## 5 Discussion  
In the experiments, we test four scheduling strategies (LPs 1-4) and identify the parameter variation ranges within which feasible schedules can be found as shown in Figures 7-9 and Table 1. The experimental results demonstrate that each strategy has its strengths under specific parameter conditions. Based on the parameter variation ranges, we can recommend the best scheduling strategy for different scenarios.  
OBS Strategy: As shown in Figures 7 and 9, when the processing time for Steps 1 and 3 is short, OBS can find feasible schedules. We can see in Figure 9 that when \(\alpha_{3}\) is in the range of [110; 140], OBS still achieves the shortest cycle time and the largest schedulable range. In Figure 2, we can see that when all four scheduling strategies are feasible, with \(\alpha_{1}=80\) and \(\psi_{1}=97\), we have that \(\psi_{1}\) is 2.02% shorter than the second smallest \(\psi_{2}\) (99). Within the parameter variation range, OBS covers the widest scheduling range, indicating that OBS is the best choice in cases where task complexity is low and processing time is short. Additionally, as processing time increases, OBS still achieves a relatively short cycle time. Overall, OBS is very effective in scenarios with a shorter wafer-processing time, especially when resource constraints are minimal.  
OHTS Strategy: OHTS performs well in certain cases, particularly as task complexity increases. As seen in Figures 7 and 9, as processing time increases, OHTS can also find feasible schedules. When the processing time for Step 2 is short, OHTS achieves feasible schedules with the shortest cycle time (Figure 9). Therefore, OHTS is the preferred strategy when the task-processing time is longer and higher resource utilization is required.  
TBS and THTS Strategies: As shown in Figure 10 and Table 1, the feasible scheduling ranges for TBS and THTS are relatively smaller, and the cycle time achieved is longer. As a result, OBS and OHTS are generally better in most cases. However, when the internal chamber rotation time of the PM increases, TBS achieves a shorter cycle time. Therefore, TBS is better in scenarios where the internal rotation time of the PM is long, while OHTS performs better when the rotation time is shorter.  
From the above analysis, it is not difficult to see that the four scheduling strategies proposed in this paper also have certain limitations. When the RTC is too short, all four scheduling strategies become very difficult to find a feasible schedule due to the tool property of multi-space PMs. Similarly, when considering the RTC, if the robot's loading/unloading or movement time is too long, none of the four scheduling methods can be applied. The experimental results can provide a guide for the practitioners. Especially, when a feasible schedule cannot be found by applying all the strategies, an engineer can think about how to redesign the process such that the fabrication of a type of wafers can be completed.  
It also needs to be pointed out that this study is conducted based on the specific application of wafer fabrication for the addressed cluster tools. This may be the limitation of the proposed method. Nevertheless, the idea may be useful for scheduling other cluster tools and for other similar scheduling problems.