## Materials and Methods
### Firefly Algorithm and Its Improvements
#### 2.1.3 Simulation Analysis  
In order to verify the superiority of IFA, this paper selects 8 standard test functions to test standard FA, LFA and IFA, respectively, as shown in Table 1. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \begin{tabular}{c} **Function** \\ **Name** \\ \end{tabular} & \begin{tabular}{c} **Dimension** \\ **Value Range** \\ \end{tabular} & \begin{tabular}{c} **Independent Variable** \\ **Value Range** \\ \end{tabular} & \
\begin{tabular}{c} **Function Formula** \\ **Value** \\ \end{tabular} &
\begin{tabular}{c} **Theoretical** \\ **Value** \\ \end{tabular} \\ \hline \(f_{1}\) & Ackley & 2 & [–32.768, 32.768] & \(f(x)=-20\exp\!\left(-0.2\sqrt{\frac{1}{2}\sum\limits_{i=1}^{d}x_{i}^{2}}\right)\) & \(f_{\min}=0\) \\  & & & \(-\exp\!\left(\frac{1}{2}\sum\limits_{i=1}^{d}\cos(2\pi x_{i})\right)+\exp(1)\) & \(f(x)=-0.0001(+1)^{\alpha 1}\) \\ \(f_{2}\) & Cross in & 2 & [–10, 10] & \(t=\left|\sin(x_{1})\sin(x_{2})\exp\!\left(\left|100.0-\frac{\sqrt{1}+x_{i}^{2} }{\alpha}\right|\right)\right|\) & \(f_{\min}=-2.06261\) \\ \(f_{3}\) & Drop & 2 & [–5.12, 5.12] & \(f(x)=-\frac{1+\alpha(12\sqrt{\frac{1}{2}+\alpha)}}{\alpha(5\sqrt{\frac{1}{2}+ \alpha)}+2}\) & \(f_{\min}=-1\) \\ \(f_{4}\) & Griewank & 2 & [–600, 600] & \(f(x)=\frac{d}{\sum\limits_{i=1}^{d}\frac{x_{i}^{2}}{\alpha 600}}-\frac{1}{1-1}\cos\left(\frac{x_{i}}{ \sqrt{\frac{1}{2}+\alpha)}+1}\) & \(f_{\min}=0\) \\ \(f_{5}\) & Levy 13 & 2 & [–10, 10] & \(f(x)=\sin^{2}(3\pi x_{1})+(x_{1}-1)^{\frac{1}{2}}[1+\sin^{2}(3\pi x_{2})]\) & \(f_{\min}=0\) \\  & & & \(+(x_{2}-1)^{\frac{1}{2}}[1+\sin^{2}(2\pi x_{2})]\) & \(f_{\min}=0\) \\ \(f_{6}\) & Rastrigin & 2 & [–5.12, 5.12] & \(f(x)=10d+\frac{d}{\sum\limits_{i=1}^{d}\left[x_{i}^{2}-10\cos(2\pi x_{i}) \right]}\) & \(f_{\min}=0\) \\ \(f_{7}\) & Booth & 2 & [–10, 10] & \(f(x)=(x_{1}+2x_{2}-7)^{2}+(2x_{1}+x_{2}-5)^{2}\) & \(f_{\min}=0\) \\ \(f\) & Three & & & & \\ \(f\)s & Flump & 2 & [–5, 5] & \(f(x)=2x_{1}^{2}-1.05x_{1}^{4}+\frac{\alpha}{6}+x_{1}x_{2}+x_{2}^{2}\) & \(f_{\min}=0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Standard test functions. Figure 1: Improved firefly algorithm flow. The unified maximum iteration number is set to 200, absorption coefficient \(\gamma=1\), initial step size \(s=0.5\), initial attraction \(\beta_{0}\)= 1, and the number of random fireflies is 20. The test experiments are conducted 30 times for each algorithm, and the best value, the best value variance and the average running times are recorded. The iteration curves are shown in Figure 2, and the simulation results are shown in Tables 2 and 3. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Function**} & \multirow{2}{*}{**Parameter**} & \multicolumn{2}{c}{**Optimal Value**} & \multicolumn{3}{c}{**Optimal Value Variance**} \\  & & **FA** & **LFA** & **FA** & **LFA** & **IFA** \\ \hline \(f_{1}\) & \(1.02\times 10^{-4}\) & \(0.00452104\) & \(5.92\times 10^{-4}\) & \(0.97116936\) & \(0.13558556\) & \(1.83\times 10^{-5}\) \\ \(f_{2}\) & \(-2.06261\) & \(-2.06261\) & \(-2.06261\) & \(6.21\times 10^{-6}\) & \(2.18\times 10^{-7}\) & \(3.43\times 10^{-13}\) \\ \(f_{3}\) & \(-1.00\) & \(-0.999951\) & \(-0.999984\) & \(0.0280539\) & \(8.28\times 10^{-4}\) & \(9.32\times 10^{-4}\) \\ \(f_{4}\) & \(7.79\times 10^{-8}\) & \(2.45\times 10^{-5}\) & \(7.11\times 10^{-8}\) & \(1.37\times 10^{-5}\) & \(1.33\times 10^{-5}\) & \(1.36\times 10^{-5}\) \\ \(f_{5}\) & \(1.83\times 10^{-7}\) & \(3.24\times 10^{-5}\) & \(2.80\times 10^{-8}\) & \(0.858863\) & \(9.74\times 10^{-4}\) & \(2.31\times 10^{-11}\) \\ \(f_{6}\) & \(7.60\times 10^{-5}\) & \(0.001572\) & \(1.94\times 10^{-7}\) & \(1.33\) & \(0.15732\) & \(0.0329\) \\ \(f_{7}\) & \(2.80\times 10^{-8}\) & \(3.57\times 10^{-5}\) & \(1.73\times 10^{-7}\) & \(0.010723\) & \(0.006544\) & \(2.11\times 10^{-11}\) \\ \(f_{8}\) & \(6.35\times 10^{-8}\) & \(8.27\times 10^{-6}\) & \(3.36\times 10^{-7}\) & \(0.005363\) & \(3.27\times 10^{-6}\) & \(1.42\times 10^{-8}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of optimal values and variance.