2.765

2022影响因子

(CJCR)

  • 中文核心
  • EI
  • 中国科技核心
  • Scopus
  • CSCD
  • 英国科学文摘

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

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, 2023, 49(8): 1688−1706 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, 2023, 49(8): 1688−1706 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, 2023, 49(8): 1688−1706 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, 2023, 49(8): 1688−1706 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: 国家重点研发计划(2021YFB2900800), 国家自然科学基金(61871272), 广东省自然科学基金(2020A1515010479, 2021A1515011911), 深圳市科技计划项目(20200811181752003, JCYJ20220531102617039)资助
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年获得武汉大学博士学位. 主要研究方向为计算智能和大数据分析与应用. E-mail: liangzp@szu.edu.cn

    李辉才:深圳大学计算机与软件学院硕士研究生. 主要研究方向为计算智能与应用. 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, 大数据分析与应用和多媒体技术与应用. E-mail: wangzq@szu.edu.cn

    胡凯峰:深圳大学信息中心工程师. 2019年获得深圳大学硕士学位. 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教授. 2008年获得南洋理工大学博士学位. 主要研究方向为计算智能, 机器学习和生物信息学. 本文通信作者. E-mail: zhuzx@szu.edu.cn

Dynamic Multi-objective Evolutionary Algorithm With Adaptive Change Response

Funds: Supported by National Key Research and Development Program of China (2021YFB2900800), National Natural Science Foundation of China (61871272), Natural Science Foundation of Guangdong (2020A1515010479, 2021A1515011911), and Shenzhen Science & Technology Project (20200811181752003, JCYJ20220531102617039)
More Information
    Author Bio:

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

    LI Hui-Cai Master student at the College of Computer Science and Software Engineering, Shenzhen University. His main research interest is computational intelligence & applications

    WANG Zhi-Qiang Professor at the College of Computer Science and Software Engineering, Shenzhen University. His research interest covers computational intelligence, big data analysis & applications, and multi-media technology & applications

    HU Kai-Feng Engineer at the Information Center, Shenzhen University. He received his master degree from Shenzhen University in 2019. His main research interest is computational intelligence & applications

    ZHU Ze-Xuan Professor at the College of Computer Science and Software Engineering, Shenzhen University. He received his Ph.D. degree from Nanyang Technological University in 2008. His research interest covers computational intelligence, machine learning, and bioinformatics. Corresponding author of this paper

  • 摘要: 动态多目标优化问题(Dynamic multi-objective optimization problems, DMOPs)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, 导致重新初始化效果尚存在较大改进空间. 为此, 提出一种对不同个体、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(Dynamic multi-objective evolutionary algorithm with adaptive change response, DMOEA-ACR). 该算法包括两个核心部分: 1)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; 2)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. 为验证DMOEA-ACR的有效性, 在最新测试问题集SDP和DF上, 将其与动态多目标优化领域的6种先进算法进行对比. 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 response

    图  3  基于存档中心进行预测策略示意图

    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测试问题集上的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测试问题集上得到的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测试问题集上得到的PF图

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

    图  7  $ \tau t $分别为5、10、20时DMOEA-ACR在SDP1 (2目标)、SDP5 (2目标)、SDP6 (2目标)、DF3、DF5和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 goals), SDP5 (2 goals), SDP6 (2 goals), DF3, DF5, and DF8 where $\tau_t $ is 5, 10, 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

    问题集评价指标DNSGA-II-ADNSGA-II-BMOEA/D-KFSGEATr-DMOEAMOEA/D-MoEDMOEA-ACR
    SDP1 (2目标)

    SDP1 (3目标)
    均值
    标准差
    2.03 × 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
    均值
    标准差
    2.26 × 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
    SDP2 (2目标)

    SDP2 (3目标)
    均值
    标准差
    1.75 × 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
    均值
    标准差
    2.87 × 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
    SDP3 (2目标)

    SDP3 (3目标)
    均值
    标准差
    1.76 × 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
    均值
    标准差
    7.01 × 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
    SDP4 (2目标)

    SDP4 (3目标)
    均值
    标准差
    1.14 × 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
    均值
    标准差
    2.44 × 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
    SDP5 (2目标)

    SDP5 (3目标)
    均值
    标准差
    9.40 × 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
    均值
    标准差
    1.41 × 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
    SDP6 (2目标)

    SDP6 (3目标)
    均值
    标准差
    2.47 × 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
    均值
    标准差
    1.61 × 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
    SDP7 (2目标)

    SDP7 (3目标)
    均值
    标准差
    5.15 × 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
    均值
    标准差
    2.41 × 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
    SDP8 (2目标)

    SDP8 (3目标)
    均值
    标准差
    1.40 × 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
    均值
    标准差
    3.45 × 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
    SDP9 (2目标)

    SDP9 (3目标)
    均值
    标准差
    8.95 × 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
    均值
    标准差
    4.01 × 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
    SDP10 (2目标)

    SDP10 (3目标)
    均值
    标准差
    8.67 × 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
    均值
    标准差
    3.05 × 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
    SDP11 (2目标)

    SDP11 (3目标)
    均值
    标准差
    1.61 × 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
    均值
    标准差
    1.49 × 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
    SDP12 (2目标)

    SDP12 (3目标)
    均值
    标准差
    2.20 × 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
    均值
    标准差
    2.22 × 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
    “+/−/≈”合计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

    问题集评价指标DNSGA-II-ADNSGA-II-BMOEA/D-KFSGEATr-DMOEAMOEA/D-MoEDMOEA-ACR
    DF1 (2目标)均值
    标准差
    3.31 × 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
    DF2 (2目标)均值
    标准差
    5.75 × 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
    DF3 (2目标)均值
    标准差
    9.48 × 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
    DF4 (2目标)均值
    标准差
    2.78 × 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
    DF5 (2目标)均值
    标准差
    8.05 × 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
    DF6 (2目标)均值
    标准差
    2.33 × 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
    DF7 (2目标)均值
    标准差
    1.56 × 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
    DF8 (2目标)均值
    标准差
    8.82 × 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
    DF9 (2目标)均值
    标准差
    7.82 × 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
    DF10 (3目标)均值
    标准差
    2.88 × 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
    DF11 (3目标)均值
    标准差
    5.77 × 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
    DF12 (3目标)均值
    标准差
    2.18 × 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
    DF13 (3目标)均值
    标准差
    1.80 × 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
    DF14 (3目标)均值
    标准差
    1.35 × 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
    “+/−/≈”合计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)

    算法SDP1SDP2SDP3SDP4SDP5SDP6SDP7SDP8SDP9SDP10SDP11SDP12平均
    排序
    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

    算法DF1DF2DF3DF4DF5DF6DF7DF8DF9DF10DF11DF12DF13DF14平均
    排序
    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和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

    问题集评价指标DMOEA-ACR-D1DMOEA-ACR-D2DMOEA-ACR-D3DMOEA-ACR
    DF1 (2目标)均值
    标准差
    1.99 × 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
    DF2 (2目标)均值
    标准差
    1.22 × 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
    DF3 (2目标)均值
    标准差
    1.99 × 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
    DF4 (2目标)均值
    标准差
    5.28 × 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
    DF5 (2目标)均值
    标准差
    1.74 × 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
    DF6 (2目标)均值
    标准差
    2.88 × 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
    DF7 (2目标)均值
    标准差
    1.85 × 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
    DF8 (2目标)均值
    标准差
    1.65 × 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
    DF9 (2目标)均值
    标准差
    4.15 × 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
    DF10 (3目标)均值
    标准差
    3.01 × 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
    DF11 (3目标)均值
    标准差
    6.65 × 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
    DF12 (3目标)均值
    标准差
    2.24 × 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
    DF13 (3目标)均值
    标准差
    2.36 × 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
    DF14 (3目标)均值
    标准差
    7.19 × 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
    “+/−/≈”合计1/13/04/10/00/14/0
    下载: 导出CSV

    表  6  DMOEA-ACR-D1、DMOEA-ACR-D2、DMOEA-ACR-D3和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

    问题集评价指标DMOEA-ACR-D1DMOEA-ACR-D2DMOEA-ACR-D3DMOEA-ACR
    SDP1 (2目标)

    SDP1 (3目标)
    均值
    标准差
    2.51 × 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
    均值
    标准差
    1.68 × 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
    SDP2 (2目标)

    SDP2 (3目标)
    均值
    标准差
    6.47 × 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
    均值
    标准差
    4.05 × 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
    SDP3 (2目标)

    SDP3 (3目标)
    均值
    标准差
    8.91 × 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
    均值
    标准差
    3.91 × 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
    SDP4 (2目标)

    SDP4 (3目标)
    均值
    标准差
    6.96 × 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
    均值
    标准差
    2.10 × 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
    SDP5 (2目标)

    SDP5 (3目标)
    均值
    标准差
    7.76 × 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
    均值
    标准差
    6.85 × 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
    SDP6 (2目标)

    SDP6 (3目标)
    均值
    标准差
    7.98 × 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
    均值
    标准差
    6.05 × 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
    SDP7 (2目标)

    SDP7 (3目标)
    均值
    标准差
    2.70 × 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
    均值
    标准差
    2.62 × 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
    SDP8 (2目标)

    SDP8 (3目标)
    均值
    标准差
    9.54 × 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
    均值
    标准差
    3.46 × 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
    SDP9 (2目标)

    SDP9 (3目标)
    均值
    标准差
    1.47 × 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
    均值
    标准差
    3.66 × 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
    SDP10 (2目标)

    SDP10 (3目标)
    均值
    标准差
    6.36 × 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
    均值
    标准差
    1.68 × 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
    SDP11 (2目标)

    SDP11 (3目标)
    均值
    标准差
    3.80 × 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
    均值
    标准差
    1.47 × 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
    SDP12 (2目标)

    SDP12 (3目标)
    均值
    标准差
    4.53 × 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
    均值
    标准差
    1.62 × 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
    “+/−/≈”合计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和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

    问题集目标数DMOEA-ACR-P1DMOEA-ACR-P2DMOEA-ACR-P3DMOEA-ACR-P4DMOEA-ACR
    SDP1 (2目标)

    SDP1 (3目标)
    均值
    标准差
    7.80 × 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
    均值
    标准差
    6.69 × 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
    SDP2 (2目标)

    SDP2 (3目标)
    均值
    标准差
    1.78 × 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
    均值
    标准差
    5.82 × 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
    SDP3 (2目标)

    SDP3 (3目标)
    均值
    标准差
    5.15 × 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
    均值
    标准差
    1.99 × 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
    SDP4 (2目标)

    SDP4 (3目标)
    均值
    标准差
    4.67 × 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
    均值
    标准差
    1.76 × 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
    SDP5 (2目标)

    SDF5 (3目标)
    均值
    标准差
    3.66 × 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
    均值
    标准差
    3.96 × 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
    SDP6 (2目标)

    SDP6 (3目标)
    均值
    标准差
    6.48 × 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
    均值
    标准差
    1.51 × 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
    SDP7 (2目标)

    SDP7 (3目标)
    均值
    标准差
    1.60 × 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
    均值
    标准差
    5.98 × 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
    SDP8 (2目标)

    SDP8 (3目标)
    均值
    标准差
    1.23 × 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
    均值
    标准差
    1.37 × 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
    SDP9 (2目标)

    SDP9 (3目标)
    均值
    标准差
    1.85 × 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
    均值
    标准差
    4.94 × 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
    SDP10 (2目标)

    SDP10 (3目标)
    均值
    标准差
    1.12 × 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
    均值
    标准差
    1.87 × 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
    SDP11 (2目标)

    SDP11 (3目标)
    均值
    标准差
    2.28 × 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
    均值
    标准差
    8.83 × 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
    SDP12 (2目标)

    SDP12 (3目标)
    均值
    标准差
    4.52 × 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
    均值
    标准差
    2.17 × 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
    “+/−/≈”合计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和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

    问题集评价指标DMOEA-ACR-P1DMOEA-ACR-P2DMOEA-ACR-P3DMOEA-ACR-P4DMOEA-ACR
    DF1 (2目标)均值
    标准差
    9.28 × 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
    DF2 (2目标)均值
    标准差
    1.01 × 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
    DF3 (2目标)均值
    标准差
    2.64 × 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
    DF4 (2目标)均值
    标准差
    2.38 × 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
    DF5 (2目标)均值
    标准差
    8.51 × 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
    DF6 (2目标)均值
    标准差
    2.26 × 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
    DF7 (2目标)均值
    标准差
    1.62 × 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
    DF8 (2目标)均值
    标准差
    1.65 × 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
    DF9 (2目标)均值
    标准差
    9.75 × 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
    DF10 (2目标)均值
    标准差
    1.92 × 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
    DF11 (2目标)均值
    标准差
    7.43 × 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
    DF12 (2目标)均值
    标准差
    1.75 × 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
    DF13 (2目标)均值
    标准差
    1.19 × 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
    DF14 (2目标)均值
    标准差
    4.43 × 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
    “+/−/≈”合计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, 20, respectively

    问题集$\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
    DF52/12/02/11/11/13/01/13/00/14/01/13/0+/−/≈
    102/11/12/12/01/13/00/14/01/13/01/13/0
    202/12/02/12/01/13/02/12/01/12/11/12/1
    下载: 导出CSV
  • [1] Cruz A R, Cardoso R T N, Takahashi R H C. Multi-objective dynamic optimization of vaccination campaigns using convex quadratic approximation local search. In: Proceedings of the 6th International Conference on Evolutionary Multi-Criterion Optimization. Ouro Preto, Brazil: 2011. 404−417
    [2] Bui L T, Michalewicz Z, Parkinson E, Abello M B. Adaptation in dynamic environments: a case study in mission planning. IEEE Transactions on Evolutionary Computation, 2012, 16(2): 190-209. doi: 10.1109/TEVC.2010.2104156
    [3] Zhong X, He Y, Du Z. Downlink power allocation in distributed satellite system based on dynamic multi-objective optimization. In: Proceedings of the International Conference on Wireless Communications & Signal Processing. Nanjing, China: 2015. 1−5
    [4] Ding J L, Yang C, Xiao Q, Chai T Y, Jin Y C. Dynamic evolutionary multiobjective optimization for raw ore allocation in mineral processing. IEEE Transactions on Emerging Topics in Computational Intelligence, 2019, 3(1): 36-48.
    [5] Jiang S Y, Yang S X, Yao X, Tan K C, Kaiser M, Krasnogor N. Benchmark Functions for the CEC 2018 Competition on Dynamic Multiobjective Optimization, Technical Report, Institute of Artificial Intelligence, Newcastle University, UK, 2018
    [6] Jiang S Y, Kaiser M, Yang S X, Kollias S D, Krasnogor N. A scalable test suite for continuous dynamic multiobjective optimization. IEEE Transactions on Cybernetics, 2020, 50(6): 2814-2826. doi: 10.1109/TCYB.2019.2896021
    [7] Deb K, Agarwal S, Pratap A, Meyarivan T. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation, 2002, 6(2): 182-197. doi: 10.1109/4235.996017
    [8] Zhang Q F, Li H. MOEA/D: A multiobjective evolutionary algorithm based on decomposition. IEEE Transactions on Evolutionary Computation, 2007, 11(6): 712-731. doi: 10.1109/TEVC.2007.892759
    [9] Yang S, Yao X. Experimental study on population-based incremental learning algorithms for dynamic optimization problems. Soft Computing, 2005, 9(11): 815-834. doi: 10.1007/s00500-004-0422-3
    [10] Deb K, Rao U B, Karthik S. Dynamic multi-objective optimization and decision-making using modified NSGA-II: A case study on hydro-thermal power scheduling. In: Proceedings of the 4th International Conference on Evolutionary Multi-Criterion Optim-ization. Matsushima, Japan: 2007. 803−817
    [11] Zhou A M, Jin Y C, Zhang Q F. A population prediction strategy for evolutionary dynamic multiobjective optimization. IEEE Transactions on Cybernetics, 2014, 44(1): 40-53. doi: 10.1109/TCYB.2013.2245892
    [12] 丁进良, 杨翠娥, 陈立鹏, 柴天佑等. 基于参考点预测的动态多目标优化算法. 自动化学报, 2017, 43(2): 313-320.

    Ding Jing-Liang, Yang Cui-er, Chen Li-Peng, Chai Tian-You. Dynamic multi-objective optimization algorithm based on reference point prediction. Acta Automatica Sinica, 2017, 43(2): 313-320 (in Chinese)
    [13] 陈美蓉, 郭一楠, 巩敦卫, 杨振. 一类新型动态多目标鲁棒进化优化方法. 自动化学报, 2017, 43(11): 2014-2032.

    Chen Mei-Rong, Guo Yi-Nan, Gong Dun-Wei, Yang Zhen. A novel dynamic multi-objective robust evolutionary optimization method. Acta Automatica Sinica, 2017, 43(11): 2014-2032 (in Chinese)
    [14] Zeng S Y, Chen G, Zheng L, Shi H, Garis H D, Ding L X X, et al. A dynamic multi-objective evolutionary algorithm based on an orthogonal design. In: Proceedings of the 6th IEEE Congress on Evolutionary Computation. Vancouver, Canada: 2006. 573− 580
    [15] Goh C K, Tan K C. A competitive-cooperative coevolutionary paradigm for dynamic multiobjective optimization. IEEE Transactions on Evolutionary Computation, 2009, 13(1): 103-127. doi: 10.1109/TEVC.2008.920671
    [16] Yen G G, Leong W F, Dynamic multiple swarms in multiobjective particle swarm optimization. IEEE Transactions on Systems, Man, and Cybernetics, Part A, 2009, 39(4): 890-911. doi: 10.1109/TSMCA.2009.2013915
    [17] Shang R, Jiao L, Ren Y, Li L, Wang L. Quantum immune clonal co-evolutionary algorithm for dynamic multi-objective optimization. Soft Computing, 2014, 18(4): 743-756. doi: 10.1007/s00500-013-1085-8
    [18] Liu R C, Li J X, Fan J, Mu C H, Jiao L C. A coevolutionary technique based on multi-swarm particle swarm optimization for dynamic multi-objective optimization. European Journal of Operational Research, 2017, 261(3): 1028-1051. doi: 10.1016/j.ejor.2017.03.048
    [19] Zhang K, Shen C, Liu X, Yen G G. Multiobjective evolution strategy for dynamic multiobjective optimization. IEEE Transactions on Evolutionary Computation, 2020, 24(5): 974-988. doi: 10.1109/TEVC.2020.2985323
    [20] Wang Y, Li B. Investigation of memory-based multi-objective optimization evolutionary algorithm in dynamic environment. In: Proceedings of the 9th IEEE Congress on Evolutionary Computation. Trondheim, Norway: 2009. 630−637
    [21] Zhang Z, Qian S. Artificial immune system in dynamic environments solving time-varying non-linear constrained multi-objective problems. Soft Computing, 2011, 15(7): 1333-1349. doi: 10.1007/s00500-010-0674-z
    [22] Sahmoud S, Topcuoglu H R. A memory-based NSGA-II algo-rithm for dynamic multi-objective optimization problems. In: Proceedings of the 19th European Conference on Applications of Evolutionary Computation. Porto, Portugal: 2016. 296−310
    [23] Hatzakis I, Wallace D. Dynamic multi-objective optimization with evolutionary algorithms: A forward-looking approach. In: Proceedings of the 8th Annual Conference on Genetic and Evolutionary Computation. Seattle, USA: 2006. 1201−1208
    [24] 郑金华, 彭舟, 邹娟, 申瑞珉. 基于引导个体的预测策略求解动态多目标优化问题. 电子学报, 2015, 43(9): 1816-1825. doi: 10.3969/j.issn.0372-2112.2015.09.021

    Zheng Jin-Hua, Peng Zhou, Zou Juan, Shen Rrui-Min. A prediction strategy based on guide-individual for dynamic multi-objective optimization. Acta Electronica Sinic, 2015, 43(9): 1816-1825 (in Chinese) doi: 10.3969/j.issn.0372-2112.2015.09.021
    [25] Liu R C, Niu X, Fan J, Mu C H, Jiao L C. An orthogonal predictive model-based dynamic multi-objective optimization algorithm. Soft Computing, 2015, 19(11): 3083-3107. doi: 10.1007/s00500-014-1470-y
    [26] Wu Y, Jin Y C, Liu X X. A directed search strategy for evolutionary dynamic multiobjective optimization, Soft Computing, 2015, 19(11): 3221-3235. doi: 10.1007/s00500-014-1477-4
    [27] Muruganantham A, Tan K. C, Vadakkepat P. Evolutionary dynamic multiobjective optimization via Kalman filter prediction, IEEE Transactions on Cybernetics, 2016, 46(12): 2862-2873. doi: 10.1109/TCYB.2015.2490738
    [28] Jiang S Y, Yang S X. A steady-state and generational evolutionary algorithm for dynamic multiobjective optimization, IEEE Transactions on Evolutionary Computation, 2017, 21(1): 65-82. doi: 10.1109/TEVC.2016.2574621
    [29] Rong M, Gong D W, Zhang Y, Jin Y C, Pedrycz W. Multi-directional prediction approach for dynamic multi-objective optimization problems. IEEE Transactions on Cybernetics, 2018, 49(9): 3362-3374.
    [30] Cao L L, Xu L H, Goodman E D, Bao C T, Zhu S W. Evolutionary dynamic multiobjective optimization assisted by a support vector regression predictor, IEEE Transactions on Evolutionary Computation, 2019, 24(2): 305-319.
    [31] Rong M, Gong D W. A multi-model prediction method for dynamic multi-objective evolutionary optimization. IEEE Transactions on Evolutionary Computation, 2020, 24(2): 290-304. doi: 10.1109/TEVC.2019.2925358
    [32] Wang C F, Yen G G, Jiang M, A grey prediction-based evolutionary algorithm for dynamic multiobjective optimization. Swarm and Evolutionary Computation, 2020, 56: 100695. doi: 10.1016/j.swevo.2020.100695
    [33] Gee S B, Tan K C, Alippi C. Solving multi-objective optimization problems in unknown dynamic environments: an inverse modeling approach. IEEE Transactions on Cybernetics, 2017, 47(12): 4223-4234. doi: 10.1109/TCYB.2016.2602561
    [34] Jiang M, Huang Z Q, Qiu L M, Huang W Z, Yen G G. Transfer learning-based dynamic multiobjective optimization algorithms, IEEE Transactions on Evolutionary Computation, 2018, 22(4): 501-514. doi: 10.1109/TEVC.2017.2771451
    [35] Jiang M, Hu W Z, Qiu L M, Shi M H, Tan K C. Solving dynamic multi-objective optimization problems via support vector machine. In: Proceedings of the 10th International Conference on Advanced Computational Intelligence. Xiamen, China: 2018. 819−824
    [36] Jiang M, Wang Z Z, Hong H K, Yen G G. Knee point based imbalanced transfer learning for dynamic multi-objective optimization. IEEE Transactions on Evolutionary Computation, to be published
    [37] Zhao Q, Yan B, Shi Y, Middendorf M. Evolutionary dynamic multi-objective optimization via learning from historical search process. IEEE Transactions on Cybernetics, 2021: Article No. 3059252
    [38] Jiang M, Wang Z, Guo S, Gao X, Tan K C. Individual-based transfer learning for dynamic multi-objective optimization. IEEE Transactions on Cybernetics, doi: 10.1109/TCYB.2020.3017049.
    [39] Peng Z, Zheng J H, Zou J, Liu Min. Novel Prediction and memory strategies for dynamic multiobjective optimization. Soft Computing, 2015, 19(9): 2633-2653. doi: 10.1007/s00500-014-1433-3
    [40] Azzouz R, Bechikh S, Said L B. A dynamic multi-objective evolutionary algorithm using a change severity-based adaptive population management strategy. Soft Computing, 2017, 21(4): 885-906. doi: 10.1007/s00500-015-1820-4
    [41] Zou J, Li Q Y, Yang S X, Bai H, Zheng J H, A prediction strategy based on center points and knee points for evolutionary dynamic multi-objective optimization, Applied Soft Computing, 2017, 61: 806-818. doi: 10.1016/j.asoc.2017.08.004
    [42] Liang Z P, Zheng S X, Zhu Z X, Yang S X. Hybrid of memory and prediction strategies for dynamic multiobjective optimization. Information Sciences, 2019, 485: 200-218. doi: 10.1016/j.ins.2019.01.066
    [43] Rambabu R, Vadakkepat P, Tan K C, Jiang M. A mixture-of-experts prediction framework for evolutionary dynamic multiobjective optimization. IEEE Transactions on Cybernetics, 2019, 50(12): 5099-5112.
    [44] Zhang Q Y, Yang S X, Jiang S Y, Wang R G, Li X L. Novel prediction strategies for dynamic multiobjective optimization, IEEE Transactions on Evolutionary Computation, 2020, 24(2): 260-274. doi: 10.1109/TEVC.2019.2922834
    [45] Gong D W, Xu B, Zhang Y, Guo Y N, Yang S X. A similarity-based cooperative co-evolutionary algorithm for dynamic interval multiobjective optimization problems. IEEE Transactions on Evolutionary Computation, 2020, 24(1): 142-156. doi: 10.1109/TEVC.2019.2912204
    [46] Feng L, Zhou W, Liu W, Ong Y S, Tan K C. Solving dynamic multi-objective problem via autoencoding evolutionary search. IEEE Transactions on Cybernetics, 2020, 3017017
    [47] Liang Z P, Wu T C, Ma X L, Zhu Z X, Yang S X. A dynamic multi-objective evolutionary algorithm based on decision variable classification. IEEE Transactions on Cybernetics, 2022, 52(3):1602−1615
    [48] 刘若辰, 李建霞, 刘静, 焦李成. 动态多目标优化研究 综述. 计算机学报, 2020, 43(07): 1246-1278.

    Liu Ruo-Chen, Li Jian-Xia, Liu Jing, Jiao Li-Cheng. A survey on dynamic multi-objective optimization. Chinese Journal of Computers, 2020, 43(07): 1246-1278 (in Chinese)
    [49] 马永杰, 陈敏, 龚影, 程时升, 王甄延. 动态多目标优化进化算法研究进展. 自动化学报, 2020, 46(11): 2302-2318.

    Ma Yong-Jie, Chen Min, Gong Ying, Cheng Shi-Sheng, Wang Zeng-Yan. Research progress of dynamic multi-objective optimization evolutionary algorithm. Acta Automatica Sinica, 2020, 46(11): 2302-2318 (in Chinese)
    [50] Woldesenbet Y G, Yen G G. Dynamic evolutionary algorithm with variable relocation, IEEE Transactions on Evolutionary Computation, 2009, 13(3): 500-513. doi: 10.1109/TEVC.2008.2009031
    [51] Koo, W T, Goh C K, Tan K C. A predictive gradient strategy for multi-objective evolutionary algorithms in a fast changing environment. Memetic Computing, 2010, 2(2): 87-110. doi: 10.1007/s12293-009-0026-7
    [52] Sierra M R, Coello C A C. Improving PSO-based multi-objective optimization using cowding, mutation and epsilon-dominance. In: Proceedings of the 3rd International Conference on Evolutionary Multi-Criterion Optimization. Guanajuato, Mexico: 2005. 505−519
    [53] Wilcoxon F. Individual comparisons by ranking methods. Breakthroughs in Statistics: Methodology and distribution. New York: Springer, 1992. 196−202
    [54] Friedman M. The use of ranks to avoid the assumption of normality implicit in the analysis of variance. Journal of the American Statistical Association, 1937, 32(200): 675-701. doi: 10.1080/01621459.1937.10503522
  • 加载中
图(7) / 表(9)
计量
  • 文章访问数:  1401
  • HTML全文浏览量:  668
  • PDF下载量:  290
  • 被引次数: 0
出版历程
  • 收稿日期:  2021-02-05
  • 录用日期:  2021-06-25
  • 网络出版日期:  2021-07-30
  • 刊出日期:  2023-08-21

目录

    /

    返回文章
    返回