## 4 Numerical Experiments
### Extended Experiments  
The FJSP can be categorized into total FJSP and partial FJSP according to the machine compatibility. Cases 5 and 6 represent the partial FJSP, where each operation can be processed only on some machines [18], whereas case 7 represents the total FJSP, where each operation can be processed on all the machines. For extended experiments, we used a simple BOM structure, as in Figure 6a, and increased the problem size including the numbers of final items, machines, and workers. In case 5, the data used for the experiments are same as those used in the previous section except for order data. In the cases of GA and GA-VNS, 10 iterations were performed with 20 populations due to the limitations in the computational time, unlike in previous studies that performed more than 20 iterations with over 100 populations. The other parameters are similar to those in the study conducted by [PERSON] et al. [6] (ratio of crossover: 0.5; ratio of mutation: 0.5; ratio of elite: 0.2; ratio of VNS: 0.5). Table 5 presents the results of a numerical experiment in case 5. In this case, GA exhibits the best performance, as presented in Table 5.  
Case 6 represents a larger problem in comparison with case 5, with 20 final items, 10 workers, and 10 machines. The environment contains more flexible processes, and the machines can process an average of 50% of the items. Due to the practical constraints on the computation time, the experiment was conducted by reducing the scale to 10 populations and 10 iterations. Additionally, the proposed algorithm was terminated after 7200 s. Table 6 summarizes the results of the numerical experiment in case 6. As the problem becomes larger, the strategy limiting the search space becomes more efficient, which is different from the results of case 5. The LSS algorithm with LSFS and the LSS with LSIS are better than the GA-based algorithms in terms of all performance measures. The results demonstrate a good performance with a significant difference in terms of lateness in a shorter computation time. For example, at low capacities, the total delayed time units was 9 for the LSFS and LSIS with a computation time of 7200 s. However, the total delayed time units was 70 and 52 for the GA and GA-VNS, respectively, with a computation time of over 9000 s. At high capacities, the lateness is reduced in all algorithms; however, the LSS-based algorithms perform better than the GA-based algorithms. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{4}{c}{**Low Capacity**} & \multicolumn{4}{c}{**High Capacity**} \\ \cline{2-9}  & **G1** & **G2** & **G3** & **Time (s)** & **G1** & **G2** & **G3** & **Time (s)** \\ \hline GA & 31 & 70,146 & 13,346 & 9102 & 29 & 70,421 & 9432 & 9121 \\ GA-VNS & 29 & 67,457 & 19,511 & 25,124 & 30 & 70,150 & 21,991 & 24,444 \\ LSFS (w/o init) & 35 & 69,979 & 18,473 & 7200 & 35 & 76,612 & 19,561 & 7200 \\ LSIS (w/o init) & 35 & 70,219 & 18,818 & 7200 & 35 & 76,612 & 19,111 & 7200 \\ LSFS & 30 & 40,542 & 24,581 & 7200 & 29 & 44,108 & 27,505 & 7200 \\ LSIS & 30 & 40,542 & 24,641 & 7200 & 29 & 44,108 & 27,105 & 7200 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Result of Case 4 (low capacity: 2000; high capacity: 4500). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{4}{c}{**Low Capacity**} & \multicolumn{4}{c}{**High Capacity**} \\ \cline{2-9}  & **G1** & **G2** & **G3** & **Time (s)** & **G1** & **G2** & **G3** & **Time (s)** \\ \hline GA & 4 & 7696.6 & 3279.2 & 2662.8 & 0 & 8215 & 3782.5 & 2953.5 \\ GA-VNS & 8 & 9200.8 & 3421.6 & 9475.7 & 0 & 12,121.7 & 3062.5 & 21,155.3 \\ LSFS & 6 & 8683.3 & 3541.6 & 9357.8 & 2 & 7899.2 & 2909.2 & 16,097.2 \\ LSIS & 7 & 8785 & 3503 & 4090 & 1 & 9135 & 4092.5 & 1532.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Result of Case 5 (low capacity: 2000; high capacity: 4500). Case 7 assumes the most flexible environment, in which all the machines can operate at all items.