Skip to main content
SpringerLink
Log in

An improved approximation scheme for the Johnson problem with parallel machines

  • Published:
Journal of Applied and Industrial Mathematics Aims and scope Submit manuscript

Abstract

An extension of the NP-hard Johnson problem to the case of identical parallel machines at each stage is considered. When the number of stages is bounded by a constant and the number of machines is part of the input, a new polynomial time approximation scheme with an improved bound on the running time is designed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. E. Buten and V. Y. Shen, “A Scheduling Model for Computer Systems with Two Classes of Processors,” in Proceedings of the 1973 Sagamore Comput. Conference Parallel Processing (1973), pp. 130–138.

  2. B. Chen, “Scheduling Multiprocessor Flow Shops,” in Advances in Optimization and Approximation (Kluwer Acad., Dordrecht, 1994), pp. 1–8.

    Google Scholar 

  3. B. Chen, “Analysis of Classes of Heuristics for Scheduling a Two-Stage Flow Shop with Parallel Machines at One Stage,” J. Oper. Res. Soc. 46(2), 234–244 (1995).

    Article  MATH  Google Scholar 

  4. B. Chen, C. N. Potts, and G. J. Woeginger, “A Review of Machine Scheduling: Complexity, Algorithms and Approximability,” in Handbook of Combinatorial Optimization, Vol. 3 (Kluwer Acad., Boston, 1998), pp. 21–169.

    Google Scholar 

  5. M. R. Garey and D. S. Johnson, “Strong NP-Completeness Results: Motivation, Examples and Implications,” J. Assoc. Comput. Mach. 25(3), 499–508 (1978).

    MATH  MathSciNet  Google Scholar 

  6. M. R. Garey, D. S. Johnson, and R. Sethi, “The Complexity of Flowshop and Jobshop Scheduling,” Math. Oper. Res. 1(2), 117–129 (1976).

    MATH  MathSciNet  Google Scholar 

  7. J. N. D. Gupta, “Two-Stage Hybrid Flowshop Scheduling Problem,” J. Oper. Res. Soc. 39(4), 359–364 (1988).

    Article  MATH  Google Scholar 

  8. L. A. Hall, “Approximability of Flow Shop Scheduling,” Math. Programming, Series B 82(1–2), 175–190 (1998).

    MathSciNet  Google Scholar 

  9. D. S. Hochbaum and D. B. Shmoys, “Using Dual Approximation Algorithms for Scheduling Problems: Theoretical and Practical Results,” J. Assoc. Comput. Mach. 34(1), 144–162 (1987).

    MathSciNet  Google Scholar 

  10. J. A. Hoogeveen, J. K. Lenstra, and B. Veltman, “Preemptive Scheduling in a Two-Stage Multiprocessor Flow Shop is NP-Hard,” Eur. J. Oper. Res. 89(1–2), 172–175 (1996).

    MATH  Google Scholar 

  11. K. Jansen and M. I. Sviridenko, “Polynomial Time Approximation Schemes for the Multiprocessor Open and Flow Shop Scheduling Problem,” in Lecture Notes in Computer Science, Vol. 1770: 17th Annual Symposium on Theoretical Aspects of Computer Science, STACS 2000 (Springer, Berlin, 2000), pp. 455–565.

    Google Scholar 

  12. K. Jansen, R. Solis-Oba, and M. I. Sviridenko, “Makespan Minimization in Job Shops: a Linear Time Approximation Scheme,” SIAMJ. Discrete Math. 16(2), 288–300 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  13. C. Koulamas and G. J. Kyparisis, “Asymptotically Optimal Linear Time Algorithms for Two-Stage and Three-Stage Flexible Flow Shops,” Naval Res. Logist. 47(3), 259–268 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  14. M. A. Langston, “Interstage Transportation Planning in the Deterministic Flow-Shop Environment,” Oper. Res. 35(4), 556–564 (1987).

    Google Scholar 

  15. C. Y. Lee and G. L. Vairaktarakis, “Minimizing Makespan in Hybrid Flowshops,” Oper. Res. Lett. 16(3), 149–158 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  16. S. Sahni, “Algorithms for Scheduling Independent Tasks,” J. Assoc. Comput. Mach. 23(1), 116–127 (1976).

    MATH  MathSciNet  Google Scholar 

  17. P. Schuurman and G. J. Woeginger, “A Polynomial Time Approximation Scheme for the Two-Stage Multiprocessor Flow Shop Problem,” Theoret. Comput. Sci. 237(1–2), 105–122 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  18. S. V. Sevastyanov, “Some Generalizations of the Johnson Problem,” Upravlyaemye Sistemy 21, 45–61 (1981).

    MathSciNet  Google Scholar 

  19. S. V. Sevastyanov, “Construction of an Approximate Schedule for a Flow-Type System,” Upravlyaemye Sistemy 31, 66–71 (1993).

    Google Scholar 

  20. S. V. Sevastyanov, “Geometrical Heuristics for Multiprocessor Flowshop Scheduling with Uniform Machines at Each Stage,” J. Sched. 5(3), 205–225 (2002).

    Article  MathSciNet  Google Scholar 

  21. C. Sriskandarajah and S. P. Sethi, “Scheduling Algorithms for Flexible Flow Shops:Worst and Average Case Performance,” European J. Oper. Res. 43(2), 143–160 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  22. D. P. Williamson, L. A. Hall, J. A. Hoogeveen, C. A. J. Hurkens, J. K. Lenstra, S. V. Sevastyanov, and D. B. Shmoys, “Short Shop Schedules,” Oper. Res. 45(2), 288–294 (1997).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090, Russia

    S. V. Sevastyanov

Authors
  1. S. V. Sevastyanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. V. Sevastyanov.

Additional information

Original Russian Text © S.V. Sevastyanov, 2007, published in Diskretnyi Analiz i Issledovanie Operatsii, Ser. 1, 2007, Vol. 14, No. 2, pp. 25–46.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sevastyanov, S.V. An improved approximation scheme for the Johnson problem with parallel machines. J. Appl. Ind. Math. 2, 406–420 (2008). https://doi.org/10.1134/S1990478908030113

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1990478908030113

Keywords

Search

Navigation