## 4 Numerical Experiments
### Extended Experiments
The number of final items, workers, and machines are 30, 10, and 10, respectively. Experiments are conducted under the same conditions as those of case 6. Table 7 presents the results of a numerical experiment in case 7. The result of case 7 is similar to that of case 6. Both LSS-based algorithms outperform the GA-based algorithms in a shorter time. LSIS is more efficient than LSFS even in an environment with low capacity, which explores more search space in a shorter time as the problem becomes more complicated. Although the search space is limited, LSFS determines all the possible candidate solutions at each iteration. Therefore, it takes a longer time and does not perform well for more complex problems. Figures 7 and 8 illustrate the relative performance in comparison with the best solution, as the problem size increases in each capacity condition. The relative performance is calculated using Equation (4). The objective function values are normalized, and the best and worst solutions become 1 and 0, respectively. In terms of lateness, the objective function with the highest priority, the GA algorithm generates the best solution when the problem size is small, and the performance of LSIS is not the worst. However, as the problem size increases, LSIS generates the best solution or one that is very close to the best solution, and the performance of the GA algorithm becomes the worst. The performance of GA-VNS is similar to that of GA for cases 6 and 7. With respect to the makespan, the trend of the result is similar to that of the lateness, when the capacity is low. At high capacities, the LSS-based algorithms generate the best solutions or those that are very close to the best solution in all the cases. As for the third objective function, none of the algorithms outperform the others because this study considers a preemptive multiobjective approach, wherein the performance depends on the first and second objective functions. \[Relative\;performance\;=\;\frac{Current\;value\;-\;Minimum\;value}{Maximum\;value\;-\;Minimum\;value} \tag{4}\] \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 & 70 & 16,235.8 & 6049.5 & 9068.7 & 6 & 18,694 & 5599.4 & 6978.4 \\ GA-VNS & 52 & 14,304.8 & 5231.6 & 19,612 & 6 & 16,531.9 & 5824.5 & 17,293 \\ LSFS & 9 & 9110 & 3258 & 7200 & 0 & 11,850.6 & 6863.6 & 7200 \\ LSIS & 9 & 9179 & 3258 & 7200 & 0 & 10,795.7 & 3826.5 & 7200 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Result of Case 6 (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 & 82 & 30,953.5 & 5686.7 & 15,064 & 13 & 30,808.7 & 5096.6 & 14,714.3 \\ GA-VNS & 79 & 28,964.9 & 4801.3 & 38,014 & 12 & 28,862.3 & 5899.6 & 38,677.4 \\ LSFS & 31 & 19,396 & 7247.4 & 7200 & 0 & 27,981.7 & 7639.4 & 7200 \\ LSIS & 29 & 19,396 & 6849 & 7200 & 0 & 27,981.0 & 7223.0 & 7200 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Result of Case 7 (low capacity: 4500, high capacity: 9000). Figure 7: Comparison of optimal performance at low capacity.