自适应变化响应的动态多目标进化算法

梁正平; 李辉才; 王志强; 胡凯峰; 朱泽轩

doi:10.16383/j.aas.c210121

自适应变化响应的动态多目标进化算法

doi: 10.16383/j.aas.c210121

梁正平^1,,
李辉才^1,,
王志强^{1, 2,},
胡凯峰^{1, 2,},
朱泽轩^1,

1.
深圳大学计算机与软件学院深圳 518060
2.
深圳大学信息中心深圳 518060

基金项目: 国家重点研发计划(2021YFB2900800), 国家自然科学基金(61871272), 广东省自然科学基金(2020A1515010479, 2021A1515011911), 深圳市科技计划项目(20200811181752003, JCYJ20220531102617039)资助

详细信息

作者简介:
梁正平：深圳大学计算机与软件学院副教授. 2006年获得武汉大学博士学位. 主要研究方向为计算智能和大数据分析与应用. E-mail: liangzp@szu.edu.cn

李辉才：深圳大学计算机与软件学院硕士研究生. 主要研究方向为计算智能与应用. E-mail: 1810273028@email.szu.edu.cn

王志强：深圳大学计算机与软件学院教授. 主要研究方向为计算智能, 大数据分析与应用和多媒体技术与应用. E-mail: wangzq@szu.edu.cn

胡凯峰：深圳大学信息中心工程师. 2019年获得深圳大学硕士学位. 主要研究方向为计算智能及其应用. E-mail: kaifeng@szu.edu.cn

朱泽轩：深圳大学计算机与软件学院教授. 2008年获得南洋理工大学博士学位. 主要研究方向为计算智能, 机器学习和生物信息学. 本文通信作者. E-mail: zhuzx@szu.edu.cn

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出版历程
- 收稿日期: 2021-02-05
- 录用日期: 2021-06-25
- 网络出版日期: 2021-07-30
- 刊出日期: 2023-08-21

Dynamic Multi-objective Evolutionary Algorithm With Adaptive Change Response

LIANG Zheng-Ping^1
,,
LI Hui-Cai^1
,,
WANG Zhi-Qiang^{1, 2
,},
HU Kai-Feng^{1, 2
,},
ZHU Ze-Xuan^1
,

1.
College of Computer Science and Software Engineering, Shenzhen University, Shenzhen 518060
2.
Information Center, Shenzhen University, Shenzhen 518060

Funds: Supported by National Key Research and Development Program of China (2021YFB2900800), National Natural Science Foundation of China (61871272), Natural Science Foundation of Guangdong (2020A1515010479, 2021A1515011911), and Shenzhen Science & Technology Project (20200811181752003, JCYJ20220531102617039)

More Information

Author Bio:
LIANG Zheng-Ping　Associate professor at the College of Computer Science and Software Engineering, Shenzhen University. He received his Ph.D. degree from Wuhan University in 2006. His research interest covers computational intelligence and big data analysis & application

LI Hui-Cai　Master student at the College of Computer Science and Software Engineering, Shenzhen University. His main research interest is computational intelligence & applications

WANG Zhi-Qiang　Professor at the College of Computer Science and Software Engineering, Shenzhen University. His research interest covers computational intelligence, big data analysis & applications, and multi-media technology & applications

HU Kai-Feng　Engineer at the Information Center, Shenzhen University. He received his master degree from Shenzhen University in 2019. His main research interest is computational intelligence & applications

ZHU Ze-Xuan　Professor at the College of Computer Science and Software Engineering, Shenzhen University. He received his Ph.D. degree from Nanyang Technological University in 2008. His research interest covers computational intelligence, machine learning, and bioinformatics. Corresponding author of this paper

摘要

摘要: 动态多目标优化问题(Dynamic multi-objective optimization problems, DMOPs)的目标函数发生变化时, 需要采取变化响应策略对种群进行重新初始化, 以快速追踪新环境中的最优解集. 现有动态多目标优化算法对不同个体、不同维度的决策变量缺乏针对性的变化响应, 导致重新初始化效果尚存在较大改进空间. 为此, 提出一种对不同个体、不同维度的决策变量分别进行自适应变化响应的动态多目标进化算法(Dynamic multi-objective evolutionary algorithm with adaptive change response, DMOEA-ACR). 该算法包括两个核心部分: 1)对$t $时间步最优种群和$t-1 $时间步最优种群中对应个体各维度决策变量之间的差异进行计算, 自适应选择变异策略或预测策略重新初始化不同个体、不同维度的决策变量; 2)在每轮迭代或重新初始化后, 对非支配个体进行存档, 基于存档中心构建预测策略. 为验证DMOEA-ACR的有效性, 在最新测试问题集SDP和DF上, 将其与动态多目标优化领域的6种先进算法进行对比. 实验结果表明, DMOEA-ACR在求解动态多目标优化问题时, 具有明显优势.
- 动态多目标优化 /
- 进化算法 /
- 自适应变化响应 /
- 预测 /
- 存档
Abstract: When the objective functions of dynamic multi-objective optimization problems (DMOPs) change, it is necessary to adopt a correspondent response strategy to reinitialize the population so that an optimal solution set in the new environment can be quickly tracked. The reinitialization effect of existing dynamic multi-objective optimization algorithms still leaves much room for improvement due to the lack of customized change response to different decision variables of different individuals. This paper proposes a dynamic multi-objective evolutionary algorithm with adaptive change response (DMOEA-ACR), which can adaptively reinitialize different decision variables of different individuals. DMOEA-ACR consists of two essential components. One is the adaptive change response strategy, which can adaptively choose the mutation strategy or the prediction strategy to reinitialize different decision variables of different individuals based on the correspondent decision variable difference between the t time step optimal population and the $t-1 $ time step optimal population. The other is the prediction strategy based on the archive core of non-dominant individuals. In order to verify its effectiveness, DMOEA-ACR is compared with six state-of-the-art algorithms in dynamic multi-objective optimization on the latest test suites SDP and DF. The experimental results show that DMOEA-ACR has obvious advantages in solving dynamic multi-objective optimization problems.
- Dynamic multi-objective optimization /
- evolutionary algorithm /
- adaptive change response /
- prediction /
- archive

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图 1 二维决策空间最理想情形下的重新初始化示意图

Fig. 1 Reinitialization illustration of the most ideal situation in 2D decision space

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图 2 自适应变化响应示意图

Fig. 2 Illustration of adaptive change response

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图 3 基于存档中心进行预测策略示意图

Fig. 3 Illustration of prediction strategy based on the central of archive

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图 4 DNSGA-II-A、DNSGA-II-B、MOEA/D-KF、SGEA、MOEA/D-MoE和DMOEA-ACR在SDP1、SDP5、SDP6、DF3、DF5、DF8测试问题集上的log(IGD)均值变化比较

Fig. 4 Comparison of average log(IGD) trends of DNSGA-II-A, DNSGA-II-B, MOEA/D-KF, SGEA, MOEA/D-MoE, and DMOEA-ACR on SDP1, SDP5, SDP6, DF3, DF5, DF8

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图 5 DNSGA-II-A、DNSGA-II-B、MOEA/D-KF和SGEA在SDP1、SDP6、DF3、DF8测试问题集上得到的PF图

Fig. 5 PF graph obtained by DNSGA-II-A, DNSGA-II-B, MOEA/D-KF, and SGEA on SDP1, SDP6, DF3, DF8

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图 6 Tr-DMOEA、MOEA/D-MoE和DMOEA-ACR在SDP1、SDP6、DF3、DF8测试问题集上得到的PF图

Fig. 6 PF graph obtained by Tr-DMOEA, MOEA/D-MoE, and DMOEA-ACR on SDP1, SDP6, DF3, DF8

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图 7 $ \tau t $分别为5、10、20时DMOEA-ACR在SDP1 (2目标)、SDP5 (2目标)、SDP6 (2目标)、DF3、DF5和DF8问题上的log(IGD)均值变化比较

Fig. 7 Comparison of average log(IGD) trends of DMOEA-ACR on SDP1 (2 goals), SDP5 (2 goals), SDP6 (2 goals), DF3, DF5, and DF8 where $\tau_t $ is 5, 10, 20, respectively

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表 1 DNSGA-II-A、DNSGA-II-B、MOEA/D-KF、SGEA、Tr-DMOEA、MOEA/D-MoE和DMOEA-ACR在SDP上获得的MIGD均值和标准差

Table 1 The mean and standard deviation of MIGD of DNSGA-II-A, DNSGA-II-B, MOEA/D-KF, SGEA, Tr-DMOEA, MOEA/D-MoE, and DMOEA-ACR were obtained on SDP

问题集	评价指标	DNSGA-II-A	DNSGA-II-B	MOEA/D-KF	SGEA	Tr-DMOEA	MOEA/D-MoE	DMOEA-ACR
SDP1 (2目标) SDP1 (3目标)	均值标准差	2.03 × 10⁻² 6.73 × 10⁻³ (−)	2.01 × 10⁻² 5.34 × 10⁻³ (−)	6.69 × 10⁻¹ 2.61 × 10⁻³ (−)	1.66 × 10⁻² 5.12 × 10⁻³ (≈)	2.53 × 10⁻¹ 2.65 × 10⁻² (−)	5.78 × 10⁻¹ 3.13 × 10⁻⁴ (−)	1.65 × 10⁻² 3.67 × 10⁻³
SDP1 (2目标) SDP1 (3目标)	均值标准差	2.26 × 10⁻¹ 3.02 × 10⁻³ (−)	2.32 × 10⁻¹ 6.78 × 10⁻³ (−)	7.74 × 10⁻¹ 6.35 × 10⁻³ (−)	1.40 × 10⁻¹ 4.74 × 10⁻³ (−)	1.44 × 10⁻¹ 7.32 × 10⁻² (−)	7.46 × 10⁻¹ 2.14 × 10⁻³) (−)	1.32 × 10⁻¹ 5.44 × 10⁻³
SDP2 (2目标) SDP2 (3目标)	均值标准差	1.75 × 10⁻² 7.58 × 10⁻³ (+)	1.74 × 10⁻² 7.05 × 10⁻³ (+)	4.38 × 10⁻² 4.34 × 10⁻³ (+)	1.00 × 10⁰ 3.17 × 10⁻³ (−)	4.93 × 10⁰ 5.43 × 10⁻¹ (−)	3.23 × 10⁻² 8.38 × 10⁻³ (+)	1.14 × 10⁻¹ 4.35 × 10⁻³
SDP2 (2目标) SDP2 (3目标)	均值标准差	2.87 × 10⁻¹ 7.18 × 10⁻³ (−)	3.60 × 10⁻¹ 8.31 × 10⁻³ (−)	3.36 × 10⁻¹ 7.67 × 10⁻⁴ (−)	9.23 × 10⁻¹ 1.59 × 10⁻² (−)	4.80 × 10⁻¹ 6.61 × 10⁻² (−)	3.34 × 10⁻¹ 1.39 × 10⁻³ (−)	2.42 × 10⁻¹ 7.12 × 10⁻³
SDP3 (2目标) SDP3 (3目标)	均值标准差	1.76 × 10⁰ 6.74× 10⁻² (+)	1.81 × 10⁻¹ 3.97 × 10⁻²) (+)	6.37 × 10⁻² 4.18 × 10⁻³ (+)	2.07 × 10⁻¹ 2.04 × 10⁻³ (+)	1.45 × 10⁺¹ 3.41 × 10⁻¹ (+)	4.58 × 10⁻¹ 1.30 × 10⁻³ (+)	4.74 × 10⁰ 6.38 × 10⁻²
SDP3 (2目标) SDP3 (3目标)	均值标准差	7.01 × 10⁰ 8.34 × 10⁻² (−)	8.51 × 10⁰ 7.37 × 10⁻² (−)	8.54 × 10⁻¹ 1.43 × 10⁻² (+)	3.74 × 10⁻¹ 3.55 × 10⁻³ (+)	2.36 × 10⁻¹ 3.67 × 10⁻² (+)	5.47 × 10⁻¹ 4.42 × 10⁻³ (+)	3.33 × 10⁰ 8.53 × 10⁻²
SDP4 (2目标) SDP4 (3目标)	均值标准差	1.14 × 10⁻¹ 6.19 × 10⁻³ (−)	7.86 × 10⁻² 3.07 × 10⁻² (−)	5.63 × 10⁻² 1.01 × 10⁻³ (−)	5.56 × 10⁻² 4.76 × 10⁻³ (−)	3.65 × 10⁰ 4.09 × 10⁻¹ (−)	5.54 × 10⁻² 3.61 × 10⁻³ (−)	5.08 × 10⁻² 5.03 × 10⁻³
SDP4 (2目标) SDP4 (3目标)	均值标准差	2.44 × 10⁻¹ 5.08 × 10⁻³ (−)	2.26 × 10⁻¹ 7.48 × 10⁻² (−)	1.81 × 10⁻¹ 3.28 × 10⁻³ (−)	1.95 × 10⁻¹ 2.43 × 10⁻³ (−)	1.74 × 10⁻¹ 5.48 × 10⁻² (−)	1.79 × 10⁻¹ 2.71 × 10⁻² (−)	1.69 × 10⁻¹ 8.25 × 10⁻³
SDP5 (2目标) SDP5 (3目标)	均值标准差	9.40 × 10⁻² 4.21 × 10⁻³ (−)	6.27 × 10⁻¹ 8.05 × 10⁻³) (−)	9.74 × 10⁻² 4.07 × 10⁻³ (−)	7.19 × 10⁻¹ 2.79 × 10⁻² (−)	8.35 × 10⁻¹ 7.87 × 10⁻³ (−)	9.86 × 10⁻² 4.37 × 10⁻⁴ (−)	6.81 × 10⁻³ 4.92 × 10⁻⁴
SDP5 (2目标) SDP5 (3目标)	均值标准差	1.41 × 10⁻¹ 6.08 × 10⁻² (−)	5.24 × 10⁻¹ 3.43 × 10⁻²(−)	1.54 × 10⁻¹ 7.81 × 10⁻³ (−)	8.68 × 10⁻² 7.15 × 10⁻³ (−)	1.46 × 10⁻¹ 6.43 × 10⁻² (−)	1.50 × 10⁻¹ 7.61 × 10⁻³ (−)	6.38 × 10⁻² 6.06 ×10⁻³
SDP6 (2目标) SDP6 (3目标)	均值标准差	2.47 × 10⁻² 1.53 × 10⁻³ (−)	3.14 × 10⁻² 5.92 × 10⁻³ (−)	2.45 × 10⁻² 3.69 × 10⁻³(−)	8.56 × 10⁻² 2.72 × 10⁻³ (−)	4.28 × 10⁻¹ 5.97 × 10⁻³ (−)	2.41 × 10⁻² 5.81 × 10⁻³ (−)	4.26 × 10⁻³ 7.43 × 10⁻⁴
SDP6 (2目标) SDP6 (3目标)	均值标准差	1.61 × 10⁻¹ 6.44 × 10⁻² (−)	1.79 × 10⁻¹ 6.45 × 10⁻² (−)	1.03 × 10⁻¹ 6.61 × 10⁻³ (−)	5.38 × 10⁻² 6.05 × 10⁻³ (−)	6.60 × 10⁻¹ 6.06 × 10⁻³ (−)	5.33 × 10⁻² 3.86 × 10⁻² (−)	5.17 × 10⁻² 5.86 × 10⁻³
SDP7 (2目标) SDP7 (3目标)	均值标准差	5.15 × 10⁻² 2.46 × 10⁻³ (+)	2.96 × 10⁻² 7.28 × 10⁻³ (+)	3.49 × 10⁻¹ 3.54 × 10⁻³ (−)	2.67 × 10⁻¹ 6.73 × 10⁻³ (−)	8.12 × 10⁻¹ 3.78 × 10⁻³ (−)	3.11 × 10⁻¹ 6.65 × 10⁻³ (−)	2.15 × 10⁻¹ 1.68 × 10⁻²
SDP7 (2目标) SDP7 (3目标)	均值标准差	2.41 × 10⁻¹ 3.12 × 10⁻³ (−)	2.87 × 10⁻¹ 8.30 × 10⁻² (−)	3.76 × 10⁻¹ 5.71 × 10⁻³ (−)	2.59 × 10⁻¹ 2.72 × 10⁻² (−)	3.71 × 10⁻¹ 4.97 × 10⁻² (−)	3.55 × 10⁻¹ 8.36 × 10⁻² (−)	2.24 × 10⁻¹ 5.11 × 10⁻³
SDP8 (2目标) SDP8 (3目标)	均值标准差	1.40 × 10⁻¹ 6.12 × 10⁻² (−)	1.56 × 10⁻¹ 6.91 × 10⁻³ (−)	1.15 × 10⁻¹ 1.31 × 10⁻³ (−)	1.27 × 10⁻¹ 4.95 × 10⁻³ (−)	2.77 × 10⁻¹ 5.62 × 10⁻² (−)	1.07 × 10⁻¹ 1.63 × 10⁻³ (−)	3.14 × 10⁻² 4.26 × 10⁻³
SDP8 (2目标) SDP8 (3目标)	均值标准差	3.45 × 10⁻¹ 8.76 × 10⁻² (−)	2.97 × 10⁻¹ 3.58 × 10⁻³ (−)	1.36 × 10⁻¹ 4.87 × 10⁻³ (−)	1.21 × 10⁻¹ 6.82 × 10⁻³ (+)	1.39 × 10⁻¹ 9.61 × 10⁻² (−)	1.03 × 10⁻¹ 4.73 × 10⁻³ (+)	1.29 × 10⁻¹ 6.43 × 10⁻³
SDP9 (2目标) SDP9 (3目标)	均值标准差	8.95 × 10⁻² 7.09 × 10⁻³ (−)	6.67 × 10⁻¹ 5.94 × 10⁻³ (−)	1.64 × 10⁻¹ 8.52 × 10⁻⁴ (−)	2.45 × 10⁻¹ 2.06 × 10⁻³ (−)	1.35 × 10⁻¹ 3.25 × 10⁻² (−)	1.56 × 10⁻¹ 6.23 × 10⁻³ (−)	8.13 × 10⁻² 7.92 × 10⁻³
SDP9 (2目标) SDP9 (3目标)	均值标准差	4.01 × 10⁻¹ 8.98 × 10⁻³ (−)	3.67 × 10⁻¹ 4.08 × 10⁻³ (−)	4.93 × 10⁻¹ 6.03 × 10⁻³ (−)	3.61 × 10⁻¹ 3.74 × 10⁻² (≈)	4.28 × 10⁻¹ 4.56 × 10⁻² (−)	4.45 × 10⁻¹ 9.13 × 10⁻³ (−)	3.60 × 10⁻¹ 8.03 × 10⁻²
SDP10 (2目标) SDP10 (3目标)	均值标准差	8.67 × 10⁻² 7.72 × 10⁻² (−)	1.15 × 10⁻¹ 8.31 × 10⁻² (−)	9.39 × 10⁻² 4.94 × 10⁻³ (−)	2.23 × 10⁻¹ 1.75 × 10⁻² (≈)	2.35 × 10⁰ 3.74 × 10⁻¹ (−)	9.19 × 10⁻² 7.02 × 10⁻⁴ (−)	2.21 × 10⁻² 1.67 × 10⁻³
SDP10 (2目标) SDP10 (3目标)	均值标准差	3.05 × 10⁻¹ 5.73 × 10⁻² (−)	2.56 × 10⁻¹ 9.76 × 10⁻² (−)	1.85 × 10⁻¹ 6.51 × 10⁻³ (−)	4.51 × 10⁻¹ 8.06 × 10⁻² (−)	3.48 × 10⁻¹ 7.43 × 10⁻² (−)	1.79 × 10⁻¹ 3.63 × 10⁻³ (−)	1.74 × 10⁻¹ 3.58 × 10⁻³
SDP11 (2目标) SDP11 (3目标)	均值标准差	1.61 × 10⁻² 6.36 × 10⁻³ (−)	7.40 × 10⁻³ 7.67 × 10⁻⁴ (+)	3.02 × 10⁻² 3.14 × 10⁻³ (−)	3.36 × 10⁻² 6.91 × 10⁻³ (−)	9.18 × 10⁻¹ 3.64 × 10⁻³ (−)	2.43 × 10⁻² 3.26 × 10⁻³ (−)	1.36 × 10⁻² 8.16 × 10⁻³
SDP11 (2目标) SDP11 (3目标)	均值标准差	1.49 × 10⁻¹ 7.48 × 10⁻³ (−)	1.23 × 10⁻¹ 5.98 × 10⁻² (−)	9.77 × 10⁻² 2.81 × 10⁻² (−)	1.47 × 10⁻¹ 1.08 × 10⁻² (−)	8.98 × 10⁻² 6.05 × 10⁻³ (−)	9.57 × 10⁻² 5.15 × 10⁻³ (−)	8.77 × 10⁻² 9.65 × 10⁻³
SDP12 (2目标) SDP12 (3目标)	均值标准差	2.20 × 10⁻² 4.03 × 10⁻³ (−)	2.52 × 10⁻² 3.08 × 10⁻³ (−)	9.16 × 10⁻³ 2.78 × 10⁻³ (−)	4.11 × 10⁻³ 5.38 × 10⁻² (−)	8.09 × 10⁻² 4.65 × 10⁻² (−)	6.38 × 10⁻³ 8.83 × 10⁻⁴ (−)	4.04 × 10⁻³ 2.17 × 10⁻⁴
SDP12 (2目标) SDP12 (3目标)	均值标准差	2.22 × 10⁻¹ 8.66 × 10⁻² (−)	2.15 × 10⁻¹ 6.79 × 10⁻² (−)	8.66 × 10⁻² 3.59 × 10⁻³ (−)	9.97 × 10⁻² 7.02 × 10⁻² (−)	7.66 × 10⁻² 6.93 × 10⁻² (−)	8.13 × 10⁻² 4.16 × 10⁻³ (−)	7.63 × 10⁻² 5.63 × 10⁻⁴
“+/−/≈”合计		3/21/0	4/20/0	3/21/0	3/18/3	2/22/0	4/20/0

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表 2 DNSGA-II-A、DNSGA-II-B、MOEA/D-KF、SGEA、Tr-DMOEA、MOEA/D-MoE和DMOEA-ACR在DF上获得的MIGD均值和标准差

Table 2 The mean and standard deviation of MIGD of DNSGA-II-A, DNSGA-II-B, MOEA/D-KF, SGEA, Tr-DMOEA, MOEA/D-MoE, and DMOEA-ACR were obtained on DF

问题集	评价指标	DNSGA-II-A	DNSGA-II-B	MOEA/D-KF	SGEA	Tr-DMOEA	MOEA/D-MoE	DMOEA-ACR
DF1 (2目标)	均值标准差	3.31 × 10⁻² 8.35 × 10⁻³ (−)	4.24 × 10⁻² 3.91 × 10⁻³ (−)	1.85 × 10⁻² 4.47 × 10⁻³ (−)	1.22 × 10⁻² 1.38 × 10⁻³ (−)	6.42 × 10⁻² 3.34 × 10⁻³ (−)	1.54 × 10⁻² 3.97 × 10⁻³ (−)	9.15 × 10⁻³ 3.67 × 10⁻³
DF2 (2目标)	均值标准差	5.75 × 10⁻³ 7.61 × 10⁻⁴ (+)	5.76 × 10⁻³ 2.08 × 10⁻⁴ (+)	3.81 × 10⁻² 5.02 × 10⁻³ (+)	1.22 × 10⁻¹ 2.76 × 10⁻² (−)	5.46 × 10⁻³ 5.21 × 10⁻³ (+)	3.10 × 10⁻² 5.23 × 10⁻³ (+)	5.80 × 10⁻² 7.85 × 10⁻³
DF3 (2目标)	均值标准差	9.48 × 10⁻² 5.32 × 10⁻³ (−)	1.48 × 10⁻¹ 7.35 × 10⁻³ (−)	2.85 × 10⁻² 3.32 × 10⁻³ (−)	2.61 × 10⁻¹ 4.58 × 10⁻² (−)	3.90 × 10⁻¹ 7.31 × 10⁻³ (−)	2.10 × 10⁻² 7.52 × 10⁻³ (−)	1.99 × 10⁻² 3.20 × 10⁻³
DF4 (2目标)	均值标准差	2.78 × 10⁻¹ 7.38 × 10⁻² (−)	3.81 × 10⁻¹ 2.79 × 10⁻² (−)	9.85 × 10⁻² 4.59 × 10⁻³ (−)	5.56 × 10⁻² 7.88 × 10⁻³ (−)	8.67 × 10⁻¹ 6.82 × 10⁻² (−)	9.09 × 10⁻² 2.81 × 10⁻³ (−)	2.89 × 10⁻² 5.19 × 10⁻³
DF5 (2目标)	均值标准差	8.05 × 10⁻² 6.39 × 10⁻² (−)	8.11 × 10⁻² 5.65 × 10⁻³ (−)	2.20 × 10⁻² 4.03 × 10⁻³ (−)	1.45 × 10⁻² 4.31 × 10⁻³ (−)	2.99 × 10⁻² 4.34 × 10⁻³ (−)	2.13 × 10⁻² 1.24 × 10⁻³ (−)	9.32 × 10⁻³ 6.01 × 10⁻⁴
DF6 (2目标)	均值标准差	2.33 × 10⁻¹ 8.01 × 10⁻³ (+)	2.40 × 10⁻¹ 8.18 × 10⁻³ (+)	5.04 × 10⁰ 9.92 × 10⁻² (−)	1.51 × 10⁰ 8.39 × 10⁻² (−)	5.32 × 10⁰ 7.31 × 10⁻¹ (−)	2.98 × 10⁰ 7.68 × 10⁻¹ (−)	1.14 × 10⁰ 4.23 × 10⁻¹
DF7 (2目标)	均值标准差	1.56 × 10⁻² 4.33 × 10⁻³ (≈)	1.85 × 10⁻² 6.02 × 10⁻³ (−)	3.66 × 10⁻² 6.39 × 10⁻³ (−)	2.08 × 10⁻¹ 5.04 × 10⁻³ (−)	6.54 × 10⁰ 5.37 × 10⁻¹ (−)	3.72 × 10⁻² 2.29 × 10⁻³ (−)	1.57 × 10⁻² 3.67 × 10⁻³
DF8 (2目标)	均值标准差	8.82 × 10⁻² 7.13 × 10⁻² (−)	8.17 × 10⁻² 7.14 × 10⁻³ (−)	7.94 × 10⁻² 3.68 × 10⁻³ (−)	2.07 × 10⁻² 5.98 × 10⁻³ (−)	2.85 × 10⁻¹ 6.81 × 10⁻² (−)	7.73 × 10⁻² 6.64 × 10⁻³ (−)	1.70 × 10⁻² 5.68 × 10⁻³
DF9 (2目标)	均值标准差	7.82 × 10⁻² 9.75 × 10⁻³ (−)	8.69 × 10⁻² 9.98 × 10⁻³ (−)	9.52 × 10⁻² 7.87 × 10⁻³ (−)	2.57 × 10⁻¹ 6.30 × 10⁻² (−)	5.33 × 10⁻¹ 2.76 × 10⁻² (−)	7.09 × 10⁻² 5.31 × 10⁻² (−)	6.87 × 10⁻² 4.97 × 10⁻³
DF10 (3目标)	均值标准差	2.88 × 10⁻¹ 8.10 × 10⁻³ (−)	2.76 × 10⁻¹ 6.31 × 10⁻³ (−)	1.80 × 10⁻¹ 3.51 × 10⁻² (−)	1.19 × 10⁻¹ 5.26 × 10⁻² (−)	8.51 × 10⁻¹ 8.61 × 10⁻² (−)	1.86 × 10⁻¹ 2.21 × 10⁻² (−)	1.05 × 10⁻¹ 8.18 × 10⁻²
DF11 (3目标)	均值标准差	5.77 × 10⁻¹ 7.57 × 10⁻² (−)	5.80 × 10⁻¹ 3.59 × 10⁻³ (−)	1.53 × 10⁻¹ 4.06 × 10⁻²) (−)	8.20 × 10⁻² 3.14 × 10⁻² (−)	6.53 × 10⁻² 4.05 × 10⁻² (−)	1.50 × 10⁻¹ (3.72 × 10⁻³) (−)	6.38 × 10⁻² 5.07 × 10⁻³
DF12 (3目标)	均值标准差	2.18 × 10⁻¹ 8.61 × 10⁻² (−)	2.34 × 10⁻¹ 2.36 × 10⁻² (−)	1.41 × 10⁻¹ 3.38 × 10⁻² (−)	1.47 × 10⁻¹ 2.78 × 10⁻² (−)	4.25 × 10⁻¹ 7.63 × 10⁻³ (−)	1.00 × 10⁻¹ 6.98 × 10⁻² (−)	9.49 × 10⁻² 6.53 × 10⁻³
DF13 (3目标)	均值标准差	1.80 × 10⁻¹ 5.32 × 10⁻² (−)	1.87 × 10⁻¹ 8.95 × 10⁻² (−)	2.79 × 10⁻¹ 5.72 × 10⁻² (−)	1.28 × 10⁻¹ 6.52 × 10⁻² (−)	1.37 × 10⁰ 2.71 × 10⁻¹ (−)	2.68 × 10⁻¹ 1.25 × 10⁻² (−)	1.15 × 10⁻¹ 4.38 × 10⁻²
DF14 (3目标)	均值标准差	1.35 × 10⁻¹ 5.31 × 10⁻² (−)	1.22 × 10⁻¹ 4.32 × 10⁻² (−)	6.91 × 10⁻² 3.31 × 10⁻² (−)	5.19 × 10⁻² 4.06 × 10⁻² (−)	1.15 × 10⁰ 3.59 × 10⁻² (−)	6.88 × 10⁻² 5.81 × 10⁻² (−)	4.28 × 10⁻² 5.56 × 10⁻³
“+/−/≈”合计		2/11/1	2/12/0	1/13/0	0/14/0	1/13/0	1/13/0

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表 3 DNSGA-II-A、DNSGA-II-B、MOEA/D-KF、SGEA、Tr-DMOEA、MOEA/D-MoE和DMOEA-ACR在SDP (包括2目标和3目标)上的性能综合排名

Table 3 Performance comprehensive ranking of DNSGA-II-A, DNSGA-II-B, MOEA/D-KF, SGEA, Tr-DMOEA, MOEA/D-MoE, and DMOEA-ACR on SDP (including 2 goals and 3 goals)

算法	SDP1	SDP2	SDP3	SDP4	SDP5	SDP6	SDP7	SDP8	SDP9	SDP10	SDP11	SDP12	平均排序
DNSGA-II-A	3	1	7	7	2	5	2	6	2	4	5	7	4
DNSGA-II-B	4	3	4	6	7	6	3	5	5	5	2	6	6
MOEA/D-KF	7	5	1	4	5	3	6	4	7	3	4	4	5
SGEA	2	6	2	3	3	4	4	3	4	6	7	3	3
Tr-DMOEA	5	7	5	5	6	7	7	7	3	7	6	5	7
MOEA/D-MoE	6	4	3	2	4	2	5	1	6	2	3	2	2
DMOEA-ACR	1	2	6	1	1	1	1	2	1	1	1	1	1

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表 4 DNSGA-II-A、DNSGA-II-B、MOEA/D-KF、SGEA、Tr-DMOEA、MOEA/D-MoE和DMOEA-ACR在DF的性能综合排名

Table 4 Performance comprehensive ranking of DNSGA-II-A, DNSGA-II-B, MOEA/D-KF, SGEA, Tr-DMOEA, MOEA/D-MoE, and DMOEA-ACR algorithms on DF

算法	DF1	DF2	DF3	DF4	DF5	DF6	DF7	DF8	DF9	DF10	DF11	DF12	DF13	DF14	平均排序
DNSGA-II-A	5	2	4	5	6	1	1	6	3	6	6	5	3	6	4
DNSGA-II-B	6	3	5	6	7	2	3	5	4	5	7	6	4	5	6
MOEA/D-KF	4	5	3	4	4	6	4	4	5	3	5	3	6	4	5
SGEA	2	7	6	2	2	4	6	2	6	2	3	4	2	2	3
Tr-DMOEA	7	1	7	7	5	7	7	7	7	7	2	7	7	7	7
MOEA/D-MoE	3	4	2	3	3	5	5	3	2	4	4	2	5	3	2
DMOEA-ACR	1	6	1	1	1	3	2	1	1	1	1	1	1	1	1

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表 5 DMOEA-ACR-D1、DMOEA-ACR-D2、DMOEA-ACR-D3和DMOEA-ACR在DF上获得的MIGD均值和标准差

Table 5 The mean and standard deviation of MIGD of DMOEA-ACR-D1, DMOEA-ACR-D2, DMOEA-ACR-D3, and DMOEA-ACR were obtained on DF

问题集	评价指标	DMOEA-ACR-D1	DMOEA-ACR-D2	DMOEA-ACR-D3	DMOEA-ACR
DF1 (2目标)	均值标准差	1.99 × 10⁻² 1.62 × 10⁻² (−)	2.58 × 10⁻² 1.51 × 10⁻² (−)	1.50 × 10⁻² 4.12 × 10⁻³ (−)	9.15 × 10⁻³ 3.67 × 10⁻³
DF2 (2目标)	均值标准差	1.22 × 10⁻¹ 7.57 × 10⁻³ (−)	2.54 × 10⁻² 4.08 × 10⁻³ (+)	5.93 × 10⁻² 4.18 × 10⁻² (−)	5.80 × 10⁻² 7.85 × 10⁻³
DF3 (2目标)	均值标准差	1.99 × 10⁻¹ 2.61 × 10⁻³ (−)	2.33 × 10⁻¹ 4.52 × 10⁻³ (−)	4.44 × 10⁻² 1.04 × 10⁻³ (−)	1.99 × 10⁻² 3.20 × 10⁻³
DF4 (2目标)	均值标准差	5.28 × 10⁻² 7.05 × 10⁻³ (−)	2.70 × 10⁻² 6.82 × 10⁻³ (+)	4.17 × 10⁻² 5.17 × 10⁻³ (−)	2.89 × 10⁻² 5.19 × 10⁻³
DF5 (2目标)	均值标准差	1.74 × 10⁻² 6.41 × 10⁻³ (−)	5.05 × 10⁻² 3.69 × 10⁻³ (−)	2.09 × 10⁻² 3.65 × 10⁻⁴ (−)	9.32 × 10⁻³ 6.01 × 10⁻⁴
DF6 (2目标)	均值标准差	2.88 × 10⁰ 8.39 × 10⁻¹ (−)	6.44 × 10⁰ 5.07 × 10⁻¹ (−)	1.23 × 10⁰ 7.47 × 10⁻¹ (−)	1.14 × 10⁰ 4.23 × 10⁻¹
DF7 (2目标)	均值标准差	1.85 × 10⁻¹ 2.66 × 10⁻³ (−)	1.42 × 10⁻² 7.06 × 10⁻³ (+)	1.99 × 10⁻² 2.61 × 10⁻³ (−)	1.57 × 10⁻² 3.67 × 10⁻³
DF8 (2目标)	均值标准差	1.65 × 10⁻² 7.09 × 10⁻³ (+)	1.56 × 10⁻² 5.13 × 10⁻³ (+)	1.91 × 10⁻² 6.05 × 10⁻³ (−)	1.70 × 10⁻² 5.68 × 10⁻³
DF9 (2目标)	均值标准差	4.15 × 10⁻¹ 4.72 × 10⁻² (−)	1.21 × 10⁻¹ 4.81 × 10⁻³ (−)	1.11 × 10⁻¹ 5.82 × 10⁻³ (−)	6.87 × 10⁻² 4.97 × 10⁻³
DF10 (3目标)	均值标准差	3.01 × 10⁻¹ 5.11 × 10⁻² (−)	1.48 × 10⁻¹ 1.42 × 10⁻² (−)	2.43 × 10⁻¹ 3.47 × 10⁻² (−)	1.05 × 10⁻¹ 8.18 × 10⁻²
DF11 (3目标)	均值标准差	6.65 × 10⁻² 5.31 × 10⁻³ (−)	7.76 × 10⁻² 6.37 × 10⁻³ (−)	6.85 × 10⁻² 4.21 × 10⁻³ (−)	6.38 × 10⁻² 5.07 × 10⁻³
DF12 (3目标)	均值标准差	2.24 × 10⁻¹ 6.99 × 10⁻² (−)	2.62 × 10⁻¹ 8.70 × 10⁻² (−)	1.84 × 10⁻¹ 7.17 × 10⁻² (−)	9.49 × 10⁻² 6.53 × 10⁻³
DF13 (3目标)	均值标准差	2.36 × 10⁻¹ 5.77 × 10⁻² (−)	1.62 × 10⁻¹ 3.91 × 10⁻² (−)	1.36 × 10⁻¹ 5.32 × 10⁻² (−)	1.15 × 10⁻¹ 4.38 × 10⁻²
DF14 (3目标)	均值标准差	7.19 × 10⁻² 9.54 × 10⁻³ (−)	6.25 × 10⁻² 4.32 × 10⁻³ (−)	5.29 × 10⁻² 8.02 × 10⁻³ (−)	4.28 × 10⁻² 5.56 × 10⁻³
“+/−/≈”合计		1/13/0	4/10/0	0/14/0

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表 6 DMOEA-ACR-D1、DMOEA-ACR-D2、DMOEA-ACR-D3和DMOEA-ACR在SDP上获得的MIGD均值和标准差

Table 6 The mean and standard deviation of MIGD of DMOEA-ACR-D1, DMOEA-ACR-D2, DMOEA-ACR-D3, and DMOEA-ACR were obtained on SDP

问题集	评价指标	DMOEA-ACR-D1	DMOEA-ACR-D2	DMOEA-ACR-D3	DMOEA-ACR
SDP1 (2目标) SDP1 (3目标)	均值标准差	2.51 × 10⁻² 4.25 × 10⁻³ (−)	1.78 × 10⁻² 3.03 × 10⁻³ (−)	1.67 × 10⁻² 5.12 × 10⁻³ (≈)	1.65 × 10⁻² 3.67 × 10⁻³
SDP1 (2目标) SDP1 (3目标)	均值标准差	1.68 × 10⁻¹ 5.34 × 10⁻² (−)	1.54 × 10⁻¹ 4.31 × 10⁻² (−)	1.41 × 10⁻¹ 4.08 × 10⁻² (−)	1.32 × 10⁻¹ 5.44 × 10⁻³
SDP2 (2目标) SDP2 (3目标)	均值标准差	6.47 × 10⁻¹ 2.27 × 10⁻² (−)	1.12 × 10⁻¹ 6.26 × 10⁻² (≈)	2.15 × 10⁻¹ 3.38 × 10⁻² (−)	1.14 × 10⁻¹ 4.35 × 10⁻³
SDP2 (2目标) SDP2 (3目标)	均值标准差	4.05 × 10⁻¹ 3.15 × 10⁻² (−)	3.94 × 10⁻¹ 7.58 × 10⁻² (−)	5.64 × 10⁻¹ 4.15 × 10⁻² (−)	2.42 × 10⁻¹ 7.12 × 10⁻³
SDP3 (2目标) SDP3 (3目标)	均值标准差	8.91 × 10⁰ 6.55 × 10⁻¹ (−)	4.81 × 10⁰ 1.06 × 10⁻² (−)	4.99 × 10⁰ 7.21 × 10⁻¹ (−)	4.74 × 10⁰ 6.38 × 10⁻²
SDP3 (2目标) SDP3 (3目标)	均值标准差	3.91 × 10⁰ 7.60 × 10⁻¹ (−)	4.40 × 10⁰ 4.96 × 10⁻¹ (−)	4.47 × 10⁰ 6.06 × 10⁻¹ (−)	3.33 × 10⁰ 8.53 × 10⁻²
SDP4 (2目标) SDP4 (3目标)	均值标准差	6.96 × 10⁻² 6.16 × 10⁻³ (−)	7.85 × 10⁻² 6.31 × 10⁻² (−)	1.60 × 10⁻¹ 5.51 × 10⁻² (−)	5.08 × 10⁻² 5.03 × 10⁻³
SDP4 (2目标) SDP4 (3目标)	均值标准差	2.10 × 10⁻¹ 4.01 × 10⁻² (−)	2.68 × 10⁻¹ 3.07 × 10⁻² (−)	1.89 × 10⁻¹ 4.19 × 10⁻² (−)	1.69 × 10⁻¹ 8.25 × 10⁻³
SDP5 (2目标) SDP5 (3目标)	均值标准差	7.76 × 10⁻² 6.63 × 10⁻³ (−)	7.40 × 10⁻³ 2.25 × 10⁻³ (−)	9.52 × 10⁻³ 5.12 × 10⁻³ (−)	6.81 × 10⁻³ 4.92 × 10⁻⁴
SDP5 (2目标) SDP5 (3目标)	均值标准差	6.85 × 10⁻² 9.08 × 10⁻³ (−)	6.71 × 10⁻² 6.91 × 10⁻³ (−)	7.26 × 10⁻² 7.61 × 10⁻³ (−)	6.38 × 10⁻² 6.06 × 10⁻³
SDP6 (2目标) SDP6 (3目标)	均值标准差	7.98 × 10⁻³ 4.37 × 10⁻³ (−)	6.84 × 10⁻³ 4.66 × 10⁻³ (−)	6.83 × 10⁻³ 2.03 × 10⁻³ (−)	4.26 × 10⁻³ 7.43 × 10⁻⁴
SDP6 (2目标) SDP6 (3目标)	均值标准差	6.05 × 10⁻² 8.12 × 10⁻³ (−)	1.16 × 10⁻¹ 3.30 × 10⁻³ (−)	5.14 × 10⁻² 4.60 × 10⁻³ (+)	5.17 × 10⁻² 5.86 × 10⁻³
SDP7 (2目标) SDP7 (3目标)	均值标准差	2.70 × 10⁻¹ 5.59 × 10⁻² (−)	2.34 × 10⁻¹ 7.57 × 10⁻² (−)	2.25 × 10⁻¹ 4.89 × 10⁻² (−)	2.15 × 10⁻¹ 1.68 × 10⁻²
SDP7 (2目标) SDP7 (3目标)	均值标准差	2.62 × 10⁻¹ 3.67 × 10⁻² (−)	1.14 × 10⁰ 3.86 × 10⁻² (−)	2.17 × 10⁻¹ 2.61 × 10⁻² (+)	2.24 × 10⁻¹ 5.11 × 10⁻³
SDP8 (2目标) SDP8 (3目标)	均值标准差	9.54 × 10⁻² 6.63 × 10⁻³ (−)	6.28 × 10⁻² 6.73 × 10⁻³ (−)	4.53 × 10⁻² 5.15 × 10⁻³ (−)	3.14 × 10⁻² 4.26 × 10⁻³
SDP8 (2目标) SDP8 (3目标)	均值标准差	3.46 × 10⁻¹ 3.10 × 10⁻² (−)	2.79 × 10⁻¹ 2.50 × 10⁻² (−)	3.21 × 10⁻¹ 2.07 × 10⁻² (−)	1.29 × 10⁻¹ 6.43 × 10⁻³
SDP9 (2目标) SDP9 (3目标)	均值标准差	1.47 × 10⁻¹ 5.06 × 10⁻² (−)	1.28 × 10⁻¹ 4.65 × 10⁻² (−)	1.35 × 10⁻¹ 3.23 × 10⁻² (−)	8.13 × 10⁻² 7.92 × 10⁻³
SDP9 (2目标) SDP9 (3目标)	均值标准差	3.66 × 10⁻¹ 8.15 × 10⁻² (−)	4.01 × 10⁻¹ 4.08 × 10⁻² (−)	3.54 × 10⁻¹ 6.61 × 10⁻² (+)	3.60 × 10⁻¹ 8.03 × 10⁻²
SDP10 (2目标) SDP10 (3目标)	均值标准差	6.36 × 10⁻² 2.03 × 10⁻³ (−)	2.61 × 10⁻² 7.38 × 10⁻³ (−)	3.75 × 10⁻² 2.25 × 10⁻³ (−)	2.21 × 10⁻² 1.67 × 10⁻³
SDP10 (2目标) SDP10 (3目标)	均值标准差	1.68 × 10⁻¹ 6.22 × 10⁻² (+)	2.41 × 10⁻¹ 1.36 × 10⁻² (−)	3.86 × 10⁻¹ 5.46 × 10⁻² (−)	1.74 × 10⁻¹ 3.58 × 10⁻³
SDP11 (2目标) SDP11 (3目标)	均值标准差	3.80 × 10⁻² 7.32 × 10⁻³ (−)	4.83 × 10⁻² 5.21 × 10⁻³ (−)	3.26 × 10⁻² 8.06 × 10⁻³ (−)	1.36 × 10⁻² 8.16 × 10⁻³
SDP11 (2目标) SDP11 (3目标)	均值标准差	1.47 × 10⁻¹ 9.08 × 10⁻³ (−)	9.68 × 10⁻² 6.52 × 10⁻³ (−)	1.96 × 10⁻¹ 6.21 × 10⁻³ (−)	8.77 × 10⁻² 9.65 × 10⁻³
SDP12 (2目标) SDP12 (3目标)	均值标准差	4.53 × 10⁻³ 3.18 × 10⁻³ (−)	1.62 × 10⁻² 2.06 × 10⁻³ (−)	1.40 × 10⁻² 1.32 × 10⁻³ (−)	4.04 × 10⁻³ 2.17 × 10⁻⁴
SDP12 (2目标) SDP12 (3目标)	均值标准差	1.62 × 10⁻¹ 5.34 × 10⁻² (−)	1.96 × 10⁻¹ 3.73 × 10⁻² (−)	1.02 × 10⁻¹ 2.17 × 10⁻² (−)	7.63 × 10⁻² 5.63 × 10⁻⁴
“+/−/≈”合计		1/23/0	0/23/1	3/20/1

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表 7 DMOEA-ACR-P1、DMOEA-ACR-P2、DMOEA-ACR-P3、DMOEA-ACR-P4和DMOEA-ACR在SDP上获得的MIGD的均值和标准差

Table 7 The mean and standard deviation of MIGD of DMOEA-ACR-P1, DMOEA-ACR-P2, DMOEA-ACR-P3, DMOEA-ACR-P4, and DMOEA-ACR were obtained on SDP

问题集	目标数	DMOEA-ACR-P1	DMOEA-ACR-P2	DMOEA-ACR-P3	DMOEA-ACR-P4	DMOEA-ACR
SDP1 (2目标) SDP1 (3目标)	均值标准差	7.80 × 10⁻² 2.17 × 10⁻³ (−)	1.87 × 10⁻² 7.05 × 10⁻³ (−)	6.34 × 10⁻² 4.24 × 10⁻³ (−)	1.73 × 10⁻² 5.76 × 10⁻³ (−)	1.65 × 10⁻² 3.67 × 10⁻³
SDP1 (2目标) SDP1 (3目标)	均值标准差	6.69 × 10⁻¹ 4.82 × 10⁻³ (−)	1.53 × 10⁻¹ 3.62 × 10⁻³ (−)	5.50 × 10⁻¹ 4.01 × 10⁻³ (−)	1.81 × 10⁻² 6.20 × 10⁻³ (−)	1.32 × 10⁻¹ 5.44 × 10⁻³
SDP2 (2目标) SDP2 (3目标)	均值标准差	1.78 × 10⁻¹ 3.16 × 10⁻³ (−)	2.02 × 10⁻¹ 4.91 × 10⁻³ (−)	2.25 × 10⁻¹ 3.97 × 10⁻³ (−)	2.36 × 10⁻¹ 6.32 × 10⁻³ (−)	1.14 × 10⁻¹ 4.35 × 10⁻³
SDP2 (2目标) SDP2 (3目标)	均值标准差	5.82 × 10⁻¹ 8.04 × 10⁻³ (−)	3.25 × 10⁻¹ 6.51 × 10⁻³ (−)	3.11 × 10⁻¹ 5.26 × 10⁻³ (−)	3.21 × 10⁻¹ 4.58 × 10⁻³ (−)	2.42 × 10⁻¹ 7.12 × 10⁻³
SDP3 (2目标) SDP3 (3目标)	均值标准差	5.15 × 10⁰ 3.66 × 10⁻² (−)	4.85 × 10⁰ 1.35 × 10⁻² (−)	5.14 × 10⁰ 6.59 × 10⁻² (−)	4.83 × 10⁰ 4.76 × 10⁻² (−)	4.74 × 10⁰ 6.38 × 10⁻²
SDP3 (2目标) SDP3 (3目标)	均值标准差	1.99 × 10⁰ 5.14 × 10⁻² (+)	3.83 × 10⁰ 2.94 × 10⁻² (−)	3.88 × 10⁰ 5.68 × 10⁻² (−)	3.91 × 10⁰ 6.20 × 10⁻² (−)	3.33 × 10⁰ 8.53 × 10⁻²
SDP4 (2目标) SDP4 (3目标)	均值标准差	4.67 × 10⁻² 5.16 × 10⁻³ (+)	8.35 × 10⁻² 5.91 × 10⁻³ (−)	1.02 × 10⁻¹ 4.85 × 10⁻³ (−)	7.58 × 10⁻² 6.24 × 10⁻³ (−)	5.08 × 10⁻² 5.03 × 10⁻³
SDP4 (2目标) SDP4 (3目标)	均值标准差	1.76 × 10⁻¹ 4.21 × 10⁻³ (−)	2.13 × 10⁻¹ 3.89 × 10⁻³ (−)	1.65 × 10⁻¹ 3.56 × 10⁻³ (+)	2.02 × 10⁻¹ 5.81 × 10⁻³ (−)	1.69 × 10⁻¹ 8.25 × 10⁻³
SDP5 (2目标) SDF5 (3目标)	均值标准差	3.66 × 10⁻² 5.41 × 10⁻³ (−)	7.63 × 10⁻³ 3.00 × 10⁻⁴ (−)	7.13 × 10⁻³ 6.35 × 10⁻⁴ (−)	8.08 × 10⁻³ 6.10 × 10⁻⁴ (−)	6.81 × 10⁻³ 4.92 × 10⁻⁴
SDP5 (2目标) SDF5 (3目标)	均值标准差	3.96 × 10⁻¹ 8.79 × 10⁻² (−)	6.43 × 10⁻² 6.84 × 10⁻³ (−)	6.49 × 10⁻² 2.96 × 10⁻³ (−)	6.57 × 10⁻² 5.39 × 10⁻³ (−)	6.38 × 10⁻² 6.06 × 10⁻³
SDP6 (2目标) SDP6 (3目标)	均值标准差	6.48 × 10⁻³ 5.61 × 10⁻⁴ (−)	4.62 × 10⁻³ 8.91 × 10⁻⁴ (−)	4.36 × 10⁻³ 6.03 × 10⁻⁴ (−)	4.31 × 10⁻³ 5.28 × 10⁻⁴ (−)	4.26 × 10⁻³ 7.43 × 10⁻⁴
SDP6 (2目标) SDP6 (3目标)	均值标准差	1.51 × 10⁻¹ 7.36 × 10⁻³ (−)	5.27 × 10⁻² 7.18 × 10⁻³ (−)	5.43 × 10⁻² 6.23 × 10⁻³ (−)	5.20 × 10⁻² 5.98 × 10⁻³ (−)	5.17 × 10⁻² 5.86 × 10⁻³
SDP7 (2目标) SDP7 (3目标)	均值标准差	1.60 × 10⁻¹ 5.31 × 10⁻² (+)	3.26 × 10⁻¹ 6.94 × 10⁻² (−)	2.32 × 10⁻¹ 3.28 × 10⁻² (−)	2.46 × 10⁻¹ 4.51 × 10⁻² (−)	2.15 × 10⁻¹ 1.68 × 10⁻²
SDP7 (2目标) SDP7 (3目标)	均值标准差	5.98 × 10⁻² 6.60 × 10⁻³ (+)	2.56 × 10⁻¹ 6.59 × 10⁻² (−)	2.43 × 10⁻¹ 4.67 × 10⁻³ (−)	2.53 × 10⁻¹ 5.81 × 10⁻³ (−)	2.24 × 10⁻¹ 5.11 × 10⁻³
SDP8 (2目标) SDP8 (3目标)	均值标准差	1.23 × 10⁻¹ 6.57 × 10⁻² (−)	4.70 × 10⁻² 5.48 × 10⁻³ (−)	2.70 × 10⁻² 4.90 × 10⁻³ (+)	2.55 × 10⁻² 5.84 × 10⁻³ (+)	3.14 × 10⁻² 4.26 × 10⁻³
SDP8 (2目标) SDP8 (3目标)	均值标准差	1.37 × 10⁻¹ 4.96 × 10⁻³ (−)	1.81 × 10⁻¹ 3.69 × 10⁻³ (−)	1.55 × 10⁻¹ 1.86 × 10⁻³ (−)	1.40 × 10⁻¹ 7.82 × 10⁻³ (−)	1.29 × 10⁻¹ 6.43 × 10⁻³
SDP9 (2目标) SDP9 (3目标)	均值标准差	1.85 × 10⁻¹ 8.80 × 10⁻³ (−)	1.29 × 10⁻¹ 7.61 × 10⁻³ (−)	1.50 × 10⁻¹ 8.55 × 10⁻³ (−)	1.32 × 10⁻¹ 6.27 × 10⁻³ (−)	8.13 × 10⁻² 7.92 × 10⁻³
SDP9 (2目标) SDP9 (3目标)	均值标准差	4.94 × 10⁻¹ 9.43 × 10⁻² (−)	3.79 × 10⁻¹ 8.99 × 10⁻² (−)	3.67 × 10⁻¹ 9.08 × 10⁻² (−)	3.69 × 10⁻¹ 4.20 × 10⁻² (−)	3.60 × 10⁻¹ 8.03 × 10⁻²
SDP10 (2目标) SDP10 (3目标)	均值标准差	1.12 × 10⁻¹ 3.13 × 10⁻³ (−)	3.28 × 10⁻² 4.10 × 10⁻³ (−)	2.23 × 10⁻² 2.15 × 10⁻³ (−)	3.63 × 10⁻² 2.23 × 10⁻³ (−)	2.21 × 10⁻² 1.67 × 10⁻³
SDP10 (2目标) SDP10 (3目标)	均值标准差	1.87 × 10⁻¹ 5.78 × 10⁻² (−)	2.80 × 10⁻¹ 3.65 × 10⁻² (−)	1.79 × 10⁻¹ 6.03 × 10⁻² (−)	1.92 × 10⁻¹ 2.65 × 10⁻³ (−)	1.74 × 10⁻¹ 3.58 × 10⁻³
SDP11 (2目标) SDP11 (3目标)	均值标准差	2.28 × 10⁻² 9.14 × 10⁻³ (−)	3.60 × 10⁻² 7.61 × 10⁻³ (−)	2.38 × 10⁻² 9.03 × 10⁻³ (−)	3.34 × 10⁻² 8.16 × 10⁻³ (−)	1.36 × 10⁻² 8.16 × 10⁻³
SDP11 (2目标) SDP11 (3目标)	均值标准差	8.83 × 10⁻² 8.87 × 10⁻³ (−)	1.04 × 10⁻¹ 8.17 × 10⁻³ (−)	8.79 × 10⁻² 5.96 × 10⁻³ (−)	1.18 × 10⁻¹ 9.08×10⁻³ (−)	8.77 × 10⁻² 9.65 × 10⁻³
SDP12 (2目标) SDP12 (3目标)	均值标准差	4.52 × 10⁻³ 3.75 × 10⁻³ (−)	4.94 × 10⁻³ 3.65 × 10⁻³ (−)	4.37 × 10⁻³ 7.29 × 10⁻³ (−)	4.60 × 10⁻³ 6.94 × 10⁻³ (−)	4.04 × 10⁻³ 2.17 × 10⁻⁴
SDP12 (2目标) SDP12 (3目标)	均值标准差	2.17 × 10⁻¹ 5.02 × 10⁻² (−)	2.65 × 10⁻¹ 4.10 × 10⁻² (−)	1.06 × 10⁻¹ 3.08 × 10⁻² (−)	3.23 × 10⁻¹ 5.24 × 10⁻² (−)	7.63 × 10⁻² 5.63 × 10⁻⁴
“+/−/≈”合计		4/20/0	0/24/0	2/22/0	1/23/0

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表 8 DMOEA-ACR-P1、DMOEA-ACR-P2、DMOEA-ACR-P3、DMOEA-ACR-P4和DMOEA-ACR在DF上获得的MIGD的均值和标准差

Table 8 The mean and standard deviation of MIGD of DMOEA-ACR-P1, DMOEA-ACR-P2, DMOEA-ACR-P3, DMOEA-ACR-P4, and DMOEA-ACR were obtained on DF

问题集	评价指标	DMOEA-ACR-P1	DMOEA-ACR-P2	DMOEA-ACR-P3	DMOEA-ACR-P4	DMOEA-ACR
DF1 (2目标)	均值标准差	9.28 × 10⁻³ 6.51 × 10⁻³ (−)	1.82 × 10⁻² 1.39 × 10⁻² (−)	9.47 × 10⁻³ 8.94 × 10⁻³ (−)	1.73 × 10⁻² 5.06 × 10⁻³ (−)	9.15 × 10⁻³ 3.67 × 10⁻³
DF2 (2目标)	均值标准差	1.01 × 10⁻¹ 3.16 × 10⁻² (−)	6.98 × 10⁻² 2.48 × 10⁻³ (−)	1.16 × 10⁻¹ 3.90 × 10⁻² (−)	5.17 × 10⁻² 8.26 × 10⁻³ (+)	5.80 × 10⁻² 7.85 × 10⁻³
DF3 (2目标)	均值标准差	2.64 × 10⁻² 4.09 × 10⁻³ (−)	2.89 × 10⁻² 6.71 × 10⁻³ (−)	2.04 × 10⁻² 5.66 × 10⁻³ (≈)	2.91 × 10⁻² 4.18 × 10⁻³ (−)	1.99 × 10⁻² 3.20 × 10⁻³
DF4 (2目标)	均值标准差	2.38 × 10⁻² 6.23 × 10⁻³ (+)	3.99 × 10⁻² 4.20 × 10⁻³ (−)	2.94 × 10⁻² 6.81 × 10⁻³ (−)	2.77 × 10⁻² 7.11 × 10⁻³ (+)	2.89 × 10⁻² 5.19 × 10⁻³
DF5 (2目标)	均值标准差	8.51 × 10⁻³ 8.38 × 10⁻³ (+)	2.48 × 10⁻² 4.82 × 10⁻³ (−)	8.83 × 10⁻³ 6.22 × 10⁻⁴ (+)	2.27 × 10⁻² 3.10 × 10⁻³ (−)	9.32 × 10⁻³ 6.01 × 10⁻⁴
DF6 (2目标)	均值标准差	2.26 × 10⁰ 7.82 × 10⁻¹ (−)	1.46 × 10⁰ 6.33 × 10⁻¹ (−)	1.16 × 10⁰ 5.67 × 10⁻¹ (−)	1.60 × 10⁰ 4.28 × 10⁻¹ (−)	1.14 × 10⁰ 4.23 × 10⁻¹
DF7 (2目标)	均值标准差	1.62 × 10⁻² 4.61 × 10⁻³ (−)	1.84 × 10⁻² 5.14 × 10⁻³ (−)	1.45 × 10⁻² 7.12 × 10⁻³ (+)	1.48 × 10⁻² 5.04 × 10⁻³ (+)	1.57 × 10⁻² 3.67 × 10⁻³
DF8 (2目标)	均值标准差	1.65 × 10⁻² 4.77 × 10⁻³ (+)	2.87 × 10⁻² 4.00 × 10⁻³ (−)	1.71 × 10⁻² 6.54 × 10⁻³ (≈)	1.68 × 10⁻² 5.26 × 10⁻³ (+)	1.70 × 10⁻² 5.68 × 10⁻³
DF9 (2目标)	均值标准差	9.75 × 10⁻² 4.63 × 10⁻² (−)	9.32 × 10⁻² 3.91 × 10⁻³ (−)	6.93 × 10⁻² 5.73 × 10⁻³ (−)	1.10 × 10⁻¹ 6.05 × 10⁻³ (−)	6.87 × 10⁻² 4.97 × 10⁻³
DF10 (2目标)	均值标准差	1.92 × 10⁻¹ 9.39 × 10⁻² (−)	4.59 × 10⁻¹ 8.35 × 10⁻² (−)	2.37 × 10⁻¹ 3.07 × 10⁻² (−)	1.51 × 10⁻¹ 7.99 × 10⁻² (−)	1.05 × 10⁻¹ 8.18 × 10⁻²
DF11 (2目标)	均值标准差	7.43 × 10⁻² 5.86 × 10⁻³ (−)	8.51 × 10⁻² 4.81 × 10⁻³ (−)	6.42 × 10⁻² 6.03 × 10⁻³ (−)	6.57 × 10⁻² 5.87 × 10⁻³ (−)	6.38 × 10⁻² 5.07 × 10⁻³
DF12 (2目标)	均值标准差	1.75 × 10⁻¹ 3.23 × 10⁻² (−)	3.29 × 10⁻¹ 2.02 × 10⁻² (−)	2.60 × 10⁻¹ 6.17 × 10⁻² (−)	1.16 × 10⁻¹ 4.03 × 10⁻² (−)	9.49 × 10⁻² 6.53 × 10⁻³
DF13 (2目标)	均值标准差	1.19 × 10⁻¹ 4.91 × 10⁻² (−)	2.57 × 10⁻¹ 2.68 × 10⁻² (−)	1.20 × 10⁻¹ 3.16 × 10⁻² (−)	2.48 × 10⁻¹ 4.82 × 10⁻² (−)	1.15 × 10⁻¹ 4.38 × 10⁻²
DF14 (2目标)	均值标准差	4.43 × 10⁻² 6.09 × 10⁻³ (−)	6.32 × 10⁻² 4.17 × 10⁻³ (−)	4.59 × 10⁻² 3.21 × 10⁻³ (−)	5.95 × 10⁻² 6.61 × 10⁻³ (−)	4.28 × 10⁻² 5.56 × 10⁻³
“+/−/≈”合计		3/11/0	0/14/0	2/10/2	4/10/0

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表 9 $\tau_t$分别为5、10、20时DNSGA-II-A、DNSGA-II-B、MOEA/D-KF、SGEA、Tr-DMOEA、MOEA/D-MoE和DMOEA-ACR在SDP和DF上获得的显著差异统计结果

Table 9 Significant difference statistical results of DNSGA-II-A, DNSGA-II-B, MOEA/D-KF, SGEA, Tr-DMOEA, MOEA/D-MoE, and DMOEA-ACR were obtained on SDP and DF where$\tau_t $is 5, 10, 20, respectively

问题集	$\tau_t $	DNSGA-II-A	DNSGA-II-B	MOEA/D- KF	SGEA	Tr-DMOEA	MOEA/D- MoE	DMOEA- ACR
SDP	5	4/19/1	4/20/0	3/21/0	2/22/0	0/24/0	1/23/0	+/−/≈
	10	3/21/0	4/20/0	3/21/0	3/18/3	2/22/0	4/20/0
	20	3/21/0	5/19/0	1/21/2	5/18/1	2/22/0	3/21/0
DF	5	2/12/0	2/11/1	1/13/0	1/13/0	0/14/0	1/13/0	+/−/≈
	10	2/11/1	2/12/0	1/13/0	0/14/0	1/13/0	1/13/0
	20	2/12/0	2/12/0	1/13/0	2/12/0	1/12/1	1/12/1

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