{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T01:59:26Z","timestamp":1775527166648,"version":"3.50.1"},"reference-count":64,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2019,5,14]],"date-time":"2019-05-14T00:00:00Z","timestamp":1557792000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Global optimization, especially on a large scale, is challenging to solve due to its nonlinearity and multimodality. In this paper, in order to enhance the global searching ability of the firefly algorithm (FA) inspired by bionics, a novel hybrid meta-heuristic algorithm is proposed by embedding the cross-entropy (CE) method into the firefly algorithm. With adaptive smoothing and co-evolution, the proposed method fully absorbs the ergodicity, adaptability and robustness of the cross-entropy method. The new hybrid algorithm achieves an effective balance between exploration and exploitation to avoid falling into a local optimum, enhance its global searching ability, and improve its convergence rate. The results of numeral experiments show that the new hybrid algorithm possesses more powerful global search capacity, higher optimization precision, and stronger robustness.<\/jats:p>","DOI":"10.3390\/e21050494","type":"journal-article","created":{"date-parts":[[2019,5,14]],"date-time":"2019-05-14T10:42:33Z","timestamp":1557830553000},"page":"494","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":22,"title":["A Novel Hybrid Meta-Heuristic Algorithm Based on the Cross-Entropy Method and Firefly Algorithm for Global Optimization"],"prefix":"10.3390","volume":"21","author":[{"given":"Guocheng","family":"Li","sequence":"first","affiliation":[{"name":"School of Finance and Mathematics, West Anhui University, Lu\u2019an 237012, China"},{"name":"Institute of Financial Risk Intelligent Control and Prevention, West Anhui University, Lu\u2019an 237012, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pei","family":"Liu","sequence":"additional","affiliation":[{"name":"College of Computer Science, Sichuan University, Chengdu 610065, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chengyi","family":"Le","sequence":"additional","affiliation":[{"name":"School of Economic &amp; Management, East China Jiaotong University, Nanchang 330013, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Benda","family":"Zhou","sequence":"additional","affiliation":[{"name":"School of Finance and Mathematics, West Anhui University, Lu\u2019an 237012, China"},{"name":"Institute of Financial Risk Intelligent Control and Prevention, West Anhui University, Lu\u2019an 237012, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,5,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Horst, R., and Pardalos, P.M. (1995). Handbook of Global Optimization, Springer.","DOI":"10.1007\/978-1-4615-2025-2"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/s10898-017-0589-7","article-title":"GOSH: Derivative-free global optimization using multi-dimensional space-filling curves","volume":"71","author":"Lera","year":"2018","journal-title":"J. Glob. Optim."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1038\/s41598-017-18940-4","article-title":"On the efficiency of nature-inspired metaheuristics in expensive global optimization with limited budget","volume":"8","author":"Sergeyev","year":"2018","journal-title":"Sci. Rep."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.4249\/scholarpedia.11472","article-title":"Metaheuristic Optimization","volume":"6","author":"Yang","year":"2011","journal-title":"Scholarpedia"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"541","DOI":"10.2307\/1909768","article-title":"Maximization by quadratic hill-climbing","volume":"34","author":"Goldfeld","year":"1966","journal-title":"Econometrica"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"887","DOI":"10.1016\/S0096-3003(03)00282-0","article-title":"Improving Newton\u2013Raphson method for nonlinear equations by modified Adomian decomposition method","volume":"145","author":"Abbasbandy","year":"2003","journal-title":"Appl. Math. Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1007\/BF00941892","article-title":"Lipschitzian optimization without the Lipschitz constant","volume":"79","author":"Jones","year":"1993","journal-title":"J. Optim. Theory Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1007\/s10898-009-9508-x","article-title":"An information global minimization algorithm using the local improvement technique","volume":"481","author":"Lera","year":"2010","journal-title":"J. Glob. Optim."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1007\/s10957-016-0947-5","article-title":"Derivative-Free Local Tuning and Local Improvement Techniques Embedded in the Univariate Global Optimization","volume":"171","author":"Sergeyev","year":"2016","journal-title":"J. Optim. Theory Appl."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Zhang, L.N., Liu, L.Q., Yang, X.S., and Dai, Y.T. (2016). A novel hybrid firefly algorithm for global optimization. PLoS ONE, 11.","DOI":"10.1371\/journal.pone.0163230"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/BF00175354","article-title":"A genetic algorithm tutorial","volume":"4","author":"Whitley","year":"1994","journal-title":"Stat. Comput."},{"key":"ref_12","unstructured":"Kennedy, J., and Eberhart, R.C. (December, January 27). Particle swarm optimization. Proceedings of the 1995 IEEE International Conference on Neural Networks, Perth, Australia."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1109\/MCI.2006.329691","article-title":"Ant colony optimization","volume":"1","author":"Dorigo","year":"2006","journal-title":"IEEE Comput. Intell. Mag."},{"key":"ref_14","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_15","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1177\/003754970107600201","article-title":"A New Heuristic Optimization Algorithm: Harmony Search","volume":"76","author":"Geem","year":"2001","journal-title":"Simulation"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1109\/MCS.2002.1004010","article-title":"Biomimicry of bacterial foraging for distributed optimization and control","volume":"22","author":"Passino","year":"2002","journal-title":"IEEE Control Syst. Mag."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1007\/s11269-005-9001-3","article-title":"Honey Bees Mating Optimization (HBMO) Algorithm: A New Heuristic Approach for Water Resources Optimization","volume":"20","author":"Hadad","year":"2006","journal-title":"Water Resour. Manag."},{"key":"ref_18","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_19","doi-asserted-by":"crossref","first-page":"702","DOI":"10.1109\/TEVC.2008.919004","article-title":"Biogeography-Based Optimization","volume":"12","author":"Simon","year":"2008","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2232","DOI":"10.1016\/j.ins.2009.03.004","article-title":"GSA: A gravitational search algorithm","volume":"179","author":"Rashedi","year":"2009","journal-title":"Inf. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Yang, X.S. (2009). Firefly algorithms for multimodal optimization. International Symposium on Stochastic Algorithms, Springer.","DOI":"10.1007\/978-3-642-04944-6_14"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Yang, X.S., and Deb, S. (2009, January 9\u201311). Cuckoo Search via L\u00e9vy flights. Proceedings of the 2009 World Congress on Nature & Biologically Inspired Computing (NaBIC), Coimbatore, India.","DOI":"10.1109\/NABIC.2009.5393690"},{"key":"ref_23","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_24","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_25","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.advengsoft.2015.01.010","article-title":"The ant lion optimizer","volume":"83","author":"Mirjalili","year":"2015","journal-title":"Adv. Eng. Softw."},{"key":"ref_26","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_27","doi-asserted-by":"crossref","first-page":"1053","DOI":"10.1007\/s00521-015-1920-1","article-title":"Dragonfly algorithm: A new meta-heuristic optimization technique for solving single-objective, discrete, and multi-objective problems","volume":"27","author":"Mirjalili","year":"2016","journal-title":"Neural Comput. Appl."},{"key":"ref_28","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_29","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_30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.compstruc.2016.03.001","article-title":"A novel metaheuristic method for solving constrained engineering optimization problems: crow search algorithm","volume":"169","author":"Askarzadeh","year":"2016","journal-title":"Comput. Struct."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Po\u0142ap, D. (2017). Polar bear optimization algorithm: Meta-heuristic with fast population movement and dynamic birth and death mechanism. Symmetry, 9.","DOI":"10.3390\/sym9100203"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1016\/j.engappai.2018.04.021","article-title":"Tree Growth Algorithm (TGA): A novel approach for solving optimization problems","volume":"72","author":"Cheraghalipour","year":"2018","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1007\/s00500-018-3102-4","article-title":"Butterfly optimization algorithm: A novel approach for global optimization","volume":"23","author":"AArora","year":"2019","journal-title":"Soft Comput."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1109\/4235.585893","article-title":"No Free Lunch Theorems for Optimization","volume":"1","author":"Wolpert","year":"1997","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_35","unstructured":"Lai, X.S., and Zhang, M.Y. (2009, January 8\u201311). An Efficient Ensemble of GA and PSO for Real Function Optimization. Proceedings of the 2009 2nd IEEE International Conference on Computer Science and Information Technology, Beijing, China."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Song, X.H., Zhou, W., Li, Q., Zou, S.C., and Liang, J. (2009, January 25\u201326). Hybrid particle swarm and ant colony optimization for Surface Wave Analysis. Proceedings of the 2009 International Conference on Information Technology and Computer Science, Kiev, Ukraine.","DOI":"10.1109\/ITCS.2009.81"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Mirjalili, S., and Hashim, S.Z.M. (2010, January 3\u20135). A New Hybrid PSOGSA Algorithm for Function Optimization. Proceedings of the 2010 International Conference on Computer and Information Application (2010 ICCIA), Tianjin, China.","DOI":"10.1109\/ICCIA.2010.6141614"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Abdullah, A., Deris, S., Mohamad, M.S., and Hashim, S.Z.M. (2012). A New Hybrid Firefly Algorithm for Complex and Nonlinear Problem. Distributed Computing and Artificial Intelligence, Springer.","DOI":"10.1007\/978-3-642-28765-7_81"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1016\/j.amc.2013.07.092","article-title":"Hybridizing Ant Colony Optimization with Firefly Algorithm for Unconstrained Optimization Problems","volume":"224","author":"Zaki","year":"2013","journal-title":"Appl. Math. Comput."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1016\/j.ins.2014.06.002","article-title":"A Hybrid Firefly-Genetic Algorithm for the Capacitated Facility Location Problem","volume":"283","author":"Rahmani","year":"2014","journal-title":"Inf. Sci."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1007\/s00521-013-1518-4","article-title":"Bat algorithm based on simulated annealing and Gaussian perturbations","volume":"25","author":"He","year":"2013","journal-title":"Neural Comput. Appl."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1007\/s00500-014-1502-7","article-title":"Hybridizing harmony search algorithm with cuckoo search for global numerical optimization","volume":"20","author":"Wang","year":"2016","journal-title":"Soft Comput."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1007\/s11771-016-3061-9","article-title":"A novel hybrid algorithm based on a harmony search and artificial bee colony for solving a portfolio optimization problem using a mean-semi variance approach","volume":"23","author":"Seyedhosseini","year":"2016","journal-title":"J. Cent. South Univ."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"302","DOI":"10.1016\/j.neucom.2017.04.053","article-title":"Hybrid Whale Optimization Algorithm with simulated annealing for feature selection","volume":"260","author":"Mafarja","year":"2017","journal-title":"Neurocomputing"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/S0377-2217(96)00385-2","article-title":"Optimization of Computer Simulation Models with Rare Events","volume":"99","author":"Rubinstein","year":"1997","journal-title":"Eur. J. Oper. Res."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1023\/A:1010091220143","article-title":"The Cross-Entropy Method for Combinatorial and Continuous Optimization","volume":"1","author":"Rubinstein","year":"1999","journal-title":"Methodol. Comput. Appl. Probab."},{"key":"ref_47","unstructured":"Rubinstein, R.Y., and Kroese, D.P. (2004). The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte Carlo Simulation and Machine Learning, Springer."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1007\/s10479-005-5724-z","article-title":"A Tutorial on the Cross-Entropy Method","volume":"134","author":"Boer","year":"2005","journal-title":"Ann. Oper. Res."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1007\/s11009-006-9753-0","article-title":"The Cross-Entropy Method for Continuous Multi-extremal Optimization","volume":"8","author":"Kroese","year":"2006","journal-title":"Methodol. Comput. Appl. Probab."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Tang, R., Fong, S., Dey, N., Wong, R., and Mohammed, S. (2017). Cross entropy method based hybridization of dynamic group optimization algorithm. Entropy, 19.","DOI":"10.3390\/e19100533"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1007\/s10479-005-5729-7","article-title":"Solving the vehicle routing problem with stochastic demands using the cross-entropy method","volume":"134","author":"Chepuri","year":"2005","journal-title":"Ann. Oper. Res."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1109\/TMAG.2011.2175437","article-title":"Multiobjective Optimization of Inverse Problems Using a Vector Cross Entropy Method","volume":"48","author":"Ho","year":"2012","journal-title":"IEEE Trans. Magnet."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1007\/s10696-015-9210-x","article-title":"An estimation of distribution algorithm and new computational results for the stochastic resource-constrained project scheduling problem","volume":"7","author":"Fang","year":"2015","journal-title":"Flex. Serv. Manuf."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"737","DOI":"10.1137\/17M1122992","article-title":"Multifidelity preconditioning of the cross-entropy method for rare event simulation and failure probability estimation","volume":"6","author":"Peherstorfer","year":"2018","journal-title":"SIAM\/ASA J. Uncertain. Quantif."},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Yang, X.S. (2009). Firefly Algorithm, L\u00e9vy Flights and Global Optimization. Research and Development in Intelligent Systems XXVI, Springer.","DOI":"10.1007\/978-1-84882-983-1_15"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1007\/s00366-012-0254-1","article-title":"Multiobjective firefly algorithm for continuous optimization","volume":"29","author":"Yang","year":"2013","journal-title":"Eng. Comput."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1504\/IJBIC.2010.032124","article-title":"Firefly algorithm, stochastic test functions and design optimization","volume":"2","author":"Yang","year":"2010","journal-title":"Int. J. Bio-Inspired Comput. Arch."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1109\/TEVC.2013.2240304","article-title":"A discrete firefly algorithm for the multi-objective hybrid flowshop scheduling problems","volume":"28","author":"Marichelvam","year":"2014","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"1180","DOI":"10.1016\/j.asoc.2011.09.017","article-title":"Firefly algorithm for solving non-convex economic dispatch problems with valve loading effect","volume":"12","author":"Yang","year":"2012","journal-title":"Appl. Soft Comput."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1016\/j.asoc.2015.06.056","article-title":"Adaptive Firefly Algorithm with Chaos for Mechanical Design Optimization Problems","volume":"36","author":"Ozsoydan","year":"2015","journal-title":"Appl. Soft Comput."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/j.ins.2013.06.022","article-title":"Binary real coded firefly algorithm for solving unit commitment problem","volume":"249","author":"Chandrasekaran","year":"2013","journal-title":"Inf. Sci."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"8221","DOI":"10.1016\/j.eswa.2015.06.024","article-title":"A Highly Accurate Firefly Based Algorithm for Heart Disease Prediction","volume":"42","author":"Long","year":"2015","journal-title":"Expert Syst. Appl."},{"key":"ref_63","first-page":"35","article-title":"On Evolutionary Exploration and Exploitation","volume":"35","author":"Eiben","year":"1998","journal-title":"Fund. Inform."},{"key":"ref_64","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."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/5\/494\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:51:50Z","timestamp":1760187110000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/5\/494"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,14]]},"references-count":64,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2019,5]]}},"alternative-id":["e21050494"],"URL":"https:\/\/doi.org\/10.3390\/e21050494","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,5,14]]}}}