基于SOM聚类和自适应算子选择的高维多目标进化算法

doi:10.12263/DZXB.20201438

电子学报 ›› 2022, Vol. 50 ›› Issue (8): 1959-1974.DOI: 10.12263/DZXB.20201438

基于SOM聚类和自适应算子选择的高维多目标进化算法

钟沛龙¹^,², 黎明¹^,²(), 何超³, 陈昊¹^,²

^1.南昌航空大学信息工程学院，江西南昌 330063
^2.南昌航空大学无损检测技术教育部重点实验室，江西南昌 330063
^3.南京航空航天大学自动化学院，江苏南京 211106

收稿日期:2020-12-15 修回日期:2021-06-04 出版日期:2022-08-25
- 通讯作者:
- 黎明
- 作者简介:
- 钟沛龙男，1997年生，江西赣州人.南昌航空大学硕士研究生，主要研究方向为进化算法.E-mail: zplong33@163.com
  黎明（通讯作者）男，1965年生，江西樟树人.南昌航空大学信息工程学院教授，南京航空航天大学博士生导师.主要研究方向为图像处理与模式识别、智能计算等.
  何超男，1992年生，浙江诸暨人.南京航空航天大学博士研究生.主要研究方向为进化算法与图像处理、模式识别等.E-mail: hechao92918@163.com
  陈昊男，1982年生，山东平度人.南昌航空大学信息工程学院教授.主要研究方向为进化算法理论与应用、图像处理与模式识别等.E-mail: chenhaoshl@163.com
- 基金资助:
- 国家自然科学基金 (61772255); 江西省教育厅科学技术项目 (GJJ170608); 江西省优势科技创新团队计划项目 (20181BCB24008); 江西省自然科学基金 (20181BAB202025); 无损检测教育部重点实验室开放基金 (EW201708505); 江西省研究生创新专项资金项目 (YC2020-S520)

Many-Objective Evolutionary Algorithm Based on SOM Clustering and Adaptive Operator Selection

ZHONG Pei-long¹^,², LI Ming¹^,²(), HE Chao³, CHEN Hao¹^,²

^1.School of Information Engineering，Nanchang Hangkong University，Nanchang，Jiangxi 330063，China
^2.Key Laboratory of Nondestructive Testing，Ministry of Education，Nanchang Hangkong University，Nanchang，Jiangxi 330063，China
^3.School of Automation，Nanjing University of Aeronautics and Astronautics，Nanjing，Jiangsu 211106，China

Received:2020-12-15 Revised:2021-06-04 Online:2022-08-25 Published:2022-09-08
- Corresponding author:
- LI Ming

摘要/Abstract

摘要：

在高维多目标进化算法中，通常利用重组算子产生优质子代来引导种群搜索，已有研究表明，利用相似个体进行重组可以提高子代个体质量.由于自组织映射（Self-Organizing Mapping，SOM）网络能够通过聚类的方式保持种群个体原有的拓扑逻辑关系并获得个体的相似信息，因此本文提出一种基于SOM聚类和自适应算子选择的高维多目标进化算法（Many-Objective Evolutionary Algorithm based on SOM Clustering and Adaptive Operator Selection，MaOEA-SCAOS）.本文首先通过自组织映射网络进行种群分类，提取个体数据结构信息，并利用相似性构建邻域交配池；然后根据类内个体支配信息进行自适应算子选择，提高算法搜索和收敛性能；最后，采用环境选择策略对种群进行多样性管理以保证种群在帕累托前沿均匀分布.仿真结果表明，本文提出的基于SOM聚类和自适应算子选择（SOM Clustering and Adaptive Operator Selection，SCAOS）方法在处理高维多目标优化问题时具有较强的竞争力并且性能指标整体优于其他方法.

关键词: 高维多目标优化, 自组织映射网络, 聚类, 自适应选择, 进化算法

Abstract:

In the many-objective evolutionary algorithm, recombination operators are usually used to generate high-quality offspring to guide the population search. Previous studies have shown that using similar individuals to reorganize can improve the quality of individual offspring. Since the self-organizing maping(SOM) network can maintain the original topological relationship of the population individuals and obtain the similar information of the individuals through clustering, this paper proposes a many-objective evolutionary algorithm based on SOM clustering and adaptive operator selection(MaOEA-SCAOS). First, the proposed method use self-organizing mapping network to classify the population, extract individual data structure information, and use similarity to build a neighborhood mating pool. Then the method select the adaptive operator based on the individual dominance information in the class to improve the search and convergence performance. Finally, the environmental selection strategy is adopted to manage the diversity of the population to ensure that the population is evenly distributed in the Pareto front. The experimental simulation results show that the SOM clustering and adaptive operator selection(SCAOS) method proposed in this paper has strong competitiveness while dealing with many-objective optimization problems, and the overall performance index is better than other methods.

Key words: many-objective optimization, self-organizing mapping network, clustering, adaptive selection, evolutionary algorithm

中图分类号:

TP18

钟沛龙, 黎明, 何超, 陈昊. 基于SOM聚类和自适应算子选择的高维多目标进化算法[J]. 电子学报, 2022, 50(8): 1959-1974.

ZHONG Pei-long, LI Ming, HE Chao, CHEN Hao. Many-Objective Evolutionary Algorithm Based on SOM Clustering and Adaptive Operator Selection[J]. Acta Electronica Sinica, 2022, 50(8): 1959-1974.

图/表 11

图1 典型的SOM拓扑结构

表1 种群大小设置

目标维数M	$H 1, H 2$	种群大小N
3	(13,0)	105
5	(5,0)	126
8	(3,2)	156
10	(3,2)	275

表1 种群大小设置

目标维数M	$H 1, H 2$	种群大小N
3	(13,0)	105
5	(5,0)	126
8	(3,2)	156
10	(3,2)	275

图2 不同交配池大小在DTLZ测试问题独立运行20次平均IGD指标值

图3 不同阈值大小在DTLZ测试问题独立运行20次平均IGD指标值

表2 6种算法在WFG1~WFG9测试问题上获得的HV指标值的实验结果

8.850 9e-1

(2.54e-2)

6.233 9e-1

(3.99e-2) -

4.837 1e-2

(9.38e-3) -

9.239 5e-1

(6.92e-3) +

8.955 7e-1

(1.38e-2) =

9.001 2e-1

(2.13e-2) +

8.206 1e-1

(3.39e-2)

4.404 2e-1

(3.23e-2) -

1.750 7e-1

(1.00e-2) -

9.341 4e-1

(2.66e-2) +

9.187 6e-1

(2.86e-2) +

8.595 6e-1

(4.73e-2) +

9.946 4e-1

(1.69e-2)

4.361 5e-1

(4.08e-2) -

2.232 1e-1

(3.16e-3) -

9.954 0e-1

(1.29e-3) +

9.949 8e-1

(1.02e-3) +

9.990 3e-1

(2.73e-4) +

9.989 7e-1

(4.24e-4)

3.649 3e-1

(3.15e-2) -

2.164 5e-1

(1.69e-3) -

9.974 8e-1

(9.71e-4) -

9.967 4e-1

(6.15e-4) -

9.992 9e-1

(2.98e-4) +

WFG2

9.322 9e-1

(9.15e-4)

9.132 1e-1

(2.07e-3) -

8.828 5e-1

(4.04e-3) -

9.301 6e-1

(2.23e-3) -

8.926 9e-1

(1.66e-2) -

9.322 5e-1

(6.85e-4) =

9.931 6e-1

(1.14e-3)

9.679 0e-1

(3.51e-3) -

7.956 4e-1

(1.47e-2) -

9.887 9e-1

(1.52e-3) -

9.407 3e-1

(1.16e-2) -

9.926 6e-1

(1.02e-3) =

9.969 3e-1

(1.86e-3)

9.854 4e-1

(3.51e-3) -

6.869 3e-1

(2.84e-2) -

9.937 4e-1

(1.03e-3) -

9.689 4e-1

(7.11e-3) -

9.974 4e-1

(8.93e-4) =

9.975 9e-1

(1.31e-3)

9.838 3e-1

(3.57e-3) -

6.377 8e-1

(2.43e-2) -

9.950 2e-1

(7.25e-4) -

9.814 8e-1

(5.32e-3) -

9.976 1e-1

(1.05e-3) =

WFG3

3.885 9e-1

(5.09e-3)

3.885 2e-1

(2.79e-3) =

3.072 0e-1

(7.73e-3) -

3.782 2e-1

(7.57e-3) -

2.209 1e-1

(3.36e-2) -

3.893 7e-1

(2.92e-3) =

1.551 0e-1

(1.89e-2)

1.749 2e-1

(1.77e-2) +

1.890 3e-2

(1.06e-2) -

6.319 9e-2

(2.39e-2) -

0.000 0e+0

(0.00e+0) -

1.474 3e-1

(1.55e-2) =

5.004 1e-2

(2.05e-2)

3.658 6e-2

(1.70e-2) -

0.000 0e+0

(0.00e+0) -

0.000 0e+0

(0.00e+0) -

0.000 0e+0

(0.00e+0) -

4.985 0e-2

(2.03e-2) =

1.889 5e-3

(6.17e-3)

0.000 0e+0

(0.00e+0) =

0.000 0e+0

(0.00e+0) =

0.000 0e+0

(0.00e+0) =

0.000 0e+0

(0.00e+0) =

6.262 2e-3

(1.28e-2) =

WFG4

5.592 2e-1

(6.52e-4)

5.011 3e-1

(4.60e-3) -

4.585 9e-1

(4.60e-3) -

5.410 7e-1

(2.59e-3) -

4.992 9e-1

(1.30e-2) -

5.588 3e-1

(8.39e-4) =

7.844 1e-1

(1.76e-3)

6.289 6e-1

(1.12e-2) -

4.765 2e-1

(1.07e-2) -

7.653 9e-1

(2.89e-3) -

5.340 7e-1

(3.20e-2) -

7.826 5e-1

(2.16e-3) -

9.167 3e-1

(1.19e-2)

6.362 9e-1

(2.47e-2) -

4.978 7e-1

(1.07e-2) -

9.079 7e-1

(3.59e-3) -

5.433 8e-1

(2.31e-2) -

9.147 0e-1

(1.00e-2) -

9.652 8e-1

(1.10e-3)

6.287 9e-1

(1.84e-2) -

5.290 0e-1

(9.13e-3) -

9.621 6e-1

(8.11e-4) -

5.936 2e-1

(3.51e-2) -

9.581 4e-1

(7.71e-3) -

WFG5

5.216 6e-1

(8.52e-5)

4.812 8e-1

(5.66e-3) -

4.517 4e-1

(8.00e-3) -

5.046 6e-1

(3.90e-3) -

4.800 5e-1

(8.31e-3) -

5.216 3e-1

(1.18e-4) =

7.429 2e-1

(5.39e-4)

5.810 9e-1

(1.44e-2) -

4.892 2e-1

(1.23e-2) -

7.247 8e-1

(3.03e-3) -

5.599 0e-1

(1.56e-2) -

7.424 7e-1

(8.36e-4) =

8.631 7e-1

(2.43e-4)

5.533 0e-1

(2.78e-2) -

5.445 8e-1

(8.27e-3) -

8.410 3e-1

(5.56e-3) -

5.891 5e-1

(2.68e-2) -

8.627 9e-1

(3.62e-4) -

9.039 5e-1

(1.86e-4)

5.758 0e-1

(2.46e-2) -

6.253 3e-1

(8.52e-3) -

8.978 5e-1

(8.76e-4) -

6.397 9e-1

(2.30e-2) -

9.034 6e-1

(3.26e-4) -

WFG6

5.098 9e-1

(1.04e-2)

4.526 0e-1

(1.36e-2) -

4.481 2e-1

(1.49e-2) -

4.846 9e-1

(9.70e-3) -

4.491 6e-1

(1.82e-2) -

5.117 1e-1

(1.31e-2) =

7.310 8e-1

(1.45e-2)

5.478 0e-1

(2.81e-2) -

4.363 7e-1

(1.37e-2) -

6.951 0e-1

(1.82e-2) -

4.260 9e-1

(4.71e-2) -

7.185 1e-1

(1.69e-2) -

8.450 3e-1

(1.55e-2)

5.665 0e-1

(3.85e-2) -

5.027 1e-1

(1.32e-2) -

7.996 5e-1

(2.02e-2) -

3.752 0e-1

(3.62e-2) -

8.492 6e-1

(1.56e-2) =

8.843 2e-1

(1.04e-2)

6.120 6e-1

(3.36e-2) -

5.759 6e-1

(1.31e-2) -

8.735 1e-1

(1.84e-2) -

4.085 6e-1

(3.04e-2) -

8.767 2e-1

(1.62e-2) =

WFG7

5.599 5e-1

(4.22e-4)

5.086 9e-1

(3.87e-3) -

4.562 3e-1

(6.39e-3) -

5.469 2e-1

(2.73e-3) -

4.979 6e-1

(1.45e-2) -

5.594 6e-1

(4.26e-4) -

7.881 6e-1

(8.79e-4)

6.192 6e-1

(2.02e-2) -

4.061 0e-1

(1.48e-2) -

7.730 3e-1

(3.13e-3) -

4.723 5e-1

(3.30e-2) -

7.859 4e-1

(1.65e-3) -

9.203 1e-1

(5.66e-4)

5.960 6e-1

(2.99e-2) -

4.273 5e-1

(1.39e-2) -

8.931 1e-1

(6.83e-3) -

3.914 4e-1

(5.96e-2) -

9.189 6e-1

(8.64e-4) -

9.658 9e-1

(4.26e-3)

6.274 9e-1

(2.54e-2) -

4.562 0e-1

(1.11e-2) -

9.581 1e-1

(5.19e-3) -

4.733 6e-1

(2.52e-2) -

9.596 2e-1

(1.43e-2) -

WFG8

4.786 7e-1

(2.51e-3)

4.297 2e-1

(3.45e-3) -

3.785 9e-1

(8.33e-3) -

4.505 2e-1

(4.87e-3) -

3.989 0e-1

(1.81e-2) -

4.742 7e-1

(3.21e-3) -

6.835 7e-1

(7.80e-3)

5.033 5e-1

(1.25e-2) -

3.418 6e-1

(1.37e-2) -

6.321 8e-1

(5.46e-3) -

3.550 9e-1

(5.73e-2) -

6.735 9e-1

(3.61e-3) -

8.110 6e-1

(3.72e-2)

5.608 5e-1

(1.43e-2) -

3.741 3e-1

(1.09e-2) -

7.614 4e-1

(3.24e-2) -

2.195 5e-1

(4.90e-2) -

7.939 3e-1

(2.18e-2) =

8.855 0e-1

(2.73e-2)

5.912 5e-1

(1.26e-2) -

4.075 0e-1

(1.17e-2) -

8.334 5e-1

(5.79e-2) -

3.541 4e-1

(5.37e-2) -

8.758 3e-1

(1.47e-2) =

WFG9

5.187 3e-1

(4.20e-2)

4.933 2e-1

(7.48e-3) -

4.755 9e-1

(1.26e-2) -

5.169 0e-1

(3.83e-2) -

4.867 0e-1

(1.05e-2) -

5.357 9e-1

(4.20e-3) =

7.321 4e-1

(3.06e-2)

5.434 2e-1

(4.40e-2) -

3.846 6e-1

(1.92e-2) -

7.416 0e-1

(3.06e-3) =

5.749 8e-1

(2.21e-2) -

7.158 4e-1

(3.95e-2) -

8.430 6e-1

(4.63e-2)

5.159 2e-1

(5.19e-2) -

4.066 5e-1

(1.31e-2) -

8.639 5e-1

(3.94e-3) =

6.442 7e-1

(2.24e-2) -

8.289 9e-1

(5.46e-2) -

8.850 6e-1

(5.12e-2)

5.942 2e-1

(1.54e-2) -

4.721 1e-1

(8.66e-3) -

9.199 3e-1

(2.48e-3) +

6.894 0e-1

(3.50e-2) -

8.842 8e-1

(3.67e-2) -

表2 6种算法在WFG1~WFG9测试问题上获得的HV指标值的实验结果

问题

MaOEA-SCAOS

MOCell

RM-MEDA

KnEA

MaOEA-CSS

NSGA-III

WFG1

8.850 9e-1

(2.54e-2)

6.233 9e-1

(3.99e-2) -

4.837 1e-2

(9.38e-3) -

9.239 5e-1

(6.92e-3) +

8.955 7e-1

(1.38e-2) =

9.001 2e-1

(2.13e-2) +

8.206 1e-1

(3.39e-2)

4.404 2e-1

(3.23e-2) -

1.750 7e-1

(1.00e-2) -

9.341 4e-1

(2.66e-2) +

9.187 6e-1

(2.86e-2) +

8.595 6e-1

(4.73e-2) +

9.946 4e-1

(1.69e-2)

4.361 5e-1

(4.08e-2) -

2.232 1e-1

(3.16e-3) -

9.954 0e-1

(1.29e-3) +

9.949 8e-1

(1.02e-3) +

9.990 3e-1

(2.73e-4) +

9.989 7e-1

(4.24e-4)

3.649 3e-1

(3.15e-2) -

2.164 5e-1

(1.69e-3) -

9.974 8e-1

(9.71e-4) -

9.967 4e-1

(6.15e-4) -

9.992 9e-1

(2.98e-4) +

WFG2

9.322 9e-1

(9.15e-4)

9.132 1e-1

(2.07e-3) -

8.828 5e-1

(4.04e-3) -

9.301 6e-1

(2.23e-3) -

8.926 9e-1

(1.66e-2) -

9.322 5e-1

(6.85e-4) =

9.931 6e-1

(1.14e-3)

9.679 0e-1

(3.51e-3) -

7.956 4e-1

(1.47e-2) -

9.887 9e-1

(1.52e-3) -

9.407 3e-1

(1.16e-2) -

9.926 6e-1

(1.02e-3) =

9.969 3e-1

(1.86e-3)

9.854 4e-1

(3.51e-3) -

6.869 3e-1

(2.84e-2) -

9.937 4e-1

(1.03e-3) -

9.689 4e-1

(7.11e-3) -

9.974 4e-1

(8.93e-4) =

9.975 9e-1

(1.31e-3)

9.838 3e-1

(3.57e-3) -

6.377 8e-1

(2.43e-2) -

9.950 2e-1

(7.25e-4) -

9.814 8e-1

(5.32e-3) -

9.976 1e-1

(1.05e-3) =

WFG3

3.885 9e-1

(5.09e-3)

3.885 2e-1

(2.79e-3) =

3.072 0e-1

(7.73e-3) -

3.782 2e-1

(7.57e-3) -

2.209 1e-1

(3.36e-2) -

3.893 7e-1

(2.92e-3) =

1.551 0e-1

(1.89e-2)

1.749 2e-1

(1.77e-2) +

1.890 3e-2

(1.06e-2) -

6.319 9e-2

(2.39e-2) -

0.000 0e+0

(0.00e+0) -

1.474 3e-1

(1.55e-2) =

5.004 1e-2

(2.05e-2)

3.658 6e-2

(1.70e-2) -

0.000 0e+0

(0.00e+0) -

0.000 0e+0

(0.00e+0) -

0.000 0e+0

(0.00e+0) -

4.985 0e-2

(2.03e-2) =

1.889 5e-3

(6.17e-3)

0.000 0e+0

(0.00e+0) =

0.000 0e+0

(0.00e+0) =

0.000 0e+0

(0.00e+0) =

0.000 0e+0

(0.00e+0) =

6.262 2e-3

(1.28e-2) =

WFG4

5.592 2e-1

(6.52e-4)

5.011 3e-1

(4.60e-3) -

4.585 9e-1

(4.60e-3) -

5.410 7e-1

(2.59e-3) -

4.992 9e-1

(1.30e-2) -

5.588 3e-1

(8.39e-4) =

7.844 1e-1

(1.76e-3)

6.289 6e-1

(1.12e-2) -

4.765 2e-1

(1.07e-2) -

7.653 9e-1

(2.89e-3) -

5.340 7e-1

(3.20e-2) -

7.826 5e-1

(2.16e-3) -

9.167 3e-1

(1.19e-2)

6.362 9e-1

(2.47e-2) -

4.978 7e-1

(1.07e-2) -

9.079 7e-1

(3.59e-3) -

5.433 8e-1

(2.31e-2) -

9.147 0e-1

(1.00e-2) -

9.652 8e-1

(1.10e-3)

6.287 9e-1

(1.84e-2) -

5.290 0e-1

(9.13e-3) -

9.621 6e-1

(8.11e-4) -

5.936 2e-1

(3.51e-2) -

9.581 4e-1

(7.71e-3) -

WFG5

5.216 6e-1

(8.52e-5)

4.812 8e-1

(5.66e-3) -

4.517 4e-1

(8.00e-3) -

5.046 6e-1

(3.90e-3) -

4.800 5e-1

(8.31e-3) -

5.216 3e-1

(1.18e-4) =

7.429 2e-1

(5.39e-4)

5.810 9e-1

(1.44e-2) -

4.892 2e-1

(1.23e-2) -

7.247 8e-1

(3.03e-3) -

5.599 0e-1

(1.56e-2) -

7.424 7e-1

(8.36e-4) =

8.631 7e-1

(2.43e-4)

5.533 0e-1

(2.78e-2) -

5.445 8e-1

(8.27e-3) -

8.410 3e-1

(5.56e-3) -

5.891 5e-1

(2.68e-2) -

8.627 9e-1

(3.62e-4) -

9.039 5e-1

(1.86e-4)

5.758 0e-1

(2.46e-2) -

6.253 3e-1

(8.52e-3) -

8.978 5e-1

(8.76e-4) -

6.397 9e-1

(2.30e-2) -

9.034 6e-1

(3.26e-4) -

WFG6

5.098 9e-1

(1.04e-2)

4.526 0e-1

(1.36e-2) -

4.481 2e-1

(1.49e-2) -

4.846 9e-1

(9.70e-3) -

4.491 6e-1

(1.82e-2) -

5.117 1e-1

(1.31e-2) =

7.310 8e-1

(1.45e-2)

5.478 0e-1

(2.81e-2) -

4.363 7e-1

(1.37e-2) -

6.951 0e-1

(1.82e-2) -

4.260 9e-1

(4.71e-2) -

7.185 1e-1

(1.69e-2) -

8.450 3e-1

(1.55e-2)

5.665 0e-1

(3.85e-2) -

5.027 1e-1

(1.32e-2) -

7.996 5e-1

(2.02e-2) -

3.752 0e-1

(3.62e-2) -

8.492 6e-1

(1.56e-2) =

8.843 2e-1

(1.04e-2)

6.120 6e-1

(3.36e-2) -

5.759 6e-1

(1.31e-2) -

8.735 1e-1

(1.84e-2) -

4.085 6e-1

(3.04e-2) -

8.767 2e-1

(1.62e-2) =

WFG7

5.599 5e-1

(4.22e-4)

5.086 9e-1

(3.87e-3) -

4.562 3e-1

(6.39e-3) -

5.469 2e-1

(2.73e-3) -

4.979 6e-1

(1.45e-2) -

5.594 6e-1

(4.26e-4) -

7.881 6e-1

(8.79e-4)

6.192 6e-1

(2.02e-2) -

4.061 0e-1

(1.48e-2) -

7.730 3e-1

(3.13e-3) -

4.723 5e-1

(3.30e-2) -

7.859 4e-1

(1.65e-3) -

9.203 1e-1

(5.66e-4)

5.960 6e-1

(2.99e-2) -

4.273 5e-1

(1.39e-2) -

8.931 1e-1

(6.83e-3) -

3.914 4e-1

(5.96e-2) -

9.189 6e-1

(8.64e-4) -

9.658 9e-1

(4.26e-3)

6.274 9e-1

(2.54e-2) -

4.562 0e-1

(1.11e-2) -

9.581 1e-1

(5.19e-3) -

4.733 6e-1

(2.52e-2) -

9.596 2e-1

(1.43e-2) -

WFG8

4.786 7e-1

(2.51e-3)

4.297 2e-1

(3.45e-3) -

3.785 9e-1

(8.33e-3) -

4.505 2e-1

(4.87e-3) -

3.989 0e-1

(1.81e-2) -

4.742 7e-1

(3.21e-3) -

6.835 7e-1

(7.80e-3)

5.033 5e-1

(1.25e-2) -

3.418 6e-1

(1.37e-2) -

6.321 8e-1

(5.46e-3) -

3.550 9e-1

(5.73e-2) -

6.735 9e-1

(3.61e-3) -

8.110 6e-1

(3.72e-2)

5.608 5e-1

(1.43e-2) -

3.741 3e-1

(1.09e-2) -

7.614 4e-1

(3.24e-2) -

2.195 5e-1

(4.90e-2) -

7.939 3e-1

(2.18e-2) =

8.855 0e-1

(2.73e-2)

5.912 5e-1

(1.26e-2) -

4.075 0e-1

(1.17e-2) -

8.334 5e-1

(5.79e-2) -

3.541 4e-1

(5.37e-2) -

8.758 3e-1

(1.47e-2) =

WFG9

5.187 3e-1

(4.20e-2)

4.933 2e-1

(7.48e-3) -

4.755 9e-1

(1.26e-2) -

5.169 0e-1

(3.83e-2) -

4.867 0e-1

(1.05e-2) -

5.357 9e-1

(4.20e-3) =

7.321 4e-1

(3.06e-2)

5.434 2e-1

(4.40e-2) -

3.846 6e-1

(1.92e-2) -

7.416 0e-1

(3.06e-3) =

5.749 8e-1

(2.21e-2) -

7.158 4e-1

(3.95e-2) -

8.430 6e-1

(4.63e-2)

5.159 2e-1

(5.19e-2) -

4.066 5e-1

(1.31e-2) -

8.639 5e-1

(3.94e-3) =

6.442 7e-1

(2.24e-2) -

8.289 9e-1

(5.46e-2) -

8.850 6e-1

(5.12e-2)

5.942 2e-1

(1.54e-2) -

4.721 1e-1

(8.66e-3) -

9.199 3e-1

(2.48e-3) +

6.894 0e-1

(3.50e-2) -

8.842 8e-1

(3.67e-2) -

+/-/=

1/34/1

0/35/1

4/29/3

2/32/2

4/15/17

表3 6种算法在WFG1~WFG9测试问题上获得的IGD指标值的实验结果

2.170 8e-1

(3.79e-2)

6.962 4e-1

(8.67e-2) -

2.032 0e+0

(3.09e-2) -

2.119 5e-1

(2.71e-2) =

2.970 0e-1

(3.23e-2) -

1.983 3e-1

(3.07e-2) =

6.312 3e-1

(3.99e-2)

1.709 9e+0

(9.63e-2) -

2.470 4e+0

(3.72e-2) -

5.095 7e-1

(2.04e-2) +

6.655 5e-1

(4.10e-2) -

5.736 5e-1

(6.24e-2) +

8.934 7e-1

(3.37e-2)

2.176 2e+0

(1.01e-1) -

3.029 0e+0

(3.62e-2) -

9.425 6e-1

(2.18e-2) -

1.296 3e+0

(7.21e-2) -

8.854 6e-1

(2.19e-2) =

9.757 4e-1

(2.01e-2)

2.475 2e+0

(1.36e-1) -

3.344 2e+0

(2.93e-2) -

1.018 3e+0

(1.51e-2) -

1.409 1e+0

(6.05e-2) -

9.581 9e-1

(1.56e-2) +

WFG2

1.503 3e-1

(1.37e-3)

2.115 7e-1

(8.97e-3) -

2.475 6e-1

(1.11e-2) -

1.845 5e-1

(7.77e-3) -

2.242 6e-1

(2.41e-2) -

1.500 9e-1

(1.14e-3) =

4.678 2e-1

(2.37e-3)

8.048 0e-1

(5.82e-2) -

1.068 9e+0

(1.01e-1) -

5.334 9e-1

(2.58e-2) -

6.836 0e-1

(4.20e-2) -

4.681 1e-1

(2.29e-3) =

1.024 6e+0

(1.38e-1)

1.466 8e+0

(7.29e-2) -

2.068 9e+0

(1.06e-1) -

1.063 1e+0

(2.96e-2) =

1.571 9e+0

(8.43e-2) -

9.880 8e-1

(1.54e-1) =

1.163 3e+0

(1.23e-1)

1.532 7e+0

(7.02e-2) -

2.400 1e+0

(8.09e-2) -

1.184 3e+0

(3.52e-2) =

1.680 4e+0

(8.61e-2) -

1.183 4e+0

(1.42e-1) =

WFG3

1.002 3e-1

(1.01e-2)

1.146 0e-1

(1.12e-2) -

2.798 5e-1

(2.26e-2) -

1.064 2e-1

(1.14e-2) =

6.106 2e-1

(1.69e-1) -

9.752 5e-2

(6.74e-3) =

5.372 8e-1

(6.52e-2)

5.272 2e-1

(8.51e-2) =

1.083 7e+0

(7.96e-2) -

5.426 1e-1

(7.19e-2) =

2.171 1e+0

(3.23e-1) -

5.766 4e-1

(5.62e-2) -

1.527 8e+0

(3.03e-1)

1.237 9e+0

(2.01e-1) +

1.765 5e+0

(1.35e-1) -

1.419 6e+0

(2.55e-1) =

5.612 5e+0

(4.85e-1) -

1.686 3e+0

(2.20e-1) -

1.812 7e+0

(2.34e-1)

1.411 2e+0

(2.16e-1) +

2.035 3e+0

(1.93e-1) -

1.830 1e+0

(2.39e-1) =

7.141 3e+0

(4.67e-1) -

1.968 2e+0

(3.01e-1) =

WFG4

2.042 6e-1

(1.43e-4)

2.849 3e-1

(1.03e-2) -

3.383 9e-1

(1.51e-2) -

2.489 6e-1

(8.80e-3) -

3.021 6e-1

(3.60e-2) -

2.044 7e-1

(1.83e-4) -

1.174 8e+0

(1.43e-3)

1.229 6e+0

(2.49e-2) -

1.490 0e+0

(2.24e-2) -

1.232 3e+0

(1.98e-2) -

1.793 6e+0

(1.95e-1) -

1.174 6e+0

(1.83e-3) =

2.970 1e+0

(3.88e-2)

3.369 4e+0

(5.47e-2) -

3.463 7e+0

(2.29e-2) -

3.450 1e+0

(5.66e-2) -

4.525 1e+0

(1.70e-1) -

2.978 0e+0

(7.20e-2) =

4.547 4e+0

(4.68e-3)

4.489 2e+0

(3.76e-2) +

4.505 3e+0

(2.58e-2) +

4.578 4e+0

(3.80e-2) -

6.216 5e+0

(2.23e-1) -

4.547 6e+0

(2.88e-2) -

WFG5

2.145 1e-1

(1.09e-4)

2.799 8e-1

(9.11e-3) -

3.122 2e-1

(1.14e-2) -

2.606 0e-1

(1.10e-2) -

2.810 3e-1

(1.42e-2) -

2.145 4e-1

(1.03e-4) =

1.162 9e+0

(5.70e-4)

1.209 8e+0

(1.80e-2) -

1.399 1e+0

(3.04e-2) -

1.211 1e+0

(1.88e-2) -

1.547 2e+0

(7.30e-2) -

1.162 7e+0

(1.37e-3) =

2.941 0e+0

(1.31e-3)

3.423 5e+0

(6.63e-2) -

3.623 6e+0

(4.71e-2) -

3.338 1e+0

(2.98e-2) -

4.286 7e+0

(2.09e-1) -

2.941 3e+0

(1.32e-3) =

4.531 1e+0

(3.32e-3)

4.599 6e+0

(5.91e-2) -

4.764 3e+0

(3.95e-2) -

4.565 3e+0

(3.36e-2) -

5.903 8e+0

(1.79e-1) -

4.530 0e+0

(3.41e-3) =

WFG6

2.239 9e-1

(7.93e-3)

3.326 4e-1

(1.70e-2) -

3.455 4e-1

(1.49e-2) -

2.893 7e-1

(1.31e-2) -

3.865 9e-1

(7.01e-2) -

2.230 2e-1

(1.08e-2) =

1.164 7e+0

(2.80e-3)

1.316 0e+0

(3.06e-2) -

1.597 0e+0

(3.88e-2) -

1.246 2e+0

(2.03e-2) -

2.180 0e+0

(2.28e-1) -

1.162 7e+0

(1.80e-3) +

2.947 7e+0

(5.27e-3)

3.506 5e+0

(4.39e-2) -

3.582 9e+0

(2.48e-2) -

3.490 8e+0

(6.50e-2) -

5.515 6e+0

(2.01e-1) -

2.953 0e+0

(4.60e-3) -

4.544 7e+0

(6.62e-3)

4.617 3e+0

(4.44e-2) -

4.555 9e+0

(2.39e-2) -

4.669 0e+0

(4.48e-2) -

7.288 3e+0

(2.25e-1) -

4.543 9e+0

(8.18e-3) =

WFG7

2.047 4e-1

(2.06e-4)

2.826 6e-1

(1.04e-2) -

3.363 6e-1

(1.00e-2) -

2.399 1e-1

(8.46e-3) -

3.223 3e-1

(6.99e-2) -

2.047 8e-1

(2.62e-4) =

1.176 3e+0

(8.69e-4)

1.255 4e+0

(3.07e-2) -

1.583 9e+0

(2.76e-2) -

1.230 3e+0

(1.81e-2) -

2.314 8e+0

(1.73e-1) -

1.175 6e+0

(1.12e-3) +

2.967 7e+0

(7.16e-3)

3.499 1e+0

(9.59e-2) -

3.614 3e+0

(3.45e-2) -

3.324 5e+0

(4.54e-2) -

4.919 2e+0

(2.46e-1) -

2.968 8e+0

(6.33e-3) =

4.547 8e+0

(2.02e-2)

4.584 2e+0

(4.81e-2) -

4.630 3e+0

(3.39e-2) -

4.431 9e+0

(3.86e-2) +

6.175 7e+0

(2.19e-1) -

4.644 6e+0

(2.51e-1) =

WFG8

2.671 6e-1

(3.98e-3)

3.834 3e-1

(5.79e-3) -

4.457 8e-1

(1.56e-2) -

3.322 4e-1

(1.14e-2) -

4.431 3e-1

(8.88e-2) -

2.751 7e-1

(5.22e-3) -

1.148 0e+0

(1.28e-3)

1.446 1e+0

(2.49e-2) -

1.773 6e+0

(2.73e-2) -

1.282 7e+0

(2.25e-2) -

1.960 5e+0

(2.06e-1) -

1.154 7e+0

(8.40e-3) -

3.198 2e+0

(2.62e-1)

3.695 7e+0

(4.49e-2) -

3.769 1e+0

(2.87e-2) -

3.533 6e+0

(6.48e-2) -

5.381 5e+0

(3.58e-1) -

3.236 4e+0

(2.01e-1) -

4.490 8e+0

(2.05e-1)

4.841 5e+0

(3.77e-2) -

4.766 5e+0

(2.84e-2) -

4.703 7e+0

(5.82e-2) -

7.363 7e+0

(2.50e-1) -

4.486 6e+0

(2.20e-1) =

WFG9

2.239 3e-1

(4.25e-2)

2.682 4e-1

(1.08e-2) -

2.794 4e-1

(1.74e-2) -

2.363 5e-1

(3.60e-2) -

2.774 2e-1

(1.65e-2) -

2.072 8e-1

(2.21e-3) =

1.126 1e+0

(3.98e-3)

1.276 2e+0

(4.07e-2) -

1.678 9e+0

(4.22e-2) -

1.165 3e+0

(1.73e-2) -

1.495 6e+0

(9.89e-2) -

1.121 7e+0

(5.43e-3) +

2.931 5e+0

(7.68e-3)

3.766 6e+0

(7.03e-2) -

4.105 7e+0

(7.16e-2) -

3.295 2e+0

(3.07e-2) -

4.016 2e+0

(1.54e-1) -

2.930 5e+0

(5.78e-3) =

4.441 3e+0

(3.80e-2)

4.869 4e+0

(5.63e-2) -

5.380 5e+0

(8.20e-2) -

4.487 4e+0

(5.71e-2) -

5.439 9e+0

(1.51e-1) -

4.438 3e+0

(4.17e-2) =

表3 6种算法在WFG1~WFG9测试问题上获得的IGD指标值的实验结果

问题

MaOEA-SCAOS

MOCell

RM-MEDA

KnEA

MaOEA-CSS

NSGA-III

WFG1

2.170 8e-1

(3.79e-2)

6.962 4e-1

(8.67e-2) -

2.032 0e+0

(3.09e-2) -

2.119 5e-1

(2.71e-2) =

2.970 0e-1

(3.23e-2) -

1.983 3e-1

(3.07e-2) =

6.312 3e-1

(3.99e-2)

1.709 9e+0

(9.63e-2) -

2.470 4e+0

(3.72e-2) -

5.095 7e-1

(2.04e-2) +

6.655 5e-1

(4.10e-2) -

5.736 5e-1

(6.24e-2) +

8.934 7e-1

(3.37e-2)

2.176 2e+0

(1.01e-1) -

3.029 0e+0

(3.62e-2) -

9.425 6e-1

(2.18e-2) -

1.296 3e+0

(7.21e-2) -

8.854 6e-1

(2.19e-2) =

9.757 4e-1

(2.01e-2)

2.475 2e+0

(1.36e-1) -

3.344 2e+0

(2.93e-2) -

1.018 3e+0

(1.51e-2) -

1.409 1e+0

(6.05e-2) -

9.581 9e-1

(1.56e-2) +

WFG2

1.503 3e-1

(1.37e-3)

2.115 7e-1

(8.97e-3) -

2.475 6e-1

(1.11e-2) -

1.845 5e-1

(7.77e-3) -

2.242 6e-1

(2.41e-2) -

1.500 9e-1

(1.14e-3) =

4.678 2e-1

(2.37e-3)

8.048 0e-1

(5.82e-2) -

1.068 9e+0

(1.01e-1) -

5.334 9e-1

(2.58e-2) -

6.836 0e-1

(4.20e-2) -

4.681 1e-1

(2.29e-3) =

1.024 6e+0

(1.38e-1)

1.466 8e+0

(7.29e-2) -

2.068 9e+0

(1.06e-1) -

1.063 1e+0

(2.96e-2) =

1.571 9e+0

(8.43e-2) -

9.880 8e-1

(1.54e-1) =

1.163 3e+0

(1.23e-1)

1.532 7e+0

(7.02e-2) -

2.400 1e+0

(8.09e-2) -

1.184 3e+0

(3.52e-2) =

1.680 4e+0

(8.61e-2) -

1.183 4e+0

(1.42e-1) =

WFG3

1.002 3e-1

(1.01e-2)

1.146 0e-1

(1.12e-2) -

2.798 5e-1

(2.26e-2) -

1.064 2e-1

(1.14e-2) =

6.106 2e-1

(1.69e-1) -

9.752 5e-2

(6.74e-3) =

5.372 8e-1

(6.52e-2)

5.272 2e-1

(8.51e-2) =

1.083 7e+0

(7.96e-2) -

5.426 1e-1

(7.19e-2) =

2.171 1e+0

(3.23e-1) -

5.766 4e-1

(5.62e-2) -

1.527 8e+0

(3.03e-1)

1.237 9e+0

(2.01e-1) +

1.765 5e+0

(1.35e-1) -

1.419 6e+0

(2.55e-1) =

5.612 5e+0

(4.85e-1) -

1.686 3e+0

(2.20e-1) -

1.812 7e+0

(2.34e-1)

1.411 2e+0

(2.16e-1) +

2.035 3e+0

(1.93e-1) -

1.830 1e+0

(2.39e-1) =

7.141 3e+0

(4.67e-1) -

1.968 2e+0

(3.01e-1) =

WFG4

2.042 6e-1

(1.43e-4)

2.849 3e-1

(1.03e-2) -

3.383 9e-1

(1.51e-2) -

2.489 6e-1

(8.80e-3) -

3.021 6e-1

(3.60e-2) -

2.044 7e-1

(1.83e-4) -

1.174 8e+0

(1.43e-3)

1.229 6e+0

(2.49e-2) -

1.490 0e+0

(2.24e-2) -

1.232 3e+0

(1.98e-2) -

1.793 6e+0

(1.95e-1) -

1.174 6e+0

(1.83e-3) =

2.970 1e+0

(3.88e-2)

3.369 4e+0

(5.47e-2) -

3.463 7e+0

(2.29e-2) -

3.450 1e+0

(5.66e-2) -

4.525 1e+0

(1.70e-1) -

2.978 0e+0

(7.20e-2) =

4.547 4e+0

(4.68e-3)

4.489 2e+0

(3.76e-2) +

4.505 3e+0

(2.58e-2) +

4.578 4e+0

(3.80e-2) -

6.216 5e+0

(2.23e-1) -

4.547 6e+0

(2.88e-2) -

WFG5

2.145 1e-1

(1.09e-4)

2.799 8e-1

(9.11e-3) -

3.122 2e-1

(1.14e-2) -

2.606 0e-1

(1.10e-2) -

2.810 3e-1

(1.42e-2) -

2.145 4e-1

(1.03e-4) =

1.162 9e+0

(5.70e-4)

1.209 8e+0

(1.80e-2) -

1.399 1e+0

(3.04e-2) -

1.211 1e+0

(1.88e-2) -

1.547 2e+0

(7.30e-2) -

1.162 7e+0

(1.37e-3) =

2.941 0e+0

(1.31e-3)

3.423 5e+0

(6.63e-2) -

3.623 6e+0

(4.71e-2) -

3.338 1e+0

(2.98e-2) -

4.286 7e+0

(2.09e-1) -

2.941 3e+0

(1.32e-3) =

4.531 1e+0

(3.32e-3)

4.599 6e+0

(5.91e-2) -

4.764 3e+0

(3.95e-2) -

4.565 3e+0

(3.36e-2) -

5.903 8e+0

(1.79e-1) -

4.530 0e+0

(3.41e-3) =

WFG6

2.239 9e-1

(7.93e-3)

3.326 4e-1

(1.70e-2) -

3.455 4e-1

(1.49e-2) -

2.893 7e-1

(1.31e-2) -

3.865 9e-1

(7.01e-2) -

2.230 2e-1

(1.08e-2) =

1.164 7e+0

(2.80e-3)

1.316 0e+0

(3.06e-2) -

1.597 0e+0

(3.88e-2) -

1.246 2e+0

(2.03e-2) -

2.180 0e+0

(2.28e-1) -

1.162 7e+0

(1.80e-3) +

2.947 7e+0

(5.27e-3)

3.506 5e+0

(4.39e-2) -

3.582 9e+0

(2.48e-2) -

3.490 8e+0

(6.50e-2) -

5.515 6e+0

(2.01e-1) -

2.953 0e+0

(4.60e-3) -

4.544 7e+0

(6.62e-3)

4.617 3e+0

(4.44e-2) -

4.555 9e+0

(2.39e-2) -

4.669 0e+0

(4.48e-2) -

7.288 3e+0

(2.25e-1) -

4.543 9e+0

(8.18e-3) =

WFG7

2.047 4e-1

(2.06e-4)

2.826 6e-1

(1.04e-2) -

3.363 6e-1

(1.00e-2) -

2.399 1e-1

(8.46e-3) -

3.223 3e-1

(6.99e-2) -

2.047 8e-1

(2.62e-4) =

1.176 3e+0

(8.69e-4)

1.255 4e+0

(3.07e-2) -

1.583 9e+0

(2.76e-2) -

1.230 3e+0

(1.81e-2) -

2.314 8e+0

(1.73e-1) -

1.175 6e+0

(1.12e-3) +

2.967 7e+0

(7.16e-3)

3.499 1e+0

(9.59e-2) -

3.614 3e+0

(3.45e-2) -

3.324 5e+0

(4.54e-2) -

4.919 2e+0

(2.46e-1) -

2.968 8e+0

(6.33e-3) =

4.547 8e+0

(2.02e-2)

4.584 2e+0

(4.81e-2) -

4.630 3e+0

(3.39e-2) -

4.431 9e+0

(3.86e-2) +

6.175 7e+0

(2.19e-1) -

4.644 6e+0

(2.51e-1) =

WFG8

2.671 6e-1

(3.98e-3)

3.834 3e-1

(5.79e-3) -

4.457 8e-1

(1.56e-2) -

3.322 4e-1

(1.14e-2) -

4.431 3e-1

(8.88e-2) -

2.751 7e-1

(5.22e-3) -

1.148 0e+0

(1.28e-3)

1.446 1e+0

(2.49e-2) -

1.773 6e+0

(2.73e-2) -

1.282 7e+0

(2.25e-2) -

1.960 5e+0

(2.06e-1) -

1.154 7e+0

(8.40e-3) -

3.198 2e+0

(2.62e-1)

3.695 7e+0

(4.49e-2) -

3.769 1e+0

(2.87e-2) -

3.533 6e+0

(6.48e-2) -

5.381 5e+0

(3.58e-1) -

3.236 4e+0

(2.01e-1) -

4.490 8e+0

(2.05e-1)

4.841 5e+0

(3.77e-2) -

4.766 5e+0

(2.84e-2) -

4.703 7e+0

(5.82e-2) -

7.363 7e+0

(2.50e-1) -

4.486 6e+0

(2.20e-1) =

WFG9

2.239 3e-1

(4.25e-2)

2.682 4e-1

(1.08e-2) -

2.794 4e-1

(1.74e-2) -

2.363 5e-1

(3.60e-2) -

2.774 2e-1

(1.65e-2) -

2.072 8e-1

(2.21e-3) =

1.126 1e+0

(3.98e-3)

1.276 2e+0

(4.07e-2) -

1.678 9e+0

(4.22e-2) -

1.165 3e+0

(1.73e-2) -

1.495 6e+0

(9.89e-2) -

1.121 7e+0

(5.43e-3) +

2.931 5e+0

(7.68e-3)

3.766 6e+0

(7.03e-2) -

4.105 7e+0

(7.16e-2) -

3.295 2e+0

(3.07e-2) -

4.016 2e+0

(1.54e-1) -

2.930 5e+0

(5.78e-3) =

4.441 3e+0

(3.80e-2)

4.869 4e+0

(5.63e-2) -

5.380 5e+0

(8.20e-2) -

4.487 4e+0

(5.71e-2) -

5.439 9e+0

(1.51e-1) -

4.438 3e+0

(4.17e-2) =

+/-/=

3/32/1

1/35/0

2/27/7

0/36/0

5/8/23

图4 WFG4问题3目标上获得的非支配解

图5 WFG6问题8目标上获得的非支配解

表4 6种算法在DTLZ1~DTLZ7测试问题上获得的HV指标值的实验结果

8.442 1e-1

(2.45e-4)

8.241 4e-1

(3.87e-3) -

0.000 0e+0

(0.00e+0) -

7.176 4e-1

(8.20e-2) -

8.078 8e-1

(1.10e-2) -

8.436 0e-1

(1.01e-3) -

9.745 0e-1

(3.96e-4)

2.319 1e-1

(3.24e-1) -

0.000 0e+0

(0.00e+0) -

5.602 7e-1

(1.61e-1) -

9.073 1e-1

(1.02e-2) -

9.744 6e-1

(3.50e-4) =

9.626 6e-1

(1.08e-1)

0.000 0e+0

(0.00e+0) -

0.000 0e+0

(0.00e+0) -

6.661 1e-1

(3.18e-1) -

9.455 0e-1

(7.01e-3) -

9.972 5e-1

(1.46e-3) =

9.901 7e-1

(2.91e-2)

0.000 0e+0

(0.00e+0) -

0.000 0e+0

(0.00e+0) -

8.793 8e-4

(3.93e-3) -

9.803 1e-1

(2.66e-3) -

9.807 0e-1

(5.89e-2) =

DTLZ2

5.630 0e-1

(1.16e-5)

5.251 6e-1

(3.33e-3) -

5.213 2e-1

(4.32e-3) -

5.428 2e-1

(4.55e-3) -

5.538 6e-1

(3.14e-3) -

5.629 9e-1

(1.98e-5) -

7.947 3e-1

(5.06e-4)

5.278 2e-1

(3.84e-2) -

5.517 1e-2

(2.08e-2) -

7.679 4e-1

(5.15e-3) -

7.557 5e-1

(8.62e-3) -

7.946 1e-1

(3.92e-4) =

9.145 0e-1

(2.31e-2)

2.298 2e-5

(1.03e-4) -

5.360 4e-3

(1.09e-2) -

8.827 2e-1

(4.24e-3) -

8.810 1e-1

(7.35e-3) -

9.125 9e-1

(3.01e-2) =

DTLZ2

9.625 1e-1

(1.81e-2)

1.667 5e-3

(4.99e-3) -

1.339 2e-4

(3.29e-4) -

9.587 1e-1

(1.60e-3) -

9.349 5e-1

(4.71e-3) -

9.621 1e-1

(1.94e-2) =

DTLZ3

5.527 4e-1

(6.27e-3)

5.131 6e-1

(1.33e-2) -

0.000 0e+0

(0.00e+0) -

4.790 1e-1

(5.20e-2) -

5.496 0e-1

(4.09e-3) =

5.401 1e-1

(1.18e-2) -

5.980 4e-1

(2.92e-1)

0.000 0e+0

(0.00e+0) -

0.000 0e+0

(0.00e+0) -

3.993 6e-1

(1.89e-1) -

7.464 4e-1

(1.22e-2) =

5.835 5e-1

(3.08e-1) =

8.908 0e-1

(7.45e-2)

0.000 0e+0

(0.00e+0) -

0.000 0e+0

(0.00e+0) -

0.000 0e+0

(0.00e+0) -

8.555 0e-1

(1.92e-2) -

8.515 2e-1

(2.08e-1) -

8.364 6e-1

(3.00e-1)

0.000 0e+0

(0.00e+0) -

0.000 0e+0

(0.00e+0) -

0.000 0e+0

(0.00e+0) -

9.232 0e-1

(7.64e-3) +

9.298 0e-1

(5.81e-2) =

DTLZ4

5.628 7e-1

(2.01e-4)

5.295 6e-1

(3.19e-3) -

5.167 0e-1

(7.45e-3) -

5.228 7e-1

(1.02e-1) -

5.585 2e-1

(1.69e-3) -

4.950 7e-1

(1.31e-1) =

7.946 0e-1

(5.00e-4)

5.993 8e-1

(3.19e-2) -

2.438 9e-1

(7.33e-2) -

7.744 1e-1

(5.25e-3) -

7.797 2e-1

(3.98e-3) -

7.567 6e-1

(6.02e-2) -

9.071 9e-1

(3.34e-2)

0.000 0e+0

(0.00e+0) -

7.206 2e-3

(1.31e-2) -

9.019 9e-1

(4.47e-3) -

9.070 8e-1

(3.48e-3) -

9.117 5e-1

(3.02e-2) =

9.696 2e-1

(1.40e-4)

0.000 0e+0

(0.00e+0) -

1.727 4e-3

(3.86e-3) -

9.555 7e-1

(5.48e-3) -

9.543 4e-1

(2.40e-3) -

9.676 9e-1

(8.56e-3) =

DTLZ5

1.946 2e-1

(8.08e-4)

1.983 8e-1

(2.55e-4) +

1.991 7e-1

(1.72e-4) +

1.940 7e-1

(2.10e-3) =

1.853 6e-1

(3.13e-3) -

1.948 6e-1

(8.93e-4) =

6.439 3e-2

(4.97e-2)

1.017 7e-1

(8.56e-3) +

2.007 9e-2

(1.59e-2) =

5.697 8e-2

(3.16e-2) =

9.805 2e-2

(1.92e-2) +

3.765 8e-2

(4.51e-2) =

9.427 7e-2

(2.59e-3)

5.347 5e-2

(3.13e-2) -

7.120 6e-4

(2.16e-3) -

6.680 9e-2

(1.92e-2) -

5.538 6e-2

(2.73e-2) -

9.418 0e-2

(2.39e-3) =

9.133 6e-2

(1.92e-3)

3.539 7e-2

(2.62e-2) -

2.841 8e-10

(1.27e-9) -

5.344 1e-2

(1.99e-2) -

4.673 7e-2

(3.13e-2) -

9.105 2e-2

(1.82e-3) =

DTLZ6

1.920 1e-1

(1.24e-3)

1.998 4e-1

(1.05e-4) +

4.921 1e-3

(2.20e-2) -

1.902 9e-1

(8.85e-3) =

1.855 2e-1

(1.58e-3) -

1.918 3e-1

(9.85e-4) =

6.864 7e-2

(3.84e-2)

0.000 0e+0

(0.00e+0) -

4.773 4e-2

(4.96e-2) =

6.607 7e-2

(3.92e-2) =

8.530 1e-2

(2.83e-2) =

6.287 1e-2

(4.21e-2) =

8.183 2e-2

(2.80e-2)

0.000 0e+0

(0.00e+0) -

8.766 4e-2

(2.07e-2) +

3.929 9e-2

(4.16e-2) -

4.301 1e-2

(4.10e-2) -

8.190 0e-2

(2.80e-2) =

7.721 1e-2

(3.33e-2)

0.000 0e+0

(0.00e+0) -

9.211 3e-2

(8.47e-4) +

6.357 8e-11

(1.55e-10) -

2.527 4e-2

(2.91e-2) -

8.653 3e-2

(2.04e-2) =

DTLZ7

2.697 4e-1

(1.55e-3)

2.578 8e-1

(1.10e-2) -

2.183 9e-1

(1.36e-2) -

2.760 2e-1

(8.07e-3) +

2.521 8e-1

(1.13e-2) -

2.712 7e-1

(1.44e-3) +

2.260 4e-1

(8.01e-3)

1.493 5e-1

(9.49e-3) -

2.898 3e-5

(4.21e-5) -

2.514 7e-1

(7.86e-3) +

1.340 8e-1

(1.51e-2) -

2.216 4e-1

(1.01e-2) =

1.950 7e-1

(4.04e-3)

5.276 9e-4

(8.36e-4) -

3.964 5e-7

(1.38e-6) -

1.034 8e-1

(9.82e-3) -

1.134 5e-2

(1.93e-3) -

2.038 6e-1

(2.27e-3) +

1.808 2e-1

(6.46e-3)

9.885 1e-6

(2.38e-5) -

1.246 0e-8

(4.35e-8) -

9.423 6e-2

(2.58e-2) -

2.379 9e-4

(1.68e-4) -

1.830 2e-1

(8.59e-3) =

表4 6种算法在DTLZ1~DTLZ7测试问题上获得的HV指标值的实验结果

问题

MaOEA-SCAOS

MOCell

RM-MEDA

KnEA

MaOEA-CSS

NSGA-III

DTLZ1

8.442 1e-1

(2.45e-4)

8.241 4e-1

(3.87e-3) -

0.000 0e+0

(0.00e+0) -

7.176 4e-1

(8.20e-2) -

8.078 8e-1

(1.10e-2) -

8.436 0e-1

(1.01e-3) -

9.745 0e-1

(3.96e-4)

2.319 1e-1

(3.24e-1) -

0.000 0e+0

(0.00e+0) -

5.602 7e-1

(1.61e-1) -

9.073 1e-1

(1.02e-2) -

9.744 6e-1

(3.50e-4) =

9.626 6e-1

(1.08e-1)

0.000 0e+0

(0.00e+0) -

0.000 0e+0

(0.00e+0) -

6.661 1e-1

(3.18e-1) -

9.455 0e-1

(7.01e-3) -

9.972 5e-1

(1.46e-3) =

9.901 7e-1

(2.91e-2)

0.000 0e+0

(0.00e+0) -

0.000 0e+0

(0.00e+0) -

8.793 8e-4

(3.93e-3) -

9.803 1e-1

(2.66e-3) -

9.807 0e-1

(5.89e-2) =

DTLZ2

5.630 0e-1

(1.16e-5)

5.251 6e-1

(3.33e-3) -

5.213 2e-1

(4.32e-3) -

5.428 2e-1

(4.55e-3) -

5.538 6e-1

(3.14e-3) -

5.629 9e-1

(1.98e-5) -

7.947 3e-1

(5.06e-4)

5.278 2e-1

(3.84e-2) -

5.517 1e-2

(2.08e-2) -

7.679 4e-1

(5.15e-3) -

7.557 5e-1

(8.62e-3) -

7.946 1e-1

(3.92e-4) =

9.145 0e-1

(2.31e-2)

2.298 2e-5

(1.03e-4) -

5.360 4e-3

(1.09e-2) -

8.827 2e-1

(4.24e-3) -

8.810 1e-1

(7.35e-3) -

9.125 9e-1

(3.01e-2) =

DTLZ2

9.625 1e-1

(1.81e-2)

1.667 5e-3

(4.99e-3) -

1.339 2e-4

(3.29e-4) -

9.587 1e-1

(1.60e-3) -

9.349 5e-1

(4.71e-3) -

9.621 1e-1

(1.94e-2) =

DTLZ3

5.527 4e-1

(6.27e-3)

5.131 6e-1

(1.33e-2) -

0.000 0e+0

(0.00e+0) -

4.790 1e-1

(5.20e-2) -

5.496 0e-1

(4.09e-3) =

5.401 1e-1

(1.18e-2) -

5.980 4e-1

(2.92e-1)

0.000 0e+0

(0.00e+0) -

0.000 0e+0

(0.00e+0) -

3.993 6e-1

(1.89e-1) -

7.464 4e-1

(1.22e-2) =

5.835 5e-1

(3.08e-1) =

8.908 0e-1

(7.45e-2)

0.000 0e+0

(0.00e+0) -

0.000 0e+0

(0.00e+0) -

0.000 0e+0

(0.00e+0) -

8.555 0e-1

(1.92e-2) -

8.515 2e-1

(2.08e-1) -

8.364 6e-1

(3.00e-1)

0.000 0e+0

(0.00e+0) -

0.000 0e+0

(0.00e+0) -

0.000 0e+0

(0.00e+0) -

9.232 0e-1

(7.64e-3) +

9.298 0e-1

(5.81e-2) =

DTLZ4

5.628 7e-1

(2.01e-4)

5.295 6e-1

(3.19e-3) -

5.167 0e-1

(7.45e-3) -

5.228 7e-1

(1.02e-1) -

5.585 2e-1

(1.69e-3) -

4.950 7e-1

(1.31e-1) =

7.946 0e-1

(5.00e-4)

5.993 8e-1

(3.19e-2) -

2.438 9e-1

(7.33e-2) -

7.744 1e-1

(5.25e-3) -

7.797 2e-1

(3.98e-3) -

7.567 6e-1

(6.02e-2) -

9.071 9e-1

(3.34e-2)

0.000 0e+0

(0.00e+0) -

7.206 2e-3

(1.31e-2) -

9.019 9e-1

(4.47e-3) -

9.070 8e-1

(3.48e-3) -

9.117 5e-1

(3.02e-2) =

9.696 2e-1

(1.40e-4)

0.000 0e+0

(0.00e+0) -

1.727 4e-3

(3.86e-3) -

9.555 7e-1

(5.48e-3) -

9.543 4e-1

(2.40e-3) -

9.676 9e-1

(8.56e-3) =

DTLZ5

1.946 2e-1

(8.08e-4)

1.983 8e-1

(2.55e-4) +

1.991 7e-1

(1.72e-4) +

1.940 7e-1

(2.10e-3) =

1.853 6e-1

(3.13e-3) -

1.948 6e-1

(8.93e-4) =

6.439 3e-2

(4.97e-2)

1.017 7e-1

(8.56e-3) +

2.007 9e-2

(1.59e-2) =

5.697 8e-2

(3.16e-2) =

9.805 2e-2

(1.92e-2) +

3.765 8e-2

(4.51e-2) =

9.427 7e-2

(2.59e-3)

5.347 5e-2

(3.13e-2) -

7.120 6e-4

(2.16e-3) -

6.680 9e-2

(1.92e-2) -

5.538 6e-2

(2.73e-2) -

9.418 0e-2

(2.39e-3) =

9.133 6e-2

(1.92e-3)

3.539 7e-2

(2.62e-2) -

2.841 8e-10

(1.27e-9) -

5.344 1e-2

(1.99e-2) -

4.673 7e-2

(3.13e-2) -

9.105 2e-2

(1.82e-3) =

DTLZ6

1.920 1e-1

(1.24e-3)

1.998 4e-1

(1.05e-4) +

4.921 1e-3

(2.20e-2) -

1.902 9e-1

(8.85e-3) =

1.855 2e-1

(1.58e-3) -

1.918 3e-1

(9.85e-4) =

6.864 7e-2

(3.84e-2)

0.000 0e+0

(0.00e+0) -

4.773 4e-2

(4.96e-2) =

6.607 7e-2

(3.92e-2) =

8.530 1e-2

(2.83e-2) =

6.287 1e-2

(4.21e-2) =

8.183 2e-2

(2.80e-2)

0.000 0e+0

(0.00e+0) -

8.766 4e-2

(2.07e-2) +

3.929 9e-2

(4.16e-2) -

4.301 1e-2

(4.10e-2) -

8.190 0e-2

(2.80e-2) =

7.721 1e-2

(3.33e-2)

0.000 0e+0

(0.00e+0) -

9.211 3e-2

(8.47e-4) +

6.357 8e-11

(1.55e-10) -

2.527 4e-2

(2.91e-2) -

8.653 3e-2

(2.04e-2) =

DTLZ7

2.697 4e-1

(1.55e-3)

2.578 8e-1

(1.10e-2) -

2.183 9e-1

(1.36e-2) -

2.760 2e-1

(8.07e-3) +

2.521 8e-1

(1.13e-2) -

2.712 7e-1

(1.44e-3) +

2.260 4e-1

(8.01e-3)

1.493 5e-1

(9.49e-3) -

2.898 3e-5

(4.21e-5) -

2.514 7e-1

(7.86e-3) +

1.340 8e-1

(1.51e-2) -

2.216 4e-1

(1.01e-2) =

1.950 7e-1

(4.04e-3)

5.276 9e-4

(8.36e-4) -

3.964 5e-7

(1.38e-6) -

1.034 8e-1

(9.82e-3) -

1.134 5e-2

(1.93e-3) -

2.038 6e-1

(2.27e-3) +

1.808 2e-1

(6.46e-3)

9.885 1e-6

(2.38e-5) -

1.246 0e-8

(4.35e-8) -

9.423 6e-2

(2.58e-2) -

2.379 9e-4

(1.68e-4) -

1.830 2e-1

(8.59e-3) =

+/-/=

3/25/0

3/24/1

2/22/4

1/25/2

2/5/21

表5 6种算法在DTLZ1~DTLZ7测试问题上获得的IGD指标值的实验结果

1.898 8e-2

(1.72e-5)

2.558 3e-2

(8.91e-4) -

1.616 9e+1

(2.13e+0) -

6.464 5e-2

(4.02e-2) -

2.287 3e-2

(9.44e-4) -

1.904 4e-2

(1.19e-4) -

6.352 9e-2

(2.56e-4)

5.903 2e-1

(6.19e-1) -

2.291 8e+1

(2.68e+0) -

2.222 5e-1

(7.45e-2) -

6.881 5e-2

(2.31e-3) -

6.362 0e-2

(3.00e-4) =

1.248 6e-1

(6.29e-2)

1.358 3e+1

(4.95e+0) -

2.288 8e+1

(2.43e+0) -

3.005 5e-1

(1.05e-1) -

1.177 1e-1

(2.30e-3) +

9.761 0e-2

(2.17e-3) =

1.216 8e-1

(3.33e-2)

2.019 9e+1

(8.38e+0) -

2.258 6e+1

(2.18e+0) -

4.941 6e+0

(6.16e+0) -

1.153 0e-1

(1.16e-3) +

1.277 9e-1

(5.52e-2) =

DTLZ2

5.030 3e-2

(1.80e-6)

6.906 6e-2

(2.08e-3) -

6.775 9e-2

(2.26e-3) -

6.786 7e-2

(4.56e-3) -

5.235 5e-2

(5.80e-4) -

5.030 5e-2

(5.07e-6) -

1.949 0e-1

(3.23e-5)

2.741 3e-1

(1.52e-2) -

6.591 8e-1

(4.02e-2) -

2.163 1e-1

(5.70e-3) -

1.883 0e-1

(1.79e-3) +

1.949 0e-1

(1.53e-5) =

3.348 8e-1

(4.82e-2)

1.907 0e+0

(3.08e-1) -

1.251 6e+0

(1.14e-1) -

3.828 3e-1

(4.75e-3) -

3.480 3e-1

(2.17e-3) -

3.421 0e-1

(6.93e-2) =

4.384 0e-1

(4.34e-2)

1.408 8e+0

(1.56e-1) -

1.463 7e+0

(8.52e-2) -

4.313 6e-1

(2.42e-3) +

3.949 0e-1

(1.53e-3) +

4.384 5e-1

(4.23e-2) =

DTLZ3

5.369 3e-2

(4.37e-3)

7.404 3e-2

(6.00e-3) -

1.804 6e+2

(9.51e+0) -

1.241 0e-1

(6.88e-2) -

5.665 6e-2

(1.22e-3) -

5.689 8e-2

(9.79e-3) -

3.821 1e-1

(3.68e-1)

1.684 8e+1

(1.24e+1) -

1.838 0e+2

(1.48e+1) -

5.693 2e-1

(2.57e-1) -

1.973 8e-1

(3.52e-3) +

5.549 8e-1

(8.36e-1) =

DTLZ3

3.614 9e-1

(8.87e-2)

8.341 6e+2

(2.95e+2) -

2.494 8e+2

(3.44e+1) -

6.097 4e+1

(2.89e+1) -

3.575 7e-1

(4.43e-3) +

4.571 9e-1

(4.76e-1) =

6.049 4e-1

(4.70e-1)

8.070 4e+2

(2.43e+2) -

2.914 8e+2

(4.04e+1) -

2.898 8e+2

(9.54e+1) -

4.009 9e-1

(3.07e-3) +

4.781 8e-1

(7.83e-2) =

DTLZ4

5.032 0e-2

(3.16e-5)

6.867 5e-2

(2.50e-3) -

7.734 7e-2

(5.29e-3) -

1.091 8e-1

(1.97e-1) -

5.312 1e-2

(6.82e-4) -

1.933 0e-1

(2.67e-1) -

1.949 4e-1

(5.05e-5)

2.551 9e-1

(1.11e-2) -

5.180 0e-1

(3.40e-2) -

2.159 5e-1

(6.93e-3) -

1.926 9e-1

(1.75e-3) +

2.666 9e-1

(1.13e-1) =

3.550 4e-1

(7.29e-2)

1.986 8e+0

(2.19e-1) -

1.023 4e+0

(5.03e-2) -

3.752 6e-1

(4.26e-3) -

3.524 1e-1

(1.51e-3) +

3.441 9e-1

(7.18e-2) =

4.215 5e-1

(4.84e-4)

1.952 7e+0

(2.43e-1) -

1.183 1e+0

(4.93e-2) -

4.449 0e-1

(2.04e-2) -

4.009 5e-1

(1.08e-3) +

4.269 3e-1

(2.86e-2) -

DTLZ5

1.247 0e-2

(1.63e-3)

6.037 4e-3

(3.21e-4) +

5.005 8e-3

(2.59e-4) +

9.000 8e-3

(1.74e-3) +

2.128 4e-2

(2.06e-3) -

1.179 4e-2

(1.61e-3) =

3.475 4e-1

(2.29e-1)

1.114 2e-1

(3.53e-2) +

2.875 8e-1

(7.26e-2) =

2.901 1e-1

(1.09e-1) =

4.876 8e-2

(7.45e-3) +

3.580 7e-1

(2.25e-1) =

3.043 8e-1

(9.17e-2)

3.275 9e-1

(1.94e-1) =

5.066 9e-1

(1.12e-1) -

2.764 0e-1

(5.07e-2) =

7.282 7e-2

(9.40e-3) +

2.481 4e-1

(9.17e-2) =

3.537 4e-1

(7.28e-2)

3.692 9e-1

(1.55e-1) =

6.323 1e-1

(8.11e-2) -

3.147 2e-1

(6.40e-2) =

8.768 5e-2

(2.53e-2) +

3.361 6e-1

(6.68e-2) =

DTLZ6

1.747 9e-2

(2.32e-3)

5.203 2e-3

(1.86e-4) +

1.624 4e+0

(5.69e-1) -

1.492 7e-2

(1.03e-2) =

2.735 0e-2

(1.90e-3) -

1.718 6e-2

(1.62e-3) =

3.584 4e-1

(1.17e-1)

5.779 2e+0

(1.02e+0) -

8.456 0e-1

(4.97e-1) -

3.727 5e-1

(1.70e-1) =

7.821 4e-2

(2.68e-2) +

3.801 4e-1

(2.62e-1) =

4.492 3e-1

(1.89e-1)

7.077 2e+0

(7.94e-1) -

4.219 7e-1

(2.26e-1) =

5.159 2e-1

(9.26e-2) =

1.602 4e-1

(6.88e-2) +

5.202 7e-1

(3.35e-1) =

5.141 3e-1

(3.32e-1)

7.735 9e+0

(6.48e-1) -

2.918 5e-1

(9.12e-2) +

1.141 5e+0

(5.23e-1) -

1.446 5e-1

(3.90e-2) +

3.718 4e-1

(1.37e-1) =

DTLZ7

7.374 7e-2

(3.23e-3)

1.316 5e-1

(1.12e-1) -

1.550 3e-1

(3.16e-2) -

8.140 6e-2

(6.36e-2) -

9.806 6e-2

(1.18e-2) -

7.021 2e-2

(2.81e-3) +

3.690 6e-1

(4.61e-2)

4.223 0e-1

(1.47e-2) -

1.724 0e+0

(5.51e-1) -

2.956 4e-1

(1.35e-2) +

3.953 6e-1

(1.98e-2) -

3.642 0e-1

(5.04e-2) =

8.402 7e-1

(5.15e-2)

1.352 6e+0

(5.77e-2) -

1.842 5e+0

(2.58e-1) -

6.689 2e-1

(2.46e-2) +

1.212 8e+0

(1.40e-1) -

7.792 1e-1

(2.88e-2) +

1.033 3e+0

(9.12e-2)

2.230 2e+0

(3.15e-1) -

2.195 8e+0

(2.55e-1) -

8.508 1e-1

(7.46e-3) +

2.309 0e+0

(4.14e-1) -

9.696 0e-1

(6.91e-2) +

表5 6种算法在DTLZ1~DTLZ7测试问题上获得的IGD指标值的实验结果

问题

MaOEA-SCAOS

MOCell

RM-MEDA

KnEA

MaOEA-CSS

NSGA-III

DTLZ1

1.898 8e-2

(1.72e-5)

2.558 3e-2

(8.91e-4) -

1.616 9e+1

(2.13e+0) -

6.464 5e-2

(4.02e-2) -

2.287 3e-2

(9.44e-4) -

1.904 4e-2

(1.19e-4) -

6.352 9e-2

(2.56e-4)

5.903 2e-1

(6.19e-1) -

2.291 8e+1

(2.68e+0) -

2.222 5e-1

(7.45e-2) -

6.881 5e-2

(2.31e-3) -

6.362 0e-2

(3.00e-4) =

1.248 6e-1

(6.29e-2)

1.358 3e+1

(4.95e+0) -

2.288 8e+1

(2.43e+0) -

3.005 5e-1

(1.05e-1) -

1.177 1e-1

(2.30e-3) +

9.761 0e-2

(2.17e-3) =

1.216 8e-1

(3.33e-2)

2.019 9e+1

(8.38e+0) -

2.258 6e+1

(2.18e+0) -

4.941 6e+0

(6.16e+0) -

1.153 0e-1

(1.16e-3) +

1.277 9e-1

(5.52e-2) =

DTLZ2

5.030 3e-2

(1.80e-6)

6.906 6e-2

(2.08e-3) -

6.775 9e-2

(2.26e-3) -

6.786 7e-2

(4.56e-3) -

5.235 5e-2

(5.80e-4) -

5.030 5e-2

(5.07e-6) -

1.949 0e-1

(3.23e-5)

2.741 3e-1

(1.52e-2) -

6.591 8e-1

(4.02e-2) -

2.163 1e-1

(5.70e-3) -

1.883 0e-1

(1.79e-3) +

1.949 0e-1

(1.53e-5) =

3.348 8e-1

(4.82e-2)

1.907 0e+0

(3.08e-1) -

1.251 6e+0

(1.14e-1) -

3.828 3e-1

(4.75e-3) -

3.480 3e-1

(2.17e-3) -

3.421 0e-1

(6.93e-2) =

4.384 0e-1

(4.34e-2)

1.408 8e+0

(1.56e-1) -

1.463 7e+0

(8.52e-2) -

4.313 6e-1

(2.42e-3) +

3.949 0e-1

(1.53e-3) +

4.384 5e-1

(4.23e-2) =

DTLZ3

5.369 3e-2

(4.37e-3)

7.404 3e-2

(6.00e-3) -

1.804 6e+2

(9.51e+0) -

1.241 0e-1

(6.88e-2) -

5.665 6e-2

(1.22e-3) -

5.689 8e-2

(9.79e-3) -

3.821 1e-1

(3.68e-1)

1.684 8e+1

(1.24e+1) -

1.838 0e+2

(1.48e+1) -

5.693 2e-1

(2.57e-1) -

1.973 8e-1

(3.52e-3) +

5.549 8e-1

(8.36e-1) =

DTLZ3

3.614 9e-1

(8.87e-2)

8.341 6e+2

(2.95e+2) -

2.494 8e+2

(3.44e+1) -

6.097 4e+1

(2.89e+1) -

3.575 7e-1

(4.43e-3) +

4.571 9e-1

(4.76e-1) =

6.049 4e-1

(4.70e-1)

8.070 4e+2

(2.43e+2) -

2.914 8e+2

(4.04e+1) -

2.898 8e+2

(9.54e+1) -

4.009 9e-1

(3.07e-3) +

4.781 8e-1

(7.83e-2) =

DTLZ4

5.032 0e-2

(3.16e-5)

6.867 5e-2

(2.50e-3) -

7.734 7e-2

(5.29e-3) -

1.091 8e-1

(1.97e-1) -

5.312 1e-2

(6.82e-4) -

1.933 0e-1

(2.67e-1) -

1.949 4e-1

(5.05e-5)

2.551 9e-1

(1.11e-2) -

5.180 0e-1

(3.40e-2) -

2.159 5e-1

(6.93e-3) -

1.926 9e-1

(1.75e-3) +

2.666 9e-1

(1.13e-1) =

3.550 4e-1

(7.29e-2)

1.986 8e+0

(2.19e-1) -

1.023 4e+0

(5.03e-2) -

3.752 6e-1

(4.26e-3) -

3.524 1e-1

(1.51e-3) +

3.441 9e-1

(7.18e-2) =

4.215 5e-1

(4.84e-4)

1.952 7e+0

(2.43e-1) -

1.183 1e+0

(4.93e-2) -

4.449 0e-1

(2.04e-2) -

4.009 5e-1

(1.08e-3) +

4.269 3e-1

(2.86e-2) -

DTLZ5

1.247 0e-2

(1.63e-3)

6.037 4e-3

(3.21e-4) +

5.005 8e-3

(2.59e-4) +

9.000 8e-3

(1.74e-3) +

2.128 4e-2

(2.06e-3) -

1.179 4e-2

(1.61e-3) =

3.475 4e-1

(2.29e-1)

1.114 2e-1

(3.53e-2) +

2.875 8e-1

(7.26e-2) =

2.901 1e-1

(1.09e-1) =

4.876 8e-2

(7.45e-3) +

3.580 7e-1

(2.25e-1) =

3.043 8e-1

(9.17e-2)

3.275 9e-1

(1.94e-1) =

5.066 9e-1

(1.12e-1) -

2.764 0e-1

(5.07e-2) =

7.282 7e-2

(9.40e-3) +

2.481 4e-1

(9.17e-2) =

3.537 4e-1

(7.28e-2)

3.692 9e-1

(1.55e-1) =

6.323 1e-1

(8.11e-2) -

3.147 2e-1

(6.40e-2) =

8.768 5e-2

(2.53e-2) +

3.361 6e-1

(6.68e-2) =

DTLZ6

1.747 9e-2

(2.32e-3)

5.203 2e-3

(1.86e-4) +

1.624 4e+0

(5.69e-1) -

1.492 7e-2

(1.03e-2) =

2.735 0e-2

(1.90e-3) -

1.718 6e-2

(1.62e-3) =

3.584 4e-1

(1.17e-1)

5.779 2e+0

(1.02e+0) -

8.456 0e-1

(4.97e-1) -

3.727 5e-1

(1.70e-1) =

7.821 4e-2

(2.68e-2) +

3.801 4e-1

(2.62e-1) =

4.492 3e-1

(1.89e-1)

7.077 2e+0

(7.94e-1) -

4.219 7e-1

(2.26e-1) =

5.159 2e-1

(9.26e-2) =

1.602 4e-1

(6.88e-2) +

5.202 7e-1

(3.35e-1) =

5.141 3e-1

(3.32e-1)

7.735 9e+0

(6.48e-1) -

2.918 5e-1

(9.12e-2) +

1.141 5e+0

(5.23e-1) -

1.446 5e-1

(3.90e-2) +

3.718 4e-1

(1.37e-1) =

DTLZ7

7.374 7e-2

(3.23e-3)

1.316 5e-1

(1.12e-1) -

1.550 3e-1

(3.16e-2) -

8.140 6e-2

(6.36e-2) -

9.806 6e-2

(1.18e-2) -

7.021 2e-2

(2.81e-3) +

3.690 6e-1

(4.61e-2)

4.223 0e-1

(1.47e-2) -

1.724 0e+0

(5.51e-1) -

2.956 4e-1

(1.35e-2) +

3.953 6e-1

(1.98e-2) -

3.642 0e-1

(5.04e-2) =

8.402 7e-1

(5.15e-2)

1.352 6e+0

(5.77e-2) -

1.842 5e+0

(2.58e-1) -

6.689 2e-1

(2.46e-2) +

1.212 8e+0

(1.40e-1) -

7.792 1e-1

(2.88e-2) +

1.033 3e+0

(9.12e-2)

2.230 2e+0

(3.15e-1) -

2.195 8e+0

(2.55e-1) -

8.508 1e-1

(7.46e-3) +

2.309 0e+0

(4.14e-1) -

9.696 0e-1

(6.91e-2) +

+/-/=

3/23/2

2/24/2

5/17/6

17/11/0

3/5/20

图6 DTLZ4问题10目标上获得的非支配解

参考文献 37

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基于SOM聚类和自适应算子选择的高维多目标进化算法

Many-Objective Evolutionary Algorithm Based on SOM Clustering and Adaptive Operator Selection

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