Schedule restoration for single-armed cluster tools

The idle time between two consecutive tasks may seriously affect the wafer quality in some wafer fabrication processes requiring high temperature and pressure. Hence, a cluster tool provides high wafer quality and predictability when operated at a steady schedule, which means that every cycle repeats the same pattern considering not only a sequence but also timing. However, multiple process modules per step can cause the fluctuation of wafer quality even under the same sequence since different modules have different wafer delays. It is known that a dual-armed cluster tool can always have a steady schedule that can be established via a timed event graph and (max, +) algebra. We extend this result to a single-armed cluster tool and generalize the method to establish its steady schedule. Although a cluster tool is initially set on a steady schedule, the schedule can become unsteady after disruptive events, such as wafer alignment failures, and it cannot always return to it. We develop strategies that change the system dynamics of a cluster tool by appropriately delaying some tasks such that any disrupted schedule pattern is stabilized in a finite time. Finally, we examine how they help to flatten wafer delays after a disruption or when tasks have stochastic time fluctuation.