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自适应变化响应的动态多目标进化算法

梁正平 李辉才 王志强 胡凯峰 朱泽轩

梁正平, 李辉才, 王志强, 胡凯峰, 朱泽轩. 自适应变化响应的动态多目标进化算法. 自动化学报, 2021, x(x): 1001−1018 doi: 10.16383/j.aas.c210121
引用本文: 梁正平, 李辉才, 王志强, 胡凯峰, 朱泽轩. 自适应变化响应的动态多目标进化算法. 自动化学报, 2021, x(x): 1001−1018 doi: 10.16383/j.aas.c210121
Liang Zheng-Ping, Li Hui-Cai, Wang Zhi-Qiang, Hu Kai-Feng, Zhu Ze-Xuan. Dynamic multi-objective evolutionary algorithm with adaptive change response. Acta Automatica Sinica, 2021, x(x): 1001−1018 doi: 10.16383/j.aas.c210121
Citation: Liang Zheng-Ping, Li Hui-Cai, Wang Zhi-Qiang, Hu Kai-Feng, Zhu Ze-Xuan. Dynamic multi-objective evolutionary algorithm with adaptive change response. Acta Automatica Sinica, 2021, x(x): 1001−1018 doi: 10.16383/j.aas.c210121

自适应变化响应的动态多目标进化算法

doi: 10.16383/j.aas.c210121
基金项目: 国家自然科学基金(61871272); 广东省自然科学基金(2020A1515010479, 2021A1515011911); 深圳市稳定支持面上项目(20200811181752003)
详细信息
    作者简介:

    梁正平:深圳大学计算机与软件学院副教授. 2006年获武汉大学博士学位. 主要研究方向为计算智能, 大数据分析与应用等. 本文通讯作者. E-mail: liangzp@szu.edu.cn

    李辉才:深圳大学计算机与软件学院硕士生. 主要研究方向为计算智能及应用. E-mail: 1810273028@email.szu.edu.cn

    王志强:深圳大学计算机与软件学院教授. 主要研究方向为计算智能, 大数据分析与应用, 多媒体技术与应用等. E-mail: wangzq@szu.edu.cn

    胡凯峰:深圳大学信息中心工程师. 2019年获深圳大学硕士学位. 主要研究方向为计算智能及其应用. E-mail: kaifeng@szu.edu.cn

    朱泽轩:深圳大学计算机与软件学院副教授. 2006年获武汉大学博士学位. 主要研究方向为计算智能, 大数据分析与应用等. 本文通讯作者. E-mail: zhuzx@szu.edu.cn

Dynamic Multi-objective Evolutionary Algorithm with Adaptive Change Response

Funds: National Natural Science Foundation of China (61871272); Natural Science Foundation of Guangdong, China (2020A1515010479, 2021A1515011911); Shenzhen Fundamental Research Program (20200811181752003)
More Information
    Author Bio:

    LIANG Zheng-Ping Associate professor at the School of Computer Science & Software Engineering, Shenzhen University. He received his Ph. D. degree from Wuhan University in 2006. His research interests include computational intelligence, big data analysis and application

    LI Hui-Cai Master student at the School of Computer Science & Software Engineering, Shenzhen University. His research interests include computational intelligence and applications

    WANG Zhi-Qiang Professor at the School of Computer Science & Software Engineering, Shenzhen University. His research interests include computational intelligence, big data analysis and applications, multimedia technology and applications

    HU Kai-Feng Engineer at Information Center of Shenzhen University. He received his master degree from Shenzhen University in 2019. His research interests include computational intelligence and applications

    ZHU Ze-Xuan Professor at the School of Computer Science & Software Engineering, Shenzhen University. He received his Ph. D. degree from Nanyang Technological University in 2008. His research interests include computational intelligence, machine learning and bioinformatics

  • 摘要: 动态多目标优化问题的目标函数发生变化时, 需要采取变化响应策略对种群进行重新初始化, 以快速追踪新环境中的最优解集. 现有动态多目标优化算法对不同个体不同维度的决策变量缺乏针对性的变化响应, 导致重新初始化的效果尚存在较大改进空间. 为此, 本文提出了一种对不同个体不同维度的决策变量分别进行自适应变化响应的动态多目标进化算法(DMOEA-ACR). 该算法包括两个核心部分, 首先是对t时间步最优种群和t−1时间步最优种群中对应个体各维度决策变量之间的差异进行计算, 自适应选择变异策略或预测策略重新初始化不同个体不同维度的决策变量. 其次, 在每轮迭代或重新初始化后, 对非支配个体进行存档, 基于存档中心构建预测策略. 为了验证DMOEA-ACR的有效性, 将其与动态多目标优化领域的6种先进算法在最新测试问题集SDP和DF上进行对比, 实验结果表明DMOEA-ACR在求解动态多目标优化问题时具有明显的优势.
  • 图  1  二维决策空间最理想情形下的重新初始化示意图

    Fig.  1  Reinitialization illustration of the most ideal situation in 2D decision space

    图  2  自适应变化响应示意图

    Fig.  2  Illustration of adaptive change respond

    图  3  基于存档中心的预测策略示意图

    Fig.  3  Illustration of prediction strategy based on the central of archive

    图  4  DNSGA-II-A, DNSGA-II-B, MOEA/D-KF, SGEA, MOEA/D-MoE和DMOEA-ACR在SDP1, SDP5,SDP6, DF3, DF5, DF8问题上的log(IGD)均值变化比较

    Fig.  4  Comparison of average log(IGD) trends of DNSGA-II-A, DNSGA-II-B, MOEA/D-KF, SGEA, MOEA/D-MoE and DMOEA-ACR on SDP1, SDP5, SDP6, DF3, DF5, DF8

    图  5  DNSGA-II-A, DNSGA-II-B, MOEA/D-KF和SGEA在SDP1, SDP6, DF3, DF8问题上得到的PF图

    Fig.  5  PF graph obtained by DNSGA-II-A, DNSGA-II-B, MOEA/D-KF and SGEA on SDP1, SDP6, DF3, DF8

    图  6  Tr-DMOEA, MOEA/D-MoE和DMOEA-ACR在SDP1, SDP6, DF3, DF8问题上得到的PF图

    Fig.  6  PF graph obtained by Tr-DMOEA, MOEA/D-MoE and DMOEA-ACR on SDP1, SDP6, DF3, DF8

    图  7  τt 分别为5,10,20时DMOEA-ACR在SDP1(2), SDP5(2),SDP6(2), DF3, DF5,DF8问题上的log(IGD)均值变化比较

    Fig.  7  Comparison of average log(IGD) trends of DMOEA-ACR on SDP1(2), SDP5(2), SDP6(2), DF3, DF5, DF8 where τt is 5, 10 and 20 respectively

    表  1  DNSGA-II-A, DNSGA-II-B, MOEA/D-KF, SGEA, Tr-DMOEA, MOEA/D-MoE和DMOEA-ACR在SDP上获得的MIGD均值和标准差

    Table  1  The mean and standard deviation of MIGD of DNSGA-II-A, DNSGA-II-B, MOEA/D-KF, SGEA, Tr-DMOEA, MOEA/D-MoE and DMOEA-ACR were obtained on SDP

    ProblemsMDNSGA-II-ADNSGA-II-BMOEA/D-KFSGEATr-DMOEAMOEA/D-MoEDMOEA-ACR
    SDP122.03×10−2
    (6.73×10−3)(−)
    2.01×10−2
    (5.34×10−3)(−)
    6.69×10−1
    (2.61×10−3)(−)
    1.66×10−2
    (5.12×10−3)(≈)
    2.53×10−1
    (2.65×10−2)(−)
    5.78×10−1
    (3.13×10−4)(−)
    1.65×10−2
    (3.67×10−3)
    32.26×10−1
    (3.02×10−3)(−)
    2.32×10−1
    (6.78×10−3)(−)
    7.74×10−1
    (6.35×10−3)(−)
    1.40×10−1
    (4.74×10−3)(−)
    1.44×10−1
    (7.32×10−2)(−)
    7.46×10−1
    (2.14×10−3)(−)
    1.32×10−1
    (5.44×10−3)
    SDP221.75×10−2
    (7.58×10−3)(+)
    1.74×10−2
    (7.05×10−3)(+)
    4.38×10−2
    (4.34×10−3)(+)
    1.00×100
    (3.17×10−3)(−)
    4.93×100
    (5.43×10−1)(−)
    3.23×10−2
    (8.38×10−3))(+)
    1.14×10−1
    (4.35×10−3)
    32.87×10−1
    (7.18×10−3)(−)
    3.60×10−1
    (8.31×10−3)(−)
    3.36×10−1
    (7.67×10−4)(−)
    9.23×10−1
    (1.59×10−2)(−)
    4.80×10−1
    (6.61×10−2)(−)
    3.34×10−1
    (1.39×10−3)(−)
    2.42×10−1
    (7.12×10−3)
    SDP321.76×100
    (6.74×10−2)(+)
    1.81×10−1
    (3.97×10−2)(+)
    6.37×10−2
    (4.18×10−3)(+)
    2.07×10−1
    (2.04×10−3)(+)
    1.45×10+1
    (3.41×10−1)(+)
    4.58×10−1
    (1.30×10−3)(+)
    4.74×100
    (6.38×10−2)
    37.01×100
    (8.34×10−2)(−)
    8.51×100
    (7.37×10−2)(−)
    8.54×10−1
    (1.43×10−2)(+)
    3.74×10−1
    (3.55×10−3)(+)
    2.36×10−1
    (3.67×10−2)(+)
    5.47×10−1
    (4.42×10−3)(+)
    3.33×100
    (8.53×10−2)
    SDP421.14×10−1
    (6.19×10−3)(−)
    7.86×10−2
    (3.07×10−2)(−)
    5.63×10−2
    (1.01×10−3)(−)
    5.56×10−2
    (4.76×10−3)(−)
    3.65×100
    (4.09×10−1)(−)
    5.54×10−2
    (3.61×10−3)(−)
    5.08×10−2
    (5.03×10−3)
    32.44×10−1
    (5.08×10−3)(−)
    2.26×10−1
    (7.48×10−2)(−)
    1.81×10−1
    (3.28×10−3)(−)
    1.95×10−1
    (2.43×10−3)(−)
    1.74×10−1
    (5.48×10−2)(−)
    1.79×10−1
    (2.71×10−2)(−)
    1.69×10−1
    (8.25×10−3)
    SDP529.40×10−2
    (4.21×10−3)(−)
    6.27×10−1
    (8.05×10−3)(−)
    9.74×10−2
    (4.07×10−3)(−)
    7.19×10−1
    (2.79×10−2)(−)
    8.35×10−1
    (7.87×10−3)(−)
    9.86×10−2
    (4.37×10−4)(−)
    6.81×10−3
    (4.92×10−4)
    31.41×10−1
    (6.08×10−2)(−)
    5.24×10−1
    (3.43×10−2)(−)
    1.54×10−1
    (7.81×10−3)(−)
    8.68×10−2
    (7.15×10−3)(−)
    1.46×10−1
    (6.43×10−2)(−)
    1.50×10−1
    (7.61×10−3)(−)
    6.38×10−2
    (6.06×10−3)
    SDP622.47×10−2
    (1.53×10−3)(−)
    3.14×10−2
    (5.92×10−3)(−)
    2.45×10−2
    (3.69×10−3)(−)
    8.56×10−2
    (2.72×10−3)(−)
    4.28×10−1
    (5.97×10−3)(−)
    2.41×10−2
    (5.81×10−3)(−)
    4.26×10−3
    (7.43×10−4)
    31.61×10−1
    (6.44×10−2)(−)
    1.79×10−1
    (6.45×10−2)(−)
    1.03×10−1
    (6.61×10−3)(−)
    5.38×10−2
    (6.05×10−3)(−)
    6.60×10−1
    (6.06×10−3)(−)
    5.33×10−2
    (3.86×10−2)(−)
    5.17×10−2
    (5.86×10−3)
    SDP725.15×10−2
    (2.46×10−3)(+)
    2.96×10−2
    (7.28×10−3)(+)
    3.49×10−1
    (3.54×10−3)(−)
    2.67×10−1
    (6.73×10−3)(−)
    8.12×10−1
    (3.78×10−3)(−)
    3.11×10−1
    (6.65×10−3)(−)
    2.15×10−1
    (1.68×10−2)
    32.41×10−1
    (3.12×10−3)(−)
    2.87×10−1
    (8.30×10−2)(−)
    3.76×10−1
    (5.71×10−3)(−)
    2.59×10−1
    (2.72×10−2)(−)
    3.71×10−1
    (4.97×10−2)(−)
    3.55×10−1
    (8.36×10−2)(−)
    2.24×10−1
    (5.11×10−3)
    SDP821.40×10−1
    (6.12×10−2)(−)
    1.56×10−1
    (6.91×10−3)(−)
    1.15×10−1
    (1.31×10−3)(−)
    1.27×10−1
    (4.95×10−3)(−)
    2.77×10−1
    (5.62×10−2)(−)
    1.07×10−1
    (1.63×10−3)(−)
    3.14×10−2
    (4.26×10−3)
    33.45×10−1
    (8.76×10−2)(−)
    2.97×10−1
    (3.58×10−3)(−)
    1.36×10−1
    (4.87×10−3)(−)
    1.21×10−1
    (6.82×10−3)(+)
    1.39×10−1
    (9.61×10−2)(−)
    1.03×10−1
    (4.73×10−3)(+)
    1.29×10−1
    (6.43×10−3)
    SDP928.95×10−2
    (7.09×10−3)(−)
    6.67×10−1
    (5.94×10−3)(−)
    1.64×10−1
    (8.52×10−4)(−)
    2.45×10−1
    (2.06×10−3)(−)
    1.35×10−1
    (3.25×10−2)(−)
    1.56×10−1
    (6.23×10−3)(−)
    8.13×10−2
    (7.92×10−3)
    34.01×10−1
    (8.98×10−3)(−)
    3.67×10−1
    (4.08×10−3)(−)
    4.93×10−1
    (6.03×10−3)(−)
    3.61×10−1
    (3.74×10−2)(≈)
    4.28×10−1
    (4.56×10−2)(−)
    4.45×10−1
    (9.13×10−3)(−)
    3.60×10−1
    (8.03×10−2)
    SDP1028.67×10−2
    (7.72×10−2)(−)
    1.15×10−1
    (8.31×10−2)(−)
    9.39×10−2
    (4.94×10−3)(−)
    2.23×10−1
    (1.75×10−2)(≈)
    2.35×100
    (3.74×10−1)(−)
    9.19×10−2
    (7.02×10−4)(−)
    2.21×10−2
    (1.67×10−3)
    33.05×10−1
    (5.73×10−2)(−)
    2.56×10−1
    (9.76×10−2)(−)
    1.85×10−1
    (6.51×10−3)(−)
    4.51×10−1
    (8.06×10−2)(−)
    3.48×10−1
    (7.43×10−2)(−)
    1.79×10−1
    (3.63×10−3)(−)
    1.74×10−1
    (3.58×10−3)
    SDP1121.61×10−2
    (6.36×10−3)(−)
    7.40×10−3
    (7.67×10−4)(+)
    3.02×10−2
    (3.14×10−3)(−)
    3.36×10−2
    (6.91×10−3)(−)
    9.18×10−1
    (3.64×10−3)(−)
    2.43×10−2
    (3.26×10−3)(−)
    1.36×10−2
    (8.16×10−3)
    31.49×10−1
    (7.48×10−3)(−)
    1.23×10−1
    (5.98×10−2)(−)
    9.77×10−2
    (2.81×10−2)(−)
    1.47×10−1
    (1.08×10−2)(−)
    8.98×10−2
    (6.05×10−3)(−)
    9.57×10−2
    (5.15×10−3)(−)
    8.77×10−2
    (9.65×10−3)
    SDP1222.20×10−2
    (4.03×10−3)(−)
    2.52×10−2
    (3.08×10−3)(−)
    9.16×10−3
    (2.78×10−3)(−)
    4.11×10−3
    (5.38×10−2)(−)
    8.09×10−2
    (4.65×10−2)(−)
    6.38×10−3
    (8.83×10−4)(−)
    4.04×10−3
    (2.17×10−4)
    32.22×10−1
    (8.66×10−2)(−)
    2.15×10−1
    (6.79×10−2)(−)
    8.66×10−2
    (3.59×10−3)(−)
    9.97×10−2
    (7.02×10−2)(−)
    7.66×10−2
    (6.93×10−2)(−)
    8.13×10−2
    (4.16×10−3)(−)
    7.63×10−2
    (5.63×10−4)
    +/−/≈3/21/04/20/03/21/03/18/32/22/04/20/0
    下载: 导出CSV

    表  2  DNSGA-II-A, DNSGA-II-B, MOEA/D-KF, SGEA, Tr-DMOEA, MOEA/D-MoE和DMOEA-ACR在DF上获得的MIGD均值和标准差

    Table  2  The mean and standard deviation of MIGD of DNSGA-II-A, DNSGA-II-B, MOEA/D-KF, SGEA, Tr-DMOEA, MOEA/D-MoE and DMOEA-ACR were obtained on DF

    ProblemsMDNSGA-II-ADNSGA-II-BMOEA/D-KFSGEATr-DMOEAMOEA/D-MoEDMOEA-ACR
    DF123.31×10−2
    (8.35×10−3)(−)
    4.24×10−2
    (3.91×10−3)(−)
    1.85×10−2
    (4.47×10−3)(−)
    1.22×10−2
    (1.38×10−3)(−)
    6.42×10−2
    (3.34×10−3)(−)
    1.54×10−2
    (3.97×10−3)(−)
    9.15×10−3
    (3.67×10−3)
    DF225.75×10−3
    (7.61×10−4)(+)
    5.76×10−3
    (2.08×10−4)(+)
    3.81×10−2
    (5.02×10−3)(+)
    1.22×10−1
    (2.76×10−2)(−)
    5.46×10−3
    (5.21×10−3)(+)
    3.10×10−2
    (5.23×10−3)(+)
    5.80×10−2
    (7.85×10−3)
    DF329.48×10−2
    (5.32×10−3)(−)
    1.48×10−1
    (7.35×10−3)(−)
    2.85×10−2
    (3.32×10−3)(−)
    2.61×10−1
    (4.58×10−2)(−)
    3.90×10−1
    (7.31×10−3)(−)
    2.10×10−2
    (7.52×10−3)(−)
    1.99×10−2
    (3.20×10−3)
    DF422.78×10−1
    (7.38×10−2)(−)
    3.81×10−1
    (2.79×10−2)(−)
    9.85×10−2
    (4.59×10−3)(−)
    5.56×10−2
    (7.88×10−3)(−)
    8.67×10−1
    (6.82×10−2)(−)
    9.09×10−2
    (2.81×10−3)(−)
    2.89×10−2
    (5.19×10−3)
    DF528.05×10−2
    (6.39×10−2)(−)
    8.11×10−2
    (5.65×10−3)(−)
    2.20×10−2
    (4.03×10−3)(−)
    1.45×10−2
    (4.31×10−3)(−)
    2.99×10−2
    (4.34×10−3)(−)
    2.13×10−2
    (1.24×10−3)(−)
    9.32×10−3
    (6.01×10−4)
    DF622.33×10−1
    (8.01×10−3)(+)
    2.40×10−1
    (8.18×10−3)(+)
    5.04×100
    (9.92×10−2)(−)
    1.51×100
    (8.39×10−2)(−)
    5.32×100
    (7.31×10−1)(−)
    2.98×100
    (7.68×10−1)(−)
    1.14×100
    (4.23×10−1)
    DF721.56×10−2
    (4.33×10−3)(≈)
    1.85×10−2
    (6.02×10−3)(−)
    3.66×10−2
    (6.39×10−3)(−)
    2.08×10−1
    (5.04×10−3)(−)
    6.54×100
    (5.37×10−1)(−)
    3.72×10−2
    (2.29×10−3)(−)
    1.57×10−2
    (3.67×10−3)
    DF828.82×10−2
    (7.13×10−2)(−)
    8.17×10−2
    (7.14×10−3)(−)
    7.94×10−2
    (3.68×10−3)(−)
    2.07×10−2
    (5.98×10−3)(−)
    2.85×10−1
    (6.81×10−2)(−)
    7.73×10−2
    (6.64×10−3)(−)
    1.70×10−2
    (5.68×10−3)
    DF927.82×10−2
    (9.75×10−3)(−)
    8.69×10−2
    (9.98×10−3)(−)
    9.52×10−2
    (7.87×10−3)(−)
    2.57×10−1
    (6.30×10−2)(−)
    5.33×10−1
    (2.76×10−2)(−)
    7.09×10−2
    (5.31×10−2)(−)
    6.87×10−2
    (4.97×10−3)
    DF1032.88×10−1
    (8.10×10−3)(−)
    2.76×10−1
    (6.31×10−3)(−)
    1.80×10−1
    (3.51×10−2)(−)
    1.19×10−1
    (5.26×10−2)(−)
    8.51×10−1
    (8.61×10−2)(−)
    1.86×10−1
    (2.21×10−2)(−)
    1.05×10−1
    (8.18×10−2)
    DF1135.77×10−1
    (7.57×10−2)(−)
    5.80×10−1
    (3.59×10−3)(−)
    1.53×10−1
    (4.06×10−2)(−)
    8.20×10−2
    (3.14×10−2)(−)
    6.53×10−2
    (4.05×10−2)(−)
    1.50×10−1
    (3.72×10−3)(−)
    6.38×10−2
    (5.07×10−3)
    DF1232.18×10−1
    (8.61×10−2)(−)
    2.34×10−1
    (2.36×10−2)(−)
    1.41×10−1
    (3.38×10−2)(−)
    1.47×10−1
    (2.78×10−2)(−)
    4.25×10−1
    (7.63×10−3)(−)
    1.00×10−1
    (6.98×10−2)(−)
    9.49×10−2
    (6.53×10−3)
    DF1331.80×10−1
    (5.32×10−2)(−)
    1.87×10−1
    (8.95×10−2)(−)
    2.79×10−1
    (5.72×10−2)(−)
    1.28×10−1
    (6.52×10−2)(−)
    1.37×100
    (2.71×10−1)(−)
    2.68×10−1
    (1.25×10−2)(−)
    1.15×10−1
    (4.38×10−2)
    DF1431.35×10−1
    (5.31×10−2)(−)
    1.22×10−1
    (4.32×10−2)(−)
    6.91×10−2
    (3.31×10−2)(−)
    5.19×10−2
    (4.06×10−2)(−)
    1.15×100
    (3.59×10−2)(−)
    6.88×10−2
    (5.81×10−2)(−)
    4.28×10−2
    (5.56×10−3)
    +/−/≈2/11/12/12/01/13/00/14/01/13/01/13/0
    下载: 导出CSV

    表  3  DNSGA-II-A, DNSGA-II-B, MOEA/D-KF, SGEA, Tr-DMOEA, MOEA/D-MoE和DMOEA-ACR在SDP(包括2目标和3目标)上的性能综合排名

    Table  3  Performance comprehensive ranking of DNSGA-II-A, DNSGA-II-B, MOEA/D-KF, SGEA, Tr-DMOEA, MOEA/D-MoE and DMOEA-ACR on SDP (including 2 goals and 3 goals)

    AlgorithmsSDP1SDP2SDP3SDP4SDP5SDP6SDP7SDP8SDP9SDP10SDP11SDP12Average
    rank
    DNSGA-II-A3177252624574
    DNSGA-II-B4346763555266
    MOEA/D-KF7514536473445
    SGEA2623344346733
    Tr-DMOEA5755677737657
    MOEA/D-MoE6432425162322
    DMOEA-ACR1261111211111
    下载: 导出CSV

    表  4  DNSGA-II-A, DNSGA-II-B, MOEA/D-KF, SGEA, Tr-DMOEA, MOEA/D-MoE和DMOEA-ACR在DF的性能综合排名

    Table  4  Performance comprehensive ranking of DNSGA-II-A, DNSGA-II-B, MOEA/D-KF, SGEA, Tr-DMOEA, MOEA/D-MoE and DMOEA-ACR algorithms on DF

    AlgorithmsDF1DF2DF3DF4DF5DF6DF7DF8DF9DF10DF11DF12DF13DF14Average
    rank
    DNSGA-II-A524561163665364
    DNSGA-II-B635672354576456
    MOEA/D-KF453446445353645
    SGEA276224626234223
    Tr-DMOEA717757777727777
    MOEA/D-MoE342335532442532
    DMOEA-ACR161113211111111
    下载: 导出CSV

    表  5  DMOEA-ACR-D1, DMOEA-ACR-D2, DMOEA-ACR-D3与DMOEA-ACR在DF上获得的MIGD均值和标准差

    Table  5  The mean and standard deviation of MIGD of DMOEA-ACR-D1, DMOEA-ACR-D2, DMOEA-ACR-D3 and DMOEA-ACR were obtained on DF

    ProblemsMDMOEA-ACR-D1DMOEA-ACR-D2DMOEA-ACR-D3DMOEA-ACR
    DF121.99×10−2
    (1.62×10−2)(−)
    2.58×10−2
    (1.51×10−2)(−)
    1.50×10−2
    (4.12×10−3)(−)
    9.15×10−3
    (3.67×10−3)
    DF221.22×10−1
    (7.57×10−3)(−)
    2.54×10−2
    (4.08×10−3)(+)
    5.93×10−2
    (4.18×10−2)(−)
    5.80×10−2
    (7.85×10−3)
    DF321.99×10−1
    (2.61×10−3))(−)
    2.33×10−1
    (4.52×10−3))(−)
    4.44×10−2
    (1.04×10−3))(−)
    1.99×10−2
    (3.20×10−3)
    DF425.28×10−2
    (7.05×10−3)(−)
    2.70×10−2
    (6.82×10−3)(+)
    4.17×10−2
    (5.17×10−3)(−)
    2.89×10−2
    (5.19×10−3)
    DF521.74×10−2
    (6.41×10−3)(−)
    5.05×10−2
    (3.69×10−3)(−)
    2.09×10−2
    (3.65×10−4)(−)
    9.32×10−3
    (6.01×10−4)
    DF622.88×100
    (8.39×10−1)(−)
    6.44×100
    (5.07×10−1)(−)
    1.23×100
    (7.47×10−1)(−)
    1.14×100
    (4.23×10−1)
    DF721.85×10−1
    (2.66×10−3)(−)
    1.42×10−2
    (7.06×10−3)(+)
    1.99×10−2
    (2.61×10−3)(−)
    1.57×10−2
    (3.67×10−3)
    DF821.65×10−2
    (7.09×10−3)(+)
    1.56×10−2
    (5.13×10−3)(+)
    1.91×10−2
    (6.05×10−3)(−)
    1.70×10−2
    (5.68×10−3)
    DF924.15×10−1
    (4.72×10−2)(−)
    1.21×10−1
    (4.81×10−3)(−)
    1.11×10−1
    (5.82×10−3)(−)
    6.87×10−2
    (4.97×10−3)
    DF1033.01×10−1
    (5.11×10−2)(−)
    1.48×10−1
    (1.42×10−2)(−)
    2.43×10−1
    (3.47×10−2)(−)
    1.05×10−1
    (8.18×10−2)
    DF1136.65×10−2
    (5.31×10−3)(−)
    7.76×10−2
    (6.37×10−3)(−)
    6.85×10−2
    (4.21×10−3)(−)
    6.38×10−2
    (5.07×10−3)
    DF1232.24×10−1
    (6.99×10−2)(−)
    2.62×10−1
    (8.70×10−2)(−)
    1.84×10−1
    (7.17×10−2)(−)
    9.49×10−2
    (6.53×10−3)
    DF1332.36×10−1
    (5.77×10−2)(−)
    1.62×10−1
    (3.91×10−2)(−)
    1.36×10−1
    (5.32×10−2)(−)
    1.15×10−1
    (4.38×10−2)
    DF1437.19×10−2
    (9.54×10−3)(−)
    6.25×10−2
    (4.32×10−3)(−)
    5.29×10−2
    (8.02×10−3)(−)
    4.28×10−2
    (5.56×10−3)
    +/−/≈1/13/04/10/00/14/0
    下载: 导出CSV

    表  6  DMOEA-ACR-D1, DMOEA-ACR-D2, DMOEA-ACR-D3与DMOEA-ACR在SDP上获得的MIGD均值和标准差

    Table  6  The mean and standard deviation of MIGD of DMOEA-ACR-D1, DMOEA-ACR-D2, DMOEA-ACR-D3 and DMOEA-ACR were obtained on SDP

    ProblemsMDMOEA-ACR-D1DMOEA-ACR-D2DMOEA-ACR-D3DMOEA-ACR
    SDP122.51×10−2
    (4.25×10−3)(−)
    1.78×10−2
    (3.03×10−3)(−)
    1.67×10−2
    (5.12×10−3)(≈)
    1.65×10−2
    (3.67×10−3)
    31.68×10−1
    (5.34×10−2)(−)
    1.54×10−1
    (4.31×10−2)(−)
    1.41×10−1
    (4.08×10−2)(−)
    1.32×10−1
    (5.44×10−3)
    SDP226.47×10−1
    (2.27×10−2)(−)
    1.12×10−1
    (6.26×10−2)(≈)
    2.15×10−1
    (3.38×10−2)(−)
    1.14×10−1
    (4.35×10−3)
    34.05×10−1
    (3.15×10−2)(−)
    3.94×10−1
    (7.58×10−2)(−)
    5.64×10−1
    (4.15×10−2)(−)
    2.42×10−1
    (7.12×10−3)
    SDP328.91×100
    (6.55×10−1)(−)
    4.81×100
    (1.06×10−2)(−)
    4.99×100
    (7.21×10−1)(−)
    4.74×100
    (6.38×10−2)
    33.91×100
    (7.60×10−1)(−)
    4.40×100
    (4.96×10−1)(−)
    4.47×100
    (6.06×10−1)(−)
    3.33×100
    (8.53×10−2)
    SDP426.96×10−2
    (6.16×10−3)(−)
    7.85×10−2
    (6.31×10−2)(−)
    1.60×10−1
    (5.51×10−2)(−)
    5.08×10−2
    (5.03×10−3)
    32.10×10−1
    (4.01×10−2)(−)
    2.68×10−1
    (3.07×10−2)(−)
    1.89×10−1
    (4.19×10−2)(−)
    1.69×10−1
    (8.25×10−3)
    SDP527.76×10−2
    (6.63×10−3)(−)
    7.40×10−3
    (2.25×10−3)(−)
    9.52×10−3
    (5.12×10−3)(−)
    6.81×10−3
    (4.92×10−4)
    36.85×10−2
    (9.08×10−3)(−)
    6.71×10−2
    (6.91×10−3)(−)
    7.26×10−2
    (7.61×10−3)(−)
    6.38×10−2
    (6.06×10−3)
    SDP627.98×10−3
    (4.37×10−3)(−)
    6.84×10−3
    (4.66×10−3)(−)
    6.83×10−3
    (2.03×10−3)(−)
    4.26×10−3
    (7.43×10−4)
    36.05×10−2
    (8.12×10−3)(−)
    1.16×10−1
    (3.30×10−3)(−)
    5.14×10−2
    (4.60×10−3)(+)
    5.17×10−2
    (5.86×10−3)
    SDP722.70×10−1
    (5.59×10−2)(−)
    2.34×10−1
    (7.57×10−2)(−)
    2.25×10−1
    (4.89×10−2)(−)
    2.15×10−1
    (1.68×10−2)
    32.62×10−1
    (3.67×10−2)(−)
    1.14×100
    (3.86×10−2)(−)
    2.17×10−1
    (2.61×10−2)(+)
    2.24×10−1
    (5.11×10−3)
    SDP829.54×10−2
    (6.63×10−3)(−)
    6.28×10−2
    (6.73×10−3)(−)
    4.53×10−2
    (5.15×10−3)(−)
    3.14×10−2
    (4.26×10−3)
    33.46×10−1
    (3.10×10−2)(−)
    2.79×10−1
    (2.50×10−2)(−)
    3.21×10−1
    (2.07×10−2)(−)
    1.29×10−1
    (6.43×10−3)
    SDP921.47×10−1
    (5.06×10−2)(−)
    1.28×10−1
    (4.65×10−2)(−)
    1.35×10−1
    (3.23×10−2)(−)
    8.13×10−2
    (7.92×10−3)
    33.66×10−1
    (8.15×10−2)(−)
    4.01×10−1
    (4.08×10−2)(−)
    3.54×10−1
    (6.61×10−2)(+)
    3.60×10−1
    (8.03×10−2)
    SDP1026.36×10−2
    (2.03×10−3)(−)
    2.61×10−2
    (7.38×10−3)(−)
    3.75×10−2
    (2.25×10−3)(−)
    2.21×10−2
    (1.67×10−3)
    31.68×10−1
    (6.22×10−2)(+)
    2.41×10−1
    (1.36×10−2)(−)
    3.86×10−1
    (5.46×10−2)(−)
    1.74×10−1
    (3.58×10−3)
    SDP1123.80×10−2
    (7.32×10−3)(−)
    4.83×10−2
    (5.21×10−3)(−)
    3.26×10−2
    (8.06×10−3)(−)
    1.36×10−2
    (8.16×10−3)
    31.47×10−1
    (9.08×10−3)(−)
    9.68×10−2
    (6.52×10−3)(−)
    1.96×10−1
    (6.21×10−3)(−)
    8.77×10−2
    (9.65×10−3)
    SDP1224.53×10−3
    (3.18×10−3)(−)
    1.62×10−2
    (2.06×10−3)(−)
    1.40×10−2
    (1.32×10−3)(−)
    4.04×10−3
    (2.17×10−4)
    31.62×10−1
    (5.34×10−2)(−)
    1.96×10−1
    (3.73×10−2)(−)
    1.02×10−1
    (2.17×10−2)(−)
    7.63×10−2
    (5.63×10−4)
    +/−/≈1/23/00/23/13/20/1
    下载: 导出CSV

    表  7  DMOEA-ACR-P1, DMOEA-ACR-P2, DMOEA-ACR-P3, DMOEA-ACR-P4与DMOEA-ACR在SDP上获得的MIGD的均值和标准差

    Table  7  The mean and standard deviation of MIGD of DMOEA-ACR-P1, DMOEA-ACR-P2, DMOEA-ACR-P3, DMOEA-ACR-P4 and DMOEA-ACR were obtained on SDP

    ProblemsMDMOEA-ACR-P1DMOEA-ACR-P2DMOEA-ACR-P3DMOEA-ACR-P4DMOEA-ACR
    SDP127.80×10−2
    (2.17×10−3)(−)
    1.87×10−2
    (7.05×10−3)(−)
    6.34×10−2
    (4.24×10−3)(−)
    1.73×10−2
    (5.76×10−3)(−)
    1.65×10−2
    (3.67×10−3)
    36.69×10−1
    (4.82×10−3)(−)
    1.53×10−1
    (3.62×10−3)(−)
    5.50×10−1
    (4.01×10−3)(−)
    1.81×10−2
    (6.20×10−3)(−)
    1.32×10−1
    (5.44×10−3)
    SDP221.78×10−1
    (3.16×10−3)(−)
    2.02×10−1
    (4.91×10−3)(−)
    2.25×10−1
    (3.97×10−3)(−)
    2.36×10−1
    (6.32×10−3)(−)
    1.14×10−1
    (4.35×10−3)
    35.82×10−1
    (8.04×10−3)(−)
    3.25×10−1
    (6.51×10−3)(−)
    3.11×10−1
    (5.26×10−3)(−)
    3.21×10−1
    (4.58×10−3)(−)
    2.42×10−1
    (7.12×10−3)
    SDP325.15×100
    (3.66×10−2)(−)
    4.85×100
    (1.35×10−2)(−)
    5.14×100
    (6.59×10−2)(−)
    4.83×100
    (4.76×10−2)(−)
    4.74×100
    (6.38×10−2)
    31.99×100
    (5.14×10−2)(+)
    3.83×100
    (2.94×10−2)(−)
    3.88×100
    (5.68×10−2)(−)
    3.91×100
    (6.20×10−2)(−)
    3.33×100
    (8.53×10−2)
    SDP424.67×10−2
    (5.16×10−3)(+)
    8.35×10−2
    (5.91×10−3)(−)
    1.02×10−1
    (4.85×10−3)(−)
    7.58×10−2
    (6.24×10−3)(−)
    5.08×10−2
    (5.03×10−3)
    31.76×10−1
    (4.21×10−3)(−)
    2.13×10−1
    (3.89×10−3)(−)
    1.65×10−1
    (3.56×10−3)(+)
    2.02×10−1
    (5.81×10−3)(−)
    1.69×10−1
    (8.25×10−3)
    SDP523.66×10−2
    (5.41×10−3)(−)
    7.63×10−3
    (3.00×10−4)(−)
    7.13×10−3
    (6.35×10−4)(−)
    8.08×10−3
    (6.10×10−4)(−)
    6.81×10−3
    (4.92×10−4)
    33.96×10−1
    (8.79×10−2)(−)
    6.43×10−2
    (6.84×10−3)(−)
    6.49×10−2
    (2.96×10−3)(−)
    6.57×10−2
    (5.39×10−3)(−)
    6.38×10−2
    (6.06×10−3)
    SDP626.48×10−3
    (5.61×10−4)(−)
    4.62×10−3
    (8.91×10−4)(−)
    4.36×10−3
    (6.03×10−4)(−)
    4.31×10−3
    (5.28×10−4)(−)
    4.26×10−3
    (7.43×10−4)
    31.51×10−1
    (7.36×10−3)(−)
    5.27×10−2
    (7.18×10−3)(−)
    5.43×10−2
    (6.23×10−3)(−)
    5.20×10−2
    (5.98×10−3)(−)
    5.17×10−2
    (5.86×10−3)
    SDP721.60×10−1
    (5.31×10−2)(+)
    3.26×10−1
    (6.94×10−2)(−)
    2.32×10−1
    (3.28×10−2)(−)
    2.46×10−1
    (4.51×10−2)(−)
    2.15×10−1
    (1.68×10−2)
    35.98×10−2
    (6.60×10−3)(+)
    2.56×10−1
    (6.59×10−2)(−)
    2.43×10−1
    (4.67×10−3)(−)
    2.53×10−1
    (5.81×10−3)(−)
    2.24×10−1
    (5.11×10−3)
    SDP821.23×10−1
    (6.57×10−2)(−)
    4.70×10−2
    (5.48×10−3)(−)
    2.70×10−2(
    4.90×10−3)(+)
    2.55×10−2
    (5.84×10−3)(+)
    3.14×10−2
    (4.26×10−3)
    31.37×10−1
    (4.96×10−3)(−)
    1.81×10−1
    (3.69×10−3)(−)
    1.55×10−1
    (1.86×10−3)(−)
    1.40×10−1
    (7.82×10−3)(−)
    1.29×10−1
    (6.43×10−3)
    SDP921.85×10−1
    (8.80×10−3)(−)
    1.29×10−1
    (7.61×10−3)(−)
    1.50×10−1
    (8.55×10−3)(−)
    1.32×10−1
    (6.27×10−3)(−)
    8.13×10−2
    (7.92×10−3)
    34.94×10−1
    (9.43×10−2)(−)
    3.79×10−1
    (8.99×10−2)(−)
    3.67×10−1
    9.08×10−2)(−)
    3.69×10−1
    (4.20×10−2)(−)
    3.60×10−1
    (8.03×10−2)
    SDP1021.12×10−1
    (3.13×10−3)(−)
    3.28×10−2
    (4.10×10−3)(−)
    2.23×10−2
    (2.15×10−3)(−)
    3.63×10−2
    (2.23×10−3)(−)
    2.21×10−2
    (1.67×10−3)
    31.87×10−1
    (5.78×10−2)(−)
    2.80×10−1
    (3.65×10−2)(−)
    1.79×10−1
    (6.03×10−2)(−)
    1.92×10−1
    (2.65×10−3)(−)
    1.74×10−1
    (3.58×10−3)
    SDP1122.28×10−2
    (9.14×10−3)(−)
    3.60×10−2
    (7.61×10−3)(−)
    2.38×10−2
    (9.03×10−3)(−)
    3.34×10−2
    (8.16×10−3)(−)
    1.36×10−2
    (8.16×10−3)
    38.83×10−2
    (8.87×10−3)(−)
    1.04×10−1
    (8.17×10−3)(−)
    8.79×10−2
    (5.96×10−3)(−)
    1.18×10−1
    (9.08×10−3)(−)
    8.77×10−2
    (9.65×10−3)
    SDP1224.52×10−3
    (3.75×10−3)(−)
    4.94×10−3(
    3.65×10−3)(−)
    4.37×10−3
    (7.29×10−3)(−)
    4.60×10−3
    (6.94×10−3)(−)
    4.04×10−3
    (2.17×10−4)
    32.17×10−1
    (5.02×10−2)(−)
    2.65×10−1
    (4.10×10−2)(−)
    1.06×10−1
    (3.08×10−2)(−)
    3.23×10−1
    (5.24×10−2)(−)
    7.63×10−2
    (5.63×10−4)
    +/−/≈4/20/00/24/02/22/01/23/0
    下载: 导出CSV

    表  8  DMOEA-ACR-P1, DMOEA-ACR-P2, DMOEA-ACR-P3, DMOEA-ACR-P4与DMOEA-ACR在DF上获得的MIGD的均值和标准差

    Table  8  The mean and standard deviation of MIGD of DMOEA-ACR-P1, DMOEA-ACR-P2, DMOEA-ACR-P3, DMOEA-ACR-P4 and DMOEA-ACR were obtained on DF

    ProblemsMDMOEA-ACR-P1DMOEA-ACR-P2DMOEA-ACR-P3DMOEA-ACR-P4DMOEA-ACR
    DF129.28×10−3
    (6.51×10−3)(−)
    1.82×10−2
    (1.39×10−2)(−)
    9.47×10−3
    (8.94×10−3)(−)
    1.73×10−2
    (5.06×10−3)(−)
    9.15×10−3
    (3.67×10−3)
    DF221.01×10−1
    (3.16×10−2)(−)
    6.98×10−2
    (2.48×10−3)(−)
    1.16×10−1
    (3.90×10−2)(−)
    5.17×10−2
    (8.26×10−3)(+)
    5.80×10−2
    (7.85×10−3)
    DF322.64×10−2
    (4.09×10−3)(−)
    2.89×10−2
    (6.71×10−3)(−)
    2.04×10−2
    (5.66×10−3)(≈)
    2.91×10−2
    (4.18×10−3)(−)
    1.99×10−2
    (3.20×10−3)
    DF422.38×10−2
    (6.23×10−3)(+)
    3.99×10−2
    (4.20×10−3)(−)
    2.94×10−2
    (6.81×10−3)(−)
    2.77×10−2
    (7.11×10−3)(+)
    2.89×10−2
    (5.19×10−3)
    DF528.51×10−3
    (8.38×10−3)(+)
    2.48×10−2
    (4.82×10−3)(−)
    8.83×10−3
    (6.22×10−4)(+)
    2.27×10−2
    (3.10×10−3)(−)
    9.32×10−3
    (6.01×10−4)
    DF622.26×100
    (7.82×10−1)(−)
    1.46×100
    (6.33×10−1)(−)
    1.16×100
    (5.67×10−1)(−)
    1.60×100
    (4.28×10−1)(−)
    1.14×100
    (4.23×10−1)
    DF721.62×10−2
    (4.61×10−3)(−)
    1.84×10−2
    (5.14×10−3)(−)
    1.45×10−2
    (7.12×10−3)(+)
    1.48×10−2
    (5.04×10−3)(+)
    1.57×10−2
    (3.67×10−3)
    DF821.65×10−2
    (4.77×10−3)(+)
    2.87×10−2
    (4.00×10−3)(−)
    1.71×10−2
    (6.54×10−3)(≈)
    1.68×10−2
    (5.26×10−3)(+)
    1.70×10−2
    (5.68×10−3)
    DF929.75×10−2
    (4.63×10−2)(−)
    9.32×10−2
    (3.91×10−3)(−)
    6.93×10−2
    (5.73×10−3)(−)
    1.10×10−1
    (6.05×10−3)(−)
    6.87×10−2
    (4.97×10−3)
    DF1031.92×10−1
    (9.39×10−2)(−)
    4.59×10−1
    (8.35×10−2)(−)
    2.37×10−1
    (3.07×10−2)(−)
    1.51×10−1
    (7.99×10−2)(−)
    1.05×10−1
    (8.18×10−2)
    DF1137.43×10−2
    (5.86×10−3)(−)
    8.51×10−2
    (4.81×10−3)(−)
    6.42×10−2
    (6.03×10−3)(−)
    6.57×10−2
    (5.87×10−3)(−)
    6.38×10−2
    (5.07×10−3)
    DF1231.75×10−1
    (3.23×10−2)(−)
    3.29×10−1
    (2.02×10−2)(−)
    2.60×10−1
    (6.17×10−2)(−)
    1.16×10−1
    (4.03×10−2)(−)
    9.49×10−2
    (6.53×10−3)
    DF1331.19×10−1
    (4.91×10−2)(−)
    2.57×10−1
    (2.68×10−2)(−)
    1.20×10−1
    (3.16×10−2)(−)
    2.48×10−1
    (4.82×10−2)(−)
    1.15×10−1
    (4.38×10−2)
    DF1434.43×10−2
    (6.09×10−3)(−)
    6.32×10−2
    (4.17×10−3)(−)
    4.59×10−2
    (3.21×10−3)(−)
    5.95×10−2
    (6.61×10−3)(−)
    4.28×10−2
    (5.56×10−3)
    +/−/≈3/11/00/14/02/10/24/10/0
    下载: 导出CSV

    表  9  $\tau_t $ 分别为5,10,20时DNSGA-II-A, DNSGA-II-B, MOEA/D-KF, SGEA, Tr-DMOEA, MOEA/D-MoE和DMOEA-ACR在SDP和DF上获得的显著差异统计结果

    Table  9  Significant difference statistical results of DNSGA-II-A, DNSGA-II-B, MOEA/D-KF, SGEA, Tr-DMOEA, MOEA/D-MoE and DMOEA-ACR were obtained on SDP and DF where $\tau_t $ is 5, 10 and 20 respectively

    Problems$\tau_t $DNSGA-II-ADNSGA-II-BMOEA/D-
    KF
    SGEATr-DMOEAMOEA/D-
    MoE
    DMOEA-
    ACR
    SDP54/19/14/20/03/21/02/22/00/24/01/23/0+/−/≈
    103/21/04/20/03/21/03/18/32/22/04/20/0
    203/21/05/19/01/21/25/18/12/22/03/21/0
    DF52/12/02002-11-11/13/01/13/00/14/01/13/0+/−/≈
    102002-11-12/12/01/13/00/14/01/13/01/13/0
    202/12/02/12/01/13/02/12/02001-12-12001-12-1
    下载: 导出CSV
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