Machine learning techniques for scheduling jobs with incompatible families and unequal ready times on parallel batch machines

doi:10.1016/j.engappai.2005.10.001

Engineering Applications of Artificial Intelligence

Volume 19, Issue 3, April 2006, Pages 235-245

https://doi.org/10.1016/j.engappai.2005.10.001 Get rights and content

Abstract

This research is motivated by a scheduling problem found in the diffusion and oxidation areas of semiconductor wafer fabrication facilities, where the machines can be modeled as parallel batch processors. Total weighted tardiness on parallel batch machines with incompatible job families and unequal ready times of the jobs is attempt to minimize. Given that the problem is NP hard, a simple heuristic based on the Apparent Tardiness Cost (ATC) Dispatching Rule is suggested. Using this rule, a look-ahead parameter has to be chosen. Because of the appearance of unequal ready times and batch machines it is hard to develop a closed formula to estimate this parameter. The use of inductive decision trees and neural networks from machine learning is suggested to tackle the problem of parameter estimation. The results of computational experiments based on stochastically generated test data are presented. The results indicate that a successful choice of the look-ahead parameter is possible by using the machine learning techniques.

Introduction

Wafer fabrication in semiconductor manufacturing is often characterized by hundreds of steps, reentrant flows, sequence dependent setups, diversity of product mix and batch processing. Because of the complexity of this type of manufacturing systems meeting customer due-dates with different priorities is still a challenging task. This research focuses on scheduling of batch-processing machines found in the diffusion and oxidation areas of a wafer fabrication facility. The processing times of these operations are extremely long (10 h) when compared to other operations (1–2 h). Mehta and Uzsoy (1998) state that the effective scheduling of these operations is important to achieving good overall system performance. Though several jobs can be processed simultaneously on these batch-processing machines, process restrictions require that only jobs belonging to the same family be processed together at one time. In addition, the jobs to be processed have different priorities/weights, due-dates and ready times. In the presence of unequal ready times, it is sometimes advantageous to form a non-full batch; in other situations it is a better strategy to wait for future job arrivals in order to increase the fullness of the batch, i.e., allow for delayed schedules.

The diffusion and oxidation operations are modeled as parallel batch processing machines with incompatible job families. The performance measure of interest to be minimized is total weighted tardiness. Total weighted tardiness is the summation of the weighted tardiness over all jobs $j = 1, 2, 3 \dots n$ , w_jT_j where $T_{j} = \max (0, C_{j} - d_{j})$ and where w_j is the weight (priority), C_j is the completion time and d_j the due-date of job j. Furthermore, the more realistic case of dynamic job arrivals is considered, i.e., unequal ready times of the jobs are allowed. Since this problem is NP-Hard (by reduction to $1 ∥ Σ w_{j} T_{j}$ which is NP-hard by Lawler, 1977) this research suggests the use of a rather simple list based scheduling approach based on the Apparent Tardiness Cost (ATC) rule (Vepsalainen and Morton, 1987; Pinedo, 2002). Applying this dispatching rule requires the choice of a look-ahead parameter. Two machine learning techniques are suggested to estimate the look-ahead parameter.

The paper is organized as follows. In Section 2, previous work related to the topic of this paper is summarized. In Section 3, the problem is described and the used notation is introduced. The suggested scheduling heuristic is described in Section 4. Factors that have an impact on the choice of the look-ahead parameter are suggested. In Section 5, the methodology behind inductive decision trees and neural networks is explained. Furthermore, the application of the machine learning techniques to the scheduling problem is described. In Section 6, the used experimental design is explained and the results of computational experiments are presented.

Section snippets

Scheduling of batch machines

Scheduling problems are usually represented in the form (α∣β∣γ) (Graham et al., 1979). Here, the α field describes the machine environment, the β field is used as a notation for process specifics and the γ field indicates the used performance measure. This notation will be used throughout the rest of the paper. Many researchers have addressed problems related to batching machines. Perez (1999) provides a detailed review and classification of papers that have dealt with deterministic scheduling

Assumptions and notation used for the scheduling problem

The assumptions involved in the scheduling of parallel batch processing machines with incompatible jobs families and unequal ready times of the jobs to minimize total weighted tardiness are:

1.
Jobs of the same family have the same processing times.
2.
All the batch-processing machines are identical in nature.
3.
Once a batch-processing machine is started, it cannot be interrupted. No preemption is allowed.

The following notation is used throughout the rest of the paper.

1.
Jobs fall into different incompatible

Scheduling heuristic

The well-known ATC heuristic suggested by Vepsalainen and Morton (1987) is used in this research as a dispatching rule to solve the parallel machine scheduling problem. At every point of time t when a machine becomes free, one batch from each family is chosen and of all the considered batches one is selected and scheduled on the machine. A time window (t,t+Δt) is considered. The set of unscheduled jobs of family j with arrival time less than the upper boundary of the time window interval is

Machine learning techniques applied to parameter setting in scheduling heuristics

In this section, neural networks and inductive decision trees are discussed as methods for parameter estimation. The work of Aytuk et al. (1994) provides a survey of other machine learning techniques applied to scheduling problems. Jain and Meeran (1998) discuss the usage of neural networks for scheduling and related literature.

Training data generation scheme

Jobs with the following attributes are generated. The due dates are uniformly distributed according to $d_{j} \sim U ((1 - T) μ \bar{p} (1 - R / 2), (1 - T) μ \bar{p} (1 + R / 2)) .$ The ready time are chosen in an analogous way from $r_{j} \sim U ((1 - \tilde{T}) μ \bar{p} (1 - \tilde{R} / 2), (1 - \tilde{T}) μ \bar{p} (1 + \tilde{R} / 2)) .$ The discrete values for $T, R, \tilde{T}, \tilde{R}$ , μ are summarized in Table 1.

In order to make sure that the ready time of the jobs are smaller than the due dates of the jobs T is added to each discrete value of $\tilde{T}$ . The processing time p_j for family j is given by the following

Conclusions and future work

In this paper, two different machine learning approaches for choosing the look-ahead parameter in a ATC-type dispatching rule applied to scheduling jobs with incompatible job families and unequal ready times on parallel batch machines are studied. The first approach uses neural networks in order to estimate an appropriate k parameter. Inductive decision trees are the essence of the second approach. Reports on the performance of the two approaches with respect to solution quality and time

Acknowledgments

This research was partially supported by a research grant of the Deutsche Forschungsgemeinschaft (DFG).

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Lars Mönch is an Assistant Professor in the Department of Information Systems at the Technical University of Ilmenau, Germany. He received a master's degree in applied mathematics, and a Ph.D. in the same subject from the University of Göttingen, Germany and a habilitation degree in information systems from Technical University of Ilmenau. After his Ph.D. he worked in the area of object-oriented software development for two years. His current research interests are in simulation-based production control of semiconductor wafer fabrication facilities, applied optimization and artificial intelligence applications in manufacturing. He is a member of GI (German Chapter of the ACM), GOR (German Operations Research Society), SCS and INFORMS.

Jens Zimmermann is a Ph.D. student in the Department of System Analysis at the Technical University of Ilmenau, Germany. He received a master's degree in information systems from the Technical University of Ilmenau. He is interested in semiconductor manufacturing, simulation and machine learning. He is a member of GI.

Peter Otto is an Associate Professor in the Department of System Analysis at the Technical University of Ilmenau, Germany. He received a master's degree in computer science from the Moscow State University and a Ph.D. in the same subject from the Technical University of Ilmenau, Germany. His current research interests are in machine learning techniques applied to control problems.

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Machine learning techniques for scheduling jobs with incompatible families and unequal ready times on parallel batch machines

Abstract

Introduction

Section snippets

Scheduling of batch machines

Assumptions and notation used for the scheduling problem

Scheduling heuristic

Machine learning techniques applied to parameter setting in scheduling heuristics

Training data generation scheme

Conclusions and future work

Acknowledgments

Computers & Industrial Engineering

Computers & Operations Research

Annals of Discrete Mathematics

Annals of Discrete Mathematics

Computers & Operations Research

Computers & Industrial Engineering

Computers & Operations Research

A review of machine learning in scheduling

IEEE Transactions of Engineering Management

Rolling horizon procedures for the single machinedeterministic total completion time scheduling problem with release dates

Annals of Operations Research

A parametric adjustment method for dispatching. Technical Report

A new dynamic look-ahead scheduling procedure for batching machines

Journal of Scheduling

Minimizing total weighted tardiness on a single batch processing machine with incompatible job families

Proceedings Symposium on Operations Research

Stochastic scheduling of a batch processing machine with incompatible job families

Annals of Operations Research

Real-time control of multiproduct bulk-service semiconductor manufacturing processes

IEEE Transactions on Semiconductor Manufacturing

Control of multi-product bulk-server diffusion/oxidation processes part two: multiple servers

IIE Transactions on Scheduling and Logistics

Dynamic batching heuristics for simultaneous processing

IEEE Transactions on Semiconductor Manufacturing

Job-shop scheduling using neural networks

International Journal of Production Research