# FMS Scheduling under Availability Constraint with Supervisor Based on Timed Petri Nets
## 8 Numerical Examples
### System Description  
The manufacturing system consists of three machines, an initial buffer and a final buffer, which are arranged in series around a central conveyor (Figure 11). Thereby, every machine is designed to realize an operation on a glass part. Each machine function is described as follows:  
Figure 10: Optimal manufacturing scheduling.  
Figure 9: The controlled timed Petri net.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(j_{E_{\text{reg}}}\) & \(j_{Ass}\) & \(j_{Pac}\) \\ \hline
**Red glass part** & 3 & 2 & 3 \\
**Blue glass part** & 2 & 3 & 3 \\
**Green glass part** & 4 & 4 & 2 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Operating time.  
Figure 11: FMS real case environment.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & \(B_{j}\) & \(E_{j}\) \\ \hline \(j_{Eng}\) & 2 & 7 \\ \(j_{Ass}\) & 0 & 6 \\ \(j_{Pac}\) & 10 & 13 \\ \hline \hline \end{tabular}
\end{table}
Table 5: The preventive maintenance periods.  
Table 6 illustrates the sub-optimal value for decision variables and firing sequence.  
In the same way as the previous described example, the computational results (Table 3) are used to determine supervisor parameters as shown in Figure 12.  
Table 7 illustrates decision variables resulting from the Algorithm 2 resolution for a set of different jobs and machines. In the fourth and fifth column, we have considered new jobs: \(i_{b1}\) and \(i_{b2}\), having the same operating time as blue glass \(i_{b}\), \(i_{g^{1}}\) and \(i_{g^{2}}\) as green glass \(i_{g}\).  
The effectiveness of the genetic algorithm is tested by evaluating the makespan computed with that determined by the mathematical model (through the Xpress optimizer). In order to make the comparison, we have computed the makespan for two machines as a function of the job number (Figure 13).  
On the basis of the results shown in Table 7 and Figure 13, the GA yields a good average makespan compared to the optimal solutions given by the mathematic model. The actual computing time for each problem was nearly the same for the two resolutions of the frameworks.  
\begin{table}
\begin{tabular}{c} \hline \hline
**The Optimal Firing Sequence** \\ \hline \(\sigma=t_{b,0}t_{b,Eng}t_{b,Ass}t_{g,Eng}t_{b,pac}t_{g,Ass}t_{r,Eng}t_{g,Pac}t_{r,Ass}t_{r,Pac}\) \\ \hline The resulting decision variables values \\ \(H\big{(}t_{b,0}\big{)}=0\); \(H\big{(}t_{b,Eng}\big{)}=7\); \(H\big{(}t_{b,Ass}\big{)}=13\); \(H\big{(}t_{g,0}\big{)}=9\); \(H\big{(}t_{g,Eng}\big{)}=13\); \(H\big{(}t_{b,Pac}\big{)}=16\); \(H\big{(}t_{g,Ass}\big{)}=18\); \\ \(H(t_{r,0})=14\); \(H\big{(}t_{r,Eng}\big{)}=18\); \(H\big{(}t_{g,Pac}\big{)}=20\); \(H(t_{r,Ass})=21\)\(H_{max}=H(t_{r,Pac})=24\); \(A\big{(}t_{b,Eng},t_{g,0}\big{)}=1\); \\ \(A\big{(}t_{g,Eng},t_{r,0}\big{)}=1\); \(Z_{b,Ass}=1\); \(Z_{g,Eng}=1\); \(Z_{b,Pac}=1\). \\ \hline \hline \end{tabular}
\end{table}
Table 6: The optimal solution of three jobs and three machines.  
Figure 12: The resulting controlled TPN using GA.  
of our approach could also be highlighted by the fact that [21] assume that the order of jobs could differ in machines (more relaxed).