Advertisement

Automation and Remote Control

, Volume 71, Issue 10, pp 2038–2057 | Cite as

Minimizing total weighted completion time with uncertain data: A stability approach

  • Yu. N. Sotskov
  • N. G. Egorova
  • F. Werner
Scheduling Problems on a Single Machine

Abstract

A single-machine scheduling problem is investigated provided that the input data are uncertain: The processing time of a job can take any real value from the given segment. The criterion is to minimize the total weighted completion time for the n jobs. As a solution concept to such a scheduling problem with an uncertain input data, it is reasonable to consider a minimal dominant set of job permutations containing an optimal permutation for each possible realization of the job processing times. To find an optimal or approximate permutation to be realized, we look for a permutation with the largest stability box being a subset of the stability region. We develop a branch-and-bound algorithm to construct a permutation with the largest volume of a stability box. If several permutations have the same volume of a stability box, we select one of them due to one of two simple heuristics. The efficiency of the constructed permutations (how close they are to a factually optimal permutation) and the efficiency of the developed software (average CPU-time used for an instance) are demonstrated on a wide set of randomly generated instances with 5 ≤ n ≤ 100.

Keywords

Schedule Problem Remote Control Completion Time Total Completion Time Optimal Permutation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Pinedo, M., Scheduling: Theory, Algorithms, and Systems, New Jersey: Prentice Hall, 2002.zbMATHGoogle Scholar
  2. 2.
    Aytug, H., Lawley, M.A., McKay, K., Mohan, S., and Uzsoy, R., Executing Production Schedules in the Face of Uncertainties: A Review and Some Future Directions, Eur. J. Oper. Res., 2005, vol. 161, pp. 86–110.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Sabuncuoglu, I. and Goren, S., Hedging Production Schedules Against Uncertainty in Manufacturing Environment with a Review of Robustness and Stability Research, Int. J. Comput. Integrated Manufacturing, 2009, vol. 22, no. 2, pp. 138–157.CrossRefGoogle Scholar
  4. 4.
    Daniels, R.L. and Kouvelis, P., Robust Scheduling to Hedge Against Processing Time Uncertainty in Single-Stage Production, Manage. Sci., 1995, vol. 41, no. 2, pp. 363–376.zbMATHCrossRefGoogle Scholar
  5. 5.
    Kouvelis, P. and Yu, G., Robust Discrete Optimization and Its Applications, Boston: Kluwer, 1997.zbMATHGoogle Scholar
  6. 6.
    Yang, J. and Yu, G., On the Robust Single Machine Scheduling Problem, J. Combinat. Optim., 2002, vol. 6, pp. 17–33.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Lai, T.-C., Sotskov, Y.N., Sotskova, N., and Werner, F., Optimal Makespan Scheduling with Given Bounds of Processing Times, Math. Comput. Model., 1997, vol. 26, pp. 67–86.zbMATHMathSciNetGoogle Scholar
  8. 8.
    Lai, T.-C. and Sotskov, Y.N., Sequencing with Uncertain Numerical Data for Makespan Minimization, J. Oper. Res. Soc., 1999, vol. 50, pp. 230–243.zbMATHGoogle Scholar
  9. 9.
    Sotskov, Y.N. and Sotskova, N.Y., Teoriya raspisanii: sistemy s neopredelennymi chislovymi parametrami (Scheduling Theory: Systems with Uncertain Numerical Parameters), Minsk: Natl. Acad. Sci. Belarus, United Inst. of Informatics Problems, 2004.Google Scholar
  10. 10.
    Matsveichuk, N.M., Sotskov, Y.N., Egorova, N.G., and Lai, T.-C., Schedule Execution for Two-Machine Flow-Shop with Interval Processing Times, Math. Comput. Modelling, 2009, vol. 49, nos. 5–6, pp. 991–1011.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Sotskov, Y.N., Egorova, N.G., and Lai, T.-C., Minimizing Total Weighted Flow Time of a Set of Jobs with Interval Processing Times, Math. Comput. Modelling, 2009, vol. 50, pp. 556–573.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lai, T-C, Sotskov, Y.N., Sotskova, N.Y., and Werner, F., Mean Flow Time Minimization with Given Bounds of Processing Times, Eur. J. Oper. Res., 2004, vol. 159, no. 3, pp. 558–573.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Sotskov, Y.N., Wagelmans, A.P.M., and Werner, F., On the Calculation of the Stability Radius of an Optimal or an Approximate Schedule, Ann. Oper. Res., 1998, vol. 83, pp. 213–252.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Sotskov, Y.N., Sotskova, N.Y., and Werner, F., Stability of an Optimal Schedule in a Job Shop, Omega, 1997, vol. 25, no. 4, pp. 397–414.CrossRefGoogle Scholar
  15. 15.
    Graham, R.L., Lawler, E.L., Lenstra, J.K., and Rinnooy Kan, A.H.G., Optimization and Approximation in Deterministic Sequencing and Scheduling. A Survey, Ann. Discr. Math., 1976, vol. 5, pp. 287–326.CrossRefMathSciNetGoogle Scholar
  16. 16.
    Smith, W.E., Various Optimizers for Single-Stage Production, Nav. Res. Logist. Quarterly, 1956, vol. 3, no. 1, pp. 59–66.CrossRefGoogle Scholar
  17. 17.
    Kasperski, A. and Zielinski, P., A 2-approximation Algorithm for Interval Data Minmax Regret Sequencing Problems with Total Flow Time Criterion, Oper. Res. Lett., 2008, vol. 36, pp. 343–344.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Montemanni, R., A Mixed Integer Programming Formulation for the Total Flow Time Single Machine Robust Scheduling Problem with Interval Data, J. Math. Model. Algorith., 2007, vol. 6, pp. 287–296.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Sotskov, Y.N., Dolgui, A., and Portmann, M.-C., Stability Analysis of Optimal Balance for Assembly Line with Fixed Cycle Time, Eur. J. Oper. Res., 2006, vol. 168, no. 3, pp. 783–797.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Sotskov, Y.N., Stability of an Optimal Schedule, Eur. J. Oper. Res., 1991, vol. 55, pp. 91–102.zbMATHCrossRefGoogle Scholar
  21. 21.
    Sotskov, Y.N., Tanaev, V.S., and Werner, F., Stability Radius of an Optimal Schedule: A Survey and Recent Developments, in Industr. Appl. Combinat. Optim., Yu, G., Ed., Boston: Kluwer, 1998, pp. 72–108.Google Scholar
  22. 22.
    Sotskova, N.Y. and Tanaev, V.S., About the Realization of an Optimal Schedule with Operation Processing Times under Conditions of Uncertainty, Dokl. Natl. Akad. Nauk Belarusi, 1998, vol. 42, no. 5, pp. 8–12.zbMATHMathSciNetGoogle Scholar
  23. 23.
    Allahverdi, A. and Sotskov, Y.N., Two-machine Flowshop Minimum-length Scheduling Problem with Random and Bounded Processing Times, Int. Transactions Oper. Res., 2003, vol. 10, pp. 65–76.zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Allahverdi, A., Aldowaisan, T., and Sotskov, Y.N., Two-machine Flowshop Scheduling Problem to Minimize Makespan or Total Completion Time with Random and Bounded Setup Times, Int. J. Math. Math. Sci., 2003, vol. 39, pp. 2475–2486.CrossRefMathSciNetGoogle Scholar
  25. 25.
    Ng, C.T., Matsveichuk, N.M., Sotskov, Y.N., and Cheng, T.C.E., Two-machine Flow-shop Minimumlength Scheduling with Interval Processing Times, Asia-Pacific J. Oper. Res., 2009, vol. 26, no. 6, pp. 1–20.MathSciNetGoogle Scholar
  26. 26.
    Sotskov, Y.N., Allahverdi, A., and Lai, T.-C., Flowshop Scheduling Problem to Minimize Total Completion Time with Random and Bounded Processing Times, J. Oper. Res. Soc., 2004, vol. 55, pp. 277–286.zbMATHCrossRefGoogle Scholar
  27. 27.
    Computer and Job-Shop Scheduling Theory, Coffman, E.G., Ed., New York: Wiley, 1976.zbMATHGoogle Scholar
  28. 28.
    Sotskov, Y.N. and Lai, T.-C., Minimizing Total Weighted Completion Time under Uncertainty Using Stability Box and Dominance, Comput. Oper. Res. (submitted).Google Scholar
  29. 29.
    Emelichev, V.A., Girlich, E.N., Nikulin, Y.V., and Podkopaev, D.P., Stability and Regularization Radius of Vector Problems of Integer Linear Programming. Optimization, 2002, vol. 51, no. 4, pp. 645–676.zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Emelichev, V.A., Krichko, V.N., and Nikulin, Y.V., The Stability Radius of an Efficient Solution in Mimimax Boolean Programming Problem. Control Cybernet., 2004, vol. 33, no. 1, pp. 127–132.zbMATHMathSciNetGoogle Scholar
  31. 31.
    Emelichev, V.A., Kuz’min, K.G., and Leonovich, A.M., Stability in the Combinatorial Vector Optimization Problems. Autom. Remote Control, 2004, vol. 65, no. 2, pp. 227–240.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  • Yu. N. Sotskov
    • 1
  • N. G. Egorova
    • 1
  • F. Werner
    • 2
  1. 1.United Institute of Informatics ProblemsBelarussian National Academy of SciencesMinskBelarus
  2. 2.Otto-von-Guericke-UniversityMagdeburgGermany

Personalised recommendations