# 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
The test results of the multi-modal benchmark function are shown in Table 7.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Functions** & **Dimension** & **Results** & **PSO** & **FA** & **FOA** & **GWO** & **CGWO** \\ \hline \multirow{4}{*}{\(f_{8}\)} & \multirow{4}{*}{30} & Best & \(-7.7963\times 10^{3}\) & \(-8.5822\times 10^{3}\) & \(-6.3676\times 10^{3}\) & \(-6.8795\times 10^{3}\) & \(\mathbf{-1.0661\times 10^{4}}\) \\  & & Mean & \(-6.2455\times 10^{3}\) & \(-6.6183\times 10^{3}\) & \(-5.3782\times 10^{3}\) & \(-5.6016\times 10^{3}\) & \(\mathbf{-7.1879\times 10^{3}}\) \\  & & Std & \(8.1381\times 10^{2}\) & \(1.3152\times 10^{3}\) & \(\mathbf{4.1550\times 10^{2}}\) & \(6.1556\times 10^{2}\) & \(1.3569\times 10^{3}\) \\ \cline{2-7}  & \multirow{4}{*}{100} & Best & \(-2.2110\times 10^{4}\) & \(-2.3127\times 10^{4}\) & \(-1.0540\times 10^{4}\) & \(-1.9276\times 10^{4}\) & \(\mathbf{-2.8872\times 10^{4}}\) \\  & & Mean & \(\mathbf{-1.8511\times 10^{4}}\) & \(-1.6545\times 10^{4}\) & \(-9.3053\times 10^{3}\) & \(-1.5222\times 10^{4}\) & \(-1.7414\times 10^{4}\) \\  & & Std & \(2.2875\times 10^{3}\) & \(2.8187\times 10^{3}\) & \(8.2188\times 10^{2}\) & \(2.3207\times 10^{3}\) & \(\mathbf{2.0124\times 10^{3}}\) \\ \hline \multirow{4}{*}{\(f_{9}\)} & \multirow{4}{*}{30} & Best & \(4.0431\times 10^{2}\) & \(2.5167\times 10^{2}\) & \(2.4458\times 10^{2}\) & **0** & **0** \\  & & Mean & \(4.5035\times 10^{2}\) & \(3.0055\times 10^{2}\) & \(3.2312\times 10^{2}\) & 1.8663 & **0** \\  & & Std & \(19.4269\) & \(28.6191\) & \(24.4103\) & 3.6846 & **0** \\ \cline{2-7}  & \multirow{4}{*}{100} & Best & \(1.5358\times 10^{3}\) & \(1.2318\times 10^{3}\) & \(1.2741\times 10^{3}\) & 7.9581 \(\times 10^{-13}\) & **0** \\  & & Mean & \(1.6923\times 10^{3}\) & \(1.3428\times 10^{3}\) & \(1.3394\times 10^{3}\) & 3.2508 & **0** \\  & & Std & \(61.4329\) & \(57.5745\) & \(40.5915\) & 4.8984 & **0** \\ \hline \multirow{4}{*}{\(f_{10}\)} & \multirow{4}{*}{30} & Best & \(16.2719\) & \(10.3920\) & \(18.0503\) & \(2.9309\times 10^{-14}\) & \(\mathbf{8.8818\times 10^{-16}}\) \\  & & Mean & \(17.5696\) & \(15.5009\) & \(19.0642\) & \(4.0086\times 10^{-14}\) & \(\mathbf{8.8818\times 10^{-16}}\) \\  & & Std & \(0.7005\) & \(2.0238\) & \(0.4556\) & \(4.1182\times 10^{-15}\) & **0** \\ \cline{2-7}  & \multirow{4}{*}{100} & Best & \(19.4492\) & \(19.2102\) & \(19.8643\) & \(4.5148\times 10^{-11}\) & \(\mathbf{8.8818\times 10^{-16}}\) \\  & & Mean & \(20.0768\) & \(20.0651\) & \(20.1739\) & \(1.1835\times 10^{-10}\) & \(\mathbf{8.8818\times 10^{-16}}\) \\  & & Std & \(0.1701\) & \(0.4020\) & \(0.1366\) & \(7.6198\times 10^{-11}\) & **0** \\ \hline \multirow{4}{*}{\(f_{11}\)} & \multirow{4}{*}{30} & Best & \(1.3112\) & \(1.4391\) & \(1.8285\times 10^{2}\) & **0** & **0** \\  & & Mean & \(1.4247\) & \(6.8053\) & \(2.8923\times 10^{2}\) & \(7.0916\times 10^{-3}\) & **0** \\  & & Std & \(6.2396\times 10^{-2}\) & \(8.7207\) & \(50.5289\) & \(1.4254\times 10^{-2}\) & **0** \\ \cline{2-7}  & \multirow{4}{*}{100} & Best & \(4.4673\) & \(20.8527\) & \(1.2152\times 10^{3}\) & **0** & **0** \\  & & Mean & \(5.3452\) & \(1.8094\times 10^{2}\) & \(1.4094\times 10^{3}\) & \(3.5592\times 10^{-3}\) & **0** \\  & & Std & \(0.3802\) & \(1.3518\times 10^{2}\) & \(1.1289\times 10^{2}\) & \(9.3449\times 10^{-3}\) & **0** \\ \hline \multirow{4}{*}{\(f_{12}\)} & \multirow{4}{*}{30} & Best & \(2.0331\times 10^{4}\) & \(9.7452\times 10^{4}\) & \(1.8072\times 10^{7}\) & \(3.9722\times 10^{-2}\) & \(\mathbf{2.0805\times 10^{-6}}\) \\  & & Mean & \(2.8855\times 10^{5}\) & \(1.2464\times 10^{7}\) & \(4.6900\times 10^{7}\) & \(0.1109\) & \(\mathbf{8.7659\times 10^{-4}}\) \\  & & Std & \(1.9070\times 10^{5}\) & \(1.4527\times 10^{7}\) & \(1.2284\times 10^{7}\) & \(9.3102\times 10^{-2}\) & \(\mathbf{3.8057\times 10^{-3}}\) \\ \cline{2-7}  & \multirow{4}{*}{100} & Best & \(7.7862\times 10^{7}\) & \(6.1173\times 10^{7}\) & \(4.6337\times 10^{8}\) & \(0.3356\) & \(\mathbf{6.1477\times 10^{-6}}\) \\  & & Mean & \(1.6346\times 10^{8}\) & \(5.7079\times 10^{8}\) & \(8.2877\times 10^{8}\) & \(0.4794\) & \(\mathbf{3.8124\times 10^{-3}}\) \\  & & Std & \(4.2783\times 10^{7}\) & \(2.4257\times 10^{8}\) & \(1.8180\times 10^{8}\) & \(8.4667\times 10^{-2}\) & \(\mathbf{5.3842\times 10^{-3}}\) \\ \hline \multirow  
By analyzing Figure 8, it can be found that the convergence speed of the CGWO algorithm is fast, especially for unimodal benchmark functions \(f_{1}\) to \(f_{4}\) and multi-modal benchmark functions \(f_{9}\) to \(f_{11}\); CGWO convergence speed is much faster than other algorithms.