Tae-Eog Lee

Scheduling analysis of time-constrained dual-armed cluster tools

Cluster tools, each of which consists of several single-wafer processing chambers and a wafer handling robot, have been increasingly used for diverse wafer fabrication processes. Processes such as some low pressure chemical vapor deposition processes require strict timing control. Unless a wafer processed at a chamber for such a process leaves the chamber within a specified time limit, the wafer is subject to quality problems due to residual gases and heat. We address the scheduling problem for such time-constrained dual-armed cluster tools that have diverse wafer flow patterns. We propose a systematic method of determining the schedulable process time range for which there exists a feasible schedule that satisfies the time constraints. We explain how to select the desirable process times within the schedulable process time range. We present a method of determining the tool operation schedule. For more flexible scheduling under the time constraints, we propose a modification of the conventional swap operation in order to allow wafer delay on a robot arm during a swap operation. We compare the performance of the new swap strategy with that of the conventional swap strategy.

An extended event graph with negative places and tokens for time window constraints

We introduce places with negative holding times and tokens with negative token counts into a timed event graph in order to model and analyze time window constraints. We extend the enabling and firing rules for such an extended event graph named a negative event graph (NEG). We develop necessary and sufficient conditions based on the circuits for which the NEG is live, that is, an infinite sequence of feasible firing epochs exist for each transition. We prove that the minimum cycle time is the same as the maximum circuit ratio of the circuits with positive token counts. We also show that when there exists circuits with negative token counts, the maximum cycle time is bounded and the same as the minimum circuit ratio of such circuits. A scheduling example for a robot-based cluster tool with wafer residency time constraints for semiconductor manufacturing is explained. Note to Practitioners-Scheduling and control problems for modern man-made systems, including automated manufacturing systems such as cluster tools for semiconductor manufacturing, microcircuits, and real-time software systems, are usually modeled as discrete event systems. Such systems often have strict time constraints on timings of some events. Our results can be used for identifying whether there can be a feasible schedule that satisfies all time constraints, computing the range of the feasible cycle times, and determining a steady schedule with the minimum cycle time. By using the feasibility condition, we also can accommodate the system configuration, the task times, and the task sequence so that the system can satisfy the time constraints while meeting the target cycle time. Such practice is already used for real cluster tool engineering. We have more results on implementing a real-time scheduler and controller for time constrained systems.

Scheduling single-armed cluster tools with reentrant wafer flows

A cluster tool for semiconductor manufacturing consists of several single-wafer processing chambers and a wafer-handling robot in a closed environment. The use of cluster tools is extended to reentrant processes such as atomic layer deposition, where a wafer visits a processing chamber more than once. Such a reentrant wafer How complicates scheduling and control of the cluster tool and often causes deadlocks. We examine the scheduling problem for a single-armed cluster tool with various reentrant wafer flows. We develop a convenient method of modeling tool operational behavior with reentrant wafer flows using Petri nets. By examining the net model, we then develop a necessary and sufficient condition for preventing a deadlock. We also show that the cycle time for the asymmetric choice Petri net model for a reentrant wafer How can be easily computed by using the equivalent event graph model. From the results, we systematically develop a mixed integer programming model for determining the optimal tool operation sequence, schedule, and cycle time. We also extend a workload measure for cluster tools with reentrant wafer flows. Finally, we discuss how our results can be used for engineering a cluster tool. We compare two proposed strategies, sharing and dedicating, of operating the parallel processing chambers for identical process steps.

Schedulability Analysis of Time-Constrained Cluster Tools With Bounded Time Variation by an Extended Petri Net

Cluster tools for some wafer fabrication processes such as low-pressure chemical vapor deposition have strict wafer delay constraints. A wafer that completes processing in a processing chamber should leave the chamber within a specified time limit. Otherwise, the wafer suffers from severe quality troubles due to residual gases and heat within the chamber. An important engineering problem is to verify whether for given task times there exists a tool operation schedule that satisfies the wafer delay limit. There have been studies on the problem, which all assume deterministic task times. However, in reality, the task times are subject to random variation. In this paper, we develop a systematic method of determining schedulability of time-constrained decision-free discrete-event systems, where time variation can be confined within finite intervals. To do this, we propose an extended Petri net for modeling such systems. We then develop a necessary and sufficient condition for which there always exists a feasible schedule and one for which there never exists any feasible schedule. We develop a graph-based computational procedure for verifying the schedulability conditions and determining the worst-case task delay. We demonstrate how the procedure can be used for cluster tool engineering to control wafer delays against wafer alignment failures and time variation.

A review of scheduling theory and methods for semiconductor manufacturing cluster tools

Cluster tools, which combine several single-wafer processing modules with wafer handling robots in a closed environment, have been increasingly used for most wafer fabrication processes. We review tool architectures, operational issues, and scheduling requirements. We then explain recent progress in tool science and engineering for scheduling and control of cluster tools.

Modeling and implementing a real-time scheduler for dual-armed cluster tools

Abstract As application of cluster tools is being extended to more wafer fabrication processes including chemical vapor deposition processes as well as etching processes, cluster tool configurations and wafer flow patterns tend to be more complicated. Frequent recipe changes require prompt adaptation of the scheduling logic to cope with changed configurations and wafer flow patterns. The scheduler should be responsive to occurrences of various events within the tool and should be able to coordinate and control complicated activities and tasks of the process modules and the wafer-handling robot. Moreover, the scheduler should meet a critical timing constraint that a wafer should leave the process module within a specified time limit after processing. Therefore, it is essential to have a systematic, flexible and agile method of developing a robust real-time scheduler. To do this, we propose a model-based approach based on finite state machines (FSMs) to develop a real-time scheduler for dual-armed cluster tools. We propose the scheduling decisions for normal operation. We then develop a scheduler model based on FSMs that specifies the scheduling decisions for normal operation and encodes real-time control decisions to cope with disturbances. We then verify the scheduler model and transform the model into an executable control code.

Performance Measures and Schedules in Periodic Job Shops

This paper discusses the periodic job shop scheduling problem, a problem where an identical mixture of items, called a minimal part set MPS, is repetitively produced. The performance and behavior of schedules are discussed. Two basic performance measures, cycle time and makespan, are shown to be closely related. The minimum cycle time is identified as a circuit measure in a directed graph. We establish that there exists a class of schedules that minimizes cycle time and repeats an identical timing pattern every MPS. An algorithm is developed to construct such schedules. We show that minimizing the makespan as a secondary criterion, minimizes several other performance measures. For makespan minimization, we examine earliest starting schedules where each operation starts as soon as possible. We characterize the cases where after a finite number of MPSs, the earliest starting schedule repeats an identical timing pattern every fixed number of MPSs. We also develop a modification to an earliest starting schedule that repeats an identical timing pattern every MPS when the beginning operations on the machines are delayed.

Petri net modeling and scheduling for cyclic job shops with blocking

Abstract Cyclic scheduling is an effective scheduling method in the repetitive discrete manufacturing environment. We investigate the scheduling problem for general cyclic job shops with blocking where each machine has an input buffer of finite capacity. We develop Petri net models for the shops. We propose a sequential buffer control policy that restricts the jobs to enter the input buffer of the next machine in a specified sequence. We show that the scheduling model of a cyclic shop with finite buffers under such a buffer control policy can be transformed into a scheduling model of a cyclic shop with no buffer that can be modeled as a timed marked graph. In addition, we characterize the structural properties for deadlock detection. Finally, we present a mixed integer programming model to find an optimal deadlock-free schedule that minimizes the cycle time.

Scheduling Cluster Tools With Ready Time Constraints for Consecutive Small Lots

In the semiconductor manufacturing industry, the lot size currently tends to be extremely small, even being only 5-8 wafers, whereas conventional lots have 25 identical wafers. The smaller lot size is made because customers demand extremely small lots, and the number of chips in a large 300 mm wafer has increased. Cyclic scheduling is not applicable for such small lot production because the number of identical work cycles accounts for a small proportion of scheduling as compared to the lengths of the starting and closing transient periods. We therefore examine a new noncyclic scheduling problem of cluster tools for small lot production that considers ready time constraints on the chambers and the robot. The ready times are the epochs when the resources are freed from processing the preceding lot. To solve the scheduling problem, we develop a Petri net model which is a graphical and mathematical method for discrete event dynamic systems. Based on the Petri net model, we also develop a mixed integer programming (MIP) model and a branch and bound (B&B) algorithm for determining an optimal schedule. The B&B algorithm solves lots with up to 25 wafers and eight wafers within 500 s for a single-armed cluster tool and a dual-armed cluster tool, respectively, when three process steps are considered. Therefore, we propose an approximation method for the dual-armed cluster tool that schedules only the first few wafers with the B&B algorithm and the succeeding wafers with a well-known cyclic sequence. From experiments, we conclude that the difference between the approximation method and an optimal makespan is less than 1%. The methods we propose can be used for general noncyclic scheduling problems that can be modeled by Petri nets.

An Efficient Mixed Integer Programming Model Based on Timed Petri Nets for Diverse Complex Cluster Tool Scheduling Problems

Cluster tools are automated production cells which are largely used for semiconductor manufacturing. They consist of several processing modules (PMs) and a transportation robot. Since cluster tools have limited buffers and diverse scheduling requirements such as complex wafer flow patterns, parallel PMs, wafer residency time constraints, and dual-arm robot, and so on, their scheduling problems are difficult. Due to the diversity of scheduling problems, dealing with those problems one by one may be impractical. Computational complexity is another difficulty. In this paper, we propose an efficient scheduling method to deal with diverse complex cluster tool scheduling problems by using timed Petri nets (TPN). We propose TPN models of cluster tools with various scheduling requirements. Then, based on the TPN models and their state equations, we develop a new mixed integer programming (MIP) model that can efficiently determine the optimal cyclic schedules. We show that many kinds of scheduling requirements such as parallel, reentrant and multiple material flows, a dual-armed robot, and time constrained PMs can be dealt with by the MIP model. Through experiments, we also show that the MIP model can efficiently solve most practical cluster tool scheduling problems.