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Minimizing expected makespan in stochastic open shops

Published online by Cambridge University Press:  01 July 2016

Michael L. Pinedo  and
Sheldon M. Ross
Michael L. Pinedo
Affiliation:
Georgia Institute of Technology
Sheldon M. Ross*
Affiliation:
University of California, Berkeley
*
∗∗Postal address: Department of Industrial Engineering and Operations Research, University of California, Berkeley, CA 94720, U.S.A.
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Abstract

Suppose that two machines are available to process n tasks. Each task has to be processed on both machines; the order in which this happens is immaterial. Task j has to be processed on machine 1 (2) for random time Xj (Yj) with distribution Fj (Gj). This kind of model is usually called an open shop. The time that it takes to process all tasks is normally called the makespan. Every time a machine finishes processing a task the decision-maker has to decide which task to process next. Assuming that Xj and Yj have the same exponential distribution we show that the optimal policy instructs the decision-maker, whenever a machine is freed, to start processing the task with longest expected processing time among the tasks still to be processed on both machines. If all tasks have been processed at least once, it does not matter what the decision-maker does, as long as he keeps the machines busy. We then consider the case of n identical tasks and two machines with different speeds. The time it takes machine 1 (2) to process a task has distribution F (G). Both distributions F and G are assumed to be new better than used (NBU) and we show that the decision-maker stochastically minimizes the makespan when he always gives priority to those tasks which have not yet received processing on either machine.

Keywords

SCHEDULINGEXPONENTIAL DISTRIBUTION
Type
Research Article
Information
Advances in Applied Probability , Volume 14 , Issue 4 , December 1982 , pp. 898 - 911
Copyright
Copyright © Applied Probability Trust 1982 

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Footnotes

a

Present address: Department of Industrial Engineering and Operations Research, S. W. Mudd Building, Columbia University, New York, NY 10027, U.S.A.

Partially supported by the Office of Naval Research, Center for Production and Distribution Research, Georgia Institute of Technology and by the National Science Foundation under grant ECS-8115344 with the Georgia Institute of Technology.

Partially supported by the Air Force Office of Scientific Research (AFSC), USAF, under Grant AFOSR with the University of California, Berkeley.

References

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