## 5 Numerical Results  
### Strategy Parameter Analysis  
As discussed in Section 3, delivery deviation tolerance and operation-based window size are important parameters in the event-driven rescheduling strategy. To determine appropriate parameter values, the numeric tests listed in Table 2 are presented.  
For these numeric tests, a static scheduling result is first obtained by taking the whole planning horizon as a special rescheduling window and using the GRN-based method. A real-coded genetic algorithm is specifically used in this method to optimize regulatory parameters, in which the population size is 200, the maximum generation is 50, the crossover possibility is 0.8, and the mutation possibility is 0.1. Figure 7 illustrates the Gantt chart of static scheduling results (makespan = 392 s). In this diagram, white rectangles represent processing times and black ones denote changeover times. Moreover, the numbers within each rectangle represents the batch of a job type. For instance, \"1, 3\" represents the third batch of the first job type.  
\begin{table}
\begin{tabular}{l c} \hline \hline
**Problem Parameter** & **Numerical Range** \\ \hline Number of production stages (\(I\)) & 4 \\ Number of parallel machines at each stage (\(K_{i}\)) & U[2; 4] \\ Number of job types (\(M\)) & 4 \\ Production volume of each job type (\(d_{m}\)) & 8 \\ Processing time (\(t_{mik}\)) (s) & U[20; 30] \\ Changeover time between same job types (\(c_{mnik}\)) (s) & U[1; 3] \\ Changeover time between different job types (\(c_{mnrk}\)) (s) & U[5; 7.5] \\ \hline \hline \end{tabular}
\end{table}
Table 2: Strategy parameter analysis experiment.  
Figure 7: Gantt chart of a static scheduling.  
Assuming that the processing times follow a 4-order Erlang distribution, the rescheduling strategies with different deviation tolerances and window sizes are then used. Table 3 lists the makespan, rescheduling times and computational time of strategies with different deviation tolerances while the window size is 20 operations. In this table, the strategy with the delivery time deviation tolerance of 0.125 minimizes the makespan. Table 4 thus lists dynamic scheduling results with different window sizes while the delivery time deviation tolerance is 0.125. Based on these results, the delivery deviation tolerance of 0.125 and the window size of 77 operations are adopted to generate dynamic scheduling results, as shown in Figure 8.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Rolling Window Size** & **Makespan** & **Rescheduling** & **Computational Time** \\ **(Operation)** & **(s)** & **Times** & **(ms)** \\ \hline
13 & 394.54 & 2 & 3216 \\
26 & 397.55 & 3 & 4997 \\
39 & 400.78 & 4 & 7914 \\
51 & 394.56 & 4 & 9305 \\
64 & 398.19 & 3 & 7219 \\
77 & 386.78 & 4 & 13,006 \\
90 & 390.33 & 2 & 6363 \\
102 & 393.33 & 2 & 7519 \\
115 & 389.03 & 4 & 18,545 \\
128 & 390.54 & 2 & 10,400 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Results of rolling window size experiments.  
Figure 8: Gantt chart of a dynamic scheduling.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Delivery Time** & **Makespan** & **Rescheduling** & **Computational Time** \\
**Deviation Tolerance** & **(s)** & **Times** & **(ms)** \\ \hline
0.025 & 408.08 & 39 & 65,708 \\
0.05 & 400.55 & 15 & 23,842 \\
0.075 & 401.03 & 7 & 11,346 \\
0.1 & 407.20 & 5 & 7245 \\
0.125 & 394.34 & 2 & 3243 \\
0.15 & 394.93 & 2 & 3193 \\
0.2 & 395.22 & 1 & 1602 \\
0.25 & 395.22 & 1 & 1602 \\
0.3 & 439.46 & 0 & 16 \\
0.35 & 439.46 & 0 & 19 \\
0.4 & 439.46 & 0 & 18 \\
0.45 & 439.46 & 0 & 21 \\
0.5 & 439.46 & 0 & 19 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Results of delivery time deviation tolerance experiments.