# Pressure Vessel Design Problem Using Improved Gray Wolf Optimizer Based on Cauchy Distribution
## 4 Simulation Experiment and Results Analysis
### Selection of Parameters  
#### 4.2.1 Selection of the Scale Parameter \(\gamma\) in the Cauchy Distribution  
The scale parameter in the Cauchy distribution controls the shape of the Cauchy distribution, which has a certain impact on the exploration ability or development ability of the algorithm. Therefore, selecting the appropriate scale parameter is very important for the improvement effect. In order to obtain a reasonable value of \(\gamma\), in this paper, a series of extensive experiments were carried out on the test functions in the table, and relatively consistent results were obtained. In brief, the mean values of the solution results of the algorithms with different values of \(\gamma\) on the five functions are given here. Among these four functions, \(f_{1}\) and \(f_{7}\) are unimodal benchmark functions, \(f_{10}\) and \(f_{12}\) are multimodal benchmark functions, and \(f_{15}\) is a fixed-dimension, multimodal benchmark function. The experimental results are shown in Table 4, and the best experimental results are shown in bold.  
From the analysis of the results in Table 4, for \(f_{1}\) and \(f_{10}\), when the parameters are different, the results of this algorithm are consistent and good. For functions \(f_{7}\), \(f_{12}\), and \(f_{15}\), when the parameters are different, the results of this algorithm are not very different. When the value of \(\gamma\) is 1, the results of each function are good and the results are consistent. Therefore, the scale parameter of the Cauchy distribution is chosen as 1, which is also the parameter of the standard Cauchy distribution.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function** & **Dim** & **Search Ranges** & **Optimal Solutions** \\ \hline \(f_{14}(x)=[\frac{1}{80}+\sum\limits_{j=1}^{25}\frac{1}{j+\sum\limits_{j=1}^{2 }\left(x_{i}-a_{ij}\right)^{5}}]^{-1}\) & 2 & \([-65,65]\) & 1 \\ \(f_{15}(x)=\frac{1}{i=1}\left[a_{i}-\frac{x_{i}(f_{1}^{2}+bx_{i})}{b_{i}^{2}+bx _{i}x_{i}}\right]^{2}\) & 4 & \([-5,5]\) & 0.0003 \\ \(f_{16}(x)=4x_{1}^{2}-2.1x_{1}^{4}+\frac{1}{3}x_{1}^{4}+x_{1}x_{1}-24x_{2}^{2}+ 4x_{2}^{4}\) & 2 & \([-5,5]\) & \(-1.03162\) \\ \(f_{17}(x)=(x_{2}-\frac{5}{47^{2}}x_{1}^{2}+\frac{5}{8}x_{1}-6)^{2}+10(1-\frac{ 1}{87^{2}})\cos x_{1}+10\) & 2 & \([-5,5]\) & 0.398 \\ \(f_{18}(x)=\left[1+(x_{1}+x_{2}+1)^{2}(19-14x_{1}+3x_{1}^{2}-14x_{2}\right.\) & 2 & \([-2,2]\) & 3 \\ \(\left.+6x_{1}x_{2}+3x_{2}^{2}\right]\right]\times\left[30+(2x_{1}-3x_{2})^{2}( 18-32x_{1}\) & & & \\ \(\left.+12x_{1}^{2}+48x_{2}-36x_{1}x_{2}+27x_{2}^{2})\right]\) & & & \\ \(f_{19}(x)=-\sum\limits_{i=1}^{4}c_{i}\exp[-\sum\limits_{j=1}^{3}a_{ij}(x_{j}-p _{ij})^{2}]\) & 3 & \([0,1]\) & \(-3.86\) \\ \(f_{20}(x)=-\sum\limits_{i=1}^{4}c_{i}\exp[-\sum\limits_{j=1}^{6}a_{ij}(x_{j}-p _{ij})^{2}]\) & 6 & \([0,1]\) & \(-3.32\) \\ \(f_{21}(x)=-\sum\limits_{i=1}^{5}\left[(x-a_{i})(x-a_{i})^{T}+c_{i}\right]^{-1}\) & 4 & \([0,10]\) & \(-10.1532\) \\ \(f_{22}(x)=-\sum\limits_{i=1}^{7}\left[(x-a_{i})(x-a_{i})^{T}+c_{i}\right]^{-1}\) & 4 & \([0,10]\) & \(-10.4028\) \\ \(f_{23}(x)=-\sum\limits_{i=1}^{10}\left[(x-a_{i})(x-a_{i})^{T}+c_{i}\right]^{-1}\) & 4 & \([0,10]\) & \(-10.5363\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Fixed dimensional multimodal reference function.  
In addition, in order to intuitively show the influence of different values on CGWO performance, line charts showing the mean value of the algorithm solution results of \(f_{12}\) and \(f_{15}\) are presented in Figure 6:  
As can be seen from Figure 6, for \(f_{12}\), when \(\gamma\in[0.8,\,1.3]\), the results of CGWO are more stable and accurate. For \(f_{15}\), when \(\gamma=0.6\) and \(\gamma=1\), the results are good. Therefore, it is appropriate to set \(\gamma=1\) in the CGWO algorithm in this paper as the algorithm can better balance the abilities of exploration and development. It is also the parameter of the standard Cauchy distribution.