# Pressure Vessel Design Problem Using Improved Gray Wolf Optimizer Based on Cauchy Distribution
## 4 Simulation Experiment and Results Analysis
### Effectiveness Test of CGWO: Comparison with Other Optimization Algorithms
For \(f_{5}\), the performance of CGWO and GWO is ahead of other algorithms, and there is little difference between CGWO and GWO  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Functions** & **Dimension** & **Results** & **PSO** & **FA** & **FOA** & **GWO** & **CGWO** \\ \hline \multirow{6}{*}{\(f_{1}\)} & \multirow{3}{*}{30} & Best & \(1.2110\times 10^{3}\) & \(7.1812\times 10^{2}\) & \(2.1618\times 10^{4}\) & \(3.6446\times 10^{-37}\) & **0** \\  & & Mean & \(1.6135\times 10^{3}\) & \(2.6262\times 10^{3}\) & \(2.9903\times 10^{4}\) & \(5.2291\times 10^{-35}\) & **0** \\  & & Std & \(2.0078\times 10^{2}\) & \(1.8835\times 10^{3}\) & \(5.2243\times 10^{3}\) & \(1.5900\times 10^{-34}\) & **0** \\ \cline{2-7}  & \multirow{3}{*}{100} & Best & \(1.5010\times 10^{4}\) & \(2.0700\times 10^{4}\) & \(1.2612\times 10^{5}\) & \(1.2832\times 10^{-19}\) & **0** \\  & & Mean & \(1.7413\times 10^{4}\) & \(5.0758\times 10^{4}\) & \(1.5794\times 10^{5}\) & \(2.6498\times 10^{-18}\) & **0** \\  & & Std & \(2.1027\times 10^{3}\) & \(1.6339\times 10^{4}\) & \(1.3272\times 10^{4}\) & \(3.2259\times 10^{-18}\) & **0** \\ \hline \multirow{6}{*}{\(f_{2}\)} & \multirow{3}{*}{30} & Best & \(1.6057\times 10^{7}\) & \(32.5437\) & \(1.7440\times 10^{2}\) & \(4.0221\times 10^{-22}\) & **0** \\  & & Mean & \(1.4651\times 10^{13}\) & \(62.4849\) & \(2.9393\times 10^{6}\) & \(3.3652\times 10^{-21}\) & **0** \\  & & Std & \(6.0585\times 10^{13}\) & \(23.0419\) & \(6.1964\times 10^{6}\) & \(2.9677\times 10^{-21}\) & **0** \\ \cline{2-7}  & \multirow{3}{*}{100} & Best & \(1.9403\times 10^{44}\) & \(1.7270\times 10^{12}\) & \(7.5431\times 10^{23}\) & \(5.2952\times 10^{-12}\) & **0** \\  & & Mean & \(4.6752\times 10^{52}\) & \(5.9032\times 10^{33}\) & \(6.6341\times 10^{44}\) & \(1.3235\times 10^{-11}\) & **0** \\  & & Std & \(2.5087\times 10^{33}\) & \(3.2334\times 10^{34}\) & \(2.1916\times 10^{35}\) & \(5.0888\times 10^{-12}\) & **0** \\ \hline \multirow{6}{*}{\(f_{3}\)} & \multirow{3}{*}{30} & Best & \(3.4135\times 10^{3}\) & \(1.9285\times 10^{4}\) & \(3.8240\times 10^{4}\) & \(1.2897\times 10^{-10}\) & **0** \\  & & Mean & \(5.7925\times 10^{3}\) & \(3.4836\times 10^{4}\) & \(6.5574\times 10^{4}\) & \(3.3327\times 10^{-5}\) & **0** \\  & & Std & \(2.2659\times 10^{3}\) & \(1.3002\times 10^{4}\) & \(1.3840\times 10^{4}\) & \(1.5905\times 10^{-4}\) & **0** \\ \cline{2-7}  & \multirow{3}{*}{100} & Best & \(8.4662\times 10^{4}\) & \(2.5906\times 10^{5}\) & \(3.8038\times 10^{5}\) & \(11.0536\) & **0** \\  & & Mean & \(1.2955\times 10^{5}\) & \(3.8818\times 10^{5}\) & \(9.2044\times 10^{5}\) & \(4.2216\times 10^{2}\) & **0** \\  & & Std & \(2.8592\times 10^{4}\) & \(1.1679\times 10^{5}\) & \(3.2277\times 10^{5}\) & \(4.6099\times 10^{2}\) & **0** \\ \hline \multirow{6}{*}{\(f_{4}\)} & \multirow{3}{*}{30} & Best & \(12.5449\) & \(54.8374\) & \(54.8327\) & \(7.5164\times 10^{-10}\) & **0** \\  & & Mean & \(16.9799\) & \(77.3924\) & \(62.2328\) & \(1.3451\times 10^{-8}\) & **0** \\  & & Std & \(1.2794\) & \(8.1602\) & \(3.6718\) & \(1.6314\times 10^{-8}\) & **0** \\ \cline{2-7}  & \multirow{3}{*}{100} & Best & \(37.2184\) & \(91.8108\) & \(75.5407\) & \(2.6104\times 10^{-2}\) & **0** \\  & & Mean & \(46.4379\) & \(96.3814\) & \(79.8232\) & \(1.2104\) & **0** \\  & & Std & \(5.2803\) & \(1.8427\) & \(2.4394\) & \(1.2159\) & **0** \\ \hline \multirow{6}{*}{\(f_{5}\)} & \multirow{3}{*}{30} & Best & \(1.1218\times 10^{7}\) & \(7.9773\times 10^{5}\) & \(2.1455\times 10^{7}\) & **26.0577** & 26.3554 \\  & & Mean & \(2.2226\times 10^{7}\) & \(1.0845\times 10^{7}\) & \(4.9234\times 10^{7}\) & \(27.7243\) & **27.6375** \\  & & Std & \(4.5716\times 10^{6}\) & \(1.1956\times 10^{7}\) & \(1.5647\times 10^{7}\) & \(0.8793\) & **0.6257** \\ \cline{2-7}  & \multirow{3}{*}{100} & Best & \(4.6114\times 10^{8}\) & \(1.9222\times 10^{8}\) & \(3.2673\times 10^{8}\) & \(97.0488\) & **96.9381** \\  & & Mean & \(6.5344\times 10^{8}\) & \(4.0775\times 10^{8}\) & \(4.77034\times 10^{8}\) & **98.2656** & 98.3083 \\  & & Std & \(7.9990\times 10^{7}\) & \(1.6320\times 10^{8}\) & \(8.2852\times 10^{7}\) & \(0.4232\) & **0.3999** \\ \hline \multirow{6}{*}{\(f_{6}\)} & \multirow{3}{*}{30} & Best & \(1.2093\times 10^{3}\) & \(3.4252\times 10^{2}\) & \(2.0372\times 10^{4}\) & \(0.5023\) & **9.4244 \(\times\ 10^{-7}\)** \\  & & Mean & \(1.6729\times 10^{3}\) & \(1.782 in terms of solution accuracy, but the standard deviation of CGWO is better, indicating that the CGWO algorithm is more stable.