A modified butterfly optimization algorithm for mechanical design optimization problems


This paper presents a modified butterfly optimization algorithm (MBOA) for solving mechanical design optimization problems. The modification is focused on an additional intensive exploitation phase which provides more chance to solutions to improve itself. The performance of the proposed algorithm is validated on fifteen benchmark test functions and three engineering design problems which have different natures of objective functions, constraints and decision variables. The experimental results are analyzed in comparison with those reported in the literature. The results indicate that the MBOA provides very competitive results in comparison with other existing optimization algorithms.

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b :

Bar’s thickness in welded beam design problem (inch)


The lower boundary of x


The upper boundary of x

c :

Sensory modality

d :

Wire diameter in tension/compression spring design problem

D :

Mean coil diameter in tension/compression spring design problem

E :

Modulus of elasticity (Young’s modulus) in welded beam design problem (psi)

f :

Benchmark function (herein f1 to f15)

f i :

Perceived magnitude of fragrance of ith butterfly

G :

Modulus of rigidity or shear modulus in welded beam design problem (psi)


Objective function, x = (x1x d )

g * :

Fittest butterfly/solution vector

H :

Hessian (a square matrix of second-order partial derivatives of a scalar-valued function)

h :

Weld thickness in welded beam design problem (inch)

I :

Stimulus intensity

i :

Butterfly index number

J :

Jacobian (all first-order partial derivatives of a vector-valued function)

L :

Overhang or cantilever length of the member in welded beam design problem (inch)

l :

Length of the bar attached to the weld in welded beam design problem

n :

Number of initial values in vector of x0

N :

Number of active coils in tension/compression spring design problem

p :

Switch probability within [0, 1]

P c :

Bar buckling load in welded beam design problem (lb)

r :

Random number (\({\text{rand}}_{1}\) and \({\text{rand}}_{2}\)) from a uniform distribution [0, 1]

T :

Number of teeth on gears (herein gears A, B, D, F) in gear train design problem

t :

Iteration number (see Eq. (2)) or bar’s height in welded beam design problem

x i :

Initial butterfly population (i = 1, 2…n)

\(\varvec{x}_{i}^{t}\) :

Solution vector \(\varvec{x}_{i}\)

\(\varvec{x}_{j}^{t}\) :

jth butterfly from the solution space from the current population

\(\varvec{x}_{k}^{t}\) :

kth butterfly from the solution space from the current population

x0 :

Vector of the initial points randomly generated

α :

Power exponent in BOA or significance level in Wilcoxon test

β 0 :

Firefly algorithm parameter


Gradient of f computed at x

γ :

Firefly algorithm parameter

η :

Decision variable (defined as η(x i ), where i = 1–4) in gear train design problem

δ :

Beam deflection in welded beam design problem (inch)

δ max :

Maximum beam deflection in welded beam design problem (inch)

σ max :

Design normal stress for the beam material in welded beam design problem (psi)

τ :

Shear stress in welded beam design problem (psi)

τ max :

Design stress of the weld in welded beam design problem (psi)


  1. 1.

    Glover F, Laguna M (1997) Tabu search. Kluwer Academic Publishers, Boston

    Google Scholar 

  2. 2.

    Wright J, Alajmi A (2016) Efficient Genetic Algorithm sets for optimizing constrained building design problem. Int J Sustain Built Environ 5:123–131

    Article  Google Scholar 

  3. 3.

    Pagani Jr CC, Trindade MA (2009) Optimization of modal filters based on arrays of piezoelectric sensors. Smart Materials Struct 18(9):1–12

    Article  Google Scholar 

  4. 4.

    Garg H (2015) A hybrid GA-GSA algorithm for optimizing the performance of an industrial system by utilizing uncertain data. In: Handbook of research on artificial intelligence techniques and algorithms. IGI Global Press, pp 620–654

  5. 5.

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

    Article  Google Scholar 

  6. 6.

    Toksari MD (2006) Ant colony optimization for finding the global minimum. Appl Math Comput 176:308–316

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Satya EJ, Venkateswarlu C (2013) Evaluation of anaerobic biofilm reactor kinetic parameters using ant colony optimization. Environ Eng Sci 30:527–535

    Article  Google Scholar 

  8. 8.

    Yuguang Z, Bo A, Yong Z (2016) A PSO algorithm for multi-objective hull assembly line balancing using the stratified optimization strategy. Comput Ind Eng 98:53–62

    Article  Google Scholar 

  9. 9.

    Garg H, Sharma SP (2013) Multi-objective reliability-redundancy allocation problem using particle swarm optimization. Comput Ind Eng 64(1):247–255

    Article  Google Scholar 

  10. 10.

    Garg H, Rani M, Sharma SP, Vishwakarma Y (2014) Bi-objective optimization of the reliability-redundancy allocation problem for series-parallel system. J Manuf Syst 33:353–367

    Article  Google Scholar 

  11. 11.

    Jadon SS, Bansal JC, Tiwari R (2016) Escalated convergent artificial bee colony. J Exp Theor Artif Intell 28:181–200

    Article  Google Scholar 

  12. 12.

    Garg H, Rani M, Sharma SP (2013) An efficient two phase approach for solving reliability–redundancy allocation problem using artificial bee colony technique. Comput Oper Res 40:2961–2969

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Kaveh A, Share MAM, Moslehi M (2013) Magnetic charged system search: a new meta-heuristic algorithm for optimization. Acta Mech 224:1–23

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Kaveh A, Talatahari S (2010) A novel heuristic optimization method: charged system search. Acta Mech 213(3):267–289

    Article  MATH  Google Scholar 

  15. 15.

    Arora S, Singh S (2013) The firefly optimization algorithm: convergence analysis and parameter selection. Int J Comput Appl 69:48–52

    Google Scholar 

  16. 16.

    Fister I, Fister I Jr, Yang X-S, Brest J (2013) A comprehensive review of firefly algorithms. Swarm Evol Comput 13:34–46

    Article  Google Scholar 

  17. 17.

    Yu S, Zhu S, Ma Y, Mao D (2015) A variable step size firefly algorithm for numerical optimization. Appl Math Comput 263:214–220

    MathSciNet  Google Scholar 

  18. 18.

    Gupta A, Padhy PK (2016) Modified Firefly Algorithm based controller design for integrating and unstable delay processes. Eng Sci Technol 19:548–558

    Google Scholar 

  19. 19.

    Singh S, Tayal S, Sachdeva G (2012) Evolutionary performance of BBO and PSO algorithms for Yagi–Uda antenna design optimization. In: 2012 World Congress on information and communication technologies (WICT). New Delhi, India, IEEE Press, pp 861–865

  20. 20.

    Alroomi AR, Albasri FA, Talaq JH (2013) Solving the associated weakness of biogeography-based optimization algorithm. Int J Soft Comput 4:1–20

    Article  Google Scholar 

  21. 21.

    Garg H (2015) An efficient biogeography based optimization algorithm for solving reliability optimization problems. Swarm Evol Comput 24:1–10

    Article  Google Scholar 

  22. 22.

    Yang XS (2010) A new metaheuristic bat-inspired algorithm. In: Gonzalez JR, et al (eds) Nature inspired cooperative strategies for optimization (NISCO 2010). Studies in computational intelligence. Springer, Berlin, vol 284, pp 65–74

  23. 23.

    Garg H (2015) An approach for solving constrained reliability-redundancy allocation problems using cuckoo search algorithm. Beni-Suef Univ J Basic Appl Sci 4:14–25

    Article  Google Scholar 

  24. 24.

    Arora S, Singh S (2017) An improved butterfly optimization algorithm with chaos. J Intell Fuzzy Syst 32:1079–1088

    Article  MATH  Google Scholar 

  25. 25.

    Arora S, Singh S (2015) A conceptual model of butterfly algorithm. In: Proceedings of national conference on latest initiatives and innovations in communication and electronics (IICE 2015). Chandigarh University, Mohali, Punjab, India, pp 69–72

  26. 26.

    Arora S, Singh S (2015) Butterfly algorithm with L’evy flights for global optimization. In: International conference on signal processing, computing and control (ISPCC). Jaypee University of Information Technology, Solan, India, pp 20–224

  27. 27.

    Yang XS (2010) Nature-inspired metaheuristic algorithms. Luniver Press, Bristol

    Google Scholar 

  28. 28.

    Blair RB, Launer AE (1997) Butterfly diversity and human land use: species assemblages along an urban gradient. Biol Conserv 80:113–125

    Article  Google Scholar 

  29. 29.

    Zwislocki JJ (2009) Sensory neuroscience: four laws of psychophysics. Springer Science & Business Media, Berlin

    Google Scholar 

  30. 30.

    Stevens SS (1975) Psychophysics. Transaction Publishers, Piscataway

    Google Scholar 

  31. 31.

    Wolpert DH, Macready WG (1997) No free lunch theorems for optimization. IEEE Trans Evol Comput 1:67–82

    Article  Google Scholar 

  32. 32.

    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 24:853–871

    Article  Google Scholar 

  33. 33.

    Eberhart RC, Kennedy J (1995) A new optimizer using particle swarm theory. In: Proceedings of the sixth international symposium on micro machine and human science, vol 1. Piscataway, NJ, Nagoya, Japan, IEEE Press, pp 39–43

  34. 34.

    Garg H, Rani M, Sharma SP, Vishwakarma Y (2014) Intuitionistic fuzzy optimization technique for solving multi-objective reliability optimization problems in interval environment. Expert Syst Appl 41:3157–3167

    Article  Google Scholar 

  35. 35.

    Hansen N, Sibylle D, Petros K (2003) Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (CMA-ES). Evolut Comput 11(1):1–18

    Article  Google Scholar 

  36. 36.

    Brest J et al (2006) Self-adapting control parameters in differential evolution: a comparative study on numerical benchmark problems. IEEE Trans Evol Comput 10(6):646–657

    Article  Google Scholar 

  37. 37.

    Shi Y, Eberhart RC (2001) Fuzzy adaptive particle swarm optimization. In: Proceedings of the 2001 Congress on evolutionary computation, vol 1. World Trade Center, Seoul, Korea, IEEE Press, pp 101–106

  38. 38.

    Yang XS (2010) Engineering optimization: an introduction with metaheuristic applications. Wiley, New York

    Google Scholar 

  39. 39.

    Tomassini M, Antonioni A, Daolio F, Buesser P (eds) (2013) Adaptive and natural computing algorithms. In: 11th international conference, ICANNGA 2013, Lausanne, Switzerland, April 4–6, 2013, proceedings. Springer, Berlin, p 506

  40. 40.

    Arora J (2004) Introduction to optimum design. Elsevier Academic Press, San Diego

    Google Scholar 

  41. 41.

    Belegundu A, Arora J (1985) A study of mathematical programming methods for structural optimization. Part I: theory. Int J Numer Methods Eng 21(9):1583–1599

    Article  MATH  Google Scholar 

  42. 42.

    Garg H (2014) Solving structural engineering design optimization problems using an artificial bee colony algorithm. J Ind Manag Optim 10:777–794

    MathSciNet  Article  MATH  Google Scholar 

  43. 43.

    Mirjalili S, Mirjalili S, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61

    Article  Google Scholar 

  44. 44.

    Mirjalili S, Lewis A (2014) Adaptive gbest-guided gravitational search algorithm. Neural Comput Appl 25(7):1569–1584

    Article  Google Scholar 

  45. 45.

    He Q, Wang L (2007) An effective co-evolutionary particle swarm optimization for constrained engineering design problems. Eng Appl Artif Intell 20(1):89–99

    MathSciNet  Article  Google Scholar 

  46. 46.

    Mezura-Montes E, Coello C (2008) An empirical study about the usefulness of evolution strategies to solve constrained optimization problems. Int J Gen Syst 37(4):443–473

    MathSciNet  Article  MATH  Google Scholar 

  47. 47.

    Coello C (2000) Use of a self-adaptive penalty approach for engineering optimization problems. Comput Ind 41(2):113–127

    Article  Google Scholar 

  48. 48.

    Mahdavi M, Fesanghary M, Damangir E (2007) An improved harmony search algorithm for solving optimization problems. Appl Math Comput 188(2):1567–1579

    MathSciNet  MATH  Google Scholar 

  49. 49.

    Huang F, Wang L, He Q (2007) An effective co-evolutionary differential evolution for constrained optimization. Appl Math Comput 186(1):340–356

    MathSciNet  MATH  Google Scholar 

  50. 50.

    Coello F (2000) Constraint-handling using an evolutionary multiobjective optimization technique. Civil Eng Syst 17(4):319–346

    Article  Google Scholar 

  51. 51.

    Ragsdell K, Phillips D (1976) Optimal design of a class of welded structures using geometric programming. J Manuf Sci Eng 98(3):1021–1025

    Google Scholar 

  52. 52.

    Sandgren E (1990) Nonlinear integer and discrete programming in mechanical design optimization. J Mech Des 112(2):223–229

    Article  Google Scholar 

  53. 53.

    Deb K (1991) Optimal design of a welded beam via genetic algorithms. AIAA J 29(11):2013–2015

    Article  Google Scholar 

  54. 54.

    Deb K (2000) An efficient constraint handling method for genetic algorithms. Comput Methods Appl Mech Eng 186(2):311–338

    Article  MATH  Google Scholar 

  55. 55.

    Lee K, Geem Z (2005) A new meta-heuristic algorithm for continuous engineering optimization: harmony search theory and practice. Comput Methods Appl Mech Eng 194(36):3902–3933

    Article  MATH  Google Scholar 

  56. 56.

    Wu S, Chow P (1995) Genetic algorithms for nonlinear mixed discrete-integer optimization problems via meta-genetic parameter optimization. Eng Optim 24(2):137–159

    Article  Google Scholar 

  57. 57.

    Deb K, Goyal M (1996) A combined genetic adaptive search (GeneAS) for engineering design. Comput Sci Inform 26:30–45

    Google Scholar 

  58. 58.

    Gandomi A, Yang XS, Alavi A (2013) Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems. Eng Comput 29(1):17–35

    Article  Google Scholar 

  59. 59.

    Parsopoulos K, Vrahatis M (2005) Unified particle swarm optimization for solving constrained engineering optimization problems. In: Advances in natural computation. Springer, Berlin, Heidelberg, pp 582–591

  60. 60.

    Gandomi A, Yun G, Yang X, Talatahari S (2013) Chaos-enhanced accelerated particle swarm optimization. Commun Nonlinear Sci Numer Simul 18(2):327–340

    MathSciNet  Article  MATH  Google Scholar 

  61. 61.

    Zhang C, Wang H (1993) Mixed-discrete nonlinear optimization with simulated annealing. Eng Optim 21:27791

    Article  Google Scholar 

  62. 62.

    Kannan B, Kramer S (1994) An augmented lagrange multiplier based method for mixed integer discrete continuous optimization and its applications to mechanical design. J Mech Des 116(2):405–411

    Article  Google Scholar 

  63. 63.

    Fu J, Fenton R, Cleghorn W (1991) A mixed integer-discrete-continuous programming method and its application to engineering design optimization. Eng Optim 17(4):263–280

    Article  Google Scholar 

  64. 64.

    Cao Y, Wu Q (1997) Mechanical design optimization by mixed variable evolutionary programming. In: IEEE conference on evolutionary computation. University Place Hotel, Indianapolis, Indiana, USA, IEEE Press, p 4436

  65. 65.

    Loh H, Papalambros P (1991) A sequential linearization approach for solving mixed-discrete nonlinear design optimization problems. J Mech Des 113(3):325–334

    Article  Google Scholar 

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Correspondence to Sankalap Arora.

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Technical Editor: Marcelo A. Trindade.

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Arora, S., Singh, S. & Yetilmezsoy, K. A modified butterfly optimization algorithm for mechanical design optimization problems. J Braz. Soc. Mech. Sci. Eng. 40, 21 (2018). https://doi.org/10.1007/s40430-017-0927-1

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  • Butterfly optimization algorithm
  • Intensive exploitation
  • Benchmark functions
  • Engineering design problems