# An Advanced Crow Search Algorithm for Solving Global Optimization Problem
## 4 Numerical Examples
### Benchmark Function Problems  
The algorithms used to compare the convergence performance of the ACS algorithm were the conventional CS algorithm, HS, DE, the grasshopper optimization (GO) algorithm, the salp swarm (SS) algorithm, and GA. Table 5 presents the parameters used in each algorithm. \(t_{max}\), \(N\), and Dim used 2000, 30, and 30, respectively, and each analysis was repeated a total of 30 times. Figure 8 is a graph representing the convergence of each algorithm, and the red line is the result of the ACS algorithm. In all of the benchmark functions except for five (\(f\)14, \(f\)20, \(f\)21, \(f\)22, and \(f\)23), it can be seen that the ACS algorithm finds the value closest to the **Min** the fastest. Table 6 presents the analysis results of each algorithm, and the last row shows the ranking using the BF of each algorithm. The ACS algorithm has the best convergence performance on unimodal (\(f\)1-\(f\)7) and multimodal (\(f\)8-\(f\)13) functions. In the fixed-dimension multimodal function (\(f\)14-\(f\)23), \(f\)15-\(f\)19 confirmed the best convergence performance, but \(f\)14 and \(f\)20-\(f\)23 did not. However, the ACS algorithm showed better convergence performance than the conventional CS algorithm. As a result of using the rankings of BF and MF, the ACS algorithm was derived as 1.65 and 1.78, confirming that it was the best. Therefore, it can be seen that the ACS algorithm has improved exploitation and exploration capabilities compared to the conventional CS algorithm. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Function** & **Equation** & **B** & **Min** \\ \hline \(f\)14 & \(f(\mathbf{x})=\left(\frac{1}{500}+\sum_{j=1}^{28}\frac{1}{j+\sum_{j=1}^{2}(1- \mathbf{x}_{j})^{5}}\right)^{-1}\) & \([-65\;65]^{2}\) & 1 \\ \(f\)15 & \(f(\mathbf{x})=\prod_{i=1}^{41}\left[a_{i}-\frac{\mathbf{x}_{i}(1+a_{i})x_{i}}{ \mathbf{x}_{i}+\mathbf{x}_{i}+\mathbf{x}_{i}}\right]^{2}\) & \([-5\;5]^{4}\) & 0.0003 \\ \(f\)16 & \(f(\mathbf{x})=4x_{1}^{2}-2.14x_{1}^{4}+\frac{1}{3}x_{1}^{4}+x_{1}^{2}-4x_{2}^ {2}+4x_{2}^{4}\) & \([-5\;5]^{2}\) & \(-1.0316\) \\ \(f\)17 & \(f(\mathbf{x})=\left(x_{2}-\frac{51}{42}x_{1}^{2}+\frac{5}{n}x_{1}-6\right)+10 \left(1-\frac{1}{8\pi}\right)\cos{\mathbf{r}}_{1}+10\) & \([-5\;5]^{2}\) & 0.398 \\ \(f\)18 & \(f(\mathbf{x})=\left[1+(x_{1}+x_{2}+1)^{2}\left(19-14x_{1}+3x_{1}^{2}-14x_{2}+6 x_{1}x_{2}+3x_{2}^{2}\right)\right]\) & \([-2\;2]^{2}\) & 3 \\ \(f\)19 & \(f(\mathbf{x})=-\sum_{i=1}^{41}c_{i}\exp\left(-\sum_{j=1}^{41}a_{ij}(x_{j}-p_{ ij})^{2}\right)\) & \([1\;3]^{3}\) & \(-3.86\) \\ \(f\)20 & \(f(\mathbf{x})=-\sum_{i=1}^{41}c_{i}\exp\left(-\sum_{j=1}^{41}a_{ij}(x_{j}-p_{ ij})^{2}\right)\) & \([0\;1]^{6}\) & \(-3.32\) \\ \(f\)21 & \(f(\mathbf{x})=-\sum_{i=1}^{58}\left[(X-a_{i})(X-a_{i})^{T}+c_{i}\right]^{-1}\) & \([0\;10]^{4}\) & \(-10.1532\) \\ \(f\)22 & \(f(\mathbf{x})=-\sum_{i=1}^{7}\left[(X-a_{i})(X-a_{i})^{T}+c_{i}\right]^{-1}\) & \([0\;10]^{4}\) & \(-10.4028\) \\ \(f\)23 & \(f(\mathbf{x})=-\sum_{i=1}^{70}\left[(X-a_{i})(X-a_{i})^{T}+c_{i}\right]^{-1}\) & \([0\;10]^{4}\) & \(-10.5363\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Fixed-dimension multimodal benchmark function for comparison. \begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithm** & **Parameters** \\ \hline ACS & \(f\)1 = 2, \(AP_{max}\) = 0.4, \(AP_{min}\) = 0.01, \(FAR\) = 0.4 \\ Conventional CS & \(f\)1 = 2, \(AP\) = 0.1 \\ HS & \(HMCR\) = 0.9, \(PAR\) = 0.1, \(bru\) = 0.03 \\ DE & \(PC_{r}\) = 0.5, \(F\) = 0.2 \\ GO & \(C_{max}\) = 1.0, \(C_{min}\) = 0.00001 \\ SS & \(Non\) = \(parameters\) \\ GA & \(P_{m}\) = 0.005, \(P_{c}\) = 0.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameters for benchmark function problems. Figure 8: _Cont_.