A modified butterfly optimization algorithm for mechanical design optimization problems

Technical Paper
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Abstract

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.

Keywords

Butterfly optimization algorithm Intensive exploitation Benchmark functions Engineering design problems 

List of symbols

b

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

b1

The lower boundary of x

b2

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)

fi

Perceived magnitude of fragrance of ith butterfly

G

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

f(x)

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]

Pc

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

xi

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

Greek symbols

α

Power exponent in BOA or significance level in Wilcoxon test

β0

Firefly algorithm parameter

f(x)

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)

δmax

Maximum beam deflection in welded beam design problem (inch)

σmax

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

τ

Shear stress in welded beam design problem (psi)

τmax

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
  • 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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