## 2 Chimp Optimization Algorithm
### Mathematical Model: Chimp Optimization
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**ID** & \multicolumn{2}{c}{**Function**} & **dim** & **Range** & \(f_{min}\) & **Type** \\ \hline F1 & \(F_{1}(x)=\sum_{x=1}^{x}x_{1}^{2}\) & & 30/50/100 & \([-100,100]\) & 0 & Unimodal \\ F2 & \(F_{2}(x)=\sum_{x=1}[x|i+1]\) & 30/50/100 & \([-10,10]\) & 0 & Unimodal \\ F3 & \(F_{3}(x)=\sum_{x=1}^{x}\left(\sum_{1}^{x}\sum_{1}^{x}\right)^{2}\) & 30/50/100 & \([-100,100]\) & 0 & Unimodal \\ F4 & \(F_{4}(x)=max_{x}\left[\left|x_{i}\right|,1\leq i\leq n\right)\) & 30/50/100 & \([-100,100]\) & 0 & Unimodal \\ F5 & \(F_{5}(x)=\sum_{x=1}^{x}\left[\left|100\left(x_{i+1}-x_{i}\right)^{2}\right|+ \left(x_{i}-1\right)^{2}\right]\) & 30/50/100 & \([-30,30]\) & 0 & Unimodal \\ F6 & \(F_{6}(x)=\sum_{x=1}^{x}\left(\left|x_{i}+0.5\right|\right)^{2}\) & 30/50/100 & \([-100,100]\) & 0 & Unimodal \\ F7 & \(F_{7}(x)=\sum_{x=1}^{x}ix^{4}+random(0,1]\) & 30/50/100 & \([-1.28,1.28]\) & 0 & Unimodal \\ F8 & \(F_{8}(x)=\sum_{x=1}^{x}-x\sin\left(\sqrt{|x_{1}|}\right)\) & 30/50/100 & \([-500,500]\) & \(-418.9829\times\text{d}\) & Multimodal \\ F9 & \(F_{9}(x)=\sum_{x=1}^{x}\left[x_{i}^{2}-10\cos(2\pi x_{i})+10\right]\) & 30/50/100 & \([-5.12,5.12]\) & 0 & Multimodal \\ F10 & \(F_{10}(x)=-20\exp\left(-0.2\sqrt{\frac{1}{x}\sum_{x=1}^{x}x_{i}^{2}}\right)- \exp\left(\frac{1}{x}\sum_{i=1}\cos(2\pi x_{i})\right)\) & 30/50/100 & \([-32,32]\) & 0 & Multimodal \\ F11 & \(F_{11}(x)=\frac{10}{200}\sum_{i=1}^{x}x_{i}^{2}-\prod_{i=1}^{x}\cos\left(\frac{ x_{i}}{\sqrt{x_{i}}}\right)+1\) & 30/50/100 & \([-600,600]\) & 0 & Multimodal \\ F12 & \(F_{12}(x)=\frac{\pi}{2}\left\{\text{loasin}(\pi y_{1})+\sum_{i=1}^{x}\left(y_{i}- 1\right)^{2}\left[1+10\text{sin}^{2}(\pi y_{i+1})\right]+\left(y_{n}-1\right)^{ 2}\right\}\) & 30/50/100 & \([-50,50]\) & 0 & Multimodal \\  & \(F_{13}(x)=0.1\left\{\text{sin}^{2}(3\pi x_{1})+\sum_{i=1}^{x}\left(x_{i}-1 \right)^{2}\left[1+\text{sin}^{2}(3\pi x_{i}+1)\right]\right.\) & & & \\ F13 & \(\left.\begin{array}{c}+(x_{n}-1)^{2}\left[1+\text{sin}^{2}(1\pi x_{n})\right] \end{array}\right\}\) & 30/50/1200 & \([-50,50]\) & 0 & Multimodal \\  & \(+\sum_{n=1}^{x}u\left(x_{i},5,100,4\right)\) & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Benchmarking functions. for functions F1-F13 and recorded the best (best), average (mean), and standard deviation (STD). When solving the minimum problem, the smaller the mean, the better the performance of the algorithm, and the smaller the standard deviation, the more stable the algorithm. Therefore, we used the mean and standard deviation of the best values to evaluate the performance of the 13 algorithms. ### Analysis of the Influence of Different Improvement Strategies on Algorithm Performance

The optimal value and the mean value are used to reflect the ability and convergence accuracy of the algorithm in finding the best performance. First, SDChOA obtains the optimal theoretical values when solving the F9 and F11 functions. For functions F3-F5, its shape is similar to a parabola, and there are a large number of local optimal values. At the same time, F7 is a multimodal function with a valley shape, and its global optimal value is located at the lower end of the mountain, which is difficult to find. As shown in Table 3, SDChOA outperforms other algorithms. Peak functions F1-F7 can be used to research and develop capability algorithms. According to the results in Table 3, six out of seven cases of the proposed SDChOA have the best results, with obvious advantages over competitors. Multimodal functions F8-F13 have a large number of local optima, and the global optimum becomes more difficult to obtain as the search space dimension increases. The results in Table 3 show that the proposed SDChOA achieved the optimal solution in six cases and achieved the best effect in nine cases. This shows that the proposed improvement strategy has a significant optimization effect on the standard ChOA. Integrating these strategies into the SDChOA improves its performance in terms of search accuracy and stability, showing certain advantages.