## 4 Simulation Results and Analysis  
In order to discuss the performance influence of FA-SFLA, four benchmark functions (Sphere, Griewank, Rosenbrock and Rrastrigin) shown in Table 1 are adopted to carry out the simulation experiments with FA and SFLA. The algorithm convergence, robustness, efficiency and performance of the proposed improved FA are compared with FA and SFLA on two performance indexes (average cost function and corresponding standard deviation) after simulation 30 times. The initialization of algorithm parameters are shown in Table 2. The simulation results are shown in Table 3, and the simulation curves under FA and FA-SFLA are shown in Figure 5a-d.  
From the simulation results, it can be seen that the operation time of the proposed FA-SFLA hybrid algorithm was less than FA and SFLA. On the other hand, the optimal solution is less than or equal to the optimal solution of SFLA. This indicates that the FA-SFLA has feasibility and high efficiency for optimal seeking. By comparing the simulation results between FA and FA-SFLA, for the first three kinds of single peak function, the convergence velocity of FA-SFLA algorithm is relatively large. Except for the Griewank function, three other functions have the delay phenomena in later iterations but did not affect the polymorphic Rastrigin function converging to the optimum. The convergence velocity of the sphere function in FA and FA-SFLA is almost simultaneous, and Rosenbrock function with the use of the FA-SFLA algorithm exists apparent hysteretic convergence phenomena. Figure 4: Flowchart of FA-SFLA algorithm. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function Name** & **Expression** & **Scope** & **Dimension** \\ \hline F1 Sphere & \(\sum\limits_{i=1}^{n}x_{i}\) & [–100,100]\({}^{\text{n}}\) & 30 \\ F11 Griewank & \(\sum\limits_{i=1}^{n}\dfrac{x_{i}^{2}}{4000}-\prod\limits_{i=1}^{n}\cos\left( \dfrac{x_{i}}{\sqrt{i}}\right)+\ 1\) & [–32,32]\({}^{\text{n}}\) & 30 \\ F5 Rosenbrock & \(\sum\limits_{i=1}^{n-1}\left[100(x_{i}^{2}-x_{i+1})^{2}+(x_{i}-1)^{2}\right]\) & [–30,30]\({}^{\text{n}}\) & 30 \\ F9 Rrastrigin & \(\sum\limits_{i=1}^{n}\left[x_{i}^{2}-10\cos(2\pi x_{i})+10\right]\) & [–5.12,5.12]\({}^{\text{n}}\) & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Testing functions. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters Set-Up of FA and FA-SFLA** & \multicolumn{2}{c}{**Parameters Set-Up of SFLA**} \\ \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Number of fireworks & 8 & Number of population grouping & 8 \\ \hline Number of offspring & 64 &
\begin{tabular}{c} Number of frogs in each group \\ contains a number of frogs \\ \end{tabular} & 12 \\ \hline Dimension of optimized problem & 30 & Iteration number in inner group & 15 \\ \hline Number of maximum iterations & 100 & Maximum step-length & 100 \\ \hline Number of maximum evaluations & 10,000 & Amount of total population evolution & 10,000 \\ \hline Coefficient factor & 1.5 & Dimension of optimized problem & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter set-up of three optimization algorithms. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function** & **Algorithms** & **Average Cost Function** & **Standard Deviation** \\ \hline \multirow{3}{*}{Sphere} & FA & \(2.62\times 10^{-18}\) & \(8.75\times 10^{-19}\) \\  & FA-SFLA & \(1.38\times 10^{-30}\) & \(5.21\times 10^{-30}\) \\  & SFLA & \(8.25\times 10^{-14}\) & \(6.73\times 10^{-14}\) \\ \hline \multirow{3}{*}{Griewank} & FA & \(0.006437\) & \(0.008332\) \\  & FA-SFLA & \(0.000352\) & \(0.002783\) \\  & SFLA & \(0.02414\) & \(0.03564\) \\ \hline \multirow{3}{*}{[PERSON]} & FA & \(35.65643\) & \(15.53476\) \\  & FA-SFLA & \(28.54633\) & \(0.84352\) \\  & SFLA & \(58.23524\) & \(46.23514\) \\ \hline \multirow{3}{*}{Rrastrigin} & FA & \(3.84563\) & \(2.45623\) \\  & FA-SFLA & \(0\) & \(0\) \\ \cline{1-1}  & SFLA & \(45.43652\) & \(13.32451\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Simulation results under three optimization algorithms.