## 3 Methodology
### Performance Analysis of the WOA  
In this study, we employed a population size of 100 and a maximum number of 250 iterations to determine the average minimum accuracy of the AWOA, WOA, gray wolf optimization (GWO) algorithm, and particle swarm optimization (PSO) algorithm on various benchmark test functions. These test functions included both single-peaked and multi-peaked parts, which evaluate the ability of the algorithms to solve for extreme values and jump out of local optima.  
Formulas (19)-(21) are three unimodal benchmark functions that have only one optimal value. The variable \(x_{i}\) denotes the range of the optimization algorithm at each position, which is [\(-\)100, 100].  
\[f_{1}(x)=\sum_{i=1}^{50}x_{i}^{2},-100\leq x_{i}\leq 100 \tag{19}\]  
\[f_{2}(x)=\max_{i}\{|x_{i}|,1\leq i\leq 50\},-100\leq x_{i}\leq 100 \tag{20}\]  
\[f_{3}(x)=\sum_{i=1}^{50}(|x_{i}+0.5|)^{2},\ -100\leq x_{i}\leq 100 \tag{21}\]  
Formulas (22)-(24) represent three benchmark multi-modal functions with multiple optimal values. The variable \(x_{i}\) denotes the weight range of the optimization algorithm in each function.  
\[f_{4}(x)=-\sum_{i=1}^{50}\biggl{(}x_{i}\sin\biggl{(}\sqrt{|x_{i}|}\biggr{)} \biggr{)},\ -500\leq x_{i}\leq 500 \tag{22}\]  
\[f_{5}(x)=\sum_{i=1}^{50}\biggl{[}x_{i}^{2}-10\cos(2\pi x_{i})+10\biggr{]},\ -5.12\leq x_{i}\leq 5.12 \tag{23}\]  
\[f_{6}(x)=-20\exp\left(-0.2\sqrt{\frac{1}{30}\sum_{i=1}^{30}x_{i}^{2}}\right) -\exp\left(\frac{1}{30}\sum_{i=1}^{30}\cos 2\pi x_{i}\right)+20+c,\ -32\leq x_{i}\leq 32 \tag{24}\]  
In all benchmark functions, the optimization algorithm explores the entire range of \(x_{i}\). The algorithm calculates a value for each iteration and compares it with previous iterations to select the optimal solution. In (24), the variable \(c\) is a constant. Since the WOA has inherent advantages in finding the minimum value, the AWOA can enhance the search range of the WOA by adjusting its control parameters, resulting in improved performance in the selected test function. By comparing the results with other algorithms, the AWOA achieves the best performance, as shown in Table 1.  
Figure 5 shows a visualization of the AWOA and its comparison with other algorithms. It shows that the AWOA reaches the optimal value after 50 iterations. In addition, AWOA achieves the smallest optimal value at the beginning of the first iteration compared with the other algorithms. Considering both the optimal value and the number of iterations, it is clear that the AWOA has a superior convergence speed and ability to find optimal solutions.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Function** & **AWOA** & **WOA** & **GWO** & **PSO** \\ \hline \(f_{1}(x)\) & \(1.92\times 10^{-3}\) & \(1.53\times 10^{0}\) & \(2.37\times 10^{-1}\) & \(1.03\times 10^{3}\) \\ \(f_{2}(x)\) & \(1.54\times 10^{-10}\) & \(7.81\times 10^{1}\) & \(3.427\times 10^{-6}\) & \(9.21\times 10^{0}\) \\ \(f_{3}(x)\) & \(1.01\times 10^{-1}\) & \(1.60\times 10^{-1}\) & \(5.54\times 10^{0}\) & \(8.96\times 10^{3}\) \\ \(f_{4}(x)\) & \(-2.09\times 10^{4}\) & \(1.99\times 10^{3}\) & \(-5.27\times 10^{3}\) & \(-3.15\times 10^{3}\) \\ \(f_{5}(x)\) & \(0\) & \(0\) & \(1.11\times 10^{-8}\) & \(3.14\times 10^{1}\) \\ \(f_{6}(x)\) & \(8.88\times 10^{-16}\) & \(3.46\times 10^{-10}\) & \(2.96\times 10^{-6}\) & \(2.47\times 10^{1}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Results comparing four kinds of algorithms.