{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,20]],"date-time":"2026-01-20T10:02:25Z","timestamp":1768903345980,"version":"3.49.0"},"reference-count":28,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,5,20]],"date-time":"2023-05-20T00:00:00Z","timestamp":1684540800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Faisal University","award":["3426"],"award-info":[{"award-number":["3426"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Differential evolution is an evolutionary algorithm that is used to solve complex numerical optimization problems. Differential evolution balances exploration and exploitation to find the best genes for the objective function. However, finding this balance is a challenging task. To overcome this challenge, we propose a clustering-based mutation strategy called Agglomerative Best Cluster Differential Evolution (ABCDE). The proposed model converges in an efficient manner without being trapped in local optima. It works by clustering the population to identify similar genes and avoids local optima. The adaptive crossover rate ensures that poor-quality genes are not reintroduced into the population. The proposed ABCDE is capable of generating a population efficiently where the difference between the values of the trial vector and objective vector is even less than 1% for some benchmark functions, and hence it outperforms both classical mutation strategies and the random neighborhood mutation strategy. The optimal and fast convergence of differential evolution has potential applications in the weight optimization of artificial neural networks and in stochastic and time-constrained environments such as cloud computing.<\/jats:p>","DOI":"10.3390\/sym15051120","type":"journal-article","created":{"date-parts":[[2023,5,22]],"date-time":"2023-05-22T01:31:46Z","timestamp":1684719106000},"page":"1120","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Differential Evolution and Agglomerative-Clustering-Based Mutation Strategy for Complex Numerical Optimization Problems"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-6196-7230","authenticated-orcid":false,"given":"Tassawar","family":"Ali","sequence":"first","affiliation":[{"name":"Department of Computer Science, COMSATS University Islamabad Wah Campus, Wah Cantt 47040, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8178-6652","authenticated-orcid":false,"given":"Hikmat Ullah","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Computer Science, COMSATS University Islamabad Wah Campus, Wah Cantt 47040, Pakistan"},{"name":"Department of Computer Science, Namal University, Mianwali 42001, Pakistan"}]},{"given":"Tasswar","family":"Iqbal","sequence":"additional","affiliation":[{"name":"Department of Computer Science, COMSATS University Islamabad Wah Campus, Wah Cantt 47040, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6598-6240","authenticated-orcid":false,"given":"Fawaz Khaled","family":"Alarfaj","sequence":"additional","affiliation":[{"name":"Department of Management Information Systems (MIS), School of Business, King Faisal University (KFU), Hofuf 31982, Al-Ahsa, Saudi Arabia"}]},{"given":"Abdullah Mohammad","family":"Alomair","sequence":"additional","affiliation":[{"name":"Department of Quantitative Methods, School of Business, King Faisal University (KFU), Hofuf 31982, Al-Ahsa, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2400-616X","authenticated-orcid":false,"given":"Naif","family":"Almusallam","sequence":"additional","affiliation":[{"name":"Department of Management Information Systems (MIS), School of Business, King Faisal University (KFU), Hofuf 31982, Al-Ahsa, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"108209","DOI":"10.1016\/j.knosys.2022.108209","article-title":"Differential Evolution with Two-Level Adaptive Mechanism for Numerical Optimization","volume":"241","author":"Yan","year":"2022","journal-title":"Knowl.-Based Syst."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3831","DOI":"10.1016\/j.aej.2021.09.013","article-title":"Differential Evolution: A Recent Review Based on State-of-the-Art Works","volume":"61","author":"Ahmad","year":"2022","journal-title":"Alex. Eng. J."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Strnad, I., and Marseti\u010d, R. (2023). Differential Evolution Based Numerical Variable Speed Limit Control Method with a Non-Equilibrium Traffic Model. Mathematics, 11.","DOI":"10.3390\/math11020265"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1016\/j.swevo.2018.03.008","article-title":"Algorithmic Design Issues in Adaptive Differential Evolution Schemes: Review and Taxonomy","volume":"43","author":"Neri","year":"2018","journal-title":"Swarm Evol. Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"103479","DOI":"10.1016\/j.engappai.2020.103479","article-title":"Differential Evolution: A Review of More than Two Decades of Research","volume":"90","author":"Pant","year":"2020","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_6","unstructured":"Price, K.V., Storn, R.M., and Lampinen, J.A. (2005). Differential Evolution. A Practical Approach to Global Optimization, Springer Science & Business Media."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"88517","DOI":"10.1109\/ACCESS.2019.2926422","article-title":"N-CODE: A Differential Evolution with n-Cauchy Operator for Global Numerical Optimization","volume":"7","author":"Deng","year":"2019","journal-title":"IEEE Access"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1353","DOI":"10.1109\/TCYB.2018.2801287","article-title":"Differential Evolution with Underestimation-Based Multimutation Strategy","volume":"49","author":"Zhou","year":"2019","journal-title":"IEEE Trans. Cybern."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2166","DOI":"10.1109\/TCYB.2017.2728725","article-title":"Distributed Differential Evolution Based on Adaptive Mergence and Split for Large-Scale Optimization","volume":"48","author":"Ge","year":"2018","journal-title":"IEEE Trans. Cybern."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/j.asoc.2017.04.018","article-title":"Hybrid Artificial Bee Colony Algorithm with Differential Evolution","volume":"58","author":"Jadon","year":"2017","journal-title":"Appl. Soft Comput. J."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1066","DOI":"10.1109\/TCBB.2014.2322360","article-title":"A Parameter Estimationmethod for Biological Systems Modelled by ODE\/DDE Models Using Splineapproximation and Differential Evolution Algorithm","volume":"11","author":"Zhan","year":"2014","journal-title":"IEEE\/ACM Trans. Comput. Biol. Bioinform."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1016\/j.ins.2014.02.039","article-title":"Power System Fault Diagnosis Based on History Driven Differential Evolution and Stochastic Time Domain Simulation","volume":"275","author":"Zhao","year":"2014","journal-title":"Inf. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.patrec.2015.07.007","article-title":"Simultaneous Feature Selection and Weighting\u2014An Evolutionary Multi-Objective Optimization Approach","volume":"65","author":"Paul","year":"2015","journal-title":"Pattern Recognit. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1016\/j.jocs.2017.07.010","article-title":"Enhancing Differential Evolution with Random Neighbors Based Strategy","volume":"26","author":"Peng","year":"2018","journal-title":"J. Comput. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1200","DOI":"10.1016\/j.asoc.2010.05.012","article-title":"A 2-Opt Based Differential Evolution for Global Optimization","volume":"10","author":"Chiang","year":"2010","journal-title":"Appl. Soft Comput. J."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"791","DOI":"10.1287\/opre.6.6.791","article-title":"A Method for Solving Traveling-Salesman Problems","volume":"6","author":"Croes","year":"1958","journal-title":"Oper. Res."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1109\/TEVC.2010.2083670","article-title":"Enhancing Differential Evolution Utilizing Proximity-Based Mutation Operators","volume":"15","author":"Epitropakis","year":"2011","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2205","DOI":"10.1016\/j.cam.2010.10.018","article-title":"Differential Evolution with Generalized Differentials","volume":"235","author":"Ali","year":"2011","journal-title":"J. Comput. Appl. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"390","DOI":"10.1016\/j.asoc.2012.08.014","article-title":"A Differential Evolution Algorithm with Intersect Mutation Operator","volume":"13","author":"Zhou","year":"2013","journal-title":"Appl. Soft Comput. J."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.knosys.2019.01.006","article-title":"PaDE: An Enhanced Differential Evolution Algorithm with Novel Control Parameter Adaptation Schemes for Numerical Optimization","volume":"168","author":"Meng","year":"2019","journal-title":"Knowl.-Based Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1016\/j.ins.2017.09.053","article-title":"Ensemble of Differential Evolution Variants","volume":"423","author":"Wu","year":"2018","journal-title":"Inf. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1016\/j.ins.2020.09.008","article-title":"Double-Layer-Clustering Differential Evolution Multimodal Optimization by Speciation and Self-Adaptive Strategies","volume":"545","author":"Liu","year":"2021","journal-title":"Inf. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"100699","DOI":"10.1016\/j.swevo.2020.100699","article-title":"Self-Organizing Neighborhood-Based Differential Evolution for Global Optimization","volume":"56","author":"Cai","year":"2020","journal-title":"Swarm Evol. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"575","DOI":"10.1007\/s10586-022-03608-0","article-title":"Design of Cultural Emperor Penguin Optimizer for Energy-Efficient Resource Scheduling in Green Cloud Computing Environment","volume":"26","author":"Mansour","year":"2023","journal-title":"Clust. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1999","DOI":"10.1080\/0305215X.2021.1969560","article-title":"A Hybrid Whale Optimization Algorithm with Differential Evolution Optimization for Multi-Objective Virtual Machine Scheduling in Cloud Computing","volume":"54","author":"Rana","year":"2022","journal-title":"Eng. Optim."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1016\/j.ins.2022.06.036","article-title":"An Ensemble of Differential Evolution and Adam for Training Feed-Forward Neural Networks","volume":"608","author":"Xue","year":"2022","journal-title":"Inf. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1109\/4235.771163","article-title":"Evolutionary Programming Made Faster","volume":"3","author":"Yao","year":"1999","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_28","unstructured":"Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A., and Tiwari, S. (2023, March 01). Problem Definitions and Evaluation Criteria for the CEC 2005 Special Session on Real-Parameter Optimization. Technical Report, Nanyang Technological University, Singapore and KanGAL Report Number 2005005 (Kanpur Genetic Algorithms Laboratory, IIT Kanpur). May 2005. Available online: http:\/\/www.cmap.polytechnique.fr\/~nikolaus.hansen\/Tech-Report-May-30-05.pdf."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/5\/1120\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:39:07Z","timestamp":1760125147000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/5\/1120"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,20]]},"references-count":28,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2023,5]]}},"alternative-id":["sym15051120"],"URL":"https:\/\/doi.org\/10.3390\/sym15051120","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,20]]}}}