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浙江大学学报(工学版)
浙江大学学报(工学版)  2024, Vol. 58 Issue (6): 1107-1120    DOI: 10.3785/j.issn.1008-973X.2024.06.002
计算机技术     
多目标粒子群优化算法及其应用研究综述
叶倩琳1(),王万良1,*(),王铮2
1. 浙江工业大学 计算机科学与技术学院，浙江 杭州 310023
2. 浙大城市学院 计算机与计算科学学院，浙江 杭州 310015
Survey of multi-objective particle swarm optimization algorithms and their applications
Qianlin YE1(),Wanliang WANG1,*(),Zheng WANG2
1. College of Computer Science and Technology, Zhejiang University of Technology, Hangzhou 310023, China
2. School of Computer and Computational Sciences, Hangzhou City University, Hangzhou 310015, China
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摘要：

现有研究较少涵盖最先进的多目标粒子群优化（MOPSO）算法. 本研究介绍了多目标优化问题（MOPs）的研究背景，阐述了MOPSO的基本理论. 根据特征将其分为基于Pareto支配、基于分解和基于指标的3类MOPSO算法，介绍了现有的经典算法. 介绍相关评价指标，并选取7个有代表性的算法进行性能分析. 实验结果展示了传统MOPSO和3类改进的MOPSO算法各自的优势与不足，其中，基于指标的MOPSO在收敛性和多样性方面表现较优. 对MOPSO算法在生产调度、图像处理和电力系统等领域的应用进行简要介绍. 并探讨了MOPSO算法用于求解复杂优化问题的局限性及未来的研究方向.
关键词： 粒子群优化;  多目标优化;  Pareto解集;  收敛性;  多样性    
Abstract:

Few existing studies cover the state-of-the-art multi-objective particle swarm optimization (MOPSO) algorithms. To fill the gap in this area, the research background of multi-objective optimization problems (MOPs) was introduced, and the fundamental theories of MOPSO were described. The MOPSO algorithms were divided into three categories according to their features: Pareto-dominated-based MOPSO, decomposition-based MOPSO, and indicator-based MOPSO, and a detailed description of their existing classical algorithms was also developed. Next, relevant evaluation indicators were described, and seven representative algorithms were selected for performance analysis. The experimental results demonstrated the strengths and weaknesses of each of the traditional MOPSO and three categories of improved MOPSO algorithms. Among them, the indicator-based MOPSO performed better in terms of convergence and diversity. Then, the applications of MOPSO algorithms in production scheduling, image processing, and power systems were briefly introduced. Finally, the limitations and future research directions of the MOPSO algorithm for solving complex optimization problems were discussed.
Key words: particle swarm optimization    multi-objective optimization    Pareto solution set    convergence    diversity
收稿日期: 2023-08-08 出版日期: 2024-05-25
CLC:  TP 393  
基金资助: 国家自然科学基金资助项目(61873240，51875524)；浙江省重点研发计划资助项目(领雁计划)(2023C01168)；数字化制造装备与技术国家重点实验室基金资助项目(2023C01168）.
通讯作者: 王万良     E-mail: yql@zjut.edu.cn;zjutwwl@zjut.edu.cn
作者简介: 叶倩琳（1999—），女，博士生，从事多目标优化算法研究. orcid.org/0009-0008-4651-5006. E-mail：yql@zjut.edu.cn
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引用本文:

叶倩琳,王万良,王铮. 多目标粒子群优化算法及其应用研究综述[J]. 浙江大学学报(工学版), 2024, 58(6): 1107-1120.

Qianlin YE,Wanliang WANG,Zheng WANG. Survey of multi-objective particle swarm optimization algorithms and their applications. Journal of ZheJiang University (Engineering Science), 2024, 58(6): 1107-1120.

链接本文:

https://www.zjujournals.com/eng/CN/10.3785/j.issn.1008-973X.2024.06.002    或    https://www.zjujournals.com/eng/CN/Y2024/V58/I6/1107

图 1  多目标优化问题示意图
图 2  粒子群和多目标粒子群优化流程图
图 3  多目标粒子群优化分类示意图
图 4  基于增强网格机制的映射示意图
图 5  HV指标和HV贡献示意图
问题mnkPF形状
ZDT1230—convex
ZDT2230—concave
ZDT3230—disconnected
ZDT4210—convex
ZDT6210—concave
DTLZ13m?1+k5linear
DTLZ23m?1+k10concave
DTLZ43m?1+k10concave
DTLZ63m?1+k10concave
UF2230—convex
UF3230—convex
UF4230—concave
UF5230—disconnected
UF8330—concave
UF9330—linear
UF10330—concave
表 1  ZDT、DTLZ和UF测试问题的特征
问题传统基于Pareto支配基于分解基于指标
MOPSONMPSOCMOPSOdMOPSOEGCCMOPSOESMOPSOR2HMOPSO
ZDT11.64×100
(9.58×10?2)		3.16×10?2
(9.92×10?3)		4.11×10?3
(7.60×10?5)		3.90×10?3
(3.84×10?5)		3.91×10?3
(4.89×10?5)		3.86×10?3
(6.05×10?7)		3.90×10?3
(6.21×10?5)
ZDT23.15×100
(1.57×10?1)		6.25×10?2
(1.32×10?1)		4.08×10?3
(7.02×10?5)		6.44×10?2
(1.85×10?1)		3.88×10?3
(3.86×10?5)		3.80×10?3
(9.45×10?7)		3.83×10?3
(4.20×10?5)
ZDT31.19×100
(8.60×10?2)		9.79×10?2
(7.02×10?3)		4.66×10?3
(6.74×10?5)		1.06×10?2
(7.10×10?4)		4.57×10?3
(4.83×10?5)		9.99×10?3
(3.59×10?4)		6.12×10?2
(1.58×10?1)
ZDT41.02×101
(4.52×100)		1.37×10?2
(2.71×10?1)		3.84×10?3
(2.96×10?5)		5.97×100
(4.48×100)		3.87×10?3
(3.87×10?5)		3.77×10?3
(6.15×10?6)		4.52×10?3
(3.73×10?3)
ZDT65.38×100
(5.10×10?1)		2.21×10?3
(1.60×10?4)		3.74×10?3
(1.39×10?4)		1.88×10?3
(8.55×10?6)		3.07×10?3
(1.75×10?5)		1.90×10?3
(7.21×10?7)		1.89×10?3
(4.33×10?5)
DTLZ11.55×100
(1.02×100)		4.25×101
(6.56×100)		3.25×100
(4.50×100)		1.53×101
(1.18×101)		2.47×100
(3.46×100)		1.28×10?2
(7.85×10?6)		1.99×10?2
(2.08×10?3)
DTLZ21.57×10?1
(3.83×10?2)		7.68×10?2
(2.69×10?3)		6.71×10?2
(1.94×10?3)		4.72×10?2
(1.49×10?3)		5.48×10?2
(5.68×10?4)		3.56×10?2
(1.67×10?4)		4.37×10?2
(1.20×10?3)
DTLZ43.61×10?1
(2.70×10?1)		1.07×10?1
(1.18×10?1)		1.29×10?1
(2.22×10?1)		1.18×10?1
(7.63×10?2)		5.63×10?2
(8.29×10?4)		4.68×10?2
(9.35×10?4)		5.20×10?2
(2.01×10?3)
DTLZ69.21×100
(6.60×10?2)		7.57×10?2
(3.11×10?3)		5.31×10?1
(2.52×10?4)		1.27×10?1
(6.73×10?3)		4.18×10?3
(4.54×10?5)		7.85×10?2
(3.16×10?3)		7.04×10?2
(8.44×10?3)
UF21.26×10?1
(1.41×10?2)		8.19×10?2
(7.06×10?3)		6.38×10?2
(4.76×10?3)		4.73×10?2
(8.12×10?3)		5.83×10?2
(7.68×10?3)		1.54×10?2
(2.55×10?3)		1.84×10?2
(7.15×10?3)
UF34.89×10?1
(2.35×10?2)		3.64×10?1
(5.59×10?2)		3.97×10?1
(2.56×10?2)		5.01×10?1
(2.56×10?1)		3.45×10?1
(7.31×10?2)		2.59×10?1
(7.62×10?2)		6.20×10?1
(1.77×10?1)
UF41.73×10?1
(6.65×10?3)		6.39×10?2
(8.78×10?3)		1.09×10?1
(1.12×10?2)		9.21×10?2
(4.96×10?3)		7.97×10?2
(5.31×10?3)		3.28×10?2
(2.90×10?3)		3.76×10?2
(2.64×10?3)
UF52.43×100
(3.40×10?1)		1.69×100
(4.19×10?1)		8.45×10?1
(1.75×10?1)		7.01×10?1
(1.73×10?1)		8.27×10?1
(3.27×10?1)		1.62×10?1
(7.93×10?3)		1.75×10?1
(2.73×10?2)
UF84.65×10?1
(3.84×10?2)		4.48×10?1
(1.18×10?1)		5.54×10?1
(1.05×10?1)		1.97×10?1
(3.14×10?2)		2.68×10?1
(3.84×10?3)		1.81×10?1
(6.89×10?2)		4.14×10?1
(5.22×10?2)
UF96.12×10?1
(4.81×10?2)		4.70×10?1
(6.12×10?2)		8.45×10?1
(1.14×10?1)		1.14×10?1
(2.88×10?2)		4.52×10?1
(2.49×10?2)		6.29×10?2
(5.04×10?3)		2.36×10?1
(5.91×10?2)
UF101.32×100
(1.94×10?1)		1.50×100
(3.41×100)		4.51×100
(4.79×10?1)		1.01×100
(1.74×10?1)		3.30×10?1
(2.03×10?2)		2.26×10?1
(1.06×10?1)		2.38×10?1
(5.69×10?2)
表 2  7个MOPSO算法在3类基准问题上的IGD
图 6  多目标粒子群优化算法的平均运行时间
图 7  多目标粒子群优化的应用领域展示图
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