# Hybrid Multi-Strategy Improved Butterfly Optimization Algorithm
## 4 Simulations and Analyses
### Analysis of Function Optimization Results
This analysis suggests that IBOA not only converges quickly, as evidenced by the small average values, but also demonstrates a high level of stability and reliability, as indicated by the low standard deviations. The fact that the standard deviation is zero for several functions implies that IBOA consistently finds the optimal solution or a solution very close to it across multiple runs, which is a strong indicator of the algorithm's robustness in these cases. For comparison with Reference [21], under the conditions of 50 dimensions and a maximum of 500 iterations, all the test functions used in Reference [21] are included in the test functions of this paper. The results for solving F1, F3, F8, and F10 functions are the same for both, with both achieving the theoretical optimal values. The computational results for F2, F4, and F6 are better than those in Reference [21], indicating that the optimization accuracy of the algorithm in this paper is higher than that of Reference [21]. Compared with [22], the seven test functions used in [22] are included in the test functions in this paper. Compared with these seven test functions, F1, F3, F5, and F7 functions all reach the theoretical optimal value. For F2 function, although IBOA does not reach the theoretical optimal value, its accuracy still reaches 3.1931 \(\times\) 10\({}^{-320}\). For F4 function, IBOA's optimization accuracy and standard deviation are better than those in the [22]. When compared with [23], for the F6 and F9 functions, IBOA demonstrates better average values and standard deviations than those reported in [23]. This clearly illustrates the superiority of the algorithm presented in this paper. The total optimization time results for 50 runs of each algorithm on the benchmark function are shown in Table 4. Overall, the running time of the IBOA algorithm is higher than that of the BOA, but with the improved strategy, the increase in running time is within an acceptable range. In summary, when optimizing the ten benchmark functions, the performance of the IBOA algorithm proposed in this paper has been significantly enhanced. Specifically, compared to the traditional BOA, IBOA has achieved higher optimization accuracy, and the convergence speed of the algorithm has also been improved, demonstrating a stronger optimization capability. When compared to the other three improved BOA algorithms, IBOA has also made significant progress in terms of optimization precision and stability. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Function** & **IBOA** & **BOA** & **PSO** & **SIBOA** & **PWMBOA** & **CFSSBOA** \\ \hline F1 & 15.294 & 12.261 & 2.3235 & 12.53 & 18.911 & 12.6485 \\ \hline F2 & 15.5355 & 12.8045 & 2.27845 & 12.521 & 19.1 & 13.047 \\ \hline F3 & 42.7375 & 39.4865 & 15.7745 & 39.4415 & 59.995 & 39.7375 \\ \hline F4 & 15.0395 & 12.23 & 2.3733 & 12.3705 & 18.652 & 12.5055 \\ \hline F5 & 14.389 & 11.5475 & 2.2595 & 11.8115 & 18.2135 & 12.0475 \\ \hline F6 & 31.4935 & 28.432 & 10.3595 & 28.5615 & 43.485 & 28.8715 \\ \hline F7 & 18.9825 & 16.9365 & 4.4275 & 16.802 & 24.422 & 15.6975 \\ \hline F8 & 16.0525 & 15.5525 & 3.5334 & 13.3875 & 22.4415 & 16.1945 \\ \hline F9 & 15.7025 & 14.211 & 3.77885 & 13.352 & 22.0775 & 15.5895 \\ \hline F10 & 17.8185 & 16.015 & 5.6445 & 15.121 & 25.3055 & 17.717 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The running time of test function.