# An Improved Butterfly Optimization Algorithm for Engineering Design Problems Using the Cross-Entropy Method

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Butterfly Optimization Algorithm

Algorithm 1: Butterfly Optimization Algorithm |

Begin |

Objective function $f\left(x\right)$, $x={({x}_{1},{x}_{2},\dots ,{x}_{d})}^{T}$, here d represents the number of dimensions. |

Generate initial population P containing n butterflies $po{p}_{i}(i=1,2,\dots ,n)$. |

Stimulus ${I}_{i}$ intensity at is $po{p}_{i}$ determined by the fitness value $f\left(po{p}_{i}\right)$. |

Define sensor modality c, power exponent a, and switch probability p. |

while stop criteria not met do |

for each butterfly in the population P do |

Calculate fragrance f using Equation (5). |

end for |

Evaluate and rank the population P, and find the best butterfly. |

for each butterfly in the population P do |

Generate a random number $r\sim U[0,1]$. |

if ($r<p$) |

Implement global search using Equation (6). |

else |

Implement local search using Equation (7). |

end if |

end for |

Update the value of the power exponent a. |

end while |

Output the best solution and optimal value. |

End |

#### 2.2. The Cross-Entropy (CE) Method

Algorithm 2: Cross-entropy (CE) for optimization problems |

Begin |

Set $t=0$. Initialize the probability parameter ${\widehat{v}}_{k}$. |

while stop criteria for CE not met do |

Generate ${S}_{1},{S}_{2},\dots ,{S}_{N}{\sim}_{iid}f(x;{\widehat{v}}_{k})$. Evaluate and rank the sample S. |

Solve the problem given in Equation (11) based on the sample S. Denote the solution by $\tilde{v}$. |

Update the parameter ${\widehat{v}}_{k}$ using $\tilde{v}$. |

Set $t=t+1$. |

end while |

Output the best solution and optimal value. |

End |

## 3. Hybrid BOA-CE Method

Algorithm 3: BOA-CE algorithm |

Begin |

Objective function $f\left(x\right)$, $x={({x}_{1},{x}_{2},\dots ,{x}_{d})}^{T}$; here, d represents the number of dimensions. |

Generate initial population P containing n butterflies $po{p}_{i}(i=1,2,\dots ,n)$. |

Stimulus ${I}_{i}$ intensity at is $po{p}_{i}$ determined by the fitness value $f\left(po{p}_{i}\right)$. |

Define sensor modality c, power exponent a, and switch probability p. |

while stop criteria not met do |

for each butterfly in the population P do |

Calculate fragrance f using Equation (5). |

end for |

Evaluate, rank P, and find the best butterfly. |

for each butterfly in the population P do |

Generate a random number $r\sim U[0,1]$. |

if ($r<p$) |

Implement global search using Equation (6). |

else |

Implement local search using Equation (7). |

end if |

end for |

Update the value of the power exponent a. |

Evaluate and rank the population P. |

Initialize the probability parameter ${\widehat{v}}_{k}$ using the population P. |

while stop criteria for CE not met do |

Generate ${S}_{1},{S}_{2},\dots ,{S}_{N}{\sim}_{iid}f(x;{\widehat{v}}_{k})$, evaluate the sample S. |

Rank the population P and the sample S together, co-update P and S, update the best butterfly. |

Calculate the probability parameter $\tilde{v}$ by the elite sample ${S}_{e}$. |

Update the probability parameters ${\widehat{v}}_{k}$ via Equation (12). |

end while |

end while |

Output the best solution and optimal value. |

End |

## 4. Experiment and Results

#### 4.1. Results of the Algorithms Using Unimodal Test Functions

#### 4.2. Results of the Algorithms Using Multimodal Test Functions

#### 4.3. Results of the Algorithms on Composite Test Functions

#### 4.4. Analysis of the Hybrid BOA-CE Algorithm

- The BOA, which mimics the food foraging behavior of butterflies in nature, has the advantage of a fast convergence rate. Based on co-updating, the BOA-CE uses the excellent individuals obtained by the BOA operator to update the CE operator’s probability parameters during the iterative process, which speeds up the convergence rate of the CE operator.
- The CE method is a global stochastic optimization method based on statistical model, and has the advantages of randomness, adaptability, and robustness, which bring a good population diversity to the BOA operator so that it can effectively avoid falling into a local optimum and improve its global search capability.
- The BOA-CE algorithm employs a co-evolutionary technique to co-update the BOA operator’s population and the CE operator’s probability parameters. This enables the improved method to obtain an appropriate balance between exploration and exploitation and have more superior performance in terms of exploitation, exploration, and local optima avoidance when solving complex optimization problems.

## 5. Using the BOA-CE Algorithm for Classical Engineering Design Problems

#### 5.1. Tension/Compression Spring Design

#### 5.2. Welded Beam Design

#### 5.3. Pressure Vessel Design

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Mathematical formulation of the primitive functions in Table 3.

Name | Formulation |
---|---|

Sphere | $f\left(x\right)={\sum}_{i=1}^{D}{x}_{i}^{2}$ |

Ackley | $f\left(x\right)=-20\mathrm{exp}(-0.2\sqrt{\frac{1}{D}{\sum}_{i=1}^{D}{x}_{i}^{2}})-\mathrm{exp}\left(\frac{1}{D}{\sum}_{i=1}^{D}\mathrm{cos}\left(2\pi {x}_{i}\right)\right)+20+e$ |

Griewank | $f\left(x\right)=\frac{1}{4000}{\sum}_{i=1}^{D}{x}_{i}^{2}-{\prod}_{i=1}^{D}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1$ |

Weierstrass | $f\left(x\right)={\sum}_{i=1}^{D}\left({\sum}_{k=0}^{{k}_{max}}\left[{a}^{k}\mathrm{cos}\left(2\pi {b}^{k}({x}_{i}+0.5)\right)\right]\right)-D{\sum}_{k=0}^{{k}_{max}}\left[{a}^{k}\mathrm{cos}(2\pi {b}^{k}\xb70.5)\right],a=0.5,b=3,{k}_{max}=20$ |

Rastrigin | $f\left(x\right)={\sum}_{i=1}^{D-1}[{x}_{i}^{2}-10\mathrm{cos}\left(2\pi {x}_{i}\right)+10]$ |

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**Figure 2.**The BOA operator and CE operator co-update the current best ${f}^{\ast}$ of the 2D Sphere function.

Function | Dim | Range | ${\mathit{F}}_{\mathit{min}}$ |
---|---|---|---|

${F}_{1}\left(x\right)={\sum}_{i=1}^{n}{x}_{i}^{2}$ | 30 | $[-100,100]$ | 0 |

${F}_{2}\left(x\right)={\sum}_{i=1}^{n}\left|{x}_{i}\right|+{\prod}_{i=1}^{n}\left|{x}_{i}\right|$ | 30 | $[-10,10]$ | 0 |

${F}_{3}\left(x\right)={\sum}_{i=1}^{n}{\left({\sum}_{j=1}^{i}{x}_{j}\right)}^{2}$ | 30 | $[-100,100]$ | 0 |

${F}_{4}\left(x\right)=\underset{i}{\mathrm{max}}\{\left|{x}_{i}\right|,1\le i\le n\}$ | 30 | $[-100,100]$ | 0 |

${F}_{5}\left(x\right)={\sum}_{i=1}^{n-1}[100{({x}_{i+1}-{x}_{i}^{2})}^{2}+{({x}_{i}-1)}^{2}]$ | 30 | $[-30,30]$ | 0 |

${F}_{6}\left(x\right)={\sum}_{i=1}^{n}{\left([{x}_{i}+0.5]\right)}^{2}$ | 30 | $[-100,100]$ | 0 |

${F}_{7}\left(x\right)={\sum}_{i=1}^{n}i{x}_{i}^{4}+random[0,1)$ | 30 | $[-1.28,1.28]$ | 0 |

Function | Dim | Range | ${\mathit{F}}_{\mathit{min}}$ |
---|---|---|---|

${F}_{8}\left(x\right)={\sum}_{i=1}^{n}-{x}_{i}\mathrm{sin}\left(\sqrt{\left|{x}_{i}\right|}\right)$ | 30 | $[-500,500]$ | $-\mathrm{12,569.487}$ |

${F}_{9}\left(x\right)={\sum}_{i=1}^{n-1}[{x}_{i}^{2}-10\mathrm{cos}\left(2\pi {x}_{i}\right)+10]$ | 30 | $[-5.12,5.12]$ | 0 |

${F}_{10}\left(x\right)=-20\mathrm{exp}(-0.2\sqrt{\frac{1}{n}{\sum}_{i=1}^{n}{x}_{i}^{2}})-\mathrm{exp}\left(\frac{1}{n}{\sum}_{i=1}^{n}\mathrm{cos}\left(2\pi {x}_{i}\right)\right)$ | 30 | [−32, 32] | 0 |

$+20+e$ | |||

${F}_{11}\left(x\right)=\frac{1}{4000}{\sum}_{i=1}^{n}{x}_{i}^{2}-{\prod}_{i=1}^{n}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1$ | 30 | [−600, 600] | 0 |

${F}_{12}\left(x\right)=\frac{\pi}{n}\{10{\mathrm{sin}}^{2}\left(\pi {y}_{1}\right)+{\sum}_{i=1}^{n-1}{({y}_{i}-1)}^{2}[1+10{\mathrm{sin}}^{2}\left(\pi {y}_{i+1}\right)$ | 30 | [−50, 50] | 0 |

$+{({y}_{n}+1)}^{2}\left]\right\}+{\sum}_{i=1}^{n}u({x}_{i},10,100,4)$ | |||

${y}_{i}=1+\frac{{x}_{i}+1}{4}$ | |||

$u({x}_{i},a,k,m)=\left\{\begin{array}{cc}k{({x}_{i}-a)}^{m},\hfill & {x}_{i}>a\hfill \\ 0,\hfill & -a\le {x}_{i}\le a\hfill \\ k{(-{x}_{i}-a)}^{m},\hfill & {x}_{i}<-a\hfill \end{array}\right.$ | |||

${F}_{13}\left(x\right)=0.1\{{\mathrm{sin}}^{2}\left(3\pi {x}_{1}\right)+{\sum}_{i=1}^{n}{({x}_{i}-1)}^{2}[1+{\mathrm{sin}}^{2}(3\pi {x}_{i}+1)]\}$ | 30 | [−50, 50] | 0 |

$+{\sum}_{i=1}^{n}u({x}_{i},5,100,4)$ |

${\mathbf{Function}}^{\phantom{\rule{0.166667em}{0ex}}1}$ | Dim | Range | ${\mathit{F}}_{\mathit{min}}$ |
---|---|---|---|

${F}_{14}$(CF1) | |||

${f}_{1},{f}_{2},\dots ,{f}_{10}=$ Sphere Function, | 10 | $[-5,5]$ | 0 |

$[{\lambda}_{1},{\lambda}_{2},\dots ,{\lambda}_{10}]=[1,1,\dots ,1],[{\sigma}_{1},{\sigma}_{2},\dots ,{\sigma}_{10}]=[1/100,5/100,\dots ,5/100]$ | |||

${F}_{15}$(CF2) | |||

${f}_{1},{f}_{2},\dots ,{f}_{10}=$ Griewank’s Function, | 10 | $[-5,5]$ | 0 |

$[{\lambda}_{1},{\lambda}_{2},\dots ,{\lambda}_{10}]=[1,1,\dots ,1],[{\sigma}_{1},{\sigma}_{2},\dots ,{\sigma}_{10}]=[1/100,5/100,\dots ,5/100]$ | |||

${F}_{16}$(CF3) | |||

${f}_{1},{f}_{2},\dots ,{f}_{10}=$ Griewank’s Function, | 10 | $[-5,5]$ | 0 |

$[{\lambda}_{1},{\lambda}_{2},\dots ,{\lambda}_{10}]=[1,1,\dots ,1],[{\sigma}_{1},{\sigma}_{2},\dots ,{\sigma}_{10}]=[1,1,\dots ,1]$ | |||

${F}_{17}$(CF4) | |||

${f}_{1},{f}_{2}=$ Ackley’s Function, ${f}_{3},{f}_{4}=$ Rastrigin’s Function, | |||

${f}_{5},{f}_{6}=$ Weierstrass Function, ${f}_{7},{f}_{8}=$ Griewank’s Function, | 10 | $[-5,5]$ | 0 |

${f}_{9},{f}_{10}=$ Sphere Function, $[{\lambda}_{1},{\lambda}_{2},\dots ,{\lambda}_{10}]=[1,1,\dots ,1]$, | |||

$[{\sigma}_{1},{\sigma}_{2},\dots ,{\sigma}_{10}]=[\frac{5}{32},\frac{5}{32},1,1,\frac{5}{0.5},\frac{5}{0.5},\frac{5}{100},\frac{5}{100},\frac{5}{100},\frac{5}{100}]$ | |||

${F}_{18}$(CF5) | |||

${f}_{1},{f}_{2}=$ Rastrigin’s Function, ${f}_{3},{f}_{4}=$ Weierstrass Function, | |||

${f}_{5},{f}_{6}=$ Griewank’s Function, ${f}_{7},{f}_{8}=$ Ackley’sGriewank’s Function, | 10 | $[-5,5]$ | 0 |

${f}_{9},{f}_{10}=$ Sphere Function, $[{\lambda}_{1},{\lambda}_{2},\dots ,{\lambda}_{10}]=[1,1,\dots ,1]$, | |||

$[{\sigma}_{1},{\sigma}_{2},\dots ,{\sigma}_{10}]=[\frac{1}{5},\frac{1}{5},\frac{5}{0.5},\frac{5}{0.5},\frac{5}{100},\frac{5}{100},\frac{5}{32},\frac{5}{32},\frac{5}{100},\frac{5}{100}]$ | |||

${F}_{19}$(CF6) | |||

${f}_{1},{f}_{2}=$ Rastrigin’s Function, ${f}_{3},{f}_{4}=$ Weierstrass Function, | |||

${f}_{5},{f}_{6}=$ Griewank’s Function, ${f}_{7},{f}_{8}=$ Ackley’sGriewank’s Function, | |||

${f}_{9},{f}_{10}=$ Sphere Function, $[{\lambda}_{1},{\lambda}_{2},\dots ,{\lambda}_{10}]=[1,1,\dots ,1]$, | 10 | $[-5,5]$ | 0 |

$[{\lambda}_{1},{\lambda}_{2},\dots ,{\lambda}_{10}]=[0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1]$, | |||

$[{\sigma}_{1},{\sigma}_{2},\dots ,{\sigma}_{10}]=[0.1\times \frac{1}{5},0.2\times \frac{1}{5},0.3\times 0.4\times \frac{5}{0.5},0.5\times \frac{5}{0.5}$, | |||

$0.6\times \frac{5}{100},0.7\times \frac{5}{100},0.8\times \frac{5}{32},0.9\times \frac{5}{32},1\times \frac{5}{100},\frac{5}{100}]$ |

Fun. | Metr. | GA | PSO | BA | GWO | CSA | SSA | BOA | BOA-CE |
---|---|---|---|---|---|---|---|---|---|

F_{1} | Mean | $1.44\times {10}^{-07}$ | $1.43\times {10}^{-15}$ | $1.46\times {10}^{-04}$ | $1.97\times {10}^{-90}$ | $1.20\times {10}^{-03}$ | $6.70\times {10}^{-09}$ | $1.57\times {10}^{-14}$ | $\mathbf{1.26}\times {\mathbf{10}}^{-\mathbf{95}}$ |

SD | $4.32\times {10}^{-07}$ | $3.42\times {10}^{-15}$ | $7.98\times {10}^{-04}$ | $5.36\times {10}^{-90}$ | $5.64\times {10}^{-04}$ | $1.23\times {10}^{-09}$ | $7.69\times {10}^{-16}$ | $\mathbf{9.78}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{96}}$ | |

F_{2} | Mean | $1.48\times {10}^{-02}$ | $7.50\times {10}^{-07}$ | $2.57\times {10}^{+01}$ | $\mathbf{2.92}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{52}}$ | $7.16\times {10}^{-01}$ | $6.00\times {10}^{-06}$ | $9.14\times {10}^{-12}$ | $3.60\times {10}^{-47}$ |

SD | $6.05\times {10}^{-02}$ | $3.30\times {10}^{-06}$ | $3.93\times {10}^{+01}$ | $\mathbf{3.35}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{52}}$ | $4.51\times {10}^{-01}$ | $1.33\times {10}^{-06}$ | $1.37\times {10}^{-12}$ | $2.50\times {10}^{-47}$ | |

F_{3} | Mean | $5.20\times {10}^{-01}$ | $2.30\times {10}^{+00}$ | $2.42\times {10}^{+03}$ | $\mathbf{1.74}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{26}}$ | $8.30\times {10}^{+00}$ | $8.33\times {10}^{-10}$ | $1.60\times {10}^{-14}$ | $2.44\times {10}^{-09}$ |

SD | $1.95\times {10}^{-01}$ | $1.18\times {10}^{+00}$ | $2.18\times {10}^{+03}$ | $\mathbf{5.09}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{26}}$ | $4.74\times {10}^{+00}$ | $3.05\times {10}^{-10}$ | $9.73\times {10}^{-16}$ | $1.83\times {10}^{-10}$ | |

F_{4} | Mean | $1.88\times {10}^{-01}$ | $2.01\times {10}^{-01}$ | $3.56\times {10}^{+01}$ | $\mathbf{1.74}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{21}}$ | $1.35\times {10}^{+00}$ | $1.23\times {10}^{-05}$ | $1.09\times {10}^{-11}$ | $8.82\times {10}^{-07}$ |

SD | $4.63\times {10}^{-01}$ | $6.33\times {10}^{-02}$ | $6.27\times {10}^{-13}$ | $\mathbf{2.81}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{21}}$ | $6.84\times {10}^{-01}$ | $1.99\times {10}^{-06}$ | $6.72\times {10}^{-13}$ | $1.51\times {10}^{-07}$ | |

F_{5} | Mean | $1.87\times {10}^{+01}$ | $4.28\times {10}^{+01}$ | $1.47\times {10}^{+02}$ | $2.62\times {10}^{+01}$ | $4.19\times {10}^{+01}$ | $\mathbf{1.26}\mathbf{\times}{\mathbf{10}}^{\mathbf{+}\mathbf{01}}$ | $2.89\times {10}^{+01}$ | $2.79\times {10}^{+01}$ |

SD | $2.82\times {10}^{+01}$ | $3.07\times {10}^{+01}$ | $2.71\times {10}^{+02}$ | $6.30\times {10}^{-01}$ | $2.82\times {10}^{+01}$ | $2.86\times {10}^{+01}$ | $\mathbf{3.95}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{02}}$ | $2.56\times {10}^{-01}$ | |

F_{6} | Mean | $1.56\times {10}^{-07}$ | $1.47\times {10}^{-15}$ | $1.06\times {10}^{-14}$ | $1.10\times {10}^{-01}$ | $9.65\times {10}^{-04}$ | $4.74\times {10}^{-10}$ | $5.05\times {10}^{+00}$ | 0 |

SD | $2.37\times {10}^{-07}$ | $2.20\times {10}^{-15}$ | $5.40\times {10}^{-14}$ | $1.58\times {10}^{-01}$ | $4.19\times {10}^{-04}$ | $1.44\times {10}^{-10}$ | $5.34\times {10}^{-01}$ | 0 | |

F_{7} | Mean | $3.79\times {10}^{-01}$ | $3.22\times {10}^{-02}$ | $1.25\times {10}^{-01}$ | $3.99\times {10}^{-04}$ | $9.87\times {10}^{-03}$ | $2.06\times {10}^{-03}$ | $6.68\times {10}^{-04}$ | $\mathbf{3.29}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{04}}$ |

SD | $9.79\times {10}^{-02}$ | $1.12\times {10}^{-02}$ | $4.30\times {10}^{-02}$ | $1.65\times {10}^{-04}$ | $4.50\times {10}^{-03}$ | $1.22\times {10}^{-03}$ | $3.25\times {10}^{-04}$ | $\mathbf{1.52}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{04}}$ |

Fun. | Metr. | GA | PSO | BA | GWO | CSA | SSA | BOA | BOA-CE |
---|---|---|---|---|---|---|---|---|---|

F_{8} | Mean | $\mathbf{-}\mathbf{10,486.64}$ | $-6500.75$ | $-3525.02$ | $-6941.03$ | $-7294.61$ | $-2824.35$ | $-4246.74$ | $-4017.15$ |

SD | $626.18$ | $819.29$ | $\mathbf{192.65}$ | $641.14$ | $801.93$ | $211.32$ | $350.36$ | $323.25$ | |

F_{9} | Mean | $1.33\times {10}^{+00}$ | $3.24\times {10}^{-01}$ | $1.35\times {10}^{+02}$ | $3.45\times {10}^{+01}$ | $1.87\times {10}^{+01}$ | $1.21\times {10}^{+01}$ | $\mathbf{3.03}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{14}}$ | $1.01\times {10}^{+01}$ |

SD | $1.23\times {10}^{+00}$ | $1.78\times {10}^{+00}$ | $3.38\times {10}^{+01}$ | $1.14\times {10}^{+01}$ | $7.82\times {10}^{+00}$ | $5.86\times {10}^{+00}$ | $\mathbf{1.46}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{13}}$ | $3.46\times {10}^{+01}$ | |

F_{10} | Mean | $1.76\times {10}^{-04}$ | $1.10\times {10}^{-14}$ | $1.64\times {10}^{+01}$ | $1.89\times {10}^{-08}$ | $2.56\times {10}^{+00}$ | $4.24\times {10}^{-01}$ | $1.01\times {10}^{-11}$ | $\mathbf{4.44}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{15}}$ |

SD | $7.59\times {10}^{-05}$ | $3.11\times {10}^{-15}$ | $2.30\times {10}^{+00}$ | $1.72\times {10}^{-08}$ | $6.99\times {10}^{-01}$ | $6.81\times {10}^{-01}$ | $1.10\times {10}^{-12}$ | 0 | |

F_{11} | Mean | $1.23\times {10}^{-03}$ | $8.70\times {10}^{-04}$ | $1.22\times {10}^{+02}$ | $1.29\times {10}^{-02}$ | $2.89\times {10}^{-02}$ | $2.47\times {10}^{-01}$ | $5.59\times {10}^{-16}$ | 0 |

SD | $3.41\times {10}^{-03}$ | $2.68\times {10}^{-03}$ | $5.54\times {10}^{+01}$ | $1.45\times {10}^{-02}$ | $1.34\times {10}^{-02}$ | $1.46\times {10}^{-01}$ | $5.64\times {10}^{-16}$ | 0 | |

F_{12} | Mean | $5.19\times {10}^{-02}$ | $1.35\times {10}^{-02}$ | $1.82\times {10}^{+01}$ | $\mathbf{9.96}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{18}}$ | $6.25\times {10}^{-01}$ | $8.53\times {10}^{-02}$ | $3.84\times {10}^{-01}$ | $2.02\times {10}^{-05}$ |

SD | $1.12\times {10}^{-01}$ | $9.37\times {10}^{-03}$ | $5.84\times {10}^{+00}$ | $\mathbf{1.13}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{17}}$ | $5.39\times {10}^{-01}$ | $2.11\times {10}^{-01}$ | $1.13\times {10}^{-01}$ | $8.76\times {10}^{-05}$ | |

F_{13} | Mean | $5.09\times {10}^{-03}$ | $1.51\times {10}^{-01}$ | $3.85\times {10}^{+01}$ | $1.10\times {10}^{-03}$ | $1.20\times {10}^{-02}$ | $7.32\times {10}^{-04}$ | $2.24\times {10}^{+00}$ | $\mathbf{1.35}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{32}}$ |

SD | $1.23\times {10}^{-02}$ | $1.43\times {10}^{-01}$ | $1.62\times {10}^{+01}$ | $3.35\times {10}^{-03}$ | $1.51\times {10}^{-02}$ | $2.79\times {10}^{-030}$ | $4.04\times {10}^{-01}$ | $\mathbf{5.57}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{48}}$ |

Fun. | Metr. | GA | PSO | BA | GWO | CSA | SSA | BOA | BOA-CE |
---|---|---|---|---|---|---|---|---|---|

F_{14} | Mean | $46.67$ | $70.00$ | $70.00$ | $45.90$ | $26.67$ | $43.33$ | $270.28$ | $\mathbf{23.33}$ |

SD | $50.74$ | $95.23$ | $83.67$ | $71.84$ | $44.98$ | $67.89$ | $58.82$ | $\mathbf{43.02}$ | |

F_{15} | Mean | $57.91$ | $128.09$ | $139.61$ | $115.17$ | $116.59$ | $\mathbf{27.77}$ | $262.27$ | $104.76$ |

SD | $62.53$ | $76.97$ | $95.29$ | $82.14$ | $61.98$ | $\mathbf{17.39}$ | $110.66$ | $72.26$ | |

F_{16} | Mean | $154.18$ | $172.96$ | $348.41$ | $160.34$ | $240.86$ | $195.18$ | $374.19$ | $\mathbf{105.92}$ |

SD | $42.63$ | $75.64$ | $111.53$ | $48.84$ | $66.57$ | $\mathbf{38.64}$ | $57.17$ | $59.17$ | |

F_{17} | Mean | $305.43$ | $378.64$ | $452.73$ | $360.09$ | $424.94$ | $319.45$ | $602.76$ | $\mathbf{264.51}$ |

SD | $35.95$ | $128.17$ | $114.34$ | $105.96$ | $92.62$ | $\mathbf{27.58}$ | $85.08$ | $96.10$ | |

F_{18} | Mean | $53.09$ | $92.41$ | $85.22$ | $56.75$ | $\mathbf{9.49}$ | $14.10$ | $130.84$ | $71.00$ |

SD | $79.02$ | $136.96$ | $150.90$ | $62.29$ | $\mathbf{17.69}$ | $29.53$ | $59.39$ | $47.42$ | |

F_{19} | Mean | $627.64$ | $765.17$ | $745.07$ | $698.91$ | $\mathbf{497.13}$ | $55.26$ | $819.40$ | $790.79$ |

SD | $192.16$ | $191.11$ | $198.03$ | $201.69$ | $\mathbf{17.93}$ | $152.72$ | $112.13$ | $153.62$ |

Fun. | GA | PSO | BA | GWO | CSA | SSA | BOA | BOA-CE |
---|---|---|---|---|---|---|---|---|

F_{1} | 8.96 | 0.20 | 0.33 | 0.55 | 0.10 | 0.66 | 0.46 | 0.41 |

F_{2} | 8.80 | 0.21 | 0.35 | 0.57 | 0.11 | 0.43 | 0.49 | 0.42 |

F_{3} | 10.75 | 0.76 | 2.52 | 1.13 | 2.02 | 0.56 | 1.61 | 1.15 |

F_{4} | 8.44 | 0.20 | 0.41 | 0.55 | 0.14 | 0.38 | 0.46 | 0.40 |

F_{5} | 9.11 | 0.27 | 0.59 | 0.62 | 0.29 | 0.45 | 0.59 | 0.51 |

F_{6} | 8.57 | 0.20 | 0.34 | 0.55 | 0.10 | 0.38 | 0.45 | 0.40 |

F_{7} | 9.29 | 0.54 | 0.91 | 1.14 | 0.58 | 0.52 | 0.84 | 0.90 |

F_{8} | 8.72 | 0.31 | 0.87 | 0.65 | 0.19 | 0.44 | 0.68 | 0.62 |

F_{9} | 8.81 | 0.26 | 0.41 | 0.57 | 0.15 | 0.40 | 0.58 | 0.48 |

F_{10} | 8.66 | 0.26 | 0.43 | 0.59 | 0.17 | 0.41 | 0.55 | 0.45 |

F_{11} | 8.97 | 0.31 | 0.69 | 0.64 | 0.34 | 0.47 | 0.66 | 0.54 |

F_{12} | 10.14 | 1.17 | 1.62 | 1.52 | 1.16 | 0.91 | 2.45 | 1.67 |

F_{13} | 10.10 | 1.17 | 1.62 | 1.52 | 1.17 | 0.91 | 2.46 | 1.66 |

F_{14} | 264.89 | 249.55 | 256.03 | 304.78 | 260.24 | 266.09 | 263.94 | 265.80 |

F_{15} | 261.13 | 232.07 | 261.98 | 300.36 | 258.01 | 255.73 | 263.67 | 259.32 |

F_{16} | 259.28 | 210.24 | 269.61 | 250.40 | 263.24 | 258.70 | 259.05 | 252.60 |

F_{17} | 313.15 | 212.06 | 287.05 | 273.28 | 282.19 | 283.91 | 286.06 | 282.38 |

F_{18} | 293.31 | 262.41 | 289.38 | 272.57 | 286.17 | 286.81 | 288.63 | 286.88 |

F_{19} | 287.42 | 172.65 | 278.74 | 288.92 | 282.58 | 297.25 | 291.06 | 284.31 |

Total | 1798.50 | 1344.82 | 1653.88 | 1700.91 | 1638.95 | 1655.41 | 1664.70 | 1640.89 |

Ranking | 8 | 1 | 5 | 7 | 2 | 6 | 4 | 3 |

Design Problem | BOA Operator | CE Operator | ||
---|---|---|---|---|

$\mathit{N}$ | ${\mathit{Iter}}_{\mathit{max}}$ | $\mathit{N}$ | ${\mathit{Iter}}_{\mathit{max}}$ | |

Tension/compression spring | 100 | 50 | 100 | 50 |

Welded beam | 100 | 100 | 100 | 50 |

Pressure vessel | 100 | 100 | 100 | 50 |

Algorithm | Optimum Variables | Optimum Weight | ||
---|---|---|---|---|

$\mathit{d}$ | $\mathit{D}$ | $\mathit{N}$ | ||

GA [48] | 0.051480 | 0.351661 | 11.632201 | 0.0127048 |

PSO [49] | 0.051728 | 0.357644 | 11.244543 | 0.012675 |

BA [50] | 0.051690 | 0.356730 | 11.288500 | 0.012670 |

GWO [15] | 0.051690 | 0.356737 | 11.288850 | 0.012666 |

HS [51] | 0.051154 | 0.349871 | 12.076432 | 0.012671 |

CSA [19] | 0.051689 | 0.356717 | 11.289012 | 0.012665 |

SSA [20] | 0.051207 | 0.345215 | 12.004032 | 0.012676 |

BOA-CE | 0.051618 | 0.355004 | 11.390144 | 0.012665 |

Algorithm | Optimum Variables | Optimum Cost | |||
---|---|---|---|---|---|

$\mathit{h}$ | $\mathit{l}$ | $\mathit{t}$ | $\mathit{b}$ | ||

GA [48] | 0.205986 | 3.471328 | 9.020224 | 0.206480 | 1.728226 |

PSO [49] | 0.202369 | 3.544214 | 9.048210 | 0.205723 | 1.731485 |

BA [50] | 0.2015 | 3.562 | 9.0414 | 0.2057 | 1.7312 |

GWO [15] | 0.205676 | 3.478377 | 9.03681 | 0.205778 | 1.72624 |

HS [51] | 0.2442 | 6.2231 | 8.2915 | 0.2443 | 2.3807 |

WOA [18] | 0.205396 | 3.484293 | 9.037426 | 0.206276 | 1.730499 |

CSA [19] | 0.205730 | 3.470489 | 9.036624 | 0.205730 | 1.724852 |

BOA-CE | 0.205730 | 3.470481 | 9.036611 | 0.205730 | 1.724854 |

Algorithm | Optimum Variables | Optimum Cost | |||
---|---|---|---|---|---|

${\mathit{T}}_{\mathit{s}}$ | ${\mathit{T}}_{\mathit{h}}$ | $\mathit{R}$ | $\mathit{L}$ | ||

GA [48] | 0.8125 | 0.4375 | 42.097398 | 176.654050 | 6059.9463 |

PSO [49] | 0.8125 | 0.4375 | 42.091266 | 176.746500 | 6061.0777 |

BA [50] | 0.8125 | 0.4375 | 42.0984456 | 176.636596 | 6059.7143 |

GWO [15] | 0.8125 | 0.4345 | 42.089181 | 176.758731 | 6051.5639 |

HS [ [51] | 1.1250 | 0.6250 | 58.2789 | 43.7549 | 7198.433 |

WOA [18] | 0.8125 | 0.4375 | 42.0982699 | 176.638998 | 6059.7410 |

DE [52] | 0.8125 | 0.4375 | 42.098411 | 176.63769 | 6059.7341 |

BOA-CE | 0.8125 | 0.4375 | 42.0984456 | 176.6365958 | 6059.7143 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, G.; Shuang, F.; Zhao, P.; Le, C.
An Improved Butterfly Optimization Algorithm for Engineering Design Problems Using the Cross-Entropy Method. *Symmetry* **2019**, *11*, 1049.
https://doi.org/10.3390/sym11081049

**AMA Style**

Li G, Shuang F, Zhao P, Le C.
An Improved Butterfly Optimization Algorithm for Engineering Design Problems Using the Cross-Entropy Method. *Symmetry*. 2019; 11(8):1049.
https://doi.org/10.3390/sym11081049

**Chicago/Turabian Style**

Li, Guocheng, Fei Shuang, Pan Zhao, and Chengyi Le.
2019. "An Improved Butterfly Optimization Algorithm for Engineering Design Problems Using the Cross-Entropy Method" *Symmetry* 11, no. 8: 1049.
https://doi.org/10.3390/sym11081049