Noncyclic Scheduling of Cluster Tools With a Branch and Bound Algorithm

Cluster tools, each of which consists of multiple processing modules, one material handling robot, and loadlocks, are widely used for wafer fabrication processes, such as lithography, etching, and deposition. There have been many approaches and algorithms for cyclic scheduling of cluster tools in which the robot repeats a specified sequence for processing identical wafers. However, the lot order size has recently been decreasing due to the larger wafer size and circuit width reductions. In modern fabs, each wafer lot can have different flow patterns and process times for the same process step, and heterogeneous lots are processed consecutively in a tool. Even some tools in a fab have idle time waiting for wafer lots depending on the work-in-process fluctuations. Such different wafer lots and frequent tool state changes cannot be handled with cyclic scheduling methods, and accordingly noncyclic scheduling methods for such cases are required. Therefore, we develop an efficient branch and bound (B&B) procedure for noncyclic scheduling problems of cluster tools to minimize the makespan. Since a timed Petri net (TPN) is known for its powerful modeling ability and analysis capability, the algorithm is developed based on a TPN. We verify the efficiency of the B&B procedure with various cluster tool scheduling problems. There have been many studies on scheduling cluster tools, but most of them utilize different scheduling approaches or develop problem specialized properties. It is impractical to implement all the different methods to a tool scheduler because the scheduling requirements continuously change depending on wafer types and tool architectures. Hence, it is required to have an efficient solution method to address diverse cluster tool scheduling problems in fabs especially for frequent lot switchings and tool state changes due to larger wafer size and smaller lot order size. Therefore, we develop an efficient branch and bound procedure for noncyclic scheduling of a cluster tool with the makespan measurement. Since TPNs have the powerful modeling ability and analysis capability, the algorithm is developed based on a TPN. From the experimental results, we observe that one lot with 25 wafers can be easily solved in a reasonable time. The proposed method can be used for many different scheduling problems of a cluster tool by generating each corresponding TPN model.