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STOCHASTIC SCHEDULING WITH ASYMMETRIC EARLINESS AND TARDINESS PENALTIES UNDER RANDOM MACHINE BREAKDOWNS

Published online by Cambridge University Press:  19 September 2006

Xiaoqiang Cai  and
Xian Zhou
Xiaoqiang Cai
Affiliation:
Department of Systems Engineering & Engineering Management, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, E-mail: [email protected]
Xian Zhou
Affiliation:
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, E-mail: [email protected]
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Abstract

We study a stochastic scheduling problem of processing a set of jobs on a single machine. Each job has a random processing time Pi and a random due date Di, which are independently and exponentially distributed. The machine is subject to stochastic breakdowns in either preempt-resume or preempt-repeat patterns, with the uptimes following an exponential distribution and the downtimes (repair times) following a general distribution. The problem is to determine an optimal sequence for the machine to process all jobs so as to minimize the expected total cost comprising asymmetric earliness and tardiness penalties, in the form of E[[sum ]αi max{0,Di − Ci} + βi max{0,Ci − Di}]. We find sufficient conditions for the optimal sequences to be V-shaped with respect to {E(Pi)/αi} and {E(Pi)/βi}, respectively, which cover previous results in the literature as special cases. We also find conditions under which optimal sequences can be derived analytically. An algorithm is provided that can compute the best V-shaped sequence.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 20 , Issue 4 , October 2006 , pp. 635 - 654
Copyright
© 2006 Cambridge University Press

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References

REFERENCES

Adiri, I., Bruno, J., Frostig, E., & Rinnooy Kan, A.H.G. (1989). Single machine flowtime scheduling with a single breakdown. Acta Informatica 26: 679–696.Google Scholar
Adiri, I., Frostig, E., & Rinnooy Kan, A.H.G. (1991). Scheduling on a single machine with a single breakdown to minimize stochastically the number of tardy jobs. Naval Research Logistics 38: 261–271.Google Scholar
Baker, K.R. & Scudder, G.D. (1990). Sequencing with earliness and tardiness penalties: A review. Operations Research 38: 22–36.Google Scholar
Birge, J., Frenk, J.B.G., Mittenthal, J., & Rinnooy Kan, A.H.G. (1990). Single-machine scheduling subject to stochastic breakdown. Naval Research Logistics 37: 661–677.Google Scholar
Cai, X., Sun, X.Q., & Zhou, X. (2003). Stochastic scheduling with preemptive-repeat machine breakdowns to minimize the expected weighted flowtime. Probability in the Engineering and Informational Sciences 17: 467–485.Google Scholar
Cai, X. & Tu, F.S. (1996). Scheduling jobs with random processing times on a single machine subject to stochastic breakdowns to minimize early-tardy penalties. Naval Research Logistics 43: 1127–1146.Google Scholar
Cai, X. & Zhou, X. (1997). Scheduling stochastic jobs with asymmetric earliness and tardiness penalties. Naval Research Logistics 44: 531–557.Google Scholar
Cai, X. & Zhou, X. (1999). Stochastic scheduling on parallel machines subject to random breakdowns to minimize expected costs for earliness and tardy jobs. Operations Research 47: 422–437.Google Scholar
Cai, X. & Zhou, X. (2000). Asymmetric earliness and tardiness scheduling with exponential processing times on an unreliable machine. Annals of Operations Research 98: 313–331.Google Scholar
Chakravarthy, S. (1986). A single machine scheduling problem with random processing times. Naval Research Logistics Quarterly 33: 391–397.Google Scholar
Eilon, S. & Chowdhury, I.G. (1977). Minimizing waiting time variance in the single machine problem. Management Science 23: 567–575.Google Scholar
Forst, F.G. (1993). Stochastic sequencing on one machine with earliness and tardiness penalties. Probability in the Engineering and Informational Sciences 7: 291–300.Google Scholar
Frenk, J.B.G. (1991). A general framework for stochastic one-machine scheduling problems with zero release times and no partial ordering. Probability in the Engineering and Informational Sciences 5: 297–315.Google Scholar
Frostig, E. (1991). A note on stochastic scheduling on a single machine subject to breakdown: The preemptive repeat model. Probability in the Engineering and Informational Sciences 5: 349–354.Google Scholar
Garey, M.R., Tarjan, R.E., & Wilfong, G.T. (1988). One-processor scheduling with symmetric earliness and tardiness penalties. Mathematics of Operations Research 13: 330–348.Google Scholar
Hall, N.G. & Posner, M.E. (1991). Earliness-tardiness scheduling problems, I: Weighted deviation of completion time about a common due date. Operations Research 39: 836–846.Google Scholar
Jia, C.F. (2001). Stochastic single machine scheduling with an exponentially distributed due date. Operations Research Letters 28: 199–203.Google Scholar
Kovalyov, M.Y. & Kubiak, W. (1999). A fully polynomial approximation scheme for the weighted earliness–tardiness problem. Operations Research 47: 757–761.Google Scholar
Luh, P.B., Chen, D., & Thakur, L.S. (1999). An effective approach for job-shop scheduling with uncertain processing requirements. IEEE Transactions on Robotics and Automation 15: 328–339.Google Scholar
Mittenthal, J. & Raghavachari, M. (1993). Stochastic single machine scheduling with quadratic early-tardy penalties. Operations Research 41: 786–796.Google Scholar
Pinedo, M. (1983). Stochastic scheduling with release dates and due dates. Operations Research 31: 559–572.Google Scholar
Pinedo, M. & Rammouz, E. (1988). A note on stochastic scheduling on a single machine subject to breakdown and repair. Probability in the Engineering and Informational Sciences 2: 41–49.Google Scholar
Qi, X.D., Yin, G., & Birge, J.R. (2000). Scheduling problems with random processing times under expected earliness/tardiness costs. Stochastic Analysis and Applications 18: 453–473.Google Scholar
Sen, T., Sulek, J.M., & Dileepan, P. (2003). Static scheduling research to minimize weighted and unweighted tardiness: A state-of-the-art survey. International Journal of Production Economics 83: 1–12.Google Scholar
Soroush, H.M. & Fredendall, L.D. (1994). The stochastic single machine scheduling problem with earliness and tardiness costs. European Journal of Operational Research 77: 287–302.Google Scholar
Vani, V. & Raghavachari, M. (1987). Deterministic and random single machine sequencing with variance minimization. Operations Research 35: 111–120.Google Scholar