{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T06:40:26Z","timestamp":1781073626296,"version":"3.54.1"},"reference-count":31,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2017,1,20]],"date-time":"2017-01-20T00:00:00Z","timestamp":1484870400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Nature Science Foundation of China","award":["No. 51575264"],"award-info":[{"award-number":["No. 51575264"]}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["No.NS2015050"],"award-info":[{"award-number":["No.NS2015050"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100013088","name":"Qing Lan Project","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100013088","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A novel orthogonal multi-swarm cooperative particle swarm optimization (PSO) algorithm with a particle trajectory knowledge base is presented in this paper. Different from the traditional PSO algorithms and other variants of PSO, the proposed orthogonal multi-swarm cooperative PSO algorithm not only introduces an orthogonal initialization mechanism and a particle trajectory knowledge base for multi-dimensional optimization problems, but also conceives a new adaptive cooperation mechanism to accomplish the information interaction among swarms and particles. Experiments are conducted on a set of benchmark functions, and the results show its better performance compared with traditional PSO algorithm in aspects of convergence, computational efficiency and avoiding premature convergence.<\/jats:p>","DOI":"10.3390\/sym9010015","type":"journal-article","created":{"date-parts":[[2017,1,20]],"date-time":"2017-01-20T10:10:12Z","timestamp":1484907012000},"page":"15","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["An Orthogonal Multi-Swarm Cooperative PSO Algorithm with a Particle Trajectory Knowledge Base"],"prefix":"10.3390","volume":"9","author":[{"given":"Jun","family":"Yang","sequence":"first","affiliation":[{"name":"College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5967-8914","authenticated-orcid":false,"given":"Haihua","family":"Zhu","sequence":"additional","affiliation":[{"name":"College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yingcong","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2017,1,20]]},"reference":[{"key":"ref_1","unstructured":"Eberhart, R., and Kennedy, J. (1995, January 4\u20136). A new optimizer using particle swarm theory. Proceedings of the International Symposium on MICRO Machine and Human Science, Nagoya, Japan."},{"key":"ref_2","unstructured":"Kennedy, J., and Eberhart, R. (December, January 27). Particle swarm optimization. Proceedings of the IEEE International Conference on Neural Networks, Perth, WA, Australia."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"432","DOI":"10.5545\/sv-jme.2015.2478","article-title":"Sensitivity-based Multidisciplinary Optimal Design of a Hydrostatic Rotary Table with Particle Swarm Optimization","volume":"61","author":"Cheng","year":"2015","journal-title":"Strojniski Vestnik"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2781","DOI":"10.1007\/s00500-015-1681-x","article-title":"Quantum-inspired firefly algorithm with particle swarm optimization for discrete optimization problems","volume":"20","author":"Zouache","year":"2016","journal-title":"Soft Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"626","DOI":"10.1109\/JPHOTOV.2013.2297513","article-title":"A Maximum Power Point Tracking Method Based on Perturb-and-Observe Combined With Particle Swarm Optimization","volume":"4","author":"Lian","year":"2014","journal-title":"IEEE J. Photovolt."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Bonyadi, M.R., and Michalewicz, Z. (2015). Locating Potentially Disjoint Feasible Regions of a Search Space with a Particle Swarm Optimizer, Springer.","DOI":"10.1007\/978-81-322-2184-5_8"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1002\/cplx.21601","article-title":"A hybrid particle swarm optimization and bacterial foraging for power system stability enhancement","volume":"21","author":"Ali","year":"2015","journal-title":"Complexity"},{"key":"ref_8","unstructured":"Tran, Q.A., Quan, D.D., and Jiang, F. (2015). Knowledge and Systems Engineering, Springer International Publishing."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2144","DOI":"10.1016\/j.asoc.2012.12.020","article-title":"A novel multi-swarm algorithm for optimization in dynamic environments based on particle swarm optimization","volume":"13","author":"Yazdani","year":"2013","journal-title":"Appl. Soft Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/j.engappai.2015.06.013","article-title":"A novel parallel multi-swarm algorithm based on comprehensive learning particle swarm optimization","volume":"45","author":"Kodaz","year":"2015","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Dey, S., Bhattacharyya, S., and Maulik, U. (2014, January 14\u201316). Quantum Behaved Multi-objective PSO and ACO Optimization for Multi-level Thresholding. Proceedings of the International Conference on Computational Intelligence and Communication Networks, Bhopal, India.","DOI":"10.1109\/CICN.2014.63"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"662","DOI":"10.4028\/www.scientific.net\/AMM.433-435.662","article-title":"A Study on Particle Trajectory of Particle Swarm Optimization","volume":"433\u2013435","author":"An","year":"2013","journal-title":"Appl. Mech. Mater."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"814","DOI":"10.1109\/TEVC.2015.2508101","article-title":"Stability analysis of the particle swarm optimization without stagnation assumption","volume":"20","author":"Bonyadi","year":"2016","journal-title":"IEEE Trans. Evolut. Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.asoc.2008.03.001","article-title":"Particle swarm optimization with adaptive population size and its application","volume":"9","author":"Chen","year":"2009","journal-title":"Appl. Soft Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1587\/nolta.6.181","article-title":"Particle swarm optimization with switched topology","volume":"6","author":"Sano","year":"2015","journal-title":"Nonlinear Theory Its Appl. Ieice"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"18508","DOI":"10.4236\/jbise.2012.54020","article-title":"A new hybrid particle swarm optimization for multimodal brain image registration","volume":"5","author":"Ayatollahi","year":"2012","journal-title":"J. Biomed. Sci. Eng."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1109\/TEVC.2004.826069","article-title":"A Cooperative approach to particle swarm optimization","volume":"8","author":"Engelbrecht","year":"2004","journal-title":"IEEE Trans. Evolut. Comput."},{"key":"ref_18","unstructured":"Liang, J.J., and Suganthan, P.N. (2005, January 8\u201310). Dynamic multi-swarm particle swarm optimizer. Proceedings of the Swarm Intelligence Symposium, Pasadena, CA, USA."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Cheng, S., and Shi, Y. (2011, January 12\u201315). Diversity control in particle swarm optimization. Proceedings of the Second International Conference on Swarm Intelligence, Chongqing, China.","DOI":"10.1109\/SIS.2011.5952581"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1016\/S0020-0190(02)00447-7","article-title":"The particle swarm optimization algorithm: convergence analysis and parameter selection","volume":"85","author":"Trelea","year":"2003","journal-title":"Inf. Process. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.asoc.2015.03.036","article-title":"An integrated PSO for parameter determination and feature selection of ELM and its application in classification of power system disturbances","volume":"32","author":"Ahila","year":"2015","journal-title":"Appl. Soft Comput."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Cazzaniga, P., Nobile, M.S., and Besozzi, D. (2015, January 12\u201315). The impact of particles initialization in PSO: Parameter estimation as a case in point. Proceedings of the IEEE Conference on Computational Intelligence in Bioinformatics and Computational Biology, Niagara Falls, ON, Canada.","DOI":"10.1109\/CIBCB.2015.7300288"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1109\/TEVC.2005.857610","article-title":"Comprehensive learning particle swarm optimizer for global optimization of multimodal functions","volume":"10","author":"Liang","year":"2006","journal-title":"IEEE Trans. Evolut. Comput."},{"key":"ref_24","unstructured":"Branke, J., Kau\u00dfler, T., Smidt, C., and Schmeck, H. (2000). Evolutionary Design and Manufacture, Springer."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Li, C., and Yang, S. (2008, January 18\u201320). Fast Multi-Swarm Optimization for Dynamic Optimization Problems. Proceedings of the 2008 Fourth International Conference on Natural Computation, ICNC \u201908, Jinan, China.","DOI":"10.1109\/ICNC.2008.313"},{"key":"ref_26","unstructured":"Mohais, A.S. (2008). Random Dynamic Neighborhood Structures in Particle Swarm Optimization. [Ph.D. Thesis, University of the West Indies]."},{"key":"ref_27","unstructured":"Huang, T., and Mohan, A.S. (2005, January 4\u20137). Significance of neighborhood topologies for the reconstruction of microwave images using particle swarm optimization. Proceedings of the 2005 Asia-Pacific Microwave Conference, Suzhou International Conference Center, Suzhou, China."},{"key":"ref_28","unstructured":"Kennedy, J. (1999, January 6\u20139). Small worlds and mega-minds: Effects of neighborhood topology on particle swarm performance. Proceedings of the 1999 Congress on Evolutionary Computation, CEC \u201999, Washington, DC, USA."},{"key":"ref_29","unstructured":"Kennedy, J., and Mendes, R. (2002, January 12\u201317). Population structure and particle swarm performance. Proceedings of the 2002 Congress on Evolutionary Computation, CEC \u201902, Honolulu, HI, USA."},{"key":"ref_30","unstructured":"Hu, X., and Eberhart, R. (2002, January 12\u201317). Multiobjective optimization using dynamic neighborhood particle swarm optimization. Proceedings of the 2002 Congress on Evolutionary Computation, CEC \u201902, Honolulu, HI, USA."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"192","DOI":"10.5897\/AJPP2015.4464","article-title":"Orthogonal test design for optimization of the isolation and purification of total favonoids from Artemisia frigida Willd using macroporous resin chromatography","volume":"10","author":"Wang","year":"2016","journal-title":"Afr. J. Pharm. 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