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Monarch butterfly optimization


In nature, the eastern North American monarch population is known for its southward migration during the late summer/autumn from the northern USA and southern Canada to Mexico, covering thousands of miles. By simplifying and idealizing the migration of monarch butterflies, a new kind of nature-inspired metaheuristic algorithm, called monarch butterfly optimization (MBO), a first of its kind, is proposed in this paper. In MBO, all the monarch butterfly individuals are located in two distinct lands, viz. southern Canada and the northern USA (Land 1) and Mexico (Land 2). Accordingly, the positions of the monarch butterflies are updated in two ways. Firstly, the offsprings are generated (position updating) by migration operator, which can be adjusted by the migration ratio. It is followed by tuning the positions for other butterflies by means of butterfly adjusting operator. In order to keep the population unchanged and minimize fitness evaluations, the sum of the newly generated butterflies in these two ways remains equal to the original population. In order to demonstrate the superior performance of the MBO algorithm, a comparative study with five other metaheuristic algorithms through thirty-eight benchmark problems is carried out. The results clearly exhibit the capability of the MBO method toward finding the enhanced function values on most of the benchmark problems with respect to the other five algorithms. Note that the source codes of the proposed MBO algorithm are publicly available at GitHub (, C++/MATLAB) and MATLAB Central (, MATLAB).

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  1. 1.

    Cui Z, Gao X (2012) Theory and applications of swarm intelligence. Neural Comput Appl 21(2):205–206

    Article  Google Scholar 

  2. 2.

    Cui Z, Fan S, Zeng J, Shi Z (2013) APOA with parabola model for directing orbits of chaotic systems. Int J Bio-Inspired Comput 5(1):67–72

    Article  Google Scholar 

  3. 3.

    Gao XZ, Wang X, Jokinen T, Ovaska SJ, Arkkio A, Zenger K (2012) A hybrid PBIL-based harmony search method. Neural Comput Appl 21(5):1071–1083. doi:10.1007/s00521-011-0675-6

    Article  Google Scholar 

  4. 4.

    Gao XZ, Ovaska SJ, Wang X, Chow MY (2007) A neural networks-based negative selection algorithm in fault diagnosis. Neural Comput Appl 17(1):91–98. doi:10.1007/s00521-007-0092-z

    Article  Google Scholar 

  5. 5.

    Fister Jr I, Yang X-S, Fister I, Brest J, Fister D (2013) A brief review of nature-inspired algorithms for optimization. arXiv:1307.4186

  6. 6.

    Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Paper presented at the proceeding of the IEEE international conference on neural networks, Perth, Australia, 27 November-1 December

  7. 7.

    Ram G, Mandal D, Kar R, Ghoshal SP (2014) Optimal design of non–uniform circular antenna arrays using PSO with wavelet mutation. Int J Bio-Inspired Comput 6(6):424–433

    Article  Google Scholar 

  8. 8.

    Mirjalili S, Wang G-G, Coelho LdS (2014) Binary optimization using hybrid particle swarm optimization and gravitational search algorithm. Neural Comput Appl 25(6):1423–1435. doi:10.1007/s00521-014-1629-6

    Article  Google Scholar 

  9. 9.

    Wang G-G, Gandomi AH, Alavi AH, Deb S (2015) A hybrid method based on krill herd and quantum-behaved particle swarm optimization. Neural Comput Appl. doi:10.1007/s00521-015-1914-z

    Google Scholar 

  10. 10.

    Wang G-G, Gandomi AH, Yang X-S, Alavi AH (2014) A novel improved accelerated particle swarm optimization algorithm for global numerical optimization. Eng Comput 31(7):1198–1220. doi:10.1108/EC-10-2012-0232

    Article  Google Scholar 

  11. 11.

    Dorigo M, Maniezzo V, Colorni A (1996) Ant system: optimization by a colony of cooperating agents. IEEE Trans Syst Man Cybern B Cybern 26(1):29–41. doi:10.1109/3477.484436

    Article  Google Scholar 

  12. 12.

    Krynicki K, Jaen J, Mocholí JA (2014) Ant colony optimisation for resource searching in dynamic peer-to-peer grids. Int J Bio-Inspired Comput 6(3):153–165. doi:10.1504/IJBIC.2014.062634

    Article  Google Scholar 

  13. 13.

    Karaboga D, Basturk B (2007) A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm. J Global Optim 39(3):459–471. doi:10.1007/s10898-007-9149-x

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Li X, Yin M (2012) Self-adaptive constrained artificial bee colony for constrained numerical optimization. Neural Comput Appl 24(3–4):723–734. doi:10.1007/s00521-012-1285-7

    Google Scholar 

  15. 15.

    Yang XS, Deb S Cuckoo search via Lévy flights. In: Abraham A, Carvalho A, Herrera F, Pai V (eds) Proceeding of world congress on nature & biologically inspired computing (NaBIC 2009), Coimbatore, December 2009. IEEE Publications, USA, pp 210–214

  16. 16.

    Ouaarab A, Ahiod B, Yang X-S (2014) Discrete cuckoo search algorithm for the travelling salesman problem. Neural Comput Appl 24(7–8):1659–1669. doi:10.1007/s00521-013-1402-2

    Article  Google Scholar 

  17. 17.

    Yang X-S, Deb S (2013) Cuckoo search: recent advances and applications. Neural Comput Appl 24(1):169–174. doi:10.1007/s00521-013-1367-1

    Article  Google Scholar 

  18. 18.

    Li X, Wang J, Yin M (2013) Enhancing the performance of cuckoo search algorithm using orthogonal learning method. Neural Comput Appl 24(6):1233–1247. doi:10.1007/s00521-013-1354-6

    Article  Google Scholar 

  19. 19.

    Wang G-G, Gandomi AH, Zhao X, Chu HCE (2014) Hybridizing harmony search algorithm with cuckoo search for global numerical optimization. Soft Comput. doi:10.1007/s00500-014-1502-7

    Google Scholar 

  20. 20.

    Yang XS (2010) Nature-inspired metaheuristic algorithms, 2nd edn. Luniver Press, Frome

    Google Scholar 

  21. 21.

    Mirjalili S, Mirjalili SM, Yang X-S (2013) Binary bat algorithm. Neural Comput Appl 25(3–4):663–681. doi:10.1007/s00521-013-1525-5

    Google Scholar 

  22. 22.

    Fister Jr I, Fong S, Brest J, Fister I Towards the self-adaptation in the bat algorithm. In: Proceedings of the 13th IASTED international conference on artificial intelligence and applications, 2014. doi:10.2316/P.2014.816-011

  23. 23.

    Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61. doi:10.1016/j.advengsoft.2013.12.007

    Article  Google Scholar 

  24. 24.

    Saremi S, Mirjalili SZ, Mirjalili SM (2014) Evolutionary population dynamics and grey wolf optimizer. Neural Comput Appl. doi:10.1007/s00521-014-1806-7

    Google Scholar 

  25. 25.

    Mirjalili S (2015) The ant lion optimizer. Adv Eng Softw 83:80–98. doi:10.1016/j.advengsoft.2015.01.010

    Article  Google Scholar 

  26. 26.

    Gandomi AH, Yang X-S, Alavi AH (2011) Mixed variable structural optimization using firefly algorithm. Comput Struct 89(23–24):2325–2336. doi:10.1016/j.compstruc.2011.08.002

    Article  Google Scholar 

  27. 27.

    Yang XS (2010) Firefly algorithm, stochastic test functions and design optimisation. Int J Bio-Inspired Comput 2(2):78–84. doi:10.1504/IJBIC.2010.032124

    Article  Google Scholar 

  28. 28.

    Wang G-G, Guo L, Duan H, Wang H (2014) A new improved firefly algorithm for global numerical optimization. J Comput Theor Nanos 11(2):477–485. doi:10.1166/jctn.2014.3383

    Article  Google Scholar 

  29. 29.

    Guo L, Wang G-G, Wang H, Wang D (2013) An effective hybrid firefly algorithm with harmony search for global numerical optimization. Sci World J 2013:1–10. doi:10.1155/2013/125625

    Google Scholar 

  30. 30.

    Meng X, Liu Y, Gao X, Zhang H (2014) A new bio-inspired algorithm: chicken swarm optimization. In: Tan Y, Shi Y, Coello CC (eds) Advances in swarm intelligence, vol 8794. Lecture notes in computer science. Springer, New York, pp 86-94. doi:10.1007/978-3-319-11857-4_10

  31. 31.

    Gandomi AH, Alavi AH (2012) Krill herd: a new bio-inspired optimization algorithm. Commun Nonlinear Sci Numer Simulat 17(12):4831–4845. doi:10.1016/j.cnsns.2012.05.010

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Li J, Tang Y, Hua C, Guan X (2014) An improved krill herd algorithm: krill herd with linear decreasing step. Appl Math Comput 234:356–367. doi:10.1016/j.amc.2014.01.146

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Wang G-G, Gandomi AH, Alavi AH (2013) A chaotic particle-swarm krill herd algorithm for global numerical optimization. Kybernetes 42(6):962–978. doi:10.1108/K-11-2012-0108

    MathSciNet  Article  Google Scholar 

  34. 34.

    Goldberg DE (1998) Genetic algorithms in search, optimization and machine learning. Addison-Wesley, New York

    Google Scholar 

  35. 35.

    Gao XZ, Ovaska SJ (2002) Genetic algorithm training of Elman neural network in motor fault detection. Neural Comput Appl 11(1):37–44. doi:10.1007/s005210200014

    Article  MATH  Google Scholar 

  36. 36.

    Bäck T (1996) Evolutionary algorithms in theory and practice: evolution strategies, evolutionary programming, genetic algorithms. Oxford University Press, Oxford

    MATH  Google Scholar 

  37. 37.

    Zhao X, Gao X-S, Hu Z-C (2007) Evolutionary programming based on non-uniform mutation. Appl Math Comput 192(1):1–11. doi:10.1016/j.amc.2006.06.107

    MathSciNet  Article  MATH  Google Scholar 

  38. 38.

    Hand D (1992) Genetic programming: on the programming of computers by means of natural selection. MIT Press, Cambridge

    Google Scholar 

  39. 39.

    Beyer H, Schwefel H (2002) Natural computing. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  40. 40.

    Storn R, Price K (1997) Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces. J Global Optim 11(4):341–359. doi:10.1023/A:1008202821328

    MathSciNet  Article  MATH  Google Scholar 

  41. 41.

    Wang G-G, Gandomi AH, Alavi AH, Hao G-S (2014) Hybrid krill herd algorithm with differential evolution for global numerical optimization. Neural Comput Appl 25(2):297–308. doi:10.1007/s00521-013-1485-9

    Article  Google Scholar 

  42. 42.

    Khatib W, Fleming P (1998) The stud GA: A mini revolution? In: Eiben A, Back T, Schoenauer M, Schwefel H (eds) Proceedings of the 5th international conference on parallel problem solving from nature, New York, 1998. Parallel problem solving from nature. Springer, London, pp 683–691

  43. 43.

    Wang G-G, Gandomi AH, Alavi AH (2014) Stud krill herd algorithm. Neurocomputing 128:363–370. doi:10.1016/j.neucom.2013.08.031

    Article  Google Scholar 

  44. 44.

    Gandomi AH, Alavi AH (2011) Multi-stage genetic programming: a new strategy to nonlinear system modeling. Inf Sci 181(23):5227–5239. doi:10.1016/j.ins.2011.07.026

    Article  Google Scholar 

  45. 45.

    Simon D (2008) Biogeography-based optimization. IEEE Trans Evolut Comput 12(6):702–713. doi:10.1109/TEVC.2008.919004

    Article  Google Scholar 

  46. 46.

    Saremi S, Mirjalili S, Lewis A (2014) Biogeography-based optimisation with chaos. Neural Comput Appl 25(5):1077–1097. doi:10.1007/s00521-014-1597-x

    Article  Google Scholar 

  47. 47.

    Li X, Yin M (2013) Multiobjective binary biogeography based optimization for feature selection using gene expression data. IEEE Trans Nanobiosci 12(4):343–353. doi:10.1109/TNB.2013.2294716

    Article  Google Scholar 

  48. 48.

    Wang G-G, Gandomi AH, Alavi AH (2014) An effective krill herd algorithm with migration operator in biogeography-based optimization. Appl Math Model 38(9–10):2454–2462. doi:10.1016/j.apm.2013.10.052

    MathSciNet  Article  MATH  Google Scholar 

  49. 49.

    Wang G, Guo L, Duan H, Wang H, Liu L, Shao M (2013) Hybridizing harmony search with biogeography based optimization for global numerical optimization. J Comput Theor Nanosci 10(10):2318–2328. doi:10.1166/jctn.2013.3207

    Google Scholar 

  50. 50.

    Li X, Zhang J, Yin M (2014) Animal migration optimization: an optimization algorithm inspired by animal migration behavior. Neural Comput Appl 24(7–8):1867–1877. doi:10.1007/s00521-013-1433-8

    Article  Google Scholar 

  51. 51.

    Garber SD (1998) The Urban Naturalist. Dover Publications, Mineola

    Google Scholar 

  52. 52.

    Klots AB (1978) Field guide to the butterflies of North America, East of the great plains. Peterson Field Guides, Boston, USA

    Google Scholar 

  53. 53.

    Breed GA, Severns PM, Edwards AM (2015) Apparent power-law distributions in animal movements can arise from intraspecific interactions. J R Soc Interface 12(103):20140927. doi:10.1098/rsif.2014.0927

    Article  Google Scholar 

  54. 54.

    Yao X, Liu Y, Lin G (1999) Evolutionary programming made faster. IEEE Trans Evolut Comput 3(2):82–102

    Article  Google Scholar 

  55. 55.

    Yang X-S, Cui Z, Xiao R, Gandomi AH, Karamanoglu M (2013) Swarm intelligence and bio-inspired computation. Elsevier, Waltham

    Book  Google Scholar 

  56. 56.

    Wang G, Guo L, Wang H, Duan H, Liu L, Li J (2014) Incorporating mutation scheme into krill herd algorithm for global numerical optimization. Neural Comput Appl 24(3–4):853–871. doi:10.1007/s00521-012-1304-8

    Article  Google Scholar 

  57. 57.

    Wang G-G, Guo L, Gandomi AH, Hao G-S, Wang H (2014) Chaotic krill herd algorithm. Inf Sci 274:17–34. doi:10.1016/j.ins.2014.02.123

    MathSciNet  Article  Google Scholar 

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This work was supported by Research Fund for the Doctoral Program of Jiangsu Normal University (No. 13XLR041).

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Correspondence to Gai-Ge Wang.

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Wang, GG., Deb, S. & Cui, Z. Monarch butterfly optimization. Neural Comput & Applic 31, 1995–2014 (2019).

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  • Evolutionary computation
  • Monarch butterfly optimization
  • Migration
  • Butterfly adjusting operator
  • Benchmark problems