# Hybrid Multi-Strategy Improved Butterfly Optimization Algorithm
## 4 Simulations and Analyses
### Analysis of Function Optimization Results
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Function** & **Algorithm** & **Optimal Value** & **Worst Value** & **Average Value** &
\begin{tabular}{c} **Standard** \\ **Deviation** \\ \end{tabular} \\ \hline \multirow{4}{*}{F4} & IBOA & 0 & 0 & 0 & 0 \\  & BOA & 3.5748 \(\times 10^{-5}\) & 9.2498 \(\times 10^{-5}\) & 6.9172 \(\times 10^{-5}\) & 1.2262 \(\times 10^{-5}\) \\  & PSO & 0.11773 & 0.99664 & 0.27003 & 0.14619 \\  & SIBOA & \(4.9814\times 10^{-178}\) & 4.7077 \(\times 10^{-173}\) & 2.4259 \(\times 10^{-174}\) & 0 \\  & PWMBOA & 0 & 0 & 0 & 0 \\  & CFSSBOA & \(3.1176\times 10^{-76}\) & 9.7379 \(\times 10^{-57}\) & 1.977 \(\times 10^{-58}\) & 1.3768 \(\times 10^{-57}\) \\ \hline \multirow{4}{*}{F5} & IBOA & 0 & 0 & 0 & 0 \\  & BOA & 3.4381 & 5.2922 & 4.3713 & 0.44399 \\  & PSO & 0.0092025 & 0.031153 & 0.016502 & 0.0048471 \\  & SIBOA & 0.59594 & 1.3643 & 0.95436 & 0.15895 \\  & PWMBOA & 2.1359 & 5.745 & 4.0601 & 0.77525 \\  & CFSSBOA & 2.8042 \(\times 10^{-5}\) & 0.0045236 & 0.0011309 & 0.00096006 \\ \hline \multirow{4}{*}{F6} & IBOA & \(3.6308\times 10^{-8}\) & 7.4089\(\times 10^{-5}\) & 2.0451 \(\times 10^{-5}\) & 1.8865 \(\times 10^{-5}\) \\  & BOA & 0.00041255 & 0.0038854 & 0.0016045 & 0.00064071 \\  & PSO & 0.016347 & 0.086173 & 0.04925 & 0.016063 \\  & SIBOA & \(1.2574\times 10^{-6}\) & 0.00016457 & 3.204 \(\times 10^{-5}\) & 3.4709 \(\times 10^{-5}\) \\  & PWMBOA & \(7.9534\times 10^{-7}\) & \(7.3818\times 10^{-5}\) & 2.1097 \(\times 10^{-5}\) & 1.9239 \(\times 10^{-5}\) \\  & CFSSBOA & \(2.2182\times 10^{-6}\) & 0.00038427 & 0.00011991 & 9.2574 \(\times 10^{-5}\) \\ \hline \multirow{4}{*}{F7} & IBOA & \(-12.569\).4593 & \(-12.448\).9448 & \(-12.547\).1233 & 27.4743 \\  & BOA & \(-5110.4061\) & \(-3576.605\) & \(-3928\).5789 & 317.0166 \\  & PSO & \(-4115.5193\) & \(-2551.9592\) & \(-3242.5415\) & 346.589 \\  & SIBOA & \(-12,285.712\) & \(-7986.9681\) & \(-9830.8295\) & 898.1468 \\  & PWMBOA & \(-12,569.4865\) & \(-9780.4795\) & \(-11,605.3028\) & 864.0931 \\  & CFSSBOA & \(-12,569.4835\) & \(-12,567.3264\) & \(-12,569.2966\) & 0.37961 \\ \hline \multirow{4}{*}{F8} & IBOA & 0 & 0 & 0 & 0 \\  & BOA & 0 & \(1.5558\times 10^{-10}\) & 6.0982 \(\times 10^{-12}\) & 2.2353 \(\times 10^{-11}\) \\  & PSO & 7.2816 & 60.3612 & 24.1695 & 10.5269 \\  & SIBOA & 0 & 0 & 0 & 0 \\  & PWMBOA & 0 & 0 & 0 & 0 \\  & CFSSBOA & 0 & 0 & 0 & 0 \\ \hline \multirow{4}{*}{F9} & IBOA & \(4.4409\times 10^{-16}\) & \(4.4409\times 10^{-16}\) & 4.4409 \(\times 10^{-16}\) & 0 \\  & BOA & \(2.9769\times 10^{-5}\) & \(6.9054\times 10^{-5}\) & 4.9166 \(\times 10^{-5}\) & 7.7772\(\times 10^{-6}\) \\ \cline{1-1}  & PSO & 0.083137 & 5.4548 & 2.8906 & 0.78444 \\ \cline{1-1}  & SIBOA & \(4.4409\times 10^{-16}\) & \(4.4409\times 10^{-16}\) & 4.4409 \(\times 10^{-16}\) & 0 \\ \cline{1-1}  & PWMBOA & \(4.4409\times 10^{-16}\) & \(4.4409\times 10^{-16}\) & 4.4409 \(\times 10^{-16}\) & 0 \\ \cline{1-1}  & CFSSBOA & \(4.4409\times 10^{-16}\) & \(4.4409\times 10^{-16}\) & 4.4409 \(\times 10^{-16}\) & 0 \\ \hline \multirow{4}{*}{F10} & IBOA & 0 & 0 & 0 & 0 \\  & BOA & \(2.1432\times 10^{-8}\) & \(2.5276\times 10^{-7}\) & \(7.5599\times 10^{-8}\) & \(4.2717\times 10^{-8}\) \\ \cline{1-1}  & PSO & 132.403 & 210.8344 & 180.0418 & 13.9491 \\ \cline{1-1}  & SIBOA & 0 & 0 & 0 & 0 \\ \cline{1-1}  & PWMBOA & 0 & 0 & 0 & 0 \\ \cline{1-1}  & CFSSBOA & 0 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._From the perspective of average values and standard deviations, the standard deviation reflects the stability and reliability of an algorithm, while the average value reflects the convergence speed. According to the data in Table 3, when solving the F6 and F7 functions, IBOA has the smallest average value and a relatively small standard deviation. For the F1, F2, F3, F4, F8, F9, and F10 functions, the standard deviation of IBOA is 0, which indicates that the high-precision results obtained by IBOA on the benchmark functions are not accidental events.