# Dynamic Job and Conveyor-Based Transport Joint Scheduling in Flexible Manufacturing Systems
## 4 Exploring the Solution Space
### Detailed Solution
**Running time**: Figure 5 shows the running time (in minutes) required to find the solutions of the 50 instances in [16]. The instances are ordered by increasing the number of tasks (shown at the top of the graph), e.g., instance 50 has almost 1000 tasks. Even with a large number of tasks, Sched-T is able to obtain a solution that is comparable to the results found in the literature in less than 10 min. **Sensitivity analysis**: Sched-T has a set of parameters that tune the depth of the exploration phase. At the task level, we can exchange tasks belonging to the same job or to the next or previous job by considering the current scheduling order. At the job level, we can exchange jobs. During our tests, we noted that switching jobs provides most of the gain, while at the task level, the impact of changing the order on the makespan is less significant. Therefore, the two main parameters we consider are the number of moves we explore \(T_{\max}\) and the number of neighbors we evaluate in each iteration, \(R\). These parameters influence the time required to compute the solution. Figure 6 shows two views for analyzing the impact of these parameters on the normalized makespan, i.e., the makespan found with a combination of \(T_{\max}\) and \(R\) divided by the makespan found with the highest values of \(T_{\max}\) and \(R\) used in our test. If we maintain \(T_{\max}\) as fixed and we increase \(R\) (Figure 6, left), when

Figure 4: Normalized makespan for the set of the 50 + 50 large instances in [12; 16]. The shaded area represents the gap (minimum and maximum makespan) found with different heuristics in [12], while the orange points are the results of Sched-T.

\(R\) is sufficiently large (\(R>10\)), no additional improvement is observed. However, even for smaller values of \(R\), the makespan only slightly increases (2-5%) with respect to the best makespan found. This is also confirmed if we keep \(R\) fixed and vary \(T_{\text{max}}\) (Figure 6, right); the two figures refer to two different instances, and they are representative of the general behavior that we observed for all the instances. Therefore, the two main parameters we consider are the number of moves we explore, denoted as \(T_{\text{max}}\), and the number of neighbors we evaluate in each iteration, denoted as \(R\). These parameters influence the time required to compute the solution. Figure 6 provides two views for analyzing the impact of these parameters on the normalized makespan. In other words, it shows the makespan found with a combination of \(T_{\text{max}}\) and \(R\) divided by the makespan found with the highest values of \(T_{\text{max}}\) and \(R\) used in our test. If we keep \(T_{\text{max}}\) fixed and increase \(R\) (Figure 6, left), when \(R\) is sufficiently large (\(R>10\)), no additional improvement is observed. Even for smaller values of \(R\), the makespan only slightly increases (2-5%) compared to the best makespan found. This observation holds even if we keep \(R\) fixed and vary \(T_{\text{max}}\) (Figure 6, right). The two figures represent different instances but are indicative of the general behavior observed for all instances. Overall, Sched-T is not extremely sensitive to parameter settings and provides good results across a wide range of values. In terms of computational time, using small \(T_{\text{max}}\) and \(R\) allows for obtaining a solution in less than 30 s, while larger \(T_{\text{max}}\) and \(R\) may take up to 10 min. However, since large \(T_{\text{max}}\) and \(R\) do not significantly improve the results, the use

Figure 5: Sched-T running time for finding the solution is shown in Figure 4. The top axis indicates the number of tasks associated with each instance. Figure 6: Sensitivity analysis of Sched-T: impact of the parameters on the makespan (makespan normalized to the best value found). **Left** and **right** figures refer to different instances. of a small \(T_{\text{max}}\) and \(R\) is appropriate, especially in dynamic contexts where scheduling is recomputed when new jobs arrive.