## 2 The Processes and Scheduling Strategies
### Robot Task Sequences  
Assume that there are \(n\) (\(\geq\)2) steps for an SF\({}^{3}\)-CT. To maximize the productivity of an SF\({}^{3}\)-CT, under the steady state, each PM should be full of wafers. To maximize the productivity and ensure the feasibility (i.e., meet RTCs) in operating an SF\({}^{3}\)-CT, it is vital to properly determine a robot task sequence that can be performed periodically. In [11; 12], the authors prove that backward and swap are the optimal scheduling strategies for single-arm and dual-arm cluster tools, respectively. With clean and dirty robot fingers, in terms of operation strategies, an SF\({}^{3}\)-CT is neither a single-arm cluster tool nor a dual-arm cluster tool but a hybrid one that is equivalent to a combination of them; at some steps, swap operations can be applied, while they are not applicable at some other steps. Due to this property, there are two types of operation strategies for an SF\({}^{3}\)-CT, one is the backward strategy and the other is hybrid strategy, which combines the backward and swap strategies. Next, we present four possible robot task sequences as the operation strategies. Note that the following robot task sequences are presented based on the assumption that the tool operates under a steady state with each PM full of wafers and the clean robot arm \(F_{2}\) is at a PM of Step \(n\). The first robot task sequence is a backward strategy, called one-time backward sequence (OBS). For OBS, the clean robot arm \(F_{2}\) first performs task sequence \(\sigma_{1}=\langle U_{n}\to M_{n,n+1}\to L_{n+1}\to M_{n+1,n}\to U_{n}\to M_{n,n+1} \to L_{n+1}\to M_{n+1,n-1}\to U_{n-1}\to M_{n-1,n}\to L_{n}\to M_{n,n-1} \to U_{n-1}\to U_{n-1,n}\to L_{n-1,n}\to M_{n-1,n}\to L_{n}\to M_{n,n-2}\to \ldots\to M_{3,1}\to U_{1}\to M_{1,2}\to L_{2}\to M_{2,1}\to U_{1}\to M_{1,2} \to L_{2}\rangle\) to handle clean wafers. Then, the dirty robot arm \(F_{1}\) performs robot task sequence \(\sigma_{2}=\langle M_{2,0}\to U_{0}\to M_{0,1}\to L_{1}\to M_{1,0}\to U_{0}\to M _{0,1}\to L_{1}\to M_{1,n}\rangle\) to load raw wafers to the system. Note that, by \(M_{1,n}\) in \(\sigma_{2}\), a state is reached such that \(F_{2}\) is at a PM of Step \(n\). Thus, by sequentially performing \(\sigma_{1}\) and \(\sigma_{2}\), a production cycle is finished, with four wafers being processed. Notice that, by this strategy, after \(\sigma_{1}\) is performed, a PM at Step \(1\) is emptied. The second robot task sequence is a hybrid strategy called one-time hybrid task sequence (OHTS). For OHTS, the clean arm \(F_{2}\) first performs a robot task sequence \(\sigma_{3}=\langle U_{n}\to M_{n,n+1}\to L_{n+1}\to M_{n+1,n}\to U_{n}\to M_{n,n+1} \to L_{n+1}\to M_{n+1,n-1}\to U_{n-1,n}\to M_{n-1,n}\to L_{n}\to M_{n,n-1} \to U_{n-1,n}\to L_{n}\to M_{n-1,n}\to U_{n-1}\to M_{n-1,n}\to L_{n}\to M_{n-1,n}\to L_{n} \to M_{n-1,n}\to L_{n}\to M_{2,3}\to L_{3}\rangle\) to handle clean wafers. Then, arms \(F_{1}\) and \(F_{2}\) collaboratively perform robot task sequence \(\sigma_{4}=\langle M_{3,0}\to U_{0}\to M_{0,1}\to\) a swap operation at Step \(1\to M_{1,2}\to L_{2}\to M_{2,0}\to U_{0}\to M_{0,1}\to\) a swap operation at Step \(1\to M_{1,2}\to L_{2}\to M_{2,n}\rangle\) to load raw wafers into the system. Note that, by \(\sigma_{4}\), the system reaches a state such that \(F_{2}\) is at a PM of Step \(n\). Thus, by sequentially performing \(\sigma_{3}\) and \(\sigma_{4}\), a production cycle is finished with four wafers being processed. Different from OBS, after performing \(\sigma_{3}\), a PM at Step \(2\) is emptied, but no PM at Step \(1\) is emptied. The third robot task sequence is also a backward strategy, called two-time backward sequence (TBS). For TBS, \(F_{2}\) first performs robot task sequence \(\sigma_{5}=\langle U_{n}\to M_{n,n+1}\to L_{n+1}\to M_{n+1,n-1}\to U_{n-1}\to M_{n -1,n}\to L_{n}\to M_{n,n-2}\to\ldots\to M_{3,1}\to U_{1}\to M_{1,2}\to L_{2}\rangle\) to handle clean wafers.