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Annals of Operations Research

, Volume 196, Issue 1, pp 247–261 | Cite as

Transforming a pseudo-polynomial algorithm for the single machine total tardiness maximization problem into a polynomial one

  • Evgeny R. Gafarov
  • Alexander A. Lazarev
  • Frank WernerEmail author
Article

Abstract

We consider the problem of maximizing total tardiness on a single machine, where the first job starts at time zero and idle times between the processing of jobs are not allowed. We present a modification of an exact pseudo-polynomial algorithm based on a graphical approach, which has a polynomial running time. This result settles the complexity status of the problem under consideration which was open.

Keywords

Scheduling Single machine problems Maximization problems Total tardiness Polynomial algorithm 

Notes

Acknowledgements

This work was partially supported by DAAD (Deutscher Akademischer Austausch- dienst): A/08/80442/Ref. 325. The authors are grateful to the referees for their constructive comments, which substantially improved the presentation of the results.

References

  1. Aloulou, M. A., & Artigues, C. (2010). Flexible solutions in disjunctive scheduling: general formulation and study of the flow-shop case. Computers & Operations Research, 37(5), 890–898. CrossRefGoogle Scholar
  2. Aloulou, M. A., Kovalyov, M. Y., & Portmann, M.-C. (2004). Maximization problems in single machine scheduling. Annals of Operations Research, 129, 21–32. CrossRefGoogle Scholar
  3. Aloulou, M. A., Kovalyov, M. Y., & Portmann, M.-C. (2007). Evaluation flexible solutions in single machine scheduling via objective function maximization: the study of computational complexity. RAIRO. Recherche Opérationnelle, 41, 1–18. CrossRefGoogle Scholar
  4. Babu, P., Peridy, L., & Pinson, E. (2004). A branch and bound algorithm to minimize total weighted tardiness on a single processor. Annals of Operations Research, 129, 33–46. CrossRefGoogle Scholar
  5. Du, J., & Leung, J. Y.-T. (1990). Minimizing total tardiness on one processor is NP-hard. Mathematics of Operations Research, 15, 483–495. CrossRefGoogle Scholar
  6. Gafarov, E. R., Lazarev, A. A., & Werner, F. (2010a). Algorithms for maximizing the number of tardy jobs or total tardiness on a single machine. Automation and Remote Control, 71(10), 2070–2084. CrossRefGoogle Scholar
  7. Gafarov, E. R., Lazarev, A. A., & Werner, F. (2010b). A modification of dynamic programming algorithms to reduce the running time or/and complexity. Preprint 20/10, FMA, Otto-von-Guericke-Universität Magdeburg. Google Scholar
  8. Gafarov, E. R., Lazarev, A. A., & Werner, F. (2010c). Classical combinatorial and single machine scheduling problems with opposite optimality criteria. Preprint 11/10, FMA, Otto-von-Guericke-Universität Magdeburg. Google Scholar
  9. Lawler, E. L. (1977). A pseudopolynomial algorithm for sequencing jobs to minimize total tardiness. Annals of Discrete Mathematics, 1, 331–342. CrossRefGoogle Scholar
  10. Lawler, E. L., & Moore, J. M. (1969). A functional equation and its application to resource allocation and sequencing problems. Management Science, 16(1), 77–84. CrossRefGoogle Scholar
  11. Lazarev, A. A., & Gafarov, E. R. (2006a). Special case of the single-machine total tardiness problem is NP-hard. Journal of Computer and Systems Sciences International, 45(3), 450–458. CrossRefGoogle Scholar
  12. Lazarev, A. A., & Gafarov, E. R. (2006b). Scheduling theory. Total tardiness problem. Moscow: Computing Center of the Russian Academy of Sciences, 128 pp. (in Russian). Google Scholar
  13. Lazarev, A. A., & Werner, F. (2009a). A graphical realization of the dynamic programming method for solving NP-hard combinatorial problems. Computers and Mathematics with Applications, 58, 619–631. CrossRefGoogle Scholar
  14. Lazarev, A. A., & Werner, F. (2009b). Algorithms for special cases of the single machine total tardiness problem and an application to the even-odd partition problem. Mathematical and Computer Modelling, 49(9–10), 2061–2072. CrossRefGoogle Scholar
  15. Matsuo, H., Suh, C. J., & Sullivan, R. S. (1989). A controlled search simulated annealing method for the single machine weighted tardiness problem. Annals of Operations Research, 21, 85–108. CrossRefGoogle Scholar
  16. Posner, M. E. (1990). Reducibility among weighted completion time scheduling problems. Annals of Operations Research, 26, 91–101. CrossRefGoogle Scholar
  17. Potts, C. N., & Van Wassenhove, L. N. (1982). A decomposition algorithm for the single machine total tardiness problem. Operations Research Letters, 1, 363–377. CrossRefGoogle Scholar
  18. Szwarc, W., Della Croce, F., & Grosso, A. (1999). Solution of the single machine total tardiness problem. Journal of Scheduling, 2, 55–71. CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Evgeny R. Gafarov
    • 1
  • Alexander A. Lazarev
    • 1
  • Frank Werner
    • 2
    Email author
  1. 1.Institute of Control Sciences of the Russian Academy of SciencesMoscowRussia
  2. 2.Fakultät für MathematikOtto-von-Guericke-Universität MagdeburgMagdeburgGermany

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