## 3 Improved Snake Optimization Algorithm for Melting Point Optimization Solution
### Analysis of Improved Snake Optimization Algorithm  
To validate the optimization performance of the proposed ISO algorithm, this study compares it with Grey Wolf Optimization (GWO), Harris Hawks Optimization (HHO), Seagull Optimization Algorithm (SOA), Snake Optimization Algorithm (SO), and Sine Cosine Algorithm (SCA). Ten typical test functions are selected for experimentation, comprising four unimodal functions \(F_{1}\)-\(F_{4}\) three multimodal functions \(F_{5}\)-\(F_{7}\), and three fixed-dimensional multimodal functions \(F_{8}\)-\(F_{10}\). The population size is set to \(n=30\), and each algorithm runs independently for 50 iterations. The main parameters of the algorithms are presented in Table 5, while the test functions are listed in Table 6. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Function** & **Value Range** & **Minimum** \\ \hline \(F_{1}(x)=\sum_{i=1}^{n}x_{i}^{2}\) & [\(-\)100,100] & 0 \\ \(F_{2}(x)=\sum_{i=1}^{n}|x_{i}|+\prod_{i=1}^{n}|x_{i}|\) & [\(-\)1.28,1.28] & 0 \\ \(F_{3}(x)=\sum_{i=1}^{d}\left(\sum_{j=1}^{i}x_{j}\right)^{2}\) & [\(-\)100,100] & 0 \\ \(F_{4}(x)=\max_{i}\left\{|x_{i}|,1\leq i\leq n\right\}\) & [\(-\)100,100] & 0 \\ \(F_{5}(x)=\sum_{i=1}^{n}\left[x_{i}^{2}-10\cos(2\pi x_{i})+10\right]\) & [\(-\)5.12,5.12] & 0 \\ \(F_{6}(x)=-20\exp(-0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_{i}^{2}})-\exp(\frac{1} {n}\sum_{i=1}^{n}\cos(2\pi x_{i}))\) & [\(-\)32,32] & 0 \\ \(F_{7}(x)=\frac{1}{4000}\sum_{i=1}^{n}x_{i}^{2}-\prod_{i=1}^{n}\cos(\frac{x_{i }}{\sqrt{i}})+1\) & [\(-\)600,600] & 0 \\ \(F_{8}(x)=-\sum_{i=1}^{4}c_{i}\exp(-\sum_{j=1}^{d}a_{ij}(x_{j}-p_{ij})^{2})\) & [0,1] & \(-\)3.32 \\ \(F_{9}(x)=-\sum_{i=1}^{5}\left[(X-a_{i})(X-a_{i})^{T}+c_{i}\right]^{-1}\) & [0,10] & \(-\)10.1532 \\ \(F_{10}(x)=\sum_{i=1}^{11}\left(a_{i}-\frac{x_{1}(b_{i}^{2}+b_{i}x_{2})}{b_{i}^ {2}+b_{i}x_{3}+x_{4}}\right)^{2}\) & [\(-\)5,5] & 0.0003 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Test Functions. \begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithm** & **Parameters** \\ \hline ISO & \(c_{1}=0.5,\ c_{2}=0.05,\ c_{3}=2\) \\ SO & \(c_{1}=0.5,\ c_{2}=0.05,\ c_{3}=2\) \\ GWO & \(\alpha=2\sim 0\) \\ HHO & \(J\in[0,2]\) \\ SOA & \(F_{c}=2\sim 0\) \\ SCA & - \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Algorithm Main Parameters. The experimental results comparing the six algorithms across the 10 test functions are presented in Table 7, with the evaluation criteria being the mean (Mean) and standard deviation (std):  
From Table 5 and Figure 5, it is evident that when solving functions F1-F9, the ISO algorithm exhibits superior optimization results and stability compared to other algorithms. It consistently converges to the theoretical optimum value of 0 across several functions. For single-peak functions F1-F4 and multi-peak functions F5 and F7, the ISO algorithm consistently converges to the theoretical optimum value of 0, showing a significantly larger magnitude difference compared to other algorithms. Regarding multi-peak function F6, both the ISO algorithm and the HHO algorithm achieve results closest to the theoretical optimum value, However, according to Figure 5\(f\), it is observed that the ISO algorithm exhibits significantly improved convergence speed compared to HHO, enabling it to rapidly attain the optimal solution. For multi-peak function F8, the ISO algorithm is closer to the theoretical optimum value than the other algorithms. On the fixed-dimensional multi-peak function F9, the ISO algorithm, along with the GWO and SO algorithms, achieves the theoretical optimum value. However, the ISO algorithm has a smaller standard deviation than the other two algorithms, and its convergence speed is significantly improved compared to the GWO and SO algorithms, demonstrating the stability of the ISO algorithm. These test functions adequately demonstrate the clear advantages of the ISO algorithm over other algorithms when solving benchmark test functions. Nevertheless, there is still room for improvement in the ISO algorithm.