# 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
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Functions** & **Dimension** & **Results** & **MGWO** & **HGWO** & **LGWO** & **CGWO** \\ \hline \multirow{3}{*}{\(f_{1}\)} & \multirow{3}{*}{30} & Best & \(4.2025\times 10^{-29}\) & \(1.2004\times 10^{-45}\) & \(9.2839\times 10^{-37}\) & **0** \\  & & Mean & \(4.6237\times 10^{-27}\) & \(1.5657\times 10^{-42}\) & \(1.9113\times 10^{-34}\) & **0** \\  & & Std & \(8.7003\times 10^{-27}\) & \(3.2037\times 10^{-42}\) & \(3.4335\times 10^{-34}\) & **0** \\ \hline \multirow{3}{*}{\(f_{2}\)} & \multirow{3}{*}{30} & Best & \(4.0702\times 10^{-17}\) & \(1.2409\times 10^{-26}\) & \(4.4512\times 10^{-22}\) & **0** \\  & & Mean & \(2.1404\times 10^{-16}\) & \(2.1612\times 10^{-25}\) & \(1.1233\times 10^{-20}\) & **0** \\  & & Std & \(2.4492\times 10^{-16}\) & \(3.2843\times 10^{-25}\) & \(1.5010\times 10^{-20}\) & **0** \\ \hline \multirow{3}{*}{\(f_{3}\)} & \multirow{3}{*}{30} & Best & \(4.3841\times 10^{-8}\) & \(6.9926\times 10^{-13}\) & \(4.4024\times 10^{-11}\) & **0** \\  & & Mean & \(2.5975\times 10^{-5}\) & \(1.9412\times 10^{-7}\) & \(3.9837\times 10^{-6}\) & **0** \\  & & Std & \(7.3876\times 10^{-5}\) & \(5.8421\times 10^{-7}\) & \(1.0671\times 10^{-5}\) & **0** \\ \hline \multirow{3}{*}{\(f_{4}\)} & \multirow{3}{*}{30} & Best & \(6.7139\times 10^{-8}\) & \(3.2630\times 10^{-12}\) & \(1.7452\times 10^{-10}\) & **0** \\  & & Mean & \(2.3427\times 10^{-6}\) & \(9.3791\times 10^{-11}\) & \(1.0686\times 10^{-8}\) & **0** \\  & & Std & \(9.2347\times 10^{-7}\) & \(1.3713\times 10^{-10}\) & \(1.5464\times 10^{-8}\) & **0** \\ \hline \multirow{3}{*}{\(f_{5}\)} & \multirow{3}{*}{30} & Best & \(26.1948\) & \(26.2161\) & \(26.8501\) & \(26.3554\) \\  & & Mean & \(27.9364\) & \(27.6936\) & \(27.6670\) & \(27.6375\) \\  & & Std & \(0.7722\) & \(0.7535\) & \(0.6298\) & \(0.6257\) \\ \hline \multirow{3}{*}{\(f_{6}\)} & \multirow{3}{*}{30} & Best & \(0.7616\) & \(0.4981\) & \(0.8267\) & \(9.4244\times 10^{-7}\) \\  & & Mean & \(2.1212\) & \(1.6741\) & \(1.5430\) & \(1.0569\times 10^{-4}\) \\  & & Std & \(0.5405\) & \(0.4381\) & \(0.4235\) & \(1.4880\times 10^{-4}\) \\ \hline \multirow{3}{*}{\(f_{7}\)} & \multirow{3}{*}{30} & Best & \(4.9014\times 10^{-4}\) & \(4.5224\times 10^{-4}\) & \(4.1187\times 10^{-6}\) & \(1.5969\times 10^{-6}\) \\  & & Mean & \(2.4231\times 10^{-3}\) & \(1.8675\times 10^{-3}\) & \(3.0570\times 10^{-4}\) & \(1.2706\times 10^{-4}\) \\  & & Std & \(1.2131\times 10^{-3}\) & \(1.0909\times 10^{-3}\) & \(2.7301\times 10^{-4}\) & \(2.4114\times 10^{-4}\) \\ \hline \multirow{3}{*}{\(f_{8}\)} & \multirow{3}{*}{30} & Best & \(-6.6830\times 10^{3}\) & \(-6.9852\times 10^{3}\) & \(-7.3442\times 10^{3}\) & \(-1.0661\times 10^{4}\) \\  & & Mean & \(-5.3102\times 10^{3}\) & \(-5.6269\times 10^{3}\) & \(-5.7067\times 10^{3}\) & \(-7.1879\times 10^{3}\) \\  & & Std & \(7.0398\times 10^{2}\) & \(7.9456\times 10^{2}\) & \(1.0612\times 10^{3}\) & \(1.3569\times 10^{3}\) \\ \hline \multirow{3}{*}{\(f_{9}\)} & \multirow{3}{*}{30} & Best & \(5.6843\times 10^{-14}\) & **0** & **0** & **0** \\  & & Mean & \(5.0670\) & \(0.8454\) & \(0.6781\) & **0** \\  & & Std & \(5.1697\) & \(3.1769\) & \(1.8011\) & **0** \\ \hline \multirow{3}{*}{\(f_{10}\)} & \multirow{3}{*}{30} & Best & \(8.6153\times 10^{-14}\) & \(1.1546\times 10^{-14}\) & \(2.5757\times 10^{-14}\) & \(8.8818\times 10^{-16}\) \\  & & Mean & \(1.3316\times 10^{-13}\) & \(1.9954\times 10^{-14}\) & \(3.3573\times 10^{-14}\) & \(8.8818\times 10^{-16}\) \\  & & Std & \(2.9262\times 10^{-14}\) & \(3.7879\times 10^{-15}\) & \(5.058  
As can be seen from Table 10, the CGWO algorithm proposed in this paper has the best performance for most of the unimodal benchmark functions, multi-modal benchmark functions, and fixed-dimension, multi-modal benchmark functions.