## 2 The Processes and Scheduling Strategies
### Robot Task Sequences
Then, \(\sigma_{6}=\langle M_{2,0}\to U_{0}\to M_{0,1}\to L_{1}\to M_{1,n}\rangle\) is performed by \(F_{1}\) and \(F_{2}\) together to load raw wafers into the system. After that, \(\sigma_{5}\) and \(\sigma_{6}\) are sequentially performed again. At this time, each PM in the tool is full of wafers again. Therefore, a production cycle is conducted by performing \(\sigma_{5}\to\sigma_{6}\to\sigma_{5}\to\sigma_{6}\), such that each step completes four wafers in such a cycle. By TBS, after performing \(\sigma_{5}\) for Steps \(1\)-\(n\), two processed wafers in a PM are moved to the next step for further processing, with two spaces of a PM at Step \(1\) being emptied. The last robot task sequence is also a hybrid strategy, called two-time hybrid task sequence (THTS). For THTS, \(F_{2}\) first performs robot task sequence \(\sigma_{7}=\langle U_{n}\to M_{n,n+1}\to L_{n+1}\to M_{n+1,n-1}\to U_{n-1}\to M_{n -1,n}\to L_{n}\to M_{n,n-2}\to\ldots\to M_{4,2}\to U_{2}\to M_{2,3}\to L_{3}\rangle\) to handle clean wafers. After performing \(\sigma_{7}\), \(\sigma_{8}=\langle M_{3,0}\to U_{0}\to M_{0,1}\to\) a swap operation at Step \(1\to M_{1,2}\to L_{2}\to M_{2,n}\rangle\) is performed by \(F_{1}\) and \(F_{2}\) collaboratively to load raw wafers into the system. Then, \(\sigma_{7}\) and \(\sigma_{8}\) are sequentially performed again. At this time, each PM in the tool is full of wafers again. Therefore, similar to TBS, a production cycle is conducted by sequentially performing \(\sigma_{7}\to\sigma_{8}\to\sigma_{7}\to\sigma_{8}\), such that each step completes four wafers in such a cycle. This strategy is similar to TBS; the difference is that, after performing \(\sigma_{7}\), no processed wafer at Step \(1\) is moved to Step \(2\), and two spaces of a PM at Step \(2\) are emptied. With these four robot task sequences, the clean and dirty fingers are correctly used. In \(\sigma_{1}\)-\(\sigma_{8}\), the robot-waiting time is not considered. Although a robot task sequence in each of \(\sigma_{1}\)-\(\sigma_{8}\) is determined, it does not present a schedule since the start and end times of the activities are not given. However, as discussed above, if the robot-waiting time is decided, a schedule is obtained. Thus, to find a feasible and optimal schedule under one of the four robot task sequences, we need to intendedly increase the robot waiting time taken by \(A_{i1}\). Thus, the key to scheduling an SF\({}^{3}\)-CT is to determine the robot-waiting time, such that the RTCs are met and the productivity of the system is maximized. The next Section aims to do so. In this Section, the operational processes of semiconductor fabs using cluster tools are discussed with a focus on wafer flow patterns and the robot task sequence. It describes how dirty and clean wafers are treated separately for a multi-arm and multi-finger robot and how RTC should be met to avoid contamination. It outlines the importance of properly sequencing robot tasks to ensure productivity while maintaining RTCs and provides the foundational details on how wafers are processed in parallel within multi-space process modules.