# Hybrid Flow Shop Scheduling Problems Using Improved Fireworks Algorithm for Permutation
## 3 Fireworks Algorithm
### The Improved Fireworks Algorithm
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Job**} & \multicolumn{8}{c}{**Stages**} \\ \cline{2-10}  & \multicolumn{3}{c}{**Lathe (s)**} & \multicolumn{3}{c}{**Planer (s)**} & \multicolumn{3}{c}{**Grinder (s)**} \\ \cline{2-10}  & **1** & **2** & **3** & **1** & **2** & **1** & **2** & **3** \\ \hline
1 & 24 & 75 & 51 & 84 & 62 & 39 & 54 & 11 \\
2 & 3 & 75 & 17 & 86 & 73 & 99 & 31 & 70 \\
3 & 66 & 17 & 71 & 52 & 23 & 33 & 16 & 18 \\
4 & 16 & 12 & 91 & 48 & 2 & 14 & 15 & 80 \\
5 & 80 & 17 & 22 & 89 & 14 & 38 & 14 & 51 \\
6 & 41 & 63 & 87 & 7 & 77 & 56 & 71 & 55 \\
7 & 33 & 84 & 21 & 51 & 97 & 63 & 46 & 21 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The processing times in hybrid flow shop scheduling. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Parameters** & **Value** \\ \hline  & Number of fireworks \(N\) & 5 \\  & Constant \(a\) & 0.04 \\  & Constant \(b\) & 0.8 \\ IFWA and FWA & \(A\) & 7 \\  & \(M\) & 50 \\  & \(A_{initial}\) & (ub-lb) \(\times\) 0.02 \\  & \(A_{final}\) & (ub-lb) \(\times\) 0.001 \\ \hline  & \(C_{1}\), \(C_{2}\) & 1.49445 \\ PSO & Population size & 40 \\  & Range of particle velocity & [–2,2] \\ \hline  & Population size & 40 \\  & WOA & \(b\) & 1 \\  & \(p\) & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter settings in PFSP. IFWA: improved fireworks algorithm; PSO: particle swarm optimization; WOA: whale optimization algorithm. lb: lower bound; ub: upper bound. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Algorithm** & \begin{tabular}{c} **Theoretical Optimal** \\ **Value (s)** \\ \end{tabular} & \begin{tabular}{c} **Success** \\ **Rate** \\ \end{tabular} & \begin{tabular}{c} **Optimal** \\ **Value (s)** \\ \end{tabular} & \begin{tabular}{c} **Average** \\ **(s)** \\ \end{tabular} & \begin{tabular}{c} **Wost (s)** \\ \end{tabular} & \
\begin{tabular}{c} **STD** \\ \end{tabular} &
\begin{tabular}{c} **Time (s)** \\ \end{tabular} \\ \hline IFWA & & 90\% & 6590 & 6595.3 & 6643 & 16.31 & 2.14 \\ FWA & & 75\% & 6590 & 6603.2 & 6643 & 23.54 & 3.40 \\ PSO & & 30\% & 6590 & 6665 & 7016 & 119.5 & 0.34 \\ WOA & & 55\% & 6590 & 6614 & 6643 & 27.05 & 0.26 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results of the optimizations in PFSP. The specific statistical results are compared in Table 6. Based on the accuracy of the optimization, the IFWA could find the minimum makespan. Compared with FWA, PSO, and WOA, the maximum completion time calculated by IFWA was reduced by 3.5%, 9.3%, and 4.1%, respectively, which shows its excellent optimization performance. From the perspective of standard deviation and average value, the improved algorithm had good performance. The corresponding evaluation indicators were much smaller than the other three algorithms, which means the IFWA could better find the optimal solution in each calculation. From the perspective of computing time, FWA and IFWA took a long time to optimize. Although the new selection strategy can save half the running time in IFWA, there was still a big gap compared with PSO and WOA. In general, the IFWA significantly improved the optimization accuracy and stability compared with the other algorithms. The calculation time with IFWA was much shorter than that of the FWA. The results prove that the IFWA has an excellent ability to deal with the HFSP. ## 6 Conclusions

An improved fireworks algorithm was proposed to minimize the makespan in PFSPs and HFSPs. Three improved strategies are introduced in the new algorithm. Firstly, a non-linear decreasing radius is introduced to replace the minimum fireworks radius selectively in order to avoid the situation where the search radius of the optimal fireworks is zero. Secondly, a Cauchy and Gaussian mutation operator is introduced to replace the original Gaussian mutation operator, which enhances the search ability. Finally, an elite group selection strategy is adopted to reduce the system load. In order to verify the ability to solve the FSP, the IFWA was used to solve two instances (PFSP and HFSP).