Stable schedule for a single-armed cluster tool with time constraints

The semiconductor manufacturing industry increasingly requires cluster tools, which consist of several single-wafer processing modules, a wafer-handling robot, and loadlocks. Some processes, such as low-pressure chemical vapor deposition, require strict time window constraints due to residual gases and heat in the chamber. The cluster tool should be operated at a steady schedule to minimize the maximum wafer delays and maintain a uniform quality of wafers, However, unforeseen events such as wafer alignment failures and retrials cause disruptions in the schedule and hence increase the variability of wafer delays within a process module. Therefore, it is important to determine the control strategies required to return to a steady schedule after sporadic disruptive events occur.We develop conditions to stabilize timing of a single-armed cluster tool using the convergence theory of a class of matrix power series in (max, +)-algebra. To derive the conditions, we model the cluster tool using a timed event graph and analyze the conditions for which the earliest firing schedule pattern of the event graph converges to a unique steady schedule, regardless of the initial schedule pattern or any disruption. From these conditions, we develop stabilizing strategies for the schedule by delaying certain robot tasks or increasing the process times. We experimentally demonstrate and compare the effectiveness and efficiency of our strategies.