# Hybrid Flow Shop Scheduling Problems Using Improved Fireworks Algorithm for Permutation
## 3 Fireworks Algorithm
### The Improved Fireworks Algorithm
When solving the PFSP, the IFWA had a 90% probability of obtaining the minimum completion time, which was 15% higher than the original version and far superior to the results of PSO and WOA. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Algorithm** & **Best (s)** & **Average (s)** & **Worst (s)** & **STD (s)** & **Time (s)** \\ \hline IFWA & 165 & 174 & 196 & 9.31 & 6.24 \\ FWA & 171 & 208 & 233 & 18.51 & 12.85 \\ PSO & 182 & 200 & 225 & 11.84 & 1.78 \\ WOA & 172 & 198 & 245 & 21.22 & 1.99 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparisons of the optimizations in HFSP. Figure 6: The optimal curves of the four algorithms. When solving the HFSP by IFWA, the optimal value and stability of the results were very superior to the other algorithms. Compared with FWA, PSO, and WOA, the makespan calculated by IFWA was reduced by 3.5%, 9.3%, and 4.1%, respectively, and the standard deviation was the smallest, indicating its excellent performance. Compared with FWA, the computation time of IFWA was reduced by nearly half, which undoubtedly indicates the improved efficiency of the algorithm. The results prove the feasibility and effectiveness of IFWA in dealing with the PFSP and HFSP. Compared with the original version, the IFWA significantly improved the optimization precision, stability, and speed, and could be adopted for NP-hard combination optimization problems. The IFWA is used as a reliable meta-heuristic algorithm to guide flow shop scheduling. Modern enterprise competition depends not only on advanced production technology and equipment, but also on advanced management technology. In the field of flow shops, the method proposed in this paper can provide a valuable suggestion for solving the FSP, thereby helping enterprises to find the best scheduling strategy and improve their management level. Because actual flow shop scheduling has large scale and complicated calculation, the proposed method is of great significance both in terms of academic research and practical industrial production. At the same time, this study also has limitations. Firstly, only two instances were used to validate the performance of IFWA, which is insufficient. More examples need to be introduced in future study. Secondly, compared with other classical algorithms, the calculation time of the proposed method is still longer, which limits further applications. This is determined by the design structure of FWA itself. The algorithm structure should be further studied and the structure of other excellent algorithms should also be explored. Future study will also apply FWA to other types of scheduling. Investigation, X.P. and H.X. ; Methodology, K.L. ; Writing--original draft, X.P. ; Writing--review & editing, M.-L.T. and M.K.L. All authors have read and agreed to the published version of the manuscript. This study was partially funded by the project of ministry of science and technology, grant number MOST 108-2221-E-468 -004 -MY2. The authors declare no conflict of interest. ## References

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