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基于R2指标和参考向量的高维多目标进化算法

陈国玉 李军华 黎明 陈昊

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, 2021, 47(11): 2675-2690 doi: 10.16383/j.aas.c180722
引用本文: 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, 2021, 47(11): 2675-2690 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 Automatica Sinica, 2021, 47(11): 2675-2690 doi: 10.16383/j.aas.c180722
Citation: Chen Guo-Yu, Li Jun-Hua, Li Ming, Chen Hao. An R2 indicator and reference vector based many-objective optimization evolutionary algorithm. Acta Automatica Sinica, 2021, 47(11): 2675-2690 doi: 10.16383/j.aas.c180722

基于R2指标和参考向量的高维多目标进化算法

doi: 10.16383/j.aas.c180722
基金项目: 

国家自然科学基金 61440049

国家自然科学基金 61866025

国家自然科学基金 61866026

国家自然科学基金 62066031

江西省自然科学基金 20181BAB202025

江西省优势科技创新团队计划 20181BCB24008

详细信息
    作者简介:

    陈国玉 南昌航空大学硕士研究生.主要研究方向为进化计算. E-mail: 1704081002001@stu.nchu.edu.cn

    黎明 南昌航空大学教授. 主要研究方向为图像处理和人工智能. E-mail: liming@nchu.edu.cn

    陈昊 南昌航空大学副教授. 主要研究方向为动态和高维多目标进化算法. E-mail: chenhaoshl@nchu.edu.cn

    通讯作者:

    李军华 南昌航空大学教授. 主要研究方向为进化计算和智能控制. 本文通信作者. E-mail: jhlee126@126.com

An R2 Indicator and Reference Vector Based Many-objective Optimization Evolutionary Algorithm

Funds: 

National Natural Science Foundation of China 61440049

National Natural Science Foundation of China 61866025

National Natural Science Foundation of China 61866026

National Natural Science Foundation of China 62066031

Natural Science Foundation of Jiangxi 20181BAB202025

Superiority Science and Technology Innovation Team Program of Jiangxi 20181BCB24008

More Information
    Author Bio:

    CHEN Guo-Yu Master student at Nanchang Hangkong University. His main research interest is evolutionary computation

    LI Ming Professor at Nanchang Hangkong University. His research interest covers image processing and artiflcial intelligence

    CHEN Hao Associate professor at Nanchang Hangkong University. His research interest covers dynamic and many-objective evolutionary algorithms

    Corresponding author: LI Jun-Hua Professor at Nanchang Hangkong University. His research interest covers evolutionary computation and intelligent control. Corresponding author of this paper
  • 摘要: 在高维多目标优化中, 不同的优化问题存在不同形状的Pareto前沿(PF), 而研究表明大多数多目标进化算法(Multi-objective evolutionary algorithms, MOEAs) 在处理不同的优化问题时普适性较差. 为了解决这个问题, 本文提出了一个基于R2指标和参考向量的高维多目标进化算法(An R2 indicator and reference vector based many-objective optimization evolutionary algorithm, R2-RVEA). R2-RVEA基于Pareto支配选取非支配解来指导种群进化, 仅当非支配解的数量超过种群规模时, 算法进一步采用种群分解策略和R2指标选择策略进行多样性管理. 通过大量的实验证明, 本文提出的算法在处理不同形状的PF时具有良好的性能.
    Recommended by Associate Editor WANG Ding
    1)  本文责任编委 王鼎
  • 图  1  目标问题上展示的15个参考向量

    Fig.  1  15 reference vectors are shown on 3-objective problem

    图  2  DTLZ4问题15目标上获得的非支配解

    Fig.  2  Nondominated solutions obtained on 15-objective DTLZ4

    图  3  DTLZ5问题3目标上获得的非支配解

    Fig.  3  Nondominated solutions obtained on 3-objective DTLZ5

    图  4  DTLZ7问题5目标上获得的非支配解

    Fig.  4  Nondominated solutions obtained on 5-objective DTLZ7

    图  5  R2-RVEA在3维DTLZ7上获得的不同结果

    Fig.  5  Different results obtained by R2-RVEA on 3-objective DTLZ7

    图  6  R2-RVEA在3维DTLZ7上GD和DM指标的进化轨迹

    Fig.  6  Evolutionary trajectories of GD and DM for R2-RVEA on 3-objective DTLZ7

    表  1  种群规模设置

    Table  1  Setting of population size

    $M$ $(p_1, p_2)$ 种群规模
    3 (12, 0) 105
    5 (6, 0) 126
    10 (3, 2) 275
    15 (2, 1) 135
    下载: 导出CSV

    表  2  R2-RVEA、NSGA-Ⅲ、RVEA、MOEA/DD、MOMBI-Ⅱ、KnEA和TS-R2EA在DTLZ1$-$DTLZ7上获得的HV值的统计结果(均值和标准差). 最好的结果已突出

    Table  2  The statistical results (mean and standard deviation) of the HV values obtained by R2-RVEA, NSGA-Ⅲ, RVEA, MOEA/DD, MOMBI-Ⅱ, KnEA and TS-R2EA on DTLZ1 to DTLZ7. The best results are highlighted

    问题 $M$ R2-RVEA NSGA-Ⅲ RVEA MOEA/DD MOMBI-Ⅱ KnEA TS-R2EA
    DTLZ 1 3 $8.3819\times 10^{-1}$
    $(1.31\times 10^{-2})$
    $8.4373\times 10^{-1}$
    $(1.16\times 10^{-3}) +$
    $8.4431\times 10^{-1}$
    $(1.41\times 10^{-4}) +$
    $8.4426\times 10^{-1}$
    $(2.26\times 10^{-4}) +$
    $8.4357\times 10^{-1}$
    $(9.08\times 10^{-4}) \approx$
    $7.8436\times 10^{-1}$
    $(4.64\times 10^{-2}) -$
    $8.4419\times 10^{-1}$
    $(3.44\times 10^{-4}) +$
    5 $9.6015\times 10^{-1}$
    $(4.69\times 10^{-2})$
    $9.2764\times 10^{-1}$
    $(1.43\times 10^{-1}) -$
    $9.7488\times 10^{-1}$
    $(2.27\times 10^{-4}) +$
    $9.7484\times 10^{-1}$
    $(2.75\times 10^{-4}) +$
    $9.3835\times 10^{-1}$
    $(1.26\times 10^{-1}) -$
    $7.6616\times 10^{-1}$
    $(9.09\times 10^{-2}) -$
    $9.7486\times 10^{-1}$
    $(2.27\times 10^{-4}) +$
    10 $9.9892\times 10^{-1}$
    $(5.94\times 10^{-4})$
    $9.9520\times 10^{-1}$
    $(2.43\times 10^{-2}) -$
    $9.9968\times 10^{-1}$
    $(1.88\times 10^{-5}) +$
    $9.9518\times 10^{-1}$
    $(2.46\times 10^{-2}) -$
    $9.4328\times 10^{-1}$
    $(4.34\times 10^{-2}) -$
    $4.0984\times 10^{-1}$
    $(1.92\times 10^{-1}) -$
    $9.9970\times 10^{-1}$
    $(2.11\times 10^{-5}) +$
    15 $8.7065\times 10^{-1}$
    $(2.75\times 10^{-1})$
    $4.8134\times 10^{-1}$
    $(4.84\times 10^{-1}) -$
    $9.4387\times 10^{-1}$
    $(1.71\times 10^{-1}) +$
    $9.5591\times 10^{-1}$
    $(1.36\times 10^{-1}) +$
    $7.4281\times 10^{-1}$
    $(1.91\times 10^{-1}) -$
    $6.4831\times 10^{-2}$
    $(1.30\times 10^{-1}) -$
    $6.1864\times 10^{-1}$
    $(3.77\times 10^{-1}) -$
    DTLZ 2 3 $5.6302\times 10^{-1}$
    $(5.99\times 10^{-6})$
    $5.6303\times 10^{-1}$
    $(7.97\times 10^{-7}) \approx$
    $5.6302\times 10^{-1}$
    $(6.53\times 10^{-7}) \approx$
    $5.6302\times 10^{-1}$
    $(3.98\times 10^{-7}) \approx$
    $5.6299\times 10^{-1}$
    $(2.19\times 10^{-5}) -$
    $5.2089\times 10^{-1}$
    $(1.20\times 10^{-2}) -$
    $5.6302\times 10^{-1}$
    $(2.36\times 10^{-6}) -$
    5 $7.9493\times 10^{-1}$
    $(4.55\times 10^{-4})$
    $7.9498\times 10^{-1}$
    $(3.66\times 10^{-4}) \approx$
    $7.9492\times 10^{-1}$
    $(3.75\times 10^{-4}) \approx$
    $7.9484\times 10^{-1}$
    $(3.71\times 10^{-4}) \approx$
    $7.9470\times 10^{-1}$
    $(4.13\times 10^{-4}) -$
    $7.4725\times 10^{-1}$
    $(1.23\times 10^{-2}) -$
    $7.9479\times 10^{-1}$
    $(3.90\times 10^{-4}) \approx$
    10 $9.6978\times 10^{-1}$
    $(1.74\times 10^{-4})$
    $9.4911\times 10^{-1}$
    $(3.63\times 10^{-2}) \approx$
    $9.6983\times 10^{-1}$
    $(1.64\times 10^{-4}) \approx$
    $9.6978\times 10^{-1}$
    $(1.72\times 10^{-4}) \approx$
    $9.7090\times 10^{-1}$
    $(1.49\times 10^{-3}) +$
    $9.2287\times 10^{-1}$
    $(8.30\times 10^{-3}) -$
    $9.7057\times 10^{-1}$
    $(1.66\times 10^{-4}) +$
    15 $9.9057\times 10^{-1}$
    $(4.02\times 10^{-4})$
    $9.6345\times 10^{-1}$
    $(1.75\times 10^{-2}) -$
    $9.7793\times 10^{-1}$
    $(4.01\times 10^{-2}) -$
    $9.9063\times 10^{-1}$
    $(2.44\times 10^{-4}) \approx$
    $8.1493\times 10^{-1}$
    $(9.09\times 10^{-2}) -$
    $9.6726\times 10^{-1}$
    $(9.69\times 10^{-3}) -$
    $9.8690\times 10^{-1}$
    $(2.06\times 10^{-3}) \approx$
    DTLZ 3 3 $3.1493\times 10^{-1}$
    $(2.64\times 10^{-1})$
    $3.3928\times 10^{-1}$
    $(2.64\times 10^{-1}) \approx$
    $4.4454\times 10^{-1}$
    $(2.26\times 10^{-1}) +$
    $3.8776\times 10^{-1}$
    $(2.55\times 10^{-1}) +$
    $4.5790\times 10^{-1}$
    $(2.08\times 10^{-1}) +$
    $3.8655\times 10^{-1}$
    $(1.84\times 10^{-1}) \approx$
    $4.6543\times 10^{-1}$
    $(2.12\times 10^{-1}) \approx$
    5 $3.7400\times 10^{-1}$
    $(3.68\times 10^{-1})$
    $1.5350\times 10^{-1}$
    $(2.96\times 10^{-1}) -$
    $2.8806\times 10^{-1}$
    $(3.85\times 10^{-1}) -$
    $4.5110\times 10^{-1}$
    $(3.88\times 10^{-1}) \approx$
    $3.9824\times 10^{-1}$
    $(3.80\times 10^{-1}) \approx$
    $3.5245\times 10^{-1}$
    $(3.42\times 10^{-1}) \approx$
    $7.9253\times 10^{-1}$
    $(2.14\times 10^{-3}) +$
    10 $5.7253\times 10^{-1}$
    $(2.29\times 10^{-1})$
    $1.2957\times 10^{-1}$
    $(3.35\times 10^{-1}) -$
    $8.7219\times 10^{-1}$
    $(2.96\times 10^{-1}) +$
    $7.7563\times 10^{-1}$
    $(3.94\times 10^{-1}) +$
    $7.9552\times 10^{-1}$
    $(2.33\times 10^{-1}) +$
    $9.4859\times 10^{-2}$
    $(1.25\times 10^{-1}) -$
    $8.0885\times 10^{-1}$
    $(3.68\times 10^{-1}) +$
    15 $1.8380\times 10^{-1}$
    $(3.12\times 10^{-1})$
    $0.0000\times 10^{0}$
    $(0.00\times 10^{0}) -$
    $1.5858\times 10^{-1}$
    $(3.61\times 10^{-1}) \approx$
    $6.5490\times 10^{-2}$
    $(2.49\times 10^{-1}) -$
    $1.2904\times 10^{-1}$
    $(1.89\times 10^{-1}) \approx$
    $0.0000\times 10^{0}$
    $(0.00\times 10^{0}) -$
    $1.2832\times 10^{-1}$
    $(3.33\times 10^{-1}) \approx$
    DTLZ 4 3 $5.5571\times 10^{-1}$
    $(4.00\times 10^{-2})$
    $4.9607\times 10^{-1}$
    $(1.04\times 10^{-1}) -$
    $5.6302\times 10^{-1}$
    $(1.76\times 10^{-6}) \approx$
    $5.6302\times 10^{-1}$
    $(9.77\times 10^{-7}) \approx$
    $5.4835\times 10^{-1}$
    $(5.57\times 10^{-2}) -$
    $5.2680\times 10^{-1}$
    $(8.24\times 10^{-2}) -$
    $5.6302\times 10^{-1}$
    $(3.79\times 10^{-6}) \approx$
    5 $7.9489\times 10^{-1}$
    $(3.32\times 10^{-4})$
    $7.8461\times 10^{-1}$
    $(3.14\times 10^{-2}) \approx$
    $7.9488\times 10^{-1}$
    $(4.10\times 10^{-4}) \approx$
    $7.9488\times 10^{-1}$
    $(3.94\times 10^{-4}) \approx$
    $7.5533\times 10^{-1}$
    $(4.54\times 10^{-2}) -$
    $7.7479\times 10^{-1}$
    $(4.58\times 10^{-3}) -$
    $7.9462\times 10^{-1}$
    $(3.32\times 10^{-4}) -$
    10 $9.6980\times 10^{-1}$
    $(1.43\times 10^{-4})$
    $9.5960\times 10^{-1}$
    $(2.48\times 10^{-2}) \approx$
    $9.6983\times 10^{-1}$
    $(1.87\times 10^{-4}) \approx$
    $9.6977\times 10^{-1}$
    $(1.78\times 10^{-4}) \approx$
    $9.7012\times 10^{-1}$
    $(4.66\times 10^{-3}) +$
    $9.5269\times 10^{-1}$
    $(2.39\times 10^{-3}) -$
    $9.7122\times 10^{-1}$
    $(1.77\times 10^{-4}) +$
    15 $9.9066\times 10^{-1}$
    $(8.86\times 10^{-5})$
    $9.7301\times 10^{-1}$
    $(1.36\times 10^{-2}) -$
    $9.8919\times 10^{-1}$
    $(2.57\times 10^{-3}) \approx$
    $9.8905\times 10^{-1}$
    $(3.03\times 10^{-3}) -$
    $9.7826\times 10^{-1}$
    $(1.03\times 10^{-2}) -$
    $9.7895\times 10^{-1}$
    $(2.52\times 10^{-3}) -$
    $9.8939\times 10^{-1}$
    $(2.01\times 10^{-3}) -$
    DTLZ 5 3 $1.9397\times 10^{-1}$
    $(1.20\times 10^{-4})$
    $1.9223\times 10^{-1}$
    $(1.03\times 10^{-3}) -$
    $1.6284\times 10^{-1}$
    $(3.84\times 10^{-3}) -$
    $1.8376\times 10^{-1}$
    $(3.37\times 10^{-4}) -$
    $1.9214\times 10^{-1}$
    $(3.58\times 10^{-7}) -$
    $1.7826\times 10^{-1}$
    $(1.21\times 10^{-2}) -$
    $1.9283\times 10^{-1}$
    $(7.92\times 10^{-4}) -$
    5 $9.2958\times 10^{-2}$
    $(3.69\times 10^{-3})$
    $8.9794\times 10^{-2}$
    $(2.82\times 10^{-2}) \approx$
    $1.0617\times 10^{-1}$
    $(4.51\times 10^{-3}) +$
    $1.1298\times 10^{-1}$
    $(4.15\times 10^{-4}) +$
    $9.0987\times 10^{-2}$
    $(2.37\times 10^{-4}) -$
    $4.3747\times 10^{-2}$
    $(4.20\times 10^{-2}) -$
    $1.0603\times 10^{-1}$
    $(1.47\times 10^{-3}) +$
    10 $9.3278\times 10^{-2}$
    $(1.61\times 10^{-3})$
    $6.7279\times 10^{-3}$
    $(1.68\times 10^{-2}) -$
    $9.1255\times 10^{-2}$
    $(4.75\times 10^{-4}) -$
    $9.4192\times 10^{-2}$
    $(2.56\times 10^{-4}) +$
    $9.1736\times 10^{-2}$
    $(1.49\times 10^{-3}) -$
    $8.5522\times 10^{-2}$
    $(1.49\times 10^{-2}) -$
    $9.1402\times 10^{-2}$
    $(5.49\times 10^{-4}) -$
    15 $9.2486\times 10^{-2}$
    $(5.85\times 10^{-4})$
    $8.9842\times 10^{-2}$
    $(2.02\times 10^{-3}) -$
    $9.1006\times 10^{-2}$
    $(4.04\times 10^{-4}) -$
    $9.2026\times 10^{-2}$
    $(4.99\times 10^{-4}) -$
    $9.1297\times 10^{-2}$
    $(3.76\times 10^{-4}) -$
    $2.7606\times 10^{-2}$
    $(3.86\times 10^{-2}) -$
    $9.1402\times 10^{-2}$
    $(5.49\times 10^{-4}) -$
    DTLZ 6 3 $1.9395\times 10^{-1}$
    $(1.70\times 10^{-4})$
    $1.9013\times 10^{-1}$
    $(1.46\times 10^{-3}) -$
    $1.5924\times 10^{-1}$
    $(5.93\times 10^{-3}) -$
    $1.8333\times 10^{-1}$
    $(1.24\times 10^{-4}) -$
    $1.9214\times 10^{-1}$
    $(4.32\times 10^{-7}) -$
    $1.7447\times 10^{-1}$
    $(1.64\times 10^{-2}) -$
    $1.9123\times 10^{-1}$
    $(1.38\times 10^{-3}) -$
    5 $9.5588\times 10^{-2}$
    $(5.77\times 10^{-3})$
    $7.5292\times 10^{-2}$
    $(3.47\times 10^{-2}) -$
    $1.0069\times 10^{-1}$
    $(2.20\times 10^{-2}) +$
    $1.1288\times 10^{-1}$
    $(2.53\times 10^{-4}) +$
    $9.0926\times 10^{-2}$
    $(2.81\times 10^{-4}) -$
    $6.7599\times 10^{-2}$
    $(3.79\times 10^{-2}) -$
    $1.0428\times 10^{-1}$
    $(4.98\times 10^{-3}) +$
    10 $9.3189\times 10^{-2}$
    $(1.68\times 10^{-3})$
    $1.5582\times 10^{-2}$
    $(3.43\times 10^{-2}) -$
    $8.8629\times 10^{-2}$
    $(1.68\times 10^{-2}) -$
    $9.4287\times 10^{-2}$
    $(2.17\times 10^{-4}) \approx$
    $9.2894\times 10^{-2}$
    $(9.94\times 10^{-4}) \approx$
    $9.1003\times 10^{-2}$
    $(3.74\times 10^{-4}) -$
    $9.4393\times 10^{-2}$
    $(9.71\times 10^{-4}) \approx$
    15 $9.1311\times 10^{-2}$
    $(5.95\times 10^{-4})$
    $7.9684\times 10^{-3}$
    $(2.45\times 10^{-2}) -$
    $9.0626\times 10^{-2}$
    $(2.20\times 10^{-3}) \approx$
    $9.2156\times 10^{-2}$
    $(2.50\times 10^{-4}) +$
    $9.1576\times 10^{-2}$
    $(5.24\times 10^{-4}) +$
    $8.4945\times 10^{-2}$
    $(2.31\times 10^{-2}) -$
    $8.7853\times 10^{-2}$
    $(1.70\times 10^{-2}) -$
    DTLZ 7 3 $2.7062\times 10^{-1}$
    $(1.53\times 10^{-2})$
    $2.7423\times 10^{-1}$
    $(1.54\times 10^{-3}) +$
    $2.6741\times 10^{-1}$
    $(1.06\times 10^{-3}) -$
    $2.1817\times 10^{-1}$
    $(1.52\times 10^{-2}) -$
    $2.7096\times 10^{-1}$
    $(1.47\times 10^{-2}) \approx$
    $2.6196\times 10^{-1}$
    $(7.35\times 10^{-3}) -$
    $2.7791\times 10^{-1}$
    $(4.93\times 10^{-4}) \approx$
    5 $2.5738\times 10^{-1}$
    $(8.66\times 10^{-3})$
    $2.3857\times 10^{-1}$
    $(7.76\times 10^{-3}) -$
    $2.0656\times 10^{-1}$
    $(9.68\times 10^{-4}) -$
    $9.3745\times 10^{-2}$
    $(1.55\times 10^{-2}) -$
    $2.4331\times 10^{-1}$
    $(6.98\times 10^{-3}) -$
    $2.4015\times 10^{-1}$
    $(2.17\times 10^{-2}) -$
    $2.5103\times 10^{-1}$
    $(1.19\times 10^{-3}) -$
    10 $1.9133\times 10^{-1}$
    $(2.17\times 10^{-3})$
    $1.9640\times 10^{-1}$
    $(5.63\times 10^{-3}) +$
    $1.7601\times 10^{-1}$
    $(1.34\times 10^{-2}) -$
    $5.6536\times 10^{-5}$
    $(3.83\times 10^{-5}) -$
    $1.5718\times 10^{-1}$
    $(9.13\times 10^{-3}) -$
    $7.5216\times 10^{-2}$
    $(3.47\times 10^{-2}) -$
    $1.8524\times 10^{-1}$
    $(4.94\times 10^{-3}) -$
    15 $1.4909\times 10^{-1}$
    $(2.38\times 10^{-3})$
    $1.4039\times 10^{-1}$
    $(1.21\times 10^{-2}) -$
    $1.0542\times 10^{-1}$
    $(3.45\times 10^{-2}) -$
    $1.5473\times 10^{-7}$
    $(5.89\times 10^{-8}) -$
    $9.7466\times 10^{-2}$
    $(4.12\times 10^{-2}) -$
    $5.6787\times 10^{-4}$
    $(1.49\times 10^{-3}) -$
    $1.0880\times 10^{-1}$
    $(1.06\times 10^{-2}) -$
    $+$ / $\approx$ / $-$ 3 / 7 / 18 8 / 9 / 11 9 / 9 / 10 5 / 5 / 18 2 / 0 / 26 9 / 7 / 12
    $+$, $\approx$和$-$分别表示获得的结果与R2-RVEA相比更好, 相似和更差.
    下载: 导出CSV

    表  3  R2-RVEA、NSGA-Ⅲ、RVEA、MOEA/DD、MOMBI-Ⅱ、KnEA和TS-R2EA在DTLZ1~DTLZ7上获得的IGD+值的统计结果(均值和标准差). 最好的结果已突出

    Table  3  The statistical results (mean and standard deviation) of the IGD+ values obtained by R2-RVEA, NSGA-Ⅲ, RVEA, MOEA/DD, MOMBI-Ⅱ, KnEA and TS-R2EA on DTLZ1 to DTLZ7. The best results are highlighted

    问题 $M$ R2-RVEA NSGA-Ⅲ RVEA MOEA/DD MOMBI-Ⅱ KnEA TS-R2EA
    DTLZ 1 3 $1.6235\times 10^{-2}$
    $(7.90\times 10^{-3})$
    $1.3651\times 10^{-2}$
    $(3.29\times 10^{-4}) +$
    $1.3492\times 10^{-2}$
    $(1.25\times 10^{-4}) +$
    $1.3523\times 10^{-2}$
    $(1.48\times 10^{-4}) +$
    $1.3709\times 10^{-2}$
    $(2.71\times 10^{-4}) \approx$
    $2.6216\times 10^{-2}$
    $(9.65\times 10^{-3}) -$
    $1.3571\times 10^{-2}$
    $(3.33\times 10^{-4}) +$
    5 $5.0560\times 10^{-2}$
    $(6.33\times 10^{-3})$
    $7.3388\times 10^{-2}$
    $(7.15\times 10^{-2}) \approx$
    $4.6106\times 10^{-2}$
    $(3.46\times 10^{-4}) +$
    $4.5988\times 10^{-2}$
    $(1.35\times 10^{-4}) +$
    $4.8230\times 10^{-2}$
    $(5.92\times 10^{-3}) \approx$
    $9.3292\times 10^{-2}$
    $(4.42\times 10^{-2}) -$
    $5.3386\times 10^{-2}$
    $(4.06\times 10^{-2}) -$
    10 $1.0287\times 10^{-1}$
    $(8.43\times 10^{-3})$
    $9.0516\times 10^{-2}$
    $(5.74\times 10^{-2}) +$
    $7.5684\times 10^{-2}$
    $(3.45\times 10^{-2}) +$
    $6.9285\times 10^{-2}$
    $(4.56\times 10^{-4}) +$
    $1.3489\times 10^{-1}$
    $(3.62\times 10^{-2}) -$
    $2.7999\times 10^{-1}$
    $(5.22\times 10^{-2}) -$
    $7.9575\times 10^{-2}$
    $(3.48\times 10^{-2}) +$
    15 $2.8426\times 10^{-1}$
    $(1.82\times 10^{-1})$
    $5.2357\times 10^{-1}$
    $(3.38\times 10^{-1}) -$
    $2.0300\times 10^{-1}$
    $(1.82\times 10^{-1}) +$
    $1.2789\times 10^{-1}$
    $(8.21\times 10^{-2}) +$
    $2.8165\times 10^{-1}$
    $(1.00\times 10^{-1}) \approx$
    $9.0915\times 10^{-1}$
    $(7.61\times 10^{-1}) -$
    $2.7442\times 10^{-1}$
    $(3.09\times 10^{-1}) \approx$
    DTLZ 2 3 $2.0861\times 10^{-2}$
    $(5.70\times 10^{-6})$
    $2.0859\times 10^{-2}$
    $(8.08\times 10^{-7}) \approx$
    $2.0859\times 10^{-2}$
    $(5.36\times 10^{-7}) \approx$
    $2.0859\times 10^{-2}$
    $(2.57\times 10^{-7}) \approx$
    $2.0870\times 10^{-2}$
    $(8.77\times 10^{-6}) -$
    $3.6494\times 10^{-2}$
    $(5.13\times 10^{-3}) -$
    $2.0861\times 10^{-2}$
    $(1.10\times 10^{-6}) -$
    5 $7.1279\times 10^{-2}$
    $(7.27\times 10^{-6})$
    $7.1282\times 10^{-2}$
    $(1.80\times 10^{-6}) -$
    $7.1283\times 10^{-2}$
    $(5.39\times 10^{-6}) -$
    $7.1283\times 10^{-2}$
    $(6.73\times 10^{-7}) -$
    $7.1348\times 10^{-2}$
    $(4.27\times 10^{-5}) -$
    $8.9581\times 10^{-2}$
    $(4.33\times 10^{-3}) -$
    $7.1291\times 10^{-2}$
    $(4.32\times 10^{-6}) -$
    10 $1.7493\times 10^{-1}$
    $(2.35\times 10^{-5})$
    $1.9840\times 10^{-1}$
    $(4.82\times 10^{-2}) -$
    $1.6809\times 10^{-1}$
    $(2.66\times 10^{-5}) +$
    $1.7493\times 10^{-1}$
    $(4.25\times 10^{-6}) \approx$
    $1.6874\times 10^{-1}$
    $(1.06\times 10^{-3}) +$
    $1.8012\times 10^{-1}$
    $(5.20\times 10^{-3}) -$
    $1.7207\times 10^{-1}$
    $(3.32\times 10^{-4}) \approx$
    15 $2.3684\times 10^{-1}$
    $(4.10\times 10^{-4})$
    $2.9249\times 10^{-1}$
    $(1.82\times 10^{-2}) -$
    $2.6116\times 10^{-1}$
    $(8.95\times 10^{-2}) -$
    $2.3623\times 10^{-1}$
    $(2.02\times 10^{-4}) \approx$
    $4.0323\times 10^{-1}$
    $(5.54\times 10^{-2}) -$
    $2.4998\times 10^{-1}$
    $(1.79\times 10^{-2}) -$
    $2.3856\times 10^{-1}$
    $(1.96\times 10^{-3}) -$
    DTLZ 3 3 $3.9763\times 10^{-1}$
    $(5.49\times 10^{-1})$
    $4.5849\times 10^{-1}$
    $(6.21\times 10^{-1}) \approx$
    $3.0314\times 10^{-1}$
    $(5.74\times 10^{-1}) \approx$
    $1.8948\times 10^{-1}$
    $(3.71\times 10^{-1}) +$
    $3.3114\times 10^{-1}$
    $(6.46\times 10^{-1}) \approx$
    $4.0938\times 10^{-1}$
    $(4.78\times 10^{-1}) -$
    $2.2115\times 10^{-1}$
    $(4.00\times 10^{-1}) +$
    5 $3.9944\times 10^{-1}$
    $(5.79\times 10^{-1})$
    $1.7578\times 10^{0}$
    $(1.67\times 10^{0}) -$
    $7.4422\times 10^{-1}$
    $(5.86\times 10^{-1}) \approx$
    $6.5811\times 10^{-1}$
    $(5.59\times 10^{-1}) \approx$
    $3.7339\times 10^{-1}$
    $(5.22\times 10^{-1}) +$
    $9.7538\times 10^{-1}$
    $(1.31\times 10^{0}) -$
    $4.6878\times 10^{-1}$
    $(5.40\times 10^{-1}) -$
    10 $3.7208\times 10^{-1}$
    $(4.00\times 10^{-1})$
    $3.2503\times 10^{0}$
    $(3.09\times 10^{0}) -$
    $4.5932\times 10^{-1}$
    $(5.80\times 10^{-1}) -$
    $4.9410\times 10^{-1}$
    $(5.27\times 10^{-1}) -$
    $3.9303\times 10^{-1}$
    $(3.16\times 10^{-1}) \approx$
    $1.1755\times 10^{1}$
    $(5.73\times 10^{1}) -$
    $3.9429\times 10^{-1}$
    $(4.89\times 10^{-1}) -$
    15 $2.6642\times 10^{0}$
    $(1.48\times 10^{0})$
    $9.7830\times 10^{0}$
    $(5.23\times 10^{0}) -$
    $5.2490\times 10^{0}$
    $(4.15\times 10^{0}) \approx$
    $2.7498\times 10^{0}$
    $(2.09\times 10^{0}) -$
    $3.1399\times 10^{0}$
    $(2.60\times 10^{0}) \approx$
    $1.3145\times 10^{1}$
    $(8.69\times 10^{0}) -$
    $3.3117\times 10^{0}$
    $(2.18\times 10^{0}) -$
    DTLZ 4 3 $2.0868\times 10^{-2}$
    $(1.87\times 10^{-5})$
    $7.5928\times 10^{-2}$
    $(9.29\times 10^{-2}) \approx$
    $2.0859\times 10^{-2}$
    $(4.09\times 10^{-7}) \approx$
    $2.0859\times 10^{-2}$
    $(2.72\times 10^{-7}) +$
    $4.1428\times 10^{-2}$
    $(6.27\times 10^{-2}) -$
    $7.7586\times 10^{-2}$
    $(1.50\times 10^{-1}) -$
    $2.0862\times 10^{-2}$
    $(1.74\times 10^{-6}) \approx$
    5 $7.1276\times 10^{-2}$
    $(1.24\times 10^{-5})$
    $9.3899\times 10^{-2}$
    $(5.00\times 10^{-2}) -$
    $7.4060\times 10^{-2}$
    $(1.52\times 10^{-2}) -$
    $7.4382\times 10^{-2}$
    $(1.70\times 10^{-2}) -$
    $8.2438\times 10^{-2}$
    $(2.87\times 10^{-2}) -$
    $8.0641\times 10^{-2}$
    $(1.73\times 10^{-3}) -$
    $7.4077\times 10^{-2}$
    $(1.52\times 10^{-3}) -$
    10 $1.7497\times 10^{-1}$
    $(4.67\times 10^{-5})$
    $1.7695\times 10^{-1}$
    $(2.02\times 10^{-2}) -$
    $1.6849\times 10^{-1}$
    $(1.77\times 10^{-3}) \approx$
    $1.7081\times 10^{-1}$
    $(3.43\times 10^{-3}) \approx$
    $1.6532\times 10^{-1}$
    $(6.13\times 10^{-3}) +$
    $1.6217\times 10^{-1}$
    $(2.21\times 10^{-3}) +$
    $1.6820\times 10^{-1}$
    $(5.03\times 10^{-4}) \approx$
    15 $2.3729\times 10^{-1}$
    $(3.58\times 10^{-4})$
    $2.6180\times 10^{-1}$
    $(2.54\times 10^{-2}) \approx$
    $2.3958\times 10^{-1}$
    $(4.59\times 10^{-3}) \approx$
    $2.4240\times 10^{-1}$
    $(6.98\times 10^{-3}) -$
    $2.6824\times 10^{-1}$
    $(2.33\times 10^{-2}) -$
    $2.3292\times 10^{-1}$
    $(3.23\times 10^{-3}) +$
    $2.4469\times 10^{-1}$
    $(8.10\times 10^{-3}) -$
    DTLZ 5 3 $6.4126\times 10^{-3}$
    $(1.23\times 10^{-4})$
    $7.5607\times 10^{-3}$
    $(8.01\times 10^{-4}) -$
    $3.3584\times 10^{-2}$
    $(4.19\times 10^{-3}) -$
    $1.3345\times 10^{-2}$
    $(3.02\times 10^{-4}) -$
    $7.6588\times 10^{-3}$
    $(3.38\times 10^{-7}) -$
    $1.4326\times 10^{-2}$
    $(7.18\times 10^{-3}) -$
    $7.1355\times 10^{-3}$
    $(5.09\times 10^{-4}) -$
    5 $8.2311\times 10^{-2}$
    $(1.90\times 10^{-2})$
    $2.3956\times 10^{-1}$
    $(2.11\times 10^{-1}) -$
    $1.7521\times 10^{-1}$
    $(5.25\times 10^{-2}) -$
    $7.2412\times 10^{-2}$
    $(6.75\times 10^{-3}) \approx$
    $9.4200\times 10^{-2}$
    $(3.23\times 10^{-5}) -$
    $2.4871\times 10^{-1}$
    $(1.59\times 10^{-1}) -$
    $3.8136\times 10^{-2}$
    $(4.16\times 10^{-3}) +$
    10 $1.2919\times 10^{-1}$
    $(4.14\times 10^{-2})$
    $4.5326\times 10^{-1}$
    $(1.29\times 10^{-1}) -$
    $1.7827\times 10^{-1}$
    $(6.47\times 10^{-2}) -$
    $9.8182\times 10^{-2}$
    $(1.79\times 10^{-2}) +$
    $3.5774\times 10^{-1}$
    $(4.93\times 10^{-2}) -$
    $8.4408\times 10^{-2}$
    $(9.10\times 10^{-3}) +$
    $1.0774\times 10^{-1}$
    $(1.94\times 10^{-2}) \approx$
    15 $1.5729\times 10^{-1}$
    $(9.70\times 10^{-2})$
    $1.5942\times 10^{-1}$
    $(7.13\times 10^{-2}) \approx$
    $3.5936\times 10^{-1}$
    $(5.81\times 10^{-2}) -$
    $5.1183\times 10^{-2}$
    $(1.74\times 10^{-3}) +$
    $3.7512\times 10^{-1}$
    $(2.29\times 10^{-3}) -$
    $3.3561\times 10^{-1}$
    $(2.97\times 10^{-1}) -$
    $1.3229\times 10^{-1}$
    $(6.34\times 10^{-2}) \approx$
    DTLZ 6 3 $6.4261\times 10^{-3}$
    $(8.57\times 10^{-5}$
    $9.1282\times 10^{-3}$
    $(1.10\times 10^{-3}) -$
    $3.3761\times 10^{-2}$
    $(8.06\times 10^{-3}) -$
    $1.3955\times 10^{-2}$
    $(5.10\times 10^{-4}) -$
    $7.6585\times 10^{-3}$
    $(2.94\times 10^{-7}) -$
    $6.0051\times 10^{-2}$
    $(1.07\times 10^{-2}) -$
    $7.4448\times 10^{-3}$
    $(7.09\times 10^{-4}) -$
    5 $8.1474\times 10^{-2}$
    $(1.79\times 10^{-2})$
    $2.3445\times 10^{-1}$
    $(1.04\times 10^{-1}) -$
    $1.0892\times 10^{-1}$
    $(1.71\times 10^{-2}) -$
    $7.1693\times 10^{-2}$
    $(3.07\times 10^{-3}) +$
    $8.4850\times 10^{-2}$
    $(3.26\times 10^{-4}) -$
    $2.0779\times 10^{-1}$
    $(2.06\times 10^{-1}) -$
    $5.8034\times 10^{-2}$
    $(6.23\times 10^{-3}) +$
    10 $7.1984\times 10^{-2}$
    $(1.08\times 10^{-2})$
    $1.6206\times 10^{0}$
    $(1.01\times 10^{0}) -$
    $2.3439\times 10^{-1}$
    $(5.94\times 10^{-2}) -$
    $7.7284\times 10^{-2}$
    $(2.48\times 10^{-2}) \approx$
    $3.5367\times 10^{-1}$
    $(4.32\times 10^{-2}) -$
    $1.0960\times 10^{-1}$
    $(7.24\times 10^{-2}) -$
    $6.0540\times 10^{-2}$
    $(3.57\times 10^{-2}) +$
    15 $2.1340\times 10^{-1}$
    $(9.95\times 10^{-2})$
    $1.8744\times 10^{0}$
    $(1.17\times 10^{0}) -$
    $1.8445\times 10^{-1}$
    $(1.19\times 10^{-1}) \approx$
    $5.1647\times 10^{-2}$
    $(2.71\times 10^{-6}) +$
    $3.7468\times 10^{-1}$
    $(4.84\times 10^{-3}) -$
    $2.9365\times 10^{-1}$
    $(2.46\times 10^{-1}) \approx$
    $1.1096\times 10^{-1}$
    $(1.49\times 10^{-1}) +$
    DTLZ 7 3 $9.8407\times 10^{-2}$
    $(1.34\times 10^{-1})$
    $3.5754\times 10^{-2}$
    $(2.15\times 10^{-3}) +$
    $4.3704\times 10^{-2}$
    $(9.60\times 10^{-4}) +$
    $3.9831\times 10^{-1}$
    $(1.74\times 10^{-1}) -$
    $4.6134\times 10^{-2}$
    $(4.79\times 10^{-2}) \approx$
    $7.1748\times 10^{-2}$
    $(6.52\times 10^{-2}) +$
    $3.4202\times 10^{-2}$
    $(6.64\times 10^{-4}) +$
    5 $1.9604\times 10^{-1}$
    $(5.86\times 10^{-2})$
    $1.8190\times 10^{-1}$
    $(2.52\times 10^{-2}) \approx$
    $2.7929\times 10^{0}$
    $(5.66\times 10^{-2}) -$
    $2.5348\times 10^{-1}$
    $(1.65\times 10^{-1}) \approx$
    $1.1096\times 10^{-1}$
    $(1.49\times 10^{-1}) +$
    $2.0713\times 10^{-1}$
    $(1.30\times 10^{-1}) \approx$
    $1.4313\times 10^{-1}$
    $(9.53\times 10^{-4}) +$
    10 $1.2600\times 10^{0}$
    $(2.10\times 10^{-2})$
    $6.6488\times 10^{-1}$
    $(4.91\times 10^{-2}) +$
    $1.0133\times 10^{0}$
    $(5.93\times 10^{-2}) +$
    $1.2994\times 10^{0}$
    $(6.05\times 10^{-2}) -$
    $4.5256\times 10^{0}$
    $(8.62\times 10^{-1}) -$
    $6.6322\times 10^{-1}$
    $(9.13\times 10^{-3}) +$
    $6.9956\times 10^{-1}$
    $(1.88\times 10^{-1}) +$
    15 $8.3596\times 10^{0}$
    $(3.76\times 10^{-2})$
    $7.3318\times 10^{0}$
    $(1.12\times 10^{0}) +$
    $1.5286\times 10^{0}$
    $(8.28\times 10^{-2}) +$
    $2.0686\times 10^{0}$
    $(1.74\times 10^{-1}) +$
    $1.1458\times 10^{1}$
    $(1.70\times 10^{0}) -$
    $4.7916\times 10^{0}$
    $(9.31\times 10^{-1}) +$
    $2.0481\times 10^{0}$
    $(6.97\times 10^{-1}) +$
    $+$ / $\approx$ / $-$ 5 / 7 / 16 8 / 8 / 12 11 / 7 / 10 3 / 8 / 17 6 / 2 / 20 11 / 6 / 11
    $+$, $\approx$和$-$分别表示获得的结果与R2-RVEA相比更好, 相似和更差.
    下载: 导出CSV

    表  4  R2-RVEA、NSGA-Ⅲ、RVEA、MOEA/DD、MOMBI-Ⅱ、KnEA和TS-R2EA在WFG1-WFG9上获得的HV值的统计结果(均值和标准差). 最好的结果已突出

    Table  4  The statistical results (mean and standard deviation) of the HV values obtained by R2-RVEA, NSGA-Ⅲ, RVEA, MOEA/DD, MOMBI-Ⅱ, KnEA and TS-R2EA on WFG1 to WFG9. The best results are highlighted

    问题 $M$ R2-RVEA NSGA-Ⅲ RVEA MOEA/DD MOMBI-Ⅱ KnEA TS-R2EA
    WFG 1 3 $9.4097\times 10^{-1}$
    $(1.32\times 10^{-2})$
    $9.3384\times 10^{-1}$
    $(2.98\times 10^{-2}) -$
    $9.4502\times 10^{-1}$
    $(1.09\times 10^{-3}) +$
    $9.1553\times 10^{-1}$
    $(3.55\times 10^{-2}) -$
    $8.8329\times 10^{-1}$
    $(5.34\times 10^{-2}) -$
    $9.2399\times 10^{-1}$
    $(2.53\times 10^{-2}) -$
    $9.3418\times 10^{-1}$
    $(3.48\times 10^{-2}) -$
    5 $9.9456\times 10^{-1}$
    $(1.96\times 10^{-2})$
    $9.8643\times 10^{-1}$
    $(3.60\times 10^{-2}) -$
    $9.8018\times 10^{-1}$
    $(4.06\times 10^{-2}) -$
    $9.5451\times 10^{-1}$
    $(3.40\times 10^{-2}) -$
    $8.4739\times 10^{-1}$
    $(8.49\times 10^{-2}) -$
    $9.7995\times 10^{-1}$
    $(1.87\times 10^{-2}) -$
    $9.9075\times 10^{-1}$
    $(2.82\times 10^{-2}) -$
    10 $9.9957\times 10^{-1}$
    $(2.10\times 10^{-4})$
    $9.9912\times 10^{-1}$
    $(6.70\times 10^{-4}) -$
    $9.8000\times 10^{-1}$
    $(3.93\times 10^{-2}) -$
    $9.8795\times 10^{-1}$
    $(1.55\times 10^{-2}) -$
    $9.9607\times 10^{-1}$
    $(1.91\times 10^{-2}) -$
    $9.9527\times 10^{-1}$
    $(1.66\times 10^{-2}) -$
    $9.9888\times 10^{-1}$
    $(3.64\times 10^{-2}) -$
    15 $9.9967\times 10^{-1}$
    $(2.14\times 10^{-4})$
    $9.9949\times 10^{-1}$
    $(3.42\times 10^{-4}) -$
    $8.9713\times 10^{-1}$
    $(8.87\times 10^{-2}) -$
    $9.3763\times 10^{-1}$
    $(7.46\times 10^{-2}) -$
    $6.5027\times 10^{-1}$
    $(2.44\times 10^{-1}) -$
    $9.8089\times 10^{-1}$
    $(3.34\times 10^{-2}) -$
    $9.4320\times 10^{-1}$
    $(7.47\times 10^{-2}) -$
    WFG 2 3 $9.2138\times 10^{-1}$
    $(8.02\times 10^{-3})$
    $9.2385\times 10^{-1}$
    $(6.17\times 10^{-3}) \approx$
    $9.1683\times 10^{-1}$
    $(1.11\times 10^{-2}) \approx$
    $9.1463\times 10^{-1}$
    $(1.22\times 10^{-2}) -$
    $9.1075\times 10^{-1}$
    $(1.81\times 10^{-2}) -$
    $9.0896\times 10^{-1}$
    $(2.65\times 10^{-2}) -$
    $9.2318\times 10^{-1}$
    $(1.17\times 10^{-2}) \approx$
    5 $9.8369\times 10^{-1}$
    $(6.66\times 10^{-3})$
    $9.8358\times 10^{-1}$
    $(7.88\times 10^{-3}) \approx$
    $9.5373\times 10^{-1}$
    $(1.83\times 10^{-2}) -$
    $9.5536\times 10^{-1}$
    $(1.32\times 10^{-2}) -$
    $9.6469\times 10^{-1}$
    $(1.80\times 10^{-2}) -$
    $9.6900\times 10^{-1}$
    $(1.21\times 10^{-2}) -$
    $9.7339\times 10^{-1}$
    $(1.56\times 10^{-2}) -$
    10 $9.9168\times 10^{-1}$
    $(4.71\times 10^{-3})$
    $9.8734\times 10^{-1}$
    $(6.36\times 10^{-3}) -$
    $9.3682\times 10^{-1}$
    $(2.27\times 10^{-2}) -$
    $9.3481\times 10^{-1}$
    $(1.89\times 10^{-2}) -$
    $9.7730\times 10^{-1}$
    $(1.08\times 10^{-2}) -$
    $9.8285\times 10^{-1}$
    $(7.58\times 10^{-3}) -$
    $9.6436\times 10^{-1}$
    $(2.13\times 10^{-2}) -$
    15 $9.5064\times 10^{-1}$
    $(5.65\times 10^{-2})$
    $9.5605\times 10^{-1}$
    $(1.68\times 10^{-2}) +$
    $8.3087\times 10^{-1}$
    $(5.14\times 10^{-2}) -$
    $8.6774\times 10^{-1}$
    $(3.16\times 10^{-2}) -$
    $6.0421\times 10^{-1}$
    $(2.45\times 10^{-1}) -$
    $9.2373\times 10^{-1}$
    $(3.23\times 10^{-2}) -$
    $8.9358\times 10^{-1}$
    $(4.33\times 10^{-2}) -$
    WFG 3 3 $4.0241\times 10^{-1}$
    $(3.81\times 10^{-3})$
    $3.9175\times 10^{-1}$
    $(5.04\times 10^{-3}) -$
    $3.4039\times 10^{-1}$
    $(7.94\times 10^{-3}) -$
    $3.0897\times 10^{-1}$
    $(3.30\times 10^{-2}) -$
    $3.9392\times 10^{-1}$
    $(6.40\times 10^{-3}) -$
    $3.7363\times 10^{-1}$
    $(2.71\times 10^{-2}) -$
    $3.9530\times 10^{-1}$
    $(4.16\times 10^{-3}) -$
    5 $1.9039\times 10^{-1}$
    $(9.10\times 10^{-3})$
    $1.6387\times 10^{-1}$
    $(2.03\times 10^{-2}) -$
    $9.9662\times 10^{-2}$
    $(4.07\times 10^{-2}) -$
    $1.1461\times 10^{-1}$
    $(2.06\times 10^{-2}) -$
    $8.9617\times 10^{-2}$
    $(3.75\times 10^{-3}) -$
    $7.9870\times 10^{-2}$
    $(2.78\times 10^{-2}) -$
    $1.5854\times 10^{-1}$
    $(2.35\times 10^{-2}) -$
    10 $2.7662\times 10^{-2}$
    $(3.29\times 10^{-2})$
    $2.3762\times 10^{-2}$
    $(2.69\times 10^{-2}) \approx$
    $0.0000\times 10^{0}$
    $(0.00\times 10^{0}) -$
    $0.0000\times 10^{0}$
    $(0.00\times 10^{0}) -$
    $5.9165\times 10^{-4}$
    $(2.45\times 10^{-3}) -$
    $0.0000\times 10^{0}$
    $(0.00\times 10^{0}) -$
    $2.6821\times 10^{-2}$
    $(3.10e-2) \approx$
    15 $0.0000\times 10^{0}$
    $(0.00\times 10^{0})$
    $0.0000\times 10^{0}$
    $(0.00\times 10^{0}) \approx$
    $0.0000\times 10^{0}$
    $(0.00\times 10^{0}) \approx$
    $0.0000\times 10^{0}$
    $(0.00\times 10^{0}) \approx$
    $0.0000\times 10^{0}$
    $(0.00\times 10^{0}) \approx$
    $0.0000\times 10^{0}$
    $(0.00\times 10^{0}) \approx$
    $0.0000\times 10^{0}$
    $(0.00\times 10^{0}) \approx$
    WFG 4 3 $5.6274\times 10^{-1}$
    $(1.26\times 10^{-4})$
    $5.6274\times 10^{-1}$
    $(1.44\times 10^{-4}) \approx$
    $5.6096\times 10^{-1}$
    $(5.24\times 10^{-4}) -$
    $5.5434\times 10^{-1}$
    $(9.95\times 10^{-4}) -$
    $5.6138\times 10^{-1}$
    $(3.92\times 10^{-3}) \approx$
    $5.4879\times 10^{-1}$
    $(2.74\times 10^{-3}) -$
    $5.6251\times 10^{-1}$
    $(3.63\times 10^{-4}) -$
    5 $7.9406\times 10^{-1}$
    $(7.67\times 10^{-4})$
    $7.9388\times 10^{-1}$
    $(6.34\times 10^{-4}) \approx$
    $7.9302\times 10^{-1}$
    $(9.54\times 10^{-4}) -$
    $7.6900\times 10^{-1}$
    $(8.68\times 10^{-4}) -$
    $7.8312\times 10^{-1}$
    $(2.45\times 10^{-2}) -$
    $7.7583\times 10^{-1}$
    $(4.18\times 10^{-3}) -$
    $7.9231\times 10^{-1}$
    $(1.58\times 10^{-3}) -$
    10 $9.6852\times 10^{-1}$
    $(7.59\times 10^{-4})$
    $9.6031\times 10^{-1}$
    $(2.55\times 10^{-4}) -$
    $9.6066\times 10^{-1}$
    $(4.21\times 10^{-3}) -$
    $7.6825\times 10^{-1}$
    $(2.17\times 10^{-4}) -$
    $9.2141\times 10^{-1}$
    $(6.09\times 10^{-4}) -$
    $9.5839\times 10^{-1}$
    $(2.16\times 10^{-3}) -$
    $9.7052\times 10^{-1}$
    $(7.92\times 10^{-4}) +$
    15 $9.8874\times 10^{-1}$
    $(9.66\times 10^{-4})$
    $9.6088\times 10^{-1}$
    $(2.38\times 10^{-2}) -$
    $9.5446\times 10^{-1}$
    $(1.55\times 10^{-2}) -$
    $6.4744\times 10^{-1}$
    $(1.70\times 10^{-1}) -$
    $3.9343\times 10^{-1}$
    $(1.22\times 10^{-1}) -$
    $9.8038\times 10^{-1}$
    $(4.46\times 10^{-3}) -$
    $9.8298\times 10^{-1}$
    $(3.55\times 10^{-3}) -$
    WFG 5 3 $5.2186\times 10^{-1}$
    $(1.51\times 10^{-5})$
    $5.2185\times 10^{-1}$
    $(1.81\times 10^{-5}) -$
    $5.2114\times 10^{-1}$
    $(2.33\times 10^{-4}) -$
    $5.1405\times 10^{-1}$
    $(4.53\times 10^{-4}) -$
    $5.0698\times 10^{-1}$
    $(2.74\times 10^{-3}) -$
    $5.0698\times 10^{-1}$
    $(2.74\times 10^{-3}) -$
    $5.2191\times 10^{-1}$
    $(2.07\times 10^{-4}) \approx$
    5 $7.4397\times 10^{-1}$
    $(3.66\times 10^{-4})$
    $7.4384\times 10^{-1}$
    $(3.37\times 10^{-4}) \approx$
    $7.4391\times 10^{-1}$
    $(3.63\times 10^{-4}) \approx$
    $7.1885\times 10^{-1}$
    $(5.68\times 10^{-4}) -$
    $7.2692\times 10^{-1}$
    $(1.26\times 10^{-2}) -$
    $7.2414\times 10^{-1}$
    $(3.96\times 10^{-3}) -$
    $7.4386\times 10^{-1}$
    $(1.17\times 10^{-3}) \approx$
    10 $9.0487\times 10^{-1}$
    $(1.77\times 10^{-4})$
    $9.0484\times 10^{-1}$
    $(1.53\times 10^{-4}) \approx$
    $9.0455\times 10^{-1}$
    $(2.23\times 10^{-4}) -$
    $7.8469\times 10^{-1}$
    $(1.21\times 10^{-2}) -$
    $8.6080\times 10^{-1}$
    $(1.05\times 10^{-2}) -$
    $8.9513\times 10^{-1}$
    $(1.94\times 10^{-3}) -$
    $9.0589\times 10^{-1}$
    $(2.49\times 10^{-4}) +$
    15 $9.1757\times 10^{-1}$
    $(1.11\times 10^{-4})$
    $9.1595\times 10^{-1}$
    $(6.23\times 10^{-3}) -$
    $9.1767\times 10^{-1}$
    $(9.41\times 10^{-5}) +$
    $6.3350\times 10^{-1}$
    $(2.39\times 10^{-2}) -$
    $1.6774\times 10^{-1}$
    $(4.92\times 10^{-2}) -$
    $9.0854\times 10^{-1}$
    $(4.69\times 10^{-3}) -$
    $9.1762\times 10^{-1}$
    $(1.33\times 10^{-4}) \approx$
    WFG 6 3 $5.0818\times 10^{-1}$
    $(1.36\times 10^{-2})$
    $5.1243\times 10^{-1}$
    $(1.32\times 10^{-2}) \approx$
    $5.0823\times 10^{-1}$
    $(1.24\times 10^{-2}) \approx$
    $5.0420\times 10^{-1}$
    $(1.45\times 10^{-2}) \approx$
    $5.0752\times 10^{-1}$
    $(1.29\times 10^{-2}) \approx$
    $4.8204\times 10^{-1}$
    $(1.70\times 10^{-2}) -$
    $5.0745\times 10^{-1}$
    $(1.42\times 10^{-2}) \approx$
    5 $7.2593\times 10^{-1}$
    $(1.41\times 10^{-2})$
    $7.3070\times 10^{-1}$
    $(1.66\times 10^{-2}) \approx$
    $7.3015\times 10^{-1}$
    $(1.90\times 10^{-2}) \approx$
    $6.9856\times 10^{-1}$
    $(2.16\times 10^{-2}) -$
    $7.1566\times 10^{-1}$
    $(2.87\times 10^{-2}) \approx$
    $6.8244\times 10^{-1}$
    $(2.81\times 10^{-2}) -$
    $7.3271\times 10^{-1}$
    $(1.64\times 10^{-2}) \approx$
    10 $8.8346\times 10^{-1}$
    $(1.26\times 10^{-2})$
    $8.8339\times 10^{-1}$
    $(1.97\times 10^{-2}) \approx$
    $8.7673\times 10^{-1}$
    $(2.06\times 10^{-2}) \approx$
    $7.2135\times 10^{-1}$
    $(2.21\times 10^{-2}) -$
    $8.8534\times 10^{-1}$
    $(3.14\times 10^{-2}) \approx$
    $8.5253\times 10^{-1}$
    $(2.22\times 10^{-2}) -$
    $8.7590\times 10^{-1}$
    $(1.91\times 10^{-2}) -$
    15 $8.9305\times 10^{-1}$
    $(2.01\times 10^{-2})$
    $8.3062\times 10^{-1}$
    $(3.59\times 10^{-2}) -$
    $7.0352\times 10^{-1}$
    $(8.04\times 10^{-2}) -$
    $5.8993\times 10^{-1}$
    $(4.37\times 10^{-2}) -$
    $4.7510\times 10^{-1}$
    $(1.49\times 10^{-1}) -$
    $8.6942\times 10^{-1}$
    $(3.32\times 10^{-2}) -$
    $8.8273\times 10^{-1}$
    $(2.84\times 10^{-2}) -$
    WFG 7 3 $5.6255\times 10^{-1}$
    $(1.73\times 10^{-4})$
    $5.6251\times 10^{-1}$
    $(1.49\times 10^{-4}) \approx$
    $5.5223\times 10^{-1}$
    $(1.21\times 10^{-3}) -$
    $8.8273\times 10^{-1}$
    $(2.84\times 10^{-2}) -$
    $5.6238\times 10^{-1}$
    $(2.13\times 10^{-3}) -$
    $5.4818\times 10^{-1}$
    $(4.75\times 10^{-3}) -$
    $5.6293\times 10^{-1}$
    $(4.19\times 10^{-5}) +$
    5 $7.9328\times 10^{-1}$
    $(5.69\times 10^{-4})$
    $7.9299\times 10^{-1}$
    $(5.80\times 10^{-4}) \approx$
    $7.9151\times 10^{-1}$
    $(1.86\times 10^{-3}) -$
    $7.6426\times 10^{-1}$
    $(5.52\times 10^{-3}) -$
    $7.8989\times 10^{-1}$
    $(6.21\times 10^{-3}) \approx$
    $7.6829\times 10^{-1}$
    $(8.45\times 10^{-3}) -$
    $7.9436\times 10^{-1}$
    $(5.86\times 10^{-4}) +$
    10 $9.6815\times 10^{-1}$
    $(4.10\times 10^{-4})$
    $9.6270\times 10^{-1}$
    $(1.43\times 10^{-2}) -$
    $9.6055\times 10^{-1}$
    $(3.10\times 10^{-3}) -$
    $8.3568\times 10^{-1}$
    $(2.25\times 10^{-2}) -$
    $9.7072\times 10^{-1}$
    $(1.25\times 10^{-3}) +$
    $9.5292\times 10^{-1}$
    $(3.13\times 10^{-3}) -$
    $9.7156\times 10^{-1}$
    $(8.72\times 10^{-4}) +$
    15 $9.9017\times 10^{-1}$
    $(2.80\times 10^{-4})$
    $9.6427\times 10^{-1}$
    $(1.91\times 10^{-2}) -$
    $5.4561\times 10^{-1}$
    $(2.74\times 10^{-1}) -$
    $7.8897\times 10^{-1}$
    $(1.15\times 10^{-1}) -$
    $5.8748\times 10^{-1}$
    $(1.09\times 10^{-1}) -$
    $9.8220\times 10^{-1}$
    $(7.12\times 10^{-3}) -$
    $9.8688\times 10^{-1}$
    $(5.62\times 10^{-3}) -$
    WFG 8 3 $4.7920\times 10^{-1}$
    $(1.78\times 10^{-3})$
    $4.8088\times 10^{-1}$
    $(2.04\times 10^{-3}) +$
    $4.7329\times 10^{-1}$
    $(3.40\times 10^{-3}) -$
    $4.6512\times 10^{-1}$
    $(1.17\times 10^{-2}) -$
    $4.5390\times 10^{-1}$
    $(3.39\times 10^{-3}) -$
    $4.5340\times 10^{-1}$
    $(5.55\times 10^{-3}) -$
    $4.8156\times 10^{-1}$
    $(2.27\times 10^{-3}) +$
    5 $6.8175\times 10^{-1}$
    (1.97\times 10^{-3})$
    $6.8490\times 10^{-1}$
    $(2.98\times 10^{-3}) +$
    $6.6856\times 10^{-1}$
    $(1.16\times 10^{-2}) -$
    $6.5481\times 10^{-1}$
    $(1.50\times 10^{-2}) -$
    $3.1317\times 10^{-1}$
    $(8.51\times 10^{-3}) -$
    $6.3578\times 10^{-1}$
    $(4.91\times 10^{-3}) -$
    $6.7843\times 10^{-1}$
    $(2.33\times 10^{-3}) -$
    10 $8.8077\times 10^{-1}$
    (8.46\times 10^{-3})$
    $8.4591\times 10^{-1}$
    $(3.70\times 10^{-2}) -$
    $7.6098\times 10^{-1}$
    $(7.42\times 10^{-2}) -$
    $6.1624\times 10^{-1}$
    $(1.02\times 10^{-1}) -$
    $6.3967\times 10^{-1}$
    $(2.31\times 10^{-2}) -$
    $7.9730\times 10^{-1}$
    $(6.81\times 10^{-2}) -$
    $8.9025\times 10^{-1}$
    $(1.77\times 10^{-2}) \approx$
    15 $9.1659\times 10^{-1}$
    $(1.34\times 10^{-2})$
    $8.3508\times 10^{-1}$
    $(4.47\times 10^{-2}) -$
    $4.6196\times 10^{-1}$
    $(1.35\times 10^{-1}) -$
    $7.6426\times 10^{-1}$
    $(1.77\times 10^{-1}) -$
    $3.3899\times 10^{-1}$
    $(4.96\times 10^{-2}) -$
    $7.9415\times 10^{-1}$
    $(1.18\times 10^{-1}) -$
    $7.6663\times 10^{-1}$
    $(9.90\times 10^{-2}) -$
    WFG 9 3 $5.4672\times 10^{-1}$
    $(4.01\times 10^{-3})$
    $5.4305\times 10^{-1}$
    $(4.39\times 10^{-3}) -$
    $5.4556\times 10^{-1}$
    $(4.15\times 10^{-3}) \approx$
    $5.3250\times 10^{-1}$
    $(7.93\times 10^{-3}) -$
    $5.1727\times 10^{-1}$
    $(6.14\times 10^{-3}) -$
    $5.3388\times 10^{-1}$
    $(3.57\times 10^{-2}) -$
    $5.5169\times 10^{-1}$
    $(2.77\times 10^{-3}) +$
    5 $7.4690\times 10^{-1}$
    $(7.37\times 10^{-3})$
    $7.2964\times 10^{-1}$
    $(3.46\times 10^{-2}) -$
    $7.4974\times 10^{-1}$
    $(1.09\times 10^{-2}) \approx$
    $7.0129\times 10^{-1}$
    $(2.16\times 10^{-2}) -$
    $5.6438\times 10^{-1}$
    $(7.52\times 10^{-2}) -$
    $7.5745\times 10^{-1}$
    $(8.38\times 10^{-3}) +$
    $7.6119\times 10^{-1}$
    $(9.23\times 10^{-3}) +$
    10 $9.1417\times 10^{-1}$
    (6.91\times 10^{-2})$
    $8.8646\times 10^{-1}$
    $(5.24\times 10^{-2}) -$
    $8.6481\times 10^{-1}$
    $(5.55\times 10^{-2}) -$
    $7.2445\times 10^{-1}$
    $(3.42\times 10^{-2}) -$
    $7.8885\times 10^{-1}$
    $(2.14\times 10^{-2}) -$
    $9.2002\times 10^{-1}$
    $(3.54\times 10^{-2}) +$
    $9.0227\times 10^{-1}$
    $(3.79\times 10^{-2}) -$
    15 $9.1357\times 10^{-1}$
    $(2.00\times 10^{-2})$
    $8.8219\times 10^{-1}$
    $(7.37\times 10^{-2}) -$
    $7.4786\times 10^{-1}$
    $(6.55\times 10^{-2}) -$
    $6.3510\times 10^{-1}$
    $(1.21\times 10^{-1}) -$
    $1.5569\times 10^{-1}$
    $(5.20\times 10^{-2}) -$
    $8.5783\times 10^{-1}$
    $(6.48\times 10^{-2}) -$
    $8.3130\times 10^{-1}$
    $(6.30\times 10^{-2}) -$
    $+$ / $\approx$ / $-$ 3 / 13 / 20 2 / 8 / 26 0 / 2 / 34 1 / 6 / 29 2 / 1 / 33 8 / 9 / 19
    $+$, $\approx$和$-$分别表示获得的结果与R2-RVEA相比更好, 相似和更差.
    下载: 导出CSV

    表  5  R2-RVEA、NSGA-Ⅲ、RVEA、MOEA/DD、MOMBI-Ⅱ、KnEA和TS-R2EA在WFG1-WFG9上获得的HV值的统计结果(均值和标准差). 最好的结果已突出

    Table  5  The statistical results (mean and standard deviation) of the IGD + values obtained by R2-RVEA, NSGA-Ⅲ, RVEA, MOEA/DD, MOMBI-Ⅱ, KnEA and TS-R2EA on WFG1 to WFG9. The best results are highlighted

    问题 $M$ R2-RVEA NSGA-Ⅲ RVEA MOEA/DD MOMBI-Ⅱ KnEA TS-R2EA
    WFG 1 3 $7.5115\times 10^{-2}$
    $(4.52\times 10^{-2})$
    $8.7381\times 10^{-2}$
    $(5.15\times 10^{-2}) -$
    $6.9562\times 10^{-2}$
    $(6.53\times 10^{-3}) +$
    $1.0065\times 10^{-1}$
    $(6.77\times 10^{-2}) -$
    $1.4254\times 10^{-1}$
    $(8.81\times 10^{-2}) -$
    $5.6061\times 10^{-1}$
    $(2.31\times 10^{-1}) -$
    $7.8279\times 10^{-2}$
    $(5.68\times 10^{-2}) -$
    5 $1.8626\times 10^{-1}$
    $(9.57\times 10^{-3})$
    $2.1152\times 10^{-1}$
    $(4.66\times 10^{-2}) -$
    $2.0426\times 10^{-1}$
    $(6.98\times 10^{-2}) -$
    $1.6063\times 10^{-1}$
    $(5.98\times 10^{-2}) +$
    $5.4411\times 10^{-1}$
    $(2.28\times 10^{-1}) -$
    $1.6029\times 10^{-1}$
    $(3.40\times 10^{-2}) +$
    $2.0587\times 10^{-1}$
    $(3.98\times 10^{-2}) -$
    10 $2.1900\times 10^{-1}$
    $(9.07\times 10^{-3})$
    $2.1486\times 10^{-1}$
    $(3.68\times 10^{-2}) +$
    $1.4214\times 10^{-1}$
    $(6.67\times 10^{-2}) +$
    $1.4874\times 10^{-1}$
    $(4.57\times 10^{-2}) +$
    $3.4832\times 10^{-1}$
    $(2.62\times 10^{-2}) -$
    $2.7665\times 10^{-1}$
    $(2.23\times 10^{-2}) -$
    $1.2162\times 10^{-1}$
    $(2.67\times 10^{-2}) +$
    15 $2.8499\times 10^{-1}$
    $(3.65\times 10^{-2})$
    $4.2663\times 10^{-1}$
    $(1.78\times 10^{-1}) -$
    $3.6853\times 10^{-1}$
    $(2.46\times 10^{-1}) \approx$
    $2.7065\times 10^{-1}$
    $(1.77\times 10^{-1}) \approx$
    $4.1737\times 10^{0}$
    $(5.59\times 10^{0}) -$
    $4.8136\times 10^{-1}$
    $(1.63\times 10^{-1}) -$
    $2.5708\times 10^{-1}$
    $(1.53\times 10^{-1}) \approx$
    WFG 2 3 $8.6719\times 10^{-2}$
    $(3.01\times 10^{-2})$
    $7.6168\times 10^{-2}$
    $(9.03\times 10^{-3}) +$
    $1.0187\times 10^{-1}$
    $(2.00\times 10^{-2}) -$
    $8.9857\times 10^{-2}$
    $(1.30\times 10^{-2}) -$
    $9.6736\times 10^{-2}$
    $(1.76\times 10^{-2}) -$
    $1.0500\times 10^{-1}$
    $(4.39\times 10^{-2}) -$
    $7.6067\times 10^{-2}$
    $(1.80\times 10^{-2}) +$
    5 $1.8338\times 10^{-1}$
    $(1.20\times 10^{-2})$
    $1.9276\times 10^{-1}$
    $(1.34\times 10^{-2}) -$
    $1.8425\times 10^{-1}$
    $(4.47\times 10^{-2}) \approx$
    $2.8035\times 10^{-1}$
    $(2.30\times 10^{-2}) -$
    $2.7826\times 10^{-1}$
    $(7.03\times 10^{-2}) -$
    $1.3904\times 10^{-1}$
    $(2.72\times 10^{-2}) +$
    $2.0231\times 10^{-1}$
    $(4.36\times 10^{-2}) -$
    10 $8.9305\times 10^{-2}$
    $(1.58\times 10^{-2})$
    $4.1897\times 10^{-1}$
    $(1.87\times 10^{-1}) -$
    $2.3948\times 10^{-1}$
    $(6.82\times 10^{-2}) -$
    $3.1155\times 10^{-1}$
    $(2.39\times 10^{-2}) -$
    $1.7810\times 10^{-1}$
    $(2.42\times 10^{-1}) -$
    $1.7637\times 10^{-1}$
    $(3.53\times 10^{-2}) -$
    $1.3011\times 10^{-1}$
    $(5.09\times 10^{-2}) -$
    15 $2.1600\times 10^{-1}$
    $(5.32\times 10^{-1})$
    $7.5861\times 10^{-1}$
    $(1.48\times 10^{-1}) -$
    $5.0074\times 10^{-1}$
    $(1.80\times 10^{-1}) -$
    $4.2257\times 10^{-1}$
    $(1.08\times 10^{-1}) -$
    $4.3099\times 10^{0}$
    $(4.12\times 10^{0}) -$
    $8.7000\times 10^{-1}$
    $(3.55\times 10^{-1}) -$
    $2.8770\times 10^{-1}$
    $(1.22\times 10^{-1}) -$
    WFG 3 3 $4.2032\times 10^{-2}$
    $(5.04\times 10^{-3})$
    $6.1249\times 10^{-2}$
    $(7.76\times 10^{-3}) -$
    $1.5612\times 10^{-1}$
    $(1.82\times 10^{-2}) -$
    $1.6785\times 10^{-1}$
    $(3.80\times 10^{-2}) -$
    $6.4076\times 10^{-2}$
    $(1.72\times 10^{-2}) -$
    $2.7852\times 10^{-2}$
    $(1.39\times 10^{-2}) +$
    $4.8933\times 10^{-2}$
    $(8.39\times 10^{-3}) -$
    5 $2.8475\times 10^{-1}$
    $(1.79\times 10^{-2})$
    $3.5253\times 10^{-1}$
    $(3.53\times 10^{-2}) -$
    $4.6344\times 10^{-1}$
    $(7.97\times 10^{-2}) -$
    $4.7518\times 10^{-1}$
    $(5.46\times 10^{-2}) -$
    $9.7449\times 10^{-1}$
    $(1.26\times 10^{-1}) -$
    $3.5587\times 10^{-1}$
    $(8.30\times 10^{-2}) -$
    $3.6494\times 10^{-1}$
    $(3.92\times 10^{-2}) -$
    10 $6.0947\times 10^{0}$
    $(1.11\times 10^{0})$
    $1.3173\times 10^{0}$
    $(3.71\times 10^{-1}) +$
    $2.7301\times 10^{0}$
    $(1.39\times 10^{-1}) +$
    $2.1179\times 10^{0}$
    $(5.86\times 10^{-2}) +$
    $9.6735\times 10^{0}$
    $(4.62\times 10^{-2}) -$
    $1.2618\times 10^{0}$
    $(1.71\times 10^{-1}) +$
    $1.3693\times 10^{0}$
    $(2.64\times 10^{-1}) +$
    15 $9.9835\times 10^{0}$
    $(3.20\times 10^{0})$
    $2.3745\times 10^{0}$
    $(7.30\times 10^{-1}) +$
    $3.7988\times 10^{0}$
    $(1.10\times 10^{-1}) +$
    $3.6219\times 10^{0}$
    $(2.39\times 10^{-2}) +$
    $1.4650\times 10^{1}$
    $(4.10\times 10^{-2}) -$
    $3.4151\times 10^{0}$
    $(5.06\times 10^{-1}) +$
    $2.9664\times 10^{0}$
    $(3.46\times 10^{-1}) +$
    WFG 4 3 $7.0960\times 10^{-1}$
    $(2.95\times 10^{-4}) +$
    $7.1039\times 10^{-2}$
    $(2.66\times 10^{-4}) \approx$
    $7.3769\times 10^{-2}$
    $(9.55\times 10^{-4}) -$
    $8.7146\times 10^{-2}$
    $(1.36\times 10^{-3}) -$
    $7.3561\times 10^{-2}$
    $(6.52\times 10^{-3}) -$
    $9.2246\times 10^{-2}$
    $(3.64\times 10^{-3}) -$
    $7.1212\times 10^{-2}$
    $(5.81\times 10^{-4}) -$
    5 $3.2825\times 10^{-1}$
    $(5.42\times 10^{-4})$
    $3.2857\times 10^{-1}$
    $(9.93\times 10^{-4}) \approx$
    $3.3068\times 10^{-1}$
    $(1.33\times 10^{-3}) -$
    $3.9675\times 10^{-1}$
    $(9.57\times 10^{-4}) -$
    $3.3731\times 10^{-1}$
    $(1.57\times 10^{-2}) -$
    $3.6449\times 10^{-1}$
    $(6.75\times 10^{-3}) -$
    $3.2845\times 10^{-1}$
    $(3.57\times 10^{-3}) \approx$
    10 $8.7337\times 10^{-1}$
    $(2.69\times 10^{-3})$
    $1.0667\times 10^{0}$
    $(5.15\times 10^{-1}) -$
    $8.9065\times 10^{-1}$
    $(9.51\times 10^{-3}) -$
    $1.1221\times 10^{0}$
    $(8.06\times 10^{-3}) -$
    $2.3431\times 10^{0}$
    $(1.47\times 10^{0}) -$
    $1.0544\times 10^{0}$
    $(8.22\times 10^{-3}) -$
    $8.7977\times 10^{-1}$
    $(8.85\times 10^{-3}) -$
    15 $1.2733\times 10^{0}$
    $(2.41\times 10^{-3})$
    $3.0661\times 10^{0}$
    $(1.63\times 10^{0}) -$
    $1.3006\times 10^{0}$
    $(1.64\times 10^{-1}) \approx$
    $1.6260\times 10^{0}$
    $(6.21\times 10^{-1}) -$
    $1.7552\times 10^{1}$
    $(2.32\times 10^{0}) -$
    $1.3759\times 10^{0}$
    $(2.11\times 10^{-2}) -$
    $1.3775\times 10^{0}$
    $(4.21\times 10^{-1}) -$
    WFG 5 3 $1.2755\times 10^{-1}$
    $(4.48\times 10^{-6})$
    $1.2755\times 10^{-1}$
    $(6.82\times 10^{-6}) \approx$
    $1.2881\times 10^{-1}$
    $(4.47\times 10^{-4}) -$
    $1.4202\times 10^{-1}$
    $(1.02\times 10^{-3}) -$
    $1.3745\times 10^{-1}$
    $(1.38\times 10^{-3}) -$
    $1.9609\times 10^{-1}$
    $(6.08\times 10^{-2}) -$
    $1.2741\times 10^{-1}$
    $(3.43\times 10^{-4}) \approx$
    5 $3.8901\times 10^{-1}$
    $(2.61\times 10^{-5})$
    $3.8902\times 10^{-1}$
    $(4.13\times 10^{-5}) \approx$
    $3.8944\times 10^{-1}$
    $(1.32\times 10^{-4}) -$
    $4.5522\times 10^{-1}$
    $(4.11\times 10^{-4}) -$
    $4.0177\times 10^{-1}$
    $(1.22\times 10^{-2}) -$
    $4.2985\times 10^{-1}$
    $(8.65\times 10^{-3}) -$
    $3.8747\times 10^{-1}$
    $(3.84\times 10^{-3}) \approx$
    10 $9.3301\times 10^{-1}$
    $(9.11\times 10^{-4})$
    $9.3354\times 10^{-1}$
    $(1.06\times 10^{-3}) \approx$
    $9.4055\times 10^{-1}$
    $(4.40\times 10^{-3}) -$
    $1.1723\times 10^{0}$
    $(3.55\times 10^{-3}) -$
    $1.2453\times 10^{0}$
    $(2.17\times 10^{-2}) -$
    $1.0843\times 10^{0}$
    $(2.67\times 10^{-2}) -$
    $9.5531\times 10^{-1}$
    $(5.26\times 10^{-3}) -$
    15 $1.3309\times 10^{0}$
    $(1.47\times 10^{-3})$
    $2.3724\times 10^{0}$
    $(2.25\times 10^{0}) -$
    $1.3259\times 10^{0}$
    $(7.86\times 10^{-3}) +$
    $1.5056\times 10^{0}$
    $(9.76\times 10^{-3}) -$
    $2.3006\times 10^{1}$
    $(2.02\times 10^{0}) -$
    $1.4335\times 10^{0}$
    $(2.73\times 10^{-2}) -$
    $1.3346\times 10^{0}$
    $(2.11\times 10^{-3}) -$
    WFG 6 3 $1.4696\times 10^{-1}$
    $(1.96\times 10^{-2})$
    $1.3922\times 10^{-1}$
    $(2.02\times 10^{-2}) \approx$
    $1.5111\times 10^{-1}$
    $(1.85\times 10^{-2}) \approx$
    $1.6034\times 10^{-1}$
    $(1.84\times 10^{-2}) -$
    $1.4684\times 10^{-1}$
    $(2.31\times 10^{-2}) \approx$
    $1.9074\times 10^{-1}$
    $(1.64\times 10^{-2}) -$
    $1.4918\times 10^{-1}$
    $(2.14\times 10^{-2}) \approx$
    5 $4.1833\times 10^{-1}$
    $(2.17\times 10^{-2})$
    $4.1223\times 10^{-1}$
    $(2.09\times 10^{-2}) \approx$
    $4.1516\times 10^{-1}$
    $(2.48\times 10^{-2}) \approx$
    $4.8181\times 10^{-1}$
    $(2.89\times 10^{-2}) -$
    $4.2618\times 10^{-1}$
    $(5.46\times 10^{-2}) \approx$
    $5.0750\times 10^{-1}$
    $(2.91\times 10^{-2}) -$
    $4.0415\times 10^{-1}$
    $(2.16\times 10^{-2}) \approx$
    10 $9.5617\times 10^{-1}$
    $(1.65\times 10^{-2})$
    $9.6279\times 10^{-1}$
    $(2.14\times 10^{-2}) \approx$
    $9.9145\times 10^{-1}$
    $(3.18\times 10^{-2}) -$
    $1.2157\times 10^{0}$
    $(2.50\times 10^{-2}) -$
    $9.5786\times 10^{-1}$
    $(3.29\times 10^{-2}) \approx$
    $1.1230\times 10^{0}$
    $(3.04\times 10^{-2}) -$
    $9.7574\times 10^{-1}$
    $(2.37\times 10^{-2}) -$
    15 $1.3522\times 10^{0}$
    $(1.58\times 10^{-2})$
    $1.6632\times 10^{0}$
    $(1.10\times 10^{-1}) -$
    $1.4286\times 10^{0}$
    $(4.27\times 10^{-2}) -$
    $1.5291\times 10^{0}$
    $(2.96\times 10^{-2}) -$
    $1.5286\times 10^{1}$
    $(3.49\times 10^{0}) -$
    $1.5321\times 10^{0}$
    $(5.28\times 10^{-2}) -$
    $1.3721\times 10^{0}$
    $(3.17\times 10^{-2}) -$
    WFG 7 3 $7.1361\times 10^{-2}$
    $(3.23\times 10^{-4})$
    $7.1371\times 10^{-2}$
    $(3.07\times 10^{-4}) \approx$
    $7.3630\times 10^{-2}$
    $(5.88\times 10^{-4}) -$
    $8.9150\times 10^{-2}$
    $(1.87\times 10^{-3}) -$
    $7.3845\times 10^{-2}$
    $(8.23\times 10^{-3}) -$
    $3.2485\times 10^{-1}$
    $(1.88\times 10^{-2}) -$
    $7.0651\times 10^{-2}$
    $(7.82\times 10^{-5}) +$
    5 $3.3002\times 10^{-1}$
    $(6.74\times 10^{-4})$
    $3.3105\times 10^{-1}$
    $(1.15\times 10^{-3}) -$
    $3.3337\times 10^{-1}$
    $(1.46\times 10^{-3}) -$
    $4.0782\times 10^{-1}$
    $(9.79\times 10^{-3}) -$
    $3.2920\times 10^{-1}$
    $(3.76\times 10^{-3}) +$
    $3.8174\times 10^{-1}$
    $(1.44\times 10^{-2})$
    $3.2722\times 10^{-1}$
    $(4.37\times 10^{-4}) +$
    10 $8.8858\times 10^{-1}$
    $(4.99\times 10^{-3})$
    $9.3332\times 10^{-1}$
    $(1.58\times 10^{-1}) \approx$
    $8.9287\times 10^{-1}$
    $(4.96\times 10^{-3}) -$
    $1.1039\times 10^{0}$
    $(1.27\times 10^{-2}) -$
    $9.8755\times 10^{-1}$
    $(3.48\times 10^{-1}) -$
    $9.6895\times 10^{-1}$
    $(1.47\times 10^{-2}) -$
    $8.9890\times 10^{-1}$
    $(1.10\times 10^{-2}) -$
    15 $1.2798\times 10^{0}$
    $(7.32\times 10^{-3})$
    $2.6475\times 10^{0}$
    $(7.87\times 10^{-1}) -$
    $2.6579\times 10^{0}$
    $(6.81\times 10^{-1}) -$
    $1.6730\times 10^{0}$
    $(5.69\times 10^{-1}) -$
    $1.3576\times 10^{0}$
    $(2.40\times 10^{0}) -$
    $1.5766\times 10^{0}$
    $(5.83\times 10^{-1}) -$
    $1.6838\times 10^{0}$
    $(6.24\times 10^{-1}) -$
    WFG 8 3 $2.0276\times 10^{-1}$
    $(2.38\times 10^{-3})$
    $2.0043\times 10^{-1}$
    $(5.27\times 10^{-3}) +$
    $2.1262\times 10^{-1}$
    $(2.76\times 10^{-3}) -$
    $2.2068\times 10^{-1}$
    $(8.51\times 10^{-3}) -$
    $2.2094\times 10^{-1}$
    $(3.61\times 10^{-3}) -$
    $2.2380\times 10^{-1}$
    $(4.90\times 10^{-3}) -$
    $1.9590\times 10^{-1}$
    $(3.10\times 10^{-3}) +$
    5 $6.0819\times 10^{-1}$
    $(1.75\times 10^{-3})$
    $6.0468\times 10^{-1}$
    $(1.31\times 10^{-3}) +$
    $6.1524\times 10^{-1}$
    $(9.59\times 10^{-3}) -$
    $6.4945\times 10^{-1}$
    $(1.99\times 10^{-2}) -$
    $1.1632\times 10^{0}$
    $(3.44\times 10^{-2}) -$
    $6.9589\times 10^{-1}$
    $(2.50\times 10^{-2}) -$
    $6.1227\times 10^{-1}$
    $(2.15\times 10^{-3}) -$
    10 $1.7814\times 10^{0}$
    (2.24\times 10^{-1})$
    $3.1854\times 10^{0}$
    $(1.15\times 10^{0}) -$
    $1.2981\times 10^{0}$
    $(2.89\times 10^{-1}) +$
    $1.3753\times 10^{0}$
    $(1.37\times 10^{-1}) +$
    $3.7183\times 10^{0}$
    $(1.07\times 10^{0}) -$
    $1.4339\times 10^{0}$
    $(2.03\times 10^{-1}) +$
    $1.5086\times 10^{0}$
    $(3.02\times 10^{-1}) \approx$
    15 $4.1237\times 10^{0}$
    $(1.62\times 10^{0})$
    $8.8664\times 10^{0}$
    $(1.08\times 10^{0}) -$
    $2.7442\times 10^{0}$
    $(1.99\times 10^{0}) +$
    $3.6008\times 10^{0}$
    $(1.27\times 10^{0}) \approx$
    $1.6919\times 10^{1}$
    $(2.15\times 10^{0}) -$
    $2.2133\times 10^{0}$
    $(6.49\times 10^{-1}) +$
    $2.3419\times 10^{0}$
    $(1.23\times 10^{0}) +$
    WFG 9 3 $9.1371\times 10^{-2}$
    $(8.31\times 10^{-3})$
    $9.8060\times 10^{-2}$
    $(7.55\times 10^{-3}) -$
    $9.4218\times 10^{-2}$
    $(5.27\times 10^{-3}) -$
    $1.1083\times 10^{-1}$
    $(7.33\times 10^{-3}) -$
    $1.2051\times 10^{-1}$
    $(7.64\times 10^{-3}) -$
    $4.1313\times 10^{-1}$
    $(5.50\times 10^{-2}) -$
    $8.5058\times 10^{-2}$
    $(3.66\times 10^{-3}) +$
    5 $4.0014\times 10^{-1}$
    $(3.90\times 10^{-2})$
    $4.1637\times 10^{-1}$
    $(4.75\times 10^{-2}) -$
    $3.7828\times 10^{-1}$
    $(1.29\times 10^{-2}) +$
    $4.8038\times 10^{-1}$
    $(3.90\times 10^{-2}) -$
    $7.4087\times 10^{-1}$
    $(1.12\times 10^{-1}) -$
    $3.8673\times 10^{-1}$
    $(1.23\times 10^{-2}) \approx$
    $3.6848\times 10^{-1}$
    $(1.24\times 10^{-2}) +$
    10 $1.0158\times 10^{0}$
    (2.47\times 10^{-2})$
    $1.4614\times 10^{0}$
    $(7.52\times 10^{-1}) -$
    $9.7817\times 10^{-1}$
    $(3.09\times 10^{-2}) +$
    $1.2487\times 10^{0}$
    $(6.41\times 10^{-2}) -$
    $1.8261\times 10^{0}$
    $(4.88\times 10^{-1}) -$
    $1.0663\times 10^{0}$
    $(2.44\times 10^{-2}) -$
    $1.0275\times 10^{0}$
    $(6.41\times 10^{-2}) \approx$
    15 $1.6670\times 10^{0}$
    (1.67\times 10^{-1})$
    $5.6055\times 10^{0}$
    $(1.89\times 10^{0}) -$
    $1.5770\times 10^{0}$
    $(3.10\times 10^{-1}) +$
    $2.0273\times 10^{0}$
    $(6.11\times 10^{-1}) -$
    $2.4264\times 10^{1}$
    $(1.08\times 10^{0}) -$
    $1.7242\times 10^{0}$
    $(3.23\times 10^{-1}) \approx$
    $1.6462\times 10^{0}$
    $(3.71\times 10^{-1}) \approx$
    $+$ / $\approx$ / $-$ 6 / 10 / 20 10 / 5 / 21 5 / 2 / 29 1 / 3 / 32 7 / 2 / 27 10 / 9 / 17
    +, $\approx$和$-$分别表示获得的结果与R2-RVEA相比更好, 相似和更差.
    下载: 导出CSV

    表  6  R2-RVEA与其他算法的测试对比

    Table  6  Comparison between R2-RVEA and other algorithms

    对比算法 对比指标
    HV IGD+
    NSGA-Ⅲ 6  20  38 11  17  36
    RVEA 10  17  37 18  13  33
    MOEA/DD 9  11  44 16  9  39
    MOMBI-Ⅱ 6  11  47 4  11  49
    KnEA 4  1  59 13  4  47
    TS-R2EA 17  16  31 21  15  28
    +  $\approx$  $-$ +  $\approx$  $-$
    下载: 导出CSV
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出版历程
  • 收稿日期:  2018-11-01
  • 录用日期:  2019-03-08
  • 刊出日期:  2021-11-18

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