# Opposition-Based Adaptive Fireworks Algorithm
## 4 Simulation Studies and Discussions
### Comparison with Other Swarm Intelligence Algorithms  
Additionally, OAFWA was compared with alternative swarm intelligent algorithms, including BA, DE, self-adapting control parameters in differential evolution (jDE) [29], FA, and SPSO2011 [30]. The resulting evaluation times (iteration) and population size are the same for each function. Two cases of population size are tested, 20 and 40, respectively. The parameters are shown in Table 8.  
How to tune the parameters of the above algorithms is a challenging issue. These parameters are obtained from the literature [31] for BA, [29] for DE and jDE, and [30] for SPSO2011. For FA, reducing randomness increases the convergence. \(\alpha^{t+1}=(1-\delta)\,a^{t}\)[32] is employed to gradually decrease \(\alpha\). \(\delta\) is a small constant related to iteration. For OAFWA, \(m=2\), A = 100, M\({}_{\text{e}}\) = 20, \(S_{\text{min}}\) = 3, \(S_{\text{max}}\) = 6, \(NG=1\) (population size is equivalent to 20), and \(J_{r}=0.4\). Tables 9 and 10 present the mean errors for the six algorithms when population sizes are 20 and 40, respectively. Tables 9 and 10 indicate that certain algorithms perform well for some functions, but less well for others. Overall, OAFWA performance was shown to have more stability than the other algorithms. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Func.** & **BA** & **DE** & **jDE** & **FA** & **SPSO2011** & **OAFWA** & **Iteration** \\ \hline F1 & \(3.22\times 10^{-3}\) & \(7.77\times 10^{-10}\) & \(3.27\times 10^{-17}\) & \(2.02\times 10^{-5}\) & \(4.16\times 10^{-31}\) & 2.46 \(\times 10^{-34}\) & 65,000 \\ F2 & \(7.15\) & \(1.28\times 10^{-15}\) & \(1.60\times 10^{-15}\) & \(1.51\times 10^{-1}\) & 3.06 & 8.21 \(\times 10^{-17}\) & 100,000 \\ F3 & \(2.03\times 10^{-2}\) & \(3.83\times 10^{-5}\) & \(7.91\times 10^{-2}\) & 40.2 & 3.25 \(\times 10^{-8}\) & 3.56 \(\times 10^{-38}\) & 190,000 \\ F4 & \(1.23\times 10^{-1}\) & \(5.12\times 10^{-2}\) & \(1.38\times 10^{-37}\) & \(1.29\times 10^{-1}\) & 5.77 \(\times 10^{-6}\) & 3.28 \(\times 10^{-38}\) & 140,000 \\ F5 & \(25.1\) & \(23.9\) & \(7.27\) & \(9.04\times 10^{-2}\) & 5.99 & 7.39 \(\times 10^{-7}\) & 110,000 \\ F6 & \(8.05\times 10^{4}\) & \(2.21\times 10^{-6}\) & \(2.07\times 10^{-28}\) & \(2.26\times 10^{-3}\) & \(3.56\times 10^{3}\) & 7.17E \(\times 10^{-46}\) & 110,000 \\ F7 & \(19.2\) & \(26.9\) & \(8.08\) & \(18.4\) & 49.1 & \(1.03\times 10^{-12}\) & 80,000 \\ F8 & \(5.25\times 10^{-1}\) & \(8.39\times 10^{-3}\) & \(2.46\times 10^{-4}\) & \(1.52\times 10^{-3}\) & \(5.22\times 10^{-3}\) & \(<1\times 10^{-35}\) & 120,000 \\ F9 & \(5.53\) & \(5.39\times 10^{-1}\) & \(7.99\times 10^{-15}\) & \(3.16\times 10^{-2}\) & 1.54 & 9.59 \(\times 10^{-16}\) & 150,000 \\ F10 & \(4.62\times 10^{-1}\) & \(2.85\times 10^{-6}\) & \(1.20\times 10^{-30}\) & \(8.71\times 10^{-2}\) & 1.41 \(\times 10^{-3}\) & 2.00 \(\times 10^{-40}\) & 200,000 \\ F11 & \(3.78\) & \(2.56\times 10^{-15}\) & \(1.01\times 10^{-3}\) & \(9.77\times 10^{-1}\) & \(2.17\times 10^{-1}\) & 5.27 \(\times 10^{-17}\) & 100,000 \\ F12 & \(11.1\) & \(1.59\times 10^{-1}\) & \(9.99\times 10^{-2}\) & \(2.28\times 10^{-1}\) & \(1.11\times 10^{-1}\) & \(1.32\times 10^{-33}\) & 140,000 \\ \hline \end{tabular}
\end{table} 
Table 10: Mean errors for the six algorithms when popSize = 40. \begin{table}
\begin{tabular}{c c} \hline
**Algorithms** & **Parameters** \\ \hline BA & \(A=0.95\), \(r=0.8\),\(f_{\min}=0\), \(f=1.0\) \\ DE & \(F=0.5\), \(CR=0.9\) \\ jDE & \(\tau_{1}=\tau_{2}=0.1\), \(F\in[0.1,1.0]\), \(CR\in[0,1]\) \\ FA & \(\alpha=0.9\), \(\beta=0.25\), \(\gamma=1.0\) \\ SPSO2011 & \(w=0.7213\), \(c1=1.1931\), \(c2=1.1931\) \\ \hline \end{tabular}
\end{table} 
Table 8: Parameters of the algorithms. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Func.