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Deterministic and Random Single Machine Sequencing with Variance Minimization

Author

Listed:
  • Vina Vani

    (Gujarat University, Ahmedabad, India)

  • M. Raghavachari

    (Indian Institute of Management, Ahmedabad, India, and the University of Pennsylvania, Philadelphia, Pennsylvania)

Abstract

We discuss the problem of scheduling several jobs on a single machine so as to minimize the variance of the jobs' completion times. By exploiting the well-known pooled variance formula, we obtain a general formula for the change in variance due to the interchange of two specified jobs whose positions have not been fixed in the sequence. This result permits us to obtain explicit optimal sequences for problems with 6 or 7 jobs, and to show that Schrage's conjecture on positioning the job with the third largest processing time is true for problems with up to 18 jobs. We then develop a heuristic procedure based on a variance interchange formula and compare it with other known heuristic methods. We also consider a problem never before examined, namely, the case when processing times are not known deterministically but are random variables. We obtain some general results, under certain assumptions on the parameters of the distributions of processing times, and extend to the random situation many results applicable to the deterministic case. Finally, we give some results for the exponential distribution of processing times.

Suggested Citation

  • Vina Vani & M. Raghavachari, 1987. "Deterministic and Random Single Machine Sequencing with Variance Minimization," Operations Research, INFORMS, vol. 35(1), pages 111-120, February.
    • Handle: RePEc:inm:oropre:v:35:y:1987:i:1:p:111-120
    DOI: 10.1287/opre.35.1.111
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    Citations

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    Cited by:

    1. V. Rajendra Prasad & D. K. Manna, 1997. "Minimization of expected variance of completion times on single machine for stochastic jobs," Naval Research Logistics (NRL), John Wiley & Sons, vol. 44(1), pages 97-108, February.
    2. Manna, D. K. & Prasad, V. Rajendra, 1999. "Bounds for the position of the smallest job in completion time variance minimization," European Journal of Operational Research, Elsevier, vol. 114(2), pages 411-419, April.
    3. Cai, X., 1996. "V-shape property for job sequences that minimize the expected completion time variance," European Journal of Operational Research, Elsevier, vol. 91(1), pages 118-123, May.
    4. X. Cai & F. S. Tu, 1996. "Scheduling jobs with random processing times on a single machine subject to stochastic breakdowns to minimize early‐tardy penalties," Naval Research Logistics (NRL), John Wiley & Sons, vol. 43(8), pages 1127-1146, December.
    5. Koulamas, Christos & Kyparisis, George J., 2023. "Two-stage no-wait proportionate flow shop scheduling with minimal service time variation and optional job rejection," European Journal of Operational Research, Elsevier, vol. 305(2), pages 608-616.
    6. Cai, X., 1995. "Minimization of agreeably weighted variance in single machine systems," European Journal of Operational Research, Elsevier, vol. 85(3), pages 576-592, September.
    7. De, Prabuddha & Ghosh, Jay B. & Wells, Charles E., 1996. "Scheduling to minimize the coefficient of variation," International Journal of Production Economics, Elsevier, vol. 44(3), pages 249-253, July.
    8. Y. P. Aneja & S. N. Kabadi & A. Nagar, 1998. "Minimizing weighted mean absolute deviation of flow times in single machine systems," Naval Research Logistics (NRL), John Wiley & Sons, vol. 45(3), pages 297-311, April.
    9. Wang, Ji-Bo & Xia, Zun-Quan, 2007. "Single machine scheduling problems with controllable processing times and total absolute differences penalties," European Journal of Operational Research, Elsevier, vol. 177(1), pages 638-645, February.
    10. Adamopoulos, G. I. & Pappis, C. P., 1996. "Scheduling jobs with different, job-dependent earliness and tardiness penalties using the SLK method," European Journal of Operational Research, Elsevier, vol. 88(2), pages 336-344, January.
    11. Gowrishankar, K. & Rajendran, Chandrasekharan & Srinivasan, G., 2001. "Flow shop scheduling algorithms for minimizing the completion time variance and the sum of squares of completion time deviations from a common due date," European Journal of Operational Research, Elsevier, vol. 132(3), pages 643-665, August.
    12. Adamopoulos, G. I. & Pappis, C. P., 1995. "The CON due-date determination method with processing time-dependent lateness penalties," International Journal of Production Economics, Elsevier, vol. 40(1), pages 29-36, June.
    13. J. Steve Davis & John J. Kanet, 1993. "Single‐machine scheduling with early and tardy completion costs," Naval Research Logistics (NRL), John Wiley & Sons, vol. 40(1), pages 85-101, February.
    14. Ganesan, Viswanath Kumar & Sivakumar, Appa Iyer, 2006. "Scheduling in static jobshops for minimizing mean flowtime subject to minimum total deviation of job completion times," International Journal of Production Economics, Elsevier, vol. 103(2), pages 633-647, October.
    15. C.T. Ng & X. Cai & T.C.E. Cheng, 1999. "Probabilistic analysis of an asymptotically optimal solution for the completion time variance problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 46(4), pages 373-398, June.
    16. Weng, Xiaohua & Ventura, Jose A., 1996. "Scheduling about a given common due date to minimize mean squared deviation of completion times," European Journal of Operational Research, Elsevier, vol. 88(2), pages 328-335, January.
    17. Awi Federgruen & Gur Mosheiov, 1993. "Simultaneous optimization of efficiency and performance balance measures in single‐machine scheduling problems," Naval Research Logistics (NRL), John Wiley & Sons, vol. 40(7), pages 951-970, December.
    18. Seo, Jong Hwa & Kim, Chae-Bogk & Lee, Dong Hoon, 2001. "Minimizing mean squared deviation of completion times with maximum tardiness constraint," European Journal of Operational Research, Elsevier, vol. 129(1), pages 95-104, February.
    19. Xiaoqiang Cai & Sean Zhou, 1999. "Stochastic Scheduling on Parallel Machines Subject to Random Breakdowns to Minimize Expected Costs for Earliness and Tardy Jobs," Operations Research, INFORMS, vol. 47(3), pages 422-437, June.
    20. X. Cai & S. Zhou, 1997. "Scheduling stochastic jobs with asymmetric earliness and tardiness penalties," Naval Research Logistics (NRL), John Wiley & Sons, vol. 44(6), pages 531-557, September.
    21. Gajpal, Yuvraj & Rajendran, Chandrasekharan, 2006. "An ant-colony optimization algorithm for minimizing the completion-time variance of jobs in flowshops," International Journal of Production Economics, Elsevier, vol. 101(2), pages 259-272, June.

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