基于距离优势关系的高维多目标进化算法

doi:10.3778/j.issn.1673-9418.2103053

计算机科学与探索 ›› 2022, Vol. 16 ›› Issue (11): 2642-2652.DOI: 10.3778/j.issn.1673-9418.2103053

• 理论与算法 • 上一篇

基于距离优势关系的高维多目标进化算法

顾清华¹^,²^,³^,⁺(), 徐青松¹^,³, 李学现¹^,³

1.西安建筑科技大学管理学院，西安 710001
2.西安建筑科技大学资源工程学院，西安 710001
3.西安市智慧工业感知计算与决策重点实验室，西安 710001

收稿日期:2021-03-16 修回日期:2021-05-28 出版日期:2022-11-01 发布日期:2021-06-04
通讯作者: + E-mail: qinghuagu@126.com
作者简介:顾清华（1981—），男，山东潍坊人，博士，教授，CCF会员，主要研究方向为矿业系统工程及露天矿车辆调度。
徐青松（1996—），男，湖北孝感人，硕士研究生，主要研究方向为露天矿运输系统优化。
李学现（1993—），男，山东德州人，博士研究生，主要研究方向为露天矿运输系统优化。
基金资助:
国家自然科学基金面上项目(51974223);陕西省自然科学基金杰青项目(2020JC-44)

Many-Objective Evolutionary Algorithm Based on Distance Dominance Relation

GU Qinghua¹^,²^,³^,⁺(), XU Qingsong¹^,³, LI Xuexian¹^,³

1. School of Management, Xi’an University of Architecture and Technology, Xi’an 710001, China
2. School of Resource Engineering, Xi’an University of Architecture and Technology, Xi’an 710001, China
3. Xi’an Key Laboratory for Intelligent Industrial Perception, Calculation and Decision, Xi’an 710001, China

Received:2021-03-16 Revised:2021-05-28 Online:2022-11-01 Published:2021-06-04
About author:GU Qinghua, born in 1981, Ph.D., professor, member of CCF. His research interests include mining system engineering and open pit mine vehicle dispatch.
XU Qingsong, born in 1996, M.S. candidate. His research interest is optimization of open pit transportation system.
LI Xuexian, born in 1993, Ph.D. candidate. His research interest is optimization of open pit transportation system.
Supported by:
National Natural Science Foundation of China(51974223);Outstanding Youth Project of Natural Science Foundation of Shaanxi Province(2020JC-44)

摘要/Abstract

摘要：

收敛性与多样性是多目标进化算法的两个主要研究方面，随着多目标优化问题目标维度的增加，传统的多目标进化算法很难维持解的多样性。为了增强算法在高维多目标优化问题中的多样性，提出了一种新的距离优势关系。首先，为了保证算法的收敛性，在同一小生境内，基于距离优势关系计算候选解到理想点的距离作为适应度值，选择适应度值好的候选解作为非支配解。然后，为了增强算法的多样性，距离优势关系设定了每个候选解具有相同的小生境，并且保证在同一小生境内只保留一个最优解。最后，基于提出的距离优势关系对VaEA算法进行改进，得到的算法命名为VaEA-DDR。在5、8、10、15维目标的DTLZ及IDTLZ测试问题上将改进的算法与目前六种常用的算法进行实验对比。实验结果表明，改进后的算法具有较强的竞争性，能显著增强算法的多样性。

关键词: 高维多目标优化, 距离优势关系, 多样性, VaEA

Abstract:

There are two main aspects of research in multi-objective optimization algorithm, namely, convergence and diversity. While, it is difficult for original algorithms to maintain the diversity of solutions in the high-dimensional objective space. In order to enhance the diversity of algorithms in many-objective optimization problems, a new distance dominance relation is proposed in this paper. Firstly, in order to ensure the convergence of the algorithm, in the same niche, the distance dominance relation calculates the distance from the candidate solution to the ideal point as the fitness value, and selects the candidate solution with good fitness value as the non-dominant solution.Then, in order to enhance the diversity of the algorithm, the distance dominance relation sets each candidate solution to have the same niche and ensures that only one optimal solution is retained in the same niche. Finally, the VaEA algorithm is improved based on the proposed distance dominance relation, and the algorithm is named VaEA-DDR. On the DTLZ and IDTLZ test of 5, 8, 10, 15 dimensional objectives, the improved algorithm is compared with six commonly used algorithms. Experimental results show that the improved algorithm is highly competitive and can significantly enhance the diversity of the algorithm.

Key words: many-objective optimization, distance dominance relation, diversity, VaEA

中图分类号:

TP301

顾清华, 徐青松, 李学现. 基于距离优势关系的高维多目标进化算法[J]. 计算机科学与探索, 2022, 16(11): 2642-2652.

GU Qinghua, XU Qingsong, LI Xuexian. Many-Objective Evolutionary Algorithm Based on Distance Dominance Relation[J]. Journal of Frontiers of Computer Science and Technology, 2022, 16(11): 2642-2652.

图/表 11

图1 VaEA算法流程图

Fig.1 Flow chart of VaEA algorithm

图2 双目标空间中距离优势关系支配区域

Fig.2 Dominated area of distance dominance relationship in dual-objective space

图3 双目标空间中不同种群分布示意图

Fig.3 Schematic diagram of distribution of different populations in dual-objective space

表1 测试问题及其特征

Table 1 Test problems and their characteristics

问题	特征
DTLZ1	线性的、多模态的
DTLZ2	凹的
DTLZ3	凹的、多模态的
DTLZ4	凹的、有偏的
DTLZ5	凹的、退化的
IDTLZ1	线性的
IDTLZ2	凸的

表2 权重向量的数量和种群规模

Table 2 Number of weight vectors and population size

$m$	权重向量数量	种群规模
5	210	212
8	156(p₁=3, p₂=2)	156
10	275(p₁=3, p₂=2)	276
15	135(p₁=2, p₂=1)	136

表2 权重向量的数量和种群规模

Table 2 Number of weight vectors and population size

$m$	权重向量数量	种群规模
5	210	212
8	156(p₁=3, p₂=2)	156
10	275(p₁=3, p₂=2)	276
15	135(p₁=2, p₂=1)	136

表3 7种算法在测试问题上获得的IGD值

Table 3 IGD values obtained by 7 algorithms on test problems

Problem	$m$	NSGA-Ⅲ	MOEADD	MOMBI-Ⅱ	RPD-NSGA-Ⅱ	NSGA-Ⅱ-SDR	VaEA	VaEA-DDR
DTLZ1	5	5.792 9E-2 (4.99E-3) +	5.286 1E-2 (7.52E-4) +	5.497 7E-2 (7.83E-4) +	6.132 0E-2 (1.24E-3) =	7.347 8E-2 (9.10E-3) -	9.660 1E-2 (2.64E-2) -	6.239 0E-2 (2.27E-3)
	8	1.564 5E-1 (7.83E-2) =	9.323 6E-2 (1.06E-3) +	2.137 9E-1 (4.34E-2) -	1.462 3E-1 (2.28E-2) -	1.598 2E-1 (1.44E-2) -	5.089 9E-1 (3.03E-1) -	1.241 8E-1 (2.69E-3)
	10	1.665 6E-1 (8.52E-2) =	1.059 3E-1 (5.74E-4) +	2.111 8E-1 (4.17E-2) -	1.723 4E-1 (1.54E-2) -	1.635 3E-1 (1.57E-2) -	3.152 4E-1 (1.41E-1) -	1.215 5E-1 (1.38E-3)
	15	2.067 7E-1 (5.40E-2) -	1.440 1E-1 (9.93E-3) +	2.807 7E-1 (1.93E-2) -	1.850 1E-1 (1.10E-2) -	2.046 9E-1 (2.70E-2) -	3.866 8E-1 (2.04E-1) -	1.507 3E-1 (1.16E-3)
DTLZ2	5	1.654 0E-1 (8.25E-5) -	1.652 8E-1 (5.80E-5) -	1.700 7E-1 (1.95E-3) -	1.667 6E-1 (9.02E-4) -	1.833 1E-1 (4.89E-3) -	1.689 2E-1 (1.04E-3) -	1.642 3E-1 (1.22E-3)
	8	3.239 5E-1 (2.88E-2) +	3.155 8E-1 (1.51E-4) +	3.358 9E-1 (1.15E-2) +	3.417 7E-1 (4.81E-3) +	4.977 6E-1 (1.39E-1) -	3.654 6E-1 (2.85E-3) -	3.566 8E-1 (4.00E-3)
	10	4.316 0E-1 (2.77E-2) -	4.215 1E-1 (4.18E-4) -	4.392 3E-1 (4.23E-3) -	4.364 2E-1 (3.58E-3) -	4.345 8E-1 (5.99E-3) -	4.268 6E-1 (4.57E-3) -	4.003 2E-1 (2.49E-3)
	15	6.451 7E-1 (1.61E-2) -	6.233 8E-1 (4.49E-3) -	8.310 3E-1 (9.34E-2) -	6.300 5E-1 (1.83E-3) -	7.363 6E-1 (9.49E-2) -	6.013 5E-1 (9.34E-3) -	5.735 6E-1 (5.32E-3)
DTLZ3	5	2.713 0E+0 (1.88E+0) -	2.756 8E+0 (1.71E+0) -	3.210 4E-1 (2.22E-1) =	5.928 4E-1 (5.08E-1) -	3.774 2E-1 (3.19E-1) =	1.783 7E+0 (1.32E+0) -	5.403 8E-1 (4.60E-1)
	8	5.431 7E+0 (3.34E+0) -	1.110 3E+0 (1.03E+0) -	5.157 9E-1 (1.27E-1) -	1.253 1E+0 (7.97E-1) -	4.185 0E-1 (1.41E-1) -	1.225 9E+1 (4.99E+0) -	3.989 6E-1 (1.51E-1)
	10	4.645 3E+0 (3.41E+0) -	5.268 2E-1 (2.00E-1) -	6.881 4E-1 (1.86E-1) -	4.908 4E-1 (3.79E-2) -	4.382 8E-1 (6.30E-3) -	3.018 3E+1 (1.01E+1) -	4.022 3E-1 (2.42E-3)
	15	5.419 6E+0 (3.06E+0) -	8.665 9E-1 (5.56E-1) -	1.102 3E+0 (2.53E-2) -	1.739 3E+0 (1.35E+0) -	6.954 6E-1 (4.58E-2) -	2.004 0E+1 (7.60E+0) -	5.762 1E-1 (6.76E-3)
DTLZ4	5	1.738 1E-1 (4.59E-2) =	1.652 0E-1 (5.00E-5) =	1.735 8E-1 (2.22E-3) -	1.695 1E-1 (1.67E-3) +	6.444 6E-1 (5.62E-2) -	1.711 4E-1 (1.40E-3) +	1.732 2E-1 (4.28E-2)
	8	3.657 7E-1 (9.09E-2) -	3.240 2E-1 (2.98E-2) +	3.689 7E-1 (4.26E-2) -	3.552 8E-1 (3.52E-3) =	7.899 0E-1 (7.55E-2) -	3.697 4E-1 (4.75E-3) -	3.654 3E-1 (2.74E-2)
	10	4.376 6E-1 (2.93E-2) -	4.205 3E-1 (6.35E-4) -	4.564 7E-1 (1.32E-2) -	4.430 6E-1 (2.36E-3) -	7.746 3E-1 (5.29E-2) -	4.420 0E-1 (8.28E-3) -	4.028 0E-1 (1.11E-2)
	15	6.548 2E-1 (1.43E-2) -	6.314 4E-1 (6.69E-3) -	6.613 5E-1 (2.61E-2) -	6.336 3E-1 (8.78E-4) -	8.121 2E-1 (2.86E-2) -	5.995 4E-1 (5.12E-3) -	5.749 3E-1 (3.21E-3)
DTLZ5	5	9.873 8E-2 (3.65E-2) -	1.085 2E-1 (9.80E-3) -	2.339 1E-1 (5.44E-2) -	8.935 3E-2 (2.19E-2) -	6.701 6E-2 (1.21E-2) -	1.286 8E-1 (3.25E-2) -	4.107 0E-2 (5.81E-3)
	8	2.449 4E-1 (8.05E-2) -	1.574 5E-1 (2.78E-2) -	3.403 5E-1 (1.62E-1) -	1.452 4E-1 (2.89E-2) -	1.249 5E-1 (2.25E-2) -	3.340 6E-1 (7.35E-2) -	7.894 4E-2 (3.14E-2)
	10	3.124 9E-1 (9.17E-2) -	1.328 8E-1 (2.10E-2) -	6.807 2E-1 (8.36E-2) -	1.568 3E-1 (2.01E-2) -	1.699 8E-1 (3.45E-2) -	3.856 3E-1 (7.23E-2) -	8.440 8E-2 (1.58E-2)
	15	2.532 9E-1 (4.41E-2) -	1.524 6E-1 (9.79E-3) -	7.097 0E-1 (2.91E-2) -	1.760 0E-1 (2.16E-2) -	9.816 3E-2 (1.59E-2) =	4.582 3E-1 (1.68E-1) -	9.620 5E-2 (2.86E-2)
IDTLZ1	5	9.136 3E-2 (6.58E-3) -	9.966 0E-2 (4.14E-3) -	1.134 6E-1 (2.13E-3) -	1.130 9E-1 (8.85E-3) -	5.278 0E-2 (7.59E-4) =	5.922 4E-2 (8.13E-3) -	5.281 3E-2 (6.88E-4)
	8	1.419 7E-1 (3.59E-3) -	2.213 3E-1 (1.73E-2) -	1.743 4E-1 (8.09E-3) -	2.774 0E-1 (3.36E-2) -	1.037 4E-1 (1.36E-3) =	1.087 8E-1 (1.30E-2) -	1.030 0E-1 (1.39E-3)
	10	1.372 9E-1 (3.20E-3) -	2.292 9E-1 (1.19E-2) -	1.858 4E-1 (6.86E-3) -	3.012 3E-1 (1.84E-2) -	1.093 4E-1 (1.20E-3) -	1.146 1E-1 (1.70E-2) -	1.042 8E-1 (1.06E-3)
	15	1.724 7E-1 (5.55E-3) -	3.325 0E-1 (2.80E-2) -	2.096 1E-1 (1.07E-2) -	4.210 5E-1 (3.53E-2) -	1.566 3E-1 (3.44E-3) =	1.506 2E-1 (3.93E-3) +	1.586 7E-1 (1.34E-2)
IDTLZ2	5	2.046 7E-1 (5.93E-3) +	2.482 6E-1 (1.80E-2) -	3.176 8E-1 (1.89E-3) -	2.594 8E-1 (2.43E-3) -	2.230 0E-1 (8.76E-3) =	1.795 9E-1 (1.73E-3) +	2.320 4E-1 (8.68E-3)
	8	5.157 6E-1 (1.33E-2) -	6.496 6E-1 (1.83E-2) -	5.919 8E-1 (3.89E-3) -	5.774 3E-1 (1.01E-2) -	4.536 7E-1 (9.71E-3) -	3.734 0E-1 (1.78E-3) =	3.601 6E-1 (1.10E-2)
	10	6.036 8E-1 (1.88E-2) -	7.433 0E-1 (4.64E-3) -	6.679 3E-1 (4.94E-3) -	6.580 4E-1 (9.34E-3) -	4.690 9E-1 (7.25E-3) -	4.259 0E-1 (2.21E-3) -	3.728 4E-1 (5.87E-3)
	15	7.828 9E-1 (9.88E-3) -	9.460 8E-1 (1.96E-2) -	8.489 1E-1 (5.53E-3) -	7.923 6E-1 (1.28E-2) -	6.554 3E-1 (7.01E-3) -	5.961 9E-1 (2.97E-3) +	6.561 3E-1 (6.97E-3)
+/-/=		3/22/3	6/21/1	2/25/1	2/24/2	0/22/6	4/23/1

表3 7种算法在测试问题上获得的IGD值

Table 3 IGD values obtained by 7 algorithms on test problems

Problem	$m$	NSGA-Ⅲ	MOEADD	MOMBI-Ⅱ	RPD-NSGA-Ⅱ	NSGA-Ⅱ-SDR	VaEA	VaEA-DDR
DTLZ1	5	5.792 9E-2 (4.99E-3) +	5.286 1E-2 (7.52E-4) +	5.497 7E-2 (7.83E-4) +	6.132 0E-2 (1.24E-3) =	7.347 8E-2 (9.10E-3) -	9.660 1E-2 (2.64E-2) -	6.239 0E-2 (2.27E-3)
	8	1.564 5E-1 (7.83E-2) =	9.323 6E-2 (1.06E-3) +	2.137 9E-1 (4.34E-2) -	1.462 3E-1 (2.28E-2) -	1.598 2E-1 (1.44E-2) -	5.089 9E-1 (3.03E-1) -	1.241 8E-1 (2.69E-3)
	10	1.665 6E-1 (8.52E-2) =	1.059 3E-1 (5.74E-4) +	2.111 8E-1 (4.17E-2) -	1.723 4E-1 (1.54E-2) -	1.635 3E-1 (1.57E-2) -	3.152 4E-1 (1.41E-1) -	1.215 5E-1 (1.38E-3)
	15	2.067 7E-1 (5.40E-2) -	1.440 1E-1 (9.93E-3) +	2.807 7E-1 (1.93E-2) -	1.850 1E-1 (1.10E-2) -	2.046 9E-1 (2.70E-2) -	3.866 8E-1 (2.04E-1) -	1.507 3E-1 (1.16E-3)
DTLZ2	5	1.654 0E-1 (8.25E-5) -	1.652 8E-1 (5.80E-5) -	1.700 7E-1 (1.95E-3) -	1.667 6E-1 (9.02E-4) -	1.833 1E-1 (4.89E-3) -	1.689 2E-1 (1.04E-3) -	1.642 3E-1 (1.22E-3)
	8	3.239 5E-1 (2.88E-2) +	3.155 8E-1 (1.51E-4) +	3.358 9E-1 (1.15E-2) +	3.417 7E-1 (4.81E-3) +	4.977 6E-1 (1.39E-1) -	3.654 6E-1 (2.85E-3) -	3.566 8E-1 (4.00E-3)
	10	4.316 0E-1 (2.77E-2) -	4.215 1E-1 (4.18E-4) -	4.392 3E-1 (4.23E-3) -	4.364 2E-1 (3.58E-3) -	4.345 8E-1 (5.99E-3) -	4.268 6E-1 (4.57E-3) -	4.003 2E-1 (2.49E-3)
	15	6.451 7E-1 (1.61E-2) -	6.233 8E-1 (4.49E-3) -	8.310 3E-1 (9.34E-2) -	6.300 5E-1 (1.83E-3) -	7.363 6E-1 (9.49E-2) -	6.013 5E-1 (9.34E-3) -	5.735 6E-1 (5.32E-3)
DTLZ3	5	2.713 0E+0 (1.88E+0) -	2.756 8E+0 (1.71E+0) -	3.210 4E-1 (2.22E-1) =	5.928 4E-1 (5.08E-1) -	3.774 2E-1 (3.19E-1) =	1.783 7E+0 (1.32E+0) -	5.403 8E-1 (4.60E-1)
	8	5.431 7E+0 (3.34E+0) -	1.110 3E+0 (1.03E+0) -	5.157 9E-1 (1.27E-1) -	1.253 1E+0 (7.97E-1) -	4.185 0E-1 (1.41E-1) -	1.225 9E+1 (4.99E+0) -	3.989 6E-1 (1.51E-1)
	10	4.645 3E+0 (3.41E+0) -	5.268 2E-1 (2.00E-1) -	6.881 4E-1 (1.86E-1) -	4.908 4E-1 (3.79E-2) -	4.382 8E-1 (6.30E-3) -	3.018 3E+1 (1.01E+1) -	4.022 3E-1 (2.42E-3)
	15	5.419 6E+0 (3.06E+0) -	8.665 9E-1 (5.56E-1) -	1.102 3E+0 (2.53E-2) -	1.739 3E+0 (1.35E+0) -	6.954 6E-1 (4.58E-2) -	2.004 0E+1 (7.60E+0) -	5.762 1E-1 (6.76E-3)
DTLZ4	5	1.738 1E-1 (4.59E-2) =	1.652 0E-1 (5.00E-5) =	1.735 8E-1 (2.22E-3) -	1.695 1E-1 (1.67E-3) +	6.444 6E-1 (5.62E-2) -	1.711 4E-1 (1.40E-3) +	1.732 2E-1 (4.28E-2)
	8	3.657 7E-1 (9.09E-2) -	3.240 2E-1 (2.98E-2) +	3.689 7E-1 (4.26E-2) -	3.552 8E-1 (3.52E-3) =	7.899 0E-1 (7.55E-2) -	3.697 4E-1 (4.75E-3) -	3.654 3E-1 (2.74E-2)
	10	4.376 6E-1 (2.93E-2) -	4.205 3E-1 (6.35E-4) -	4.564 7E-1 (1.32E-2) -	4.430 6E-1 (2.36E-3) -	7.746 3E-1 (5.29E-2) -	4.420 0E-1 (8.28E-3) -	4.028 0E-1 (1.11E-2)
	15	6.548 2E-1 (1.43E-2) -	6.314 4E-1 (6.69E-3) -	6.613 5E-1 (2.61E-2) -	6.336 3E-1 (8.78E-4) -	8.121 2E-1 (2.86E-2) -	5.995 4E-1 (5.12E-3) -	5.749 3E-1 (3.21E-3)
DTLZ5	5	9.873 8E-2 (3.65E-2) -	1.085 2E-1 (9.80E-3) -	2.339 1E-1 (5.44E-2) -	8.935 3E-2 (2.19E-2) -	6.701 6E-2 (1.21E-2) -	1.286 8E-1 (3.25E-2) -	4.107 0E-2 (5.81E-3)
	8	2.449 4E-1 (8.05E-2) -	1.574 5E-1 (2.78E-2) -	3.403 5E-1 (1.62E-1) -	1.452 4E-1 (2.89E-2) -	1.249 5E-1 (2.25E-2) -	3.340 6E-1 (7.35E-2) -	7.894 4E-2 (3.14E-2)
	10	3.124 9E-1 (9.17E-2) -	1.328 8E-1 (2.10E-2) -	6.807 2E-1 (8.36E-2) -	1.568 3E-1 (2.01E-2) -	1.699 8E-1 (3.45E-2) -	3.856 3E-1 (7.23E-2) -	8.440 8E-2 (1.58E-2)
	15	2.532 9E-1 (4.41E-2) -	1.524 6E-1 (9.79E-3) -	7.097 0E-1 (2.91E-2) -	1.760 0E-1 (2.16E-2) -	9.816 3E-2 (1.59E-2) =	4.582 3E-1 (1.68E-1) -	9.620 5E-2 (2.86E-2)
IDTLZ1	5	9.136 3E-2 (6.58E-3) -	9.966 0E-2 (4.14E-3) -	1.134 6E-1 (2.13E-3) -	1.130 9E-1 (8.85E-3) -	5.278 0E-2 (7.59E-4) =	5.922 4E-2 (8.13E-3) -	5.281 3E-2 (6.88E-4)
	8	1.419 7E-1 (3.59E-3) -	2.213 3E-1 (1.73E-2) -	1.743 4E-1 (8.09E-3) -	2.774 0E-1 (3.36E-2) -	1.037 4E-1 (1.36E-3) =	1.087 8E-1 (1.30E-2) -	1.030 0E-1 (1.39E-3)
	10	1.372 9E-1 (3.20E-3) -	2.292 9E-1 (1.19E-2) -	1.858 4E-1 (6.86E-3) -	3.012 3E-1 (1.84E-2) -	1.093 4E-1 (1.20E-3) -	1.146 1E-1 (1.70E-2) -	1.042 8E-1 (1.06E-3)
	15	1.724 7E-1 (5.55E-3) -	3.325 0E-1 (2.80E-2) -	2.096 1E-1 (1.07E-2) -	4.210 5E-1 (3.53E-2) -	1.566 3E-1 (3.44E-3) =	1.506 2E-1 (3.93E-3) +	1.586 7E-1 (1.34E-2)
IDTLZ2	5	2.046 7E-1 (5.93E-3) +	2.482 6E-1 (1.80E-2) -	3.176 8E-1 (1.89E-3) -	2.594 8E-1 (2.43E-3) -	2.230 0E-1 (8.76E-3) =	1.795 9E-1 (1.73E-3) +	2.320 4E-1 (8.68E-3)
	8	5.157 6E-1 (1.33E-2) -	6.496 6E-1 (1.83E-2) -	5.919 8E-1 (3.89E-3) -	5.774 3E-1 (1.01E-2) -	4.536 7E-1 (9.71E-3) -	3.734 0E-1 (1.78E-3) =	3.601 6E-1 (1.10E-2)
	10	6.036 8E-1 (1.88E-2) -	7.433 0E-1 (4.64E-3) -	6.679 3E-1 (4.94E-3) -	6.580 4E-1 (9.34E-3) -	4.690 9E-1 (7.25E-3) -	4.259 0E-1 (2.21E-3) -	3.728 4E-1 (5.87E-3)
	15	7.828 9E-1 (9.88E-3) -	9.460 8E-1 (1.96E-2) -	8.489 1E-1 (5.53E-3) -	7.923 6E-1 (1.28E-2) -	6.554 3E-1 (7.01E-3) -	5.961 9E-1 (2.97E-3) +	6.561 3E-1 (6.97E-3)
+/-/=		3/22/3	6/21/1	2/25/1	2/24/2	0/22/6	4/23/1

图4 7种算法在15目标上获得的IGD值变化曲线

Fig.4 IGD value change curves of 7 algorithms on 15 objectives

表4 7种算法在测试问题上获得的Spread值

Table 4 Spread values obtained by 7 algorithms on test problems

Problem	$m$	NSGA-Ⅲ	MOEADD	MOMBI-Ⅱ	RPD-NSGA-Ⅱ	NSGA-Ⅱ-SDR	VaEA	VaEA-DDR
DTLZ1	5	1.856 4E-1 (1.47E-1) +	5.539 6E-2 (1.38E-2) +	1.947 4E-1 (5.28E-2) +	7.893 1E-1 (2.52E-1) -	4.043 2E-1 (4.98E-2) -	1.464 3E+0 (5.81E-1) -	2.884 6E-1 (2.81E-2)
	8	5.571 3E-1 (3.91E-1) =	1.112 8E-1 (3.72E-2) +	1.238 0E+0 (1.22E-1) -	1.761 4E+0 (4.04E-1) -	5.708 3E-1 (5.75E-2) -	1.639 6E+0 (1.85E-1) -	3.577 4E-1 (3.82E-2)
	10	7.504 8E-1 (6.28E-1) -	1.018 8E-1 (8.81E-3) +	1.277 2E+0 (1.19E-1) -	1.940 8E+0 (4.60E-2) -	5.554 0E-1 (6.00E-2) -	1.625 8E+0 (1.73E-1) -	3.116 2E-1 (3.52E-2)
	15	1.028 2E+0 (8.01E-1) =	3.256 9E-1 (1.87E-1) +	1.137 7E+0 (9.21E-2) -	1.992 6E+0 (5.37E-2) -	6.897 5E-1 (1.42E-1) -	1.505 9E+0 (2.04E-1) -	5.462 2E-1 (4.51E-2)
DTLZ2	5	1.514 0E-1 (1.25E-3) -	1.506 9E-1 (8.93E-4) -	1.873 6E-1 (1.74E-2) -	1.668 8E-1 (1.31E-2) -	5.220 3E-1 (9.01E-2) -	1.047 7E-1 (8.52E-3) =	1.090 6E-1 (1.01E-2)
	8	1.572 7E-1 (1.10E-1) -	1.376 1E-1 (1.71E-3) -	2.056 3E-1 (6.98E-2) -	2.516 7E-1 (4.92E-2) -	9.096 3E-1 (4.42E-1) -	1.272 3E-1 (9.67E-3) -	1.113 5E-1 (1.08E-2)
	10	1.938 9E-1 (1.25E-1) -	1.514 8E-1 (1.75E-3) -	2.950 0E-1 (1.13E-1) -	3.116 6E-1 (7.64E-2) -	6.750 8E-1 (1.63E-1) -	1.050 2E-1 (7.21E-3) -	8.870 1E-2 (6.60E-3)
	15	7.573 5E-1 (3.49E-1) -	3.409 8E-1 (8.21E-2) -	1.075 7E+0 (1.61E-1) -	3.926 0E-1 (3.17E-2) -	1.108 0E+0 (2.20E-1) -	2.566 6E-1 (1.23E-2) -	1.677 2E-1 (7.94E-3)
DTLZ3	5	1.154 6E+0 (3.83E-1) -	5.065 7E-1 (3.24E-1) -	7.701 9E-1 (1.67E-1) -	1.844 7E+0 (1.19E-1) -	4.306 2E-1 (1.07E-1) -	1.193 5E+0 (4.92E-1) -	1.933 2E-1 (8.16E-2)
	8	9.009 6E-1 (3.51E-1) -	5.239 0E-1 (4.07E-1) -	8.949 1E-1 (2.06E-1) -	1.927 1E+0 (3.72E-2) -	7.463 5E-1 (1.46E-1) -	8.652 8E-1 (1.29E-1) -	1.330 8E-1 (2.68E-2)
	10	1.064 9E+0 (2.93E-1) -	3.053 2E-1 (1.81E-1) -	9.650 0E-1 (2.21E-1) -	1.954 0E+0 (2.07E-2) -	7.052 3E-1 (1.12E-1) -	4.406 8E-1 (6.42E-2) -	9.885 0E-2 (5.53E-3)
	15	1.410 9E+0 (1.85E-1) -	5.018 6E-1 (4.06E-1) -	1.005 8E+0 (5.75E-3) -	1.911 9E+0 (4.63E-2) -	1.001 7E+0 (2.19E-1) -	4.894 0E-1 (5.78E-2) -	1.776 7E-1 (1.25E-2)
DTLZ4	5	1.715 1E-1 (1.19E-1) -	1.497 5E-1 (8.55E-4) -	1.866 5E-1 (1.61E-2) -	1.771 7E-1 (2.38E-2) -	7.401 0E-1 (3.25E-1) -	1.046 8E-1 (5.62E-3) -	9.932 0E-2 (1.33E-2)
	8	2.407 6E-1 (2.01E-1) -	2.016 6E-1 (2.56E-1) -	3.226 0E-1 (1.46E-1) -	3.276 8E-1 (5.37E-2) -	1.029 5E+0 (2.48E-1) -	1.288 7E-1 (9.16E-3) -	9.818 9E-2 (1.24E-2)
	10	2.113 4E-1 (1.28E-1) -	1.519 4E-1 (2.25E-3) -	2.995 5E-1 (7.89E-2) -	3.516 6E-1 (3.95E-2) -	8.621 9E-1 (3.32E-1) -	1.105 8E-1 (8.63E-3) -	7.572 6E-2 (4.68E-3)
	15	7.699 7E-1 (2.38E-1) -	3.979 0E-1 (1.38E-1) -	9.079 6E-1 (3.69E-1) -	4.732 3E-1 (1.40E-2) -	1.000 0E+0 (4.86E-1) -	2.431 7E-1 (2.34E-2) -	1.273 2E-1 (1.39E-2)
DTLZ5	5	6.696 4E-1 (5.49E-2) -	1.335 4E+0 (7.70E-2) -	1.341 0E+0 (1.91E-1) -	1.053 1E+0 (6.66E-2) -	4.278 3E-1 (5.28E-2) =	3.769 3E-1 (1.97E-2) +	4.194 1E-1 (4.29E-2)
	8	6.234 3E-1 (3.98E-2) -	1.285 7E+0 (6.48E-2) -	1.029 0E+0 (4.04E-2) -	1.162 1E+0 (1.02E-1) -	4.400 9E-1 (7.14E-2) =	2.750 7E-1 (1.92E-2) +	4.422 7E-1 (9.28E-2)
	10	5.639 0E-1 (3.09E-2) -	1.269 5E+0 (5.82E-2) -	1.003 7E+0 (1.11E-2) -	1.114 7E+0 (6.13E-2) -	4.486 9E-1 (4.44E-2) =	1.889 6E-1 (2.01E-2) +	4.925 0E-1 (9.00E-2)
	15	6.429 4E-1 (6.01E-2) =	1.613 4E+0 (1.33E-1) -	1.000 1E+0 (3.38E-4) -	1.400 2E+0 (8.79E-2) -	4.349 9E-1 (8.47E-2) +	3.668 7E-1 (5.67E-2) +	5.893 5E-1 (3.43E-1)
IDTLZ1	5	8.210 9E-1 (3.92E-1) -	1.061 4E+0 (6.74E-2) -	1.763 7E+0 (1.18E-1) -	1.780 6E+0 (3.22E-1) -	2.103 4E-1 (5.43E-2) =	7.503 9E-1 (8.19E-1) =	2.035 2E-1 (1.61E-2)
	8	7.101 2E-1 (5.57E-2) -	1.179 6E+0 (2.00E-1) -	9.937 7E-1 (2.09E-2) -	1.607 2E+0 (4.36E-1) -	2.604 2E-1 (5.72E-2) -	4.644 9E-1 (6.63E-1) -	2.327 3E-1 (2.18E-2)
	10	5.830 2E-1 (3.84E-2) -	1.093 0E+0 (7.26E-2) -	9.742 2E-1 (1.95E-2) -	1.626 0E+0 (4.32E-1) -	2.097 0E-1 (4.39E-2) -	4.229 1E-1 (5.82E-1) -	1.791 1E-1 (1.55E-2)
	15	7.410 3E-1 (4.33E-2) -	1.369 9E+0 (4.77E-1) -	9.821 0E-1 (7.24E-3) -	1.488 7E+0 (4.68E-1) -	3.850 5E-1 (1.20E-1) =	5.552 6E-1 (4.63E-1) -	3.572 5E-1 (2.40E-2)
IDTLZ2	5	6.362 7E-1 (5.44E-2) -	1.149 1E+0 (8.37E-2) -	1.460 2E+0 (2.52E-1) -	1.121 3E+0 (6.38E-2) -	2.696 2E-1 (3.25E-2) =	1.626 0E-1 (9.77E-3) +	2.612 9E-1 (3.91E-2)
	8	6.313 2E-1 (4.49E-2) -	1.101 9E+0 (6.53E-2) -	1.079 2E+0 (4.98E-2) -	1.025 4E+0 (3.53E-2) -	2.818 6E-1 (3.88E-2) -	1.932 1E-1 (1.86E-2) +	2.530 6E-1 (3.01E-2)
	10	6.448 6E-1 (7.12E-2) -	1.004 4E+0 (2.79E-3) -	9.497 7E-1 (3.73E-2) -	1.011 4E+0 (4.56E-2) -	2.347 6E-1 (3.24E-2) -	1.496 8E-1 (1.10E-2) +	1.820 1E-1 (2.18E-2)
	15	8.593 5E-1 (2.57E-2) -	1.057 5E+0 (1.63E-1) -	9.401 8E-1 (2.13E-2) -	9.476 0E-1 (6.75E-2) -	3.440 8E-1 (4.75E-2) -	4.534 2E-1 (1.41E-2) -	3.098 5E-1 (3.48E-2)
+/-/=		1/24/3	4/24/0	1/27/0	0/28/0	1/21/6	7/19/2

表4 7种算法在测试问题上获得的Spread值

Table 4 Spread values obtained by 7 algorithms on test problems

Problem	$m$	NSGA-Ⅲ	MOEADD	MOMBI-Ⅱ	RPD-NSGA-Ⅱ	NSGA-Ⅱ-SDR	VaEA	VaEA-DDR
DTLZ1	5	1.856 4E-1 (1.47E-1) +	5.539 6E-2 (1.38E-2) +	1.947 4E-1 (5.28E-2) +	7.893 1E-1 (2.52E-1) -	4.043 2E-1 (4.98E-2) -	1.464 3E+0 (5.81E-1) -	2.884 6E-1 (2.81E-2)
	8	5.571 3E-1 (3.91E-1) =	1.112 8E-1 (3.72E-2) +	1.238 0E+0 (1.22E-1) -	1.761 4E+0 (4.04E-1) -	5.708 3E-1 (5.75E-2) -	1.639 6E+0 (1.85E-1) -	3.577 4E-1 (3.82E-2)
	10	7.504 8E-1 (6.28E-1) -	1.018 8E-1 (8.81E-3) +	1.277 2E+0 (1.19E-1) -	1.940 8E+0 (4.60E-2) -	5.554 0E-1 (6.00E-2) -	1.625 8E+0 (1.73E-1) -	3.116 2E-1 (3.52E-2)
	15	1.028 2E+0 (8.01E-1) =	3.256 9E-1 (1.87E-1) +	1.137 7E+0 (9.21E-2) -	1.992 6E+0 (5.37E-2) -	6.897 5E-1 (1.42E-1) -	1.505 9E+0 (2.04E-1) -	5.462 2E-1 (4.51E-2)
DTLZ2	5	1.514 0E-1 (1.25E-3) -	1.506 9E-1 (8.93E-4) -	1.873 6E-1 (1.74E-2) -	1.668 8E-1 (1.31E-2) -	5.220 3E-1 (9.01E-2) -	1.047 7E-1 (8.52E-3) =	1.090 6E-1 (1.01E-2)
	8	1.572 7E-1 (1.10E-1) -	1.376 1E-1 (1.71E-3) -	2.056 3E-1 (6.98E-2) -	2.516 7E-1 (4.92E-2) -	9.096 3E-1 (4.42E-1) -	1.272 3E-1 (9.67E-3) -	1.113 5E-1 (1.08E-2)
	10	1.938 9E-1 (1.25E-1) -	1.514 8E-1 (1.75E-3) -	2.950 0E-1 (1.13E-1) -	3.116 6E-1 (7.64E-2) -	6.750 8E-1 (1.63E-1) -	1.050 2E-1 (7.21E-3) -	8.870 1E-2 (6.60E-3)
	15	7.573 5E-1 (3.49E-1) -	3.409 8E-1 (8.21E-2) -	1.075 7E+0 (1.61E-1) -	3.926 0E-1 (3.17E-2) -	1.108 0E+0 (2.20E-1) -	2.566 6E-1 (1.23E-2) -	1.677 2E-1 (7.94E-3)
DTLZ3	5	1.154 6E+0 (3.83E-1) -	5.065 7E-1 (3.24E-1) -	7.701 9E-1 (1.67E-1) -	1.844 7E+0 (1.19E-1) -	4.306 2E-1 (1.07E-1) -	1.193 5E+0 (4.92E-1) -	1.933 2E-1 (8.16E-2)
	8	9.009 6E-1 (3.51E-1) -	5.239 0E-1 (4.07E-1) -	8.949 1E-1 (2.06E-1) -	1.927 1E+0 (3.72E-2) -	7.463 5E-1 (1.46E-1) -	8.652 8E-1 (1.29E-1) -	1.330 8E-1 (2.68E-2)
	10	1.064 9E+0 (2.93E-1) -	3.053 2E-1 (1.81E-1) -	9.650 0E-1 (2.21E-1) -	1.954 0E+0 (2.07E-2) -	7.052 3E-1 (1.12E-1) -	4.406 8E-1 (6.42E-2) -	9.885 0E-2 (5.53E-3)
	15	1.410 9E+0 (1.85E-1) -	5.018 6E-1 (4.06E-1) -	1.005 8E+0 (5.75E-3) -	1.911 9E+0 (4.63E-2) -	1.001 7E+0 (2.19E-1) -	4.894 0E-1 (5.78E-2) -	1.776 7E-1 (1.25E-2)
DTLZ4	5	1.715 1E-1 (1.19E-1) -	1.497 5E-1 (8.55E-4) -	1.866 5E-1 (1.61E-2) -	1.771 7E-1 (2.38E-2) -	7.401 0E-1 (3.25E-1) -	1.046 8E-1 (5.62E-3) -	9.932 0E-2 (1.33E-2)
	8	2.407 6E-1 (2.01E-1) -	2.016 6E-1 (2.56E-1) -	3.226 0E-1 (1.46E-1) -	3.276 8E-1 (5.37E-2) -	1.029 5E+0 (2.48E-1) -	1.288 7E-1 (9.16E-3) -	9.818 9E-2 (1.24E-2)
	10	2.113 4E-1 (1.28E-1) -	1.519 4E-1 (2.25E-3) -	2.995 5E-1 (7.89E-2) -	3.516 6E-1 (3.95E-2) -	8.621 9E-1 (3.32E-1) -	1.105 8E-1 (8.63E-3) -	7.572 6E-2 (4.68E-3)
	15	7.699 7E-1 (2.38E-1) -	3.979 0E-1 (1.38E-1) -	9.079 6E-1 (3.69E-1) -	4.732 3E-1 (1.40E-2) -	1.000 0E+0 (4.86E-1) -	2.431 7E-1 (2.34E-2) -	1.273 2E-1 (1.39E-2)
DTLZ5	5	6.696 4E-1 (5.49E-2) -	1.335 4E+0 (7.70E-2) -	1.341 0E+0 (1.91E-1) -	1.053 1E+0 (6.66E-2) -	4.278 3E-1 (5.28E-2) =	3.769 3E-1 (1.97E-2) +	4.194 1E-1 (4.29E-2)
	8	6.234 3E-1 (3.98E-2) -	1.285 7E+0 (6.48E-2) -	1.029 0E+0 (4.04E-2) -	1.162 1E+0 (1.02E-1) -	4.400 9E-1 (7.14E-2) =	2.750 7E-1 (1.92E-2) +	4.422 7E-1 (9.28E-2)
	10	5.639 0E-1 (3.09E-2) -	1.269 5E+0 (5.82E-2) -	1.003 7E+0 (1.11E-2) -	1.114 7E+0 (6.13E-2) -	4.486 9E-1 (4.44E-2) =	1.889 6E-1 (2.01E-2) +	4.925 0E-1 (9.00E-2)
	15	6.429 4E-1 (6.01E-2) =	1.613 4E+0 (1.33E-1) -	1.000 1E+0 (3.38E-4) -	1.400 2E+0 (8.79E-2) -	4.349 9E-1 (8.47E-2) +	3.668 7E-1 (5.67E-2) +	5.893 5E-1 (3.43E-1)
IDTLZ1	5	8.210 9E-1 (3.92E-1) -	1.061 4E+0 (6.74E-2) -	1.763 7E+0 (1.18E-1) -	1.780 6E+0 (3.22E-1) -	2.103 4E-1 (5.43E-2) =	7.503 9E-1 (8.19E-1) =	2.035 2E-1 (1.61E-2)
	8	7.101 2E-1 (5.57E-2) -	1.179 6E+0 (2.00E-1) -	9.937 7E-1 (2.09E-2) -	1.607 2E+0 (4.36E-1) -	2.604 2E-1 (5.72E-2) -	4.644 9E-1 (6.63E-1) -	2.327 3E-1 (2.18E-2)
	10	5.830 2E-1 (3.84E-2) -	1.093 0E+0 (7.26E-2) -	9.742 2E-1 (1.95E-2) -	1.626 0E+0 (4.32E-1) -	2.097 0E-1 (4.39E-2) -	4.229 1E-1 (5.82E-1) -	1.791 1E-1 (1.55E-2)
	15	7.410 3E-1 (4.33E-2) -	1.369 9E+0 (4.77E-1) -	9.821 0E-1 (7.24E-3) -	1.488 7E+0 (4.68E-1) -	3.850 5E-1 (1.20E-1) =	5.552 6E-1 (4.63E-1) -	3.572 5E-1 (2.40E-2)
IDTLZ2	5	6.362 7E-1 (5.44E-2) -	1.149 1E+0 (8.37E-2) -	1.460 2E+0 (2.52E-1) -	1.121 3E+0 (6.38E-2) -	2.696 2E-1 (3.25E-2) =	1.626 0E-1 (9.77E-3) +	2.612 9E-1 (3.91E-2)
	8	6.313 2E-1 (4.49E-2) -	1.101 9E+0 (6.53E-2) -	1.079 2E+0 (4.98E-2) -	1.025 4E+0 (3.53E-2) -	2.818 6E-1 (3.88E-2) -	1.932 1E-1 (1.86E-2) +	2.530 6E-1 (3.01E-2)
	10	6.448 6E-1 (7.12E-2) -	1.004 4E+0 (2.79E-3) -	9.497 7E-1 (3.73E-2) -	1.011 4E+0 (4.56E-2) -	2.347 6E-1 (3.24E-2) -	1.496 8E-1 (1.10E-2) +	1.820 1E-1 (2.18E-2)
	15	8.593 5E-1 (2.57E-2) -	1.057 5E+0 (1.63E-1) -	9.401 8E-1 (2.13E-2) -	9.476 0E-1 (6.75E-2) -	3.440 8E-1 (4.75E-2) -	4.534 2E-1 (1.41E-2) -	3.098 5E-1 (3.48E-2)
+/-/=		1/24/3	4/24/0	1/27/0	0/28/0	1/21/6	7/19/2

表5 VaEA与VaEA-DDR算法GD值对比

Table 5 Comparison of GD values between VaEA and VaEA-DDR algorithms

Problem	$m$	VaEA	VaEA-DDR
DTLZ1	5	2.609 5E-1 (4.38E-1) -	7.039 4E-4 (1.33E-3)
	8	3.275 6E-1 (2.93E-1) -	1.247 9E-3 (4.15E-5)
	10	1.005 3E+0 (3.42E-1) -	4.162 0E-3 (1.06E-4)
	15	8.660 1E-1 (2.66E-1) -	3.113 0E-3 (4.89E-5)
DTLZ2	5	3.176 0E+0 (8.18E-1) -	5.880 4E-3 (1.25E-4)
	8	8.187 1E-4 (9.88E-5) -	5.889 5E-4 (5.72E-5)
	10	4.111 0E-3 (1.02E-4) -	4.048 4E-3 (7.35E-5)
	15	1.253 9E-2 (4.57E-4) +	1.403 5E-2 (4.70E-4)
DTLZ3	5	1.381 7E-2 (4.03E-4) +	1.408 1E-2 (2.99E-4)
	8	2.517 2E-2 (1.64E-3) +	2.738 8E-2 (6.32E-4)
	10	9.219 4E-1 (1.32E+0) -	2.955 3E-1 (2.45E-1)
	15	7.627 9E-1 (7.37E-1) -	4.775 3E-2 (5.24E-2)
DTLZ4	5	6.752 1E+0 (2.42E+0) -	1.916 5E-2 (1.72E-2)
	8	2.239 5E+1 (3.71E+0) -	1.446 9E-2 (2.66E-4)
	10	5.152 5E+1 (6.68E+0) -	2.760 8E-2 (7.68E-4)
	15	7.808 1E-4 (2.61E-4) -	5.523 9E-4 (8.37E-5)
DTLZ5	5	4.222 5E-3 (1.44E-4) -	3.781 3E-3 (1.34E-4)
	8	1.304 4E-2 (6.09E-4) -	1.261 1E-2 (4.82E-4)
	10	1.555 1E-2 (8.83E-4) -	1.351 9E-2 (2.53E-4)
	15	2.495 6E-2 (1.15E-3) +	2.694 3E-2 (8.57E-4)
IDTLZ1	5	2.534 8E-1 (3.85E-1) =	1.235 9E-3 (6.05E-5)
	8	2.138 0E-1 (4.87E-1) -	3.938 1E-3 (1.15E-4)
	10	8.780 3E-2 (2.45E-1) -	3.539 0E-3 (3.97E-5)
	15	1.354 6E-1 (4.29E-1) -	6.650 7E-3 (2.67E-4)
IDTLZ2	5	4.312 8E-3 (8.65E-5) +	4.810 9E-3 (7.89E-5)
	8	1.377 6E-2 (4.53E-4) +	1.955 0E-2 (5.47E-4)
	10	1.460 6E-2 (2.82E-4) +	1.586 2E-2 (2.29E-4)
	15	3.629 8E-2 (9.23E-4) =	3.643 7E-2 (6.97E-4)
+/-/=		7/19/2

表5 VaEA与VaEA-DDR算法GD值对比

Table 5 Comparison of GD values between VaEA and VaEA-DDR algorithms

Problem	$m$	VaEA	VaEA-DDR
DTLZ1	5	2.609 5E-1 (4.38E-1) -	7.039 4E-4 (1.33E-3)
	8	3.275 6E-1 (2.93E-1) -	1.247 9E-3 (4.15E-5)
	10	1.005 3E+0 (3.42E-1) -	4.162 0E-3 (1.06E-4)
	15	8.660 1E-1 (2.66E-1) -	3.113 0E-3 (4.89E-5)
DTLZ2	5	3.176 0E+0 (8.18E-1) -	5.880 4E-3 (1.25E-4)
	8	8.187 1E-4 (9.88E-5) -	5.889 5E-4 (5.72E-5)
	10	4.111 0E-3 (1.02E-4) -	4.048 4E-3 (7.35E-5)
	15	1.253 9E-2 (4.57E-4) +	1.403 5E-2 (4.70E-4)
DTLZ3	5	1.381 7E-2 (4.03E-4) +	1.408 1E-2 (2.99E-4)
	8	2.517 2E-2 (1.64E-3) +	2.738 8E-2 (6.32E-4)
	10	9.219 4E-1 (1.32E+0) -	2.955 3E-1 (2.45E-1)
	15	7.627 9E-1 (7.37E-1) -	4.775 3E-2 (5.24E-2)
DTLZ4	5	6.752 1E+0 (2.42E+0) -	1.916 5E-2 (1.72E-2)
	8	2.239 5E+1 (3.71E+0) -	1.446 9E-2 (2.66E-4)
	10	5.152 5E+1 (6.68E+0) -	2.760 8E-2 (7.68E-4)
	15	7.808 1E-4 (2.61E-4) -	5.523 9E-4 (8.37E-5)
DTLZ5	5	4.222 5E-3 (1.44E-4) -	3.781 3E-3 (1.34E-4)
	8	1.304 4E-2 (6.09E-4) -	1.261 1E-2 (4.82E-4)
	10	1.555 1E-2 (8.83E-4) -	1.351 9E-2 (2.53E-4)
	15	2.495 6E-2 (1.15E-3) +	2.694 3E-2 (8.57E-4)
IDTLZ1	5	2.534 8E-1 (3.85E-1) =	1.235 9E-3 (6.05E-5)
	8	2.138 0E-1 (4.87E-1) -	3.938 1E-3 (1.15E-4)
	10	8.780 3E-2 (2.45E-1) -	3.539 0E-3 (3.97E-5)
	15	1.354 6E-1 (4.29E-1) -	6.650 7E-3 (2.67E-4)
IDTLZ2	5	4.312 8E-3 (8.65E-5) +	4.810 9E-3 (7.89E-5)
	8	1.377 6E-2 (4.53E-4) +	1.955 0E-2 (5.47E-4)
	10	1.460 6E-2 (2.82E-4) +	1.586 2E-2 (2.29E-4)
	15	3.629 8E-2 (9.23E-4) =	3.643 7E-2 (6.97E-4)
+/-/=		7/19/2

表6 7种算法在汽车碰撞实验获得的IGD与HV值

Table 6 IGD and HV values obtained by 7 algorithms in car crash experiment

Algorithm	IGD	HV
NSGA-Ⅲ	0.927 7	0.012 0
MOEADD	1.045 4	0.008 5
MOMBI-Ⅱ	3.081 8	0.008 8
RPD-NSGA-Ⅱ	0.935 5	0.011 1
NSGA-Ⅱ-SDR	2.494 1	0.009 5
VaEA	0.905 1	0.007 8
VaEA-DDR	0.774 5	0.015 8

表7 两种算法在测试问题上获得的评价指标

Table 7 Evaluation indicators obtained by two algorithms on test problems

Problem	$m$	IGD		Spread		GD
Problem	$m$	NSGA-Ⅲ	NSGA-Ⅲ-DDR	NSGA-Ⅲ	NSGA-Ⅲ-DDR	NSGA-Ⅲ	NSGA-Ⅲ-DDR
DTLZ1	5	6.108 1E-2 (2.99E-2) +	6.1715E-2 (1.48E-3)	1.0070E-1 (6.82E-2) +	3.676 4E-1 (3.73E-2)	2.010 9E-3 (3.83E-3) =	1.244 2E-3 (3.29E-5)
	8	1.423 1E-1 (6.90E-2) =	1.233 4E-1 (2.44E-3)	4.673 9E-1 (3.12E-1) =	4.597 9E-1 (5.03E-2)	2.295 7E-2 (4.13E-2) -	4.179 3E-3 (8.24E-5)
	10	1.847 6E-1 (1.17E-1) =	1.218 1E-1 (1.70E-3)	8.367 5E-1 (6.94E-1) =	4.065 8E-1 (2.79E-2)	1.777 5E-1 (2.73E-1) -	3.159 1E-3 (4.94E-5)
	15	2.202 6E-1 (8.39E-2) -	1.508 0E-1 (1.36E-3)	1.024 6E+0 (8.17E-1) =	5.913 4E-1 (3.59E-2)	5.006 9E-1 (6.54E-1) -	5.742 6E-3 (1.27E-4)
DTLZ2	5	1.654 0E-1 (1.10E-4) -	1.639 8E-1 (1.42E-3)	1.515 8E-1 (1.28E-3) -	1.280 2E-1 (1.69E-2)	3.366 7E-3 (2.96E-5) +	4.038 9E-3 (8.84E-5)
	8	3.311 9E-1 (4.73E-2) +	3.516 6E-1 (3.72E-3)	1.809 3E-1 (1.66E-1) -	1.306 3E-1 (1.88E-2)	1.314 4E-2 (9.11E-4) +	1.394 9E-2 (4.42E-4)
	10	4.284 7E-1 (2.22E-2) -	3.956 2E-1 (2.64E-3)	1.788 1E-1 (9.68E-2) -	9.531 0E-2 (1.08E-2)	4.416 9E-3 (6.69E-4) +	1.399 7E-2 (2.10E-4)
	15	6.403 6E-1 (1.50E-2) -	5.622 3E-1 (4.63E-3)	6.856 1E-1 (3.37E-1) -	1.847 6E-1 (2.37E-2)	7.483 5E-3 (3.30E-3) +	2.679 1E-2 (8.53E-4)
DTLZ3	5	2.757 4E+0 (1.45E+0) -	7.879 0E-1 (7.83E-1)	1.151 9E+0 (4.14E-1) -	2.949 8E-1 (1.01E-1)	9.274 5E-1 (6.04E-1) -	7.823 4E-2 (8.46E-2)
	8	6.438 1E+0 (4.65E+0) -	4.304 6E-1 (2.22E-1)	8.477 9E-1 (3.60E-1) -	1.552 4E-1 (3.24E-2)	2.068 4E+0 (1.59E+0) -	2.136 0E-2 (2.10E-2)
	10	5.669 4E+0 (4.63E+0) -	4.002 1E-1 (3.08E-3)	1.052 0E+0 (3.08E-1) -	1.079 0E-1 (1.26E-2)	2.403 2E+0 (2.08E+0) -	1.444 7E-2 (3.08E-4)
	15	5.924 8E+0 (3.78E+0) -	5.805 5E-1 (1.83E-2)	1.385 1E+0 (2.64E-1) -	1.961 0E-1 (2.36E-2)	7.311 4E+0 (2.92E+0) -	2.770 2E-2 (8.86E-4)
DTLZ4	5	1.654 5E-1 (1.38E-4) =	1.659 6E-1 (1.20E-3)	1.492 9E-1 (1.88E-3) -	1.284 3E-1 (2.29E-2)	3.347 0E-3 (4.72E-5) +	3.763 2E-3 (9.15E-5)
	8	3.806 6E-1 (9.52E-2) +	4.062 5E-1 (6.54E-2)	3.098 5E-1 (2.71E-1) -	1.274 6E-1 (2.12E-2)	1.165 4E-2 (1.91E-3) +	1.207 3E-2 (4.97E-4)
	10	4.439 1E-1 (4.03E-2) -	4.058 9E-1 (2.54E-2)	2.431 7E-1 (1.86E-1) -	9.025 8E-2 (9.05E-3)	5.406 3E-3 (1.01E-3) +	1.306 7E-2 (2.58E-4)
	15	6.491 7E-1 (1.50E-2) -	5.746 0E-1 (6.66E-3)	6.695 8E-1 (2.53E-1) -	1.375 3E-1 (1.85E-2)	8.956 7E-3 (2.91E-3) +	2.596 9E-2 (1.19E-3)
DTLZ5	5	9.019 4E-2 (2.25E-2) -	4.175 5E-2 (6.29E-3)	6.710 9E-1 (4.44E-2) -	5.210 0E-1 (7.61E-2)	9.640 1E-2 (5.75E-3) -	6.223 5E-2 (7.47E-3)
	8	2.182 8E-1 (5.69E-2)-	8.351 6E-2 (3.97E-2)	5.914 8E-1 (4.41E-2) =	5.766 5E-1 (1.00E-1)	1.053 3E-1 (5.16E-3) -	8.364 8E-2 (6.68E-3)
	10	2.873 7E-1 (8.56E-2) -	8.913 7E-2 (2.67E-2)	5.807 1E-1 (4.25E-2) =	5.628 6E-1 (9.76E-2)	8.920 8E-2 (2.87E-3) +	9.934 0E-2 (1.17E-2)
	15	2.519 1E-1 (3.85E-2) -	1.004 6E-1 (4.18E-2)	6.257 8E-1 (6.13E-2) +	8.163 3E-1 (2.29E-1)	9.190 2E-2 (6.63E-3) +	1.403 7E-1 (2.18E-2)
IDTLZ1	5	9.136 3E-2 (6.58E-3) -	5.354 0E-2 (1.04E-3)	8.210 9E-1 (3.92E-1) -	3.224 0E-1 (6.99E-2)	9.046 1E-2 (2.93E-1) =	1.249 5E-3 (8.59E-5)
	8	1.419 7E-1 (3.59E-3) -	1.036 3E-1 (1.22E-3)	7.101 2E-1 (5.57E-2) -	3.287 5E-1 (5.15E-2)	3.785 2E-3 (3.33E-4) +	3.897 7E-3 (9.14E-5)
	10	1.372 9E-1 (3.20E-3) -	1.049 9E-1 (9.66E-4)	5.830 2E-1 (3.84E-2) -	2.476 5E-1 (3.36E-2)	3.524 2E-3 (1.01E-4) =	3.544 7E-3 (4.52E-5)
	15	1.724 7E-1 (5.55E-3) =	1.722 8E-1 (1.70E-2)	7.410 3E-1 (4.33E-2) -	4.229 4E-1 (6.51E-2)	7.216 2E-3 (6.94E-4) -	6.870 0E-3 (2.44E-4)
IDTLZ2	5	2.046 7E-1 (5.93E-3) +	2.314 1E-1 (8.89E-3)	6.362 7E-1 (5.44E-2) -	3.436 1E-1 (5.27E-2)	3.943 4E-3 (1.57E-4) +	4.790 7E-3 (8.24E-5)
	8	5.157 6e-1 (1.33E-2) -	4.616 9E-1 (9.56E-3)	6.313 2E-1 (4.49E-2) -	3.609 9E-1 (4.67E-2)	1.810 4E-2 (7.07E-4) +	1.953 5E-2 (5.49E-4)
	10	6.036 8e-1 (1.88E-2) -	4.773 0E-1 (7.97E-3)	6.448 6E-1 (7.12E-2) -	2.879 2E-1 (4.07E-2)	2.095 8E-2 (1.04E-3) -	1.618 0E-2 (2.01E-4)
	15	7.828 9E-1 (9.88E-3) -	6.605 2E-1 (6.45E-3)	8.593 5E-1 (2.57E-2) -	3.790 4E-1 (5.27E-2)	5.174 8E-2 (6.49E-4) -	3.723 8E-2 (5.52E-4)
+/-/=		4/20/4		2/21/5		13/12/3

表7 两种算法在测试问题上获得的评价指标

Table 7 Evaluation indicators obtained by two algorithms on test problems

Problem	$m$	IGD		Spread		GD
Problem	$m$	NSGA-Ⅲ	NSGA-Ⅲ-DDR	NSGA-Ⅲ	NSGA-Ⅲ-DDR	NSGA-Ⅲ	NSGA-Ⅲ-DDR
DTLZ1	5	6.108 1E-2 (2.99E-2) +	6.1715E-2 (1.48E-3)	1.0070E-1 (6.82E-2) +	3.676 4E-1 (3.73E-2)	2.010 9E-3 (3.83E-3) =	1.244 2E-3 (3.29E-5)
	8	1.423 1E-1 (6.90E-2) =	1.233 4E-1 (2.44E-3)	4.673 9E-1 (3.12E-1) =	4.597 9E-1 (5.03E-2)	2.295 7E-2 (4.13E-2) -	4.179 3E-3 (8.24E-5)
	10	1.847 6E-1 (1.17E-1) =	1.218 1E-1 (1.70E-3)	8.367 5E-1 (6.94E-1) =	4.065 8E-1 (2.79E-2)	1.777 5E-1 (2.73E-1) -	3.159 1E-3 (4.94E-5)
	15	2.202 6E-1 (8.39E-2) -	1.508 0E-1 (1.36E-3)	1.024 6E+0 (8.17E-1) =	5.913 4E-1 (3.59E-2)	5.006 9E-1 (6.54E-1) -	5.742 6E-3 (1.27E-4)
DTLZ2	5	1.654 0E-1 (1.10E-4) -	1.639 8E-1 (1.42E-3)	1.515 8E-1 (1.28E-3) -	1.280 2E-1 (1.69E-2)	3.366 7E-3 (2.96E-5) +	4.038 9E-3 (8.84E-5)
	8	3.311 9E-1 (4.73E-2) +	3.516 6E-1 (3.72E-3)	1.809 3E-1 (1.66E-1) -	1.306 3E-1 (1.88E-2)	1.314 4E-2 (9.11E-4) +	1.394 9E-2 (4.42E-4)
	10	4.284 7E-1 (2.22E-2) -	3.956 2E-1 (2.64E-3)	1.788 1E-1 (9.68E-2) -	9.531 0E-2 (1.08E-2)	4.416 9E-3 (6.69E-4) +	1.399 7E-2 (2.10E-4)
	15	6.403 6E-1 (1.50E-2) -	5.622 3E-1 (4.63E-3)	6.856 1E-1 (3.37E-1) -	1.847 6E-1 (2.37E-2)	7.483 5E-3 (3.30E-3) +	2.679 1E-2 (8.53E-4)
DTLZ3	5	2.757 4E+0 (1.45E+0) -	7.879 0E-1 (7.83E-1)	1.151 9E+0 (4.14E-1) -	2.949 8E-1 (1.01E-1)	9.274 5E-1 (6.04E-1) -	7.823 4E-2 (8.46E-2)
	8	6.438 1E+0 (4.65E+0) -	4.304 6E-1 (2.22E-1)	8.477 9E-1 (3.60E-1) -	1.552 4E-1 (3.24E-2)	2.068 4E+0 (1.59E+0) -	2.136 0E-2 (2.10E-2)
	10	5.669 4E+0 (4.63E+0) -	4.002 1E-1 (3.08E-3)	1.052 0E+0 (3.08E-1) -	1.079 0E-1 (1.26E-2)	2.403 2E+0 (2.08E+0) -	1.444 7E-2 (3.08E-4)
	15	5.924 8E+0 (3.78E+0) -	5.805 5E-1 (1.83E-2)	1.385 1E+0 (2.64E-1) -	1.961 0E-1 (2.36E-2)	7.311 4E+0 (2.92E+0) -	2.770 2E-2 (8.86E-4)
DTLZ4	5	1.654 5E-1 (1.38E-4) =	1.659 6E-1 (1.20E-3)	1.492 9E-1 (1.88E-3) -	1.284 3E-1 (2.29E-2)	3.347 0E-3 (4.72E-5) +	3.763 2E-3 (9.15E-5)
	8	3.806 6E-1 (9.52E-2) +	4.062 5E-1 (6.54E-2)	3.098 5E-1 (2.71E-1) -	1.274 6E-1 (2.12E-2)	1.165 4E-2 (1.91E-3) +	1.207 3E-2 (4.97E-4)
	10	4.439 1E-1 (4.03E-2) -	4.058 9E-1 (2.54E-2)	2.431 7E-1 (1.86E-1) -	9.025 8E-2 (9.05E-3)	5.406 3E-3 (1.01E-3) +	1.306 7E-2 (2.58E-4)
	15	6.491 7E-1 (1.50E-2) -	5.746 0E-1 (6.66E-3)	6.695 8E-1 (2.53E-1) -	1.375 3E-1 (1.85E-2)	8.956 7E-3 (2.91E-3) +	2.596 9E-2 (1.19E-3)
DTLZ5	5	9.019 4E-2 (2.25E-2) -	4.175 5E-2 (6.29E-3)	6.710 9E-1 (4.44E-2) -	5.210 0E-1 (7.61E-2)	9.640 1E-2 (5.75E-3) -	6.223 5E-2 (7.47E-3)
	8	2.182 8E-1 (5.69E-2)-	8.351 6E-2 (3.97E-2)	5.914 8E-1 (4.41E-2) =	5.766 5E-1 (1.00E-1)	1.053 3E-1 (5.16E-3) -	8.364 8E-2 (6.68E-3)
	10	2.873 7E-1 (8.56E-2) -	8.913 7E-2 (2.67E-2)	5.807 1E-1 (4.25E-2) =	5.628 6E-1 (9.76E-2)	8.920 8E-2 (2.87E-3) +	9.934 0E-2 (1.17E-2)
	15	2.519 1E-1 (3.85E-2) -	1.004 6E-1 (4.18E-2)	6.257 8E-1 (6.13E-2) +	8.163 3E-1 (2.29E-1)	9.190 2E-2 (6.63E-3) +	1.403 7E-1 (2.18E-2)
IDTLZ1	5	9.136 3E-2 (6.58E-3) -	5.354 0E-2 (1.04E-3)	8.210 9E-1 (3.92E-1) -	3.224 0E-1 (6.99E-2)	9.046 1E-2 (2.93E-1) =	1.249 5E-3 (8.59E-5)
	8	1.419 7E-1 (3.59E-3) -	1.036 3E-1 (1.22E-3)	7.101 2E-1 (5.57E-2) -	3.287 5E-1 (5.15E-2)	3.785 2E-3 (3.33E-4) +	3.897 7E-3 (9.14E-5)
	10	1.372 9E-1 (3.20E-3) -	1.049 9E-1 (9.66E-4)	5.830 2E-1 (3.84E-2) -	2.476 5E-1 (3.36E-2)	3.524 2E-3 (1.01E-4) =	3.544 7E-3 (4.52E-5)
	15	1.724 7E-1 (5.55E-3) =	1.722 8E-1 (1.70E-2)	7.410 3E-1 (4.33E-2) -	4.229 4E-1 (6.51E-2)	7.216 2E-3 (6.94E-4) -	6.870 0E-3 (2.44E-4)
IDTLZ2	5	2.046 7E-1 (5.93E-3) +	2.314 1E-1 (8.89E-3)	6.362 7E-1 (5.44E-2) -	3.436 1E-1 (5.27E-2)	3.943 4E-3 (1.57E-4) +	4.790 7E-3 (8.24E-5)
	8	5.157 6e-1 (1.33E-2) -	4.616 9E-1 (9.56E-3)	6.313 2E-1 (4.49E-2) -	3.609 9E-1 (4.67E-2)	1.810 4E-2 (7.07E-4) +	1.953 5E-2 (5.49E-4)
	10	6.036 8e-1 (1.88E-2) -	4.773 0E-1 (7.97E-3)	6.448 6E-1 (7.12E-2) -	2.879 2E-1 (4.07E-2)	2.095 8E-2 (1.04E-3) -	1.618 0E-2 (2.01E-4)
	15	7.828 9E-1 (9.88E-3) -	6.605 2E-1 (6.45E-3)	8.593 5E-1 (2.57E-2) -	3.790 4E-1 (5.27E-2)	5.174 8E-2 (6.49E-4) -	3.723 8E-2 (5.52E-4)
+/-/=		4/20/4		2/21/5		13/12/3

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基于距离优势关系的高维多目标进化算法

Many-Objective Evolutionary Algorithm Based on Distance Dominance Relation

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