A modified butterfly optimization algorithm for mechanical design optimization problems

  • Sankalap AroraEmail author
  • Satvir Singh
  • Kaan Yetilmezsoy
Technical Paper


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.


Butterfly optimization algorithm Intensive exploitation Benchmark functions Engineering design problems 

List of symbols


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


The lower boundary of x


The upper boundary of x


Sensory modality


Wire diameter in tension/compression spring design problem


Mean coil diameter in tension/compression spring design problem


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


Benchmark function (herein f1 to f15)


Perceived magnitude of fragrance of ith butterfly


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


Objective function, x = (x1x d )


Fittest butterfly/solution vector


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


Weld thickness in welded beam design problem (inch)


Stimulus intensity


Butterfly index number


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


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


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


Number of initial values in vector of x0


Number of active coils in tension/compression spring design problem


Switch probability within [0, 1]


Bar buckling load in welded beam design problem (lb)


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


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


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


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


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


jth butterfly from the solution space from the current population


kth butterfly from the solution space from the current population


Vector of the initial points randomly generated

Greek symbols


Power exponent in BOA or significance level in Wilcoxon test


Firefly algorithm parameter


Gradient of f computed at x


Firefly algorithm parameter


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


Beam deflection in welded beam design problem (inch)


Maximum beam deflection in welded beam design problem (inch)


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


Shear stress in welded beam design problem (psi)


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


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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  • Sankalap Arora
    • 1
    Email author
  • Satvir Singh
    • 2
  • Kaan Yetilmezsoy
    • 3
  1. 1.Department of Computer Science and EngineeringDAV UniversityJalandharIndia
  2. 2.Department of Electronics and Communication EngineeringI.K. Gujral Punjab Technical UniversityJalandharIndia
  3. 3.Department of Environmental Engineering, Faculty of Civil EngineeringYildiz Technical UniversityIstanbulTurkey

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