# Dynamic Job and Conveyor-Based Transport Joint Scheduling in Flexible Manufacturing Systems
## 4 Exploring the Solution Space
### Detailed Solution
In any case, the tuning process can be performed in a simulated environment where data collected from the production line is replayed, and different values of \(T_{\text{max}}\) and \(R\) are tested. If the production load does not vary much from one day to the next, this offline analysis should provide an indication of the best parameter settings for that specific production line. **ICE instances**: In order to test Sched-T in a real test case, we generated six instances that can be executed within our research laboratory (described in Section 2). The number of tasks and jobs for each instance are summarized in Table 2. Without considering the transport time, we were able to solve the problem using a MILP model and compare the results with Sched-T (Table 2, column **Opt**). By using Sched-T, we are able to find a solution in less than 30 s. The difference between the local optimal and the global optimal solution is less than 5% (columns **Sched-T** and **Err. ** of Table 2). As mentioned earlier, the literature deems an error gap of up to 10% acceptable, acknowledging the trade-off between accuracy and computational speed provided by the approximate solutions. Regarding the variability in error, we were unable to pinpoint any specific job characteristic that might influence such variability. It can be regarded as an inherent variability intrinsic to the considered problem. ### Instances with the Transport System

Introducing a transport system increases the complexity of the MILP model exponentially. In fact, a MILP formulation given to standard MILP solvers, such as IBM CPLEX, cannot be solved after hours of computation. As such, we do not have any values that can be used as a global optimal comparison reference to our results. The results presented in Table 3, column **Sched-T** of **Experim.-I**, report the makespan found by Sched-T. Next, we describe the test we performed on a real-world testbed to validate such results. ### Real-World Experiments

We consider the set of instances described in Table 2, and we run them on an actual production line built in our lab (described in Section 2). The production line is governed by \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Id** & **\#Jobs** & **\# Tasks** & **Opt** & **Sched-T** & **Err.** \\ \hline
1 & 5 & 32 & 900 & 900 & 0\% \\
2 & 10 & 106 & 3580 & 3650 & 2.0\% \\
3 & 15 & 100 & 2930 & 3014 & 2.9\% \\
4 & 20 & 142 & 3530 & 3698 & 4.8\% \\
5 & 25 & 174 & 4935 & 4998 & 1.3\% \\
6 & 30 & 263 & 7475 & 7570 & 1.3\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: ICE instances: characteristics (cols. 2 and 3); makespan with no transport (cols. 4, 5, and 6). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Id**} & \multicolumn{3}{c}{**Experim.-I**} & \multicolumn{3}{c}{**Experim.-II**} \\ \cline{2-7}  & **Sched-T** & **Actual** & **Err.** & **Sched-T** & **Actual** & **Err.** \\ \hline
1 & 1511 s & 1556 s & \(-\)2.89\% & 1221 s & 1555 s & \(-\)21.48\% \\
2 & 2814 s & 2861 s & \(-\)1.64\% & 2243 s & 2875 s & \(-\)21.98\% \\
3 & 3557 s & 3674 s & \(-\)3.18\% & 3321 s & 4248 s & \(-\)21.82\% \\
4 & 4654 s & 4892 s & \(-\)4.87\% & 4366 s & 5577 s & \(-\)21.71\% \\
5 & 5820 s & 5963 s & \(-\)2.40\% & 5473 s & 7012 s & \(-\)21.95\% \\
6 & 7025 s & 7227 s & \(-\)2.80\% & 6461 s & 8377 s & \(-\)22.87\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Reference layout scheduling instances with transport. service-oriented manufacturing (SOM) software architecture similar to the one presented in [33; 34], which automatically manages the production line, interacting with both the manufacturing execution system (MES) and the machines. On top, we have developed a module that implements our proposed scheduling heuristic, and the execution is forwarded to the SOM architecture. For each instance's job, we have fixed the task execution time. This allows for a fair evaluation of the accuracy of Sched-T, removing processing time variability. The case of a stochastic execution time will be considered in future works.