Please wait a minute...
浙江大学学报(工学版)
Journal of ZheJiang University (Engineering Science)  2024, Vol. 58 Issue (6): 1107-1120    DOI: 10.3785/j.issn.1008-973X.2024.06.002
    
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
Download: HTML     PDF(1559KB) HTML
Export: BibTeX | EndNote (RIS)      

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     
Received: 08 August 2023      Published: 25 May 2024
CLC:  TP 393  
Fund:  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）.
Corresponding Authors: Wanliang WANG     E-mail: yql@zjut.edu.cn;zjutwwl@zjut.edu.cn
Service
E-mail this article
Add to my bookshelf
Add to citation manager
E-mail Alert
RSS
Articles by authors
Qianlin YE
Wanliang WANG
Zheng WANG
Cite this article:

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.

URL:

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


多目标粒子群优化算法及其应用研究综述

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 
Fig.1 Illustration of multi-objective optimization problems
Fig.2 Flowcharts of particle swarm optimization and multi-objective particle swarm optimization
Fig.3 Illustration of multi-objective particle swarm optimization classification
Fig.4 Illustration of mapping based on enhanced grid mechanism
Fig.5 Illustration of HV indicator and HV contribution
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
Tab.1 Characteristics of ZDT, DTLZ and UF test problems
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)
Tab.2 IGD of seven MOPSO algorithms on three benchmark problems
Fig.6 Average runtimes of multi-objective particle swarm optimization algorithms
Fig.7 Illustration of application domain of multi-objective particle swarm optimization
[1]   王 丽, 任宇, 邱启仓, 等 多目标进化算法性能评价指标研究综述[J]. 计算机学报, 2021, 44 (8): 1590- 1619
WANG Li, Ren Yu, Qiu Qicang, et al Survey on performance indicator for multi-objective evolutionary algorithms[J]. Chinese Journal of Computers, 2021, 44 (8): 1590- 1619
doi: 10.11897/SP.J.1016.2021.01590
[2]   王万良, 金雅文, 陈嘉诚, 等 多角色多策略多目标粒子群优化算法[J]. 浙江大学学报: 工学版, 2022, 56 (3): 531- 541
WANG Wanliang, JIN Yawen, CHEN Jiachen, et al Multi-objective particle-swarm optimization algorithm with multi-role and multi-strategy[J]. Journal of Zhejiang University: Engineering Science, 2022, 56 (3): 531- 541
[3]   LUO Y, ZHANG K, YANG H, et al. A reduced mixed representation based multi-objective evolutionary algorithm for large-scale overlapping community detection [C]// 2021 IEEE Congress on Evolutionary Computation . Kraków: IEEE, 2021: 2435−2442.
[4]   王万良. 人工智能导论 第5版[M]. 北京: 高等教育出版社, 2022.
[5]   ZHOU C, DAI G, ZHANG C, et al Entropy based evolutionary algorithm with adaptive reference points for many-objective optimization problems[J]. Information Sciences, 2018, 465: 232- 247
doi: 10.1016/j.ins.2018.07.012
[6]   冯茜, 李擎, 全威, 等 多目标粒子群优化算法研究综述[J]. 工程科学学报, 2021, 43 (6): 745- 753
FENG Qian, LI Qing, QUAN Wei, et al Overview of multiobjective particle swarm optimization algorithm[J]. Chinese Journal of Engineering, 2021, 43 (6): 745- 753
[7]   YUAN Y, XU H, WANG B. An improved NSGA-III procedure for evolutionary many-objective optimization [C]// Proceedings of the 2014 Annual Conference on Genetic and Evolutionary Computation . New York: [s. n. ], 2014: 661−668.
[8]   ZHENG J, ZHOU F, ZOU J, et al A dynamic multi-objective optimization based on a hybrid of pivot points prediction and diversity strategies[J]. Swarm and Evolutionary Computation, 2023, 78: 101284
[9]   ZHANG K, SHEN C, YEN G G, et al Two-stage double niched evolution strategy for multimodal multiobjective optimization[J]. IEEE Transactionsactions on Evolutionary Computation, 2021, 25 (4): 754- 768
doi: 10.1109/TEVC.2021.3064508
[10]   MING M, TRIVEDI A, WANG R, et al A dual-population-based evolutionary algorithm for constrained multiobjective optimization[J]. IEEE Transactionsactions on Evolutionary Computation, 2021, 25 (4): 739- 753
doi: 10.1109/TEVC.2021.3066301
[11]   王万良. 人工智能及其应用 第4版[M]. 北京: 高等教育出版社, 2020.
[12]   杨辉华, 谢谱模, 张晓凤, 等 求解多目标优化问题的改进布谷鸟搜索算法[J]. 浙江大学学报: 工学版, 2015, 49 (8): 1600- 1608
YANG Huihua, XIE Pumo, ZHANG Xiaofeng, et al Improved cuckoo search algorithm for multi-objective optimization problems[J]. Journal of Zhejiang University: Engineering Science, 2015, 49 (8): 1600- 1608
[13]   鲁建厦, 翟文倩, 李嘉丰, 等 基于改进混合蛙跳算法的多约束车辆路径优化[J]. 浙江大学学报: 工学版, 2021, 55 (2): 259- 270
LU Jianxia, ZHAI Wenqian, LI Jiafeng, et al Muti-constraint vehicle routing optimization based on improved hybrid shuffled frog leaping algorithm[J]. Journal of Zhejiang University: Engineering Science, 2021, 55 (2): 259- 270
[14]   CORUS D, DANG D C, EREMEEV A V, et al Level-based analysis of genetic algorithms and other search processes[J]. IEEE Transactionsactions on Evolutionary Computation, 2018, 22 (5): 707- 719
doi: 10.1109/TEVC.2017.2753538
[15]   PIOTROWSKI A P Review of differential evolution population size[J]. Swarm and Evolutionary Computation, 2017, 32: 1- 24
doi: 10.1016/j.swevo.2016.05.003
[16]   COLORNI A, DORIGO M, MANIEZZO V, et al. Distributed optimization by ant colonies [C]// Proceedings of ECAL91-European Conference on Artificial Life . Milano: Elsevier, 1992: 134−142.
[17]   DORIGO M. Optimization, learning and natural algorithms [EB/OL]. [2024-04-09]. https://xueshu.baidu.com/usercenter/paper/show?paperid=1b9ebb7e73db6f097f286e54d5fa31db.
[18]   KENNEDY J, EBERHART R. Particle swarm optimization [C]// Proceedings of ICNN'95-International Conference on Neural Networks . Perth: IEEE, 1995: 1942−1948.
[19]   COELLO C A C, LECHUGA M S. MOPSO: a proposal for multiple objective particle swarm optimization [C]// Proceedings of the 2002 Congress on Evolutionary Computation . Honolulu: IEEE, 2002: 1051−1056.
[20]   毕晓君, 王朝 基于超平面投影的高维多目标进化算法[J]. 浙江大学学报: 工学版, 2018, 52 (7): 1284- 1293
BI Xiaojun, WANG Chao Many-objective evolutionary algorithm based on hyperplane projection[J]. Journal of Zhejiang University: Engineering Science, 2018, 52 (7): 1284- 1293
[21]   谢承旺, 潘嘉敏, 郭华, 等. 一种采用混合策略的大规模多目标进化算法. 计算机学报[EB/OL]. (2023-07-13) [2023-08-08]. https: //kns.cnki.net/kcms2/detail/11.1826.TP.20230712.1925.021.html
[22]   XU G, LUO K, JING G, et al On convergence analysis of multi-objective particle swarm optimization algorithm[J]. European Journal of Operational Research, 2020, 286 (1): 32- 38
doi: 10.1016/j.ejor.2020.03.035
[23]   HOUSSEIN E H, GAD A G, HUSSAIN K, et al Major advances in particle swarm optimization: theory, analysis, and application[J]. Swarm and Evolutionary Computation, 2021, 63: 100868- 100905
doi: 10.1016/j.swevo.2021.100868
[24]   DE CARVALHO A B, POZO A Measuring the convergence and diversity of CDAS multi-objective particle swarm optimization algorithms: a study of many-objective problems[J]. Neurocomputing, 2012, 75 (1): 43- 51
doi: 10.1016/j.neucom.2011.03.053
[25]   ZAPOTECAS MARTíNEZ S, COELLO COELLO C A. A multi-objective particle swarm optimizer based on decomposition [C]// Proceedings of the Annual Conference on Genetic and Evolutionary Computation . Dublin: IEEE, 2011: 69−76.
[26]   AL MOUBAYED N, PETROVSKI A, MCCALL J D2MOPSO: MOPSO based on decomposition and dominance with archiving using crowding distance in objective and solution spaces[J]. Evolution Computation, 2014, 22 (1): 47- 77
doi: 10.1162/EVCO_a_00104
[27]   纪昌明, 马皓宇, 李宁宁, 等 基于树形结构无界存档的多目标粒子群算法[J]. 控制与决策, 2020, 35 (11): 2657- 2686
JI Changming, MA Haoyu, LI Ningning, et al Multi-objective particle swarm optimization algorithm based on tree-structured unbounded archive[J]. Control and Decision, 2020, 35 (11): 2657- 2686
[28]   ZHANG X, ZHENG X, CHENG R, et al A competitive mechanism based multi-objective particle swarm optimizer with fast convergence[J]. Information Sciences, 2018, 427: 63- 76
doi: 10.1016/j.ins.2017.10.037
[29]   LIN Q, LIU S, ZHU Q, et al Particle swarm optimization with a balanceable fitness estimation for many-objective optimization problems[J]. IEEE Transactionsactions on Evolutionary Computation, 2018, 22 (1): 32- 46
doi: 10.1109/TEVC.2016.2631279
[30]   余伟伟, 谢承旺, 闭应洲, 等 一种基于自适应模糊支配的高维多目标粒子群算法[J]. 自动化学报, 2018, 44 (12): 2278- 2289
YU Weiwei, XIE Chengwang, BI Yingzhou, et al Many-objective particle swarm optimization based on adaptive fuzzy dominance[J]. Acta Automatica Sinica, 2018, 44 (12): 2278- 2289
[31]   谭阳, 唐德权, 曹守富 基于超球形模糊支配的高维多目标粒子群优化算法[J]. 计算机应用, 2019, 39 (11): 3233- 3241
TAN Yang, TANG Dequan, CAO Shoufu Many-objective particle swarm optimization algorithm based on hyper-spherical fuzzy dominance[J]. Journal of Computer Applications, 2019, 39 (11): 3233- 3241
[32]   TANABE R, ISHIBUCHI H A review of evolutionary multimodal multiobjective optimization[J]. IEEE Transactionsactions on Evolutionary Computation, 2020, 24 (1): 193- 200
doi: 10.1109/TEVC.2019.2909744
[33]   QU B, LI C, LIANG J, et al A self-organized speciation based multi-objective particle swarm optimizer for multimodal multi-objective problems[J]. Applied Soft Computing, 2020, 86: 105886
doi: 10.1016/j.asoc.2019.105886
[34]   顾清华, 骆家乐, 李学现 基于小生境的多目标进化算法[J]. 计算机工程与应用, 2023, 59 (1): 126- 139
GU Qinghua, LUO Jiale, LI Xuexian Evolutionary algorithm based on niche for multi-objective optimization[J]. Computer Engineering and Applications, 2023, 59 (1): 126- 139
doi: 10.3778/j.issn.1002-8331.2207-0006
[35]   公硕鹏. 基于小生境的多模态多目标优化算法研究[D]. 武汉: 华中科技大学, 2022.
GONG Shuopeng. Research on multimodel multiobjective optimization algorithm based on niching methods [D]. Wuhan: Huazhong University of Science and Technology, 2022.
[36]   ZAPOTECAS MARTíNEZ S, COELLO COELLO C A. A multi-objective particle swarm optimizer based on decomposition [C]// Proceedings of the 13th Annual Conference on Genetic and Evolutionary Computation . Dublin: [s. n.], 2011: 69−76.
[37]   ZHAN Z H, LI J, CAO J, et al Multiple populations for multiple objectives: a coevolutionary technique for solving multiobjective optimization problems[J]. IEEE Transactions on Cybernetics, 2013, 43 (2): 445- 463
doi: 10.1109/TSMCB.2012.2209115
[38]   DAI C, WANG Y, YE M A new multi-objective particle swarm optimization algorithm based on decomposition[J]. Information Sciences, 2015, 325: 541- 557
doi: 10.1016/j.ins.2015.07.018
[39]   黄佩秋, 刘建昌, 谭树彬, 等 混合多目标粒子群优化算法在热精轧负荷分配优化中的应用[J]. 控制理论与应用, 2017, 34 (1): 93- 100
HUANG Peiqiu, LIU Jianchang, TAN Shubin, et al Application of the hybrid multi-objective particle swarm optimization algorithm in load distribution of hot finishing mills[J]. Control Theory and Applications, 2017, 34 (1): 93- 100
[40]   韩红桂, 阿音嘎, 张璐, 等 自适应分解式多目标粒子群优化算法[J]. 电子学报, 2020, 48 (7): 1245- 1254
HAN Honggui, A Yinga, ZHANG Lu, et al Adaptive multiobjective particle swarm optimization based on decomposition archive[J]. Acta Electronica Sinica, 2020, 48 (7): 1245- 1254
doi: 10.3969/j.issn.0372-2112.2020.07.001
[41]   CHEN L, LIU H L A region decomposition-based multi-objective particle swarm optimization algorithm[J]. International Journal of Pattern Recognition and Artificial Intelligence, 2014, 28 (8): 1459009
doi: 10.1142/S0218001414590095
[42]   LUO J, QI Y, XIE J, et al A hybrid multi-objective PSO-EDA algorithm for reservoir flood control operation[J]. Applied Soft Computing, 2015, 34: 526- 538
doi: 10.1016/j.asoc.2015.05.036
[43]   YAO G, DING Y, JIN Y, et al Endocrine-based coevolutionary multi-swarm for multi-objective workflow scheduling in a cloud system[J]. Soft Computing, 2017, 21 (15): 4309- 4322
[44]   YANG Y, ZHANG T, YI W, et al Deployment of multistatic radar system using multi-objective particle swarm optimisation[J]. IET Radar, Sonar and Navigation, 2018, 12 (5): 485- 493
doi: 10.1049/iet-rsn.2017.0351
[45]   张庆科, 孟祥旭, 张化祥, 等 基于随机维度划分与学习的粒子群优化算法[J]. 浙江大学学报: 工学版, 2018, 52 (2): 367- 378
ZHANG Qingke, MENG Xiangxu, ZHANG Huaxiang, et al Particle swarm optimization based on random vector partition and learning[J]. Journal of Zhejiang University: Engineering Science, 2018, 52 (2): 367- 378
[46]   ZHANG W, LI G, ZHANG W, et al A cluster based PSO with leader updating mechanism and ring-topology for multimodal multi-objective optimization[J]. Swarm and Evolutionary Computation, 2019, 50: 100569
doi: 10.1016/j.swevo.2019.100569
[47]   刘彬, 刘泽仁, 赵志彪, 等 基于速度交流的多种群多目标粒子群算法研究[J]. 计量学报, 2020, 41 (8): 1002- 1011
LIU Bin, LIU Zeren, ZHAO Zhibiao, et al Research on multi-population multi-objective particle swarm optimization algorithm based on velocity communication[J]. Acta Metrologica Sinica, 2020, 41 (8): 1002- 1011
doi: 10.3969/j.issn.1000-1158.2020.08.18
[48]   QI Y, MA X, LIU F, et al MOEA/D with adaptive weight adjustment[J]. Evolution Computation, 2014, 22 (2): 231- 264
doi: 10.1162/EVCO_a_00109
[49]   LIN Q, LI J, DU Z, et al A novel multi-objective particle swarm optimization with multiple search strategies[J]. European Journal of Operational Research, 2015, 247 (3): 732- 744
[50]   ZHU Q, LIN Q, CHEN W, et al An external archive-guided multiobjective particle swarm optimization algorithm[J]. IEEE Transactions on Cybernetics, 2017, 47 (9): 2794- 2808
doi: 10.1109/TCYB.2017.2710133
[51]   杨景明, 侯新培, 崔慧慧, 等 基于融合多策略改进的多目标粒子群优化算法[J]. 控制与决策, 2018, 33 (2): 226- 234
YANG Jingming, HOU Xinpei, CUI Huihui, et al Muti-objective adoptive chaotic particle swarm optimization algorithm[J]. Control and Decision, 2018, 33 (2): 226- 234
[52]   KNOWLES J D, COME D W Approximating the nondominated front using the Pareto archived evolution strategy[J]. Evolutionary Computation, 2000, 8 (2): 149- 172
doi: 10.1162/106365600568167
[53]   李笠, 王万良, 徐新黎, 等 基于网格排序的多目标粒子群优化算法[J]. 计算机研究与发展, 2017, 54 (5): 1012- 1023
LI Li, WANG Wanliang, XU Xinli, et al Muti-objective particle swarm optimization based on grid ranking[J]. Journal of Computer Research and Development, 2017, 54 (5): 1012- 1023
[54]   LI L, WANG W, XU X Multi-objective particle swarm optimization based on global margin ranking[J]. Information Sciences, 2017, 375: 30- 47
doi: 10.1016/j.ins.2016.08.043
[55]   李笠. 基于排序策略的多目标粒子群优化: 研究与应用[D]. 杭州: 浙江工业大学, 2017.
LI Li. Ranking-based multi-objective particle swarm optimization: rearch and application [D]. Hangzhou: Zhejiang University of Technology, 2017.
[56]   LENG R, OUYANG A, LIU Y, et al A multi-objective particle swarm optimization based on grid distance[J]. International Journal of Pattern Recognition and Artificial Intelligence, 2019, 34 (3): 2059008
[57]   ZOU K, LIU Y, WANG S, et al A multiobjective particle swarm optimization algorithm based on grid technique and multistrategy[J]. Journal of Mathematics, 2021, 2021: 1- 17
[58]   吴耀威, 刘衍民 基于网格密度的混合新型多目标粒子群算法[J]. 遵义师范学院学报, 2021, 23 (5): 67- 71
WU Yaowei, LIU Yanmin Hybrid new multi-objective particle swarm optimization algorithm based on grid density[J]. Journal of Zunyi Normal University, 2021, 23 (5): 67- 71
doi: 10.3969/j.issn.1009-3583.2021.05.018
[59]   YE Q, WANG Z, ZHAO Y, et al A clustering-based competitive particle swarm optimization with grid ranking for multi-objective optimization problems[J]. Scientific Reports, 2023, 13 (1): 11754
doi: 10.1038/s41598-023-38529-4
[60]   LI G, WANG W, ZHANG W, et al Grid search based multi-population particle swarm optimization algorithm for multimodal multi-objective optimization[J]. Swarm and Evolutionary Computation, 2021, 62: 100843
doi: 10.1016/j.swevo.2021.100843
[61]   王学武, 魏建斌, 周昕, 等 一种基于超体积指标的多目标进化算法[J]. 华东理工大学学报: 自然科学版, 2020, 46 (6): 780- 791
WANG Xuewu, WEI Jianbin, ZHOU Xin, et al Hypervolume-based multi-objective evolutionary algorithm[J]. Journal of East China University of Science and Technology, 2020, 46 (6): 780- 791
[62]   郝秦霞 基于R2指标的高维多目标差分进化推荐式课程系统[J]. 计算机应用, 2020, 40 (10): 2951- 2959
HAO Qinxia Course recommendation system based on R2 index and multi-objective differential evolution[J]. Journal of Computer Applications, 2020, 40 (10): 2951- 2959
[63]   GARCíA I C, COELLO C A C, ARIAS-MONTAñO A. MOPSOhv: a new hypervolume-based multi-objective particle swarm optimizer [C]// Proceedings of the 2014 IEEE Congress on Evolutionary Computation . Beijing: IEEE, 2014: 266−273.
[64]   LIANG X, DAI C, YE N. Multi-objective particle swarm optimization algorithm based on decomposition and hypervolume for synthesis gas production [C]// Proceedings of the 2022 18th International Conference on Computational Intelligence and Security . Chengdu: IEEE, 2022: 356−359.
[65]   LI F, LIU J, TAN S, et al. R2-MOPSO: a multi-objective particle swarm optimizer based on R2-indicator and decomposition [C]// Proceedings of the 2015 IEEE congress on evolutionary computation . Sendai: IEEE, 2015: 3148−3155.
[66]   WEI L X, LI X, FAN R, et al A hybrid multiobjective particle swarm optimization algorithm based on R2 indicator[J]. IEEE Access, 2018, 6: 14710- 14721
doi: 10.1109/ACCESS.2018.2812701
[67]   LI X, LI X L, WANG K, et al A multi-objective particle swarm optimization algorithm based on enhanced selection[J]. IEEE Access, 2019, 7: 168091- 168103
doi: 10.1109/ACCESS.2019.2954542
[68]   LIU J, LI F, KONG X, et al Handling many-objective optimisation problems with R2 indicator and decomposition-based particle swarm optimiser[J]. International Journal of Systems Science, 2019, 50 (2): 320- 336
doi: 10.1080/00207721.2018.1552765
[69]   李飞, 吴紫恒, 刘阚蓉, 等 基于R2指标和目标空间分解的高维多目标粒子群优化算法[J]. 控制与决策, 2021, 36 (9): 2085- 2094
LI Fei, WU Ziheng, LIU Kanrong, et al R2 indicator and objective space partition based many-objective particle swarm optimizer[J]. Control and Decision, 2021, 36 (9): 2085- 2094
[70]   GU Q, JIANG M, JIANG S, et al Multi-objective particle swarm optimization with R2 indicator and adaptive method[J]. Complex and Intelligent Systems, 2021, 7 (5): 2697- 2710
doi: 10.1007/s40747-021-00445-3
[71]   SUN X, CHEN Y, LIU Y, et al Indicator-based set evolution particle swarm optimization for many-objective problems[J]. Soft Computing, 2016, 20 (6): 2219- 2232
doi: 10.1007/s00500-015-1637-1
[72]   WU B, HU W, HE Z, et al. A many-objective particle swarm optimization based on virtual Pareto front [C]// Proceedings of the 2018 IEEE Congress on Evolutionary Computation . Rio de Janeiro: IEEE, 2018: 1−8.
[73]   LUO J, HUANG X, YANG Y, et al A many-objective particle swarm optimizer based on indicator and direction vectors for many-objective optimization[J]. Information Sciences, 2020, 514: 166- 202
doi: 10.1016/j.ins.2019.11.047
[74]   ZAPOTECAS-MARTíNEZ S, LóPEZ-JAIMES A, GARCíA-NáJERA A LIBEA: a lebesgue indicator-based evolutionary algorithm for multi-objective optimization[J]. Swarm and Evolutionary Computation, 2019, 44: 404- 419
doi: 10.1016/j.swevo.2018.05.004
[75]   KAWAGUCHI S, FUKUYAMA Y Improved parallel reactive hybrid particle swarm optimization using improved neighborhood schedule generation method for the integrated framework of optimal production scheduling and operational planning of an energy plant in a factory[J]. Electronics and Communications in Japan, 2020, 103 (7): 37- 48
doi: 10.1002/ecj.12237
[76]   SECK-TUOH-MORA J C, MEDINA-MARIN J, MARTINEZ-GOMEZ E S, et al Cellular particle swarm optimization with a simple adaptive local search strategy for the permutation flow shop scheduling problem[J]. Archives of Control Sciences, 2019, 29 (2): 205- 226
[77]   KAYA S, GüMüŞçü A, AYDILEK İ B, et al Solution for flow shop scheduling problems using chaotic hybrid firefly and particle swarm optimization algorithm with improved local search[J]. Soft Computing, 2021, 25 (10): 7143- 7154
doi: 10.1007/s00500-021-05673-w
[78]   谢美华, 李艳武, 葛棚丹 自适应混合粒子群算法求解置换流水车间调度问题[J]. 计算机应用研究, 2023, 40 (11): 1- 8
XIE Meihua, LI Yanwu, GE Pengdan Self-adaptive hybrid particle swarm optimization for permutation flow shop scheduling problem[J]. Application Research of Computers, 2023, 40 (11): 1- 8
[79]   LONG F, JIN B, XU H, et al Research on multi-objective optimization of smart grid based on particle swarm optimization[J]. Electrica, 2023, 23 (2): 222- 230
[80]   XIONG W, GONG K, SHI W, et al Design and implementation of power mobile inspection system based on improved particle swarm optimization[J]. Journal of Physics: Conference Series, 2021, 2033 (1): 012202
doi: 10.1088/1742-6596/2033/1/012202
[81]   DENG C, ZHANG X, HUANG Y, et al Equipping seasonal exponential smoothing models with particle swarm optimization algorithm for electricity consumption forecasting[J]. Energies, 2021, 14 (13): 4036
doi: 10.3390/en14134036
[82]   KAHOULI O, ALSAIF H, BOUTERAA Y, et al Power system reconfiguration in distribution network for improving reliability using genetic algorithm and particle swarm optimization[J]. Applied Sciences, 2021, 11 (7): 3092
doi: 10.3390/app11073092
[83]   DUMITRU D, DIOȘAN L, ANDREICA A, et al A transfer learning approach on the optimization of edge detectors for medical images using particle swarm optimization[J]. Entropy, 2021, 23 (4): 414
doi: 10.3390/e23040414
[84]   HASSAN M U, BADSHAH N, ZAHIR S Automatic initialization for active contour models based on particle swarm optimization and application to medical images[J]. Journal of Mathematical and Computational Science, 2021, 11 (1): 243- 264
[85]   井向阳, 林鹏飞, 游进军, 等 岷江中游梯级闸坝联合优化调度研究[J]. 水利水电技术, 2021, 52 (3): 23- 31
JIN Xiangyang, LIN Pengfei, YOU Jinjun, et al Study on joint optimal scheduling of cascade sluice on mid-Minjiang River[J]. Water Resources and Hydropower Engineering, 2021, 52 (3): 23- 31
[1] Xue-yang TANG,Xiao-pei CAI,Wei-hua WANG,Wen-hao CHANG,Qi-hao WANG. Rail corrugation recognition based on particle probabilistic neural network algorithm[J]. Journal of ZheJiang University (Engineering Science), 2023, 57(9): 1766-1774.
[2] Dian-hai WANG,Rui XIE,Zheng-yi CAI. Prediction of urban interrupted traffic flow based on optimal convergence time interval[J]. Journal of ZheJiang University (Engineering Science), 2023, 57(8): 1607-1617.
[3] Xiao-yan CAO,Min YU,Jin ZHOU,Yun-zhi WANG. Multi-objective optimization design of adjustable rotary fluid damper parameter[J]. Journal of ZheJiang University (Engineering Science), 2023, 57(7): 1439-1449.
[4] Tong-li CHANG,Wan-bin FU. Human-robot matching design of self-aligning artificial knee joint[J]. Journal of ZheJiang University (Engineering Science), 2023, 57(4): 753-759.
[5] Ting-fang YU,Ling SONG. Performance analysis and optimization of supercritical CO2 Brayton cycle waste heat recovery system[J]. Journal of ZheJiang University (Engineering Science), 2023, 57(2): 404-414.
[6] Wan-liang WANG,Zhong-kui CHEN,Fei WU,Zheng WANG,Meng-jiao YU. Dynamic multi-objective optimization algorithm based on individual prediction[J]. Journal of ZheJiang University (Engineering Science), 2023, 57(11): 2133-2146.
[7] Zhi-tong TAO,Jian-feng TAO,Cheng-jin QIN,Cheng-liang LIU. Trajectory planning of TBM disc cutter changing robot based on time-jerk optimization[J]. Journal of ZheJiang University (Engineering Science), 2023, 57(1): 1-9.
[8] Ya-guang ZHU,Chun-chao LIU,Liang ZHANG. Behavior control of hexapod robot based on virtual motoneuron network[J]. Journal of ZheJiang University (Engineering Science), 2022, 56(6): 1107-1118.
[9] Meng-qing HONG,Meng DING,Xiu-tao GU,Yu GUO. Fixed time trajectory tracking control for dual-arm space robot[J]. Journal of ZheJiang University (Engineering Science), 2022, 56(6): 1168-1174.
[10] Wan-liang WANG,Ya-wen JIN,Jia-cheng CHEN,Guo-qing LI,Ming-zhi HU,Jian-hang DONG. Multi-objective particle swarm optimization algorithm with multi-role and multi-strategy[J]. Journal of ZheJiang University (Engineering Science), 2022, 56(3): 531-541.
[11] Rui YU,Xue-feng XU,Hua ZHOU,Hua-yong YANG. Improved switching-gain adaptation based sliding mode control for trajectory tracking of underactuated unmanned surface vessels[J]. Journal of ZheJiang University (Engineering Science), 2022, 56(3): 436-443.
[12] Jun-heng XU,Xiao-jun YANG,Bing LI. Design of wing mechanism with variable camber based on cross-spring flexural pivots[J]. Journal of ZheJiang University (Engineering Science), 2022, 56(3): 444-451, 509.
[13] Jie-guang HE,Zhi-ping PENG,De-long CUI,Qi-rui LI. Adaptive crow search algorithm based on population diversity[J]. Journal of ZheJiang University (Engineering Science), 2022, 56(12): 2426-2435.
[14] Dong-yang HAN,Ze-yu LIN,Yu ZHENG,Mei-mei ZHENG,Tang-bin XIA. Remaining useful life estimation of turbofan engine based on selective ensemble of deep neural networks[J]. Journal of ZheJiang University (Engineering Science), 2022, 56(11): 2109-2118.
[15] Qi-lin DENG,Juan LU,Yong-hui CHEN,Jian FENG,Xiao-ping LIAO,Jun-yan MA. Optimization method of CNC milling parameters based on deep reinforcement learning[J]. Journal of ZheJiang University (Engineering Science), 2022, 56(11): 2145-2155.