Soft Computing

, Volume 23, Issue 3, pp 715–734 | Cite as

Butterfly optimization algorithm: a novel approach for global optimization

  • Sankalap AroraEmail author
  • Satvir Singh


Real-world problems are complex as they are multidimensional and multimodal in nature that encourages computer scientists to develop better and efficient problem-solving methods. Nature-inspired metaheuristics have shown better performances than that of traditional approaches. Till date, researchers have presented and experimented with various nature-inspired metaheuristic algorithms to handle various search problems. This paper introduces a new nature-inspired algorithm, namely butterfly optimization algorithm (BOA) that mimics food search and mating behavior of butterflies, to solve global optimization problems. The framework is mainly based on the foraging strategy of butterflies, which utilize their sense of smell to determine the location of nectar or mating partner. In this paper, the proposed algorithm is tested and validated on a set of 30 benchmark test functions and its performance is compared with other metaheuristic algorithms. BOA is also employed to solve three classical engineering problems (spring design, welded beam design, and gear train design). Results indicate that the proposed BOA is more efficient than other metaheuristic algorithms.


Butterfly optimization algorithm Global optimization Nature inspired Metaheuristic Benchmark test functions Engineering design problems 



The authors acknowledge the contribution of I. K. Gujral Punjab Technical University, Kapurthala, Punjab, India.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Arora JS (2004) Introduction to optimum design. Elsevier Academic Press, EnglandGoogle Scholar
  2. Arora S, Singh S (2013a) The firefly optimization algorithm: convergence analysis and parameter selection. Int J Comput Appl 69(3):48–52Google Scholar
  3. Arora S, Singh S (2013b) A conceptual comparison of firefly algorithm, bat algorithm and cuckoo search. In: 2013 international conference on control computing communication and materials (ICCCCM), IEEE, pp 1–4Google Scholar
  4. Arora S, Singh S, Singh S, Sharma B (2014) Mutated firefly algorithm. In: 2014 international conference on parallel, distributed and grid computing (PDGC), IEEE, pp 33–38Google Scholar
  5. Back T (1996) Evolutionary algorithms in theory and practice. Oxford University Press, OxfordzbMATHGoogle Scholar
  6. Baird JC, Noma EJ (1978) Fundamentals of scaling and psychophysics. Wiley, HobokenGoogle Scholar
  7. Belegundu AD, Arora JS (1985) A study of mathematical programming methods for structural optimization. Part I: theory. Int J Numer Methods Eng 21(9):1583–1599zbMATHGoogle Scholar
  8. Blair RB, Launer AE (1997) Butterfly diversity and human land use: species assemblages along an urban grandient. Biol Conserv 80(1):113–125Google Scholar
  9. Brownlee J (2011) Clever algorithms: nature-inspired programming recipes, 1st edn. LuLu. ISBN 978-1-4467-8506-5Google Scholar
  10. Cao Y, Wu Q (1997) Mechanical design optimization by mixed-variable evolutionary programming. In: IEEE conference on evolutionary computation, IEEE Press, p 443–446Google Scholar
  11. Coello CAC (2000a) Use of a self-adaptive penalty approach for engineering optimization problems. Comput Ind 41(2):113–127Google Scholar
  12. Coello CA (2000b) Constraint-handling using an evolutionary multiobjective optimization technique. Civil Eng Syst 17(4):319–346Google Scholar
  13. Coello CAC, Montes EM (2002) Constraint-handling in genetic algorithms through the use of dominance-based tournament selection. Adv Eng Inform 16(3):193–203Google Scholar
  14. Deb K (1991) Optimal design of a welded beam via genetic algorithms. AIAA J 29(11):2013–2015Google Scholar
  15. Deb K (2000) An efficient constraint handling method for genetic algorithms. Comput Methods Appl Mech Eng 186(2):311–338zbMATHGoogle Scholar
  16. Deb K, Goyal M (1996) A combined genetic adaptive search (GeneAS) for engineering design. Comput Sci Inf 26:30–45Google Scholar
  17. Eberhart RC, Shi Y (2001) Particle swarm optimization: developments, applications and resources. In: Proceedings of the 2001 congress on evolutionary computation, 2001, vol 1. IEEE, pp 81–86Google Scholar
  18. Fister I Jr, Yang X-S, Fister I, Brest J, Fister D (2013) A brief review of nature-inspired algorithms for optimization. arXiv preprint arXiv:1307.4186
  19. Fu J-F, Fenton RG, Cleghorn WL (1991) A mixed integer-discrete-continuous programming method and its application to engineering design optimization. Eng Optim 17(4):263–280Google Scholar
  20. Gandomi AH, Yang X-S, Alavi AH (2013a) Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems. Eng Comput 29(1):17–35Google Scholar
  21. Gandomi AH, Yun GJ, Yang X-S, Talatahari S (2013) Chaos-enhanced accelerated particle swarm optimization. Commun Nonlinear Sci Numer Simul 18(2):327–340MathSciNetzbMATHGoogle Scholar
  22. Gazi V, Passino KM (2004) Stability analysis of social foraging swarms. Syst Man Cybern Part B: Cybern IEEE Trans 34(1):539–557Google Scholar
  23. Goldberg DE, Holland JH (1988) Genetic algorithms and machine learning. Mach Learn 3(2):95–99Google Scholar
  24. Gupta S, Arora S (2015) A hybrid firefly algorithm and social spider algorithm for multimodal function. Intell Syst Technol Appl 1:17Google Scholar
  25. He Q, Wang L (2007) An effective co-evolutionary particle swarm optimization for constrained engineering design problems. Eng Appl Artif Intell 20(1):89–99Google Scholar
  26. Holland JH (1992) Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control, and artificial intelligence. MIT press, CambridgeGoogle Scholar
  27. Huang F-Z, Wang L, He Q (2007) An effective co-evolutionary differential evolution for constrained optimization. Appl Math Comput 186(1):340–356MathSciNetzbMATHGoogle Scholar
  28. Kalra S, Arora S (2016) Firefly algorithm hybridized with flower pollination algorithm for multimodal functions. In: Proceedings of the international congress on information and communication technology, Springer, Singapore, pp 207–219Google Scholar
  29. Kannan B, Kramer SN (1994) An augmented lagrange multiplier based method for mixed integer discrete continuous optimization and its applications to mechanical design. J Mech Des 116(2):405–411Google Scholar
  30. Karaboga D, Basturk B (2007) A powerful and efficient algorithm for numerical function optimization: artificial bee colony (abc) algorithm. J Glob Optim 39(3):459–471MathSciNetzbMATHGoogle Scholar
  31. Karaboga D, Basturk B (2008) On the performance of artificial bee colony (abc) algorithm. Appl Soft Comput 8(1):687–697Google Scholar
  32. Kennedy J (2010) Particle swarm optimization. In: Sammut C, Webb GI (eds) Encyclopedia of machine learning. Springer, Boston, MAGoogle Scholar
  33. Lee KS, Geem ZW (2005) A new meta-heuristic algorithm for continuous engineering optimization: harmony search theory and practice. Comput Methods Appl Mech Eng 194(36):3902–3933zbMATHGoogle Scholar
  34. Liang JJ, Qu BY, Suganthan PN, Hernández-Díaz AG (2013) Problem definitions and evaluation criteria for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212, 3–18Google Scholar
  35. Loh HT, Papalambros PY (1991) A sequential linearization approach for solving mixed-discrete nonlinear design optimization problems. J Mech Des 113(3):325–334Google Scholar
  36. MacKay D (1963) Psychophysics of perceived intensity: a theoretical basis for fechner’s and stevens’ laws. Science 139(3560):1213–1216Google Scholar
  37. Maesono Y (1987) Competitors of the wilcoxon signed rank test. Ann Inst Stat Math 39(1):363–375MathSciNetzbMATHGoogle Scholar
  38. Mahdavi M, Fesanghary M, Damangir E (2007) An improved harmony search algorithm for solving optimization problems. Appl Math Comput 188(2):1567–1579MathSciNetzbMATHGoogle Scholar
  39. Mezura-Montes E, Coello CAC (2008) An empirical study about the usefulness of evolution strategies to solve constrained optimization problems. Int J Gen Syst 37(4):443–473MathSciNetzbMATHGoogle Scholar
  40. Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61Google Scholar
  41. Onwubolu GC, Babu B (2004) New optimization techniques in engineering, vol 141. Springer, BerlinzbMATHGoogle Scholar
  42. Parsopoulos KE, Vrahatis MN (2005) Unified particle swarm optimization for solving constrained engineering optimization problems. In: Wang L, Chen K, Ong YS (eds) Advances in natural computation. Springer, Berlin, pp 582–591Google Scholar
  43. Pollard E, Yates TJ (1994) Monitoring butterflies for ecology and conservation: the British butterfly monitoring scheme. Springer, BerlinGoogle Scholar
  44. Ragsdell K, Phillips D (1976) Optimal design of a class of welded structures using geometric programming. J Manuf Sci Eng 98(3):1021–1025Google Scholar
  45. Raguso RA (2008) Wake up and smell the roses: the ecology and evolution of floral scent. Ann Rev Ecol Evolut Syst 39:549–569Google Scholar
  46. Rashedi E, Nezamabadi-Pour H, Saryazdi S (2009) GSA: a gravitational search algorithm. Inf Sci 179(13):2232–2248zbMATHGoogle Scholar
  47. Saccheri I, Kuussaari M, Kankare M, Vikman P, Fortelius W, Hanski I (1998) Inbreeding and extinction in a butterfly metapopulation. Nature 392(6675):491–494Google Scholar
  48. Sandgren E (1990) Nonlinear integer and discrete programming in mechanical design optimization. J Mech Des 112(2):223–229Google Scholar
  49. Shilane D, Martikainen J, Dudoit S, Ovaska SJ (2008) A general framework for statistical performance comparison of evolutionary computation algorithms. Inf Sci 178(14):2870–2879Google Scholar
  50. Stevens SS (1975) Psychophysics. Transaction Publishers, RoutledgeGoogle Scholar
  51. Storn R, Price K (1997) Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim 11(4):341–359MathSciNetzbMATHGoogle Scholar
  52. Talbi E-G (2009) Metaheuristics: from design to implementation, vol 74. Wiley, HobokenzbMATHGoogle Scholar
  53. Wang G-G, Deb S, Cui Z (2015) Monarch butterfly optimization. Neural Comput Applic.
  54. 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):853–871Google Scholar
  55. Wolpert DH, Macready WG (1997) No free lunch theorems for optimization. IEEE Trans Evolut Comput 1(1):67–82Google Scholar
  56. Wyatt TD (2003) Pheromones and animal behaviour: communication by smell and taste. Cambridge University Press, CambridgeGoogle Scholar
  57. Yang X-S, Deb S (2009) Cuckoo search via lévy flights. In: World congress on nature and biologically inspired computing, NaBIC 2009, IEEE, pp 210–214Google Scholar
  58. Yang X-S (2009) Firefly algorithms for multimodal optimization. In: Watanabe O, Zeugmann T (eds) Stochastic algorithms: foundations and applications. SAGA 2009. Lecture Notes in Computer Science, vol 5792. Springer, Berlin, HeidelbergGoogle Scholar
  59. Yang X-S (2010a) Nature-inspired metaheuristic algorithms. Luniver press, BeckingtonGoogle Scholar
  60. Yang X-S (2010b) Firefly algorithm, levy flights and global optimization. In: Bramer M, Ellis R, Petridis M (eds) Research and development in intelligent systems XXVI. Springer, Berlin, pp 209–218Google Scholar
  61. Yao X, Liu Y, Lin G (1999) Evolutionary programming made faster. Evolut Comput IEEE Trans 3(2):82–102Google Scholar
  62. Zhang C, Wang H (1993) Mixed-discrete nonlinear optimization with simulated annealing. Eng Optim 21:277–91Google Scholar
  63. Zwislocki JJ (2009) Sensory neuroscience: four laws of psychophysics: four laws of psychophysics. Springer, BerlinGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.I. K. GUJRAL Punjab Technical UniversityJalandharIndia

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