## 5 Numerical Results
### Case Study  
A specific PCB assembly shop composed of four production stages (SMT chip processing stage, plug processing stage, welding processing stage, and test stage) are also investigated. At each production stage, several production lines are alternative for PCB assembly (S1, S2 and S3 at the 1 st stage; M1 and M2 at the 2 nd stage; A1 and A2 at the 3 rd stage; and T1 and T2 at the 4 th stage). Ten types of PCBs are assembled in these lines. The requirement for each PCB type is 1000. The processing times and setup times are listed in Tables 7 and 8, respectively.  
Figure 9 illustrates the Gantt chart of static scheduling results obtained by using the GRN-based method. As shown in this figure, all PCB products are divided into 100 batches, and the makespan is 3308 s. Because the starting time of operations is directly determined by the end time of the former operation on the same machine or that of the same job, the processing time variations have cumulative impacts on the completion time of jobs. Based on static scheduling results, the IACO method is used to deal with processing time variations. Figure 10 is the Gantt chart obtained by the IACO method, in which the makespan is 3415.77 s. The GRN-based method is also used to solve the dynamic scheduling problem and achieves a makespan of 3261.98 s, as shown in Figure 11. By integrating job sequencing rules and job assigning rules in a reasonable manner, the GRN-based method ensures waiting jobs to be in-time assigned to an idle machine with comparatively shorter setup time. The regulatory optimization procedure realizes the tradeoff between shorter job waiting time and less changeover activities to minimize the makespan. However, the IACO method fails to realize these targets for some rolling windows because its global search procedure can hardly search out optimal solutions for a real-time scheduling. Taking the first production stage for example, the Garnet chart in Figure 10 realizes higher utilization of Machine 1 and shorter job waiting time on Machine 3 than that in Figure 11. Consequently, the GRN-based method is a more effective scheduling method than the IACO method.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**PCB Type** & **S1** & **S2** & **S3** & **M1** & **M2** & **A1** & **A2** & **T1** & **T2** \\ \hline
**3 ET0321 AF** & 4.615 & 1 & 1.2 & 0.3 & 0.24 & 0.6 & 0.75 & 0.6 & 0.6 \\
**3 ET0322 AF** & 4.615 & 1 & 0.293 & 0.4 & 0.3 & 0.6 & 0.75 & 0.6 & 0.55 \\
**3 ET0100 CET** & 1 & 1 & 0.6 & 0.06 & 1.5 & 0.75 & 0.75 & 0.4 & 0.55 \\
**3 ET0141 CET** & 1 & 1.428 & 1 & 0.4 & 0.24 & 0.6 & 0.3 & 0.55 & 0.5 \\
**3 ET0349 CET** & 1 & 1 & 0.6 & 0.4 & 0.3 & 0.75 & 0.75 & 0.6 & 0.6 \\
**3 ET0630 CET** & 1 & 1 & 1 & 0.06 & 0.24 & 0.75 & 0.6 & 0.55 & 0.5 \\
**3 ET0631 CET** & 1 & 1 & 0.6 & 0.06 & 0.3 & 0.6 & 0.75 & 0.55 & 0.4 \\
**3 ET0741 CET** & 1 & 4.615 & 0.293 & 0.6 & 0.3 & 0.75 & 0.6 & 0.4 & 0.4 \\
**3 ET0374 TEK** & 1.428 & 0.923 & 1.2 & 0.4 & 0.3 & 0.6 & 0.3 & 0.6 & 0.55 \\
**3 ET0435 TEK** & 0.923 & 0.923 & 0.882 & 0.6 & 1.5 & 0.6 & 0.4 & 0.6 & 0.4 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Processing times (s) of different PCB (Printed Circuit Board) types in each production line.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**PCB Type** & **3 ET0** & **3 ET0** & **3 ET0** & **3 ET0** & **3 ET0** & **3 ET0** & **3 ET0** & **3 ET0** & **3 ET0** \\
**321 AF** & **322 AF** & **100 CET** & **141 CET** & **349 CET** & **630 CET** & **631 CET** & **741 CET** & **374 TEK** & **43 STEK** \\  
Figure 11: GRN-based dynamic scheduling Gantt chart.  
Figure 10: IACO (Improved Ant Colony Algorithm) dynamic scheduling Gantt chart.  
Figure 9: Static scheduling Gantt chart.