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Probl. Eng.</a> <a href="/?q=in%3A416355" title="Articles in this Issue">2016, Article ID 1519542, 7 p. (2016)</a>. </div> <div class="abstract">Summary: In this paper, an improved NSGA2 algorithm is proposed, which is used to solve the multiobjective problem. For the original NSGA2 algorithm, the paper made one improvement: joining the local search strategy into the NSGA2 algorithm. After each iteration calculation of the NSGA2 algorithm, a kind of local search strategy is performed in the Pareto optimal set to search better solutions, such that the NSGA2 algorithm can gain a better local search ability which is helpful to the optimization process. Finally, the proposed modified NSGA2 algorithm (MNSGA2) is simulated in the two classic multiobjective problems which is called KUR problem and ZDT3 problem. The calculation results show the modified NSGA2 outperforms the original NSGA2, which indicates that the improvement strategy is helpful to improve the algorithm.</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C59" title="MSC2020">90C59</a> </td> <td class="space"> Approximation methods and heuristics in mathematical programming </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C29" title="MSC2020">90C29</a> </td> <td class="space"> Multi-objective and goal programming </td> </tr></table> </div> <div class="software"> <h3>Software:</h3><a href="/software/10950">ABC</a>; <a href="/software/6686">SPEA2</a>; <a href="/software/6582">NSGA-II</a></div> <!-- Modal used to show zbmath metadata in different output formats--> <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;"> <!-- alert for connection errors etc --> </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function"> <!-- Button trigger metadata modal --> <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1400.90312" data-ciurl="/ci/06952797" data-biburl="/bibtex/06952797.bib" data-amsurl="/amsrefs/06952797.bib" data-xmlurl="/xml/06952797.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06952797.pdf" title="Zbl 1400.90312 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1155/2016/1519542" aria-label="DOI for “An improved nondominated sorting genetic algorithm for multiobjective problem”" title="10.1155/2016/1519542">DOI</a> </div> <div class="sfx" style="float: right;"> <a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" title="Open Access License" class="cc-license-link no-new-tab-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/cc.svg" alt="Creative Commons CC license icon" class="cc-license-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/by.svg" alt="Creative Commons BY license icon" class="cc-license-icon"> </a> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Razvan, C.; Lucian, G.; Ioan, M., An OOP MATLAB extensible framework for the implementation of genetic algorithms. Part I: the framework, Proceedings of the 8th International Conference Interdisciplinarity in Engineering (INTER-ENG ’14)</td> </tr><tr> <td>[2]</td> <td class="space">Kennedy, J., Particle swarm optimization, Encyclopedia of Machine Learning, 760-766, (2011), New York, NY, USA: Springer, New York, NY, USA</td> </tr><tr> <td>[3]</td> <td class="space">Qin, A. K.; Huang, V. L.; Suganthan, P. N., Differential evolution algorithm with strategy adaptation for global numerical optimization, IEEE Transactions on Evolutionary Computation, 13, 2, 398-417, (2009) · <a href="https://doi.org/10.1109/TEVC.2008.927706" class="nowrap">doi:10.1109/TEVC.2008.927706</a></td> </tr><tr> <td>[4]</td> <td class="space">Eusuff, M.; Lansey, K.; Pasha, F., Shuffled frog-leaping algorithm: a memetic meta-heuristic for discrete optimization, Engineering Optimization, 38, 2, 129-154, (2006) · <a href="https://doi.org/10.1080/03052150500384759" class="nowrap">doi:10.1080/03052150500384759</a></td> </tr><tr> <td>[5]</td> <td class="space">Karaboga, D.; Basturk, B., A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm, Journal of Global Optimization, 39, 3, 459-471, (2007) · <a href="/1149.90186" class="nowrap">Zbl 1149.90186</a> · <a href="https://doi.org/10.1007/s10898-007-9149-x" class="nowrap">doi:10.1007/s10898-007-9149-x</a></td> </tr><tr> <td>[6]</td> <td class="space">Park, Y.-B.; Yoo, J.-S.; Park, H.-S., A genetic algorithm for the vendor-managed inventory routing problem with lost sales, Expert Systems with Applications, 53, 1, 149-159, (2016) · <a href="https://doi.org/10.1016/j.eswa.2016.01.041" class="nowrap">doi:10.1016/j.eswa.2016.01.041</a></td> </tr><tr> <td>[7]</td> <td class="space">Ghosh, P.; Mitchell, M.; Tanyi, J. A.; Hung, A. Y., Incorporating priors for medical image segmentation using a genetic algorithm, Neurocomputing, 195, 181-194, (2016) · <a href="https://doi.org/10.1016/j.neucom.2015.09.123" class="nowrap">doi:10.1016/j.neucom.2015.09.123</a></td> </tr><tr> <td>[8]</td> <td class="space">Liu, Y.; Dong, H.; Lohse, N.; Petrovic, S., A multi-objective genetic algorithm for optimisation of energy consumption and shop floor production performance, International Journal of Production Economics, 179, 259-272, (2016) · <a href="https://doi.org/10.1016/j.ijpe.2016.06.019" class="nowrap">doi:10.1016/j.ijpe.2016.06.019</a></td> </tr><tr> <td>[9]</td> <td class="space">Salido, M. A.; Escamilla, J.; Giret, A.; Barber, F., A genetic algorithm for energy-efficiency in job-shop scheduling, The International Journal of Advanced Manufacturing Technology, 85, 5, 1304-1314, (2016) · <a href="https://doi.org/10.1007/s00170-015-7987-0" class="nowrap">doi:10.1007/s00170-015-7987-0</a></td> </tr><tr> <td>[10]</td> <td class="space">Abdelkhalek, O.; Krichen, S.; Guitouni, A., A genetic algorithm based decision support system for the multi-objective node placement problem in next wireless generation network, Applied Soft Computing Journal, 33, 8, 278-291, (2015) · <a href="https://doi.org/10.1016/j.asoc.2015.03.034" class="nowrap">doi:10.1016/j.asoc.2015.03.034</a></td> </tr><tr> <td>[11]</td> <td class="space">Salami, M.; Hendtlass, T., A fitness estimation strategy for genetic algorithms, Lecture Notes in Computer Science, 2358, 502-513, (2002) · <a href="/1045.68899" class="nowrap">Zbl 1045.68899</a></td> </tr><tr> <td>[12]</td> <td class="space">Korkmaz, E. E., Multi-objective genetic algorithms for grouping problems, Applied Intelligence, 33, 2, 179-192, (2010) · <a href="https://doi.org/10.1007/s10489-008-0158-3" class="nowrap">doi:10.1007/s10489-008-0158-3</a></td> </tr><tr> <td>[13]</td> <td class="space">Kaya, M., Autonomous classifiers with understandable rule using multi-objective genetic algorithms, Expert Systems with Applications, 37, 4, 3489-3494, (2010) · <a href="https://doi.org/10.1016/j.eswa.2009.10.039" class="nowrap">doi:10.1016/j.eswa.2009.10.039</a></td> </tr><tr> <td>[14]</td> <td class="space">Abido, M. A., Multiobjective optimal VAR dispatch using Strength Pareto Evolutionary Algorithm, Proceedings of the IEEE Congress on Evolutionary Computation</td> </tr><tr> <td>[15]</td> <td class="space">Eghbal, M.; Yorino, N.; El-Araby, E. E.; Zoka, Y., Optimal VAr expansion considering ATC using strength pareto evolutionary algorithm, Proceedings of the 40th North American Power Symposium (NAPS ’08), IEEE</td> </tr><tr> <td>[16]</td> <td class="space">Deb, K.; Pratap, A.; Agarwal, S.; Meyarivan, T., A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation, 6, 2, 182-197, (2002) · <a href="https://doi.org/10.1109/4235.996017" class="nowrap">doi:10.1109/4235.996017</a></td> </tr><tr> <td>[17]</td> <td class="space">Zitzler, E.; Laumanns, M.; Thiele, L., SPEA2: improving the strength Pareto evolutionary algorithm, Eurogen, 3242, 103, 95-100, (2001)</td> </tr><tr> <td>[18]</td> <td class="space">Kim, M.; Hiroyasu, T.; Miki, M., SPEA2+: improving the performance of the strength Pareto evolutionary algorithm 2, Parallel Problem Solving from Nature—PPSN VIII: 8th International Conference, Birmingham, UK, September 18–22, 2004. Proceedings, 3242, 742-751, (2004), Berlin, Germany: Springer, Berlin, Germany · <a href="https://doi.org/10.1007/978-3-540-30217-9_75" class="nowrap">doi:10.1007/978-3-540-30217-9_75</a></td> </tr><tr> <td>[19]</td> <td class="space">Deb, K.; Agrawal, S.; Pratap, A., A fast elitist non-dominated sorting genetic algorithm for multi-objective optimization: NSGA-II, Parallel Problem Solving from Nature PPSN VI: 6th International Conference Paris, France, September 18–20, 2000 Proceedings. Parallel Problem Solving from Nature PPSN VI: 6th International Conference Paris, France, September 18–20, 2000 Proceedings, Lecture Notes in Computer Science, 1917, 849-858, (2000), Berlin, Germany: Springer, Berlin, Germany · <a href="https://doi.org/10.1007/3-540-45356-3_83" class="nowrap">doi:10.1007/3-540-45356-3_83</a></td> </tr><tr> <td>[20]</td> <td class="space">Milošević, B.; Begović, M., Nondominated sorting genetic algorithm for optimal phasor measurement placement, IEEE Transactions on Power Systems, 18, 1, 69-75, (2003) · <a href="https://doi.org/10.1109/TPWRS.2002.807064" class="nowrap">doi:10.1109/TPWRS.2002.807064</a></td> </tr><tr> <td>[21]</td> <td class="space">Kuriakose, S.; Shunmugam, M. S., Multi-objective optimization of wire-electro discharge machining process by non-dominated sorting genetic algorithm, Journal of Materials Processing Technology, 170, 1-2, 133-141, (2005) · <a href="https://doi.org/10.1016/j.jmatprotec.2005.04.105" class="nowrap">doi:10.1016/j.jmatprotec.2005.04.105</a></td> </tr><tr> <td>[22]</td> <td class="space">Kanagarajan, D.; Karthikeyan, R.; Palanikumar, K.; Davim, J. P., Optimization of electrical discharge machining characteristics of WC/Co composites using non-dominated sorting genetic algorithm (NSGA-II), International Journal of Advanced Manufacturing Technology, 36, 11-12, 1124-1132, (2008) · <a href="https://doi.org/10.1007/s00170-006-0921-8" class="nowrap">doi:10.1007/s00170-006-0921-8</a></td> </tr><tr> <td>[23]</td> <td class="space">Panda, S., Multi-objective PID controller tuning for a FACTS-based damping stabilizer using non-dominated sorting genetic algorithm-II, International Journal of Electrical Power & Energy Systems, 33, 7, 1296-1308, (2011) · <a href="https://doi.org/10.1016/j.ijepes.2011.06.002" class="nowrap">doi:10.1016/j.ijepes.2011.06.002</a></td> </tr><tr> <td>[24]</td> <td class="space">Panda, S.; Yegireddy, N. K., Automatic generation control of multi-area power system using multi-objective non-dominated sorting genetic algorithm-II, International Journal of Electrical Power & Energy Systems, 53, 1, 54-63, (2013) · <a href="https://doi.org/10.1016/j.ijepes.2013.04.003" class="nowrap">doi:10.1016/j.ijepes.2013.04.003</a></td> </tr><tr> <td>[25]</td> <td class="space">Yandamuri, S. R. M.; Srinivasan, K.; Murty Bhallamudi, S., Multiobjective optimal waste load allocation models for rivers using nondominated sorting genetic algorithm-II, Journal of Water Resources Planning and Management, 132, 3, 133-143, (2006) · <a href="https://doi.org/10.1061/(asce)0733-9496(2006)132:3(133)" class="nowrap">doi:10.1061/(asce)0733-9496(2006)132:3(133)</a></td> </tr><tr> <td>[26]</td> <td class="space">Wang, L.; Wang, T.-G.; Luo, Y., Improved non-dominated sorting genetic algorithm (NSGA)-II in multi-objective optimization studies of wind turbine blades, Applied Mathematics and Mechanics, 32, 6, 739-748, (2011) · <a href="/1237.76155" class="nowrap">Zbl 1237.76155</a> · <a href="https://doi.org/10.1007/s10483-011-1453-x" class="nowrap">doi:10.1007/s10483-011-1453-x</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>