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Scheduling jobs on a machine subject to stochastic breakdowns to minimize absolute early-tardy penalties

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Abstract

This article addresses the problem of scheduling n jobs with a common due date on a machine subject to stochastic breakdowns to minimize absolute early-tardy penalties. We investigate the problem under the conditions that the uptimes follow an exponential distribution, and the objective measure in detail is to minimize the expected sum of the absolute deviations of completion times from the common due date. We proceed to study in two versions (the downtime follows an exponential distribution or is a constant entailed for the repeat model job), one of which is the so-called preempt-resume version, the other of which is the preempt-repeat version. Three terms of work have been done. (i) Formulations and Preliminaries. A few of necessary definitions, relations and basic facts are established. In particular, the conclusion that the expectation of the absolute deviation of the completion time about a job with deterministic processing time t from a due date is a semi-V-shape function in t has been proved. (ii) Properties of Optimal Solutions. A few characteristics of optimal solutions are established. Most importantly, the conclusion that optimal solutions possess semi-V-shape property has been proved. (iii) Algorithm. Some computing problems on searching for optimal solutions are discussed.

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References

  1. Birge J R, Frenk J B G, Mittenthal J, et al. Single-machine scheduling subject to stochastic breakdowns. Naval Res Logist, 37: 661–677 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  2. Mittenthal J, Raghavachair M. Stochastic single machine scheduling with quadratic early-tardy penalties. Oper Res, 41(4): 786–796 (1993)

    MATH  MathSciNet  Google Scholar 

  3. Cai X, Tu F S. Scheduling jobs with random processing times on a single machine subject to stochastic breakdowns to minimize early-tardy penalties. Naval Res Logist, 43: 1127–1146 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  4. Al-Turki U M, Mittenthal J, Raghavachari M. The single machine absolute-deviation early-tardy problem with random completion times. Naval Res Logist, 43: 573–587 (1996)

    Article  MATH  Google Scholar 

  5. Qi X D, Yin G, Birge J R. Single-machine scheduling with random machine breakdowns and randomly compressible processing times. Stoch Anal Appl, 18(4): 635–653 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cai X, Sun X, Zhou X. Stochastic scheduling with preemptive-repeat machine breakdowns to minimize the expected weighted flow time. Probab Engrg Inform Sci, 17: 467–485 (2003)

    Article  MathSciNet  Google Scholar 

  7. Zhou X, Cai X. General stochastic single-machine scheduling with regular cost functions. Math Comput Modelling, 26(3): 95–108 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Tang H Y, Zhao C L, Cheng C D. Single machine stochastic JIT scheduling problem subject to machine breakdowns. Sci China Ser A-Math, 51(2): 273–292 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Panwalkar S S, Smith M L, Seidmann A. Common due date assignment to minimize total penalty for the one machine scheduling problem. Oper Res, 30(2): 391–399 (1982)

    MATH  Google Scholar 

  10. Bagchi U, Chang Y L, Sullivan, R S. Minimizing absolute and squared deviation of completion times with different earliness and tardiness penalties and a common due date. Naval Res Logist, 34: 739–751 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  11. Bagchi U, Sullivan R S, Chang Y L. Minimizing mean squared deviation of completion times about a common due date. Management Science, 33(7): 894–906 (1987)

    MATH  MathSciNet  Google Scholar 

  12. Baker K R, Scudder G D. Sequencing with earliness and tardiness penalties: a review. Oper Res, 38(1): 22–36 (1990)

    MATH  MathSciNet  Google Scholar 

  13. Raghavachari M. Scheduling problems with nonregular penalty functions-a review. Opsearch, 25: 141–164 (1988)

    MathSciNet  Google Scholar 

  14. Seidmann A, Panwalkar S S, Smith M L. Optimal assignment of due-dates for a single processor scheduling problem. Int J Prod Res, 19(4): 393–399 (1981)

    Article  Google Scholar 

  15. Federgruen A, Mosheiov G. Single machine scheduling problems with general breakdowns, earliness and tardiness costs. Oper Res, 45(1): 66–71 (1997)

    MATH  Google Scholar 

  16. Krieger A M, Raghavachari M. V-shape property of optimal schedules with monotone penalty functions. Comput Oper Res, 19(6): 533–534 (1992)

    Article  MATH  Google Scholar 

  17. Al-Turki U M, Mittenthal J, Raghavachari M. A dominant subset of V-shaped sequences for a class of single machine sequencing problems. European J Oper Res, 88: 345–347 (1996)

    Article  MATH  Google Scholar 

  18. Ross S M. Stochastic Processes. New York: Wiley, 1983

    MATH  Google Scholar 

  19. Bagchi U, Sullivan R S, Chang Y L. Minimizing mean absolute deviation of completion times about a common due date. Naval Res Logist, 33: 227–240 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  20. Bector C, Gupta Y P, Gupta M C. V-shape property of optimal sequence of jobs about a common due date on a single machine. Comput Oper Res, 16(6): 583–588 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  21. Eilon S. Chowdhury I G. Minimizing waiting time variance in the single machine problem. Management Science, 23(7): 567–575 (1977)

    MATH  MathSciNet  Google Scholar 

  22. Raghavachari M. A V-shape property of optimal schedule of jobs about a common due date. European J Oper Res, 23: 401–402 (1986)

    Article  Google Scholar 

  23. Raghavachari M, Zhammouri M. Single machine scheduling with coefficient of variation minimization. European J Oper Res, 62: 302–310 (1992)

    Article  MATH  Google Scholar 

  24. Hall N G, Kubiak W, Sethi S P. Earliness-tardiness scheduling problems, II: deviation of completion times about a restrictive common due date. Oper Res, 39(5): 847–856 (1991)

    MATH  MathSciNet  Google Scholar 

  25. Hall N G, Posner M E. Earliness-tardiness scheduling problems, I: deviation of completion times about a common due date. Oper Res, 39(5): 847–856 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  26. Cai X, Zhou X. Stochastic scheduling with asymmetric earliness and tardiness penalties under random machine breakdowns. Probab Engrg Inform Sci, 20(4): 635–654 (2006)

    MATH  MathSciNet  Google Scholar 

  27. Glazebrook K D. Optimal scheduling of tasks when service is subject to disruption: the preempt-repeat case. Math Methods Oper Res, 61: 147–169 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  28. Liu L, Gu H Y, Xi Y G. Robust and stable scheduling of a single machine with random machine breakdowns. Int Adv Manuf Technol, 31: 645–654 (2007)

    Google Scholar 

  29. Chiu S W, Wang S L, Chiu Y S. Determining the optimal run time for EPQ model with scrap, rework, and stochastic breakdowns. European J Oper Res, 180: 664–676 (2007)

    Article  MATH  Google Scholar 

  30. Yue D Q, Tu F S. The single machine stochastic scheduling with the weighted job tardiness minimization. J Syst Sci Syst Eng, 13: 342–347 (2004)

    Article  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and Systems Science, Shenyang Normal University, Shenyang, 110034, China

    CongDian Cheng, HengYong Tang & ChuanLi Zhao

Authors
  1. CongDian Cheng
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  2. HengYong Tang
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  3. ChuanLi Zhao
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Correspondence to CongDian Cheng.

Additional information

This work was supported by the National Natural Science Foundation of China (Grant No. 10471096)

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Cheng, C., Tang, H. & Zhao, C. Scheduling jobs on a machine subject to stochastic breakdowns to minimize absolute early-tardy penalties. Sci. China Ser. A-Math. 51, 864–888 (2008). https://doi.org/10.1007/s11425-007-0193-2

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  • DOI: https://doi.org/10.1007/s11425-007-0193-2

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