2 0
    CESSA[23] 1.84 × 10−23 1.51 × 10−2 1.25 × 10−13 5.08 × 10−5 4.89 × 10−2 3.11 × 10−2 0
    EHO[5] 1.26 × 10−7 8.23 × 10−4 8.44 × 10−1 1.26 × 10−5 1.83 × 10−2 7.56 × 100 6.21 × 10−5
    EWA[6] 7.43 × 101 1.54 × 100 7.37 × 10−6 8.73 × 10−2 8.93 × 103 2.58 × 10−3 1.93 × 10−4
    MBO[7] 1.28 × 10−4 1.83 × 10−4 3.00 × 100 3.89 × 10−1 3.54 × 10−7 3.65 × 10−1 1.10 × 10−2
    MABC[24] 7.23 × 10−3 3.63 × 10−3 1.18 × 100 1.77 × 10−1 3.87 × 10−7 1.21 × 10−1 1.55 × 10−1
    MIWO[25] 4.32 × 10−6 1.28 × 10−5 2.21 × 10−12 2.63 × 10−3 7.54 × 10−1 1.82 × 10−2 2.09 × 10−2
    MPEA[26] 5.74 × 10−10 4.13 × 10−18 1.80 × 10−1 5.22 × 10−2 3.19 × 10−10 2.63−6 3.56 × 10−1
    算法 f8 f9 f10 f11 f12 f13 f14
    MSNSSA 2.95 × 10−23 3.56 × 10−8 0 0 0 3.56 × 10−34 7.31 × 10−30
    MFOA-SQP[18] 1.13 × 10−1 4.47 × 10−4 0 1.32 × 10−12 0 1.76 × 10−6 5.56 × 10−10
    CSO[19] 8.01 × 100 9.92 × 10−4 3.16 × 10−5 1.01 × 10−11 0 4.74 × 10−7 1.17 × 10−1
    HCPSO[20] 2.96 × 100 6.22 × 10−2 1.05 × 10−5 2.52 × 10−4 2.79 × 10−6 5.95 × 10−11 1.84 × 10−5
    DMS-PSO[21] 5.94 × 10−1 8.04 × 10−1 2.80 × 100 2.46 × 10−1 1.77 × 10−2 7.54 × 10−1 5.02 × 10−1
    PSO-SMS[11] 2.77 × 10−1 5.76 × 10−3 1.29 × 100 3.87 × 10−1 6.36 × 10−2 3.35 × 10−6 9.20 × 10−9
    CASSA[22] 2.82 × 10−1 2.56 × 10−5 0 9.86 × 10−32 0 8.12 × 10−18 1.35 × 10−2
    CESSA[23] 9.19×101 8.90 × 10−1 2.15 × 101 5.31 × 10−2 3.41 × 10−1 1.08 × 10−18 6.14 × 100
    EHO[5] 8.37 × 10−1 1.62 × 100 2.30 × 10−7 1.37 × 10−5 3.17 × 10−7 2.20 × 100 3.37 × 101
    EWA[6] 2.28 × 10−1 4.22 × 10−2 1.87 × 101 1.24 × 100 4.99 × 10−1 3.22 × 10−1 2.04 × 10−3
    MBO[7] 3.95 × 100 4.65 × 100 4.17 × 10−1 7.19 × 100 8.32 × 10−1 1.27 × 10−12 2.27 × 10−1
    MABC[24] 6.58 × 100 8.28 × 10−1 2.41 × 10−1 1.65 × 10−1 5.66 × 10−1 6.77 × 10−1 1.67 × 10−1
    MIWO[25] 1.92 × 101 2.07 × 10−2 1.30 × 10−1 2.68 × 100 5.31 × 10−1 2.30 × 10−1 1.42 × 10−2
    MPEA[26] 5.37 × 10−1 7.43 × 10−3 5.35 × 10−6 3.91 × 10−1 7.48 × 10−3 5.94 × 10−8 6.36 × 10−4
    下载: 导出CSV
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出版历程
  • 收稿日期:  2019-09-30
  • 录用日期:  2020-04-16
  • 网络出版日期:  2022-04-20
  • 刊出日期:  2022-05-13

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