2.793

2018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

一种基于目标空间转换权重求和的超多目标进化算法

梁正平 骆婷婷 王志强 朱泽轩 胡凯峰

梁正平, 骆婷婷, 王志强, 朱泽轩, 胡凯峰. 一种基于目标空间转换权重求和的超多目标进化算法. 自动化学报, 2021, 47(x): 1−19 doi: 10.16383/j.aas.c200483
引用本文: 梁正平, 骆婷婷, 王志强, 朱泽轩, 胡凯峰. 一种基于目标空间转换权重求和的超多目标进化算法. 自动化学报, 2021, 47(x): 1−19 doi: 10.16383/j.aas.c200483
Liang Zheng-Ping, Luo Ting-Ting, Wang Zhi-Qiang, Zhu Ze-Xuan, Hu Kai-Feng. A many-objective evolutionary algorithm based on weighted sum of objective space transformation. Acta Automatica Sinica, 2021, 47(x): 1−19 doi: 10.16383/j.aas.c200483
Citation: Liang Zheng-Ping, Luo Ting-Ting, Wang Zhi-Qiang, Zhu Ze-Xuan, Hu Kai-Feng. A many-objective evolutionary algorithm based on weighted sum of objective space transformation. Acta Automatica Sinica, 2021, 47(x): 1−19 doi: 10.16383/j.aas.c200483

一种基于目标空间转换权重求和的超多目标进化算法

doi: 10.16383/j.aas.c200483
基金项目: 国家自然科学基金(61871272); 广东省自然科学基金(2020A1515010479); 深圳市科技计划项目(JCYJ20190808173617147, GGFW2018020518310863)
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

    骆婷婷:华为技术有限公司工程师. 2020年获深圳大学硕士学位. 主要研究方向为计算智能, 自然语言处理与应用等. E-mail: luotingting2017@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

    朱泽轩:深圳大学计算机与软件学院教授. 2008年获新加坡南洋理工大学博士学位. 主要研究方向为计算智能, 机器学习与生物信息学等. 本文通讯作者. E-mail: zhuzx@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

A Many-objective Evolutionary Algorithm based on Weighted Sum of Objective Space Transformation

Funds: National Natural Science Foundation of China (61871272); Natural Science Foundation of Guangdong, China (2020A1515010479); Shenzhen Scientific Research and Development Funding Program (JCYJ20190808173617147, GGFW2018020518310863)
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 interest covers computational intelligence, big data analysis and application

    LUO Ting-Ting Engineer at Huawei Technology Co., Ltd. She received her master degree from Shenzhen University in 2020. Her research interest covers computational intelligence, natural language processing and applications

    WANG Zhi-Qiang professor at the School of Computer Science & Software Engineering, Shenzhen University. His research interest covers computational intelligence, big data analysis and applications, multimedia technology 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 interest covers computational intelligence, machine learning and bioinformatics

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

  • 摘要: 权重求和是基于分解的超多目标进化算法中常用的方法, 相比其它方法具有计算简单、搜索效率高等优点, 但难以有效处理帕累托前沿面(Pareto optimal front, PF)为非凸型的问题. 为充分发挥权重求和方法的优势, 同时又能处理好PF为非凸型的问题, 本文提出了一种基于目标空间转换权重求和的超多目标进化算法, 简称NSGAIII-OSTWS. 该算法的核心是将各种问题的PF转换为凸型曲面, 再利用权重求和方法进行优化. 具体地, 首先利用预估PF的形状计算个体到预估PF的距离; 然后, 根据该距离值将个体映射到目标空间中预估凸型曲面与理想点之间的对应位置; 最后, 采用权重求和函数计算出映射后个体的适应值, 据此实现对问题的进化优化. 为验证NSGAIII-OSTWS的有效性, 将NSGAIII-OSTWS与7个NSGAIII的变体, 以及9个具有代表性的先进超多目标进化算法在WFG、DTLZ和LSMOP基准问题上进行对比, 实验结果表明NSGAIII-OSTWS具备明显的竞争性能.
  • 图  1  分解方法WS, TCH和PBI在参考向量w上的二维示意图, 其中虚线为等高线

    Fig.  1  Illustration of the decomposition methods WS, TCH and PBI on reference vector w, where dashed lines are contour lines

    图  2  NSGAIII-WS和NSGAIII-LWS算法在ZDT1上获得的最终种群分布

    Fig.  2  The final population distribution obtained by NSGAIII-WS and NSGAIII-LWS algorithm on ZDT1

    图  3  OSTWS方法将PF形状为线形(a), 凸形(b)和凹形(c)种群中的个体转换到凸目标空间的整个过程

    Fig.  3  The whole process of transforming the population individuals from linear (a), convex (b) and concave (c) into convex objective space by OSTWS method

    图  4  NSGAIII-OSTWS, NSGAIII-LWS, NSGAIII-TCH, NSGAIII-PBI, NSGAIII-AS, NSGAIII-SS, NSGAIII-APS和NSGAIII-PaS, 在所有测试问题实例中的平均IGD+性能得分排名. 得分越小, 整体性能越好

    Fig.  4  Ranking in the average performance score over all test problem instances for the algorithms of NSGAIII-OSTWS, NSGAIII-LWS, NSGAIII-TCH, NSGAIII-PBI, NSGAIII-AS, NSGAIII-SS, NSGAIII-APS and NSGAIII-PaS. The smaller the score, the better the overall performance in terms of IGD+

    图  5  NSGAIII-OSTWS, NSGAIII-LWS, NSGAIII-TCH, NSGAIII-PBI, NSGAIII-AS, NSGAIII-SS, NSGAIII-PaS和NSGAIII-APS在10维DTLZ2问题上所获得的解集

    Fig.  5  Solution set of NSGAIII-OSTWS, NSGAIII-LWS, NSGAIII-TCH, NSGAIII-PBI, NSGAIII-AS, NSGAIII-SS, NSGAIII-PaS and NSGAIII-APS on DTLZ2 problem with 10-objectives

    图  6  SRA, SPEAR, DDEANS, HpaEA, ARMOEA, MaOEA-IT和PaRP/EA在10维DTLZ4问题上所获得的解集

    Fig.  6  Solution set of NSGAIII-OSTWS, NSGAIII, Two_arch2, SRA, SPEAR, DDEANS HpaEA, ARMOEA, MaOEA-IT and PaRP/EA on DTLZ4 problem with 10-objectives

    图  7  NSGAIII-OSTWS, NSGAIII, Two_arch2, SRA, SPEAR, DDEANS, HpaEA, ARMOEA, MaOEA-IT和PaRP/EA在所有测试问题, 即DTLZ(Dx), WFG(Wx) 和LSMOP(Lx) 上的平均GD表现分, 分值越小, 算法的整体性能越好. 通过实线连接NSGAIII-OSTWS的得分, 以便易于评估分数

    Fig.  7  Average performance score of NSGAIII-OSTWS, NSGAIII, Two_arch2, SRA, SPEAR, DDEANS, HpaEA, ARMOEA, MaOEA-IT and PaRP/EA on all test problems, namely DTLZ(Dx), WFG(Wx)and LSMOP(Lx). The smaller the score, the better the overall performance in terms of GD. The values of NSGAIII-OSTWS are connected by a solid line to easier assess the score

    图  9  不同$ C $值在WFG1, WFG4, DTLZ1和DTLZ7问题的3,5,8和10目标维度上的IGD+均值

    Fig.  9  The median IGD+ values of different $ C $ values on WFG1, WFG4, DTLZ1 and DTLZ7 problems with 3-, 5- 8- and 10-objectives

    图  8  不同预设凸曲线在参考向量w上获得的最优解示意图

    Fig.  8  The optimal solution obtained by different preset convex curves on the reference vector w

    表  1  种群大小设置

    Table  1  Setting of the population size

    目标数($ m $) 分割数($ H $) 种群大小($ N $)
    3 12 91
    5 6 210
    8 3, 2 156
    10 3, 2 275
    下载: 导出CSV

    表  2  交叉变异参数设置

    Table  2  Parameter settings for crossover and mutation

    参数名 参数值
    交叉概率($ P_c $) 1.0
    变异概率($ P_m $) 1/$ D $
    交叉分布指标($ \eta_c $) 20
    变异分布指标($ \eta_m $) 20
    下载: 导出CSV

    表  3  OSTWS, LWS, TCH, PBI, AS, SS, PaS和APS方法在框架为NSGAIII, 测试问题为DTLZ1-7上获得的GD值统计结果(均值和标准差). 每个实例算法中的最好结果以加粗突出显示

    Table  3  The statistical results (mean and standard deviation) of the GD values obtained by OSTWS, LWS, TCH, PBI, AS, SS, PaS and APS methods on the NSGAIII framework and DTLZ1-7 test problems. The best average value among the algorithms for each instance is highlighted in bold

    Problem $m$ NSGAIII-OSTWS NSGAIII-LWS NSGAIII-TCH NSGAIII-PBI NSGAIII-AS NSGAIII- SS NSGAIII-PaS NSGAIII-APS
    DTLZ1 3 6.915×101 7.734×101 7.411×101 7.140×101 7.678×101 7.749×101 7.554×101 7.376×101
    (1.2×101) (9.4×100)− (9.3×100)$\approx$ (1.1×101)$\approx$ (1.1×101)− (7.3×100)− (9.4×100)$\approx$ (9.8×100)$\approx$
    5 4.218×1017.562×101 8.102×101 7.015×101 7.870×101 1.294×102 7.993×101 6.507×101
    (4.5×100) (6.6×100)− (8.9×100)− (7.6×100)− (8.1×100)− (9.7×100)− (8.0×100)− (7.6×100)−
    8 4.881×1019.120×101 7.814×101 9.002×101 7.875×101 2.458×102 7.411×101 8.732×101
    (1.2×101) (7.3×100)− (1.1×101)− (1.1×101)− (1.2×101)− (5.8×101)− (9.0×100)− (8.8×100)−
    10 4.422×1019.338×101 7.014×101 7.334×101 6.757×101 2.672×102 7.299×101 7.538×101
    (1.8×101) (4.9×100)− (8.2×100)− (2.4×101)− (5.3×100)− (5.1×101)− (7.3×100)− (3.2×101)−
    DTLZ2 3 1.681×10−3 3.971×10−3 3.465×10−3 4.582×10−3 3.552×10−3 6.753×10−3 3.585×10−3 4.529×10−3
    (2.3×10−4) (6.1×10−4)− (3.9×10−4)− (6.8×10−4)− (4.2×10−4)− (1.1×10−3)− (5.2×10−4)− (6.4×10−4)−
    5 3.337×10−34.526×10−3 4.968×10−3 6.740×10−3 4.996×10−3 1.003×10−2 4.903×10−3 6.756×10−3
    (9.7×10−5) (3.6×10−4)− (3.0×10−4)− (5.3×10−4)− (4.3×10−4)− (1.4×10−3)− (4.1×10−4)− (4.0×10−4)−
    8 1.058×10−21.265×10−2 1.439×10−2 2.362×10−2 1.470×10−2 7.435×10−2 1.516×10−2 2.488×10−2
    (3.3×10−4) (2.5×10−3)− (1.7×10−3)− (4.6×10−3)− (1.4×10−3)− (3.5×10−2)− (2.5×10−3)− (3.2×10−3)−
    10 1.070×10−21.503×10−2 1.130×10−2 1.733×10−2 1.133×10−2 7.939×10−2 1.227×10−2 1.996×10−2
    (4.2×10−3) (6.0×10−3)− (2.7×10−3)− (7.3×10−3)− (1.8×10−3)− (5.4×10−2)− (3.5×10−3)− (6.6×10−3)−
    DTLZ3 3 8.888×101 8.622×101 8.323×101 8.322×101 8.246×101 8.259×101 8.910×101 8.705×101
    (1.2×101) (1.5×101)$\approx$ (1.5×101)$\approx$ (9.7×100)≈(1.1×101)$\approx$ (6.9×100)$\approx$ (1.3×101)$\approx$ (1.2×101)$\approx$
    5 6.174×1019.024×101 8.343×101 9.531×101 8.414×101 1.255×102 8.277×101 9.465×101
    (7.3×100) (9.1×100)− (1.0×101)− (1.3×101)− (1.1×101)− (9.7×100)− (8.8×100)− (9.7×100)−
    8 8.605×1011.266×102 1.220×102 1.536×102 1.176×102 3.001×102 1.293×102 1.434×102
    (2.0×101) (1.4×101)− (9.6×100)− (2.3×101)− (9.2×100)− (8.2×101)− (1.3×101)− (2.5×101)−
    10 7.830×1011.340×102 1.201×102 1.427×102 1.157×102 3.629×102 1.273×102 1.314×102
    (3.5×101) (2.3×101)− (1.4×101)− (4.5×101)− (7.3×100)− (7.1×101)− (2.7×101)− (2.9×101)−
    DTLZ4 3 1.918×10−3 4.026×10−3 3.588×10−3 4.068×10−3 3.455×10−3 7.379×10−3 3.242×10−3 4.393×10−3
    (3.1×10−4) (1.1×10−3)− (1.1×10−3)− (2.1×10−3)− (1.1×10−3)− (3.0×10−3)− (1.3×10−3)− (1.6×10−3)−
    5 3.506×10−35.160×10−3 5.502×10−3 8.623×10−3 5.326×10−3 7.816×10−3 5.367×10−3 8.775×10−3
    (5.0×10−4) (5.4×10−4)− (4.4×10−4)− (1.4×10−3)− (2.9×10−4)− (2.3×10−3)− (3.5×10−4)− (1.0×10−3)−
    8 1.658×10−22.169×10−2 2.858×10−2 3.486×10−2 2.597×10−2 7.637×10−2 1.887×10−2 3.592×10−2
    (2.0×10−2) (1.7×10−2)$\approx$ (1.8×10−2)− (1.6×10−2)− (1.7×10−2)− (3.6×10−2)− (5.4×10−3)− (2.2×10−2)−
    10 7.670×10−31.737×10−2 1.282×10−2 2.060×10−2 1.336×10−2 1.052×10−1 1.047×10−2 1.792×10−2
    (2.0×10−3) (1.8×10−2)$\approx$ (7.1×10−3)− (1.2×10−2)− (8.2×10−3)− (6.0×10−2)− (4.2×10−3)− (4.8×10−3)−
    DTLZ5 3 4.566×10−3 4.388×10−3 4.698×10−3 5.107×10−3 4.836×10−3 5.209×10−3 5.353×10−3 5.055×10−3
    (7.2×10−4) (7.7×10−4)≈ (6.3×10−4)$\approx$ (7.8×10−4)− (7.4×10−4)$\approx$ (8.2×10−4)− (8.9×10−4)− (5.4×10−4)−
    5 6.411×10−24.279×10−1 1.027×10−1 6.492×10−2 1.194×10−1 2.345×10−1 1.186×10−1 7.556×10−2
    (1.7×10−2) (8.2×10−2)− (1.6×10−2)− (1.5×10−2)$\approx$ (2.3×10−2)− (3.6×10−2)− (1.4×10−2)− (1.6×10−2)−
    8 2.795×10−1 5.134×10−1 4.167×10−1 4.142×10−1 5.253×10−1 1.082×100 5.319×10−1 4.527×10−1
    (5.2×10−2) (1.1×10−1)− (7.0×10−2)− (5.7×10−2)− (8.8×10−2)− (5.9×10−1)− (1.1×10−1)− (9.5×10−2)−
    10 3.845×10−11.266×100 1.283×100 8.411×10−1 1.660×100 2.054×100 1.668×100 9.914×10−1
    (2.3×10−1) (3.6×10−1)− (3.1×10−1)− (2.7×10−1)− (2.1×10−1)− (6.5×10−1)− (2.3×10−1)− (2.1×10−1)−
    DTLZ6 3 3.555×100 4.484×100 4.150×100 4.294×100 4.055×100 6.531×100 4.099×100 4.164×100
    (3.4×10−1)(3.6×10−1)− (4.0×10−1)− (4.3×10−1)− (2.3×10−1)− (2.6×10−1)− (4.0×10−1)− (4.2×10−1)−
    5 2.454×1001.135×101 8.566×100 6.659×100 8.595×100 7.759×100 8.606×100 6.597×100
    (2.8×10−1)(2.3×10−1)− (5.4×10−1)− (1.8×10−1)− (3.0×10−1)− (4.1×10−1)− (3.6×10−1)− (2.7×10−1)−
    8 1.235×1012.182×101 1.927×101 2.201×101 1.915×101 2.548×101 1.933×101 2.174×101
    (8.7×10−1)(2.3×100)− (8.5×10−1)− (3.1×100)− (9.4×10−1)− (6.2×100)− (1.1×100)− (4.0×100)−
    10 1.344×1012.871×101 2.511×101 2.395×101 2.535×101 2.887×101 2.501×101 2.203×101
    (1.6×100)(8.3×100)− (1.6×100)− (1.0×101)− (1.1×100)− (1.1×101)− (4.0×100)− (8.9×100)−
    DTLZ7 3 1.385×10−2 1.479×10−2 1.476×10−2 1.788×10−2 1.538×10−2 1.780×10−2 1.628×10−2 1.839×10−2
    (2.3×10−3)(2.5×10−3)$\approx$ (2.0×10−3)$\approx$ (2.3×10−3)− (1.9×10−3)− (3.3×10−3)− (3.7×10−3)− (2.5×10−3)−
    5 8.419×10−39.474×10−3 9.755×10−3 1.481×10−2 9.498×10−3 1.712×10−2 9.146×10−3 1.534×10−2
    (1.2×10−3)(1.4×10−3)− (1.0×10−3)− (1.6×10−3)− (1.0×10−3)− (1.6×10−3)− (1.3×10−3)$\approx$ (1.4×10−3)−
    8 2.742×10−22.987×10−2 3.999×10−2 3.594×10−2 3.700×10−2 5.275×10−2 4.093×10−2 3.597×10−2
    (1.8×10−3)(4.4×10−3)$\approx$ (3.2×10−3)− (5.5×10−3)− (5.2×10−3)− (5.6×10−3)− (5.4×10−3)− (4.7×10−3)−
    10 2.928×10−2 2.449×10−22.800×10−2 2.893×10−2 3.052×10−2 4.273×10−2 3.055×10−2 2.869×10−2
    (2.3×10−3) (3.6×10−3)+(2.5×10−3)$\approx$ (2.0×10−3)$\approx$ (1.9×10−3)$\approx$ (3.5×10−3)− (3.1×10−3)$\approx$ (3.6×10−3)$\approx$
    $+/-/\approx$ 1/21/6 0/23/5 0/24/4 0/25/3 0/27/1 0/25/3 0/24/4
    下载: 导出CSV

    表  4  OSTWS, LWS, TCH, PBI, AS, SS, PaS和APS方法在框架为NSGAIII, 测试问题集为WFG1-9上获得的GD值统计结果(均值和标准差). 每个实例算法中的最好结果以加粗突出显示

    Table  4  The statistical results (mean and standard deviation) of the GD values obtained by OSTWS, LWS, TCH, PBI, AS, SS, PaS and APS methods on the NSGAIII framework and WFG1-9 test problems. The best average value among the algorithms for each instance is highlighted in bold

    Problem $m$ NSGAIII-OSTWS NSGAIII-LWS NSGAIII-TCH NSGAIII-PBI NSGAIII-AS NSGAIII- SS NSGAIII-PaS NSGAIII-APS
    WFG1 3 4.082×10−2 4.125×10−2 4.435×10−2 4.357×10−2 4.475×10−2 4.206×10−2 4.454×10−2 4.368×10−2
    (6.0×10−4) (9.1×10−4)$\approx$ (8.9×10−4 (7.2×10−4)− (7.5×10−4)− (1.1×10−3)− (1.2×10−3)− (6.7×10−4)−
    5 2.789×10−2 2.670×10−2 3.220×10−2 2.936×10−2 3.194×10−2 2.790×10−2 3.225×10−2 2.960×10−2
    (9.2×10−4) (4.3×10−4)+ (6.2×10−4)− (3.9×10−4)− (7.8×10−4)− (7.8×10−4)− (6.2×10−4)− (3.0×10−4)−
    8 3.323×10−2 3.429×10−2 3.472×10−2 3.483×10−2 3.504×10−2 3.624×10−2 3.506×10−2 3.446×10−2
    (9.2×10−4) (1.3×10−3)− (8.6×10−4)− (9.1×10−4)− (1.3×10−3)− (3.4×10−3)− (1.5×10−3)− (1.4×10−3)−
    10 2.474×10−2 2.585×10−2 2.589×10−2 2.614×10−2 2.546×10−2 2.816×10−2 2.535×10−2 2.607×10−2
    (5.6×10−4) (5.3×10−4)− (1.2×10−3)− (9.3×10−4)− (9.5×10−4)− (1.6×10−3)− (9.3×10−4)− (8.2×10−4)−
    WFG2 3 5.354×10−3 5.103×10−3 5.846×10−3 6.124×10−3 5.965×10−3 8.529×10−3 6.085×10−3 6.088×10−3
    (6.5×10−4) (4.8×10−4)≈ (6.4×10−4)− (4.6×10−4)− (4.8×10−4)− (1.6×10−3)− (7.1×10−4)− (4.9×10−4)−
    5 4.885×10−3 5.663×10−3 7.042×10−3 5.902×10−3 7.329×10−3 6.705×10−3 6.924×10−3 6.009×10−3
    (2.2×10−4) (6.0×10−4)− (6.3×10−4)− (2.1×10−4)− (5.3×10−4)− (8.3×10−4)− (1.1×10−3)− (2.1×10−4)−
    8 8.277×10−3 1.011×10−2 9.969×10−3 9.745×10−3 1.025×10−2 1.269×10−2 1.006×10−2 1.018×10−2
    (7.0×10−4) (1.1×10−3)− (5.7×10−4)− (1.6×10−3)− (1.0×10−3)− (2.8×10−3)− (9.9×10−4)− (3.1×10−3)−
    10 1.528×10−2 1.447×10−2 1.309×10−2 1.329×10−2 1.142×10−2 1.460×10−2 1.179×10−2 1.317×10−2
    (1.8×10−3) (1.8×10−3)$\approx$ (1.9×10−3)+ (2.3×10−3)+ (1.6×10−3)+ (2.7×10−3)$\approx$ (1.1×10−3)+ (2.6×10−3)+
    WFG3 3 1.154×10−2 1.273×10−2 1.480×10−2 1.684×10−2 1.4610×10−2 2.620×10−2 1.497×10−2 1.693×10−2
    (1.5×10−3) (1.6×10−3)− (1.2×10−3)− (2.0×10−3)− (1.1×10−3)− (2.0×10−3)− (2.0×10−3)− (1.1×10−3)−
    5 3.912×10−2 3.555×10−2 1.130×10−1 8.826×10−2 1.320×10−1 5.550×10−2 1.156×10−1 7.736×10−2
    (4.2×10−3) (4.5×10−3)+ (1.1×10−2)− (2.7×10−2)− (1.2×10−2)− (7.5×10−3)− (1.5×10−2)− (2.0×10−2)−
    8 6.263×10−1 8.328×10−1 6.008×10−1 5.867×10−1 7.676×10−1 5.991×10−1 6.118×10−1 6.314×10−1
    (1.3×10−1) (8.3×10−2)− (1.2×10−1)$\approx$ (1.2×10−1)≈ (2.5×10−1)− (9.2×10−2)$\approx$ (1.2×10−1)$\approx$ (2.3×10−1)$\approx$
    10 2.374×100 3.674×100 2.840×100 1.902×100 3.110×100 2.402×100 3.252×100 1.965×100
    (8.6×10−1) (6.4×10−1)− (1.0×100)− (6.0×10−1)≈ (9.7×10−1)− (2.7×10−1)− (1.1×100)− (6.7×10−1)$\approx$
    WFG4 3 1.387×10−3 2.231×10−3 3.001×10−3 3.370×10−3 2.893×10−3 4.412×10−3 2.953×10−3 3.284×10−3
    (1.1×10−4) (1.5×10−4)− (2.5×10−4)− (2.1×10−4)− (2.3×10−4)− (3.6×10−4)− (1.5×10−4)− (2.4×10−4)−
    5 3.717×10−3 2.834×10−3 5.696×10−3 4.746×10−3 5.847×10−3 4.401×10−3 5.766×10−3 4.725×10−3
    (6.3×10−5) (7.6×10−5)+ (3.1×10−4)− (8.3×10−5)− (3.6×10−4)− (1.4×10−4)− (3.4×10−4)− (6.5×10−5)−
    8 1.263×10−2 1.244×10−2 1.543×10−2 1.501×10−2 1.497×10−2 1.462×10−2 1.517×10−2 1.437×10−2
    (3.8×10−4) (3.6×10−4)≈ (6.9×10−4)− (7.1×10−4)− (6.3×10−4)− (1.4×10−3)− (5.4×10−4)− (1.5×10−3)−
    10 7.624×10−3 1.344×10−2 1.203×10−2 8.833×10−3 1.210×10−2 1.401×10−2 1.211×10−2 8.060×10−3
    (8.9×10−4) (5.8×10−4)− (1.4×10−4)− (1.8×10−3)− (2.0×10−4)− (6.9×10−4)− (2.1×10−4)− (1.3×10−3)$\approx$
    WFG5 3 2.770×10−3 3.404×10−3 3.979×10−3 4.138×10−3 3.927×10−3 5.689×10−3 3.861×10−3 4.077×10−3
    (7.4×10−5) (2.0×10−4)− (2.2×10−4)− (1.7×10−4)− (1.8×10−4)− (5.0×10−4)− (1.9×10−4)− (1.6×10−4)−
    5 4.094×10−3 3.377×10−3 6.612×10−3 4.771×10−3 6.671×10−3 4.338×10−3 6.681×10−3 4.755×10−3
    (8.3×10−5) (5.6×10−5)+ (4.5×10−4)− (8.2×10−5)− (4.7×10−4)− (1.9×10−4)− (5.1×10−4)− (8.3×10−5)−
    8 1.288×10−2 1.281×10−2 1.747×10−2 1.543×10−2 1.767×10−2 1.385×10−2 1.737×10−2 1.548×10−2
    (2.2×10−4) (5.9×10−4)≈ (5.6×10−4)− (2.0×10−4)− (5.8×10−4)− (1.1×10−3)− (4.6×10−4)− (3.0×10−4)−
    10 9.292×10−3 1.366×10−2 1.149×10−2 8.550×10−3 1.158×10−2 1.246×10−2 1.159×10−2 8.604×10−3
    (3.8×10−4) (3.0×10−4)− (3.3×10−4)− (3.8×10−4)+ (3.0×10−4)− (8.8×10−4)− (3.2×10−4)− (4.0×10−4)+
    WFG6 3 2.151×10−3 3.052×10−3 3.902×10−3 4.170×10−3 3.933×10−3 5.274×10−3 3.913×10−3 4.134×10−3
    (1.6×10−4) (1.9×10−4)− (2.0×10−4)− (3.0×10−4)− (2.7×10−4)− (4.4×10−4)− (2.4×10−4)− (2.3×10−4)−
    5 3.999×10−3 3.168×10−3 8.235×10−3 4.969×10−3 8.134×10−3 4.872×10−3 7.986×10−3 4.941×10−3
    (9.9×10−5) (7.5×10−5)+ (9.9×10−4)− (1.2×10−4)− (8.0×10−4)− (2.0×10−4)− (8.7×10−4)− (1.1×10−4)−
    8 1.250×10−2 1.215×10−2 1.800×10−2 1.544×10−2 1.799×10−2 1.536×10−2 1.7820×10−2 1.555×10−2
    (2.4×10−4) (7.6×10−4)≈ (5.8×10−4)− (2.3×10−4)− (7.9×10−4)− (9.2×10−4)− (9.1×10−4)− (3.0×10−4)−
    10 7.483×10−3 1.238×10−2 1.230×10−2 7.812×10−3 1.241×10−2 1.463×10−2 1.244×10−2 8.120×10−3
    (5.2×10−4) (6.1×10−4)− (4.0×10−4)− (3.6×10−4)− (3.1×10−4)− (6.7×10−4)− (3.2×10−4)− (9.1×10−4)−
    WFG7 3 8.882×10−4 2.105×10−3 3.270×10−3 5.073×10−3 3.280×10−3 7.455×10−3 3.182×10−3 4.750×10−3
    (4.9×10−5) (3.2×10−4)− (7.6×10−4)− (7.9×10−4)− (3.7×10−4)− (3.0×10−3)− (4.5×10−4)− (5.8×10−4)−
    5 3.323×10−3 2.647×10−3 7.618×10−3 5.695×10−3 8.228×10−3 4.176×10−3 8.303×10−3 5.888×10−3
    (8.2×10−5) (9.6×10−5)+ (1.6×10−3)− (5.7×10−4)− (1.8×10−3)− (3.2×10−4)− (1.6×10−3)− (7.7×10−4)−
    8 1.168×10−2 1.245×10−2 1.742×10−2 1.599×10−2 1.745×10−2 1.330×10−2 1.736×10−2 1.606×10−2
    (6.5×10−4) (2.2×10−3)− (6.5×10−4)− (6.3×10−4)− (5.7×10−4)− (1.0×10−3)− (6.3×10−4)− (8.2×10−4)−
    10 6.602×10−3 1.129×10−2 1.217×10−2 8.449×10−3 1.222×10−2 1.308×10−2 1.220×10−2 8.566×10−3
    (6.2×10−4) (8.8×10−4)− (3.1×10−4)− (5.0×10−4)− (2.6×10−4)− (5.0×10−4)− (3.6×10−4)− (7.1×10−4)−
    WFG8 3 4.390×10−3 5.139×10−3 5.619×10−3 5.561×10−3 5.792×10−3 7.016×10−3 5.703×10−3 5.550×10−3
    (2.7×10−4) (2.4×10−4)− (2.6×10−4)− (2.5×10−4)− (2.5×10−4)− (4.8×10−4)− (3.5×10−4)− (3.0×10−4)−
    5 4.796×10−3 4.301×10−3 7.939×10−3 5.018×10−3 7.786×10−3 5.403×10−3 7.969×10−3 5.053×10−3
    (1.1×10−4) (1.0×10−4)+ (4.1×10−4)− (1.0×10−4)− (4.8×10−4)− (3.6×10−4)− (5.9×10−4)− (9.2×10−5)−
    8 1.307×10−2 1.308×10−2 1.832×10−2 1.569×10−2 1.834×10−2 1.524×10−2 1.828×10−2 1.560×10−2
    (3.0×10−4) (1.1×10−3)$\approx$ (5.5×10−4)− (2.5×10−4)− (5.0×10−4)− (1.1×10−3)− (4.7×10−4)− (6.4×10−4)−
    10 9.844×10−3 1.405×10−2 1.227×10−2 9.342×10−3 1.229×10−2 1.346×10−2 1.243×10−2 9.774×10−3
    (2.6×10−4) (4.3×10−4)− (4.0×10−4)− (4.8×10−4)+ (3.0×10−4)− (8.9×10−4)− (4.6×10−4)− (8.4×10−4)$\approx$
    WFG9 3 2.200×10−3 6.110×10−3 5.287×10−3 8.679×10−3 5.360×10−3 8.726×10−3 4.796×10−3 8.011×10−3
    (2.8×10−4) (6.8×10−4)− (5.0×10−4)− (1.1×10−3)− (6.0×10−4)− (9.6×10−4)− (7.7×10−4)− (6.3×10−4)−
    5 4.144×10−3 4.250×10−3 8.592×10−3 7.838×10−3 8.773×10−3 5.496×10−3 8.772×10−3 7.720×10−3
    (1.1×10−4) (2.1×10−4)− (1.3×10−3)− (3.6×10−4)− (2.0×10−3)− (4.4×10−4)− (1.3×10−3)− (4.5×10−4)−
    8 1.339×10−2 1.570×10−2 1.810×10−2 1.814×10−2 1.827×10−2 1.508×10−2 1.801×10−2 1.793×10−2
    (3.2×10−4) (1.8×10−3)− (7.5×10−4)− (5.5×10−4)− (6.2×10−4)− (1.6×10−3)− (7.3×10−4)− (4.3×10−4)−
    10 1.082×10−2 1.533×10−2 1.245×10−2 1.288×10−2 1.280×10−2 1.315×10−2 1.276×10−2 1.274×10−2
    (3.5×10−4) (3.1×10−4)− (3.2×10−4)− (4.1×10−4)− (4.6×10−4)− (8.6×10−4)− (4.3×10−4)− (3.8×10−4)−
    $+/-/\approx$ 7/22/7 1/33/2 3/31/2 1/35/0 0/32/4 1/34/1 2/30/4
    下载: 导出CSV

    表  5  OSTWS, LWS, TCH, PBI, AS, SS, PaS和APS方法在框架为NSGAIII, 测试问题集为LSMOP1-9上获得的GD值统计结果(均值和标准差). 每个实例算法中的最好结果以加粗突出显示

    Table  5  The statistical results (mean and standard deviation) of the GD values obtained by OSTWS, LWS, TCH, PBI, AS, SS, PaS and APS methods on the NSGAIII framework and LSMOP1-9 test problems. The best average value among the algorithms for each instance is highlighted in bold

    Problem m NSGAIII-OSTWS NSGAIII-LWS NSGAIII-TCH NSGAIII-PBI NSGAIII-AS NSGAIII-SS NSGAIII-PaS NSGAIII-APS
    LSMOP1 3 8.526×10−1 (1.3×10−1) 9.451×10−1 (9.7×10−2)− 8.776×10−1 (1.1×10−1)$\approx$ 9.691×10−1 (1.4×10−1)− 8.904×10−1 (1.4×10−1)$\approx$ 1.439×100 (7.8×10−1)$\approx$ 9.296×10−1 (1.4×10−1)$\approx$ 9.070×10−1 (1.3×10−1)$\approx$
    5 4.829×10−1 (2.1×10−1) 5.596×10−1 (5.7×10−2)− 5.858×10−1 (5.3×10−2)− 5.483×10−1 (9.3×10−2)− 5.800×10−1 (4.8×10−2)− 8.027×10−1 (8.0×10−2)− 4.784×10−1 (1.0×10−1)≈ 5.518×10−1 (4.5×10−2)−
    8 5.409×10−1 (1.4×10−1) 8.855×10−1 (5.9×10−2)− 8.447×10−1 (1.6×10−1)− 8.037×10−1 (2.0×10−1)− 9.307×10−1 (2.1×10−1)− 1.343×100 (1.7×10−1)− 7.875×10−1 (1.1×10−1)− 8.568×10−1 (1.4×10−1)−
    10 4.598×10−1 (1.1×10−1) 8.812×10−1 (1.01×10−1)− 8.967×10−1 (7.9×10−2)− 6.762×10−1 (1.4×10−1)− 9.021×10−1 (8.4×10−2)− 9.689×10−1 (1.1×10−1)− 6.232×10−1 (7.5×10−2)− 9.011×10−1 (1.0×10−1)−
    LSMOP2 3 7.646×10−3 (1.8×10−4) 1.026×10−2 (2.3×10−4)− 9.405×10−3 (1.3×10−4)− 9.825×10−3 (1.6×10−4)− 9.403×10−3 (1.4×10−4)− 1.105×10−2 (1.1×10−3)− 9.698×10−3 (1.7×10−4)− 9.490×10−3 (1.8×10−4)−
    5 6.197×10−3 (8.9×10-5) 8.530×10−3 (9.0×10-5)− 7.660×10−3 (7.0×10-5)− 7.866×10−3 (5.8×10-5)− 7.661×10−3 (1.6×10−4)− 9.447×10−3 (5.4×10−4)− 7.920×10−3 (5.0×10-5)− 7.623×10−3 (4.5×10-5)−
    8 1.243×10−2 (6.0×10−4) 2.217×10−2 (2.6×10−3)− 1.890×10−2 (2.2×10−3)− 1.948×10−2 (6.8×10−4)− 1.839×10−2 (1.5×10−3)− 1.792×10−2 (3.7×10−3)− 1.982×10−2 (1.5×10−3)− 1.824×10−2 (2.1×10−3)−
    10 9.467×10−3 (2.1×10−4) 1.364×10−2 (6.5×10−4)− 1.230×10−2 (1.3×10−3)− 1.271×10−2 (1.6×10−4)− 1.248×10−2 (1.1×10−3)− 1.658×10−2 (4.0×10−3)− 1.213×10−2 (3.3×10−4)− 1.239×10−2 (1.3×10−3)−
    LSMOP3 3 2.990×102 (1.32×102) 2.763×102 (7.0×101)≈ 2.993×102 (8.8×101)$\approx$ 3.366×102 (1.8×102)$\approx$ 3.316×102 (8.7×101)$\approx$ 4.780×102 (2.5×102)− 3.243×102 (1.3×102)$\approx$ 3.793×102 (1.0×102)−
    5 6.116×102 (2.8×102) 8.207×102 (1.7×102)− 9.284×102 (1.5×102)− 9.599×102 (5.5×102)− 9.630×102 (2.2×102)− 1.906×103 (4.6×102)− 9.785×102 (4.95×102)− 9.834×102 (2.7×102)−
    8 1.369×103 (5.8×102) 2.207×103 (5.8×102)− 2.607×103 (6.1×102)− 1.927×103 (6.2×102)− 3.040×103 (5.2×102)− 3.418×103 (8.0×102)− 2.167×103 (8.0×102)− 3.087×103 (8.9×102)−
    10 3.379×103 (1.7×103) 3.133×103 (1.1×103)$\approx$ 4.097×103 (9.2×102)$\approx$ 2.945×103 (1.5×103)≈ 4.655×103 (1.2×103)− 3.510×103 (7.9×102)$\approx$ 3.373×103 (1.9×103)$\approx$ 4.029×103 (9.6×102)$\approx$
    LSMOP4 3 3.073×10−2 (2.0×10−3) 3.744×10−2 (9.4×10−4)− 3.730×10−2 (1.4×10−3)− 4.045×10−2 (1.2×10−3)− 3.704×10−2 (1.1×10−3)− 4.004×10−2 (4.9×10−3)− 4.034×10−2 (1.0×10−3)− 3.742×10−2 (1.2×10−3)−
    5 2.598×10−2 (1.8×10−3) 3.942×10−2 (1.8×10−3)− 3.656×10−2 (2.1×10−3)− 3.888×10−2 (2.0×10−3)− 3.547×10−2 (1.5×10−3)− 3.750×10−2 (4.6×10−3)− 4.093×10−2 (1.6×10−3)− 3.641×10−2 (1.6×10−3)−
    8 1.995×10−2 (7.8×10−4) 3.397×10−2 (1.3×10−3)− 2.209×10−2 (3.5×10−3)− 3.297×10−2 (1.4×10−3)− 2.324×10−2 (4.2×10−3)− 5.114×10−2 (9.6×10−3)− 3.301×10−2 (1.4×10−3)− 2.257×10−2 (4.2×10−3)−
    10 1.459×10−2 (4.8×10−4) 2.521×10−2 (1.3×10−3)− 1.490×10−2 (8.0×10−4)$\approx$ 2.260×10−2 (7.0×10−4)− 1.548×10−2 (1.1×10−3)− 2.585×10−2 (6.9×10−3)− 2.283×10−2 (1.0×10−3)− 1.613×10−2 (2.4×10−3)−
    LSMOP5 3 2.605×100 (3.9×10−1) 2.694×100 (3.2×10−1)$\approx$ 2.534×100 (3.2×10−1)$\approx$ 2.780×100 (2.7×10−1)$\approx$ 2.516×100 (4.6×10−1)$\approx$ 3.782×100 (4.9×10−1)− 2.772×100 (4.1×10−1)$\approx$ 2.292×100 (2.4×10−1)+
    5 2.314×100 (1.4×100) 3.210×100 (2.3×10−1)− 3.370×100 (3.1×10−1)− 3.510×100 (8.2×10−1)− 3.075×100 (2.4×10−1)− 5.678×100 (5.9×10−1)− 3.810×100 (4.8×10−1)− 3.079×100 (3.1×10−1)−
    8 4.708×100 (2.2×100) 7.771×100 (1.1×100)− 7.613×100 (9.5×10−1)− 5.93×100 (3.6×100)$\approx$ 7.218×100 (9.6×10−1)− 7.384×100 (1.7×100)− 8.816×100 (1.8×100)− 7.327×100 (1.1×100)−
    10 5.077×100 (7.7×10−1) 6.883×100 (6.8×10−1)− 6.279×100 (9.4×10−1)− 6.707×100 (2.0×100)− 6.522×100 (9.3×10−1)− 4.854×100 (6.3×10−1)≈ 7.631×100 (2.8×10−1)− 6.118×100 (7.2×10−1)−
    LSMOP6 3 4.402×103 (2.5×103) 3.127×103 (1.4×103)$\approx$ 2.938×103 (1.1×103)≈ 3.477×103 (1.7×103)$\approx$ 3.410×103 (1.6×103)$\approx$ 4.088×103 (2.0×103)$\approx$ 4.089×103 (2.5×103)$\approx$ 3.025×103 (2.0×103)$+$
    5 5.002×103 (2.6×103) 5.108×103 (1.3×103)$\approx$ 5.354×103 (1.0×103)$\approx$ 4.366×103 (2.5×103)$\approx$ 5.637×103 (1.4×103)$\approx$ 1.202×104 (3.4×103)− 3.587×103 (1.2×103)+ 5.220×103 (1.6×103)$\approx$
    8 2.206×104 (4.9×103) 2.041×104 (5.6×103)≈ 2.484×104 (9.6×103)$\approx$ 3.608×104 (9.8×103)− 2.475×104 (8.7×103)$\approx$ 7.647×104 (1.8×104)− 3.385×104 (1.3×104)− 2.425×104 (5.4×103)$\approx$
    10 1.933×104 (4.8×103) 1.924×104 (2.7×103)≈ 2.847×104 (6.9×103)− 3.601×104 (8.8×103)− 2.661×104 (9.5×103)− 5.022×104 (8.7×103)− 3.410×104 (7.8×103)− 2.844×104 (6.2×103)−
    LSMOP7 3 5.540×102 (1.4×102) 8.520×102 (4.2×102)− 8.707×102 (3.8×102)− 9.200×102 (4.0×102)− 8.946×102 (2.5×102)− 2.598×103 (8.7×102)− 9.381×102 (2.8×102)− 1.013×103 (3.0×102)−
    5 4.597×103 (1.6×103) 4.424×103 (1.8×103)$\approx$ 5.261×103 (2.1×103)$\approx$ 4.238×103 (9.4×102)≈ 5.665×103 (1.6×103)− 1.403×104 (3.9×103)− 4.530×103 (1.8×103)$\approx$ 5.627×103 (3.4×103)$\approx$
    8 3.305×104 (8.4×103) 3.482×104 (8.4×103)$\approx$ 3.490×104 (7.6×103)$\approx$ 4.471×104 (1.8×104)− 2.847×104 (9.6×103)≈ 5.022×104 (1.0×104)− 4.770×104 (1.6×104)− 3.268×104 (7.4×103)$\approx$
    10 3.545×104 (4.5×103) 3.599×104 (6.3×103)$\approx$ 3.980×104 (8.8×103)$\approx$ 4.835×104 (8.0×103)− 3.065×104 (6.0×103)+ 3.246×104 (5.90×103)$\approx$ 5.216×104 (6.9×103)− 3.455×104 (8.2×103)$\approx$
    LSMOP8 3 3.940×10−1 (8.9×10−2) 4.540×10−1 (5.3×10−2)− 4.006×10−1 (6.0×10−2)$\approx$ 4.445×10−1 (7.2×10−2)− 4.205×10−1 (6.5×10−2)$\approx$ 1.071×100 (1.8×10−1)− 4.654×10−1 (7.6×10−2)− 4.081×10−1 (8.3×10−2)$\approx$
    5 5.216×10−1 (1.3×10−1) 6.430×10−1 (7.1×10−2)− 6.342×10−1 (1.1×10−1)− 8.661×10−1 (1.3×10−1)− 6.569×10−1 (1.2×10−1)− 2.096×100 (2.8×10−1)− 8.376×10−1 (1.3×10−1)− 6.559×10−1 (1.1×10−1)−
    8 2.282×100 (6.5×10−1) 3.190×100 (4.8×10−1)− 3.522×100 (5.1×10−1)− 4.130×100 (6.7×10−1)− 3.117×100 (5.0×10−1)− 3.437×100 (4.4×10−1)− 4.167×100 (3.1×10−1)− 3.361×100 (4.3×10−1)−
    10 2.307×100 (2.4×10−1) 2.924×100 (3.4×10−1)− 2.958×100 (3.6×10−1)− 3.363×100 (2.9×10−1)− 2.690×100 (4.0×10−1)− 2.299×100 (2.1×10−1)≈ 3.322×100 (2.9×10−1)− 2.853×100 (4.5×10−1)−
    LSMOP9 3 4.151×10−1 (8.1×10−2) 4.150×10−1 (5.9×10−2)$\approx$ 4.220×10−1 (1.0×10−1)$\approx$ 4.240×10−1 (8.7×10−2)$\approx$ 3.690×10−1 (5.8×10−2)+ 5.607×10−1 (1.3×10−1)− 4.334×10−1 (9.4×10−2)$\approx$ 4.134×10−1 (6.4×10−2)$\approx$
    5 2.658×10−1 (3.9×10−2) 2.945×10−1 (3.7×10−2)− 2.915×10−1 (5.0×10−2)$\approx$ 3.451×10−1 (5.9×10−2)− 2.772×10−1 (4.2×10−2)$\approx$ 4.072×10−1 (7.1×10−2)− 3.220×10−1 (5.3×102)− 2.750×10−1 (3.4×10−2)$\approx$
    8 1.766×100 (1.7×10−1) 2.421×100 (2.4×10−1)− 3.296×100 (3.5×10−1)− 2.265×100 (2.2×10−1)− 3.525×100 (3.4×10−1)− 4.172×100 (3.9×10−1)− 2.267×100 (2.5×10−1)− 3.801×100 (4.0×10−1)−
    10 1.789×100 (2.5×10−1) 2.766×100 (2.5×10−1)− 4.112×100 (3.2×10−1)− 2.824×100 (3.4×10−1)− 4.783×100 (3.5×10−1)− 4.820×100 (2.8×10−1)− 2.740×100 (3.5×10−1)− 4.940×100 (3.6×10−1)−
    $+/-/\approx$ 0/25/11 0/22/14 0/28/8 2/25/9 0/30/6 1/27/8 2/24/10
    下载: 导出CSV

    表  6  OSTWS, LWS, TCH, PBI, AS, SS, PaS和APS方法在框架为NSGAIII, 测试问题集为DTLZ1, DTLZ2, DTLZ5和DTLZ7上获得的CPF值统计结果(均值和标准差). 每个实例算法中的最好结果以加粗突出显示

    Table  6  The statistical results (mean and standard deviation) of the CPF values obtained by OSTWS, LWS, TCH, PBI, AS, SS, PaS and APS methods on the NSGAIII framework and DTLZ1, DTLZ2, DTLZ5 and DTLZ7 test problems. The best average value among the algorithms for each instance is highlighted in bold

    Problem $m$ NSGAIII-OSTWS NSGAIII-LWS NSGAIII-TCH NSGAIII-PBI NSGAIII-AS NSGAIII-SS NSGAIII-PaS NSGAIII-APS
    DTLZ1 3 1.374×10−3 (2.4×10−3) 5.495×10−4 (1.7×10−3)$\approx$ 8.242×10−4 (2.7×10−3)$\approx$ 4.558×10−4 (1.4×10−3)$\approx$ 1.099×10−3 (2.3×10−3)$\approx$ 0.000×100 (0.0×100)− 2.748×10−4 (1.2×10−3)$\approx$ 4.021×10−4 (1.3×10−3)$\approx$
    5 1.436×10−4 (3.5×10−4) 8.930×10−5 (2.2×10−4)$\approx$ 2.924×10−4 (9.7×10−4)$\approx$ 2.837×10−4 (5.6×10−4)$\approx$ 2.954×10−4 (5.9×10−4)$\approx$ 5.953×10−5 (2.7×10−4)$\approx$ 3.020×10−4 (5.0×10−4)≈ 2.339×10−4 (4.1×10−4)$\approx$
    8 7.099×10−5 (1.1×10−4) 3.072×10−5 (6.0×10−5)$\approx$ 1.083×10−4 (2.0×10−4)$\approx$ 3.749×10−4 (7.7×10−4)≈ 5.327×10−5 (1.1×10−4)$\approx$ 2.757×10−5 (6.4×10−5)$\approx$ 1.655×10−4 (4.5×10−4)$\approx$ 1.068×10−4 (2.0×10−4)$\approx$
    10 1.657×10−4 (4.1×10−4) 7.355×10−5 (2.2×10−4)$\approx$ 6.571×10-6 (1.7×10−5)− 1.422×10−4 (4.2×10−4)$\approx$ 4.603×10−5 (1.2×10−4)$\approx$ 1.399×10−4 (4.4×10−4)$\approx$ 7.511×10−5 (2.4×10−4)$\approx$ 8.130×10−5 (2.3×10−4)$\approx$
    DTLZ2 3 5.698×10−1 (4.3×10−2) 3.218×10−1 (2.3×10−2)− 5.792×10−1 (3.5×10−2)$\approx$ 6.891×10−1 (1.1×10−2)+ 5.427×10−1 (4.5×10−2)− 1.684×10−1 (3.6×10−2)− 5.632×10−1 (3.1×10−2)$\approx$ 6.837×10−1 (2.3×10−2)$+$
    5 5.993×10−1 (2.2×10−2) 1.585×10−1 (1.2×10−2)− 5.521×10−1 (4.1×10−2)− 7.114×10−1 (1.4×10−2)+ 5.416×10−1 (4.7×10−2)− 1.307×10−1 (3.0×10−2)− 5.433×10−1 (4.1×10−2)− 7.108×10−1 (1.5×10−2)$+$
    8 3.780×10−1 (2.8×10−2) 5.395×10−2 (1.6×10−2)− 2.871×10−1 (4.2×10−2)− 4.085×10−1 (2.8×10−2)+ 2.922×10−1 (2.4×10−2)− 3.258×10−2 (2.6×10−2)− 2.947×10−1 (2.4×10−2)− 3.682×10−1 (1.1×10−1)$\approx$
    10 2.185×10−1 (3.7×10−3) 2.729×10−2 (1.5×10−2)− 1.752×10−1 (4.3×10−2)− 1.914×10−1 (6.6×10−2)$\approx$ 1.912×10−1 (2.1×10−2)− 3.958×10−2 (1.3×10−2)− 1.855×10−1 (1.9×10−2)− 1.900×10−1 (6.3×10−2)$\approx$
    DTLZ5 3 6.043×10−1 (4.4×10−2) 5.616×10−1 (7.6×10−2)− 5.755×10−1 (7.9×10−2)$\approx$ 6.053×10−1 (4.6×10−2)$\approx$ 5.639×10−1 (5.4×10−2)− 5.423×10−1 (5.4×10−2)− 5.925×10−1 (5.9×10−2)$\approx$ 6.092×10−1 (5.3×10−2)≈
    5 5.397×10−1 (7.5×10−2) 4.781×10−1 (4.9×10−2)− 3.670×10−1 (5.6×10−2)− 4.987×10−1 (4.5×10−2)− 2.654×10−1 (6.3×10−2)− 1.838×10−1 (4.1×10−2)− 2.452×10−1 (7.8×10−2)− 4.935×10−1 (6.0×10−2)−
    8 5.903×10−1 (1.2×10−1) 4.791×10−1 (7.8×10−2)− 5.093×10−1 (6.2×10−2)− 5.213×10−1 (8.9×10−2)$\approx$ 4.770×10−1 (9.3×10−2)− 3.355×10−1 (1.3×10−1)− 3.963×10−1 (9.9×10−2)− 5.067×10−1 (8.9×10−2)−
    10 3.857×10−1 (5.1×10−2) 2.622×10−1 (5.2×10−2)− 3.066×10−1 (5.6×10−2)− 3.804×10−1 (4.8×10−2)$\approx$ 2.635×10−1 (5.7×10−2)− 3.790×10−1 (1.4×10−1)$\approx$ 2.391×10−1 (8.5×10−2)− 3.524×10−1 (4.4×10−2)−
    DTLZ7 3 2.961×10−1 (4.3×10−2) 2.502×10−1 (4.3×10−2)− 2.853×10−1 (5.1×10−2)$\approx$ 2.866×10−1 (3.9×10−2)$\approx$ 2.835×10−1 (6.6×10−2)$\approx$ 1.519×10−1 (3.6×10−2)− 2.676×10−1 (5.6×10−2)$\approx$ 2.911×10−1 (4.8×10−2)$\approx$
    5 2.716×10−1 (3.4×10−2) 1.956×10−1 (2.8×10−2)− 2.760×10−1 (2.3×10−2)$\approx$ 2.890×10−1 (4.2×10−2)$\approx$ 2.622×10−1 (2.7×10−2)$\approx$ 2.139×10−1 (3.5×10−2)− 2.530×10−1 (1.8×10−2)$\approx$ 2.974×10−1 (3.1×10−2)+
    8 5.846×10−1 (1.0×10−1) 2.044×10−1 (3.4×10−2)− 3.897×10−1 (5.4×10−2)− 5.149×10−1 (4.7×10−2)− 3.996×10−1 (6.1×10−2)− 2.534×10−1 (3.7×10−2)− 3.618×10−1 (7.0×10−2)− 5.210×10−1 (4.9×10−2)−
    10 1.3102×10−1 (4.1×10−2) 2.657×10−1 (3.2×10−2)$+$ 9.318×10−2 (2.0×10−2)− 1.994×10−1 (1.6×10−2)$+$ 9.436×10−2 (1.5×10−2)− 2.663×10−1 (3.3×10−2)+ 1.056×10−1 (2.4×10−2)− 2.015×10−1 (2.0×10−2)$+$
    $+/-/\approx$ 1/11/4 0/9/7 4/2/10 0/10/6 1/11/4 0/8/8 4/4/8
    下载: 导出CSV

    表  7  NSGAIII-OSTWS, NSGAIII, Two_arch2, SRA, SPEAR, DDEANS, HpaEA, ARMOEA, MaOEA-IT和PaRP/EA在DTLZ1-7上上获得的IGD+值的统计结果.

    Table  7  The statistical results of the IGD+ values obtained by NSGAIII-OSTWS, NSGAIII, Two_arch2, SRA, SPEAR, DDEANS, hpaEA, ARMOEA, MaOEA-IT and PaRP/EA on DTLZ1-7.

    NSGAIII-OSTWS vs NSGAIII Two_Arch2 SRA SPEAR DDEANS HpaEA ARMOEA MaOEA-IT PaRP/EA
    + 0/28 1/28 5/28 1/28 2/28 2/28 2/28 2/28 1/28
    27/28 26/28 22/28 25/28 24/28 25/28 24/28 26/28 23/28
    $\approx$ 1/28 1/28 1/28 2/28 2/28 1/28 2/28 0/28 4/28
    下载: 导出CSV

    表  8  NSGAIII-OSTWS, NSGAIII, Two_arch2, SRA, SPEAR, DDEANS, HpaEA, ARMOEA, MaOEA-IT和PaRP/EA在WFG1-9上上获得的IGD+值的统计结果.

    Table  8  The statistical results of the IGD+ values obtained by NSGAIII-OSTWS, NSGAIII, Two_arch2, SRA, SPEAR, DDEANS, HpaEA, ARMOEA, MaOEA-IT and PaRP/EA on WFG1-9.

    NSGAIII-OSTWS vs NSGAIII Two_Arch2 SRA SPEAR DDEANS HpaEA ARMOEA MaOEA-IT PaRP/EA
    + 1/36 0/36 0/36 0/36 5/36 0/36 0/36 0/36 0/36
    35/36 26/36 35/36 35/36 30/36 36/36 36/36 36/36 34/36
    $\approx$ 0/36 0/36 1/36 1/36 1/36 0/36 0/36 0/36 2/36
    下载: 导出CSV

    表  9  NSGAIII-OSTWS, NSGAIII, Two_arch2, SRA, SPEAR, DDEANS, HpaEA, ARMOEA, MaOEA-IT和PaRP/EA在LSMOP1-9上获得的IGD+值的统计结果.

    Table  9  The statistical results of the IGD+ values obtained by NSGAIII-OSTWS, NSGAIII, Two_arch2, SRA, SPEAR, DDEANS, HpaEA, ARMOEA, MaOEA-IT and PaRP/EA on LSMOP1-9.

    NSGAIII-OSTWS vs NSGAIII Two_Arch2 SRA SPEAR DDEANS HpaEA ARMOEA MaOEA-IT PaRP/EA
    + 10/36 13/36 12/36 10/36 11/36 10/36 16/36 7/36 9/36
    - 21/36 21/36 20/36 23/36 22/36 23/36 17/36 24/36 23/36
    $\approx$ 5/36 2/36 4/36 3/36 3/36 3/36 3/36 5/36 4/36
    下载: 导出CSV
  • [1] 孔维健, 柴天佑, 丁进良, 吴志伟. 镁砂熔炼过程全厂电能分配实时多目标优化方法研究. 自动化学报, 2014, 40(01): 51−61

    Kong Wei-Jian, Chai Tian-You, Ding Jin-Liang, Wu Zhi-Wei. A real-time multiobjective electric energy allocation optimization approach for the smelting process of magnesia. Acta Electronica Sinica, 2014, 40(01): 51−61
    [2] 乔俊飞, 韩改堂, 周红标. 基于知识的污水生化处理过程智能优化方法. 自动化学报, 2017, 43(06): 1038−1046

    Qiao Jun-Fei, Han Gai-Tang, Zhou Hong-Biao. Knowledge-based intelligent optimal control for wastewater biochemical treatment process. Acta Electronica Sinica, 2017, 43(06): 1038−1046
    [3] 林闯, 陈莹, 黄霁崴, 向旭东. 服务计算中服务质量的多目标优化模型与求解研究. 计算机学报, 2015, 38(10): 1907−1923 doi: 10.11897/SP.J.1016.2015.01907

    Lin Chuang, Chen Ying, Huang Ji-Wei, Xiang Xu-Dong. A survey on models and splutions of multi-objecive optimization for qoS in services computing. Chinese Journal of Computers, 2015, 38(10): 1907−1923 doi: 10.11897/SP.J.1016.2015.01907
    [4] Wang J, Zhou Y, Wang Y, Zhang J, Chen C L P, Zheng Z. Multiobjective vehicle routing problems with simultaneous delivery and pickup and time windows: formulation, instances, and algorithms. IEEE Transactions on Cybernetics, 2016, 46(3): 582−594 doi: 10.1109/TCYB.2015.2409837
    [5] Sarro F, Ferrucci F, Harman M, Manna A, Ren J. Adaptive multi-objective evolutionary algorithms for overtime planning in software projects. IEEE Transactions on Software Engineerging, 2017, 43(10): 898−917 doi: 10.1109/TSE.2017.2650914
    [6] Bin C, Weinan D, Zhihan L, Yu G, Surjit S, Pawan K. Hybrid microgrid many-objective sizing optimization with fuzzy decision. IEEE Transactions on fuzzy systems, 2020, 28(11): 2702−2710 doi: 10.1109/TFUZZ.2020.3026140
    [7] Deb K, Pratap A, Agarwal S, 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] Zitzler E, Laumanns M, Thiele L. SPEA2: improving the strength pareto evolutionary algorithm. technical report 103, Computer Engineering and Communication Networks Lab (TIK), Swiss Federal Institute of Technology (ETH), Switzerland, 2001
    [9] 封文清, 巩敦卫. 基于在线感知Pareto前沿划分目标空间的多目标进化优化. 自动化学报, 2020, 46(08): 1628−1643

    Feng Wei-Qing, Dong Dun-Wei. Multi-objective evolutionary optimization with objective space partition based on online perception of pareto front. Acta Electronica Sinica, 2020, 46(08): 1628−1643
    [10] Purshouse R C, Fleming P J. On the evolutionary optimization of many conflicting objectives. IEEE Transactions on Evolutionary Computation, 2007, 11(6): 770−784 doi: 10.1109/TEVC.2007.910138
    [11] 王丽萍, 章鸣雷, 邱飞岳, 江波. 基于角度惩罚距离精英选择策略的偏好高维目标优化算法. 计算机学报, 2018, 41(01): 236−253

    Wang Li-Ping, Zhang Ming-Lei, Qiu Fei-Yue, Jiang Bo. Many-objective optimization algorithm with perference based on the angle penalty distance elite selection strategy. Chinese Journal of Computers, 2018, 41(01): 236−253
    [12] 谢承旺, 余伟伟, 闭应洲, 汪慎文, 胡玉荣. 一种基于分解和协同的高维多目标进化算法. 软件学报, 2020, 31(02): 356−373

    Xie Cheng-Wang, Yu Wei-Wei, Bi Ying-Zhou, Wang Sheng-Wen, Hu Yu-Rong. Many-objective evolutionary algorithm based on decomposition and coevolution. Journal of Software, 2020, 31(02): 356−373
    [13] He Z N, Yen G G. Many-objective evolutionary algorithms based on coordinated selection strategy. IEEE Transactions on Evolutionary Computation, 2017, 21(02): 220−233 doi: 10.1109/TEVC.2016.2598687
    [14] Chen L, Liu H L, Tan K C, Cheung Y M, Wang Y P. Evolutionary many-objective algorithm using decomposition-based dominance relationship. IEEE Transactions on Cybernetics, 2019, 49(12): 4129−4139 doi: 10.1109/TCYB.2018.2859171
    [15] Pan L Q, He C, Tian Y, Wang H D, Zhang X Y, Jin Y C. A classification-based surrogate-assisted evolutionary algorithm for expensive many-objective optimization. IEEE Transactions on Evolutionary Computation, 2019, 23(1): 74−88 doi: 10.1109/TEVC.2018.2802784
    [16] Laumanns M, Thiele L, Deb K, Zitzler E. Combining con-vergence and diversity in evolutionary multiobjective optimization. Evolutionary Computation, 2002, 10(3): 263−282 doi: 10.1162/106365602760234108
    [17] Zou X F, Chen Y, Liu M Z, Kang L S. A new evolutionary algorithm for solving many-objective optimization problems. IEEE Transactions on Systems Man and Cybernetics Part B Cybernetics A Publication of the IEEE Systems Man and Cybernetics Society, 2008, 38(5): 1405−1412
    [18] He Z, Yen G G, Zhang J. Fuzzy-based pareto optimality for many-objective evolutionary algorithms. IEEE Transactions on Evolutionary Computation, 2014, 18(2): 269−285 doi: 10.1109/TEVC.2013.2258025
    [19] 余伟伟, 谢承旺, 闭应洲, 夏学文, 李雄, 任柯燕, 赵怀瑞, 王少锋. 一种基于自适应模糊支配的高维多目标粒子群算法. 自动化学报, 2018, 44(12): 2278−2289

    Yu Wei-Wei, Xie Cheng-Wang, Bi Ying-Zhou, Xia Xue-Wen, Li Xiong, Ren Ke-Yan, Zhao Huai-Rui, Wang Shao-Feng. Many-objective particle swarm optimization based on adaptive fuzzy dominance. Acta Electronica Sinica, 2018, 44(12): 2278−2289
    [20] Zitzler E, Künzli S. Indicator-based selection in multiobjective search. In: Proceedings of the 8th International Conference on Parallel Problem Solving from Nature. Birmingham, UK, LNCS, 2004, 3242: 832−842
    [21] Bader J, Zitzler E. HypE: An algorithm for fast hypervolume-based many-objective optimization. Evolutionary Computation, 2011, 19(1): 45−76 doi: 10.1162/EVCO_a_00009
    [22] Tian Y, Cheng R, Zhang X Y, Cheng F, Jin Y C. An indicator-based multiobjective evolutionary algorithm with reference point adaptation for better versatility. IEEE Transactions on Evolutionary Computation, 2018, 22(4): 609−622 doi: 10.1109/TEVC.2017.2749619
    [23] 陈国玉, 李军华, 黎明, 陈昊. 基于R2指标和参考向量的高维多目标进化算法. 自动化学报, 2019 doi: 10.16383/j.aas.c180722

    Chen Guo-Yu, Li Jun-Hua, Li Ming, Chen Hao. An r2 indicator and reference vector based many-objective optimization Evolutionary Algorithm. Acta Electronica Sinica, 2019 doi: 10.16383/j.aas.c180722
    [24] 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
    [25] Deb K, Jain H. An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, partI: solving problems with box constraints. IEEE Transactions on Evolutionary Computation, 2014, 18(4): 577−601 doi: 10.1109/TEVC.2013.2281535
    [26] Jiang S Y, Yang S X. A strength pareto evolutionary algorithm based on reference direction for multiobjective and many-objective optimization. IEEE Transactions on Evolutionary Computation, 2017, 21(3): 821−837
    [27] Wang H, Jiao L, Yao X. Two_arch2: an improved two-archive algorithm for many-objective optimization. IEEE Transactions on Evolutionary Computation, 2015, 19(4): 524−541 doi: 10.1109/TEVC.2014.2350987
    [28] Chen H K, Tian Y, Pedrycz W, Wu G H, Wang R, Wang L. Hyperplane assisted evolutionary algorithm for many-objective optimization problems. IEEE Transactions on Cybernetics, 2020, 50(7): 3367−3380 doi: 10.1109/TCYB.2019.2899225
    [29] Li B, Tang K, Li J, Yao X. Stochastic ranking algorithm for many-objective optimization based on multiple indicators. IEEE Transactions on Evolutionary Computation, 2016, 20(6): 924−938 doi: 10.1109/TEVC.2016.2549267
    [30] Hua Y, Jin Y, Hao K. A clustering based adaptive evolutionary algorithm for multi-objective optimization with irregular Pareto fronts. IEEE Transactions on Cybernetics, 2019, 49(7): 2758−2770 doi: 10.1109/TCYB.2018.2834466
    [31] Qi Y T, Ma X L, Liu F, Jiao L C, Sun J Y, Wu J S. MOEA/D with adaptive weight adjustment. Evolutionary Computation, 2014, 22(2): 231−264 doi: 10.1162/EVCO_a_00109
    [32] Asafuddoula M, Singh H K, Ray T. An enhanced decomposition-based evolutionary algorithm with adaptive reference vectors. IEEE Transactions on Cybernetics, 2018, 48(8): 2321−2334 doi: 10.1109/TCYB.2017.2737519
    [33] He X Y, Zhou Y R, Chen Z F, Zhang Q F. Evolutionary many-objective optimization based on dynamical decomposition. IEEE Transactions on Evolutionary Computation, 2019, 23(3): 361−375 doi: 10.1109/TEVC.2018.2865590
    [34] Ishibuchi H, Setoguchi Y, Masuda H, Nojima Y. Performance of decomposition-based many-objective algorithms strongly depends on pareto front shapes. IEEE Transactions on Evolutionary Computation, 2017, 21(2): 169−190 doi: 10.1109/TEVC.2016.2587749
    [35] Hansen M P. Use of substitute scalarizing functions to guide a local search based heuristic: The case of motsp. Journal of Heuristics, 2000, 6(3): 419−431 doi: 10.1023/A:1009690717521
    [36] Wang R, Zhang Q F, Zhang T. Decomposition based algorithms using pareto adaptive scalarizing methods. IEEE Transactions on Evolutionary Computation, 2016, 20(6): 821−837 doi: 10.1109/TEVC.2016.2521175
    [37] Ishibuchi H, Sakane Y, Tsukamoto N, Nojima Y. Adaptation of scalarizing functions in MOEAD: an adaptive scalarizing function-based multiobjective evolutionary algorithm. In: Proceedings of the 5th International Conference on Evolutionary Multi-Criterion Optimization. Nantes, France, LNCS, 2009, 5467: 438−452
    [38] Ishibuchi H, Sakane Y, Tsukamoto N, Nojima Y. Simultaneous use of different scalarizing functions in MOEA/D. In: Proceedings of the Genetic and Evolutionary Computation Conference. Portland, Oregon, USA, 2010, 519−526
    [39] Wang R, Zhou Z B, Ishibuchi H, Liao T J, Zhang T. Localized weighted sum method for many-objective optimization. IEEE Transactions on Evolutionary Computation, 2018, 22(1): 3−18 doi: 10.1109/TEVC.2016.2611642
    [40] Li K, Zhang Q F, Kwong S, Li M Q, Wang R. Stable matching-based selection in evolutionary multiobjective optimization. IEEE Transactions on Evolutionary Computation, 2014, 18(6): 909−923 doi: 10.1109/TEVC.2013.2293776
    [41] Jiang S Y, Yang S X. An improved multiobjective optimization evolutionary algorithm based on decomposition for complex pareto fronts. IEEE Transactions on Cybernetics, 2016, 46(2): 421−437 doi: 10.1109/TCYB.2015.2403131
    [42] Wang L P, Zhang Q F, Zhou A, Gong M G, Jiao L C. Constrained subproblems in a decomposition-based multiobjective evolutionary algorithm. IEEE Transactions on Evolutionary Computation, 2015, 20(3): 475−480
    [43] Ishibuchi H, DoiK, Nojima Y. Use of piecewise linear and nonlinear scalarizing functions in MOEA/D. In: Proceedings of the 14th International Conference on Parallel Problem Solving from Nature (PPSN), Edinburgh, UK, LNCS, 2016, 9921: 503−513
    [44] Sato H. Analysis of inverted PBI and comparison with other scalarizing functions in decomposition based MOEAs. Journal of Heuristics, 2015, 21(6): 819−849 doi: 10.1007/s10732-015-9301-6
    [45] Zitzler E, Thiele L. Multiobjective evolutionary algorithms: a comparative case study and the strength pareto approach. IEEE Transactions on Evolutionary Computation, 1999, 3(4): 257−271 doi: 10.1109/4235.797969
    [46] Liang Z P, Hu K F, Ma X L, Zhu Z X. A many-objective evolutionary algorithm based on a two-round selection strategy. IEEE Transactions on Cybernetics, 2019 doi: 10.1109/TCYB.2019.2918087
    [47] Yang S X, Li M Q, Liu X H, Zheng J H. A grid-based evolutionary algorithm for many-objective optimization. IEEE Transactions on Evolutionary Computation, 2013, 17(5): 721−736 doi: 10.1109/TEVC.2012.2227145
    [48] Yuan Y, Xu H, Wang B, Yao X. A new dominance relation-based evolutionary algorithm for many-objective optimization. IEEE Transactions on Evolutionary Computation, 2016, 20(1): 16−37 doi: 10.1109/TEVC.2015.2420112
    [49] Deb K, Agrawal R B. Simulated binary crossover for continuous search space. Complex Systems, 1994, 9(3): 115−148
    [50] Deb K, Goyal M. A combined genetic adaptive search (GeneAS) for Engineering design. Computer Science and Informatics, 1996, 26(4): 30−45
    [51] Huband S, Hingston P, Barone L, While L. A review of multiobjective test problems and a scalable test problem toolkit. IEEE Transactions on Evolutionary Computation, 2006, 10(5): 477−506 doi: 10.1109/TEVC.2005.861417
    [52] Deb K, Thiele L, Laumanns M, Zitzler E. Scalable test problems for evolutionary multiobjective optimization. In: Proceedings of the 2005 Evolutionary Multiobjective Optimization. London: Springer, 2005, 105−145
    [53] Cheng R, Jin Y C, Olhofer M, Sendhoff B. Test problems for large-scale multiobjective and many-objective optimization. IEEE Transactions on Cybernetics, 2017, 47(12): 4108−4121 doi: 10.1109/TCYB.2016.2600577
    [54] Bosman P A N, Thierens D. The balance between proximity and diversity in multiobjective evolutionary algorithms. IEEE Transactions on Evolutionary Computation, 2003, 7(2): 174−188 doi: 10.1109/TEVC.2003.810761
    [55] Tian Y, Cheng R, Zhang X Y, Li M Q, Jin Y C. Diversity assessment of multi-objective evolutionary algorithms: performance metric and benchmark problems. IEEE Comoutational Intelligence Magazine, 2019, 14(3): 61−74 doi: 10.1109/MCI.2019.2919398
    [56] Ishibuchi H, Masuda H, Nojima Y. A study on performance evaluation ability of a modified inverted generational distance indicator. In: Proceedings of the 2015 Annual Conference on Genetic and Evolutionary Computation, 2015, 695−702
    [57] Liang Z P, Luo T T, Hu K F, Ma X L, Zhu Z X. An indicator-based many-objective evolutionary algorithm with boundary protection. IEEE Transactions on Cybernetics, 2020 doi: 10.1109TCYB.2019.2960302
    [58] Li K, Deb K, Zhang Q F, Kwong S. An evolutionary many-objective optimization algorithm based on dominance and decomposition. IEEE Transactions on Evolutionary Computation, 2015, 19(5): 694−716 doi: 10.1109/TEVC.2014.2373386
    [59] Wilcoxon F. Individual comparisons by ranking methods. Biometrics Bulletin, 1945, 1(6): 80−83 doi: 10.2307/3001968
    [60] Wang R, Zhang Q F, Zhang T. Decomposition-based algorithms using pareto adaptive scalarizing methods. IEEE Transactions on Evolutionary Computation, 2016, 20(6): 821−837 doi: 10.1109/TEVC.2016.2521175
    [61] Yang S X, Jiang S Y, Jiang Y. Improving the multiobjective evolutionary algorithm based on decomposition with new penalty schemes. Soft Computing, 2017, 21: 4677−4691 doi: 10.1007/s00500-016-2076-3
    [62] Yu Y N, Xue B, Zhang M J, Yen G G. A new two-stage evolutionary algorithm for many-objective optimization. IEEE Transactions on Evolutionary Computation, 2019, 23(5): 748−761 doi: 10.1109/TEVC.2018.2882166
    [63] Xiang Y, Zhou Y R, Yang X W, Huang H. A many-objective evolutionary algorithm with pareto-adaptive reference points. IEEE Transactions on Evolutionary Computation, 2020, 24(1): 99−113 doi: 10.1109/TEVC.2019.2909636
  • 加载中
计量
  • 文章访问数:  51
  • HTML全文浏览量:  26
  • 被引次数: 0
出版历程
  • 收稿日期:  2020-06-30
  • 录用日期:  2021-01-26
  • 网络出版日期:  2021-03-16

目录

    /

    返回文章
    返回