{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T10:49:53Z","timestamp":1777546193437,"version":"3.51.4"},"reference-count":55,"publisher":"MDPI AG","issue":"16","license":[{"start":{"date-parts":[[2021,8,18]],"date-time":"2021-08-18T00:00:00Z","timestamp":1629244800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Applied Sciences"],"abstract":"<jats:p>Metaheuristic algorithms have been widely used to solve diverse kinds of optimization problems. For an optimization problem, population initialization plays a significant role in metaheuristic algorithms. These algorithms can influence the convergence to find an efficient optimal solution. Mainly, for recognizing the importance of diversity, several researchers have worked on the performance for the improvement of metaheuristic algorithms. Population initialization is a vital factor in metaheuristic algorithms such as PSO and DE. Instead of applying the random distribution for the initialization of the population, quasirandom sequences are more useful for the improvement the diversity and convergence factors. This study presents three new low-discrepancy sequences named WELL sequence, Knuth sequence, and Torus sequence to initialize the population in the search space. This paper also gives a comprehensive survey of the various PSO and DE initialization approaches based on the family of quasirandom sequences such as Sobol sequence, Halton sequence, and uniform random distribution. The proposed methods for PSO (TO-PSO, KN-PSO, and WE-PSO) and DE (DE-TO, DE-WE, and DE-KN) have been examined for well-known benchmark test problems and training of the artificial neural network. The finding of our techniques shows promising performance using the family of low-discrepancy sequences over uniform random numbers. For a fair comparison, the approaches using low-discrepancy sequences for PSO and DE are compared with the other family of low-discrepancy sequences and uniform random number and depict the superior results. The experimental results show that the low-discrepancy sequences-based initialization performed exceptionally better than a uniform random number. Moreover, the outcome of our work presents a foresight on how the proposed technique profoundly impacts convergence and diversity. It is anticipated that this low-discrepancy sequence comparative simulation survey would be helpful for studying the metaheuristic algorithm in detail for the researcher.<\/jats:p>","DOI":"10.3390\/app11167591","type":"journal-article","created":{"date-parts":[[2021,8,18]],"date-time":"2021-08-18T10:54:35Z","timestamp":1629284075000},"page":"7591","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":72,"title":["Comparative Analysis of Low Discrepancy Sequence-Based Initialization Approaches Using Population-Based Algorithms for Solving the Global Optimization Problems"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5797-4821","authenticated-orcid":false,"given":"Waqas Haider","family":"Bangyal","sequence":"first","affiliation":[{"name":"Department of Computer Science, University of Gujrat, Gujrat 50700, Pakistan"}]},{"given":"Kashif","family":"Nisar","sequence":"additional","affiliation":[{"name":"Faculty of Computing and Informatics, Universiti Malaysia Sabah, Jalan UMS, Kota Kinabalu 88400, Malaysia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1793-5905","authenticated-orcid":false,"given":"Ag. Asri Bin","family":"Ag. Ibrahim","sequence":"additional","affiliation":[{"name":"Faculty of Computing and Informatics, Universiti Malaysia Sabah, Jalan UMS, Kota Kinabalu 88400, Malaysia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3531-220X","authenticated-orcid":false,"given":"Muhammad Reazul","family":"Haque","sequence":"additional","affiliation":[{"name":"Faculty of Computing & Informatics, Multimedia University, Cyberjaya 63100, Malaysia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8657-3800","authenticated-orcid":false,"given":"Joel J. P. C.","family":"Rodrigues","sequence":"additional","affiliation":[{"name":"Post-Graduation Program on Electrical Engineering, Federal University of Piau\u00ed (UFPI), Teresina 64049-550, PI, Brazil"},{"name":"Instituto de Telecomunica\u00e7\u00f5es, 6201-001 Covilh\u00e3, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3638-3464","authenticated-orcid":false,"given":"Danda B.","family":"Rawat","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Science, Howard University, Washington, DC 20059, USA"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Yang, C.H., Chang, H.W., Ho, C.H., Chou, Y.C., and Chuang, L.Y. (2011). Conserved PCR primer set designing for closely-related species to complete mitochondrial genome sequencing using a sliding window-based PSO algorithm. PLoS ONE, 6.","DOI":"10.1371\/journal.pone.0017729"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"484","DOI":"10.1016\/j.asoc.2015.01.068","article-title":"A new hybrid method based on particle swarm optimization, ant colony optimization and 3-opt algorithms for traveling salesman problem","volume":"30","author":"Mahi","year":"2015","journal-title":"Appl. Soft Comput."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Rao, S.S. (2019). Engineering Optimization: Theory and Practice, John Wiley & Sons.","DOI":"10.1002\/9781119454816"},{"key":"ref_4","unstructured":"Zhang, G., Lu, J., and Gao, Y. (2021, April 15). Multi-Level Decision Making: Models, Methods and Applications. Available online: https:\/\/www.springer.com\/gp\/book\/9783662460580."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Beni, G., and Wang, J. (1993). Swarm Intelligence in Cellular Robotic Systems, in Robots and Biological Systems: Towards a New Bionics?, Springer.","DOI":"10.1007\/978-3-642-58069-7_38"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Acharya, J., Mehta, M., and Saini, B. (2016, January 21\u201322). Particle swarm optimization based load balancing in cloud computing. Proceedings of the 2016 International Conference on Communication and Electronics Systems (ICCES), Coimbatore, India.","DOI":"10.1109\/CESYS.2016.7889943"},{"key":"ref_7","first-page":"1","article-title":"An improved self-adaptive PSO algorithm with detection function for multimodal function optimization problems","volume":"2013","author":"Zhang","year":"2013","journal-title":"Math. Probl. Eng."},{"key":"ref_8","unstructured":"Eberhart, R., and Kennedy, J. (December, January 27). Particle swarm optimization. Proceedings of the IEEE International Conference on Neural Networks, Perth, WA, Australia."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Yang, X.-S. (2010). A new metaheuristic bat-inspired algorithm. Nature Inspired Cooperative Strategies for Optimization (NICSO 2010), Springer.","DOI":"10.1007\/978-3-642-12538-6_6"},{"key":"ref_10","unstructured":"Dorigo, M., and Di Caro, G. (1999, January 6\u20139). Ant colony optimization: A new meta-heuristic. Proceedings of the 1999 Congress on Evolutionary Computation\u2014CEC99, Washington, DC, USA. Cat. No. 99TH8406."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Pham, D.T., Ghanbarzadeh, A., Ko\u00e7, E., Otri, S., Rahim, S., and Zaidi, M. (2006). The bees algorithm\u2014A novel tool for complex optimisation problems. Intelligent Production Machines and Systems, Elsevier.","DOI":"10.1016\/B978-008045157-2\/50081-X"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Poli, R., Kennedy, J., and Blackwell, T. (2007). Particle Swarm Optimization, Springer.","DOI":"10.2139\/ssrn.2693499"},{"key":"ref_13","first-page":"180","article-title":"Analysis of particle swarm optimization algorithm","volume":"3","author":"Bai","year":"2010","journal-title":"Comput. Inf. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"913","DOI":"10.1109\/TEVC.2006.880326","article-title":"A survey of particle swarm optimization applications in electric power systems","volume":"13","author":"AlRashidi","year":"2008","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_15","first-page":"20","article-title":"A PSO algorithm with high speed convergence","volume":"25","author":"Zhu","year":"2010","journal-title":"Control Decis."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11432-018-9546-4","article-title":"A hybrid quantum-based PIO algorithm for global numerical optimization","volume":"62","author":"Chen","year":"2019","journal-title":"Sci. China Inf. Sci."},{"key":"ref_17","unstructured":"Shi, Y. (1999, January 6\u20139). Particle swarm optimization: Developments, applications and resources. Proceedings of the 2001 Congress on Evolutionary Computation, Washington, DC, USA. Cat. No. 01TH8546."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Chen, S., and Montgomery, J. (2013, January 20\u201323). Particle swarm optimization with thresheld convergence. Proceedings of the 2013 IEEE Congress on Evolutionary Computation, Cancun, Mexico.","DOI":"10.1109\/CEC.2013.6557611"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.epsr.2015.06.018","article-title":"A comparative study of metaheuristic optimization approaches for directional overcurrent relays coordination","volume":"128","author":"Alam","year":"2015","journal-title":"Electr. Power Syst. Res."},{"key":"ref_20","first-page":"417","article-title":"Adaptive Mutation PSO Algorithm","volume":"32","author":"Lu","year":"2004","journal-title":"Acta Electronca Sinica 3"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Rashedi, E., Nezamabadi-Pour, H., and Saryazdi, S. (2009). GSA: A Gravitational Search Algorithm, Elsevier.","DOI":"10.1016\/j.ins.2009.03.004"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Li, X., Zhuang, J., Wang, S., and Zhang, Y. (2008, January 18\u201320). A particle swarm optimization algorithm based on adaptive periodic mutation. Proceedings of the 2008 Fourth International Conference on Natural Computation, Jinan, China.","DOI":"10.1109\/ICNC.2008.36"},{"key":"ref_23","unstructured":"Song, M.-P., and Gu, G.-C. (2004, January 26\u201329). Research on particle swarm optimization: A review. Proceedings of the 2004 International Conference on Machine Learning and Cybernetics, Shanghai, China. Cat. No. 04EX826."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Maaranen, H., Miettinen, K., and Penttinen, A. (2007). On Initial Populations of a Genetic Algorithm for Continuous Optimization Problems, Springer.","DOI":"10.1007\/s10898-006-9056-6"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Pant, M., Thangaraj, R., Grosan, C., and Abraham, A. (2008, January 1\u20136). Improved particle swarm optimization with low-discrepancy sequences. Proceedings of the 2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence), Hong Kong, China.","DOI":"10.1109\/CEC.2008.4631204"},{"key":"ref_26","first-page":"1","article-title":"Initializing the particle swarm optimizer using the nonlinear simplex method","volume":"216","author":"Parsopoulos","year":"2002","journal-title":"Adv. Intell. Syst. Fuzzy Syst. Evol. Comput."},{"key":"ref_27","first-page":"2309","article-title":"Choosing a starting configuration for particle swarm optimization","volume":"25","author":"Richards","year":"2004","journal-title":"Neural Netw."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Nguyen, X.H., Nguyen, Q.U., and McKay, R.I. (2007, January 7\u201311). PSO with randomized low-discrepancy sequences. Proceedings of the 9th Annual Conference on Genetic and Evolutionary Computation\u2014GECCO \u201907, New York, NY, USA.","DOI":"10.1145\/1276958.1276987"},{"key":"ref_29","unstructured":"Uy, N.Q., Hoai, N.X., McKay, R.I., and Tuan, P.M. (2007, January 25\u201328). Initialising PSO with randomised low-discrepancy sequences: The comparative results. Proceedings of the 2007 IEEE Congress on Evolutionary Computation, Singapore."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Thangaraj, R., Pant, M., and Deep, K. (2009, January 9\u201311). Initializing PSO with probability distributions and low-discrepancy sequences: The comparative results. Proceedings of the World Congress on Nature & Biologically Inspired Computing (NaBIC), Coimbatore, India.","DOI":"10.1109\/NABIC.2009.5393814"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Thangaraj, R., Pant, M., Abraham, A., and Badr, Y. (2009, January 10\u201312). Hybrid Evolutionary Algorithm for Solving Global Optimization Problems. Proceedings of the International Conference on Hybrid Artificial Intelligence Systems, Salamanca, Spain.","DOI":"10.1007\/978-3-642-02319-4_37"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Pant, M., Thangaraj, R., Singh, V.P., and Abraham, A. (2008, January 16\u201318). Particle Swarm Optimization Using Sobol Mutation. Proceedings of the 2008 First International Conference on Emerging Trends in Engineering and Technology, Nagpur, India.","DOI":"10.1109\/ICETET.2008.35"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Du, J., Zhang, F., Huang, G., and Yang, J. (2011, January 10\u201312). A new initializing mechanism in Particle Swarm Optimization. Proceedings of the 2011 IEEE International Conference on Computer Science and Automation Engineering, Shanghai, China.","DOI":"10.1109\/CSAE.2011.5952861"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1132","DOI":"10.1049\/iet-gtd.2012.0183","article-title":"Modified particle swarm optimisation with a novel initialisation for finding optimal solution to the transmission expansion planning problem","volume":"6","author":"Murugan","year":"2012","journal-title":"IET Gener. Transm. Distrib."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Yin, L., Hu, X.-M., and Zhang, J. (2013, January 6\u201310). Space-based initialization strategy for particle swarm optimization. Proceedings of the fifteenth Annual Conference Companion on Genetic and Evolutionary Computation Conference Companion\u2014GECCO \u201913 Companion, New York, NY, USA.","DOI":"10.1145\/2464576.2464585"},{"key":"ref_36","unstructured":"Jensen, B., Bouhmala, N., and Nordli, T. (2013). A Novel Tangent based Framework for Optimizing Continuous Functions. J. Emerg. Trends Comput. Inf. Sci., 4."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"242","DOI":"10.18517\/ijaseit.7.1.1794","article-title":"A new initialization technique in polar coordinates for Particle Swarm Optimization and Polar PSO","volume":"7","author":"Shatnawi","year":"2017","journal-title":"Int. J. Adv. Sci. Eng. Inf. Technol."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Bewoor, L., Prakash, V.C., and Sapkal, S.U. (2017). Evolutionary Hybrid Particle Swarm Optimization Algorithm for Solving NP-Hard No-Wait Flow Shop Scheduling Problems. Algorithms, 10.","DOI":"10.3390\/a10040121"},{"key":"ref_39","first-page":"1026","article-title":"A hybrid particle swarm optimization\u2013back-propagation algorithm for feedforward neural network training","volume":"185","author":"Zhang","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Carvalho, M., and Ludermir, T.B. (2007, January 17\u201319). Particle swarm optimization of neural network architectures and weights. Proceedings of the 7th International Conference on Hybrid Intelligent Systems (HIS 2007), Kaiserslautern, Germany.","DOI":"10.1109\/HIS.2007.45"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"113","DOI":"10.22436\/jmcs.012.02.03","article-title":"Comparison of particle swarm optimization and backpropagation algorithms for training feed forward neural network","volume":"12","author":"Mohammadi","year":"2014","journal-title":"J. Math. Comput. Sci."},{"key":"ref_42","first-page":"79","article-title":"Hybrid algorithm for the optimization of training convolutional neural network","volume":"1","author":"Albeahdili","year":"2015","journal-title":"Int. J. Adv. Comput. Sci. Appl."},{"key":"ref_43","first-page":"95","article-title":"Simplex differential evolution","volume":"6","author":"Gudise","year":"2009","journal-title":"Acta Polytech. Hung."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Nakib, A., Daachi, B., and Siarry, P. (2012, January 21\u201325). Hybrid Differential Evolution Using Low-Discrepancy Sequences for Image Segmentation. Proceedings of the 2012 IEEE 26th International Parallel and Distributed Processing Symposium Workshops & PhD Forum, Shanghai, China.","DOI":"10.1109\/IPDPSW.2012.79"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1109\/TEVC.2013.2250977","article-title":"An Improved Differential Evolution Algorithm for Practical Dynamic Scheduling in Steelmaking-Continuous Casting Production","volume":"18","author":"Tang","year":"2013","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_46","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_47","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1016\/j.future.2017.08.041","article-title":"Fuzzy neural network optimization and network traffic forecasting based on improved differential evolution","volume":"81","author":"Hou","year":"2018","journal-title":"Future Gener. Comput. Syst."},{"key":"ref_48","first-page":"133","article-title":"A modified differential evolution algorithm trained pi-sigma neural network for pattern classification","volume":"3","author":"Panigrahi","year":"2013","journal-title":"Int. J. Soft Comput. Eng."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1145\/272991.272995","article-title":"Mersenne twister: A 623-dimensionally equidistributed uniform pseudo-random number generator","volume":"8","author":"Matsumoto","year":"1998","journal-title":"ACM Trans. Model. Comput. Simul. TOMACS"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/0041-5553(67)90144-9","article-title":"On the distribution of points in a cube and the approximate evaluation of integrals","volume":"7","year":"1967","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"701","DOI":"10.1145\/355588.365104","article-title":"Algorithm 247: Radical-inverse quasi-random point sequence","volume":"7","author":"Halton","year":"1964","journal-title":"Commun. ACM"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/1132973.1132974","article-title":"Improved long-period generators based on linear recurrences modulo 2","volume":"32","author":"Panneton","year":"2006","journal-title":"ACM Trans. Math. Softw."},{"key":"ref_53","unstructured":"Knuth, D.E. (1973). The Art of Computer Programming, Addison-Wesley."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"257","DOI":"10.2307\/3618480","article-title":"Geometries and Groups","volume":"73","author":"Williams","year":"1989","journal-title":"Math. Gaz."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1016\/j.powtec.2008.04.036","article-title":"Application of ANOVA to image analysis results of talc particles produced by different milling","volume":"188","author":"Ulusoy","year":"2008","journal-title":"Powder Technol."}],"container-title":["Applied Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2076-3417\/11\/16\/7591\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:46:51Z","timestamp":1760165211000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2076-3417\/11\/16\/7591"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,18]]},"references-count":55,"journal-issue":{"issue":"16","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["app11167591"],"URL":"https:\/\/doi.org\/10.3390\/app11167591","relation":{},"ISSN":["2076-3417"],"issn-type":[{"value":"2076-3417","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,8,18]]}}}