# Pressure Vessel Design Problem Using Improved Gray Wolf Optimizer Based on Cauchy Distribution
## 4 Simulation Experiment and Results Analysis
### Selection of Parameters
#### 4.2.2 Selection of Inertia Weight Parameter \(k\)  
The parameter \(k\) of the logarithmic function of the dynamic inertia weight in CGWO can control the decreasing trend of the inertia weight. In order to obtain the appropriate parameter \(k\) value, this paper conducts a series of extensive experiments on the test functions in the table. For brevity, the results of four typical functions are shown. Among them, functions \(f_{1}\) and \(f_{7}\) are single-peak benchmark functions, \(f_{12}\) is a multi-modal benchmark function, and \(f_{15}\) is a fixed-dimension multi-modal benchmark function. The test results for parameter \(k\) are shown in Table 5, and the best experimental results in the table are shown in bold. \begin{table}
\begin{tabular}{c c c c c c} \hline \(\gamma\) & **Results** & \(f_{1}\) & \(f_{7}\) & \(f_{10}\) & \(f_{12}\) & \(f_{15}\) \\ \hline
0.1 & Mean & **0** & \(9.522\times 10^{-5}\) & \(\textbf{8.882}\times\textbf{10}^{-16}\) & \(1.6169\times 10^{-2}\) & \(2.5387\times 10^{-3}\) \\
0.2 & Mean & **0** & \(9.105\times 10^{-5}\) & \(\textbf{8.882}\times\textbf{10}^{-16}\) & \(4.1501\times 10^{-3}\) & \(1.1452\times 10^{-3}\) \\
0.3 & Mean & **0** & \(\textbf{9.037}\times\textbf{10}^{-5}\) & \(\textbf{8.882}\times\textbf{10}^{-16}\) & \(3.2343\times 10^{-3}\) & \(1.2585\times 10^{-3}\) \\
0.4 & Mean & **0** & \(1.170\times 10^{-4}\) & \(\textbf{8.882}\times\textbf{10}^{-16}\) & \(1.4396\times 10^{-3}\) & \(1.8597\times 10^{-3}\) \\
0.5 & Mean & **0** & \(1.098\times 10^{-4}\) & \(\textbf{8.882}\times\textbf{10}^{-16}\) & \(1.2138\times 10^{-3}\) & \(1.1940\times 10^{-3}\) \\
0.6 & Mean & **0** & \(9.788\times 10^{-5}\) & \(\textbf{8.882}\times\textbf{10}^{-16}\) & \(1.9674\times 10^{-3}\) & \(4.4491\times 10^{-4}\) \\
0.7 & Mean & **0** & \(9.865\times 10^{-5}\) & \(\textbf{8.882}\times\textbf{10}^{-16}\) & \(8.2372\times 10^{-4}\) & \(3.8147\times 10^{-3}\) \\
0.8 & Mean & **0** & \(1.262\times 10^{-4}\) & \(\textbf{8.882}\times\textbf{10}^{-16}\) & \(6.5960\times 10^{-4}\) & \(1.0151\times 10^{-3}\) \\
0.9 & Mean & **0** & \(1.226\times 10^{-4}\) & \(\textbf{8.882}\times\textbf{10}^{-16}\) & \(4.2559\times 10^{-4}\) & \(1.0260\times 10^{-3}\) \\
1.0 & Mean & **0** & \(9.131\times 10^{-5}\) & \(\textbf{8.882}\times\textbf{10}^{-16}\) & \(2.2409\times 10^{-4}\) & \(\textbf{4.3320}\times\textbf{10}^{-4}\) \\
1.1 & Mean & **0** & \(1.044\times 10^{-4}\) & \(\textbf{8.882}\times\textbf{10}^{-16}\) & \(\textbf{2.2394}\times\textbf{10}^{-4}\) & \(1.0942\times 10^{-3}\) \\
1.2 & Mean & **0** & \(1.331\times 10^{-4}\) & \(\textbf{8.882}\times\textbf{10}^{-16}\) & \(5.5001\times 10^{-4}\) & \(1.0882\times 10^{-3}\) \\
1.3 & Mean & **0** & \(9.928\times 10^{-5}\) & \(\textbf{8.882}\times\textbf{10}^{-16}\) & \(8.7611\times 10^{-4}\) & \(1.7111\times 10^{-3}\) \\ \hline \end{tabular}
\end{table} 
Table 4: The test results of CGWO with different values of \(\gamma\). Figure 6: Line plots of the results with different scale parameters. From the test results in Table 5, it can be analyzed that, for function \(f_{1}\), when the value of the \(k\) parameter is different, the algorithm achieves the optimal result. For \(f_{7}\), the performance of the algorithm is relatively good after the value of \(k\) is 4.0, and the effect is optimal when \(k=4.5\). For functions \(f_{12}\) and \(f_{15}\), the algorithm performance is good in the range of the \(k\) value; therefore, the \(k\) value is determined to be 4.5. In order to see the influence of different \(k\) values more intuitively on the performance of the algorithm, a line diagram of the mean value of the algorithm solution results changing with the \(k\) value is shown in Figure 7.  
According to Figure 7 the following conclusions can be drawn. For function \(f_{7}\), the mean value solved using the algorithm shows a downward trend with the increase in the value of \(k\), and the results of the algorithm are better in the second half of the broken line, especially when \(k=4.5\).