## 3 Implementation
### Evaluating the Enhanced Algorithm Through Tests and Simulations
#### 3.1.2 Evaluating IGWO Against Competing Algorithms  
The Improved Grey Wolf Optimizer (IGWO) was assessed by calculating the mean and standard deviation of its performance on ten complex functions, both unimodal and multimodal. These metrics were then compared with those from the original Grey Wolf Optimizer (GWO), the Whale Optimization Algorithm (WOA) [39], and the Northern Goshawk Optimization (NGO) algorithms [40]. The comparative performance of these four algorithms is compiled in Table 2, where the superior outcomes are accentuated in bold. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function Name** & **Function Expression** & **Search Range** & \(\mathsf{f}_{\text{min}}\) \\ \hline Schwefel 1.2 & \(\mathsf{f}_{1}(x)=\sum\limits_{i=1}^{n}\left(\sum\limits_{j=1}^{i}x_{j}\right)^ {2}\) & \([-100,100]\) & 0 \\ Quartic & \(\mathsf{f}_{2}(x)=\sum\limits_{i=1}^{n}\left.x_{i}^{4}+\text{random}[0,1)\right.\) & \([-1.28,1.28]\) & 0 \\ Sphere & \(\mathsf{f}_{3}(x)=\sum\limits_{i=1}^{n}x_{i}^{2}\) & \([-100,100]\) & 0 \\ Schwefel 2.21 & \(\mathsf{f}_{4}(x)=\max_{i}\left\{\left|\,x_{i}\right.\right|,1\leq i\leq n \right\}\) & \([-100,100]\) & 0 \\ Schwefel 2.22 & \(\mathsf{f}_{5}(x)=\sum\limits_{i=1}^{n}x_{i}+\prod\
olimits_{i=1}^{n}\left|\,x_ {i}\right.\) & \([-10,10]\) & 0 \\ Schwefel 2.26 & \(\mathsf{f}_{6}=\sum\limits_{i=1}^{n}-x_{i}\sin(\sqrt{\left|\,x_{i}\right.})\) & \([-500,500]\) & \(-12,569.5\) \\ Rastrigin & \(\mathsf{f}_{7}(x)=\sum\limits_{i=1}^{n}\left[x_{i}^{2}-10\cos(2\pi x_{i})+10 \right]\) & \([-5.12,5.12]\) & 0 \\ Ackley & \(\mathsf{f}_{8}(x)=-20\exp(-0.2\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}x_{i}^{2}} )-\) & \([-32,32]\) & 0 \\  & \(\exp(\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}\cos(2\pi x_{i})})+20+\text{e}\) & \\ [PERSON] & \(\mathsf{f}_{9}=\frac{1}{4000}\sum\limits_{i=1}^{n}x_{i}^{2}-\prod\limits_{j=1}^{ n}\cos(\frac{x_{i}}{\sqrt{i}})+1\) & \([-50,50]\) & 0 \\ Goldstein Price & \(\mathsf{f}_{10}=\left[1+\kappa_{1}+x_{2}+1\right]\times 2\right]\times 30+\left(2 \kappa_{1}-3x_{9}\right)^{2}\)n] & \([-2,2]\) & 3 \\  & \(\mathsf{m}=\left(19-14\kappa_{1}+3x_{1}^{2}-14\kappa_{2}+6\kappa_{1}x_{2}+3x_{ 2}^{2}\right)\) & & \\  & \(\mathsf{n}=\left(18-32\kappa_{1}+12x_{1}^{2}+48\kappa_{2}-36\kappa_{1}x_{2}+27 \kappa_{2}^{2}\right)\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Set of common test functions. Analyzing the data presented in Table 2 yields several insights:
* 8. The IGWO algorithm outperforms the basic GWO, WOA, and NGO algorithms in terms of both accuracy and robustness, as evidenced by its mean values and standard deviations across all test functions. * 9. In dealing with unimodal functions (\(\mathrm{f_{1}}-\mathrm{f_{5}}\)), the IGWO achieves global optimums or near-optimums more efficiently than its counterparts. * 10. When tackling multimodal functions (\(\mathrm{f_{6}}-\mathrm{f_{10}}\)), the IGWO successfully navigates away from local optima to secure superior solutions, especially apparent in its performance on the Rastrigin (\(\mathrm{f_{7}}\)) and Griewank (\(\mathrm{f_{9}}\)) functions where it, alongside NGO, reaches the global optimum with minimal standard deviation. In contrast, the basic GWO is notably prone to settling at local optima. * 11. Overall, the IGWO algorithm's results across various test functions attest to its excellent versatility and adaptability, showcasing its effectiveness in addressing optimization challenges of varying types and complexities. For a more detailed comparison of the IGWO algorithm's optimization capabilities against those of the basic GWO, WOA, and NGO algorithms on unimodal test functions, fitness value change curves for these algorithms across functions \(\mathrm{f_{1}}-\mathrm{f_{5}}\) are illustrated in Figure 4.  
In all unimodal test functions, the IGWO algorithm consistently converges to the global optimum or very near it, with stable fitness value fluctuations.