# Hybrid Multi-Strategy Improved Butterfly Optimization Algorithm
## 4 Simulations and Analyses
### Improve Strategy Effectiveness Analysis
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Function** & **Algorithm** & **Optimal Value** & **Average Value** & **Standard Deviation** \\ \hline \multirow{4}{*}{F1} & IBOA & 0 & 0 & 0 \\  & BOA & \(1.4878\times 10^{-7}\) & \(2.4402\times 10^{-7}\) & \(5.5853\times 10^{-8}\) \\  & IBOA-1 & \(9.1267\times 10^{-8}\) & \(2.1207\times 10^{-7}\) & \(4.747\times 10^{-8}\) \\  & IBOA-2 & \(2.3502\times 10^{-87}\) & \(4.7501\times 10^{-66}\) & \(2.2556\times 10^{-65}\) \\  & IBOA-3 & \(5.0304\times 10^{-284}\) & \(1.1878\times 10^{-249}\) & 0 \\  & IBOA-4 & \(1.9111\times 10^{-11}\) & \(3.1856\times 10^{-9}\) & \(2.3931\times 10^{-9}\) \\ \hline \multirow{4}{*}{F2} & IBOA & 0 & 0 & 0 \\  & BOA & \(3.8163\times 10^{-14}\) & \(9.7811\times 10^{-12}\) & \(2.9634\times 10^{-11}\) \\  & IBOA-1 & \(4.0197\times 10^{-15}\) & \(1.6523\times 10^{-11}\) & \(2.3754\times 10^{-11}\) \\  & IBOA-2 & \(2.1354\times 10^{-205}\) & \(5.7688\times 10^{-173}\) & 0 \\  & IBOA-3 & \(5.6229\times 10^{-181}\) & \(1.6621\times 10^{-120}\) & \(1.1753\times 10^{-119}\) \\  & IBOA-4 & \(6.3567\times 10^{-7}\) & \(4.049\times 10^{-6}\) & \(1.126\times 10^{-6}\) \\ \hline \multirow{4}{*}{F3} & IBOA & 0 & 0 & 0 \\  & BOA & \(1.1995\times 10^{-7}\) & \(2.1382\times 10^{-7}\) & \(4.4449\times 10^{-8}\) \\  & IBOA-1 & \(1.0164\times 10^{-7}\) & \(2.2642\times 10^{-7}\) & \(4.7077\times 10^{-8}\) \\  & IBOA-2 & \(2.4818\times 10^{-87}\) & \(3.0135\times 10^{-67}\) & \(1.7174\times 10^{-66}\) \\  & IBOA-3 & \(1.365\times 10^{-284}\) & \(9.8693\times 10^{-250}\) & \(0\times 10^{-8}\) \\  & IBOA-4 & \(6.476\times 10^{-9}\) & \(5.825\times 10^{-8}\) & \(2.787\times 10^{-8}\) \\ \hline \multirow{4}{*}{F4} & IBOA & 0 & 0 & 0 \\  & BOA & \(3.5748\times 10^{-5}\) & \(6.9172\times 10^{-5}\) & \(1.2262\times 10^{-5}\) \\  & IBOA-1 & \(4.7552\times 10^{-5}\) & \(6.8879\times 10^{-5}\) & \(1.1012\times 10^{-5}\) \\  & IBOA-2 & \(1.4145\times 10^{-212}\) & \(1.8653\times 10^{-192}\) & 0 \\  & IBOA-3 & \(2.2966\times 10^{-309}\) & \(5.6237\times 10^{-270}\) & 0 \\  & IBOA-4 & \(1.7483\times 10^{-7}\) & \(8.0464\times 10^{-7}\) & \(3.095\times 10^{-7}\) \\ \hline \multirow{4}{*}{F5} & IBOA & \(9.7965\times 10^{-10}\) & \(4.3216\times 10^{-5}\) & \(9.5949\times 10^{-5}\) \\  & BOA & \(3.4381\) & \(4.3713\) & \(0.44399\) \\  & IBOA-1 & \(3.3771\) & \(0.016502\) & \(0.0048471\) \\  & IBOA-2 & \(1.5912\) & \(2.7275\) & \(0.63549\) \\  & IBOA-3 & \(2.1359\) & \(4.0601\) & \(0.77525\) \\  & IBOA-4 & \(9.1907\times 10^{-7}\) & \(1.2461\times 10^{-3}\) & \(2.5234\times 10^{-3}\) \\ \hline \multirow{4}{*}{F6} & IBOA & \(3.6308\times 10^{-8}\) & \(2.0451\times 10^{-5}\) & \(1.8865\times 10^{-5}\) \\  & BOA & \(0.00041255\) & \(0.0016045\) & \(0.00064071\) \\  & IBOA-1 & \(0.00038958\) & \(0.0014505\) & \(0.005137\) \\  & IBOA-2 & \(5.0144\times 10^{-8}\) & \(3.0865\times 10^{-5}\) & \(2.9218\times 10^{-5}\) \\  & IBOA-3 & \(4.023\times 10^{-7}\) & \(3.1476\times 10^{-5}\) & \(3.1526\times 10^{-5}\) \\  & IBOA-4 & \(4.0126\times 10^{-6}\) & \(8.0777\times 10^{-5}\) & \(7.089\times 10^{-5}\) \\ \hline \multirow{4}{*}{F7} & IBOA & \(-12,\)569.4593 & \(-12,\)547.1233 & \(27.4743\) \\  & BOA & \(-5110.4061\) & \(-3928.5789\) & \(317.0166\) \\  & IBOA-1 & \(-5510.3021\) & \(-4488.0117\) & \(421.6456\) \\  & IBOA-2 & \(-7036.16\) & \(-4764.2174\) & \(594.7845\) \\  & IBOA-3 & \(-5728.157\) & \(-3759.8554\) & \(451.7682\) \\  & IBOA-4 & \(-12,\)569.4835 & \(-12,\)569.2966 & \(0.37961\) \\ \hline \multirow{4}{*}{F8} & IBOA & 0 & 0 & 0 \\  & BOA & \(0\) & \(6.0982\times 10^{-12}\) & \(2.2353\times 10^{-11}\) \\  & IBOA-1 & \(0\) & \(3.6709\times 10^{-12}\) & \(8.1933\times 10^{-12}\) \\ \cline{1-1}  & IBOA-2 & \(0\) & \(0\) & \(0\) \\  & IBOA-3 & \(0\) & \(0\) & \(0\) \\  & IBOA-4 & \(7.2021\times 10^{-11}\) & \(1.5048\times 10^{-9}\) & \(1.0752\times 10^{-9}\) \\ \hline \multirow{4}{*}{F9} & IBOA & \(4.4409\times 10^{-16}\) & \(4.4409\times 10^{-16}\) & \(0\) \\  & BOA & \(2.9769\times 10^{-5}\) & \(4.9166\times 10^{-5}\) & \(7.7772\times 10^{-6}\) \\ \cline{1-1}  & IBOA-1 & \(3.3185\times 10^{-6}\) & \(4.6771\times 10^{-6}\) & \(6.6874\times 10^{-7}\) \\ \cline{1-1}  & IBOA-2 & \(4.4409\times 10^{-16}\) & \(4.4409\times 10^{-16}\) & \(0\) \\ \cline{1-1}  & IBOA-3 & \(4.4409\times 10^{-16}\) & \(4.4409\times 10^{-16}\) & \(0\) \\ \cline{1-1}  & IBOA-4 & \(6.6504\times 10^{-8}\) & \(8.265