{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T12:25:30Z","timestamp":1776342330396,"version":"3.51.2"},"reference-count":46,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2021,6,20]],"date-time":"2021-06-20T00:00:00Z","timestamp":1624147200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Open Foundation of Key Laboratory in Software Engineering of Yunnan Province","award":["2020SE307"],"award-info":[{"award-number":["2020SE307"]}]},{"name":"Open Foundation of Key Laboratory in Software Engineering of Yunnan Province","award":["2015SE204"],"award-info":[{"award-number":["2015SE204"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Swarm-based algorithm can successfully avoid the local optimal constraints, thus achieving a smooth balance between exploration and exploitation. Salp swarm algorithm (SSA), as a swarm-based algorithm on account of the predation behavior of the salp, can solve complex daily life optimization problems in nature. SSA also has the problems of local stagnation and slow convergence rate. This paper introduces an improved salp swarm algorithm, which improve the SSA by using the chaotic sequence initialization strategy and symmetric adaptive population division. Moreover, a simulated annealing mechanism based on symmetric perturbation is introduced to enhance the local jumping ability of the algorithm. The improved algorithm is referred to SASSA. The CEC standard benchmark functions are used to evaluate the efficiency of the SASSA and the results demonstrate that the SASSA has better global search capability. SASSA is also applied to solve engineering optimization problems. The experimental results demonstrate that the exploratory and exploitative proclivities of the proposed algorithm and its convergence patterns are vividly improved.<\/jats:p>","DOI":"10.3390\/sym13061092","type":"journal-article","created":{"date-parts":[[2021,6,20]],"date-time":"2021-06-20T21:50:15Z","timestamp":1624225815000},"page":"1092","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["Improved Salp Swarm Algorithm with Simulated Annealing for Solving Engineering Optimization Problems"],"prefix":"10.3390","volume":"13","author":[{"given":"Qing","family":"Duan","sequence":"first","affiliation":[{"name":"School of Software, Yunnan University, Kunming 650000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lu","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Software, Yunnan University, Kunming 650000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5466-7092","authenticated-orcid":false,"given":"Hongwei","family":"Kang","sequence":"additional","affiliation":[{"name":"School of Software, Yunnan University, Kunming 650000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yong","family":"Shen","sequence":"additional","affiliation":[{"name":"School of Software, Yunnan University, Kunming 650000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xingping","family":"Sun","sequence":"additional","affiliation":[{"name":"School of Software, Yunnan University, Kunming 650000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qingyi","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Software, Yunnan University, Kunming 650000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,6,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1515\/REVCE.2000.16.1.1","article-title":"Applications of multiobjective optimization in chemical engineering","volume":"16","author":"Bhaskar","year":"2000","journal-title":"Rev. Chem. Eng."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Sobieszczanski-Sobieski, J. (1995). Multidisciplinary Design Optimization: An Emerging New Engineering Discipline. Advances in Structural Optimization, Springer.","DOI":"10.1007\/978-94-011-0453-1_14"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Kondratenko, Y.P., and Simon, D. (2018). Structural and parametric optimization of fuzzy control and decision making systems. Recent Developments and the New Direction in Soft-Computing Foundations and Applications, Springer.","DOI":"10.1007\/978-3-319-75408-6_22"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/s10994-006-6889-7","article-title":"The max-min hill-climbing Bayesian network structure learning algorithm","volume":"65","author":"Tsamardinos","year":"2006","journal-title":"Mach. Learn."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1057\/palgrave.jors.2602535","article-title":"A tabu search algorithm for the training of neural networks","volume":"60","author":"Dengiz","year":"2009","journal-title":"J. Oper. Res. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1023\/A:1022602019183","article-title":"Genetic Algorithms and Machine Learning","volume":"3","author":"Goldberg","year":"1988","journal-title":"Mach. Learn."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"855","DOI":"10.1016\/j.eswa.2014.08.018","article-title":"Back propagation neural network with adaptive differential evolution algorithm for time series forecasting","volume":"42","author":"Wang","year":"2015","journal-title":"Expert Syst. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Kennedy, J., and Eberhart, R. (1995). Particle swarm optimization. Proceedings of ICNN\u201995-International Conference on Neural Networks, IEEE.","DOI":"10.1109\/ICNN.1995.488968"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Dorigo, M., and Di Caro, G. (1999). Ant colony optimization: A new meta-heuristic. Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406), IEEE.","DOI":"10.1109\/CEC.1999.782657"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.advengsoft.2016.01.008","article-title":"The Whale Optimization Algorithm","volume":"95","author":"Mirjalili","year":"2016","journal-title":"Adv. Eng. Softw."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1016\/j.advengsoft.2013.12.007","article-title":"Grey wolf optimizer","volume":"69","author":"Mirjalili","year":"2014","journal-title":"Adv. Eng. Softw."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1007\/s10898-007-9149-x","article-title":"A powerful and efficient algorithm for numerical function optimization: Artificial bee colony (ABC) algorithm","volume":"39","author":"Karaboga","year":"2007","journal-title":"J. Glob. Optim."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Sun, X., Wang, Y., Kang, H., Shen, Y., Chen, Q., and Wang, D. (2021). Modified Multi-Crossover Operator NSGA-III for Solving Low Carbon Flexible Job Shop Scheduling Problem. Processes, 9.","DOI":"10.3390\/pr9010062"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Liu, Y., Zhang, Z., Bo, L., and Zhu, D. (2021). Multi-Objective Optimization of a Mine Water Reuse System Based on Improved Particle Swarm Optimization. Sensors, 21.","DOI":"10.3390\/s21124114"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Shen, Y., Liang, Z., Kang, H., Sun, X., and Chen, Q. (2020). A Modified jSO Algorithm for Solving Constrained Engineering Problems. Symmetry, 13.","DOI":"10.3390\/sym13010063"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1016\/j.advengsoft.2017.07.002","article-title":"Salp Swarm Algorithm: A bio-inspired optimizer for engineering design problems","volume":"114","author":"Mirjalili","year":"2017","journal-title":"Adv. Eng. Softw."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3462","DOI":"10.1007\/s10489-018-1158-6","article-title":"A novel chaotic salp swarm algorithm for global optimization and feature selection","volume":"48","author":"Sayed","year":"2018","journal-title":"Appl. Intell."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3155","DOI":"10.1007\/s12652-018-1031-9","article-title":"Improved salp swarm algorithm based on particle swarm optimization for feature selection","volume":"10","author":"Ibrahim","year":"2019","journal-title":"J. Ambient. Intell. Humaniz. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.knosys.2018.05.009","article-title":"An efficient binary Salp Swarm Algorithm with crossover scheme for feature selection problems","volume":"154","author":"Faris","year":"2018","journal-title":"Knowl.-Based Syst."},{"key":"ref_20","first-page":"854","article-title":"The improved salp swarm algorithm is used to solve the engineering optimization design problem","volume":"4","author":"Liu","year":"2021","journal-title":"J. Syst. Simul."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2743","DOI":"10.1007\/s13369-019-04132-x","article-title":"Improved Salp Swarm Algorithm with Space Transformation Search for Training Neural Network","volume":"45","author":"Panda","year":"2019","journal-title":"Arab. J. Sci. Eng."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1007\/s007780050040","article-title":"Heuristic and randomized optimization for the join ordering problem","volume":"6","author":"Steinbrunn","year":"1997","journal-title":"VLDB J."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"119548","DOI":"10.1016\/j.jclepro.2019.119548","article-title":"Sustainable design and optimization of coal supply chain network under different carbon emission policies","volume":"250","author":"Li","year":"2020","journal-title":"J. Clean. Prod."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"99740","DOI":"10.1109\/ACCESS.2020.2997783","article-title":"Improved Salp Swarm Algorithm Based on Levy Flight and Sine Cosine Operator","volume":"8","author":"Zhang","year":"2020","journal-title":"IEEE Access"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Haupt, R.L., and Haupt, S.E. (2004). Practical Genetic Algorithms, Wiley.","DOI":"10.1002\/0471671746"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"136452","DOI":"10.1109\/ACCESS.2019.2933265","article-title":"A Comprehensive Improved Salp Swarm Algorithm on Redundant Container Deployment Problem","volume":"7","author":"Ma","year":"2019","journal-title":"IEEE Access"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1126\/science.220.4598.671","article-title":"Optimization by Simulated Annealing","volume":"220","author":"Kirkpatrick","year":"1983","journal-title":"Science"},{"key":"ref_28","unstructured":"Li, J., and Liu, F. (2012, January 7\u20139). A trifocal tensor calculation method based on simulated annealing algorithm. Proceedings of the International Conference on Information Science and Control Engineering (ICISCE), Shenzhen, China."},{"key":"ref_29","first-page":"341","article-title":"Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization","volume":"174","author":"Suganthan","year":"2005","journal-title":"KanGAL Rep."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"31243","DOI":"10.1109\/ACCESS.2019.2902306","article-title":"Chaos-Induced and Mutation-Driven Schemes Boosting Salp Chains-Inspired Optimizers","volume":"7","author":"Zhang","year":"2019","journal-title":"IEEE Access"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1016\/j.knosys.2015.07.006","article-title":"Moth-flame optimization algorithm: A novel nature-inspired heuristic paradigm","volume":"89","author":"Mirjalili","year":"2015","journal-title":"Knowl. Based Syst."},{"key":"ref_32","first-page":"1","article-title":"Statistical comparisons of classifiers over multiple data sets","volume":"7","year":"2006","journal-title":"J. Mach. Learn. Res."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Sun, X., Jiang, L., Shen, Y., Kang, H., and Chen, Q. (2020). Success History-Based Adaptive Differential Evolution Using Turning-Based Mutation. Mathematics, 8.","DOI":"10.3390\/math8091565"},{"key":"ref_34","first-page":"119","article-title":"A new evolutionary algorithm for solving constrained optimization problems","volume":"37","author":"Wang","year":"2006","journal-title":"J. Cent. South Univ. (Sci. Technol.)"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1109\/LCSYS.2018.2889963","article-title":"A Distributed Method for Linear Programming Problems With Box Constraints and Time-Varying Inequalities","volume":"3","author":"Hosseinzadeh","year":"2018","journal-title":"IEEE Control. Syst. Lett."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1932","DOI":"10.1109\/TAC.2018.2867359","article-title":"Embedding Constrained Model Predictive Control in a Continuous-Time Dynamic Feedback","volume":"64","author":"Nicotra","year":"2018","journal-title":"IEEE Trans. Autom. Control."},{"key":"ref_37","first-page":"37","article-title":"Application of Chaotic Simulated Annealing Algorithm in Numerical Fuction Optimization","volume":"38","author":"Xu","year":"2010","journal-title":"Comput. Digit. Eng."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"100693","DOI":"10.1016\/j.swevo.2020.100693","article-title":"A test-suite of non-convex constrained optimization problems from the real-world and some baseline results","volume":"56","author":"Kumar","year":"2020","journal-title":"Swarm Evol. Comput."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1007\/s10489-014-0645-7","article-title":"How effective is the Grey Wolf optimizer in training multi-layer perceptrons","volume":"43","author":"Mirjalili","year":"2015","journal-title":"Appl. Intell."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1023\/A:1008202821328","article-title":"Differential Evolution\u2014A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces","volume":"11","author":"Storn","year":"1997","journal-title":"J. Glob. Optim."},{"key":"ref_41","first-page":"693","article-title":"Multi-layer perceptron using hybrid differential evolution and biogeography-based optimization","volume":"34","author":"Juan","year":"2017","journal-title":"Appl. Res. Comput."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Gandomi, A.H., and Yang, X.-S. (2011). Benchmark Problems in Structural Optimization. Computational Optimization, Methods and Algorithms, Springer.","DOI":"10.1007\/978-3-642-20859-1_12"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"3861","DOI":"10.1007\/s11042-017-4803-x","article-title":"Krill herd algorithm based on cuckoo search for solving engineering optimization problems","volume":"78","author":"Wang","year":"2019","journal-title":"Multimed. Tools Appl."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Andrei, N. (2013). Nonlinear Optimization Applications Using the GAMS Technology, Springer.","DOI":"10.1007\/978-1-4614-6797-7"},{"key":"ref_45","first-page":"2","article-title":"Computer points way to more profits","volume":"84","author":"Sauer","year":"1964","journal-title":"Hydrocarb. Process."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/j.ins.2014.02.123","article-title":"Chaotic Krill Herd algorithm","volume":"274","author":"Wang","year":"2014","journal-title":"Inf. Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/6\/1092\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:19:36Z","timestamp":1760163576000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/6\/1092"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,20]]},"references-count":46,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2021,6]]}},"alternative-id":["sym13061092"],"URL":"https:\/\/doi.org\/10.3390\/sym13061092","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,6,20]]}}}