Automation and Remote Control

, Volume 71, Issue 10, pp 2070–2084 | Cite as

Algorithms for some maximization scheduling problems on a single machine

  • E. R. Gafarov
  • A. A. Lazarev
  • F. Werner
Scheduling Problems on a Single Machine


In this paper, we consider two scheduling problems on a single machine, where a specific objective function has to be maximized in contrast to usual minimization problems. We propose exact algorithms for the single machine problem of maximizing total tardiness 1‖max-ΣT j and for the problem of maximizing the number of tardy jobs 1‖maxΣU j . In both cases, it is assumed that the processing of the first job starts at time zero and there is no idle time between the jobs. We show that problem 1‖max-ΣT j is polynomially solvable. For several special cases of problem 1‖maxΣT j , we present exact polynomial algorithms. Moreover, we give an exact pseudo-polynomial algorithm for the general case of the latter problem and an alternative exact algorithm.


Remote Control Optimal Schedule Idle Time Single Machine Total Tardiness 
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  1. 1.
    Moore, J.M., An n Job, One Machine Sequencing Algorithm for Minimizing the Number of Late Jobs, Manage. Sci., 1968, vol. 15, no. 1, pp. 102–109.zbMATHCrossRefGoogle Scholar
  2. 2.
    Du, J. and Leung, J. Y.-T., Minimizing Total Tardiness on One Processor is NP-hard, Math. Oper. Res., 1990, vol. 15, pp. 483–495.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Gafarov, E.P. and Lazarev, A.A., A Special Case of the Single-machine Total Tardiness Problem is NP-hard, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 2006, no. 3, pp. 120–128.Google Scholar
  4. 4.
    Lawler, E.L., A Pseudopolynomial Algorithm for Sequencing Jobs to Minimize Total Tardiness, Ann. Discret. Math., 1977, vol. 1, pp. 331–342.CrossRefMathSciNetGoogle Scholar
  5. 5.
    Szwarc, W., Della Croce, F., and Grosso, A., Solution of the Single Machine Total Tardiness Problem, J. Scheduling, 1999, vol. 2, pp. 55–71.zbMATHCrossRefGoogle Scholar
  6. 6.
    Potts, C.N. and Van Wassenhove, L.N., A Decomposition Algorithm for the Single Machine Total Tardiness Problem, Oper. Res. Lett., 1982, vol. 1, pp. 363–377.CrossRefGoogle Scholar
  7. 7.
    Lazarev, A.A. and Gafarov, E.R., Teoriya raspisanii. Minimizatsiya summarnogo zapazdyvaniya dlya odnogo pribora (Scheduling Theory. Minimizing Total Delay for a Single Device), Moscow: Vychisl. Tsentr Ross. Akad. Nauk, 2006.Google Scholar
  8. 8.
    Lazarev, A., Dual of the Maximum Cost Minimization Problem, J. Math. Sci., 1989, vol. 44, no. 5, pp. 642–644.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Huang, R.H. and Yang, C.L., Single-Machine Scheduling to Minimize the Number of Early Jobs, IEEE Int. Conf. Indust. Eng. Eng. Manage., 2007, 4419333, pp. 955–957.Google Scholar
  10. 10.
    Lazarev, A.A. and Werner, F., Algorithms for Special Cases of the Single Machine Total Tardiness Problem and an Application to the Even-Odd Partition Problem, Math. Comput. Modelling, 2009, vol. 49, no. 9–10, pp. 2061–2072.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Lazarev, A.A., Kvaratskhelia, A.G., and Gafarov, E.R., Algorithms for Solving the NP-Hard Problem of Minimizing Total Tardiness for a Single Machine, Dokl. Akad. Nauk, Math., 2007, vol. 412, no. 6, pp. 739–742.Google Scholar
  12. 12.
    Gafarov, E.R., Lazarev, A.A., and Werner, F., Algorithms for Maximizing the Number of Tardy Jobs or Total Tardiness on a Single Machine, Preprint of Otto-von-Guericke-Universität, Magdeburg, 2009, no. 38/09.Google Scholar
  13. 13.
    Lazarev, A.A. and Werner, F., A Graphical Realization of the Dynamic ProgrammingMethod for Solving NP-Hard Combinatorial Problems, Comput. Math. App., 2009, vol. 58, no. 4, pp. 619–631.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  • E. R. Gafarov
    • 1
  • A. A. Lazarev
    • 1
  • F. Werner
    • 2
  1. 1.Trapeznikov Institute of Control SciencesRussian Academy of SciencesMoscowRussia
  2. 2.Otto-von-Guericke-Universität MagdeburgMagdeburgGermany

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