# Pressure Vessel Design Problem Using Improved Gray Wolf Optimizer Based on Cauchy Distribution
## 4 Simulation Experiment and Results Analysis
### Performance Comparison of CGWO with Several Other Improved GWOs
Specifically, for unimodal benchmark functions \(f_{1}\) to \(f_{7}\), the optimal value, average value, and standard deviation of the CGWO algorithm results are the best, among which, for functions \(f_{1}\) to \(f_{4}\), only CGWO can achieve the theoretical optimal value with a mean value and standard deviation of 0, and the effect is excellent. For \(f_{7}\), although CGWO is the same as MGWO, HGWO, and LGWO, all fail to reach the theoretical optimal value; however, their optimal value is the closest to the theoretical optimal value and the most stable, reflecting the excellent global search ability of CGWO. For the multi-modal benchmark functions \(f_{8}\) to \(f_{13}\), the three evaluation indexes of the CGWO results are also optimal, which reflects its strong local search ability and ability to prevent easily falling into the local optimum. For functions \(f_{9}\) to \(f_{11}\), CGWO reached the theoretical optimal value. For several other multimodal benchmark functions, the CGWO results are up to five orders of magnitude higher than other algorithms, which is not only accurate but also the most stable. For fixed-dimension, multi-modal benchmark functions \(f_{14}\) to \(f_{23}\), CGWO also has obvious advantages, with  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Functions** & **Dimension** & **Results** & **MGWO** & **HGWO** & **LGWO** & **CGWO** \\ \hline \multirow{3}{*}{\(f_{13}\)} & \multirow{3}{*}{30} & Best & 1.6705 & 0.9072 & 0.6699 & \(\mathbf{1.8532\times 10^{-5}}\) \\  & & Mean & 2.3244 & 1.2733 & 1.2143 & \(\mathbf{7.4011\times 10^{-2}}\) \\  & & Std & 0.2944 & 0.9072 & 0.6815 & \(\mathbf{0.1671}\) \\ \hline \multirow{3}{*}{\(f_{14}\)} & \multirow{3}{*}{2} & Best & \(\mathbf{0.9980}\) & \(\mathbf{0.9980}\) & \(\mathbf{0.9980}\) & \(\mathbf{0.9980}\) \\  & & Mean & 8.6374 & 7.2793 & 5.4385 & \(\mathbf{1.3287}\) \\  & & Std & 4.6651 & 4.7953 & 4.3227 & \(\mathbf{0.7521}\) \\ \hline \multirow{3}{*}{\(f_{15}\)} & \multirow{3}{*}{4} & Best & \(\mathbf{3.0749\times 10^{-4}}\) & 3.0752 \(\times 10^{-4}\) & 3.0759 \(\times\) 10\({}^{-4}\) & \(\mathbf{3.0749\times 10^{-4}}\) \\  & & Mean & 4.49267 \(\times 10^{-3}\) & 7.7792 \(\times\) 10\({}^{-3}\) & 3.2275 \(\times\) 10\({}^{-3}\) & \(\mathbf{1.1376\times 10^{-3}}\) \\  & & Std & 8.0769 \(\times 10^{-3}\) & 9.7403 \(\times\) 10\({}^{-3}\) & 6.8910 \(\times\) 10\({}^{-3}\) & \(\mathbf{3.6033\times 10^{-3}}\) \\ \hline \multirow{3}{*}{\(f_{16}\)} & \multirow{3}{*}{2} & Best & \(\mathbf{-1.0316}\) & \(\mathbf{-1.0316}\) & \(\mathbf{-1.0316}\) & \(\mathbf{-1.0316}\) \\  & & Mean & \(-1.0307\) & \(\mathbf{-1.0316}\) & \(\mathbf{-1.0316}\) & \(\mathbf{-1.0316}\) \\  & & Std & \(5.2236\times 10^{-3}\) & 6.9846 \(\times\) 10\({}^{-8}\) & \(\mathbf{5.3524\times 10^{-9}}\) & \(1.0175\times 10^{-8}\) \\ \hline \multirow{3}{*}{\(f_{17}\)} & \multirow{3}{*}{2} & Best & \(\mathbf{0.3980}\) & \(\mathbf{0.3980}\) & \(\mathbf{0.3980}\) & \(\mathbf{0.3980}\) \\  & & Mean & \(\mathbf{0.3980}\) & \(\mathbf{0.3980}\) & \(\mathbf{0.3980}\) & \(\mathbf{0.3980}\) \\  & & Std & \(3.9409\times 10^{-7}\) & 4.9314 \(\times\) 10\({}^{-7}\  
the highest solution accuracy and high stability, which proves that its solution ability is also excellent in multi-modal problems with fixed low dimensions. In general, the proposed improved CGWO outperforms the other three improved GWO algorithms in solving the unimodal benchmark function, the multi-modal benchmark function, and the fixed-dimension, multi-modal benchmark function. In order to further compare the performance of the four improved GWO algorithms, this paper draws the iterative convergence curves of the four improved algorithms under different test functions, as shown in Figure 10.  
Figure 10: _Cont._  
From Figure 10, it can be found that, for \(f_{1}\) to \(f_{4}\), \(f_{9}\), and \(f_{11}\), CGWO has the fastest iteration speed and highest accuracy, and its performance far exceeds that of the other three improved algorithms.