Journal of Scheduling

, Volume 8, Issue 3, pp 233–253 | Cite as

Pre-Emptive Scheduling Problems with Controllable Processing Times

Abstract

We consider a range of single machine and identical parallel machine pre-emptive scheduling models with controllable processing times. For each model we study a single criterion problem to minimize the compression cost of the processing times subject to the constraint that all due dates should be met. We demonstrate that each single criterion problem can be formulated in terms of minimizing a linear function over a polymatroid, and this justifies the greedy approach to its solution. A unified technique allows us to develop fast algorithms for solving both single criterion problems and bicriteria counterparts.

Keywords

single machine scheduling parallel machine scheduling controllable processing times bicriteria problems polymatroids greedy algorithms 

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References

  1. Aho, A. V., J. E. Hopcroft, and J. D. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, London, 1974.Google Scholar
  2. Balas, E. and E. Zemel, “An algorithm for large zero-one knapsack problems,” Oper. Res., 28, 1130–1154 (1980).Google Scholar
  3. Brucker, P., Scheduling Algorithms, Springer, Berlin, 2001.Google Scholar
  4. Chase, R. B., N. J. Acquilano, and F. R. Jacobs, Production and Operations Management: Manufacturing and Services, Irwin McGraw-Hill, 1998.Google Scholar
  5. Cheng, T. C. E. and A. Janiak, “Resource optimal control in some single machine scheduling problems,” IEEE Trans. Automat. Control, 39, 1243–1246 (1994).CrossRefMathSciNetGoogle Scholar
  6. Cheng, T. C. E., A. Janiak, and M. Kovalyov, “Bicriterion single machine scheduling with resource dependent processing times,” SIAM J. Optim., 8, 617–630 (1998).CrossRefGoogle Scholar
  7. Cheng, T. C. E., A. Janiak, and M. Kovalyov, “Single machine batch scheduling with resource dependent setup and processing times,” Eur. J. Oper. Res., 135, 177–183 (2001).CrossRefGoogle Scholar
  8. Edmonds, J., “Submodular functions, matroids, and certain polyhedra,” in: R. Guy, H. Hanani, N. Sauer, and J. Schonheim (eds.), Combinatorial Structures and their Applications, Gordon and Breach, New York, 1970.Google Scholar
  9. Frank, A. and E. Tardos, “Generalized polymatroids and submodular flows,” Math. Program., 42, 489–563 (1988).CrossRefGoogle Scholar
  10. Gordon, V. S. and V. S. Tanaev, “Due dates in single-stage deterministic scheduling,” in Optimization of Systems of Collecting, Transfer and Processing of Analogous and Discrete Data in Local Information Computing Systems, Institute of Engineering Cybernetics, Minsk, Belarus, 1973, pp. 54–58 (in Russian).Google Scholar
  11. Hoogeveen H. and G. J. Woeginger, “Some comments on sequencing with controllable processing times,” Computing, 68, 181–192 (2002).CrossRefGoogle Scholar
  12. Horn, W. A., “Some simple scheduling algorithms,” Naval Res. Logist. Quart., 21, 177–185 (1974).Google Scholar
  13. Janiak, A, “Minimization of the makespan in two-machine problem under given resource constraints,” Eur. J. Oper. Res., 107, 325–337 (1998).CrossRefGoogle Scholar
  14. Janiak, A. and M. Y. Kovalyov, “Single machine scheduling subject to deadlines and resource dependent processing times,” Eur. J. Oper. Res., 94, 284–291 (1996).CrossRefGoogle Scholar
  15. Janiak, A., M. Y. Kovalyov, W. Kubiak, and F. Werner, “Positive half-products and scheduling with controllable processing times,” Eur. J. Oper. Res., 165, 416–422 (2005).CrossRefMathSciNetGoogle Scholar
  16. Lawler, E. L., J. K. Lenstra, A. H. G. Rinnooy Kan, and D. B. Shmoys, “Sequencing and scheduling: Algorithms and complexity,” in S. C. Graves, A. H. G. Rinnooy Kan and P. H. Zipkin (eds.), Handbooks in Operations Research and Management Science, Vol. 4, Logistics of Production and Inventory, North-Holland, Amsterdam, 1993, pp. 445–522.Google Scholar
  17. Lenstra, J. K., A. H. G. Rinnooy Kan, and P. Brucker, “Complexity of machine scheduling problems,” Ann. Discrete Math., 1, 343–362 (1977).Google Scholar
  18. Leung, J. Y.-T., V. K. M. Yu, and W.-D. Wei, “Minimizing the weighted number of tardy task units,” Discrete Appl. Math., 51, 307–316 (1994).CrossRefGoogle Scholar
  19. McNaughton, R., “Scheduling with deadlines and loss functions,” Manage. Sci., 59, 1–12 (1959).Google Scholar
  20. Nemhauser, G. L. and L. A. Wolsey, Integer and Combinatorial Optimization, John Wiley & Sons, 1988.Google Scholar
  21. Nowicki, E. and S. Zdrzalka, “Scheduling jobs with controllable processing times as an optimal-control problem,” Int. J. Contr., 39, 839–848 (1984).Google Scholar
  22. Nowiki, E., and S. Zdrzalka, “A survey of results for sequencing problems with controllable processing times,” Discrete Appl. Math., 26, 271–287 (1990).CrossRefGoogle Scholar
  23. Nowicki, E. and S. Zdrzalka, “A bicriterion approach to pre-emptive scheduling of parallel machines with controllable job processing times,” Discrete Appl. Math., 63, 271–287 (1995).CrossRefGoogle Scholar
  24. Schrijver, A., Combinatorial Optimization: Polyhedra and Efficiency, Springer, Berlin, 2003.Google Scholar
  25. Shih W.-K., J. W. S. Liu, and J.-Y. Chung, “Fast algorithms for scheduling imprecise computations,” in Proceedings of the 10th Real-time Systems Symposium, Santa-Monica, 1989, pp. 12–19.Google Scholar
  26. Shih W.-K., J. W. S. Liu, and J.-Y. Chung, “Algorithms for scheduling imprecise computations with timing constraints,” SIAM J. Comput., 20, 537–552 (1991).CrossRefGoogle Scholar
  27. Tanaev, V. S., V. S. Gordon and Y. M. Shafransky, Scheduling Theory. Single-Stage Systems, Kluwer Academic Publishers, Dordrecht, 1994.Google Scholar
  28. T’kindt V., and J.-C. Billaut, Multicriteria Scheduling: Theory, Models and Algorithms, Springer, Berlin, 2002.Google Scholar
  29. Van Wassenhove, L. N., and K. R. Baker, “A bicriterion approach to time/cost tradeoffs in sequencing,” Eur. J. Oper. Res., 11, 48–54 (1982).Google Scholar
  30. Vickson, R. G. “Choosing the job sequence and processing times to minimize total processing plus flow cost on a single machine,” Oper. Res., 28, 1155–1167 (1980).Google Scholar
  31. Waller, D. L., Operations Management: A Supply Chain Approach, International Thomson Publishing, London, 1999.Google Scholar
  32. Wan, G., B. P. C. Yen, and C. L. Li, “Single machine scheduling to minimize total compression plus weighted flow cost is NP-hard,” Information Processing Letters, 79, 273–280 (2001).CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  • Natalia V. Shakhlevich
    • 1
  • Vitaly A. Strusevich
    • 2
  1. 1.School of ComputingUniversity of LeedsLeedsU.K.
  2. 2.School of Computing and Mathematical SciencesUniversity of GreenwichLondonU.K.

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