# Opposition-Based Adaptive Fireworks Algorithm
## 4 Simulation Studies and Discussions
### Comparison with FWA-Based Algorithms
#### 4.1.2 Comparison with FWA-Based Algorithms  
To compare performances of OAFWA, AFWA, and EFWA, EFWA was tested for twelve benchmark functions. Success rates and mean error ranks for three FWA based algorithms were obtained. Table 6 shows the statistical results for EFWA. The results from Tables 6 and 7 indicate that the mean error of AFWA is larger than that of EFWA. The Sr of AFWA is lower that of EFWA. Thus, AFWA is not better than EFWA; The Sr of OAFWA is the highest and OAFWA ranks the highest among three FWA algorithms. Thus, OAFWA greatly improved the performance of AFWA. For the runtimes of EFWA, AFWA and OAFWA, Tables 2 and 3 indicate that the time cost of OAFWA is not much different from that of AFWA. But Tables 2 and 6 indicate the time cost of OAFWA drops significantly as compared with that of EFWA. Table 7 compares three FWA-based algorithms. Figure 2 shows searching curves for EFWA, AFWA, and OAFWA. Figure 2 shows that OAFWA results in a global optimum for all twelve functions and with fast convergence. However, the alternative EFWA methods do not always locate the global optimum solutions. These include EFWA for F9 and F11. AFWA is the worst among the three algorithms. It is bad for a majority of functions. Thus, OAFWA is the best one in terms of solution accuracy. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Func.** & **Best** & **Mean** & **Median** & **Worst** & **Std. dev.** & **Time (s)** \\ \hline F1 & \(1.66\times 10^{-3}\) & \(2.63\times 10^{-3}\) & \(2.63\times 10^{-3}\) & \(3.44\times 10^{-3}\) & \(3.83\times 10^{-4}\) & \(4.34\) \\ F2 & \(2.07\times 10^{-1}\) & \(5.67\times 10^{-1}\) & \(2.75\times 10^{-1}\) & \(2.84\) & \(5.89\times 10^{-1}\) & \(8.88\) \\ F3 & \(7.52\times 10^{-3}\) & \(1.44\times 10^{-2}\) & \(1.40\times 10^{-2}\) & \(2.77\times 10^{-2}\) & \(3.67\times 10^{-3}\) & \(42.54\) \\ F4 & \(3.31\times 10^{-1}\) & \(5.22\times 10^{-1}\) & \(4.92\times 10^{-1}\) & \(8.91\times 10^{-1}\) & \(1.23\times 10^{-1}\) & \(13.55\) \\ F5 & \(1.62\times 10^{-1}\) & \(7.01\times 10^{-1}\) & \(2.18\times 10^{-1}\) & \(9.26\) & \(1.47\) & \(9.22\) \\ F6 & \(2.01\times 10^{-1}\) & \(3.28\times 10^{-1}\) & \(3.12\times 10^{-1}\) & \(4.75\times 10^{-1}\) & \(6.67\times 10^{-2}\) & \(9.11\) \\ F7 & \(3.22\times 10^{-3}\) & \(5.04\times 10^{-3}\) & \(5.07\times 10^{-3}\) & \(6.87\times 10^{-3}\) & \(6.73\times 10^{-4}\) & \(6.46\) \\ F8 & \(6.01\times 10^{-3}\) & \(1.63\times 10^{-2}\) & \(1.48\times 10^{-2}\) & \(4.91\times 10^{-2}\) & \(8.80\times 10^{-3}\) & \(12.90\) \\ F9 & \(2.39\) & \(5.05\) & \(5.08\) & \(8.51\) & \(1.20\) & \(16.82\) \\ F10 & \(1.68\times 10^{-1}\) & \(2.69\times 10^{-1}\) & \(2.64\times 10^{-1}\) & \(4.57\times 10^{-1}\) & \(5.86\times 10^{-2}\) & \(37.31\) \\ F11 & \(9.73\times 10^{-1}\) & \(4.14\) & \(3.80\) & \(13.6\) & \(2.16\) & \(8.98\) \\ F12 & \(2.00\times 10^{-1}\) & \(3.15\times 10^{-1}\) & \(3.00\times 10^{-1}\) & \(4.00\times 10^{-1}\) & \(5.39\times 10^{-2}\) & \(13.46\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Statistical results for EFWA. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **F1** & **F2** & **F3** & **F4** & **F5** & **F6** \\ \hline H & 1 & 1 & 1 & 1 & 1 & 1 \\ P & \(4.14\times 10^{-108}\) & \(3.33\times 10^{-165}\) & \(<\)1 \(\times\) 10\({}^{-315}\) & \(2.01\times 10^{-230}\) & \(1.63\times 10^{-181}\) & \(1.63\times 10^{-181}\) \\ \hline  & **F7** & **F8** & **F9** & **F10** & **F11** & **F12** \\ \hline H & 1 & 1 & 1 & 1 & 1 & 1 \\ P & \(1.39\times 10^{-132}\) & \(8.13\times 10^{-198}\) & \(1.01\times 10^{-246}\) & \(<\)1 10\({}^{-315}\) & \(3.33\times 10^{-165}\) & \(2.01\times 10^{-230}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Wilcoxon signed-rank test results for OAFWA and AFWA.