# An Improved Gray Wolf Optimization Algorithm to Solve Engineering Problems

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Gray Wolf Optimization

#### 2.1. Social Hierarchy

#### 2.2. Encircling the Prey

#### 2.3. Attacking the Prey

## 3. Improved Gray Wolf Optimization Algorithm (IGWO)

#### 3.1. Tent Chaos Initialization

#### 3.2. Gaussian Perturbation

#### 3.3. Cosine Control Factor

## 4. The Simulation Results

#### 4.1. Optimization Function and Experimental Environment

^{®}Celeron

^{®}CPU N3060 @1.60 GHz processor, 4 GB memory, and the software configuration of the simulation environment is MATLABR2016a.

#### 4.2. Analysis of Different Strategies

#### 4.3. Analysis of Experimental Results

#### 4.3.1. Compared with Other Algorithms

#### 4.3.2. Convergence Analysis

**a**–

**w**) in Figure 10, when solving the unimodal test function, the convergence curve of the IGWOS algorithm decreases to the lower right corner as the iterations increase, and the convergence accuracy is higher than other algorithms. Among the multimodal test functions, the IGWO tests other functions except F8 have better performance. In particular, F9 and F11 can quickly find the optimal solution with fast convergence speed and high precision. For the fixed dimensional multimodal functions, most of the function curve drops in a stepped way. This is because the algorithm keeps exploring in the iterative process, jumps out of the optimal local solution and looks for the optimal global solution, and its convergence accuracy is relatively good.

#### 4.3.3. Numerical Result Test

## 5. Application to Solve Engineering Optimization Problem

^{12}). $m$ and $l$ are the numbers of inequality constraints and equality constraints. Parameters ${\alpha}_{1}$ and ${\alpha}_{2}$ take 2 and 1, respectively. When dealing with the objective problem, this method excludes all the solutions that do not meet the constraints from the candidate solutions. If the search solution satisfies the constraints, then the objective function is $\tilde{f}\left(x\right)=f\left(x\right)$. When solving these engineering optimization problems, the population size n = 30, the maximum iteration number Max_iter = 500, and the results in the table are the best values obtained by running 30 times independently. The bold values indicate the best one among all methods.

#### 5.1. Pressure Vessel Design Problem

#### 5.2. Spring Design Problem

_{1}), the wire diameter D(x

_{2}), the number of effective windings P(x

_{3}). For this problem, the mathematical model of the objective function and constraint function is shown in Equation (19).

#### 5.3. Welded Beam Design Problem

#### 5.4. Three Truss Design Problem

_{1}, and the cross-sectional areas of rod 2, x

_{2}. The mathematical model is shown in Equation (21). The schematic diagram is shown in Figure 14.

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Number | Name | Benchmark | Dim | Range | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ |
---|---|---|---|---|---|

F1 | Sphere | ${f}_{1}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{n}}{x}_{i}^{2}$ | 30 | [−100, 100] | 0 |

F2 | Schwefel’problem2.22 | ${f}_{2}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{n}}\left|{x}_{i}\right|+{\displaystyle {\displaystyle \prod}_{i=1}^{n}}\left|{x}_{i}\right|$ | 30 | [−10, 10] | 0 |

F3 | Schwefel’problem1.2 | ${f}_{3}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{n}}{\left({\displaystyle {\displaystyle \sum}_{j-1}^{i}}{x}_{j}\right)}^{2}$ | 30 | [−100, 100] | 0 |

F4 | Schwefel’problem2.21 | ${f}_{4}\left(x\right)=ma{x}_{i}\left\{\left|{x}_{i}\right|,1\ll i\ll n\right\}$ | 30 | [−100, 100] | 0 |

F5 | Rosenbrock | ${f}_{5}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{n-1}}\left[100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{\left({x}_{i}-1\right)}^{2}\right]$ | 30 | [−30, 30] | 0 |

F6 | Step | ${f}_{6}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{n}}{\left(\left[{x}_{i}+0.5\right]\right)}^{2}$ | 30 | [−100, 100] | 0 |

F7 | Noise | ${f}_{7}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{n}}i{x}_{i}^{4}+random\left[0,\left.1\right)\right.$ | 30 | [−1.28, 1.28] | 0 |

Number | Name | Benchmark | Dim | Range | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ |
---|---|---|---|---|---|

F8 | Generalized Schwfel’s problem | ${f}_{8}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{n}}-{x}_{i}sin\left(\sqrt{\left|{x}_{i}\right|}\right)$ | 30 | [−500, 500] | −12, 569.5 |

F9 | Rastrigin | ${f}_{9}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{n}}\left[{x}_{i}^{2}-10cos\left(2\pi {x}_{i}+10\right)\right]$ | 30 | [−5.12, 5.12] | 0 |

F10 | Ackley | ${f}_{10}\left(x\right)=20+\mathrm{e}-20\mathrm{exp}\left(-0.2\sqrt{\frac{1}{n}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}{x}_{i}^{2}}\right)-exp\left(\frac{1}{n}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}cos\left(2\pi {x}_{i}\right)\right)$ | 30 | [−32, 32] | 0 |

F11 | Griewank | ${f}_{11}\left(x\right)=\frac{1}{4000}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}{x}_{i}-{\displaystyle {\displaystyle \prod}_{i=1}^{n}}cos\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1$ | 30 | [−600, 600] | 0 |

F12 | Generalized penalized function 1 | ${\mathrm{f}}_{12}\left(\mathrm{x}\right)=\frac{\mathsf{\pi}}{\mathrm{n}}\left\{10\mathrm{sin}\left({\mathsf{\pi}\mathrm{y}}_{1}\right)+{\displaystyle {\displaystyle \sum}_{\mathrm{i}=1}^{\mathrm{n}-1}}{\left({\mathrm{y}}_{\mathrm{i}}-1\right)}^{2}\left[1+10{\mathrm{sin}}^{2}\left({\mathsf{\pi}\mathrm{y}}_{\mathrm{i}+1}\right)\right]+{\left({\mathrm{y}}_{\mathrm{n}}-1\right)}^{2}\right\}+{\displaystyle {\displaystyle \sum}_{\mathrm{i}=1}^{\mathrm{n}}}\mathrm{u}\left({\mathrm{x}}_{\mathrm{i}},10,100,4\right)$ ${\mathrm{y}}_{\mathrm{i}}=1+\frac{{\mathrm{x}}_{\mathrm{i}}+1}{4}$ $\mathrm{u}\left({\mathrm{x}}_{\mathrm{i},}\mathrm{a},\mathrm{k},\mathrm{m}\right)=\left\{\begin{array}{c}k{\left({\mathrm{x}}_{\mathrm{i}}-\mathrm{a}\right)}^{\mathrm{m}}\text{}{\mathrm{x}}_{\mathrm{i}}a\\ 0-a{\mathrm{x}}_{\mathrm{i}}a\\ k{\left(-{\mathrm{x}}_{\mathrm{i}}-\mathrm{a}\right)}^{\mathrm{m}}\text{}{\mathrm{x}}_{\mathrm{i}}-a\end{array}\right.$ | 30 | [−50, 50] | 0 |

F13 | Generalized Penalized Function 2 | ${\mathrm{f}}_{13}\left(\mathrm{x}\right)=0.1\left\{{\mathrm{sin}}^{2}\left(3{\mathsf{\pi}\mathrm{x}}_{1}\right)+{\displaystyle {\displaystyle \sum}_{\mathrm{i}=1}^{\mathrm{n}}}{\left({\mathrm{x}}_{\mathrm{i}}-1\right)}^{2}\left[1+{\mathrm{sin}}^{2}\left(3{\mathsf{\pi}\mathrm{x}}_{\mathrm{i}}+1\right)\right]+{\left({\mathrm{x}}_{\mathrm{n}}-1\right)}^{2}\left[1+{\mathrm{sin}}^{2}\left(2{\mathsf{\pi}\mathrm{x}}_{\mathrm{n}}\right)\right]\right\}+{\displaystyle {\displaystyle \sum}_{\mathrm{i}=1}^{\mathrm{n}}}\mathrm{u}\left({\mathrm{x}}_{\mathrm{i}},5,100,4\right)$ | 30 | [−50, 50] | 0 |

Number | Name | Benchmark | Dim | Range | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ |
---|---|---|---|---|---|

F14 | Shekel’s foxholes function | ${\mathrm{f}}_{14}\left(\mathrm{x}\right)={\left(\frac{1}{500}+{\displaystyle {\displaystyle \sum}_{\mathrm{j}=1}^{25}}\frac{1}{\mathrm{j}+{{\displaystyle \sum}}_{\mathrm{i}=1}^{2}{\left({\mathrm{x}}_{\mathrm{i}}-{\mathrm{a}}_{\mathrm{ij}}\right)}^{6}}\right)}^{-1}$ | 2 | [−65.536, 65.536] | 1 |

F15 | Kowalik’s Function | ${\mathrm{f}}_{15}\left(\mathrm{x}\right)={\displaystyle {\displaystyle \sum}_{\mathrm{i}=1}^{11}}{\left[{\mathrm{a}}_{\mathrm{i}}-\frac{{\mathrm{x}}_{1}\left({\mathrm{b}}_{\mathrm{i}}^{2}+{\mathrm{b}}_{\mathrm{i}}{\mathrm{x}}_{2}\right)}{{\mathrm{b}}_{\mathrm{i}}^{2}+{\mathrm{b}}_{\mathrm{i}}{\mathrm{x}}_{3}+{\mathrm{x}}_{4}}\right]}^{2}$ | 4 | [−5, 5] | 0.0003 |

F16 | Six-hump camelback | ${\mathrm{f}}_{16}\left(\mathrm{x}\right)=4{\mathrm{x}}_{1}^{2}-2.1{\mathrm{x}}_{1}^{4}+\frac{{\mathrm{x}}_{1}^{6}}{3}+{\mathrm{x}}_{1}{\mathrm{x}}_{2}-4{\mathrm{x}}_{2}^{2}+4{\mathrm{x}}_{2}^{4}$ | 2 | [−5, 5] | −1.0316 |

F17 | Branin | ${\mathrm{f}}_{17}={\left({\mathrm{x}}_{2}-\frac{5.1}{4{\mathsf{\pi}}^{2}}{\mathrm{x}}_{1}^{2}+\frac{5}{\mathsf{\pi}}{\mathrm{x}}_{1}-6\right)}^{2}+10\left(1-\frac{1}{8\mathsf{\pi}}\right){\mathrm{cos}\mathrm{x}}_{1}+10$ | 2 | [−5, 10] [10, 15] | 0.39788 |

F18 | Goldstein–Price function | ${\mathrm{f}}_{18}\left(\mathrm{x}\right)=\left[1+{\left({\mathrm{x}}_{1}+{\mathrm{x}}_{2}+1\right)}^{2}\xb7\left(19-14{\mathrm{x}}_{1}+3{\mathrm{x}}_{1}^{2}-14{\mathrm{x}}_{2}+6{\mathrm{x}}_{1}{\mathrm{x}}_{2}+3{\mathrm{x}}_{2}^{2}\right)\right]\times \left[30+{\left(2{\mathrm{x}}_{1}-3{\mathrm{x}}_{2}^{2}\right)}^{2}\xb7\left(18-32{\mathrm{x}}_{1}+12{\mathrm{x}}_{1}^{2}+48{\mathrm{x}}_{2}^{2}-36{\mathrm{x}}_{1}{\mathrm{x}}_{2}+27{\mathrm{x}}_{2}^{2}\right)\right]$ | 2 | [−2, 2] | 3 |

F19 | Hartmann 1 | ${\mathrm{f}}_{19}\left(\mathrm{x}\right)=-{\displaystyle {\displaystyle \sum}_{\mathrm{i}=1}^{4}}{\mathrm{c}}_{\mathrm{i}}\mathrm{exp}\left(-{\displaystyle {\displaystyle \sum}_{\mathrm{j}=1}^{3}}{\mathrm{a}}_{\mathrm{ij}}{\left({\mathrm{x}}_{\mathrm{j}}-{\mathrm{p}}_{\mathrm{ij}}\right)}^{2}\right)$ | 3 | [0, 1] | −3.86 |

F20 | Hartmann 2 | ${\mathrm{f}}_{20}\left(\mathrm{x}\right)=-{\displaystyle {\displaystyle \sum}_{\mathrm{i}=1}^{4}}{\mathrm{c}}_{\mathrm{i}}\mathrm{exp}\left(-{\displaystyle {\displaystyle \sum}_{\mathrm{j}=1}^{3}}{\mathrm{a}}_{\mathrm{ij}}{\left({\mathrm{x}}_{\mathrm{j}}-{\mathrm{p}}_{\mathrm{ij}}\right)}^{2}\right)$ | 6 | [1, 6] | −3.32 |

F21 | Shekel 1 | ${\mathrm{f}}_{21}\left(\mathrm{x}\right)=-{\displaystyle \sum _{\mathrm{i}=1}^{5}}{\left[\left(\mathrm{X}-{\mathrm{a}}_{\mathrm{i}}\right){\left(\mathrm{X}-{\mathrm{a}}_{\mathrm{i}}\right)}^{\mathrm{T}}+{\mathrm{C}}_{\mathrm{i}}\right]}^{-1}$ | 4 | [0, 10] | −10.1532 |

F22 | Shekel 2 | ${\mathrm{f}}_{22}\left(\mathrm{x}\right)=-{\displaystyle {\displaystyle \sum}_{\mathrm{i}=1}^{7}}{\left[\left(\mathrm{X}-{\mathrm{a}}_{\mathrm{i}}\right){\left(\mathrm{X}-{\mathrm{a}}_{\mathrm{i}}\right)}^{\mathrm{T}}+{\mathrm{C}}_{\mathrm{I}}\right]}^{-1}$ | 4 | [0, 10] | −10.4028 |

F23 | Shekel 3 | ${\mathrm{f}}_{23}\left(\mathrm{x}\right)=-{\displaystyle {\displaystyle \sum}_{\mathrm{i}=1}^{10}}{\left[\left(\mathrm{X}-{\mathrm{a}}_{\mathrm{i}}\right){\left(\mathrm{X}-{\mathrm{a}}_{\mathrm{i}}\right)}^{\mathrm{T}}+{\mathrm{C}}_{\mathrm{I}}\right]}^{-1}$ | 4 | [0, 10] | −10.5363 |

Function | GWO | GWO1 | GWO2 | GWO3 | IGWO | |||||
---|---|---|---|---|---|---|---|---|---|---|

ave | std | ave | std | ave | std | ave | std | ave | std | |

F1 | 1.55 × 10^{−27} | 2.95 × 10^{−27} | 1.18 × 10^{−30} | 2.15 × 10^{−30} | 1.72 × 10^{−38} | 6.97 × 10^{−38} | 3.25 × 10^{−28} | 4.78 × 10^{−28} | 8.34 × 10^{−40} | 2.35 × 10^{−39} |

F2 | 9.89 × 10^{−17} | 9.40 × 10^{−17} | 8.14 × 10^{−17} | 4.84 × 10^{−18} | 9.97 × 10^{−19} | 1.92 × 10^{−18} | 1.03 × 10^{−17} | 6.89 × 10^{−18} | 9.96 × 10^{−24} | 1.23 × 10^{−23} |

F3 | 2.47 × 10^{−5} | 6.57 × 10^{−5} | 3.03 × 10^{−5} | 7.30 × 10^{−5} | 3.19 × 10^{−10} | 1.70 × 10^{−9} | 3.74 × 10^{−5} | 7.15 × 10^{−5} | 5.95 × 10^{−8} | 9.72 × 10^{−8} |

F4 | 7.54 × 10^{−7} | 1.16 × 10^{−6} | 8.47 × 10^{−9} | 9.94 × 10^{−9} | 5.21 × 10^{−15} | 1.04 × 10^{−14} | 7.45 × 10^{−8} | 5.55 × 10^{−7} | 1.78 × 10^{−11} | 3.00 × 10^{−11} |

F5 | 2.73 × 10^{1} | 7.53 × 10^{−1} | 2.98 × 10^{2} | 2.46 × 10^{−1} | 2.98 × 10^{2} | 2.39 × 10^{−1} | 2.98 × 10^{2} | 2.09 × 10^{−1} | 2.72 × 10^{1} | 7.30 × 10^{−1} |

F6 | 7.37 × 10^{−1} | 3.06 × 10^{−1} | 7.26 × 10^{−1} | 3.53 × 10^{−1} | 1.72 | 5.35 × 10^{−1} | 6.56 × 10^{−1} | 3.95 × 10^{−1} | 1.68 | 4.28 × 10^{−1} |

F7 | 2.40 × 10^{−3} | 1.40 × 10^{−3} | 2.20 × 10^{−3} | 9.42 × 10^{−4} | 7.55 × 10^{−4} | 4.52 × 10^{−4} | 2.00 × 10^{−3} | 1.60 × 10^{−3} | 8.87 × 10^{−4} | 5.52 × 10^{−4} |

F8 | 1.84 × 10^{−14} | 8.25 × 10^{2} | −6.09 × 10^{3} | 6.96 × 10^{2} | −4.44 × 10^{3} | 1.37 × 10^{3} | −5.98 × 10^{3} | 9.15 × 10^{2} | −4.64 × 10^{3} | 1.26 × 10^{3} |

F9 | 2.41 | 3.30 | 1.99 | 3.53 | 0 | 0 | 3.51 × 10^{−2} | 1.40 × 10^{−1} | 0 | 0 |

F10 | 1.03 × 10^{−13} | 1.84 × 10^{−14} | 1.06 × 10^{−15} | 2.26 × 10^{−14} | 9.30 × 10^{−15} | 2.38 × 10^{−15} | 1.66 × 10^{−13} | 5.28 × 10^{−14} | 1.88 × 10^{−14} | 4.01 × 10^{−15} |

F11 | 3.20 × 10^{−3} | 6.70 × 10^{−3} | 1.20 × 10^{−4} | 5.00 × 10^{−3} | 0 | 0 | 3.50 × 10^{−6} | 9.40 × 10^{−6} | 0 | 0 |

F12 | 4.59 × 10^{−2} | 2.13 × 10^{−2} | 5.43 × 10^{−2} | 3.12 × 10^{−2} | 1.11 × 10^{−1} | 4.30 × 10^{−2} | 3.84 × 10^{−2} | 2.28 × 10^{−2} | 1.02 × 10^{−1} | 3.95 × 10^{−2} |

F13 | 6.41 × 10^{−1} | 2.37 × 10^{−1} | 6.48 × 10^{−1} | 2.23 × 10^{−1} | 1.10 | 2.32 × 10^{−1} | 5.21 × 10^{−1} | 1.92 × 10^{−1} | 1.05 | 2.24 × 10^{−1} |

F14 | 4.55 | 4.36 | 5.02 | 4.11 | 4.98 | 4.23 | 5.04 | 4.49 | 5.08 | 4.31 |

F15 | 6.50 × 10^{−3} | 9.30 × 10^{−3} | 4.40 × 10^{−3} | 8.10 × 10^{−3} | 5.19 × 10^{−4} | 1.86 × 10^{−4} | 5.10 × 10^{−3} | 8.60 × 10^{−3} | 4.94 × 10^{−4} | 1.22 × 10^{−4} |

F16 | −1.03 | 4.65 × 10^{−8} | −1.03 | 1.69 × 10^{−8} | −1.03 | 1.04 × 10^{−5} | −1.03 | 1.86 × 10^{−11} | −1.03 | 4.84 × 10^{−8} |

F17 | 3.98 × 10^{−1} | 3.88 × 10^{−6} | 3.98 × 10^{−1} | 1.63 × 10^{−6} | 3.98 × 10^{−1} | 1.63 × 10^{−5} | 3.98 × 10^{−1} | 1.16 × 10^{−4} | 3.98 × 10^{−1} | 3.83 × 10^{−4} |

F18 | 3.00 | 2.99 × 10^{−5} | 5.70 | 1.48 × 10^{1} | 3.00 | 3.17 × 10^{−5} | 3.00 | 4.14 × 10^{−5} | 3.00× | 3.15 × 10^{−5} |

F19 | 4.56 | 2.90 × 10^{−3} | −3.86 | 2.40 × 10^{−3} | −3.86 | 2.90 × 10^{−3} | −3.86 | 2.70 × 10^{−3} | −3.86 | 2.90 × 10^{−3} |

F20 | −3.27 | 7.48 × 10^{−2} | −3.27 | 6.75 × 10^{−2} | −3.19 | 8.67 × 10^{−2} | −3.23 | 8.52 × 10^{−2} | −3.23 | 9.19 × 10^{−2} |

F21 | −9.39 | 2.00 | −8.47 | 2.68 | −8.28 | 2.49 | −9.81 | 1.29 | −7.87 | 2.69 |

F22 | −1.04 × 10^{1} | 9.22 × 10^{−4} | −1.02 × 10^{1} | 9.70 × 10^{−1} | −9.78 | 1.90 | −1.00 × 10^{1} | 1.35 | −9.69 | 1.83 |

F23 | −1.04 × 10^{1} | 9.87 × 10^{−1} | −1.05 × 10^{1} | 7.63 × 10^{−4} | −1.03 × 10^{1} | 1.07 | −1.05 × 10^{1} | 2.36 × 10^{−7} | −1.02 × 10^{1} | 1.37 |

Function | Index | IGWO | GWO | SCA | MFO | PSO | BA | FPA | SSA |
---|---|---|---|---|---|---|---|---|---|

F1 | ave | 8.34 × 10^{−40} | 1.55 × 10^{−27} | 2.70 × 10^{−12} | 2.34 × 10^{3} | 8.76 × 10^{−3} | 4.64 | 4.22 × 10^{1} | 2.67 × 10^{−7} |

std | 2.35 × 10^{−39} | 2.95 × 10^{−27} | 7.91 × 10^{−12} | 5.04 × 10^{3} | 1.32 × 10^{−2} | 1.58 | 1.61 × 10^{1} | 3.31 × 10^{−7} | |

F2 | ave | 9.96 × 10^{−24} | 9.89 × 10^{−17} | 9.26 × 10^{−10} | 2.92 × 10^{1} | 1.12 × 10^{−1} | 1.05 × 10^{1} | 8.40 | 2.72 |

std | 1.23 × 10^{−23} | 9.40 × 10^{−17} | 2.51 × 10^{−9} | 1.78 × 10^{1} | 1.01 × 10^{−1} | 2.10 | 1.65 × 10^{1} | 1.94 | |

F3 | ave | 5.95 × 10^{−8} | 2.47 × 10^{−5} | 5.49 × 10^{−1} | 2.03 × 10^{4} | 5.61 × 10^{−3} | 7.05 | 6.02 × 10^{1} | 1.38 × 10^{3} |

std | 9.72 × 10^{−8} | 6.57 × 10^{−5} | 3.00 | 1.02 × 10^{4} | 6.75 × 10^{−3} | 2.42 | 3.33 × 10^{1} | 6.63 × 10^{2} | |

F4 | ave | 1.78 × 10^{−11} | 7.54 × 10^{−7} | 1.00 × 10^{−3} | 6.78 × 10^{1} | 7.66 × 10^{−2} | 1.23 | 2.61 | 1.14 × 10^{1} |

std | 3.00 × 10^{−11} | 1.16 × 10^{−6} | 2.60 × 10^{−3} | 1.02 × 10^{1} | 8.77 × 10^{−2} | 11.36 × 10^{−1} | 5.22 × 10^{−1} | 3.18 | |

F5 | ave | 2.72 × 10^{1} | 2.73 × 10^{1} | 7.36 | 2.68 × 10^{6} | 5.38 × 10^{−3} | 4.98 × 10^{2} | 2.48 × 10^{4} | 3.11 × 10^{2} |

std | 7.30 × 10^{−1} | 7.53 × 10^{−1} | 3.81 × 10^{−1} | 1.46 × 10^{7} | 7.82 × 10^{−3} | 1.65 × 10^{2} | 2.02 × 10^{4} | 4.69 × 10^{2} | |

F6 | ave | 1.68 | 7.37 × 10^{−1} | 4.53 × 10^{−1} | 1.35 × 10^{3} | 8.57 × 10^{−3} | 6.07 | 5.04 × 10^{1} | 2.11 × 10^{−7} |

std | 4.28 × 10^{−1} | 3.06 × 10^{−1} | 1.55 × 10^{−1} | 4.37 × 10^{3} | 1.44 × 10^{−2} | 1.72 | 1.54 × 10^{1} | 3.85 × 10^{−7} | |

F7 | ave | 8.87 × 10^{−4} | 2.40 × 10^{−3} | 2.50 × 10^{−3} | 2.83 | 1.12 × 10^{−1} | 1.52 × 10^{2} | 2.99 × 10^{3} | 1.74 × 10^{−1} |

std | 5.52 × 10^{−4} | 1.40 × 10^{−3} | 2.00 × 10^{−3} | 4.45 | 1.03 × 10^{−1} | 3.51 × 10^{1} | 2.15 × 10^{3} | 7.68 × 10^{−2} |

Function | Index | IGWO | GWO | SCA | MFO | PSO | BA | FPA | SSA |
---|---|---|---|---|---|---|---|---|---|

F8 | ave | −4.64 × 10^{3} | 1.84 × 10^{−14} | −2.16 × 10^{3} | −8.61 × 10^{3} | −5.57 × 10^{2} | -Inf | −4.73 × 10^{1} | −7.40 × 10^{3} |

std | 1.26 × 10^{3} | 8.25 × 10^{2} | 1.77 × 10^{2} | 8.71 × 10^{2} | 4.79 × 10^{−3} | --- | 8.83 | 6.39 × 10^{2} | |

F9 | ave | 0 | 2.41 | 5.52 × 10^{−1} | 1.64 × 10^{2} | 5.70 × 10^{−1} | 4.22 × 10^{1} | 1.90 × 10^{2} | 5.43 × 10^{1} |

std | 0 | 3.30 | 2.92 | 2.98 × 10^{1} | 6.01 × 10^{−1} | 7.28 | 3.08 × 10^{1} | 2.24 × 10^{1} | |

F10 | ave | 1.88 × 10^{−14} | 1.03 × 10^{−13} | 5.50 × 10^{−6} | 1.59 × 10^{1} | 3.96 | 3.44 | 5.60 | 2.46 |

std | 4.01 × 10^{−15} | 1.84 × 10^{−14} | 2.76 × 10^{−5} | 6.56 | 7.34 | 2.51 × 10^{−1} | 6.66 × 10^{−1} | 8.54 × 10^{−1} | |

F11 | ave | 0 | 3.20 × 10^{−3} | 8.05 × 10^{−2} | 3.70 × 10^{1} | 3.36 × 10^{−3} | 2.32 × 10^{−1} | 8.64 × 10^{−1} | 1.95 × 10^{−2} |

std | 0 | 6.70 × 10^{−3} | 1.45 × 10^{−1} | 5.06 × 10^{1} | 3.83 × 10^{−3} | 8.29 × 10^{−2} | 1.12 × 10^{−1} | 1.67 × 10^{−2} | |

F12 | ave | 1.02 × 10^{−1} | 4.59 × 10^{−2} | 9.27 × 10^{−2} | 1.55 × 10^{2} | 2.27 × 10^{−1} | 2.33 × 10^{−1} | 2.06 | 6.16 |

std | 3.95 × 10^{−2} | 2.13 × 10^{−2} | 4.55 × 10^{−2} | 4.58 × 10^{2} | 5.16 × 10^{−1} | 9.85 × 10^{−2} | 7.12 × 10^{−1} | 2.26 | |

F13 | ave | 1.05 | 6.41 × 10^{−1} | 3.00 × 10^{−1} | 1.37 × 10^{7} | 1.38 × 10^{−2} | 2.93 | 8.20 | 1.20 × 10^{1} |

std | 2.24 × 10^{−1} | 2.37 × 10^{−1} | 1.08 × 10^{−1} | 7.49 × 10^{7} | 1.19 × 10^{−2} | 7.01 × 10^{−1} | 2.79 | 1.54 × 10^{1} |

Function | Index | IGWO | GWO | SCA | MFO | PSO | BA | FPA | SSA |
---|---|---|---|---|---|---|---|---|---|

F14 | ave | 5.08 | 4.55 | 1.94 | 3.13 | 9.98 × 10^{−1} | 1.26 × 10^{1} | 1.27 × 10^{1} | 1.26 |

std | 4.310 | 4.36 | 9.97 × 10^{−1} | 2.51 | 8.07 × 10^{−6} | 3.48 × 10^{−1} | 1.88 × 10^{−14} | 6.35 × 10^{−1} | |

F15 | ave | 4.94 × 10^{−4} | 6.50 × 10^{−3} | 1.00 × 10^{−3} | 1.06 × 10^{−3} | 5.30 × 10^{−3} | 3.70 × 10^{−3} | 6.31 × 10^{−3} | 2.20 × 10^{−3} |

std | 1.22 × 10^{−4} | 9.30 × 10^{−3} | 3.66 × 10^{−4} | 4.24 × 10^{−4} | 5.47 × 10^{−3} | 6.10 × 10^{−3} | 1.01 × 10^{−2} | 4.90 × 10^{−3} | |

F16 | ave | −1.03 | −1.03 | −1.03 | −1.03 | 1.22 × 10^{1} | −1.03 | −1.03 | −1.03 |

std | 4.84 × 10^{−8} | 4.65 × 10^{−8} | 4.28 × 10^{−5} | 6.78 × 10^{−16} | 3.29 × 10^{1} | 2.11 × 10^{−9} | 4.99 × 10^{−7} | 1.68 × 10^{−14} | |

F17 | ave | 3.98 × 10^{−1} | 3.98 × 10^{−1} | 4.01 × 10^{−1} | 3.98 × 10^{−1} | 3.20 | 3.99 × 10^{−1} | 3.98 × 10^{−1} | 3.81 |

std | 3.83 × 10^{−4} | 3.88 × 10^{−6} | 3.00 × 10^{−3} | 0 | 3.30 | 2.90 × 10^{−3} | 3.54 × 10^{−10} | 3.57 × 10^{−2} | |

F18 | ave | 3.00 | 3.00 | 3.00 | 3.00 | 8.14 × 10^{3} | 1.83 × 10^{1} | 6.60 | 3.00 |

std | 3.15 × 10^{−5} | 2.99 × 10^{−5} | 1.54 × 10^{−4} | 2.23 × 10^{−15} | 1.88 × 10^{4} | 2.53 × 10^{1} | 1.54 × 10^{1} | 2.45 × 10^{−13} | |

F19 | ave | −3.86 | 4.56 | −3.85 | −3.86 | −1.08 | −3.42 | −3.86 | −3.86 |

std | 2.90 × 10^{−3} | 2.90 × 10^{−3} | 3.10 × 10^{−3} | 2.71 × 10^{−15} | 8.23 × 10^{−1} | 9.85 × 10^{−1} | 2.86 × 10^{−7} | 5.68 × 10^{−9} | |

F20 | ave | −3.23 | −3.27 | −2.93 | −3.24 | −5.03 × 10^{−1} | −3.16 | −2.93 | −3.22 |

std | 9.19 × 10^{−2} | 7.48 × 10^{−2} | 2.17 × 10^{−1} | 7.08 × 10^{−2} | 5.34 × 10^{−1} | 3.74 × 10^{−1} | 2.17 × 10^{−1} | 6.12 × 10^{−2} | |

F21 | ave | −7.87 | −9.39 | −1.93 | −7.72 | −2.49 × 10^{−1} | −5.23 | −1.02 × 10^{1} | −7.15 |

std | 2.69 | 2.00 | 1.57 | 3.12 | 3.82 × 10^{−1} | 9.31 × 10^{−1} | 3.92 × 10^{−4} | 3.56 | |

F22 | ave | −9.69 | −1.04 × 10^{1} | −4.11 | −7.46 | −1.52 × 10^{−1} | −5.09 | −1.04 × 10^{1} | −8.92 |

std | 1.83 | 9.22 × 10^{−4} | 1.66 | 3.49 | 1.20 × 10^{−1} | 5.23 × 10^{−7} | 5.50 × 10^{−3} | 2.78 | |

F23 | ave | −1.02 × 10^{1} | −1.04 × 10^{1} | −3.82 | −7.58 | −2.45 × 10^{−1} | −5.30 | −1.05 × 10^{1} | −8.34 |

std | 1.37 | 9.87 × 10^{−1} | 1.59 | 3.74 | 2.23 × 10^{−1} | 9.38 × 10^{−1} | 3.26 × 10^{−3} | 3.46 |

Function | IGWO vs. GWO p-Value Win | IGWO vs. SCA p-Value Win | IGWO vs. MFO p-Value Win | IGWO vs. PSO p-Value Win | IGWO vs. BA p-Value Win | IGWO vs. FPA p-Value Win | 1GWO vs. SSA p-Value Win |
---|---|---|---|---|---|---|---|

F1 | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} |

F2 | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} |

F3 | 1.78 × 10^{−10} | 3.029 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} |

F4 | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.01 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} |

F5 | 4.13 × 10^{−2} | 2.58 × 10^{−11} | 2.58 × 10^{−11} | 2.58 × 10^{−11} | 8.26 × 10^{−6} | 2.57 × 10^{−11} | 1.26 × 10^{−10} |

F6 | 2.97 × 10^{−11} | 2.50 × 10^{−2} | 2.70 × 10^{−2} | 2.97 × 10^{−11} | 2.97 × 10^{−11} | 4.89 × 10^{−11} | 2.97 × 10^{−11} |

F7 | 7.27 × 10^{−6} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} |

F8 | 3.01 × 10^{−11} | 1.81 × 10^{−1} | 3.01 × 10^{−11} | 2.99 × 10^{−11} | 3.01 × 10^{−11} | 3.02 × 10^{−11} | 1.21 × 10^{−10} |

F9 | 1.19 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.20 × 10^{−12} | 1.21 × 10^{−12} |

F10 | 1.58 × 10^{−11} | 1.68 × 10^{−11} | 1.68 × 10^{−11} | 5.65 × 10^{−13} | 1.63 × 10^{−11} | 1.68 × 10^{−11} | 1.68 × 10^{−11} |

F11 | 1.10 × 10^{−2} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} |

F12 | 1.06 × 10^{−7} | 3.01 × 10^{−11} | 3.01 × 10^{−11} | 3.01 × 10^{−11} | 9.12 × 10^{−1} | 1.61 × 10^{−1}0 | 3.01 × 10^{−11} |

F13 | 4.50 × 10^{−11} | 3.01 × 10^{−11} | 3.01 × 10^{−11} | 3.01 × 10^{−11} | 8.14 × 10^{−5} | 4.07 × 10^{−11} | 1.86 × 10^{−6} |

F14 | 3.26 × 10^{−1} | 8.26 × 10^{−1} | 6.58 × 10^{−1} | 4.76 × 10^{−4} | 5.12 × 10^{−11} | 2.30 × 10^{−12} | 1.79 × 10^{−4} |

F15 | 1.84 × 10^{−2} | 7.08 × 10^{−8} | 7.64 × 10^{−8} | 5.07 × 10^{−10} | 1.10 × 10^{−4} | 9.50 × 10^{−3} | 8.84 × 10^{−7} |

F16 | 1.06 × 10^{−11} | 2.36 × 10^{−12} | 1.61 × 10^{−1} | 2.36 × 10^{−12} | 6.50 × 10^{−14} | 2.49 × 10^{−12} | 1.61 × 10^{−1} |

F17 | 8.44 × 10^{−7} | 2.04 × 10^{−9} | 8.59 × 10^{−7} | 2.42 × 10^{−11} | 1.20 × 10^{−7} | 8.59 × 10^{−7} | 3.71 × 10^{−7} |

F18 | 3.79 × 10^{−1} | 1.76 × 10^{−1} | 1.21 × 10^{−12} | 3.01 × 10^{−11} | 1.21 × 10^{−12} | 5.96 × 10^{−11} | 1.21 × 10^{−12} |

F19 | 8.00 × 10^{−3} | 1.20 × 10^{−8} | 1.20 × 10^{−12} | 1.20 × 10^{−12} | 3.71 × 10^{−7} | 7.35 × 10^{−11} | 1.20 × 10^{−12} |

F20 | 2.23 × 10^{−1} | 1.01 × 10^{−8} | 1.50 × 10^{−3} | 1.21 × 10^{−12} | 6.25 × 10^{−4} | 5.53 × 10^{−8} | 1.40 × 10^{−2} |

F21 | 5.11 × 10^{−1} | 1.95 × 10^{−10} | 2.48 × 10^{−1} | 3.01 × 10^{−11} | 2.55 × 10^{−1} | 3.79 × 10^{−1} | 1.02 × 10^{−1} |

F22 | 5.07 × 10^{−10} | 3.68 × 10^{−11} | 5.73 × 10^{−2} | 3.01 × 10^{−11} | 7.73 × 10^{−11} | 5.57 × 10^{−10} | 6.35 × 10^{−2} |

F23 | 3.01 × 10^{−11} | 4.50 × 10^{−11} | 1.70 × 10^{−1} | 3.01 × 10^{−11} | 8.82 × 10^{−10} | 3.01 × 10^{−11} | 1.23 × 10^{−9} |

Function | IGWO | GWO | SCA | MFO | PSO | BA | FPA | SSA |
---|---|---|---|---|---|---|---|---|

F1 | 1.5 | 1.5 | 3 | 8 | 5 | 6 | 7 | 4 |

F2 | 1.5 | 1.5 | 3 | 8 | 4 | 7 | 6 | 5 |

F3 | 1 | 2 | 4 | 8 | 3 | 5 | 6 | 7 |

F4 | 1 | 2 | 3 | 8 | 4 | 5 | 6 | 7 |

F5 | 3 | 4 | 2 | 8 | 1 | 6 | 7 | 5 |

F6 | 5 | 4 | 3 | 8 | 2 | 6 | 7 | 1 |

F7 | 1 | 2 | 3 | 6 | 4 | 7 | 8 | 5 |

F8 | 3 | 7 | 4 | 1 | 5 | -- | 6 | 2 |

F9 | 1 | 4 | 2 | 7 | 3 | 5 | 8 | 6 |

F10 | 1 | 2 | 3 | 8 | 6 | 5 | 7 | 4 |

F11 | 1 | 2 | 5 | 8 | 3 | 6 | 7 | 4 |

F12 | 3 | 1 | 2 | 8 | 4 | 5 | 6 | 7 |

F13 | 4 | 3 | 2 | 8 | 1 | 5 | 6 | 7 |

F14 | 6 | 5 | 3 | 4 | 1 | 7 | 8 | 2 |

F15 | 1 | 8 | 2 | 3 | 6 | 5 | 7 | 4 |

F16 | 1.5 | 5 | 5 | 5 | 8 | 5 | 5 | 1.5 |

F17 | 2.5 | 2.5 | 6 | 2.5 | 7 | 5 | 2.5 | 8 |

F18 | 3 | 3 | 3 | 3 | 8 | 7 | 6 | 3 |

F19 | 2 | 8 | 5 | 3.5 | 7 | 6 | 3.5 | 1 |

F20 | 3 | 1 | 6.5 | 2 | 8 | 5 | 6.5 | 4 |

F21 | 3 | 2 | 7 | 4 | 8 | 6 | 1 | 5 |

F22 | 3 | 1.5 | 7 | 5 | 8 | 6 | 1.5 | 4 |

F23 | 3 | 2 | 7 | 5 | 8 | 6 | 1 | 4 |

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

Ts | Th | R | L | ||

MVO [37] | 0.8125 | 0.4375 | 42.0907382 | 176.738690 | 6060.8066 |

GSA [38] | 1.125000 | 0.6250 | 55.988659 | 84.4542025 | 8538.8359 |

PSO [39] | 0.812500 | 0.437500 | 42.091266 | 176.746500 | 6061.0777 |

MSCA [40] | 0.776256 | 0.399600 | 40.325450 | 199.9213 | 5935.7161 |

GA (Coello) [41] | 0.812500 | 0.4345 | 40.323900 | 200.0000 | 6288.7445 |

GA (Coello and Montes) [42] | 0.812500 | 0.4375 | 42.097397 | 176.654050 | 6059.9463 |

GA (Deb et al.) [43] | 0.937500 | 0.50000 | 48.329000 | 112.679000 | 6410.3811 |

ES [44] | 0.812500 | 0.437500 | 42.098087 | 176.640518 | 6059.745605 |

DE (Huang et al.) [45] | 0.8125 | 0.4375 | 42.098411 | 176.637690 | 6059.7340 |

ACO (Kaveh et al.) [46] | 0.8125 | 0.4375 | 42.103624 | 176.572656 | 6059.0888 |

HIS [47] | 1.125000 | 0.625000 | 58.29015 | 43.69268 | 7197.7300 |

MFO | 0.8125 | 0.4375 | 42.098445 | 176.636596 | 6059.7143 |

WOA [48] | 0.812500 | 0.437500 | 42.098209 | 176.638998 | 6059.7410 |

IGWO [21] | 0.8125 | 0.4375 | 42.0984456 | 176.636596 | 6059.7143 |

GWO | 0.7852686 | 0.3891504 | 40.67564 | 195.6436 | 5913.8838 |

IGWO | 0.7784458 | 0.3854034 | 40.33393 | 199.8019 | 5888.6000 |

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

d | N | D | ||

Mathematical optimization method (Belegundu) [49] | 0.053396 | 0.3177 | 14.0260 | 0.0127303 |

GSA (Kaveh) [46] | 0.050000 | 0.317312 | 14.22867 | 0.0128739 |

GSA (Rashedi) [38] | 0.050276 | 0.323680 | 13.525410 | 0.0127022 |

SCA [6] | 0.050780 | 0.334779 | 12.72269 | 0.127097 |

MVO [37] | 0.05000 | 0.315956 | 14.22623 | 0.128169 |

GWO | 0.05000 | 0.31739 | 14.0351 | 0.012699 |

IGWO | 0.05159 | 0.354337 | 11.4301 | 0.012700 |

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

h | l | t | b | ||

NGS-WOA [50] | 0.202369 | 3.544214 | 9.04821 | 0.205723 | 1.72802 |

WOA [48] | 0.205396 | 33.484293 | 9.037426 | 0.206276 | 1.730499 |

RO [51] | 0.203687 | 3.528467 | 9.004233 | 0.207241 | 1.735344 |

MVO [37] | 0.20722744 | 3.393969312 | 9.018874001 | 0.207225774 | 1.7250 |

CPSO [52] | 0.205463 | 3.473193 | 9.044502 | 0.205695 | 1.72645 |

CPSO [53] | 0.202369 | 3.544214 | 9.048210 | 0.205723 | 1.73148 |

HS [54] | 0.2442 | 6.2231 | 8.2915 | 0.2433 | 2.3807 |

GSA [38] | 0.182129 | 3.856979 | 10.0000 | 0.202376 | 1.87995 |

GA [55] | 0.1829 | 4.0483 | 9.3666 | 0.2059 | 1.82420 |

GA [56] | 0.2489 | 6.1730 | 8.1789 | 0.2533 | 2.43312 |

Coello [40] | 0.208800 | 3.420500 | 8.997500 | 0.2100 | 1.74831 |

Coello and Monters [41] | 0.205986 | 3.471328 | 9.020224 | 0.206480 | 1.72822 |

GWO | 0.20527 | 3.4819 | 9.0389 | 0.20583 | 1.7269 |

IGWO | 0.20496 | 3.4872 | 9.0366 | 0.20573 | 1.7254 |

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

X1 | X2 | ||

PSO-DE [39] | 0.7886751 | 0.4082482 | 263.8958433 |

MBA [57] | 0.7885650 | 0.4085597 | 263.8958522 |

DEDS [58] | 0.78867513 | 0.40824828 | 263.8958434 |

CS [59] | 0.78867 | 0.40902 | 263.9716 |

Ray and Sain [60] | 0.795 | 0.395 | 264.3 |

Tsa [61] | 0.788 | 0.408 | 263.68 |

WOA [48] | 0.789050544 | 0.407187512 | 263.8959474 |

MFO | 0.788244770931922 | 0.409466905784741 | 263.895979682 |

GWO | 0.78769 | 0.41108 | 263.9011 |

IGWO | 0.78846 | 0.40884 | 263.8959 |

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**MDPI and ACS Style**

Li, Y.; Lin, X.; Liu, J.
An Improved Gray Wolf Optimization Algorithm to Solve Engineering Problems. *Sustainability* **2021**, *13*, 3208.
https://doi.org/10.3390/su13063208

**AMA Style**

Li Y, Lin X, Liu J.
An Improved Gray Wolf Optimization Algorithm to Solve Engineering Problems. *Sustainability*. 2021; 13(6):3208.
https://doi.org/10.3390/su13063208

**Chicago/Turabian Style**

Li, Yu, Xiaoxiao Lin, and Jingsen Liu.
2021. "An Improved Gray Wolf Optimization Algorithm to Solve Engineering Problems" *Sustainability* 13, no. 6: 3208.
https://doi.org/10.3390/su13063208