## 3 Multi-Strategy Improved Sparrow Search Algorithm
### Performance of ISSA  
To validate the improved performance and effectiveness of the ISSA, we employed the CEC-2005 test suite for evaluation. The CEC-2005 suite comprises a set of challenging benchmark functions characterized by intricate local optima, effectively simulating the complexity of real search spaces. Testing on these benchmark functions allows for a comprehensive assessment of algorithmic performance and robustness. Some of the benchmark functions are listed in Table 1.  
Figure 3: Improved sparrow search algorithm flowchart. F1-F7 represent multi-dimensional unimodal functions, which are typically difficult to solve and are suitable for evaluating the solving and optimization capabilities of algorithms. Meanwhile, F8-F13 represent multi-dimensional multimodal functions, characterized by multiple local optimal points and barriers, and are suitable for evaluating the ability of algorithms to escape local optimal values. The experiments were conducted on an Intel(R) Core(TM) i5-12600 KF processor running at 3.70 GHz, with 32.00 GB of RAM, using the Windows 10 operating system and MATLAB R2022b. During the evaluation of the ISSA algorithm, it was compared against SSA and three other widely respected metaheuristic algorithms (SA, WOA, and PSO). By statistically analyzing the results from 30 independent runs, we could gain a more accurate understanding of the ISSA's performance relative to other algorithms. Through the repeated debugging and operation of all algorithms, the parameter values are as shown in Table 2.  
The experimental results are based on datasets obtained from 30 independent runs for each benchmark test function. We utilize the average value and standard deviation for the 13 test functions as the final performance evaluation metrics. These metrics are presented in Table 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function Name** & **Search Space** & **Dimensionality** & **Optimum Value** \\ \hline F1 (Sphere) & \([-100,100]\) & 30 & 0 \\ F2 (Chwefel 2.22) & \([-10,10]\) & 30 & 0 \\ F3 (Schwefel 1.2) & \([-100,100]\) & 30 & 0 \\ F4 (Schwefel 2.21) & \([-100,100]\) & 30 & 0 \\ F5 (Generalized & \([-30,30]\) & 30 & 0 \\ Rosenbrock) & \([-100,100]\) & 30 & 0 \\ F6 (Step Function) & \([-100,100]\) & 30 & 0 \\ F7 (Quartic) & \([-1.28,1.28]\) & 30 & 0 \\ \hline F8 (Schwefel 2.26) & \([-500,500]\) & 30 & \(-12\),569.5 \\ F9 (Rastrigin) & \([-5.12,5.12]\) & 30 & 0 \\ F10 (Ackley) & \([-32,32]\) & 30 & 0 \\ F11 (Griewank) & \([-600,600]\) & 30 & 0 \\ F12 (Generalized & \([-50,50]\) & 30 & 0 \\ Penalized Function1) & \([-50,50]\) & 30 & 0 \\ F13 (Generalized & \([-50,50]\) & 30 & 0 \\ Penalized Function2) & \([-50,50]\) & 30 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Benchmark test functions. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Symbols** & **Meaning** & **ISSA** & **SSA** & **PSO** & **SA** & **WOA** \\ \hline \(N\) & Population size & 50 & 50 & 50 & / & 50 \\ \(T\) & Maximum iterations & 200 & 200 & 200 & / & 200 \\ \(PD\) & Discoverers & 0.2 & 0.2 & / & / & / \\ \(SD\) & Sentirels & 0.1 & 0.1 & / & / & / \\ \(R_{2}\) & Warming value & 0.8 & 0.8 & / & / & / \\ \(\lambda\) & Learning coefficient & [0.4, 1] & / & / & / & / \\ \(c_{1}\) & Learning coefficient 1 & / & / & 2 & / & / \\ \(c_{2}\) & Learning coefficient 2 & / & / & 2 & / & / \\ \(\omega\) & Inertia weight & / & / & 0.7 & / & / \\ \(v\) & Velocity & / & / & \([-1,11]\) & / & / \\ \(T_{0}\) & Initial temperature & / & / & / & 100 & / \\ \(T_{f}\) & Final temperature & / & / & / & 0.001 & / \\ \(\alpha\) & Annealing coefficient & / & / & / & 0.95 & / \\ \(a\) & Decreasing vector & / & / & / & / & \([2,0]\) \\ \(b\) & Constant factor & / & / & / & / & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter values of five algorithms. Table 3 presents the mean and standard deviation of results obtained from 30 independent runs of five algorithms on benchmark test functions under 30-dimensional conditions.