LMBO-DE: a linearized monarch butterfly optimization algorithm improved with differential evolution

  • Samaneh YazdaniEmail author
  • Esmaeil Hadavandi
Methodologies and Application


Monarch butterfly optimization (MBO) is a recently developed evolutionary algorithm which has been used in many optimization problems. Migration and adjusting operators of MBO have a significant effect on the performance of it. These two operators change candidate variables of each individual independently. So, they are rotationally variant and this is one of the limitations of MBO which can degrade its performance on non-separable problems. There are interactions among variables in non-separable problems and MBO’s operators have not any consideration to it. In this paper, we propose a linearized version of MBO to overcome the above-mentioned limitation of MBO. In other words, migration and adjusting operators of MBO are linearized. Moreover, DE’s mutation operator is integrated in our proposed algorithm to improve exploration of MBO. Our proposed algorithm which is a linearized and hybrid version of MBO (LMBO-DE) is validated by 18 benchmark functions in different dimensionality and is compared with original MBO, one of recently MBO’s improvements, and three other evolutionary algorithms (jDE, JADE, and CLPSO). Experimental results show that our proposed algorithm significantly outperforms the original MBO and its improvement in terms of solution quality and convergence rate. In comparison with the other three algorithms, LMBO-DE can find more accurate solutions.


Monarch butterfly optimization Differential evolution Linearized migration Wilcoxon signed-rank test 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no Conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. Brest J, Greiner S, Boskovic B, Mernik M, Zumer V (2006) Self-adapting control parameters in differential evolution: a comparative study on numerical benchmark problems. IEEE Trans Evolut Comput 10:646–657CrossRefGoogle Scholar
  2. Devikanniga D (2018) Classification of osteoporosis by artificial neural network based on monarch butterfly optimization algorithm. Healthc Technol Lett 5:70CrossRefGoogle Scholar
  3. Dorigo M (1992) Optimization, learning and natural algorithms Ph.D. Thesis, Politecnico di MilanoGoogle Scholar
  4. Eberhart R, Kennedy JA (1995) New optimizer using particle swarm theory. In: Proceedings of the sixth international symposium on micro machine and human science, 1995. MHS’95. IEEE, pp 39–43Google Scholar
  5. Eiben AE, Smith JE (2003) Introduction to evolutionary computing, vol 53. Springer, BerlinCrossRefzbMATHGoogle Scholar
  6. Faris H, Aljarah I, Mirjalili S (2018) Improved monarch butterfly optimization for unconstrained global search and neural network training. Appl Intell 48:445–464CrossRefGoogle Scholar
  7. Feng Y, Yang J, Wu C, Lu M, Zhao X-J (2016) Solving 0–1 knapsack problems by chaotic monarch butterfly optimization algorithm with Gaussian mutation. Memet Comput 10:1–16Google Scholar
  8. Feng Y, Wang G-G, Dong J, Wang L (2017) Opposition-based learning monarch butterfly optimization with Gaussian perturbation for large-scale 0-1 knapsack problem. Comput Electr Eng 67:454CrossRefGoogle Scholar
  9. García S, Molina D, Lozano M, Herrera F (2009) A study on the use of non-parametric tests for analyzing the evolutionary algorithms’ behaviour: a case study on the CEC’2005 special session on real parameter optimization. J Heuristics 15:617–644CrossRefzbMATHGoogle Scholar
  10. Ghetas M, Yong CH, Sumari P (2015) Harmony-based monarch butterfly optimization algorithm. In: 2015 IEEE international conference on control system, computing and engineering (ICCSCE). IEEE, pp 156–161Google Scholar
  11. Goldberg DE (1989) Genetic alogorithms in search optimization & machine learning. Mach Learn 32:95Google Scholar
  12. Gong W, Cai Z, Ling CX (2010) DE/BBO: a hybrid differential evolution with biogeography-based optimization for global numerical optimization. Soft Comput 15:645–665CrossRefGoogle Scholar
  13. Karaboga D, Basturk B (2007) A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm. J Global Optim 39:459–471MathSciNetCrossRefzbMATHGoogle Scholar
  14. Khatib W, Fleming PJ (1998) The stud GA: a mini revolution? In: International conference on parallel problem solving from nature. Springer, pp 683–691Google Scholar
  15. Liang JJ, Qin AK, Suganthan PN, Baskar S (2006) Comprehensive learning particle swarm optimizer for global optimization of multimodal functions. IEEE Trans Evolut Comput 10:281–295CrossRefGoogle Scholar
  16. Liang J, Qu B, Suganthan P (2013) Problem definitions and evaluation criteria for the CEC 2014 special session and competition on single objective real-parameter numerical optimization Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, SingaporeGoogle Scholar
  17. Maitre O, Krüger F, Querry S, Lachiche N, Collet P (2012) EASEA: specification and execution of evolutionary algorithms on GPGPU. Soft Comput 16:261–279. CrossRefGoogle Scholar
  18. Miller JF, Thomson P (2000) Cartesian genetic programming. In: European conference on genetic programming. Springer, pp 121–132Google Scholar
  19. Simon D (2008) Biogeography-based optimization. IEEE Trans Evolut Comput 12:702–713CrossRefGoogle Scholar
  20. Simon D, Omran MG, Clerc M (2014) Linearized biogeography-based optimization with re-initialization and local search. Inf Sci 267:140–157MathSciNetCrossRefGoogle Scholar
  21. Storn R, Price K (1997) Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. J Global Optim 11:341–359MathSciNetCrossRefzbMATHGoogle Scholar
  22. Wang G-G, Deb S, Cui Z (2015) Monarch butterfly optimization. Neural Comput Appl. Google Scholar
  23. Wang G-G, Deb S, Gandomi AH, Zhang Z, Alavi AH (2016a) Chaotic cuckoo search. Soft Comput 20:3349–3362CrossRefGoogle Scholar
  24. Wang G-G, Deb S, Zhao X, Cui Z (2016b) A new monarch butterfly optimization with an improved crossover operator. Oper Res 15:1–25Google Scholar
  25. Yazdani S, Shanbehzadeh J (2015) Balanced Cartesian Genetic Programming via migration and opposition-based learning: application to symbolic regression. Genet Program Evol Mach 16:133–150CrossRefGoogle Scholar
  26. Yazdani S, Shanbehzadeh J, Hadavandi E (2017) MBCGP-FE: a modified balanced cartesian genetic programming feature extractor. Knowl Based Syst 135:89–98CrossRefGoogle Scholar
  27. Yazdani S, Hadavandi E, Hower J, Chehreh Chelgani S (2018) A novel nature-inspired optimization based neural network simulator to predict coal grindability index. Eng Comput 35:1003CrossRefGoogle Scholar
  28. Zhang J, Sanderson AC (2009) JADE: adaptive differential evolution with optional external archive. IEEE Trans Evol Comput 13:945–958CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Computer EngineeringNorth Tehran Branch, Islamic Azad UniversityTehranIran
  2. 2.Department of Industrial EngineeringBirjand University of TechnologyBirjandIran

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