A Novel Mixed Integer Programming Model With Precedence Relation-Based Decision Variables for Non-Cyclic Scheduling of Cluster Tools

Jeongsun Ahn, Hyun-Jung Kim

Abstract: Cluster tools, which are extensively used in semiconductor and display manufacturing, offer the capability to perform multiple processing steps within a single tool. Many companies have recently been reducing wafer lot sizes due to diversified customer demands and circuit width reduction, resulting in more frequent transient and non-cyclic periods. We therefore present a novel mixed integer programming (MIP) model that can handle these non-cyclic scheduling problems of cluster tools. Our model is specifically designed to efficiently handle various wafer flows, accommodating both single- and dual-armed robots, while ensuring a shorter computation time, compared to the previous formulations. We first model cluster tools with a timed Petri net (TPN) and develop several precedence relations between transitions by analyzing the characteristics of the scheduling problems. These precedence relations are incorporated into a TPN by introducing additional places and arcs. This modification helps reduce the overall number of decision variables involved in the scheduling problem. We then develop an MIP model which specifies the precedence relations between transitions, as opposed to the position-based decision variables used in the previous studies. The proposed MIP model is capable of handling various flow scenarios encountered in cluster tools, including serial, parallel, concurrent, and re-entrant flows with time window constraints. Note to Practitioners—In response to the diverse demands of customers, the use of larger wafer sizes, and the need for smaller circuit widths, operating cluster tools in a cyclic manner has become increasingly challenging. Cyclic scheduling, where the robot repeats a fixed sequence while assuming identical wafers, is no longer viable in such scenarios. Unfortunately, previous research on cluster tool scheduling has predominantly focused on cyclic scheduling, failing to address the growing importance of non-cyclic scenarios. To address this gap, we propose an innovative and efficient mixed integer programming (MIP) model capable of handling non-cyclic scheduling problems in cluster tools. Our model accommodates various flow types, including serial, parallel, concurrent, and re-entrant flows. Through comparative analysis, we demonstrate that our proposed model outperforms the well-known MIP model that relies on position-based decision variables. We believe that our proposed model holds practical value as it exhibits relatively short computation times. Additionally, widely-used commercial solvers, such as CPLEX or Google OR Tools, can seamlessly implement the model, making it easily accessible and applicable in real-world scenarios.