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基于径向空间划分的昂贵多目标进化算法

顾清华,  周煜丰,  李学现,  阮顺领

文章导航 > 自动化学报  > 2022 >  48(10): 2564-2584
顾清华, 周煜丰, 李学现, 阮顺领. 基于径向空间划分的昂贵多目标进化算法. 自动化学报, 2022, 48(10): 2564−2584 doi: 10.16383/j.aas.c200791
引用本文: 顾清华, 周煜丰, 李学现, 阮顺领. 基于径向空间划分的昂贵多目标进化算法. 自动化学报, 2022, 48(10): 2564−2584 doi: 10.16383/j.aas.c200791
shu
Gu Qing-Hua, Zhou Yu-Feng, Li Xue-Xian, Ruan Shun-Ling. Expensive many-objective evolutionary algorithm based on radial space division. Acta Automatica Sinica, 2022, 48(10): 2564−2584 doi: 10.16383/j.aas.c200791
Citation: Gu Qing-Hua, Zhou Yu-Feng, Li Xue-Xian, Ruan Shun-Ling. Expensive many-objective evolutionary algorithm based on radial space division. Acta Automatica Sinica, 2022, 48(10): 2564−2584 doi: 10.16383/j.aas.c200791
shu

基于径向空间划分的昂贵多目标进化算法

doi: 10.16383/j.aas.c200791
  • 1.

    西安建筑科技大学管理学院 西安 710055

  • 2.

    西安建筑科技大学资源工程学院 西安 710055

基金项目: 国家自然科学基金(51774228, 51864046), 陕西省自然科学基金杰出青年项目(2020JC-44)资助
详细信息
    作者简介:

    顾清华：西安建筑科技大学教授. 主要研究方向为多目标优化, 车辆调度和复杂系统建模与仿真. 本文通信作者. E-mail: qinghuagu@126.com

    周煜丰：西安建筑科技大学硕士研究生. 主要研究方向为多目标优化和车辆调度. E-mail: zyf18215649083@163.com

    李学现：西安建筑科技大学博士研究生. 主要研究方向为群智能优化算法在采矿系统工程中的应用. E-mail: lixuexian2019@163.com

    阮顺领：西安建筑科技大学副教授. 主要研究方向为矿山智能系统和深度学习. E-mail: ruanshunling@163.com

Expensive Many-objective Evolutionary Algorithm Based on Radial Space Division

  • 1.

    School of Management, Xi＇an University of Architecture and Technology, Xi＇an 710055

  • 2.

    School of Resource Engineering, Xi＇an University of Architecture and Technology, Xi＇an 710055

Funds: Supported by National Natural Science Foundation of China (51774228, 51864046), and Outstanding Youth Project of Shaanxi Natural Science Foundation Grant (2020JC-44)
More Information
    Author Bio:

    GU Qing-Hua　Professor at Xi＇an University of Architecture and Te-chnology. His research interest covers multi-objective optimization, vehicle scheduling and complex system modeling and simulation. Corresponding author of this paper

    ZHOU Yu-Feng　Master student at Xi＇an University of Architecture and Technology. His research interest covers multi-objective optimization and vehicle scheduling

    LI Xue-Xian　Ph.D. candidate at Xi＇an University of Architecture and Technology. His main research interest is application of swarm intelligence optimization algorithm in mining system engineering

    RUAN Shun-Ling　Associate professor at Xi＇an University of Architecture and Technology. His rese-arch interest covers mine intelligent system and deep learning

    摘要
  • 摘要: 为了解决难以建立精确数学模型或者真实评估实验成本高昂的多目标优化问题, 提出了一种基于径向空间划分的昂贵多目标进化算法. 首先算法使用高斯回归作为代理模型逼近目标函数; 然后将目标空间的个体投影到径向空间, 结合目标空间和径向空间信息保留对种群贡献更高的个体; 之后由径向空间中个体的位置分布决定下一步应该选择哪些个体进行真实评估; 最后, 采用一种双档案管理策略维护代理模型的质量. 数值实验和现实问题上的结果表明, 与5种先进算法相比, 该算法在解决昂贵多目标优化问题时能够提供更高质量的解.
    Abstract: In order to solve the problem of many-objective optimization that it is difficult to establish an accurate mathematical model or the actual evaluation experiment is expensive, an expensive many-objective evolutionary algorithm based on radial space division is proposed. First, the algorithm uses Gaussian regression as a surrogate model to approximate the objective function; Second the individuals in the objective space are projected to the radial space, and the individuals with higher contributions to the population are retained through the objective space and radial space information; Third the position distribution of the individuals in the radial space determines which individuals should be selected for real evaluation in the next step; Finally, a double archives management strategy is adopted to maintain the quality of the surrogate model. The results of numerical experiments and real problems show that compared with five advanced algorithms, the algorithm proposed in this paper can provide higher quality solutions when solving expensive many-objective optimization problems.
    • HTML全文

  • 图  1  高斯过程(Kriging模型)图解

    Fig.  1  Gaussian process (Kriging model) description

    下载: 全尺寸图片 幻灯片

    图  2  径向空间中个体的位置分布情况

    Fig.  2  Individual location distribution in radial space

    下载: 全尺寸图片 幻灯片

    图  3  6种算法在求解3个目标DTLZ1问题过程中获得的最佳HV值对应的非支配解集

    Fig.  3  The non-dominated solution set corresponding to the best HV value obtained by the six algorithms in solving the three objective DTLZ1 problems

    下载: 全尺寸图片 幻灯片

    图  4  6种算法在10个目标DTLZ2测试问题上最佳解集的平行坐标图和径向坐标图

    Fig.  4  Parallel coordinate diagram and radial coordinate diagram of the best solution set of the six algorithms on 10 objective DTLZ2 test problems

    下载: 全尺寸图片 幻灯片

    图  5  K-RSEA和5种对比算法在求解3个和10个目标DTZL4问题时的IGD和HV变化

    Fig.  5  The IGD and HV changes of K-RSEA and five comparison algorithms when solving 3 and 10 objective DTZL4 problems

    下载: 全尺寸图片 幻灯片

    图  6  K-RSEA和5种对比算法在求解3个目标DTZL5问题时的IGD和HV变化

    Fig.  6  The IGD and HV changes of K-RSEA and five comparison algorithms when solving 3 objective DTZL5 problems

    下载: 全尺寸图片 幻灯片

    图  7  K-RSEA和5种对比算法求解DTLZ7问题过程中获得的最佳HV值对应的非支配前沿

    Fig.  7  The non-dominant frontier corresponding to the best HV value obtained in the process of solving the DTLZ7 problem by K-RSEA and five comparison algorithms

    下载: 全尺寸图片 幻灯片

    图  8  K-RSEA和5种对比算法求解WFG2问题过程中获得的最佳HV值对应的非支配前沿

    Fig.  8  The non-dominant frontier corresponding to the best HV value obtained in the process of solving the WFG2 problem with K-RSEA and five comparison algorithms

    下载: 全尺寸图片 幻灯片

    图  9  K-RSEA和5种对比算法在求解3个目标WFG5、WFG6和WFG8问题时的IGD变化

    Fig.  9  The IGD changes of K-RSEA and five comparison algorithms when solving three objective WFG5, WFG6 and WFG8 problems

    下载: 全尺寸图片 幻灯片

    图  10  K-RSEA和5种对比算法在求解10个目标WFG5、WFG6和WFG8问题时的HV变化

    Fig.  10  The HV changes of K-RSEA and five comparison algorithms when solving 10 objective WFG5, WFG6 and WFG8 problems

    下载: 全尺寸图片 幻灯片

    图  11  6种算法在10个目标WFG7测试问题上最佳解集的平行坐标图和径向坐标图

    Fig.  11  Parallel coordinate diagram and radial coordinate diagram of the best solution set of the six algorithms on 10 objective WFG7 test problems

    下载: 全尺寸图片 幻灯片

    图  12  6种算法分别在不同问题上的运行时间比较

    Fig.  12  Comparison of the running time of the six algorithms on different problems

    下载: 全尺寸图片 幻灯片

    图  13  算法在汽车耐撞性优化问题上的非支配解

    Fig.  13  The non-dominant solution of the algorithm in the optimization of automobile crash-worthiness

    下载: 全尺寸图片 幻灯片

    表  1  测试问题及特征

    Table  1  Test problems and their features

    问题特征
    DTLZ1, 3多模态、DTLZ1 线性
    DTLZ2, 4 ~ 6凹、DTLZ4 有偏好、DTLZ5 退化、
    DTLZ6 退化且有偏好
    DTLZ7混合、不连续、多模态
    WFG1凸的、混合有偏好
    WFG2凸的、不连续
    WFG3线性、退化
    WFG4 ~ 9凹的、WFG4 多模态、WFG5 具有欺骗性、
    WFG6 不可分、WFG7 有偏好、WFG8 不可分
    且有偏好、WFG9 多模态、有偏好
    下载: 导出CSV

    表  2  6种算法在不同维数的DTLZ测试问题上获得的IGD平均值和标准差

    Table  2  The IGD average and standard deviation obtained by the six algorithms on DTLZ test problems of different dimensions

    测试问题目标数NSGA-IIICPS-MOEACSEAK-RVEAMOEA/D-EGOK-RSEA
    DTLZ139.9972 × 101
    (2.47 × 101) −    		8.2935 × 101
    (1.74 × 101) =    		5.6789 × 101
    (1.00 × 101) +    		8.2001 × 101
    (1.84 × 101) =    		8.2705 × 101
    (1.33 × 101) =    		8.6600 × 101
    (1.91 × 101)
    47.3063 × 101
    (1.63 × 101) −    		6.4843 × 101
    (1.64 × 101) =    		4.3597 × 101
    (1.34 × 101) +    		5.6857 × 101
    (1.25 × 101) =    		6.9011 × 101
    (1.39 × 101) =    		5.9990 × 101
    (1.41 × 101)
    63.3806 × 101
    (1.24 × 101) =    		3.1427 × 101
    (7.13 × 100) =    		1.5953 × 101
    (5.45 × 100) +    		2.7870 × 101
    (1.01 × 101) =    		3.3203 × 101
    (1.04 × 101) −    		2.6831 × 101
    (7.05 × 100)
    86.7072 × 100
    (3.99 × 100) =    		1.1629 × 101
    (2.84 × 100) −    		3.6585 × 100
    (2.21 × 100) +    		7.8298 × 100
    (2.68 × 100) =    		1.0297 × 101
    (4.54 × 100) =    		8.0032 × 100
    (3.95 × 100)
    105.0209 × 10−1
    (3.59 × 10−1) −    		4.7694 × 10−1
    (1.80 × 10−1) −    		2.8192 × 10−1
    (5.67 × 10−2) =    		3.7187 × 10−1
    (8.49 × 10−2) =    		4.3847 × 10−1
    (1.13 × 10−1) −    		3.3052 × 10−1
    (1.04 × 10−1)
    DTLZ233.2991 × 10−1
    (2.63 × 10−2) −    		3.2931 × 10−1
    (2.69 × 10−2) −    		2.1298 × 10−1
    (2.81 × 10−2) −    		1.1972 × 10−1
    (1.36 × 10−2) −    		3.4065 × 10−1
    (3.28 × 10−2) −    		1.1075 × 10−1
    (7.03 × 10−2)
    43.5643 × 10−1
    (2.82 × 10−2) −    		3.7663 × 10−1
    (2.18 × 10−2) −    		2.9615 × 10−1
    (2.61 × 10−2) −    		2.2144 × 10−1
    (1.80 × 10−2) =    		3.6787 × 10−1
    (2.61 × 10−2) −    		2.3347 × 10−1
    (4.97 × 10−2)
    64.8763 × 10−1
    (2.37 × 10−2) −    		4.9201 × 10−1
    (2.47 × 10−2) −    		4.2782 × 10−1
    (4.32 × 10−2) −    		3.6589 × 10−1
    (1.98 × 10−2) =    		4.7191 × 10−1
    (2.23 × 10−2) −    		3.6494 × 10−1
    (1.93 × 10−2)
    85.8827 × 10−1
    (3.63 × 10−2) −    		5.8008 × 10−1
    (2.61 × 10−2) −    		5.8141 × 10−1
    (3.39 × 10−2) −    		4.1673 × 10−1
    (1.44 × 10−2) +    		5.3642 × 10−1
    (2.73 × 10−2) −    		4.2547 × 10−1
    (1.04 × 10−2)
    106.3227 × 10−1
    (2.04 × 10−2) −    		6.3801 × 10−1
    (1.80 × 10−2) −    		6.7240 × 10−1
    (2.38 × 10−2) −    		5.0316 × 10−1
    (1.57 × 10−2) −    		5.2187 × 10−1
    (2.24 × 10−2) −    		4.7549 × 10−1
    (7.36 × 10−3)
    DTLZ332.8787 × 102
    (6.58 × 101) −    		2.3395 × 102
    (3.82 × 101) =    		1.5653 × 102
    (3.81 × 101) +    		2.3708 × 102
    (4.75 × 101) =    		1.9962 × 102
    (2.65 × 101) +    		2.3648 × 102
    (5.58 × 101)
    42.0889 × 102
    (6.54 × 101) =    		1.6267 × 102
    (4.26 × 101) =    		1.2297 × 102
    (2.83 × 101) +    		1.8558 × 102
    (3.48 × 101) =    		1.5411 × 102
    (1.28 × 101) =    		1.8023 × 102
    (5.61 × 101)
    61.0529 × 102
    (2.46 × 101) =    		9.4164 × 101
    (1.89 × 101) =    		5.6044 × 101
    (1.64 × 101) +    		8.4557 × 101
    (2.82 × 101) =    		9.6519 × 101
    (1.60 × 101) =    		8.9132 × 101
    (3.24 × 101)
    82.6642 × 101
    (9.61 × 100) =    		2.8860 × 101
    (1.24 × 101) =    		1.3883 × 101
    (5.25 × 100) +    		2.2607 × 101
    (8.99 × 100) +    		3.7525 × 101
    (1.23 × 101) =    		3.0443 × 101
    (1.15 × 101)
    101.5073 × 100
    (4.05 × 10−1) −    		1.5000 × 100
    (3.78 × 10−1) −    		1.0257 × 100
    (2.63 × 10−1) =    		1.2960 × 100
    (3.55 × 10−1) =    		1.2942 × 100
    (3.36 × 10−1) =    		1.1642 × 100
    (2.93 × 10−1)
    DTLZ437.2107 × 10−1
    (1.19 × 10−1) −    		5.9002 × 10−1
    (3.92 × 10−2) −    		5.1951 × 10−1
    (1.51 × 10−1) =    		3.0267 × 10−1
    (7.37 × 10−2) +    		5.9687 × 10−1
    (6.57 × 10−2) −    		4.8903 × 10−1
    (1.47 × 10−1)
    47.0404 × 10−1
    (1.25 × 10−1) −    		6.2009 × 10−1
    (4.24 × 10−2) −    		4.6042 × 10−1
    (7.33 × 10−2) +    		4.0021 × 10−1
    (8.08 × 10−2) +    		6.8696 × 10−1
    (4.77 × 10−2) −    		5.5991 × 10−1
    (1.28 × 10−1)
    68.0377 × 10−1
    (7.80 × 10−2) −    		6.5279 × 10−1
    (1.84 × 10−2) =    		4.9747 × 10−1
    (5.83 × 10−2) +    		4.8631 × 10−1
    (5.15 × 10−2) +    		6.8610 × 10−1
    (2.90 × 10−2) −    		6.2727 × 10−1
    (6.36 × 10−2)
    87.4087 × 10−1
    (4.38 × 10−2) −    		6.3356 × 10−1
    (1.32 × 10−2) −    		5.8324 × 10−1
    (3.16 × 10−2) =    		5.5700 × 10−1
    (2.98 × 10−2) =    		6.5251 × 10−1
    (1.32 × 10−2) −    		5.7685 × 10−1
    (4.08 × 10−2)
    107.3581 × 10−1
    (4.34 × 10−2) −    		6.5510 × 10−1
    (1.01 × 10−2) −    		6.3597 × 10−1
    (3.28 × 10−2) −    		5.9740 × 10−1
    (2.69 × 10−2) =    		6.4268 × 10−1
    (9.34 × 10−3) −    		5.8861 × 10−1
    (2.14 × 10−2)
    DTLZ532.5926 × 10−1
    (3.51 × 10−2) −    		2.4959 × 10−1
    (2.59 × 10−2) −    		1.1067 × 10−1
    (2.85 × 10−2) −    		8.0805 × 10−2
    (2.49 × 10−2) −    		2.4856 × 10−1
    (2.24 × 10−2) −    		6.5513 × 10−2
    (4.58 × 10−2)
    41.9155 × 10−1
    (2.36 × 10−2) −    		2.0562 × 10−1
    (2.30 × 10−2) −    		1.2544 × 10−1
    (2.99 × 10−2) −    		5.9826 × 10−2
    (9.29 × 10−3) −    		2.1704 × 10−1
    (2.73 × 10−2) −    		2.6100 × 10−2
    (1.00 × 10−2)
    61.4568 × 10−1
    (2.39 × 10−2) −    		1.2304 × 10−1
    (1.85 × 10−2) −    		7.4651 × 10−2
    (2.05 × 10−2) −    		3.4219 × 10−2
    (1.06 × 10−2) −    		1.5500 × 10−1
    (1.93 × 10−2) −    		1.6379 × 10−2
    (1.12 × 10−2)
    88.7377 × 10−2
    (1.55 × 10−2) −    		6.5776 × 10−2
    (1.34 × 10−2) −    		3.8178 × 10−2
    (8.52 × 10−3) −    		2.0890 × 10−2
    (5.60 × 10−3) −    		8.2292 × 10−2
    (1.25 × 10−2) −    		1.2091 × 10−2
    (2.70 × 10−3)
    104.7648 × 10−2
    (1.45 × 10−2) −    		2.5322 × 10−2
    (4.41 × 10−3) −    		1.1891 × 10−2
    (1.17 × 10−3) −    		1.2745 × 10−2
    (2.29 × 10−3) −    		2.2116 × 10−2
    (2.45 × 10−3) −    		7.4301 × 10−3
    (1.10 × 10−3)
    DTLZ636.1232 × 100
    (2.01 × 10−1) −    		4.1051 × 100
    (4.45 × 10−1) −    		4.9049 × 100
    (6.04 × 10−1) −    		3.1198 × 100
    (3.59 × 10−1) −    		1.8762 × 100
    (5.59 × 10−1) =    		2.0558 × 100
    (4.60 × 10−1)
    45.4855 × 100
    (2.45 × 10−1) −    		3.4387 × 100
    (4.98 × 10−1) −    		4.9792 × 100
    (5.02 × 10−1) −    		2.4647 × 100
    (3.55 × 10−1) −    		1.6752 × 100
    (7.38 × 10−1) =    		2.0560 × 100
    (3.25 × 10−1)
    63.8995 × 100
    (2.21 × 10−1) −    		2.3140 × 100
    (5.21 × 10−1) −    		3.1061 × 100
    (5.07 × 10−1) −    		1.2890 × 100
    (3.22 × 10−1) =    		9.2648 × 10−1
    (3.65 × 10−1) +    		1.2599 × 100
    (3.56 × 10−1)
    82.1839 × 100
    (2.82 × 10−1) −    		9.0259 × 10−1
    (2.50 × 10−1) −    		1.4584 × 100
    (4.60 × 10−1) −    		5.3505 × 10−1
    (1.82 × 10−1) =    		5.1243 × 10−1
    (2.60 × 10−1) =    		5.9215 × 10−1
    (2.14 × 10−1)
    105.9508 × 10−1
    (2.62 × 10−1) −    		5.2061 × 10−2
    (1.55 × 10−2) +    		1.3300 × 10−1
    (9.60 × 10−2) =    		7.1410 × 10−2
    (2.06 × 10−2) =    		1.9708 × 10−1
    (7.72 × 10−2) −    		8.4350 × 10−2
    (2.65 × 10−2)
    DTLZ735.7028 × 100
    (8.46 × 10−1) −    		4.9515 × 100
    (7.40 × 10−1) −    		1.7040 × 100
    (5.07 × 10−1) −    		1.4314 × 10−1
    (4.87 × 10−2) −    		2.3750 × 10−1
    (9.67 × 10−2) −    		9.0290 × 10−2
    (6.53 × 10−2)
    47.1332 × 100
    (8.88 × 10−1) −    		5.3360 × 100
    (1.43 × 100) −    		2.6700 × 100
    (9.51 × 10−1) −    		3.7612 × 10−1
    (1.37 × 10−1) =    		5.3337 × 10−1
    (9.02 × 10−2) −    		3.3295 × 10−1
    (1.07 × 10−1)
    68.6385 × 100
    (2.00 × 100) −    		6.2498 × 100
    (1.86 × 100) −    		4.5051 × 100
    (8.80 × 10−1) −    		6.3454 × 10−1
    (8.41 × 10−2) +    		8.6377 × 10−1
    (6.08 × 10−2) +    		1.0134 × 100
    (1.85 × 10−1)
    81.0623 × 101
    (2.69 × 100) −    		4.5094 × 100
    (3.09 × 100) −    		6.1099 × 100
    (1.99 × 100) −    		8.7425 × 10−1
    (6.82 × 10−2) +    		1.0618 × 100
    (3.82 × 10−2) +    		2.2654 × 100
    (4.25 × 10−1)
    103.8760 × 100
    (1.81 × 100) −    		1.5793 × 100
    (9.05 × 10−2) +    		2.0827 × 100
    (5.27 × 10−1) =    		1.0910 × 100
    (4.35 × 10−2) +    		1.2142 × 100
    (2.27 × 10−2) +    		2.1206 × 100
    (3.77 × 10−1)
    +/−/=0/30/52/25/810/19/68/10/175/20/10
    下载: 导出CSV

    表  3  6种算法在不同维数的DTLZ测试问题上获得的HV平均值和标准差

    Table  3  The HV average and standard deviation obtained by the six algorithms on DTLZ test problems of different dimensions

    测试问题目标数NSGA-IIICPS-MOEACSEAK-RVEAMOEA/D-EGOK-RSEA
    DTLZ130.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100)
    40.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100)
    60.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100)
    80.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100)
    102.6667 × 10−1
    (2.12 × 10−1) −    		1.9822 × 10−1
    (1.68 × 10−1) −    		6.2982 × 10−1
    (1.71 × 10−1) +    		3.2562 × 10−1
    (2.12 × 10−1) =    		1.9434 × 10−1
    (1.65 × 10−1) −    		4.0899 × 10−1
    (2.13 × 10−1)
    DTLZ231.1734 × 10−1
    (2.42 × 10−2) −    		1.2929 × 10−1
    (2.86 × 10−2) −    		3.1461 × 10−1
    (5.78 × 10−2) −    		4.4429 × 10−1
    (2.12 × 10−2) −    		1.3839 × 10−1
    (4.54 × 10−2) −    		4.6981 × 10−1
    (1.16 × 10−1)
    42.1214 × 10−1
    (3.91 × 10−2) −    		2.0737 × 10−1
    (3.04 × 10−2) −    		3.5746 × 10−1
    (7.24 × 10−2) −    		5.7115 × 10−1
    (2.41 × 10−2) =    		2.2493 × 10−1
    (4.31 × 10−2) −    		5.3573 × 10−1
    (1.08 × 10−1)
    63.0640 × 10−1
    (3.14 × 10−2) −    		3.0719 × 10−1
    (3.32 × 10−2) −    		5.0211 × 10−1
    (7.01 × 10−2) −    		6.8636 × 10−1
    (3.86 × 10−2) =    		3.6893 × 10−1
    (3.19 × 10−2) −    		6.9607 × 10−1
    (4.86 × 10−2)
    84.4578 × 10−1
    (4.27 × 10−2) −    		4.3830 × 10−1
    (2.13 × 10−2) −    		5.6690 × 10−1
    (4.55 × 10−2) −    		7.6366 × 10−1
    (3.84 × 10−2) −    		5.5566 × 10−1
    (2.98 × 10−2) −    		8.3822 × 10−1
    (2.06 × 10−2)
    105.4274 × 10−1
    (2.52 × 10−2) −    		5.9705 × 10−1
    (2.81 × 10−2) −    		6.3060 × 10−1
    (2.88 × 10−2) −    		8.6321 × 10−1
    (1.23 × 10−2) −    		8.1393 × 10−1
    (1.64 × 10−2) −    		9.1251 × 10−1
    (7.76 × 10−3)
    DTLZ330.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100)
    40.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100)
    60.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100)
    80.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100)
    103.5213 × 10−2
    (4.90 × 10−2) −    		2.8103 × 10−2
    (4.73 × 10−2) −    		2.6473 × 10−1
    (1.54 × 10−1) +    		9.2144 × 10−2
    (1.22 × 10−1) =    		6.5751 × 10−2
    (7.30 × 10−2) =    		8.5208 × 10−2
    (8.29 × 10−2)
    DTLZ431.1785 × 10−2
    (2.18 × 10−2) −    		1.5375 × 10−2
    (2.50 × 10−2) −    		1.7626 × 10−1
    (8.31 × 10−2) +    		1.7423 × 10−1
    (1.08 × 10−1) +    		1.7616 × 10−2
    (2.25 × 10−2) −    		7.5949 × 10−2
    (7.70 × 10−2)
    44.9153 × 10−2
    (3.74 × 10−2) =    		2.4608 × 10−2
    (2.91 × 10−2) −    		3.1304 × 10−1
    (6.63 × 10−2) +    		2.2725 × 10−1
    (8.66 × 10−2) +    		2.8193 × 10−2
    (2.68 × 10−2) −    		1.3719 × 10−1
    (1.25 × 10−1)
    61.1881 × 10−1
    (4.18 × 10−2) −    		1.2211 × 10−1
    (3.42 × 10−2) −    		5.6483 × 10−1
    (8.07 × 10−2) +    		4.1915 × 10−1
    (1.31 × 10−1) +    		1.0424 × 10−1
    (4.08 × 10−2) −    		2.1455 × 10−1
    (9.54 × 10−2)
    83.1494 × 10−1
    (7.82 × 10−2) −    		4.3521 × 10−1
    (5.84 × 10−2) −    		6.8774 × 10−1
    (3.61 × 10−2) +    		6.2693 × 10−1
    (7.95 × 10−2) +    		3.3577 × 10−1
    (6.00 × 10−2) −    		5.4786 × 10−1
    (9.97 × 10−2)
    106.1117 × 10−1
    (5.26 × 10−2) −    		7.4214 × 10−1
    (2.37 × 10−2) −    		8.0539 × 10−1
    (3.71 × 10−2) =    		8.3652 × 10−1
    (4.00 × 10−2) =    		7.4450 × 10−1
    (2.53 × 10−2) −    		8.2411 × 10−1
    (4.08 × 10−2)
    DTLZ531.7464 × 10−2
    (8.81 × 10−3) −    		2.3213 × 10−2
    (1.15 × 10−2) −    		9.1632 × 10−2
    (2.53 × 10−2) −    		1.2641 × 10−1
    (2.94 × 10−2) −    		1.8468 × 10−2
    (2.17 × 10−2) −    		1.5632 × 10−1
    (4.42 × 10−2)
    41.8002 × 10−2
    (8.78 × 10−3) −    		1.8297 × 10−2
    (7.45 × 10−3) −    		5.6265 × 10−2
    (2.37 × 10−2) −    		1.1696 × 10−1
    (6.53 × 10−3) −    		2.6234 × 10−2
    (2.37 × 10−2) −    		1.3582 × 10−1
    (1.02 × 10−2)
    61.9682 × 10−2
    (1.21 × 10−2) −    		2.6909 × 10−2
    (1.97 × 10−2) −    		7.5674 × 10−2
    (1.88 × 10−2) −    		1.0592 × 10−1
    (3.54 × 10−3) −    		5.0133 × 10−2
    (2.49 × 10−2) −    		1.1271 × 10−1
    (8.05 × 10−3)
    85.2712 × 10−2
    (2.30 × 10−2) −    		6.5698 × 10−2
    (1.40 × 10−2) −    		9.3848 × 10−2
    (4.29 × 10−3) −    		1.0261 × 10−1
    (2.69 × 10−3) −    		8.5375 × 10−2
    (6.49 × 10−3) −    		1.0487 × 10−1
    (3.42 × 10−4)
    108.6211 × 10−2
    (1.15 × 10−2) −    		9.7441 × 10−2
    (9.74 × 10−4) −    		9.9494 × 10−2
    (5.70 × 10−4) −    		9.7695 × 10−2
    (7.46 × 10−4) −    		9.6720 × 10−2
    (7.96 × 10−4) −    		1.0031 × 10−1
    (2.98 × 10−4)
    DTLZ630.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100)
    40.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		4.7399 × 10−3
    (2.03 × 10−2) +    		0.0000 × 100
    (0.00 × 100)
    60.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		0.0000 × 100
    (0.00 × 100) =    		3.2073 × 10−2
    (4.49 × 10−2) +    		4.5458 × 10−3
    (2.03 × 10−2)
    80.0000 × 100
    (0.00 × 100) −    		1.8646 × 10−4
    (7.43 × 10−4) =    		9.2532 × 10−4
    (4.14 × 10−3) −    		2.2110 × 10−2
    (3.93 × 10−2) =    		6.0557 × 10−2
    (4.40 × 10−2) +    		8.0524 × 10−3
    (2.08 × 10−2)
    101.7423 × 10−2
    (3.52 × 10−2) −    		8.0620 × 10−2
    (2.45 × 10−2) =    		5.7994 × 10−2
    (3.98 × 10−2) −    		9.4737 × 10−2
    (1.76 × 10−3) =    		9.2210 × 10−2
    (1.35 × 10−3) −    		9.2912 × 10−2
    (1.31 × 10−2)
    DTLZ730.0000 × 100
    (0.00 × 100) −    		0.0000 × 100
    (0.00 × 100) −    		4.2044 × 10−2
    (4.24 × 10−2) −    		2.4684 × 10−1
    (5.96 × 10−3) −    		2.0656 × 10−1
    (1.66 × 10−2) −    		2.7126 × 10−1
    (8.89 × 10−3)
    40.0000 × 100
    (0.00 × 100) −    		5.5343 × 10−7
    (2.48 × 10−6) −    		3.3341 × 10−2
    (3.99 × 10−2) −    		2.3586 × 10−1
    (7.10 × 10−3) −    		7.9986 × 10−2
    (5.40 × 10−2) −    		2.5760 × 10−1
    (7.67 × 10−3)
    60.0000 × 100
    (0.00 × 100) −    		0.0000 × 100
    (0.00 × 100) −    		2.9404 × 10−2
    (3.52 × 10−2) −    		1.9617 × 10−1
    (8.41 × 10−3) −    		3.0127 × 10−2
    (3.37 × 10−2) −    		2.1700 × 10−1
    (5.99 × 10−3)
    80.0000 × 100
    (0.00 × 100) −    		2.5202 × 10−4
    (9.83 × 10−4) −    		3.0467 × 10−2
    (3.86 × 10−2) −    		1.8919 × 10−1
    (4.73 × 10−3) =    		5.7129 × 10−3
    (5.65 × 10−3) −    		1.8593 × 10−1
    (8.01 × 10−3)
    103.3199 × 10−3
    (5.02 × 10−3) −    		7.5420 × 10−4
    (8.53 × 10−4) −    		3.5750 × 10−2
    (3.02 × 10−2) −    		1.7539 × 10−1
    (4.01 × 10−3) =    		1.3231 × 10−2
    (1.67 × 10−2) −    		1.7687 × 10−1
    (4.85 × 10−3)
    +/−/=0/23/120/22/136/17/124/11/203/22/10
    下载: 导出CSV

    表  4  6种算法在不同维数的WFG测试问题上获得的IGD平均值和标准差

    Table  4  The IGD average and standard deviation obtained by the six algorithms on WFG test problems of different dimensions

    测试问题目标数NSGAIIICPS-MOEACSEAK-RVEAMOEA/D-EGOK-RSEA
    WFG132.3044 × 100
    (7.04 × 10−2) −    		2.2817 × 100
    (6.96 × 10−2) −    		1.7246 × 100
    (7.58 × 10−2) +    		1.7654 × 100
    (9.23 × 10−2) +    		2.1936 × 100
    (6.64 × 10−2) −    		1.8391 × 100
    (8.45 × 10−2)
    42.1642 × 100
    (1.32 × 10−1) =    		1.9453 × 100
    (1.71 × 10−2) +    		1.9531 × 100
    (1.11 × 10−1) +    		2.0752 × 100
    (1.55 × 10−1) =    		2.0130 × 100
    (7.58 × 10−2) +    		2.0919 × 100
    (1.31 × 10−1)
    62.7873 × 100
    (5.89 × 10−2) −    		2.8059 × 100
    (6.16 × 10−2) −    		2.4902 × 100
    (5.09 × 10−2) =    		2.4594 × 100
    (9.80 × 10−2) +    		2.7526 × 100
    (5.35 × 10−2) −    		2.5304 × 100
    (1.00 × 10−1)
    83.1194 × 100
    (5.49 × 10−2) −    		3.1420 × 100
    (4.58 × 10−2) −    		2.8827 × 100
    (6.05 × 10−2) =    		2.8566 × 100
    (8.07 × 10−2) =    		3.0855 × 100
    (5.32 × 10−2) −    		2.8335 × 100
    (2.54 × 10−1)
    103.4110 × 100
    (4.80 × 10−2) −    		3.4220 × 100
    (4.68 × 10−2) −    		3.1817 × 100
    (8.33 × 10−2) =    		3.1589 × 100
    (5.88 × 10−2) =    		3.3937 × 100
    (3.38 × 10−2) −    		3.1129 × 100
    (2.06 × 10−1)
    WFG238.8153 × 10−1
    (1.09 × 10−1) −    		7.8323 × 10−1
    (4.94 × 10−2) −    		5.3426 × 10−1
    (4.34 × 10−2) −    		3.9180 × 10−1
    (4.71 × 10−2) −    		6.9725 × 10−1
    (4.98 × 10−2) −    		3.4170 × 10−1
    (4.16 × 10−2)
    41.4386 × 100
    (2.16 × 10−1) −    		1.2247 × 100
    (1.92 × 10−1) −    		1.2237 × 100
    (3.42 × 10−1) −    		9.9551 × 10−1
    (1.34 × 10−1) −    		1.1293 × 100
    (1.61 × 10−1) −    		9.1510 × 10−1
    (1.09 × 10−1)
    61.9640 × 100
    (4.29 × 10−1) −    		1.5348 × 100
    (1.38 × 10−1) −    		1.2796 × 100
    (4.14 × 10−1) −    		7.6122 × 10−1
    (5.92 × 10−2) −    		1.3485 × 100
    (1.72 × 10−1) −    		7.1461 × 10−1
    (3.51 × 10−2)
    82.5026 × 100
    (5.75 × 10−1) −    		2.0548 × 100
    (2.06 × 10−1) −    		1.8266 × 100
    (7.53 × 10−1) −    		1.0294 × 100
    (4.52 × 10−2) =    		1.7364 × 100
    (2.53 × 10−1) −    		1.0125 × 100
    (3.41 × 10−2)
    103.8506 × 100
    (7.40 × 10−1) −    		3.0779 × 100
    (4.97 × 10−1) −    		3.1377 × 100
    (9.98 × 10−1) −    		1.1813 × 100
    (5.78 × 10−2) +    		2.1748 × 100
    (3.75 × 10−1) −    		1.2223 × 100
    (6.17 × 10−2)
    WFG336.0295 × 10−1
    (4.49 × 10−2) −    		6.0035 × 10−1
    (2.83 × 10−2) −    		4.7826 × 10−1
    (7.08 × 10−2) −    		3.8454 × 10−1
    (5.62 × 10−2) =    		6.3111 × 10−1
    (3.08 × 10−2) −    		4.2141 × 10−1
    (8.51 × 10−2)
    45.6516 × 10−1
    (5.35 × 10−2) −    		5.5932 × 10−1
    (3.92 × 10−2) −    		3.6608 × 10−1
    (6.42 × 10−2) −    		2.2293 × 10−1
    (3.65 × 10−2) +    		6.0164 × 10−1
    (4.47 × 10−2) −    		2.5478 × 10−1
    (2.73 × 10−2)
    61.0518 × 100
    (1.02 × 10−1) −    		9.8651 × 10−1
    (8.00 × 10−2) −    		7.0767 × 10−1
    (9.63 × 10−2) −    		6.6060 × 10−1
    (8.61 × 10−2) −    		9.7316 × 10−1
    (4.10 × 10−2) −    		4.9911 × 10−1
    (9.35 × 10−2)
    89.2359 × 10−1
    (1.11 × 10−1) −    		7.8556 × 10−1
    (7.35 × 10−2) −    		4.4627 × 10−1
    (1.18 × 10−1) −    		5.3840 × 10−1
    (7.52 × 10−2) −    		8.2687 × 10−1
    (8.87 × 10−2) −    		3.4794 × 10−1
    (5.25 × 10−2)
    101.0022 × 100
    (8.69 × 10−2) −    		8.7818 × 10−1
    (1.11 × 10−1) −    		5.9822 × 10−1
    (1.14 × 10−1) −    		6.3955 × 10−1
    (8.21 × 10−2) −    		9.2789 × 10−1
    (7.38 × 10−2) −    		4.2756 × 10−1
    (6.74 × 10−2)
    WFG436.3183 × 10−1
    (6.30 × 10−2) −    		5.3726 × 10−1
    (2.65 × 10−2) −    		4.4571 × 10−1
    (3.28 × 10−2) +    		4.5254 × 10−1
    (1.90 × 10−2) +    		5.8593 × 10−1
    (3.46 × 10−2) −    		5.0332 × 10−1
    (2.65 × 10−2)
    41.7199 × 100
    (1.84 × 10−1) −    		1.2114 × 100
    (8.23 × 10−2) −    		1.5483 × 100
    (2.75 × 10−1) −    		8.1670 × 10−1
    (8.21 × 10−2) =    		9.8678 × 10−1
    (7.18 × 10−2) −    		7.7364 × 10−1
    (7.65 × 10−2)
    63.6040 × 100
    (3.76 × 10−1) −    		2.5352 × 100
    (1.50 × 10−1) −    		2.9141 × 100
    (3.36 × 10−1) −    		1.7992 × 100
    (4.76 × 10−2) +    		2.1228 × 100
    (1.22 × 10−1) −    		1.8370 × 100
    (5.25 × 10−2)
    85.9740 × 100
    (4.07 × 10−1) −    		4.3118 × 100
    (2.44 × 10−1) −    		5.8308 × 100
    (4.61 × 10−1) −    		3.2283 × 100
    (2.39 × 10−1) =    		3.4432 × 100
    (1.44 × 10−1) −    		3.3000 × 100
    (2.59 × 10−1)
    109.1735 × 100
    (5.27 × 10−1) −    		7.2985 × 100
    (3.71 × 10−1) −    		8.6988 × 100
    (9.68 × 10−1) −    		5.9483 × 100
    (5.44 × 10−1) −    		5.0437 × 100
    (2.82 × 10−1) +    		5.4156 × 100
    (3.45 × 10−1)
    WFG536.9770 × 10−1
    (3.25 × 10−2) −    		5.8013 × 10−1
    (1.71 × 10−2) −    		5.2657 × 10−1
    (3.82 × 10−2) −    		4.3283 × 10−1
    (6.77 × 10−2) −    		5.8135 × 10−1
    (2.95 × 10−2) −    		3.8674 × 10−1
    (6.33 × 10−2)
    41.3558 × 100
    (1.03 × 10−1) −    		1.3003 × 100
    (4.72 × 10−2) −    		1.1067 × 100
    (1.51 × 10−1) −    		7.8509 × 10−1
    (6.09 × 10−2) −    		9.8506 × 10−1
    (4.70 × 10−2) −    		7.1114 × 10−1
    (4.18 × 10−2)
    62.7646 × 100
    (1.73 × 10−1) −    		2.5373 × 100
    (1.03 × 10−1) −    		2.4254 × 100
    (2.43 × 10−1) −    		1.7916 × 100
    (8.92 × 10−2) =    		2.1839 × 100
    (1.45 × 10−1) −    		1.8386 × 100
    (8.92 × 10−2)
    84.7298 × 100
    (2.08 × 10−1) −    		4.5985 × 100
    (1.45 × 10−1) −    		4.7238 × 100
    (4.32 × 10−1) −    		3.0908 × 100
    (7.65 × 10−2) +    		4.4346 × 100
    (2.48 × 10−1) −    		3.1917 × 100
    (1.30 × 10−1)
    107.3037 × 100
    (2.91 × 10−1) −    		7.0171 × 100
    (3.14 × 10−1) −    		7.0938 × 100
    (3.27 × 10−1) −    		4.8049 × 100
    (3.18 × 10−1) +    		6.6075 × 100
    (4.97 × 10−1) −    		5.0163 × 100
    (2.42 × 10−1)
    WFG638.0656 × 10−1
    (3.30 × 10−2) −    		7.8806 × 10−1
    (2.19 × 10−2) −    		7.1317 × 10−1
    (3.92 × 10−2) =    		7.1713 × 10−1
    (4.87 × 10−2) =    		8.0570 × 10−1
    (4.64 × 10−2) −    		7.2868 × 10−1
    (3.97 × 10−2)
    41.3827 × 100
    (8.35 × 10−2) −    		1.2283 × 100
    (5.92 × 10−2) −    		1.0173 × 100
    (8.09 × 10−2) =    		1.0307 × 100
    (9.56 × 10−2) =    		1.1020 × 100
    (4.08 × 10−2) −    		1.0481 × 100
    (4.41 × 10−2)
    62.8461 × 100
    (2.15 × 10−1) −    		2.6368 × 100
    (1.36 × 10−1) −    		2.3941 × 100
    (1.92 × 10−1) −    		2.2878 × 100
    (1.10 × 10−1) −    		2.1672 × 100
    (5.45 × 10−2) −    		2.1051 × 100
    (7.48 × 10−2)
    84.9875 × 100
    (3.24 × 10−1) −    		4.6116 × 100
    (2.17 × 10−1) −    		4.7585 × 100
    (5.04 × 10−1) −    		3.6354 × 100
    (8.70 × 10−2) −    		3.7239 × 100
    (1.43 × 10−1) −    		3.4682 × 100
    (1.18 × 10−1)
    107.4853 × 100
    (4.37 × 10−1) −    		6.9814 × 100
    (4.94 × 10−1) −    		7.2251 × 100
    (6.41 × 10−1) −    		5.1438 × 100
    (1.55 × 10−1) =    		5.3090 × 100
    (3.81 × 10−1) =    		5.0901 × 100
    (1.35 × 10−1)
    WFG736.6448 × 10−1
    (4.60 × 10−2) =    		6.3768 × 10−1
    (3.13 × 10−2) =    		5.8351 × 10−1
    (3.07 × 10−2) +    		6.0448 × 10−1
    (2.89 × 10−2) +    		6.6027 × 10−1
    (3.26 × 10−2) =    		6.5385 × 10−1
    (4.34 × 10−2)
    41.5156 × 100
    (1.31 × 10−1) −    		1.4000 × 100
    (1.10 × 10−1) −    		1.3798 × 100
    (1.35 × 10−1) −    		8.9610 × 10−1
    (6.80 × 10−2) =    		1.2373 × 100
    (1.22 × 10−1) −    		8.6343 × 10−1
    (6.67 × 10−2)
    63.0239 × 100
    (2.27 × 10−1) −    		2.6972 × 100
    (1.90 × 10−1) −    		2.5951 × 100
    (2.70 × 10−1) −    		1.9468 × 100
    (4.95 × 10−2) +    		2.4804 × 100
    (1.69 × 10−1) −    		2.0185 × 100
    (5.45 × 10−2)
    85.2874 × 100
    (2.64 × 10−1) −    		4.9740 × 100
    (4.19 × 10−1) −    		5.4691 × 100
    (4.66 × 10−1) −    		3.4310 × 100
    (1.05 × 10−1) =    		5.1211 × 100
    (3.92 × 10−1) −    		3.4353 × 100
    (7.80 × 10−2)
    108.0948 × 100
    (4.84 × 10−1) −    		7.6933 × 100
    (4.38 × 10−1) −    		8.1050 × 100
    (5.92 × 10−1) −    		5.1689 × 100
    (1.77 × 10−1) =    		6.9773 × 100
    (6.25 × 10−1) −    		5.2380 × 100
    (3.09 × 10−1)
    WFG838.7226 × 10−1
    (3.67 × 10−2) −    		8.4186 × 10−1
    (2.79 × 10−2) −    		7.3788 × 10−1
    (5.34 × 10−2) −    		7.2196 × 10−1
    (3.90 × 10−2) −    		8.6180 × 10−1
    (2.30 × 10−2) −    		6.7398 × 10−1
    (4.58 × 10−2)
    41.7785 × 100
    (1.02 × 10−1) −    		1.7130 × 100
    (8.86 × 10−2) −    		1.7122 × 100
    (1.64 × 10−1) −    		1.3654 × 100
    (5.98 × 10−2) −    		1.3509 × 100
    (3.48 × 10−2) −    		1.2069 × 100
    (5.72 × 10−2)
    63.2270 × 100
    (2.38 × 10−1) −    		2.8330 × 100
    (1.62 × 10−1) −    		3.0250 × 100
    (2.19 × 10−1) −    		2.3476 × 100
    (9.36 × 10−2) −    		2.4530 × 100
    (5.19 × 10−2) −    		2.2095 × 100
    (4.49 × 10−2)
    85.2767 × 100
    (2.91 × 10−1) −    		5.1048 × 100
    (3.42 × 10−1) −    		5.4616 × 100
    (3.15 × 10−1) −    		3.5830 × 100
    (1.18 × 10−1) =    		4.2845 × 100
    (2.58 × 10−1) −    		3.5916 × 100
    (1.02 × 10−1)
    107.8537 × 100
    (3.65 × 10−1) −    		7.3927 × 100
    (4.54 × 10−1) −    		7.9521 × 100
    (4.13 × 10−1) −    		5.0690 × 100
    (1.21 × 10−1) +    		5.6966 × 100
    (2.55 × 10−1) −    		5.1969 × 100
    (1.70 × 10−1)
    WFG938.1617 × 10−1
    (4.19 × 10−2) −    		7.8612 × 10−1
    (4.07 × 10−2) −    		6.2832 × 10−1
    (6.58 × 10−2) =    		6.6725 × 10−1
    (4.52 × 10−2) =    		7.8107 × 10−1
    (5.79 × 10−2) −    		6.6782 × 10−1
    (5.50 × 10−2)
    41.3187 × 100
    (8.63 × 10−2) −    		1.3147 × 100
    (7.88 × 10−2) −    		1.1780 × 100
    (1.25 × 10−1) −    		1.1332 × 100
    (2.03 × 10−1) =    		1.3423 × 100
    (1.28 × 10−1) −    		1.0305 × 100
    (1.72 × 10−1)
    63.0388 × 100
    (2.54 × 10−1) −    		3.0806 × 100
    (1.69 × 10−1) −    		2.9442 × 100
    (2.94 × 10−1) −    		2.1042 × 100
    (1.22 × 10−1) +    		2.7797 × 100
    (3.25 × 10−1) −    		2.4146 × 100
    (1.84 × 10−1)
    85.1537 × 100
    (3.04 × 10−1) −    		5.1279 × 100
    (3.30 × 10−1) −    		5.2404 × 100
    (4.41 × 10−1) −    		3.9706 × 100
    (6.27 × 10−1) =    		4.8820 × 100
    (4.82 × 10−1) −    		4.0674 × 100
    (5.76 × 10−1)
    107.6142 × 100
    (4.20 × 10−1) −    		7.6831 × 100
    (3.48 × 10−1) −    		7.5833 × 100
    (5.26 × 10−1) −    		6.2079 × 100
    (5.79 × 10−1) =    		7.2823 × 100
    (5.75 × 10−1) −    		6.0729 × 100
    (6.67 × 10−1)
    +/−/=0/43/21/43/14/35/612/14/192/41/2
    下载: 导出CSV

    表  5  6种算法在不同维数的WFG测试问题上获得的HV平均值和标准差

    Table  5  The HV average and standard deviation obtained by the six algorithms on WFG test problems of different dimensions

    测试问题目标数NSGA-IIICPS-MOEACSEAK-RVEAMOEA/D-EGOK-RSEA
    WFG13 0.0000 × 100
    (0.00 × 100) −    		 2.3754 × 10−3
    (5.39 × 10−3) −    		 1.5551 × 10−1
    (4.56 × 10−2) =    		 1.6255 × 10−1
    (2.88 × 10−2) =    		 5.6758 × 10−3
    (1.33 × 10−2) −    		 1.4541 × 10−1
    (3.90 × 10−2)
    4 1.9475 × 10−1
    (3.30 × 10−2) =    		 2.9110 × 10−1
    (6.17 × 10−3) +    		 2.7323 × 10−1
    (3.27 × 10−2) +    		 2.3182 × 10−1
    (4.43 × 10−2) =    		 2.7638 × 10−1
    (1.47 × 10−2) +    		 2.1878 × 10−1
    (6.03 × 10−2)
    6 3.0659 × 10−2
    (2.10 × 10−2) −    		 9.9187 × 10−2
    (1.20 × 10−2) −    		 2.0714 × 10−1
    (3.65 × 10−2) =    		 2.0241 × 10−1
    (5.23 × 10−2) =    		 1.2496 × 10−1
    (2.73 × 10−2) −    		 1.8319 × 10−1
    (6.40 × 10−2)
    8 1.0632 × 10−1
    (2.34 × 10−2) −    		 1.7126 × 10−1
    (9.62 × 10−3) =    		 2.0631 × 10−1
    (2.29 × 10−2) =    		 2.0651 × 10−1
    (3.16 × 10−2) =    		 1.8449 × 10−1
    (1.97 × 10−2) =    		 1.9834 × 10−1
    (1.06 × 10−1)
    10 1.0646 × 10−1
    (2.91 × 10−2) −    		 1.9083 × 10−1
    (6.61 × 10−3) =    		 2.1349 × 10−1
    (2.85 × 10−2) =    		 2.1469 × 10−1
    (7.69 × 10−3) =    		 1.8981 × 10−1
    (1.13 × 10−2) =    		 1.8994 × 10−1
    (6.85 × 10−2)
    WFG23 5.7155 × 10−1
    (3.23 × 10−2) −    		 6.0180 × 10−1
    (1.47 × 10−2) −    		 7.0816 × 10−1
    (2.55 × 10−2) −    		 7.6969 × 10−1
    (2.32 × 10−2) −    		 6.3021 × 10−1
    (2.06 × 10−2) −    		 7.9608 × 10−1
    (3.12 × 10−2)
    4 6.0584 × 10−1
    (3.12 × 10−2) −    		 6.5097 × 10−1
    (2.87 × 10−2) −    		 6.8373 × 10−1
    (5.07 × 10−2) −    		 7.5823 × 10−1
    (3.40 × 10−2) −    		 6.5436 × 10−1
    (3.73 × 10−2) −    		 7.8405 × 10−1
    (2.80 × 10−2)
    6 6.4455 × 10−1
    (3.97 × 10−2) −    		 6.8126 × 10−1
    (2.19 × 10−2) −    		 7.9691 × 10−1
    (6.05 × 10−2) −    		 8.6678 × 10−1
    (3.48 × 10−2) −    		 6.9228 × 10−1
    (2.95 × 10−2) −    		 9.3463 × 10−1
    (1.69 × 10−2)
    8 8.1555 × 10−1
    (6.07 × 10−2) −    		 8.6795 × 10−1
    (1.99 × 10−2) −    		 9.1947 × 10−1
    (4.73 × 10−2) −    		 9.6566 × 10−1
    (1.18 × 10−2) −    		 8.5448 × 10−1
    (3.00 × 10−2) −    		 9.9044 × 10−1
    (4.99 × 10−3)
    10 7.6764 × 10−1
    (4.56 × 10−2) −    		 8.3857 × 10−1
    (3.36 × 10−2) −    		 8.7953 × 10−1
    (5.73 × 10−2) −    		 9.6830 × 10−1
    (1.01 × 10−2) −    		 8.4508 × 10−1
    (4.68 × 10−2) −    		 9.9466 × 10−1
    (2.01 × 10−3)
    WFG33 1.5521 × 10−1
    (1.05 × 10−2) −    		 1.6033 × 10−1
    (9.18 × 10−3) −    		 1.9966 × 10−1
    (2.70 × 10−2) =    		 2.4291 × 10−1
    (2.18 × 10−2) +    		 1.4825 × 10−1
    (9.09 × 10−3) −    		 2.0974 × 10−1
    (3.45 × 10−2)
    4 8.3428 × 10−2
    (2.38 × 10−2) −    		 8.8702 × 10−2
    (2.16 × 10−2) −    		 1.6360 × 10−1
    (3.01 × 10−2) −    		 2.2785 × 10−1
    (2.52 × 10−2) +    		 8.1881 × 10−2
    (1.96 × 10−2) −    		 2.0271 × 10−1
    (2.29 × 10−2)
    6 0.0000 × 100
    (0.00 × 100) −    		 0.0000 × 100
    (0.00 × 100) −    		 6.2531 × 10−3
    (1.43 × 10−2) −    		 7.5288 × 10−3
    (1.24 × 10−2) −    		 1.2130 × 10−3
    (5.42 × 10−3) −    		 2.7253 × 10−2
    (2.58 × 10−2)
    8 4.5931 × 10−3
    (2.01 × 10−2) −    		 2.6674 × 10−4
    (9.23 × 10−4) −    		 3.3423 × 10−2
    (4.46 × 10−2) =    		 1.4620 × 10−2
    (2.21 × 10−2) −    		 9.6695 × 10−3
    (2.28 × 10−2) −    		 4.9333 × 10−2
    (3.21 × 10−2)
    10 0.0000 × 100
    (0.00 × 100) −    		 8.4506 × 10−5
    (3.78 × 10−4) =    		 2.1320 × 10−3
    (8.52 × 10−3) =    		 2.5236 × 10−3
    (1.13 × 10−2) =    		 0.0000 × 100
    (0.00 × 100) −    		 5.1517 × 10−3
    (1.19 × 10−2)
    WFG43 3.1526 × 10−1
    (1.26 × 10−2) −    		 3.3905 × 10−1
    (1.08 × 10−2) =    		 3.8760 × 10−1
    (2.00 × 10−2) +    		 3.6710 × 10−1
    (1.16 × 10−2) +    		 3.3783 × 10−1
    (1.33 × 10−2) −    		 3.4727 × 10−1
    (1.35 × 10−2)
    4 3.1231 × 10−1
    (1.20 × 10−2) −    		 3.8630 × 10−1
    (1.57 × 10−2) −    		 3.6772 × 10−1
    (3.32 × 10−2) −    		 4.7669 × 10−1
    (2.16 × 10−2) −    		 4.3451 × 10−1
    (2.48 × 10−2) −    		 4.8908 × 10−1
    (1.89 × 10−2)
    6 3.6756 × 10−1
    (2.66 × 10−2) −    		 4.6632 × 10−1
    (1.82 × 10−2) −    		 4.6846 × 10−1
    (2.89 × 10−2) −    		 5.8503 × 10−1
    (3.32 × 10−2) −    		 5.1160 × 10−1
    (2.69 × 10−2) −    		 6.3545 × 10−1
    (2.31 × 10−2)
    8 4.2581 × 10−1
    (1.93 × 10−2) −    		 5.5373 × 10−1
    (2.95 × 10−2) −    		 5.0393 × 10−1
    (4.44 × 10−2) −    		 7.0749 × 10−1
    (3.11 × 10−2) −    		 6.5901 × 10−1
    (2.55 × 10−2) −    		 7.7127 × 10−1
    (3.11 × 10−2)
    10 4.1946 × 10−1
    (2.30 × 10−2) −    		 5.4379 × 10−1
    (2.16 × 10−2) −    		 4.9750 × 10−1
    (5.18 × 10−2) −    		 6.5396 × 10−1
    (3.22 × 10−2) −    		 6.5974 × 10−1
    (3.52 × 10−2) −    		 7.6087 × 10−1
    (2.71 × 10−2)
    WFG53 2.4395 × 10−1
    (1.12 × 10−2) −    		 2.9918 × 10−1
    (7.98 × 10−3) −    		 3.4805 × 10−1
    (2.40 × 10−2) −    		 3.9021 × 10−1
    (3.64 × 10−2) −    		 3.6004 × 10−1
    (1.89 × 10−2) −    		 4.2110 × 10−1
    (3.46 × 10−2)
    4 2.7954 × 10−1
    (1.14 × 10−2) −    		 3.3688 × 10−1
    (1.26 × 10−2) −    		 3.9549 × 10−1
    (3.14 × 10−2) −    		 4.7832 × 10−1
    (2.30 × 10−2) −    		 4.0185 × 10−1
    (1.44 × 10−2) −    		 5.0051 × 10−1
    (2.40 × 10−2)
    6 3.2385 × 10−1
    (1.38 × 10−2) −    		 4.0172 × 10−1
    (1.59 × 10−2) −    		 4.9501 × 10−1
    (3.25 × 10−2) −    		 5.6239 × 10−1
    (4.28 × 10−2) −    		 5.0007 × 10−1
    (1.55 × 10−2) −    		 5.9915 × 10−1
    (3.21 × 10−2)
    8 3.6783 × 10−1
    (2.95 × 10−2) −    		 4.6548 × 10−1
    (1.24 × 10−2) −    		 5.4239 × 10−1
    (3.56 × 10−2) −    		 6.6693 × 10−1
    (4.28 × 10−2) −    		 5.3718 × 10−1
    (2.14 × 10−2) −    		 7.2794 × 10−1
    (2.07 × 10−2)
    10 3.7342 × 10−1
    (2.07 × 10−2) −    		 4.6464 × 10−1
    (1.42 × 10−2) −    		 5.2257 × 10−1
    (2.90 × 10−2) −    		 6.2569 × 10−1
    (4.07 × 10−2) −    		 5.5522 × 10−1
    (3.41 × 10−2) −    		 7.1028 × 10−1
    (3.10 × 10−2)
    WFG63 2.0832 × 10−1
    (1.01 × 10−2) −    		 2.0973 × 10−1
    (7.97 × 10−3) −    		 2.5872 × 10−1
    (1.90 × 10−2) +    		 2.5438 × 10−1
    (1.91 × 10−2) +    		 2.5210 × 10−1
    (1.88 × 10−2) +    		 2.2621 × 10−1
    (1.93 × 10−2)
    4 2.6812 × 10−1
    (1.80 × 10−2) −    		 2.9358 × 10−1
    (1.92 × 10−2) −    		 3.5227 × 10−1
    (2.85 × 10−2) +    		 3.7527 × 10−1
    (4.14 × 10−2) +    		 2.9643 × 10−1
    (1.56 × 10−2) −    		 3.3339 × 10−1
    (3.07 × 10−2)
    6 3.0489 × 10−1
    (1.75 × 10−2) −    		 3.4169 × 10−1
    (1.88 × 10−2) −    		 4.0660 × 10−1
    (3.70 × 10−2) =    		 3.9532 × 10−1
    (4.83 × 10−2) =    		 4.1998 × 10−1
    (1.48 × 10−2) =    		 4.0315 × 10−1
    (3.75 × 10−2)
    8 3.9079 × 10−1
    (4.10 × 10−2) −    		 4.6339 × 10−1
    (2.22 × 10−2) −    		 5.1016 × 10−1
    (4.92 × 10−2) −    		 6.6623 × 10−1
    (2.77 × 10−2) −    		 5.1183 × 10−1
    (2.19 × 10−2) −    		 7.3524 × 10−1
    (4.15 × 10−2)
    10 3.9579 × 10−1
    (2.55 × 10−2) −    		 4.7448 × 10−1
    (2.34 × 10−2) −    		 4.9997 × 10−1
    (4.80 × 10−2) −    		 6.7299 × 10−1
    (3.78 × 10−2) −    		 5.3339 × 10−1
    (2.18 × 10−2) −    		 7.9146 × 10−1
    (4.70 × 10−2)
    WFG73 2.7294 × 10−1
    (1.31 × 10−2) =    		 2.7768 × 10−1
    (1.21 × 10−2) =    		 3.2790 × 10−1
    (2.16 × 10−2) +    		 2.9637 × 10−1
    (1.66 × 10−2) +    		 2.8180 × 10−1
    (1.13 × 10−2) =    		 2.7832 × 10−1
    (1.80 × 10−2)
    4 3.0770 × 10−1
    (1.53 × 10−2) −    		 3.3113 × 10−1
    (1.54 × 10−2) −    		 3.6574 × 10−1
    (2.60 × 10−2) −    		 4.5508 × 10−1
    (2.66 × 10−2) =    		 3.5393 × 10−1
    (1.72 × 10−2) −    		 4.6052 × 10−1
    (2.59 × 10−2)
    6 3.5607 × 10−1
    (1.68 × 10−2) −    		 4.0546 × 10−1
    (1.42 × 10−2) −    		 4.6728 × 10−1
    (2.93 × 10−2) −    		 5.1981 × 10−1
    (3.50 × 10−2) −    		 4.1921 × 10−1
    (1.60 × 10−2) −    		 5.5770 × 10−1
    (4.09 × 10−2)
    8 4.1954 × 10−1
    (3.00 × 10−2) −    		 4.7747 × 10−1
    (2.38 × 10−2) −    		 5.3422 × 10−1
    (4.07 × 10−2) −    		 6.6874 × 10−1
    (3.33 × 10−2) −    		 4.9354 × 10−1
    (2.52 × 10−2) −    		 7.9566 × 10−1
    (1.82 × 10−2)
    10 4.3978 × 10−1
    (2.68 × 10−2) −    		 4.9236 × 10−1
    (1.55 × 10−2) −    		 5.3589 × 10−1
    (3.53 × 10−2) −    		 6.4650 × 10−1
    (4.91 × 10−2) −    		 5.2947 × 10−1
    (3.36 × 10−2) −    		 8.2131 × 10−1
    (3.53 × 10−2)
    WFG83 2.0563 × 10−1
    (9.02 × 10−3) −    		 2.1117 × 10−1
    (9.38 × 10−3) −    		 2.5649 × 10−1
    (1.96 × 10−2) =    		 2.7987 × 10−1
    (1.16 × 10−2) +    		 2.0196 × 10−1
    (9.26 × 10−3) −    		 2.5619 × 10−1
    (2.12 × 10−2)
    4 2.2504 × 10−1
    (1.58 × 10−2) −    		 2.4391 × 10−1
    (1.41 × 10−2) −    		 2.7342 × 10−1
    (2.59 × 10−2) =    		 3.0350 × 10−1
    (2.26 × 10−2) =    		 2.6327 × 10−1
    (1.60 × 10−2) −    		 2.9119 × 10−1
    (2.71 × 10−2)
    6 2.9374 × 10−1
    (2.14 × 10−2) −    		 3.1166 × 10−1
    (1.44 × 10−2) −    		 3.5815 × 10−1
    (3.05 × 10−2) −    		 3.4101 × 10−1
    (1.70 × 10−2) −    		 3.4154 × 10−1
    (1.45 × 10−2) −    		 3.8034 × 10−1
    (1.95 × 10−2)
    8 3.3827 × 10−1
    (2.25 × 10−2) −    		 3.9489 × 10−1
    (2.24 × 10−2) −    		 4.3731 × 10−1
    (2.62 × 10−2) −    		 4.9176 × 10−1
    (3.96 × 10−2) −    		 4.3251 × 10−1
    (3.33 × 10−2) −    		 5.4014 × 10−1
    (4.21 × 10−2)
    10 3.7424 × 10−1
    (1.66 × 10−2) −    		 4.1948 × 10−1
    (1.39 × 10−2) −    		 4.3470 × 10−1
    (2.64 × 10−2) −    		 5.0197 × 10−1
    (4.74 × 10−2) −    		 4.7105 × 10−1
    (2.48 × 10−2) −    		 6.1406 × 10−1
    (5.41 × 10−2)
    WFG93 2.1195 × 10−1
    (1.55 × 10−2) −    		 2.2346 × 10−1
    (2.06 × 10−2) −    		 2.8163 × 10−1
    (3.02 × 10−2) =    		 2.6005 × 10−1
    (1.97 × 10−2) =    		 2.2416 × 10−1
    (2.21 × 10−2) −    		 2.6152 × 10−1
    (3.23 × 10−2)
    4 3.1377 × 10−1
    (2.84 × 10−2) −    		 3.3327 × 10−1
    (1.77 × 10−2) −    		 3.5582 × 10−1
    (4.06 × 10−2) −    		 3.8240 × 10−1
    (6.37 × 10−2) =    		 3.0577 × 10−1
    (1.98 × 10−2) −    		 4.1389 × 10−1
    (6.77 × 10−2)
    6 3.1971 × 10−1
    (2.97 × 10−2) −    		 3.3719 × 10−1
    (1.64 × 10−2) −    		 3.8869 × 10−1
    (4.85 × 10−2) −    		 4.8276 × 10−1
    (5.15 × 10−2) =    		 3.4419 × 10−1
    (2.18 × 10−2) −    		 4.7707 × 10−1
    (6.45 × 10−2)
    8 4.5419 × 10−1
    (3.01 × 10−2) −    		 4.7014 × 10−1
    (2.00 × 10−2) −    		 5.4358 × 10−1
    (3.44 × 10−2) −    		 6.3781 × 10−1
    (6.24 × 10−2) =    		 4.9095 × 10−1
    (3.53 × 10−2) −    		 6.5968 × 10−1
    (6.23 × 10−2)
    10 4.8221 × 10−1
    (2.31 × 10−2) −    		 4.7502 × 10−1
    (1.84 × 10−2) −    		 5.6346 × 10−1
    (4.24 × 10−2) −    		 6.1737 × 10−1
    (3.46 × 10−2) −    		 5.0326 × 10−1
    (3.14 × 10−2) −    		 6.7554 × 10−1
    (4.40 × 10−2)
    +/−/=0/43/21/39/55/29/117/25/132/39/4
    下载: 导出CSV

    表  6  汽车碰撞优化设计问题上获得的IGD和HV的平均值

    Table  6  The average values of IGD and HV obtained on the car crash optimization design problem

    算法名称IGDHV
    NSGA-III2.49750.0308
    CPS-MOEA2.36650.0340
    CSEA1.31440.0368
    K-RVEA0.71420.0374
    MOEA/D-EGO0.67930.0384
    K-RSEA0.49200.0389
    下载: 导出CSV
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出版历程
  • 收稿日期:  2020-09-24
  • 录用日期:  2020-12-14
  • 网络出版日期:  2021-01-11
  • 刊出日期:  2022-10-14

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