## 4 Experimental Study
### Robustness of Interval Schedules under Different Confidence Levels  
In the proposed method, the confidence level of uncertain parameters is an important tool to control the robustness of the schedule. Based on Equation (15), different confidence levels correspond to different bounds of intervals, which make the schedule flexible to fluctuations of production parameters. Taking Case 1 as an example, the product structure and the original deterministic processing time \(PT_{i}\) are detailed in [PERSON] et al. [6]. In this study, the uncertain processing times \(\widehat{PT_{i}}\) were assumed to follow Gaussian distribution with the mean of \(PT_{i}\) and the standard deviation \(\sigma=0.2*p_{0}\). Here, \(p_{0}\) represents the standard processing time of the operation. We obtained the optimized solution (see Figure 6) of Case 1 in the deterministic scenario and in the uncertain scenario with a 80% confidence level. As can be seen from Figure 6b, each operation in the interval schedule was represented by two intervals. Taking operation 4 as an example, the lower line O4_S indicated the interval start time of operation 4, and the upper line O4_E indicated the interval completion time, which was obtained by adding the interval processing time to the interval start time.  
Figure 5: Means plot at the 95% confidence level LSD intervals for RPD metric.  
Figure 6: Gantt chart of the optimal solution for Case 1. (**a**) Optimal solution on deterministic scenario; (**b**) Optimal solution on uncertain scenario with 80% confidence level.  
Compared with the schedule for deterministic scenario, although the makespan of interval schedule increased, it is obvious that each operation reserves appropriate time redundancy based on its historical data, which is more suitable for the actual production process. From another perspective, the result also proves that the uncertainty of the processing time may cause the maximum 16.5% fluctuation in the makespan of Case 1, which can provide a reference for the delivery date of the product. In addition, the proposed method can help managers flexibly adjust the conservativeness and robustness of the schedules by setting different confidence levels. We tested the performance of interval schedules under different confidence levels and reported the results of Cases 1-5 in Figure 7.  
In Figure 7, when the confidence level was equal to 0, all parameters became deterministic values, and the interval scheduling at this time was equivalent to deterministic scheduling. With the continuous improvement of the confidence level, the interval of uncertain parameters covers more uncertain scenarios, and the value of the objective also increases. Taking Case 1 as an example, the objective value of confidence level 0 was 178, and the value of confidence level 1 was 211. This meant that the uncertainty of the processing time and setup time may cause a performance deviation of 18.5% in the makespan. In addition, for Cases 3-10, the average fluctuation of the objective value caused by different confidence levels of the processing time and setup time is 15-20%, which is basically consistent with the actual situation of the workshop.