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基于邻域竞赛的多目标优化算法

刘元 郑金华 邹娟 喻果

刘元, 郑金华, 邹娟, 喻果. 基于邻域竞赛的多目标优化算法. 自动化学报, 2018, 44(7): 1304-1320. doi: 10.16383/j.aas.2017.c160315
引用本文: 刘元, 郑金华, 邹娟, 喻果. 基于邻域竞赛的多目标优化算法. 自动化学报, 2018, 44(7): 1304-1320. doi: 10.16383/j.aas.2017.c160315
LIU Yuan, ZHENG Jin-Hua, ZOU Juan, YU Guo. An Evolutionary Algorithm Through Neighborhood Competition for Multi-objective Optimization. ACTA AUTOMATICA SINICA, 2018, 44(7): 1304-1320. doi: 10.16383/j.aas.2017.c160315
Citation: LIU Yuan, ZHENG Jin-Hua, ZOU Juan, YU Guo. An Evolutionary Algorithm Through Neighborhood Competition for Multi-objective Optimization. ACTA AUTOMATICA SINICA, 2018, 44(7): 1304-1320. doi: 10.16383/j.aas.2017.c160315

基于邻域竞赛的多目标优化算法

doi: 10.16383/j.aas.2017.c160315
基金项目: 

国家自然科学基金 61673331

国家自然科学基金 61403326

湖南省科技计划项目 2016TP1020

湖南省自然科学基金 2017JJ4001

湖南省自然科学基金 14JJ2072

国家自然科学基金 61379062

湖南省自教育厅重点项目 17A212

国家自然科学基金 61502408

赛尔网络创新项目 NGII20150302

详细信息
    作者简介:

    刘元  湘潭大学信息工程学院硕士研究生.主要研究方向为高维多目标进化算法.E-mail:liu3yuan@gmail.com

    郑金华  湘潭大学信息工程学院教授.2000年获得中南大学控制理论与控制工程专业博士学位.主要研究方向为进化计算, 多目标遗传算法, 机器学习.E-mail:jhzheng@xtu.edu.com

    喻果  湘潭大学信息工程学院硕士研究生.主要研究方向为偏好多目标优化算法.E-mail:yuguo0801@126.com

    通讯作者:

    邹娟  湘潭大学信息工程学院副教授.2014年获得湘潭大学应用与数学专业博士学位.主要研究方向为人工智能, 优化算法设计, 进化计算.本文通信作者.E-mail:zoujuan@xtu.edu.com

An Evolutionary Algorithm Through Neighborhood Competition for Multi-objective Optimization

Funds: 

National Natural Science Foundation of China 61673331

National Natural Science Foundation of China 61403326

The Science and Technology Plan Project of Hunan Province 2016TP1020

the National Natural Science Foundation of Hunan Province 2017JJ4001

the National Natural Science Foundation of Hunan Province 14JJ2072

National Natural Science Foundation of China 61379062

The Education Department Major Project 17A212

National Natural Science Foundation of China 61502408

The CERNET Innovation Project NGII20150302

More Information
    Author Bio:

     Master student at the College of Information Engineering, Xiangtan University. His research interest covers multi-objective evolutionary algorithms

     Professor at the College of Information Engineering, Xiangtan University. He received his Ph. D. degree in control theory and control engineering from Central South University in 2000. His research interest covers evolutionary computation, multi-objective genetic algorithm, machine learning

     Master student at the College of Information Engineering, Xiangtan University. His research interest covers preference for multi-objective optimization algorithms

    Corresponding author: ZOU Juan  Associate professor at the College of Information Engineering, Xiangtan University. She received her Ph. D. degree in application and mathematics from Xiangtan University in 2014. Her research interest covers artificial intelligence, optimization design, evolutionary computation. Corresponding author of this paper
  • 摘要: 传统多目标优化算法(Multi-objective evolution algorithms,MOEAs)的基本框架大致分为两部分:首先是收敛性保持,采用Pareto支配方法将种群分成若干非支配层;其次是分布性保持,在临界层中,采用分布性保持机制维持种群的分布性.然而在处理高维优化问题(Many-objective optimization problems,MOPs)(目标维数大于3)时,随着目标维数的增加,种群的收敛性和分布性的冲突加剧,Pareto支配关系比较个体优劣的能力也迅速下降,此时传统的MOEA已不再适用于高维优化问题.鉴于此,本文提出了一种基于邻域竞赛的多目标优化算法(Evolutionary algorithm based on neighborhood competition for multi-objective optimization,NCEA).NCEA首先将个体的各个目标之和作为个体的收敛性估计;然后,计算当前个体向量与收敛性最好的个体向量之间的夹角,并将其作为当前个体的邻域估计;最后,通过邻域竞赛方法将问题划分为若干个相互关联的子问题并逐步优化.为了验证NCEA的有效性,本文选取5个优秀的算法与NCEA进行对比实验.通过对比实验验证,NCEA具有较强的竞争力,能同时保持良好的收敛性和分布性.
    1)  本文责任编委 魏庆来
  • 图  1  邻域竞赛机制示意图

    Fig.  1  Sketch of neighborhood competition mechanism

    图  2  无分布性保持机制的NSGA-Ⅱ ($A$)与NCEA ($A^*$)的收敛性比较

    Fig.  2  Evolutionary trajectories of the average GD for 30 runs of the modified without the diversity maintenance mechanism (denoted as $A$) and the NCEA (denoted as $A^*$) on DTLZ2

    图  3  收敛信息对不同类型问题的影响

    Fig.  3  Impact of convergence information on different issues

    图  4  分布信息示意图

    Fig.  4  Distribution diagram

    图  5  参数与时间复杂度的关系

    Fig.  5  The relationship between parameters and time complexity

    图  6  在(0, $\pi$]内, 将$\alpha$等分成100份, 分析NCEA在不同指标下的均值

    Fig.  6  Splitting the parameter $\alpha$ into 100 parts in certain ranges, and analyzing the different indicators$'$ mean values NCEA

    图  7  各算法在3目标DTLZ3测试问题上的最终解集

    Fig.  7  The final solution set of different algorithms on 3-objective DTLZ3 test problem

    图  8  各算法在6目标DTLZ3测试问题上的最终解集

    Fig.  8  The final solution set of different algorithms on 6-objective DTLZ3 test problem

    表  1  6个算法的时间复杂度

    Table  1  The time complexity of six algorithms

    算法 NCEA $\varepsilon $-MOEA GrEA AR + DMO MSOPS NSGA-Ⅲ
    时间复杂度 O$(mn^2)$ O$(mn(n+k))$ O$(mn^2)$ O$(mn^2)$ O$(mnt \cdot {\rm log}(t))$ ($n>t$) O$(mn^2)$
    下载: 导出CSV

    表  2  DTLZ系列测试问题及相关算法参数设置

    Table  2  DTLZ series test and related algorithm parameters

    问题 目标个数 特性 $\epsilon$参数 GrEA参数
    DTLZ1 3, 4, 5, 6, 8, 10 Linear, multimodal 0.033, 0.052, 0.059, 0.0554, 0.0549, 0.0565 10, 10, 10, 10, 10, 11
    DTLZ2 3, 4, 5, 6, 8, 10 Concave 0.06, 0.1312, 0.1927, 0.234, 0.29, 0.308 10, 10, 9, 8, 7, 8
    DTLZ3 3, 4, 5, 6, 8, 10 Concave, multimodal 0.06, 0.1385, 0.2, 0.227, 0.1567, 0.85 11, 11, 11, 11, 10, 11
    DTLZ4 3, 4, 5, 6, 8, 10 Concave, biased 0.06, 0.1312, 0.1927, 0.234, 0.29, 0.308 10, 10, 9, 8, 7, 8
    DTLZ5 3, 4, 5, 6, 8, 10 Concave, degenerate 0.0052, 0.042, 0.0785, 0.11, 0.1272, 1.15, 1.45 35, 35, 29, 14, 11, 11
    DTLZ6 3, 4, 5, 6, 8, 10 Concave, degenerate, biased 0.0227, 0.12, 0.3552, 0.75, 1.15, 1.45 36, 36, 24, 50, 50, 50
    DTLZ7 3, 4, 5, 6, 8, 10 Mixed, disconnected, biased 0.048, 0.105, 0.158, 0.15, 0.225, 0.46 9, 9, 8, 6, 5, 4
    下载: 导出CSV

    表  3  Metric的网格划分数设置

    Table  3  Settings of division for diversity metric

    目标数 3 4 5 6 8 10
    网格划分数 10 6 4 3 3 3
    下载: 导出CSV

    表  4  终止条件, 以代数为单位

    Table  4  Terminate condition, in generation

    问题 DTLZ1 DTLZ2 DTLZ3 DTLZ4 DTLZ5 DTLZ6 DTLZ7
    运行代数 1 000 300 1 000 300 300 1 000 300
    下载: 导出CSV

    表  5  NCEA的参数设置

    Table  5  Settings of $\alpha$ parameter for NCEA, in degree

    目标数 问题
    DTLZ1 DTLZ2 DTLZ3 DTLZ4 DTLZ5 DTLZ6 DTLZ7
    3 0.128 0.192 0.128 0.256 0.128 0.192 0.064
    4 0.256 0.320 0.256 0.512 0.064 0.064 0.128
    5 0.385 0.385 0.385 0.385 0.064 0.064 0.064
    6 0.385 0.769 0.513 0.833 0.064 0.064 0.128
    8 0.385 0.577 0.577 0.577 0.577 0.064 0.192
    10 0.513 0.769 0.641 0.641 0.641 0.064 0.256
    下载: 导出CSV

    表  6  收敛性指标GD的统计数据(均值和方差)

    Table  6  Statistical results of the convergence indicator GD (mean and SD)

    问题 目标数 均值与方差
    NCEA $\varepsilon $-MOEA GrEA AR + DMO MSOPS NSGA-Ⅲ
    DTLZ1 3 6.5173E-04(2.16738E-03) 2.4240E-04(3.24306E-05) 1.7059E-02(8.29783E-02)$^\dagger$ 1.6188E-02(4.54827E-02) 6.9466E-03(3.67556E-02) 8.3968E-02(3.03360E-01)
    4 1.3333E-03(5.87960E-04) 1.5342E-03(1.01728E-04) 5.0108E-02(1.48793E-01) 7.1744E-03(1.27516E-02) 8.0661E-03(3.25514E-02) 2.7705E-02(7.46720E-02)
    5 2.3973E-03(3.29058E-04) 2.8019E-03(5.10129E-04) 6.5782E-02(3.16118E-01) 5.9357E-02(1.54993E-01) 2.8872E-02(9.41475E-02) 5.3457E-02(1.27134E-01)$^\dagger$
    6 3.7016E-03(8.14840E-04) 3.5723E-03(4.48798E-04) 4.1469E-02(1.31608E-01) 5.1421E-02(1.02472E-01) 3.8983E-02(1.02165E-01)$^\dagger$ 1.6179E-01(2.49415E-01)$^\dagger$
    8 5.5829E-03(1.27751E-04) 6.1082E-03(9.60869E-04) 8.6450E-02(3.30601E-01)$^\dagger$ 3.6173E-02(9.55063E-02)$^\dagger$ 9.9818E-02(1.48950E-01) $^\dagger$ 9.0905E-01(1.19337E+00)$^\dagger$
    10 8.2959E-03(6.14113E-03) $\underline{3.4608\text{E}-02{{(3.75171\text{E}-02)}^{\dagger }}}$ 4.1050E-02(2.78262E-02)$^\dagger$ 7.8265E-02(1.99814E-01) 1.2987E-01(1.80125E-01)$^\dagger$ 2.0312E-01(4.43355E-01)$^\dagger$
    DTLZ2 3 2.4938E-04(7.71848E-05) 7.5429E-04(5.67439E-05)$^\dagger$ 4.4901E-05(4.52089E-05)$^\dagger$ 4.9012E-04(1.50824E-04)$^\dagger$ $\underline{1.1257\text{E}-04{{(1.33222\text{E}-04)}^{\dagger }}}$ 3.0712E-04(2.34867E-04)
    4 3.4325E-04(1.07068E-04) 2.1259E-03(1.25929E-04)$^\dagger$ 2.4815E-04(3.10381E-04) 1.1270E-03(3.29167E-04)$^\dagger$ 2.0637E-04(1.21438E-04)$^\dagger$ 7.0224E-04(1.24904E-04)$^\dagger$
    5 2.1616E-04(4.46158E-05) 4.1994E-03(6.61445E-04)$^\dagger$ 4.6204E-04(1.75780E-04)$^\dagger$ 4.1831E-03(1.24812E-03)$^\dagger$ $\underline{3.7035\text{E}-04{{(2.29673\text{E}-04)}^{\dagger }}}$ 1.9392E-03(2.82516E-04) $^\dagger$
    6 5.7860E-04(1.84086E-04) 5.6277E-03(1.97491E-03)$^\dagger$ 6.3318E-04(1.86383E-04) 9.1966E-03(2.34274E-03) $^\dagger$ 5.2284E-04(1.85337E-04) 4.3691E-03(6.82254E-04)$^\dagger$
    8 2.8758E-04(1.06852E-04) 6.8790E-03(8.32033E-04)$^\dagger$ 2.2182E-03(8.86939E-04)$^\dagger$ 1.9660E-02(3.92533E-03)$^\dagger$ $\underline{1.0396\text{E}-03{{(2.79993\text{E}-04)}^{\dagger }}}$ 1.1352E-02(3.21989E-03)$^\dagger$
    10 3.4444E-04(1.40595E-04) 5.5698E-03(4.47660E-04)$^\dagger$ $\underline{1.7998\text{E}-03{{(3.52157\text{E}-04)}^{\dagger }}}$ 3.1090E-02(3.81003E-03)$^\dagger$ 1.6458E-03(3.34714E-04) $^\dagger$ 4.4401E-03(2.74564E-03) $^\dagger$
    DTLZ3 3 2.8456E-04(2.41286E-04) 13291E-03(4.28045E-04)$^\dagger$ 1.3041E-01(5.43435E-01)$^\dagger$ 4.0661E-03(8.97266E-03)$^\dagger$ 1.5370E-04(1.11865E-04)$^\dagger$ 8.0032E-03(3.01346E-02)
    4 3.9361E-04(2.52161E-04) $\underline{4.8620\text{E}-03{{(2.40060\text{E}-03)}^{\dagger }}}$ 9.6669E-02(4.69495E-01) 1.2016E-01(4.21198E-01) 6.6798E-03(2.22771E-02) 6.3762E-02(1.68358E-01)$^\dagger$
    5 5.6584E-04(3.77953E-04) 8.9207E-03(4.14879E-03) 1.5762E+00(2.65502E+00) 1.7752E-02(4.77234E-02)$^\dagger$ 9.4459E-02(3.57033E-01)$^\dagger$ 6.0880E-01(1.38366E+00)
    6 5.6684E-04(3.12144E-04) $\underline{1.7783\text{E}-02{{(1.10904\text{E}-02)}^{\dagger }}}$ 2.9341E+00(4.02971E+00)$^\dagger$ 8.4404E-02(2.22184E-01)$^\dagger$ 2.5392E-01(6.65749E-01)$^\dagger$ 2.7222E+00(2.01272E+00) $^\dagger$
    8 6.8971E-04(3.84881E-04) 1.5738E+00(2.37967E+00)$^\dagger$ 2.1981E+00(2.32611E+00)$^\dagger$ 1.9263E-01(6.19185E-01) 1.3256E+00(1.21451E+00)$^\dagger$ 1.8191E+01(5.70052E+00)$^\dagger$
    10 7.6931E-04(4.06091E-04) 3.1649E+00(2.87679E+00)$^\dagger$ 2.4362E-01(6.95779E-01) $\underline{1.4932\text{E}-01{{(2.98389\text{E}-01)}^{\dagger }}}$ 1.4714E+00(9.55260E-01) $^\dagger$ 1.7709E+01(1.09209E+01)$^\dagger$
    DTLZ4 3 2.1106E-04(1.30383E-04) 9.8535E-04(4.16771E-04)$^\dagger$ 1.2073E-04(2.58715E-04) 2.5681E-04(2.91554E-04) 6.5585E-05(1.65279E-04)$^\dagger$ 2.4774E-04(1.30491E-04)
    4 5.7399E-04(1.77486E-04) 2.4360E-03(5.77431E-04)$^\dagger$ $\underline{2.0198\text{E}-04{{(3.13694\text{E}-04)}^{\dagger }}}$ 1.5170E-03(3.12700E-03) 1.7653E-04(9.46011E-05)$^\dagger$ 6.7933E-04(2.68225E-04)
    5 2.8600E-04(1.56925E-04) 5.3361E-03(1.76792E-03)$^\dagger$ 4.3361E-04(1.76189E-04) $^\dagger$ 2.4495E-03(2.17001E-03) $^\dagger$ 3.5318E-04(9.82127E-05) 1.6399E-03(3.96018E-04)$^\dagger$
    6 5.9198E-04(2.68476E-04) 1.0150E-02(8.10313E-03) $^\dagger$ 8.3766E-04(3.33206E-04) $^\dagger$ 4.8072E-03(3.04760E-03) $^\dagger$ $\underline{8.3145\text{E}-04{{(5.98874\text{E}-04)}^{\dagger }}}$ 3.4386E-03(1.12700E-03) $^\dagger$
    8 4.3000E-04(2.03320E-04) 1.0441E-02(5.13729E-03) $^\dagger$ 2.4109E-03(1.05389E-03) $^\dagger$ 1.4767E-02(2.59938E-03) $^\dagger$ $\underline{1.5980\text{E}-03{{(5.54551\text{E}-04)}^{\dagger }}}$ 5.2812E-03(4.42282E-03) $^\dagger$
    10 3.3791E-04(1.63041E-04) 1.5882E-02(1.24883E-02)$^\dagger$ $\underline{1.6270\text{E}-03{{(2.48190\text{E}-04)}^{\dagger }}}$ 2.8114E-02(4.28606E-03)$^\dagger$ 2.8111E-03(9.12415E-04)$^\dagger$ 1.5193E-02(5.35355E-03)$^\dagger$
    DTLZ5 3 8.2379E-05(4.28364E-05) $\underline{6.0527\text{E}-05{{(6.42860\text{E}-06)}^{\dagger }}}$ 5.9233E-05(5.66109E-05) 8.3765E-04(1.13766E-03)$^\dagger$ 1.0821E-01(2.80056E-03)$^\dagger$ 2.0193E-04(4.83885E-05)$^\dagger$
    4 3.4898E-02(2.52789E-03) 5.0231E-02(3.20057E-03)$^\dagger$ 1.9988E-03(1.06200E-03)$^\dagger$ $\underline{1.6314\text{E}-02{{(7.87450\text{E}-03)}^{\dagger }}}$ 1.5362E-01(3.29054E-03)$^\dagger$ 3.3321E-02(1.53968E-02)
    5 1.7277E-02(9.20726E-04) 5.1506E-02(1.82190E-03)$^\dagger$ 1.8679E-02(1.93218E-02) 2.4014E-02(6.34334E-03)$^\dagger$ 1.8936E-01(3.08728E-03)$^\dagger$ 5.4787E-02(9.97725E-03)$^\dagger$
    6 1.4070E-02(6.11126E-04) 5.7760E-02(6.23864E-03)$^\dagger$ 5.6970E-02(3.73031E-03)$^\dagger$ $\underline{3.4634\text{E}-02{{(8.44098\text{E}-03)}^{\dagger }}}$ 2.0364E-01(2.87587E-03)$^\dagger$ 7.0914E-02(1.34050E-02)$^\dagger$
    8 4.8683E-02(4.12569E-03) $\underline{5.3952\text{E}-02{{(4.32586\text{E}-03)}^{\dagger }}}$ 1.0139E-01(5.95976E-03) $^\dagger$ 1.3747E-01(3.12555E-02)$^\dagger$ 2.2976E-01(2.27175E-03)$^\dagger$ 1.0419E-01(1.47761E-02) $^\dagger$
    10 5.4205E-02(5.08366E-03) $\underline{6.0848\text{E}-02{{(6.67483\text{E}-03)}^{\dagger }}}$ 1.1183E-01(7.60976E-03)$^\dagger$ 1.7080E-01(2.86372E-02)$^\dagger$ 2.3387E-01(1.92825E-03)$^\dagger$ 1.5273E-01(1.37459E-02)$^\dagger$
    DTLZ6 3 3.1665E-03(2.76698E-03) 5.2588E-03(4.90865E-04)$^\dagger$ 3.3264E-03(1.61298E-03) 5.1275E-03(3.55167E-03)$^\dagger$ 2.0425E-01(4.49689E-03)$^\dagger$ 9.8696E-03(7.13856E-03)$^\dagger$
    4 4.2131E-02(5.62475E-03) 1.1064E-01(9.59272E-03)$^\dagger$ 2.5454E-02(9.72852E-03)$^\dagger$ 1.4171E-01(2.38262E-02)$^\dagger$ 3.3404E-01(7.50779E-03) $^\dagger$ 1.6955E-01(2.03052E-02)$^\dagger$
    5 1.5250E-02(1.92197E-03) 1.5048E-01(5.64830E-03)$^\dagger$ $\underline{9.3128\text{E}-02{{(5.99215\text{E}-02)}^{\dagger }}}$ 2.9541E-01(2.07511E-02)$^\dagger$ 4.9366E-01(1.37175E-02)$^\dagger$ 6.8704E-01(2.20882E-02)$^\dagger$
    6 1.3164E-02(1.52962E-03) 2.5553E-01(2.02566E-02)$^\dagger$ $\underline{3.3090\text{E}-02{{(1.30923\text{E}-01)}^{\dagger }}}$ 3.3130E-01(3.18902E-02)$^\dagger$ 5.5517E-01(1.51144E-02) $^\dagger$ 8.7899E-01(1.26348E-03)$^\dagger$
    8 1.3972E-02(1.16462E-03) 2.9568E-01(1.46263E-01)$^\dagger$ $\underline{1.3262\text{E}-01{{(2.81758\text{E}-01)}^{\dagger }}}$ 6.3637E-01(3.31022E-02)$^\dagger$ 7.6863E-01(1.65607E-02)$^\dagger$ 8.8443E-01(6.83775E-02)$^\dagger$
    10 1.6102E-02(9.86460E-04) $\underline{3.4632\text{E}-01{{(2.74054\text{E}-01)}^{\dagger }}}$ 4.0319E-01(3.82174E-01)$^\dagger$ 7.4899E-01(5.44463E-02)$^\dagger$ 8.1216E-01(1.81888E-02) $^\dagger$ 7.6559E-01(3.71012E-02)$^\dagger$
    DTLZ7 3 1.4269E-03(6.27153E-04) 6.9496E-04(3.51603E-05)$^\dagger$ $\underline{9.5260\text{E}-04{{(7.14402\text{E}-04)}^{\dagger }}}$ 2.2605E-03(1.09999E-03)$^\dagger$ 4.4358E-03(6.88598E-04)$^\dagger$ 2.6307E-03(9.13444E-04)$^\dagger$
    4 5.6080E-03(4.49695E-04) 2.3024E-03(5.55552E-04)$^\dagger$ $\underline{4.7689\text{E}-03{{(2.24499\text{E}-04)}^{\dagger }}}$ 1.4251E-02(5.16088E-03)$^\dagger$ 6.9670E-03(1.08085E-03)$^\dagger$ 5.4934E-03(1.15755E-03)
    5 7.6494E-03(3.39809E-04) 3.3534E-03(1.06828E-03)$^\dagger$ 1.0717E-02(6.69624E-04)$^\dagger$ 8.3798E-02(2.79588E-02)$^\dagger$ 1.0825E-02(3.33610E-03) $^\dagger$ 2.0268E-02(5.51018E-03)$^\dagger$
    6 1.6013E-02(6.62155E-04) 4.6055E-03(1.70038E-03)$^\dagger$ 1.1947E-02(4.76109E-04)$^\dagger$ 1.7956E-01(4.96227E-02)$^\dagger$ 2.4724E-02(1.41937E-02)$^\dagger$ 9.1749E-02(3.63369E-02)$^\dagger$
    8 2.1716E-02(3.75157E-04) 1.4274E-02(1.09412E-02)$^\dagger$ 2.6989E-02(1.56032E-03)$^\dagger$ 3.3977E-01(7.68451E-02)$^\dagger$ 2.0888E-01(1.13986E-01)$^\dagger$ 9.5229E-01(2.31546E-01)$^\dagger$
    10 4.8018E-02(2.56958E-03) 1.5505E-02(1.25040E-02)$^\dagger$ 4.9859E-02(2.27362E-03)$^\dagger$ 6.6739E-01(1.39620E-01)$^\dagger$ 6.5667E-01(2.37870E-01)$^\dagger$ 2.6579E+00(4.98006E-01)$^\dagger$
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    表  7  收敛性指标DM的统计数据(均值和方差)

    Table  7  Statistical results of the diversity indicator DM (mean and SD)

    问题 目标数 均值与方差
    NCEA $\varepsilon $-MOEA GrEA AR + DMO MSOPS NSGA-Ⅲ
    DTLZ1 3 9.0819E-01(2.30021E-02) 1.0026E+00(9.73518E-03)$^\dagger$ 5.9153E-01(1.56974E-01)$^\dagger$ 4.8806E-01(1.41348E-01) $^\dagger$ 7.2868E-01(8.60635E-03)$^\dagger$ 6.5816E-01(1.22945E-01)$^\dagger$
    4 9.3824E-01(3.92344E-02) 9.0282E-01(9.30006E-02) 8.0118E-01(1.90615E-01)$^\dagger$ 2.7235E-01(1.05381E-01)$^\dagger$ 7.5392E-01(1.49390E-02)$^\dagger$ 1.0242E+00(2.41363E-01)
    5 8.9325E-01(5.14120E-02) 7.3063E-01(1.26330E-01)$^\dagger$ 7.7569E-01(1.75880E-01)$^\dagger$ 2.4440E-01(8.58044E-02)$^\dagger$ 8.5011E-01(1.34821E-02) $^\dagger$ 1.1297E+00(2.79471E-01)$^\dagger$
    6 9.0806E-01(7.17513E-02) 7.7961E-01(3.00338E-02)$^\dagger$ 8.9749E-01(2.66680E-01) 3.3819E-01(8.43546E-02)$^\dagger$ $\underline{9.9849\text{E}-01{{(1.02165\text{E}-01)}^{\dagger }}}$ 1.0617E+00(4.52105E-01)
    8 5.1983E-01(6.96936E-02) 5.5871E+00(1.19030E+01)$^\dagger$ 7.7478E-01(1.97824E-01)$^{\dagger }$ 2.1625E-01(6.57912E-02)$^\dagger$ 9.9818E-02(3.21026E-02)$^\dagger$ 1.3082E-01(3.92564E-01) $^\dagger$
    10 3.7010E-01(6.41427E-02) 6.7953E+00(1.34618E+01)$^\dagger$ 7.3715E-01(3.20770E-01)$^{\dagger }$ 1.5727E-01(4.25823E-02)$^\dagger$ 6.7204E-01(2.40584E-02)$^\dagger$ 3.1103E-02(4.48038E-02) $^\dagger$
    DTLZ2 3 9.8798E-01(1.24042E-02) $\underline{8.8260\text{E}-01{{(2.40016\text{E}-02)}^{\dagger }}}$ 6.8468E-01(2.64578E-02) $^\dagger$ 2.7246E-01(6.91272E-02) $^\dagger$ 4.4127E-01(2.73807E-02) $^\dagger$ 6.0373E-01(2.89793E-03)$^\dagger$
    4 1.0198E+00(6.74614E-03) 9.1986E-01(4.11008E-02)$^\dagger$ 8.7220E-01(2.73173E-02)$^\dagger$ 2.3443E-01(7.24373E-02)$^\dagger$ 5.8514E-01(1.84067E-02) $^\dagger$ 1.1324E+00(7.99399E-03)$^\dagger$
    5 1.0296E+00(7.83495E-03) 9.3372E-01(6.11612E-02) $^\dagger$ 9.5388E-01(3.10375E-02) $^\dagger$ 3.5310E-01(9.60867E-02) $^\dagger$ 6.4514E-01(2.30645E-02) $^\dagger$ 1.3012E+00(5.25382E-03)$^\dagger$
    6 1.1161E+00(2.33482E-02) 8.3860E-01(2.37732E-02) $^\dagger$ 9.5260E-01(3.97041E-02) $^\dagger$ 2.9994E-01(5.69157E-02) $^\dagger$ 6.7853E-01(3.23148E-02) $^\dagger$ 1.3499E+00(2.26676E-02)$^\dagger$
    8 9.8795E-01(6.45634E-03) 1.0236E+00(1.25375E-01) 9.1789E-01(2.48723E-02) $^\dagger$ 2.8558E-01(7.24260E-02)$^\dagger$ 6.4368E-01(2.45093E-02)$^\dagger$ 1.0534E+00(2.47743E-01)
    10 9.7058E-01(9.77224E-03) 1.1440E+00(1.84157E-01)$^\dagger$ 9.7039E-01(1.69468E-02) 2.6161E-01(5.65284E-02) $^\dagger$ 6.6149E-01(2.40152E-02)$^\dagger$ 2.1640E-01(2.45770E-01)$^\dagger$
    DTLZ3 3 9.9727E-01(1.18592E-02) $\underline{8.7523\text{E}-01{{(3.60858\text{E}-02)}^{\dagger }}}$ 5.6777E-01(1.60735E-01) $^\dagger$ 2.5400E-01(1.34175E-01) $^\dagger$ 5.8864E-01(9.58502E-03) $^\dagger$ 5.9554E-01(3.14145E-02) $^\dagger$
    4 9.7707E-01(8.21852E-03) $\underline{8.7997\text{E}-01{{(1.05353\text{E}-01)}^{\dagger }}}$ 6.3245E-01(2.74485E-01) $^\dagger$ 2.6224E-01(8.97735E-02) $^\dagger$ 6.0668E-01(1.93795E-02) $^\dagger$ 9.2399E-01(2.43623E-01)
    5 1.0168E+00(6.54401E-03) $\underline{7.4548\text{E}-01{{(1.93596\text{E}-01)}^{\dagger }}}$ 4.1279E-01(3.28220E-01)$^\dagger$ 1.9796E-01(1.09482E-01)$^\dagger$ 6.4776E-01(1.83865E-02)$^\dagger$ 2.2725E-01(3.28181E-01)$^\dagger$
    6 1.1173E+00(1.13113E-02) $\underline{8.7064\text{E}-01{{(4.34695\text{E}-01)}^{\dagger }}}$ 2.8565E-01(3.17094E-01)$^\dagger$ 1.4374E-01(8.15058E-02)$^\dagger$ 6.5772E-01(1.12801E-01)$^\dagger$ 0.0000E+00(0.00000E+00)$^\dagger$
    8 9.7677E-01(9.65714E-03) 6.2783E-01(1.34328E+00) 1.9315E-01(2.84029E-01)$^\dagger$ 1.2406E-01(4.04796E-02)$^\dagger$ 3.9382E-01(2.50102E-01)$^\dagger$ 0.0000E+00(0.00000E+00)$^\dagger$
    10 9.3645E-01(5.16393E-03) 5.7879E-03(1.54484E-02)$^\dagger$ $\underline{6.0733\text{E}-01{{(2.74915\text{E}-01)}^{\dagger }}}$ 1.3581E-01(5.93626E-02)$^\dagger$ 2.6957E-01(2.31341E-01)$^\dagger$ 0.0000E+00(0.00000E+00)$^\dagger$
    DTLZ4 3 6.6197E-01(4.27371E-01) 4.6188E-01(3.95122E-01) 6.1770E-01(1.83316E-01) 2.1352E-01(1.48506E-01)$^\dagger$ 5.7741E-01(5.28796E-03) 4.1289E-01(2.54687E-01)$^\dagger$
    4 9.1824E-01(2.05013E-01) 4.4573E-01(3.64057E-01)$^\dagger$ 7.4143E-01(2.48123E-01)$^\dagger$ 2.5653E-01(1.87011E-01)$^\dagger$ 5.7539E-01(1.76688E-02)$^\dagger$ 1.0616E+00(2.37711E-01)$^\dagger$
    5 9.5754E-01(1.68432E-01) 3.9015E-01(2.93544E-01)$^\dagger$ 8.6074E-01(1.58894E-01)$^\dagger$ 2.6176E-01(1.86893E-01)$^\dagger$ 6.3258E-01(3.13921E-02)$^\dagger$ 1.1518E+00(3.40926E-01)$^\dagger$
    6 1.0725E+00(6.38296E-02) 5.1194E-01(2.85566E-01)$^\dagger$ 9.5005E-01(3.82742E-02)$^\dagger$ 3.2241E-01(2.01053E-01)$^\dagger$ 6.8764E-01(2.43298E-02) 9.6736E-01(5.37545E-01)
    8 9.4195E-01(1.74009E-02) 7.2219E-01(3.03730E-01)$^\dagger$ $\underline{9.2790\text{E}-01{{(2.26546\text{E}-02)}^{\dagger }}}$ 3.5804E-01(7.20134E-02)$^\dagger$ 6.1996E-01(2.37531E-02)$^\dagger$ 5.2325E-01(5.52567E-01)$^\dagger$
    10 9.1121E-01(2.45957E-02) 9.6665E-01(3.79451E-01) $\underline{9.6638\text{E}-01{{(1.14391\text{E}-02)}^{\dagger }}}$ 3.2112E-01(8.25448E-02)$^\dagger$ 6.9372E-01(3.57074E-02)$^\dagger$ 1.0501E-01(2.40114E-01)$^\dagger$
    DTLZ5 3 9.3890E-01(4.68561E-02) 9.4044E-01(1.02859E-02)$^\dagger$ 9.2575E-01(3.86653E-02)$^\dagger$ $\underline{9.5395\text{E}-01{{(6.11423\text{E}-02)}^{\dagger }}}$ $\underline{1.3685\text{E}-00{{(3.30802\text{E}-02)}^{\dagger }}}$ 9.2722E-01(6.44623E-02)$^\dagger$
    4 2.1707E+00(1.27853E-01) $\underline{1.9068\text{E}-00{{(1.34963\text{E}-01)}^{\dagger }}}$ 9.9110E-01(1.54379E-01)$^\dagger$ 1.2684E+00(4.40710E-01)$^\dagger$ 1.3788E+00(8.11361E-02) $^\dagger$ 9.5825E-01(2.91647E-01)$^\dagger$
    5 1.7299E+00(1.43731E-01) $\underline{1.6524\text{E}-00{{(1.18251\text{E}-01)}^{\dagger }}}$ 1.1591E+00(1.75680E-01)$^\dagger$ 1.3607E+00(3.95131E-01)$^\dagger$ 1.2771E+00(1.41335E-01)$^\dagger$ 1.1022E+00(4.99464E-01)
    6 2.4580E+00(2.24064E-01) 2.7376E+00(3.75631E-01) 2.6813E+00(2.33371E-01) 1.4756E+00(5.42065E-01) 1.5616E+00(2.20953E-01) 2.1622E+00(9.78533E-01)
    8 7.0107E+00(5.94377E-01) $\underline{2.4153\text{E}-00{{(3.37337\text{E}-01)}^{\dagger }}}$ 2.1679E+00(6.84087E-01)$^\dagger$ 5.2367E-02(9.60686E-02)$^\dagger$ 7.2613E-01(2.18209E-01) $^\dagger$ 7.9548E-01(6.32929E-01) $^\dagger$
    10 8.5619E+00(6.27846E-01) 2.3997E+00(3.22199E-01)$^\dagger$ $\underline{2.4172\text{E}-00{{(7.50029\text{E}-01)}^{\dagger }}}$ 4.9863E-03(2.73110E-02)$^\dagger$ 4.0498E-01(1.09439E-01)$^\dagger$ 7.6460E-02(1.46121E-01)$^\dagger$
    DTLZ6 3 1.3435E+00(1.34596E-01) 1.3889E+00(8.52688E-02) 1.3396E+00(1.72888E-01) 1.1501E+00(3.41395E-01)$^\dagger$ 1.7816E+00(4.67236E-02)$^\dagger$ $\underline{1.4259\text{E}-00{{(111\text{E}-01)}^{\dagger }}}$
    4 2.4410E+00(1.26378E-01) 3.0357E+00(2.93393E-01)$^\dagger$ 1.5270E+00(4.90222E-01)$^\dagger$ 0.0000E+00(0.00000E+00)$^\dagger$ 1.5699E+00(1.05247E-01)$^\dagger$ 4.5635E-03(1.74166E-02)$^\dagger$
    5 1.7951E+00(1.42279E-01) 3.0859E-03(1.69023E-02)$^\dagger$ $\underline{1.4194\text{E}-00{{(4.09709\text{E}-01)}^{\dagger }}}$ 0.0000E+00(0.00000E+00)$^\dagger$ 1.1870E+00(1.78062E-01)$^\dagger$ 0.0000E+00(0.00000E+00)$^\dagger$
    6 2.8895E+00(3.80171E-01) 0.0000E+00(0.00000E+00)$^\dagger$ 2.5593E-01(8.02479E-02)$^\dagger$ 0.0000E+00(0.00000E+00)$^\dagger$ $\underline{9.4320\text{E}-01{{(4.62978\text{E}-01)}^{\dagger }}}$ 0.0000E+00(0.00000E+00)$^\dagger$
    8 1.9559E+00(2.16270E-01) 6.4114E-02(9.25841E-02)$^\dagger$ $\underline{2.6740\text{E}-01{{(1.18849\text{E}-01)}^{\dagger }}}$ 0.0000E+00(0.00000E+00)$^\dagger$ 0.0000E+00(0.00000E+00) $^\dagger$ 0.0000E+00(0.00000E+00)$^\dagger$
    10 1.8986E+00(2.20235E-01) 3.7771E-02(7.68760E-02)$^\dagger$ $\underline{1.4075\text{E}-01{{(1.32006\text{E}-01)}^{\dagger }}}$ 0.0000E+00(0.00000E+00)$^\dagger$ 0.0000E+00(0.00000E+00)$^\dagger$ 0.0000E+00(0.00000E+00)$^\dagger$
    DTLZ7 3 9.7062E-01(1.33304E-01) 9.9912E-01(1.44470E-01)$^\dagger$ 7.1857E-01(4.28577E-02)$^\dagger$ 3.7137E-01(1.72866E-01)$^\dagger$ 7.3749E-01(2.16690E-02)$^\dagger$ 5.6630E-01(5.12711E-02) $^\dagger$
    4 6.6394E-01(5.01756E-02) 3.2074E-01(1.09399E-01)$^\dagger$ $\underline{5.0842\text{E}-01{{(7.85438\text{E}-02)}^{\dagger }}}$ 2.3610E-01(7.93598E-02)$^\dagger$ 4.7106E-01(2.01419E-02)$^\dagger$ 4.0124E-01(1.83707E-01)$^\dagger$
    5 7.1072E-01(6.29801E-02) 1.4714E+00(6.34844E-01)$^\dagger$ $\underline{8.2760\text{E}-01{{(3.84418\text{E}-02)}^{\dagger }}}$ 3.5503E-01(1.60258E-01)$^\dagger$ 4.6132E-01(1.90079E-02) $^\dagger$ 2.4188E-01(9.36043E-02)$^\dagger$
    6 8.7237E-01(3.46582E-02) $\underline{6.5235\text{E}-01{{(4.31959\text{E}-01)}^{\dagger }}}$ 5.4844E-01(4.80601E-02)$^\dagger$ 3.3734E-01(1.55325E-01)$^\dagger$ 2.9776E-01(2.34195E-02)$^\dagger$ 1.0036E-01(6.84524E-02)$^\dagger$
    8 5.9887E-01(1.52663E-02) 2.2230E+00(2.02132E+00)$^\dagger$ $\underline{8.7016\text{E}-01{{(6.50580\text{E}-02)}^{\dagger }}}$ 6.1437E-02(6.25518E-02) $^\dagger$ 1.5046E-01(5.11722E-02)$^\dagger$ 0.0000E+00(0.00000E+00)$^\dagger$
    10 4.4481E-02(1.46695E-02) 3.2024E+00(2.16088E+00)$^\dagger$ $\underline{9.6667\text{E}-01{{(6.04123\text{E}-02)}^{\dagger }}}$ 3.4990E-03(7.53848E-03)$^\dagger$ 1.8454E-02(1.15707E-02)$^\dagger$ 0.0000E+00(0.00000E+00)$^\dagger$
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    表  8  综合性指标IGD的统计数据(均值和方差)

    Table  8  Statistical results of the integrated indicator IGD (mean and SD)

    问题 目标数 均值与方差
    NCEA $\varepsilon $-MOEA GrEA AR + DMO MSOPS NSGA-Ⅲ
    DTLZ1 3 2.2436E-02(1.63962E-03) 2.4240E-04(3.24306E-05)$^\dagger$ 1.7059E-02(8.29783E-02) $^\dagger$ 1.6188E-02(4.54827E-02)$^\dagger$ $\underline{6.9466E\text{E}-03{{(3.67556\text{E}-02)}^{\dagger }}}$ 8.3968E-02(3.03360E-01)
    4 4.5368E-02(1.22833E-03) 1.5342E-03(1.01728E-04)$^\dagger$ 5.0108E-02(1.48793E-01) $\underline{7.1744\text{E}-03{{(1.27516\text{E}-02)}^{\dagger }}}$ 8.0661E-03(3.25514E-02)$^\dagger$ 2.7705E-02(7.46720E-02)
    5 6.6923E-02(1.51909E-03) 2.8019E-03(5.10129E-04) $^\dagger$ 6.5782E-02(3.16118E-01) $^\dagger$ 5.9357E-02(1.54993E-01) 2.8872E-02(9.41475E-02) 5.3457E-02(1.27134E-01)$^\dagger$
    6 8.5221E-02(2.19744E-03) 3.5723E-03(4.48798E-04) 4.1469E-02(1.31608E-01) 5.1421E-02(1.02472E-01)$^\dagger$ $\underline{3.8983\text{E}-02{{(1.02165\text{E}-01)}^{\dagger }}}$ 1.6179E-01(2.49415E-01)$^\dagger$
    8 5.5829E-03(1.27751E-04) $\underline{6.1082\text{E}-03{{(9.60869\text{E}-04)}^{\dagger }}}$ 8.6450E-02(3.30601E-01) 3.6173E-02(9.55063E-02)$^\dagger$ 9.9818E-02(1.48950E-01) $^\dagger$ 9.0905E-01(1.19337E+00)$^\dagger$
    10 8.2959E-03(6.14113E-03) $\underline{3.4608\text{E}-02{{(3.75171\text{E}-02)}^{\dagger }}}$ 4.1050E-02(2.78262E-02)$^\dagger$ 7.8265E-02(1.99814E-01)$^\dagger$ 1.2987E-01(1.80125E-01) $^\dagger$ 2.0312E-01(4.43355E-01)$^\dagger$
    DTLZ2 3 2.4938E-04(7.71848E-05) 7.5429E-04(5.67439E-05)$^\dagger$ 4.4901E-05(4.52089E-05)$^\dagger$ 4.9012E-04(1.50824E-04)$^\dagger$ $\underline{1.1257\text{E}-04{{(1.33222\text{E}-04)}^{\dagger }}}$ 3.0712E-04(2.34867E-04)$^\dagger$
    4 3.4325E-04(1.07068E-04) 2.1259E-03(1.25929E-04)$^\dagger$ $\underline{2.4815\text{E}-04{{(3.10381\text{E}-04)}^{\dagger }}}$ 1.1270E-03(3.29167E-04)$^\dagger$ 2.0637E-04(1.21438E-04)$^\dagger$ 7.0224E-04(1.24904E-04) $^\dagger$
    5 2.1616E-04(4.46158E-05) 4.1994E-03(6.61445E-04)$^\dagger$ 4.6204E-04(1.75780E-04)$^\dagger$ 4.1831E-03(1.24812E-03)$^\dagger$ $\underline{3.7035\text{E}-04{{(2.29673\text{E}-04)}^{\dagger }}}$ 1.9392E-03(2.82516E-04)$^\dagger$
    6 5.7860E-04(1.84086E-04) 5.6277E-03(1.97491E-03) $^\dagger$ 6.3318E-04(1.86383E-04)$^\dagger$ 9.1966E-03(2.34274E-03)$^\dagger$ 5.2284E-04(1.85337E-04)$^\dagger$ 4.3691E-03(6.82254E-04)$^\dagger$
    8 2.8758E-04(1.06852E-04) 6.8790E-03(8.32033E-04)$^\dagger$ 2.2182E-03(8.86939E-04)$^\dagger$ 1.9660E-02(3.92533E-03)$^\dagger$ $\underline{1.0396\text{E}-03{{(2.79993\text{E}-04)}^{\dagger }}}$ 1.1352E-02(3.21989E-03)$^\dagger$
    10 3.4444E-04(1.40595E-04) 5.5698E-03(4.47660E-04)$^\dagger$ 1.7998E-03(3.52157E-04)$^\dagger$ 3.1090E-02(3.81003E-03) $^\dagger$ $\underline{1.6458\text{E}-03{{(3.34714E\text{E}-04)}^{\dagger }}}$ 4.4401E-03(2.74564E-03)$^\dagger$
    DTLZ3 3 2.8456E-04(2.41286E-04) 1.3291E-03(4.28045E-04)$^\dagger$ 1.3041E-01(5.43435E-01) $^\dagger$ 4.0661E-03(8.97266E-03)$^\dagger$ 1.5370E-04(1.11865E-04) $^\dagger$ 8.0032E-03(3.01346E-02) $^\dagger$
    4 3.9361E-04(2.52161E-04) $\underline{4.8620\text{E}-03{{(2.40060\text{E}-03)}^{\dagger }}}$ 9.6669E-02(4.69495E-01)$^\dagger$ 1.2016E-01(4.21198E-01)$^\dagger$ 6.6798E-03(2.22771E-02)$^\dagger$ 6.3762E-02(1.68358E-01)
    5 5.6584E-04(3.77953E-04) $\underline{8.9207\text{E}-03{{(4.14879\text{E}-03)}^{\dagger }}}$ 1.5762E+00(2.65502E+00)$^\dagger$ 1.7752E-02(4.77234E-02)$^\dagger$ 9.4459E-02(3.57033E-01)$^\dagger$ 6.0880E-01(1.38366E+00)$^\dagger$
    6 5.6684E-04(3.12144E-04) $\underline{1.7783\text{E}-02{{(1.10904\text{E}-02)}^{\dagger }}}$ 2.9341E+00(4.02971E+00)$^\dagger$ 8.4404E-02(2.22184E-01)$^\dagger$ 2.5392E-01(6.65749E-01) 2.7222E+00(2.01272E+00)
    8 6.8971E-04(3.84881E-04) 1.5738E+00(2.37967E+00)$^\dagger$ 2.1981E+00(2.32611E+00) $^\dagger$ $\underline{1.9263\text{E}-01{{(6.19185\text{E}-01)}^{\dagger }}}$ 1.3256E+00(1.21451E+00) $^\dagger$ 1.8191E+01(5.70052E+00) $^\dagger$
    10 7.6931E-04(4.06091E-04) 3.1649E+00(2.87679E+00) $^\dagger$ 2.4362E-01(6.95779E-01) $^\dagger$ $\underline{1.4932\text{E}-01{{(2.98389\text{E}-01)}^{\dagger }}}$ 1.4714E+00(9.55260E-01) $^\dagger$ 1.7709E+01(1.09209E+01) $^\dagger$
    DTLZ4 3 2.1106E-04(1.30383E-04) 9.8535E-04(4.16771E-04) 1.2073E-04(2.58715E-04) 2.5681E-04(2.91554E-04) $^\dagger$ 6.5585E-05(1.65279E-04)$^\dagger$ 2.4774E-04(1.30491E-04)
    4 5.7399E-04(1.77486E-04) 2.4360E-03(5.77431E-04) $^\dagger$ 2.0198E-04(3.13694E-04) 1.5170E-03(3.12700E-03)$^\dagger$ 1.7653E-04(9.46011E-05) 6.7933E-04(2.68225E-04)
    5 2.8600E-04(1.56925E-04) 5.3361E-03(1.76792E-03)$^\dagger$ 4.3361E-04(1.76189E-04) 2.4495E-03(2.17001E-03)$^\dagger$ 3.5318E-04(9.82127E-05) 1.6399E-03(3.96018E-04)
    6 5.9198E-04(2.68476E-04) 1.0150E-02(8.10313E-03)$^\dagger$ 8.3766E-04(3.33206E-04)$^\dagger$ 4.8072E-03(3.04760E-03)$^\dagger$ $\underline{8.3145\text{E}-04{{(5.98874\text{E}-04)}^{\dagger }}}$ 3.4386E-03(1.12700E-03)$^\dagger$
    8 4.3000E-04(2.03320E-04) 1.0441E-02(5.13729E-03)$^\dagger$ 2.4109E-03(1.05389E-03) $^\dagger$ 1.4767E-02(2.59938E-03)$^\dagger$ $\underline{1.5980\text{E}-03{{(5.54551\text{E}-04)}^{\dagger }}}$ 5.2812E-03(4.42282E-03) $^\dagger$
    10 3.3791E-04(1.63041E-04) 1.5882E-02(1.24883E-02)$^\dagger$ 1.6270E-03(2.48190E-04)$^\dagger$ 2.8114E-02(4.28606E-03)$^\dagger$ $\underline{2.8111\text{E}-03{{(9.12415\text{E}-04)}^{\dagger }}}$ 1.5193E-02(5.35355E-03) $^\dagger$
    DTLZ5 3 8.2379E-05(4.28364E-05) $\underline{6.0527\text{E}-05{{(6.42860\text{E}-06)}^{\dagger }}}$ 5.9233E-05(5.66109E-05)$^\dagger$ 8.3765E-04(1.13766E-03) 1.0821E-01(2.80056E-03)$^\dagger$ 2.0193E-04(4.83885E-05) $^\dagger$
    4 3.4898E-02(2.52789E-03) 5.0231E-02(3.20057E-03)$^\dagger$ 1.9988E-03(1.06200E-03) $\underline{1.6314\text{E}-02{{(7.87450\text{E}-03)}^{\dagger }}}$ 1.5362E-01(3.29054E-03)$^\dagger$ 3.3321E-02(1.53968E-02)$^\dagger$
    5 1.7277E-02(9.20726E-04) 5.1506E-02(1.82190E-03)$^\dagger$ $\underline{1.8679\text{E}-02{{(1.93218\text{E}-02)}^{\dagger }}}$ 2.4014E-02(6.34334E-03)$^\dagger$ 1.8936E-01(3.08728E-03)$^\dagger$ 5.4787E-02(9.97725E-03)$^\dagger$
    6 1.4070E-02(6.11126E-04) 5.7760E-02(6.23864E-03)$^\dagger$ 5.6970E-02(3.73031E-03)$^\dagger$ $\underline{3.4634\text{E}-02{{(8.44098\text{E}-03)}^{\dagger }}}$ 2.0364E-01(2.87587E-03)$^\dagger$ 7.0914E-02(1.34050E-02)$^\dagger$
    8 4.8683E-02(4.12569E-03) $\underline{5.3952\text{E}-02{{(4.32586\text{E}-03)}^{\dagger }}}$ 1.0139E-01(5.95976E-03)$^\dagger$ 1.3747E-01(3.12555E-02)$^\dagger$ 2.2976E-01(2.27175E-03) $^\dagger$ 1.0419E-01(1.47761E-02)$^\dagger$
    10 5.4205E-02(5.08366E-03) $\underline{6.0848\text{E}-02{{(6.67483\text{E}-03)}^{\dagger }}}$ 1.1183E-01(7.60976E-03) $^\dagger$ 1.7080E-01(2.86372E-02) $^\dagger$ 2.3387E-01(1.92825E-03)$^\dagger$ 1.5273E-01(1.37459E-02)$^\dagger$
    DTLZ6 3 3.1665E-03(2.76698E-03) 5.2588E-03(4.90865E-04) $^\dagger$ $\underline{3.3264\text{E}-03{{(1.61298\text{E}-03)}^{\dagger }}}$ 5.1275E-03(3.55167E-03) $^\dagger$ 2.0425E-01(4.49689E-03) $^\dagger$ 9.8696E-03(7.13856E-03) $^\dagger$
    4 4.2131E-02(5.62475E-03) 1.1064E-01(9.59272E-03) $^\dagger$ 2.5454E-02(9.72852E-03) $^\dagger$ 1.4171E-01(2.38262E-02) $^\dagger$ 3.3404E-01(7.50779E-03) $^\dagger$ 1.6955E-01(2.03052E-02) $^\dagger$
    5 1.5250E-02(1.92197E-03) 1.5048E-01(5.64830E-03) $^\dagger$ $\underline{9.3128\text{E}-02{{(5.99215\text{E}-02)}^{\dagger }}}$ 2.9541E-01(2.07511E-02) $^\dagger$ 4.9366E-01(1.37175E-02) $^\dagger$ 6.8704E-01(2.20882E-02) $^\dagger$
    6 1.3164E-02(1.52962E-03) $\underline{2.5553\text{E}-01{{(2.02566\text{E}-02)}^{\dagger }}}$ 3.3090E-02(1.30923E-01) $^\dagger$ 3.3130E-01(3.18902E-02) $^\dagger$ 5.5517E-01(1.51144E-02) $^\dagger$ 8.7899E-01(1.26348E-03) $^\dagger$
    8 1.3972E-02(1.16462E-03) $\underline{2.9568\text{E}-01{{(1.46263\text{E}-01)}^{\dagger }}}$ 7.6863E-01(1.65607E-02)$^\dagger$ 8.8443E-01(6.83775E-02)$^\dagger$ 3.2865E+00(4.21908E-01)$^\dagger$ 9.6881E+00(9.73209E-01)$^\dagger$
    10 1.6102E-02(9.86460E-04) $\underline{3.4632\text{E}-01{{(2.74054\text{E}-01)}^{\dagger }}}$ 4.0319E-01(3.82174E-01)$^\dagger$ 7.4899E-01(5.44463E-02)$^\dagger$ 8.1216E-01(1.81888E-02)$^\dagger$ 7.6559E-01(3.71012E-02)$^\dagger$
    DTLZ7 3 1.4269E-03(6.27153E-04) 6.9496E-04(3.51603E-05) 9.5260E-04(7.14402E-04) 2.2605E-03(1.09999E-03)$^\dagger$ 4.4358E-03(6.88598E-04)$^\dagger$ 2.6307E-03(9.13444E-04)$^\dagger$
    4 5.6080E-03(4.49695E-04) 2.3024E-03(5.55552E-04)$^\dagger$ 4.7689E-03(2.24499E-04) 1.4251E-02(5.16088E-03)$^\dagger$ 6.9670E-03(1.08085E-03)$^\dagger$ 5.4934E-03(1.15755E-03)$^\dagger$
    5 7.6494E-03(3.39809E-04) 3.3534E-03(1.06828E-03)$^\dagger$ 1.0717E-02(6.69624E-04) $^\dagger$ 8.3798E-02(2.79588E-02)$^\dagger$ 1.0825E-02(3.33610E-03)$^\dagger$ 2.0268E-02(5.51018E-03)$^\dagger$
    6 1.6013E-02(6.62155E-04) 4.6055E-03(1.70038E-03) 1.1947E-02(4.76109E-04)$^\dagger$ 1.7956E-01(4.96227E-02)$^\dagger$ 2.4724E-02(1.41937E-02) $^\dagger$ 9.1749E-02(3.63369E-02)$^\dagger$
    8 2.1716E-02(3.75157E-04) 1.4274E-02(1.09412E-02)$^\dagger$ 2.6989E-02(1.56032E-03)$^\dagger$ 3.3977E-01(7.68451E-02)$^\dagger$ 2.0888E-01(1.13986E-01)$^\dagger$ 9.5229E-01(2.31546E-01)$^\dagger$
    10 4.8018E-02(2.56958E-03) 1.5505E-02(1.25040E-02)$^\dagger$ 4.9859E-02(2.27362E-03)$^\dagger$ 6.6739E-01(1.39620E-01)$^\dagger$ 6.5667E-01(2.37870E-01)$^\dagger$ 2.6579E+00(4.98006E-01)$^\dagger$
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  • 收稿日期:  2016-04-09
  • 录用日期:  2017-06-22
  • 刊出日期:  2018-07-20

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