# Pressure Vessel Design Problem Using Improved Gray Wolf Optimizer Based on Cauchy Distribution
## 4 Simulation Experiment and Results Analysis
### Validity Test of CGWO Inertia Weights: Comparison with Other Inertia Weight Strategies  
In order to prove the effectiveness of the dynamic inertia weight based on the Cauchy distribution and logarithmic function proposed in this paper, three different types of inertia weights are selected from other improved Gray Wolf Optimizers (GWOs) in the literature for comparison. This paper calls the GWO algorithm improved by these three inertia weight strategies GWO1 [48], GWO2 [49], and GWO3 [50], respectively. The comparative test results of the algorithm improved by the Cauchy nonlinear weighting strategy and the GWO algorithm improved by these three weighting strategies are shown in Table 9, and the best experimental results are shown in bold. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Functions** & **Dimension** & **Results** & **GWO1** & **GWO2** & **GWO3** & **CGWO** \\ \hline \multirow{3}{*}{\(f_{1}\)} & \multirow{3}{*}{30} & Best & **0** & 6.3747 \(\times\) 10\({}^{-318}\) & 1.5510 \(\times\) 10\({}^{-73}\) & **0** \\  & & Mean & **0** & 3.8695 \(\times\) 10\({}^{-305}\) & 2.1090 \(\times\) 10\({}^{-67}\) & **0** \\  & & Std & **0** & **0** & 7.1793 \(\times\) 10\({}^{-67}\) & **0** \\ \hline \multirow{3}{*}{\(f_{2}\)} & \multirow{3}{*}{30} & Best & 6.6232 \(\times\) 10\({}^{-267}\) & 7.2792 \(\times\) 10\({}^{-158}\) & 1.0396 \(\times\) 10\({}^{-41}\) & **0** \\  & & Mean & 2.0976 \(\times\) 10\({}^{-268}\) & 3.3602 \(\times\) 10\({}^{-148}\) & 7.8699 \(\times\) 10\({}^{-40}\) & **0** \\  & & Std & **0** & 1.8404 \(\times\) 10\({}^{-147}\) & 2.1656 \(\times\) 10\({}^{-39}\) & **0** \\ \hline \multirow{3}{*}{\(f_{3}\)} & \multirow{3}{*}{30} & Best & 4.0944 \(\times\) 10\({}^{-313}\) & 1.2715 \(\times\) 10\({}^{-315}\) & 4.9458 \(\times\) 10\({}^{-25}\) & **0** \\  & & Mean & 3.2456 \(\times\) 10\({}^{-304}\) & 7.9840 \(\times\) 10\({}^{-302}\) & 1.4509 \(\times\) 10\({}^{-14}\) & **0** \\  & & Std & **0** & **0** & 4.8297 \(\times\) 10\({}^{-14}\) & **0** \\ \hline \multirow{3}{*}{\(f_{4}\)} & \multirow{3}{*}{30} & Best & 5.9069 \(\times\) 10\({}^{-179}\) & 5.0573 \(\times\) 10\({}^{-165}\) & 7.6416 \(\times\) 10\({}^{-22}\) & **0** \\  & & Mean & 1.6441 \(\times\) 10\({}^{-177}\) & 3.0469 \(\times\) 10\({}^{-163}\) & 8.3291 \(\times\) 10\({}^{-19}\) & **0** \\  & & Std & **0** & **0** & 3.4206 \(\times\) 10\({}^{-18}\) & **0** \\ \hline \multirow{3}{*}{\(f_{5}\)} & \multirow{3}{*}{30} & Best & 26.8436 & 28.0934 & 27.1557 & **26.3554** \\  & & Mean & 28.4824 & 28.5668 & 28.2155 & **27.6375** \\  & & Std & 0.3808 & **0.3204** & 0.5722 & 0.6257 \\ \hline \multirow{3}{*}{\(f_{6}\)} & \multirow{3}{*}{30} & Best & 1.9655 & 4.9234 & 2.4369 & **9.4244 \(\times\) 10\({}^{-7}\)** \\  & & Mean & 3.8131 & 5.5318 & 3.3551 & **1.0569 \(\times\) 10\({}^{-4}\)** \\  & & Std & 0.4311 & 0.2562 & 0.4350 & **1.4880 \(\times\) 10\({}^{-4}\)** \\ \hline \multirow{3}{*}{\(f_{7}\)} & \multirow{3}{*}{30} & Best & 5.2987 \(\times\) 10\({}^{-6}\) & 8.6640 \(\times\) 10\({}^{-6}\) & 5.0086 \(\times\) 10\({}^{-4}\) & **1.5969 \(\times\) 10\({}^{-6}\)** \\  & & Mean & 1.9207 \(\times\) 10\({}^{-4}\) & 2.4277 \(\times\) 10\({}^{-4}\) & 1.6574 \(\times\) 10\({}^{-3}\) & **1.2706 \(\times\) 10\({}^{-4}\)** \\  & & Std & **1.7044 \(\times\) 10\({}^{-4}\)** & 2.0526 \(\times\) 10\({}^{-4}\) & 9.2681 \(\times\) 10\({}^{-4}\) & 2.4114 \(\times\) 10\({}^{-4}\) \\ \hline \multirow{3}{*}{\(f_{8}\)} & \multirow{3}{*}{30} & Best & \(-3.3749\times\) 10\({}^{3}\) & \(-3.7980\times\) 10\({}^{3}\) & \(-6.7440\times\) 10\({}^{3}\) & \(-\)**1.0661 \(\times\) 10\({}^{4}\)** \\  & & Mean & \(-2.5126\times\) 10\({}^{3}\) & \(-3.1165\times\) 10\({}^{3}\) & \(-4.6594\times\) 10\({}^{3}\) & \(-\)**7.1879 \(\times\) 10\({}^{3}\)** \\ \cline{1-1}  & & Std & \(3.9021\times\) 10\({}^{2}\) & **2.9672 \(\times\) 10\({}^{2}\)** & \(1.1482\times\) 10\({}^{3}\) & \(1.3569\times\) 10\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The comparison results of four inertia weights. to \(f_{23}\) all reach the theoretical optimal value.