{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T15:01:52Z","timestamp":1781103712825,"version":"3.54.1"},"reference-count":28,"publisher":"Emerald","issue":"2","license":[{"start":{"date-parts":[[2018,4,3]],"date-time":"2018-04-03T00:00:00Z","timestamp":1522713600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.emerald.com\/insight\/site-policies"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["GS"],"published-print":{"date-parts":[[2018,4,3]]},"abstract":"<jats:sec><jats:title content-type=\"abstract-subheading\">Purpose<\/jats:title><jats:p>The purpose of this paper is to find out which algorithm, among Genetic Algorithm (GA), Particle Swarm Optimizer (PSO), the novel Grey Wolf Optimizer (GWO) and the novel Ant Lion Optimizer (ALO), is the best to obtain the optimal value of the nonlinear parameter<jats:italic>\u03b3<\/jats:italic>of nonlinear grey Bernoulli model (NGBM(1,1)) under different situations.<\/jats:p><\/jats:sec><jats:sec><jats:title content-type=\"abstract-subheading\">Design\/methodology\/approach<\/jats:title><jats:p>The optimization of<jats:italic>\u03b3<\/jats:italic>has been attributed to a nonlinear programming problem at first. The convergence, convergence rate, time consuming and stability of GA, PSO, GWO and ALO are compared in the numerical experiments, and in each subcase the criteria are set to be the same. Over 10,000 iterations have been run on the same environment in order to guarantee the reliability of the results.<\/jats:p><\/jats:sec><jats:sec><jats:title content-type=\"abstract-subheading\">Findings<\/jats:title><jats:p>All the selected algorithms can converge to the same optimal value with sufficient iterations. But the best algorithm should be chose under different situations.<\/jats:p><\/jats:sec><jats:sec><jats:title content-type=\"abstract-subheading\">Practical implications<\/jats:title><jats:p>The optimal value of<jats:italic>\u03b3<\/jats:italic>seems to exist uniquely due to the empirical results. And there does not exist a best algorithm for all the cases. The researchers and commercial software developers should choose a proper algorithm due to different cases.<\/jats:p><\/jats:sec><jats:sec><jats:title content-type=\"abstract-subheading\">Originality\/value<\/jats:title><jats:p>The performance of GA, PSO, GWO and ALO to compute the optimal<jats:italic>\u03b3<\/jats:italic>of NGBM(1,1) has been compared for the first time. And it is the original work which uses the GWO and ALO to optimize the NGBM(1,1).<\/jats:p><\/jats:sec>","DOI":"10.1108\/gs-01-2018-0005","type":"journal-article","created":{"date-parts":[[2018,2,14]],"date-time":"2018-02-14T19:13:30Z","timestamp":1518635610000},"page":"210-226","source":"Crossref","is-referenced-by-count":31,"title":["Comparison study on the nonlinear parameter optimization of nonlinear grey Bernoulli model (NGBM(1,1)) between intelligent optimizers"],"prefix":"10.1108","volume":"8","author":[{"given":"Lingcun","family":"Kong","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xin","family":"Ma","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"140","reference":[{"issue":"1","key":"key2020092920445607000_ref001","doi-asserted-by":"crossref","first-page":"278","DOI":"10.1016\/j.chaos.2006.08.024","article-title":"Application of the novel nonlinear grey Bernoulli model for forecasting unemployment rate","volume":"37","year":"2008","journal-title":"Chaos, Solitons & Fractals"},{"issue":"12","key":"key2020092920445607000_ref002","doi-asserted-by":"crossref","first-page":"7557","DOI":"10.1016\/j.eswa.2010.04.088","article-title":"Forecasting Taiwan\u2019s major stock indices by the Nash nonlinear grey Bernoulli model","volume":"37","year":"2010","journal-title":"Expert Systems with Applications"},{"key":"key2020092920445607000_ref003","volume-title":"Genetic Algorithms in Search, Optimization, and Machine Learning, 1989","year":"1989"},{"issue":"1","key":"key2020092920445607000_ref004","first-page":"168","article-title":"Application of game theory on parameter optimization of the novel two-stage Nash nonlinear grey Bernoulli model","volume":"27","year":"2015","journal-title":"Communications in Nonlinear Science and Numerical Simulation"},{"issue":"6","key":"key2020092920445607000_ref005","doi-asserted-by":"crossref","first-page":"4318","DOI":"10.1016\/j.eswa.2009.11.068","article-title":"A genetic algorithm based nonlinear grey Bernoulli model for output forecasting in integrated circuit industry","volume":"37","year":"2010","journal-title":"Expert systems with Applications"},{"key":"key2020092920445607000_ref006","article-title":"Elements on grey theory","volume-title":"Huazhong University of Science and Technology","year":"2002"},{"key":"key2020092920445607000_ref007","doi-asserted-by":"crossref","unstructured":"Kennedy, J. (2011), \u201cParticle swarm optimization\u201d, Encyclopedia of Machine Learning, Springer, pp. 760-766.","DOI":"10.1007\/978-0-387-30164-8_630"},{"key":"key2020092920445607000_ref008","doi-asserted-by":"crossref","first-page":"206","DOI":"10.1016\/j.neucom.2015.11.032","article-title":"An optimized nonlinear grey Bernoulli model and its applications","volume":"177","year":"2016","journal-title":"Neurocomputing"},{"key":"key2020092920445607000_ref009","first-page":"1","article-title":"Research on a novel kernel based grey prediction model and its applications","volume":"2016","year":"2016","journal-title":"Mathematical Problems in Engineering"},{"issue":"7","key":"key2020092920445607000_ref011","first-page":"4876","article-title":"Research on the novel recursive discrete multivariate grey prediction model and its applications","volume":"40","year":"2016","journal-title":"Applied Mathematical Modelling"},{"key":"key2020092920445607000_ref012","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/j.cam.2017.04.020","article-title":"Application of a novel time-delayed polynomial grey model to predict the natural gas consumption in China","volume":"324","year":"2017","journal-title":"Journal of Computational and Applied Mathematics"},{"issue":"4","key":"key2020092920445607000_ref013","first-page":"122","article-title":"The GMC(1, n) model with optimized parameters and its application","volume":"29","year":"2017","journal-title":"The Journal of Grey System"},{"issue":"2","key":"key2020092920445607000_ref010","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1007\/s00521-016-2721-x","article-title":"Predicting the oil production using the novel multivariate nonlinear model based on Arps decline model and kernel method","volume":"29","year":"2018","journal-title":"Neural Computing and Applications"},{"key":"key2020092920445607000_ref014","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.cnsns.2016.12.017","article-title":"A novel kernel regularized nonhomogeneous grey model and its applications","volume":"48","year":"2017","journal-title":"Communications in Nonlinear Science and Numerical Simulation"},{"key":"key2020092920445607000_ref015","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.advengsoft.2015.01.010","article-title":"The Ant Lion optimizer","volume":"83","year":"2015","journal-title":"Advances in Engineering Software"},{"key":"key2020092920445607000_ref016","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1016\/j.advengsoft.2013.12.007","article-title":"Grey wolf optimizer","volume":"69","year":"2014","journal-title":"Advances in Engineering Software"},{"issue":"1","key":"key2020092920445607000_ref017","doi-asserted-by":"crossref","first-page":"400","DOI":"10.1016\/j.energy.2012.01.037","article-title":"Forecasting of CO2 emissions, energy consumption and economic growth in China using an improved grey model","volume":"40","year":"2012","journal-title":"Energy"},{"key":"key2020092920445607000_ref018","doi-asserted-by":"crossref","first-page":"941","DOI":"10.1016\/j.energy.2017.09.037","article-title":"Forecasting China\u2019s natural gas demand based on optimised nonlinear grey models","volume":"140","year":"2017","journal-title":"Energy"},{"issue":"3","key":"key2020092920445607000_ref019","doi-asserted-by":"crossref","first-page":"780","DOI":"10.1016\/j.cie.2012.12.010","article-title":"An optimized Nash nonlinear grey Bernoulli model for forecasting the main economic indices of high technology enterprises in China","volume":"64","year":"2013","journal-title":"Computers & Industrial Engineering"},{"issue":"11","key":"key2020092920445607000_ref020","first-page":"5745","article-title":"An improved grey multivariable model for predicting industrial energy consumption in China","volume":"40","year":"2016","journal-title":"Applied Mathematical Modelling"},{"key":"key2020092920445607000_ref021","doi-asserted-by":"crossref","first-page":"600","DOI":"10.1016\/j.jclepro.2016.08.067","article-title":"Forecasting Chinese carbon emissions from fossil energy consumption using non-linear grey multivariable models","volume":"142","year":"2017","journal-title":"Journal of Cleaner Production"},{"issue":"4","key":"key2020092920445607000_ref022","first-page":"144","article-title":"Unbiased GM(1,1) power model and its application","volume":"19","year":"2011","journal-title":"Chinese Journal of Management Science"},{"key":"key2020092920445607000_ref023","doi-asserted-by":"crossref","first-page":"302","DOI":"10.1016\/j.apm.2017.07.003","article-title":"Grey forecasting method of quarterly hydropower production in China based on a data grouping approach","volume":"51","year":"2017","journal-title":"Applied Mathematical Modelling"},{"issue":"12","key":"key2020092920445607000_ref024","doi-asserted-by":"crossref","first-page":"5524","DOI":"10.1016\/j.apm.2011.05.022","article-title":"An optimized NGBM(1,1) model for forecasting the qualified discharge rate of industrial wastewater in China","volume":"35","year":"2011","journal-title":"Applied Mathematical Modelling"},{"key":"key2020092920445607000_ref025","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1016\/j.enpol.2017.06.050","article-title":"Decomposition of the factors influencing export fluctuation in China\u2019s new energy industry based on a constant market share model","volume":"109","year":"2017","journal-title":"Energy Policy"},{"key":"key2020092920445607000_ref026","volume-title":"Grey Systems: Modeling and Prediction","year":"2004"},{"key":"key2020092920445607000_ref027","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/j.compbiomed.2014.02.008","article-title":"An optimized Nash nonlinear grey Bernoulli model based on particle swarm optimization and its application in prediction for the incidence of hepatitis B in Xinjiang, China","volume":"49","year":"2014","journal-title":"Computers in Biology and Medicine"},{"issue":"2","key":"key2020092920445607000_ref028","doi-asserted-by":"crossref","first-page":"292","DOI":"10.1016\/j.amc.2008.10.045","article-title":"Parameter optimization of nonlinear grey Bernoulli model using particle swarm optimization","volume":"207","year":"2009","journal-title":"Applied Mathematics and Computation"}],"container-title":["Grey Systems: Theory and Application"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.emerald.com\/insight\/content\/doi\/10.1108\/GS-01-2018-0005\/full\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.emerald.com\/insight\/content\/doi\/10.1108\/GS-01-2018-0005\/full\/html","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,7,25]],"date-time":"2025-07-25T00:45:25Z","timestamp":1753404325000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.emerald.com\/gs\/article\/8\/2\/210-226\/85498"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,4,3]]},"references-count":28,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2018,4,3]]}},"alternative-id":["10.1108\/GS-01-2018-0005"],"URL":"https:\/\/doi.org\/10.1108\/gs-01-2018-0005","relation":{},"ISSN":["2043-9377"],"issn-type":[{"value":"2043-9377","type":"print"}],"subject":[],"published":{"date-parts":[[2018,4,3]]}}}