Litho Machine Scheduling With Convex Hull Analyses

The increasing pressure to meet demand are forcing semiconductor manufacturers to seek efficient scheduling methods. Lithography, with a limited number of expensive resources and the reentrant nature of the fabrication processes, is a major bottleneck. This paper presents a litho machine scheduling formulation for high-volume and low-variety manufacturing over a day, with novel modeling of resource setups, reticle expirations, and future stacking layer load balancing. The problem is believed to be NP hard. After linearization and simplification, it is solved by using the branch-and-cut method by exploiting problem linearity. Near-optimal solutions for practical problems, however, are still difficult to obtain efficiently. Through detailed analyses, it was discovered that the convex hull of the problem is difficult to delineate and many low-efficient branching operations are needed. A two-phase approach is therefore established. In the first phase, a simplified problem with certain complicating constraints dropped is efficiently solved by exploiting linearity to reduce ranges of decision variables. The problem with the full set of constraints is then solved in the second phase with a much reduced decision space. Numerical testing shows that this two-phase approach can generate near-optimal schedules within reasonable amounts of computation time. This two-phase approach is generic, and will have major implications on other semiconductor scheduling problems and beyond.