## 3 Scheduling Analysis
### Linear Programs for Solution
**_For an SF\({}^{3}\)-CT, if the tool is operated by OBS under the steady state, a linear program is established to obtain a schedule as follows:_  
\[\psi_{1}=\theta_{i}\text{, }i\in\mathbf{N}_{n}\backslash\mathbf{N}_{1}\text{ and }n>2 \tag{51}\]1  
\[\psi_{1}=\theta_{i}\text{, }i\in\mathbf{N}_{n}\backslash\mathbf{N}_{1}\text{ and }n>2 \tag{51}\]2  
In LP1, Constraints (1) and (2) guarantee that the wafer sojourn time is within the permissible time windows. Constraints (3)--(5) calculate the time required to complete four wafers at each step from Steps 1 to \(n\). Then, Constraints (6)--(8) calculate the time for robot-waiting activities \(A_{i2}\) and \(A_{i3}\) at each step. Constraints (9)--(12) determine the sojourn time of \(W_{i,12}\) and \(W_{i,34}\). Equation (15) obtains the robot cycle time under OBS. Constraints (50)--(52) come from Proposition 1, and we can obtain \(\omega_{i1}\) and \(\psi_{1}\). **Linear Program 2 (LP2). **_For an SF\({}^{3}\)-CT, if the tool is operated by OHTS under the steady state, a linear program is established to obtain a schedule as follows:_  
\[\psi_{1}=\theta_{i}\text{, }i\in\mathbf{N}_{n}\backslash\mathbf{N}_{1}\text{ and }n>2 \tag{51}\]3  
\[\psi_{1}=\theta_{i}\text{, }i\in\mathbf{N}_{n}\backslash\mathbf{N}_{1}\text{ and }n>2 \tag{51}\]4  
\[\psi_{1}=\theta_{i}\text{, }i\in\mathbf{N}_{n}\backslash\mathbf{N}_{1}\text{ and }n>2 \tag{51}\]5  
\[\psi_{1}=\theta_{i}\text{, }i\in\mathbf{N}_{n}\backslash\mathbf{N}_{1}\text{ and }n>2 \tag{51}\]6  
\[\psi_{1}=\theta_{i}\text{, }i\in\mathbf{N}_{n}\backslash\mathbf{N}_{1}\text{ and }n>2 \tag{51}\]7  
\[\psi_{1}=\theta_{i}\text{, }i\in\mathbf{N}_{n}\backslash\mathbf{N}_{1}\text{ and }n>2 \tag{51}\]8  
In LP2, similar to LP1, Constraints (1) and (2) guarantee that the RTCs are satisfied. Constraints (3), (4), (14), and (15) calculate the time required to complete four wafers at each step from Steps 1 to \(n\). Then, Constraints (6), (7), (17), and (18) calculate the time for robot-waiting activities \(A_{i2}\) and \(A_{i3}\) at each step. Constraints (9), (10), (19), and (20) determine the sojourn time of \(W_{i,12}\) and \(W_{i,34}\). Equation (20) obtains the robot cycle time under OHTS, i.e., the system cycle time. Constraints (53)-(56) come from Proposition 2, and we can obtain \(\omega_{i1}\) and \(\psi_{2}\). **Linear Program 3 (LP3). **_For an SF\({}^{3}\)-CT, if the tool is operated by TBS under the steady state, a linear program is established to obtain a schedule as follows._  
\[\psi_{1}=\theta_{i}\text{, }i\in\mathbf{N}_{n}\backslash\mathbf{N}_{1}\text{ and }n>2 \tag{51}\]9  
\[\psi_{1}=\theta_{n} \tag{52}\]0  
Similar to LP1 and LP2, Constraints (1) and (2) guarantee that the RTCs are satisfied. Constraints (22) and (23) calculate the time required to complete four wafers at each step from Steps 1 to \(n\). Then, Constraints (24)-(28) calculate the time for robot waiting activities \(A_{12}\) at each step. Constraints (29)-(32) determine the sojourn time of \(W_{i,12}\) and \(W_{i,34}\). Equation (33) obtains the robot cycle time under TBS. Constraint (57) comes from Proposition 3, and we can obtain \(\omega_{i1}\) and \(\psi_{3}\). **Linear Program 4 (LP4). **_For an SF\({}^{3}\)-CT, if the tool is operated by THTS under the steady state, a linear program is established to obtain a schedule as follows._  
Minimize \(\psi_{4}\)  
Subject to: (1), (2),  
(34)-(49), (58)-(60),  
(29) where \(i\in\mathbf{N}_{n}\backslash\mathbf{N}_{2}\) and \(n>2\),  
(30) where \(i\in\mathbf{N}_{n}\backslash\mathbf{N}_{1}\) and \(n>2\), and  
(31) in which \(n>2\). Similar to the linear programs above, Constraints (1) and (2) guarantee that the RTCs are satisfied. Constraints (34)-(38) calculate the time required to complete four wafers at each step from Steps 1 to \(n\). Then, Constraints (39)-(44) calculate the time for robot-waiting activities \(A_{12}\) at each step. Constraints (29)-(31) and (45)-(48) determine the sojourn time of \(W_{i,12}\) and \(W_{i,34}\). Equation (49) calculates the robot cycle time under THTS.