## 2 The Processes and Scheduling Strategies  
### The Operation Processes  
In semiconductor fabs, wafers are associated with recipes, specifying the wafer flow patterns (WFPs) and processing parameters. WFPs specify the processing routes of wafers. For an SF\({}^{3}\)-CT, WFP is defined as \((m_{1},\,m_{2},\,m_{3},\,\ldots,\,m_{n})\), with \(n\) being the number of process steps required to process wafers and \(m_{i}\) the number of parallel PMs at Step \(i\). LLs are treated as Steps 0 and \(n\)+1 for holding raw and completed wafers, respectively. The robot arm with two dirty fingers is denoted by \(F_{1}\), and we call this arm the dirty arm, while the other robot arm with two clean fingers is represented by \(F_{2}\), and we call it the clean arm. Two raw wafers are unloaded from an LL by \(F_{1}\) and delivered to a PM at Step \(1\) for processing. Once they are processed at Step \(1\), they are treated as clean wafers. Then, \(F_{2}\) should deliver them to Steps \(2\), \(\ldots\), \(n\) for processing sequentially. After all operations are completed, they are returned to LL by \(F_{2}\). To schedule a cluster tool, determining the robot task sequence is vitally important so as to maximize the tool's productivity and meet RTCs. The robot tasks include unloading, loading, and moving; \(U_{i}\) and \(L_{i}\) are used to denote the robot tasks of unloading and loading at a PM of Step \(i\), \(i\in\mathbf{N}_{n}=\{0,\,1,\ldots,\,n\}\). Since the time taken for a robot task is quite short in practice, the time taken to perform \(U_{i}\), \(i\in\mathbf{N}_{n}\backslash\{0\}\), and \(L_{i}\), \(i\in\mathbf{N}_{n}\), is the same and denoted by \(\beta\). Note that before the robot unloads two clean wafers from an LL, the alignment of the wafers should be carried out first, resulting in the time to perform \(U_{0}\) being relatively longer. Thus, \(\beta_{0}\) is used to denote the time taken for \(U_{0}\). The moving task from a PM at Step \(i\) (or an LL at Step \(0\)) to a PM at Step \(j\) (or an LL at Step \(0\)) is denoted by \(M_{ij}\), \(i\), \(j\), \(\in\mathbf{N}_{n}\), and the time taken for \(M_{i,j}\) is denoted by \(\mu\). Note that for \(M_{ij}\), \(i\), \(j\), \(\in\mathbf{N}_{n}\), if \(i=j\), such a robot-moving task is called a rotation task. Assume that a robot arm is in the front of a PM, if the robot performs a rotation task, then the other robot arm should rotate to the front of the PM after the rotation. Furthermore, a swap operation of the robot includes an unloading task, a rotation task, and a loading task. At the idle state, all four spaces in a PM are empty. Then, after two wafers are loaded into two of the spaces, the internal chamber of the PM rotates to make the other two empty spaces face the door, such that the robot can load another two wafers into the PM. Similarly, a rotation operation should be performed when the four wafers are processed and should be removed from the PM by the robot consecutively. Symbol \(v\) is used to denote the time taken for a rotation operation of a PM. The wafer-processing time in a PM at Step \(i\) is denoted by \(\alpha_{i}\), \(i\in\mathbf{N}_{n}\). Assume that there are four wafers in a PM at Step \(i\). Let \(W_{i,12}\) and \(W_{i,34}\) be the four wafers that are to be processed in a PM at Step \(i\) concurrently and \(W_{i,12}\) be the two wafers that are loaded into the PM earlier than \(W_{i;34}\). Notice that after \(W_{i;12}\) are loaded into the PM at Step \(i\), the internal chamber rotates such that \(W_{i;34}\) can be loaded into the PM. Then, with \(W_{i;34}\) being loaded, the rotation operation should be executed again to move the two spaces holding \(W_{i,12}\) to the door position such that the PM can start to process wafers. After the processing activity is completed, \(W_{i,12}\) can be removed from the PM earlier than \(W_{i,34}\). In this way, one can ensure that the wafers loaded into the PM earlier can be removed from the PM earlier. Note that, under a steady state, there are four wafers being processed in a PM at Step \(i\).