## 2 Virtual Wafer-Based Method
### Approach to Deal with Chamber Cleaning Requirements  
In order to ensure wafer quality, a chamber in a PM needs to be cleaned after a specified quantity of wafers are consecutively processed. In this way, residual chemicals and air particles can be removed from the chamber. Let \(m_{i}\), \(i\in\)N\({}^{+}\)n, denote the maximal number of wafers that a chamber in a PM at Step \(i\) consecutively process before the PM is required to clean, and \(C_{i}\), \(i\in\)N\({}^{+}\)n, the required that. Note that \(m_{i}\geq 1\) and \(C_{i}\), \(i\in\)N\({}^{+}\)n, are known in advance and can be obtained by statistical experiments performed by process engineers and data engineers. With chamber cleaning requirements, once the robot completes unloading a wafer from a PM at Step \(i\), the PM can start to clean its chamber. Then, by the backward strategy, the robot performs a sequence of robot tasks (i.e., \(M_{i(i+1)}\to L_{i+1}\to M_{(i+1)(i-1)}\to W_{i-1}\)\(\to U_{i-1}\to M_{(i-1)i}\)) such that it moves to the PM again for being ready to load a wafer. Note that the robot waiting time taken by \(W_{i-1}\) is determined by a one-wafer cyclic schedule in advance. If the time needed for a chamber cleaning is too long, leading to \(max(\alpha+\beta+3\mu+\omega_{i-1},C_{i})=C_{i}\), it would postpone the starting time of the subsequent robot tasks. In otherwords, the chamber cleaning operation at Step \(i\) means that the robot cannot arrive at a PM at Step \(j\), \(1\leq j\leq i-2\), for unloading a processed wafer in time, leading to a long wafer delay time, violating WRTCs. Such a long wafer delay time may cause a quality problem to the wafer. Therefore, it is very important to develop a precise scheduling and control method for an SACT with WRTCs and chamber cleaning operations. This work aims to do so. In fact, by ignoring chamber cleaning requirements, a one-wafer cyclic schedule can be found by properly regulating the robot waiting time [15]. With chamber cleaning requirements, virtual wafers are introduced into SACTs based on a one-wafer cyclic schedule such that the chambers in the tools can be cleaned in time and WRTCs are not violated. Specifically, by a one-wafer cyclic schedule, sometimes when the robot comes to LLs for unloading a raw wafer, it unloads a virtual wafer. Thus, after a real wafer is processed in a PM, if the next wafer to be processed at the PM is a virtual one, it implies that the chamber in the PM can be cleaned. Thus, when the robot task of unloading the real wafer from the PM is performed, it triggers a cleaning task for the chamber in the PM. When a virtual is loaded into the PM, its processing is not performed since it is a virtual one. Instead, the chamber performs the cleaning operation. Then, by the backward strategy and the one-wafer cyclic schedule, the robot can move to any step according to the schedule without any delay. In this way, enough time for chamber cleaning at the PM can be ensured and, at the same time, the starting time of any robot task in a cycle is not postponed such that the WRTCs can be met. However, if the required chamber cleaning time is too long, it may need to consecutively load more virtual wafers to provide enough time for chamber cleaning. In order to determine how many virtual wafers should be consecutively loaded into a PM it is necessary to know how long can be scheduled for the chamber cleaning operation if a specified number of virtual wafers are consecutively loaded into a PM. Let \(O_{i}^{d}\) denote the scheduled time for the chamber cleaning operation of a PM at Step \(i\), \(i\in\)N\({}^{+}\)n, if there are \(d\) virtual wafers being consecutively loaded into the PM. Assume that there is no virtual wafer being processed in a PM at Step \(i\), \(i\in\)N\({}^{+}\)n. For such a case, with chamber cleaning requirements, once the robot completes unloading a wafer from a PM at Step \(i\), the PM can start to clean its chamber.