## 6 Conclusions  
In this paper, we examine the scheduling problem of a four-finger four-space-PM cluster tool (SF\({}^{3}\)-CT), a typical type of cluster tool with multi-space PM and a multi-finger-arm robot adopted in modern fabs. With the configuration properties, we find that both the robot task sequences and the rotation operations of the internal chambers of PMs affect the wafer stay time in PMs. Due to the increased time within PMs by dealing with multiple spaces in a PM, fulfilling the residency time constraints becomes more challenging. Consequently, finding an optimal and feasible schedule for such tools is a significant issue. To solve this problem, four robot task sequences (OBS, OHTS, TBS, and THTS) as scheduling strategies are presented, and based on them, we establish four linear programs to find optimal and feasible schedules to minimize the cycle time. Since a linear program can be solved in polynomial time, the proposed method is computationally efficient. Also, based on the property of the method, a feasible and optimal schedule can be found if itexists. Finally, experiments are conducted with different cases to test the four strategies. It shows that for different parameter combinations, different strategies achieve different performance. Thus, given a group of parameters, one should apply all the strategies and then choose the best schedule obtained by one of the strategies.  
With continuous experimentation, we have discovered that the delay time at each step is quite long. Our future endeavors will concentrate on minimizing delay time in the scheduling problem of an SF\({}^{3}\)-CT, while making efforts to reduce cycle time and enhance production efficiency.  
In this paper, we focus on the scheduling strategies and solutions for finding feasible and optimal schedules for the addressed system. To do so, we assume that the activity time is known. However, in practice, uncertainty exists regarding the activity time, and this imposes another challenge for scheduling the addressed system. Today, a digital twin is a widely used technology for various application problems [33; 34; 35; 36; 37]. In our future work, based on the presented strategies, digital twin technology will be applied to extend the proposed method for the scheduling problem of the addressed cluster tools with activity-time uncertainty.  
Currently, high-throughput screening systems are widely used for new chemical substance discovery, and their scheduling problem is also very challenging due to the residency time constraints similar to the wafer residency time constraints here [38; 39; 40]. We will examine the possibility of extending the proposed methods to study the scheduling problem of high-throughput screening systems.  
Conceptualization, L.G. and Y.Q.; methodology, L.G. and N.W.; software, S.Z.; validation, T.L.; formal analysis, L.G.; investigation, Y.Q.; resources, T.L.; data curation, S.Z. and T.L.; writing--original draft preparation, L.G.; writing--review and editing, N.W. and S. Z.; visualization, Y.Q.; supervision, N.W.; project administration, S.Z.; funding acquisition, N.W. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Science and Technology Development Fund (FDCT), Macau SAR (File Nos. 0018/2021/A1, 0083/2021/A2 and 0011/2023/RIA1).  
All data are contained within the article.  
The authors declare no conflict of interest.