Skip to main content
SpringerLink
Log in

Lower bounds for minimizing total completion time in a two-machine flow shop

  • Papers
  • Published:
Journal of Scheduling Aims and scope Submit manuscript

Abstract

For the \(\mathcal{NP}\)-hard problem of scheduling n jobs in a two-machine flow shop so as to minimize the total completion time, we present two equivalent lower bounds that are computable in polynomial time. We formulate the problem by the use of positional completion time variables, which results in two integer linear programming formulations with O(n 2) variables and O(n) constraints. Solving the linear programming relaxation renders a very strong lower bound with an average relative gap of only 0.8% for instances with more than 30 jobs. We further show that relaxing the formulation in terms of positional completion times by applying Lagrangean relaxation yields the same bound, no matter which set of constraints we relax.

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

  • Akkan, C. and S. Karabati, “The two-machine flowshop total completion time problem: Improved lower bounds and a branch-and-bound algorithm,” European Journal of Operational Research, 159, 420–429 (2004).

    Article  Google Scholar 

  • Cadambi, B. W. and Y. S. Sathe, “Two-machine flowshop scheduling to minimise mean flow time,” Opsearch, 30, 35–41 (1993).

    Google Scholar 

  • Conway, R. W., W. L. Maxwell, and L. W. Miller, Theory of Scheduling. Addison-Wesley, Reading, MA, (1967).

    Google Scholar 

  • Della Croce, F., M. Ghirardi, and R. Tadei, “An improved branch-and-bound algorithm for the two machine total completion time flow shop problem,” European Journal of Operational Research, 139, 293–301 (2002).

    Article  Google Scholar 

  • Della Croce, F., V. Narayan, and R. Tadei, “The two-machine total completion time flow shop problem,” European Journal of Operational Research, 90, 227–237 (1996).

    Article  Google Scholar 

  • Garey, M. R., D. S. Johnson, and R. Sethi, “The complexity of flowshop and jobshop scheduling,” Mathematics of Operations Research, 13, 330–348 (1976).

    Google Scholar 

  • Hoogeveen, J. A. and S. L. Van de Velde, “Stronger Lagrangian bounds by use of slack variables: applications to machine scheduling problems,” Mathematical Programming, 70, 173–190 (1995).

    Google Scholar 

  • Hoogeveen, J. A. and S. L. Van de Velde, “Scheduling by positional completion times: Analysis of a two-stage flow shop problem with a batching machine,” Mathematical Programming, 82, 273–289 (1998).

    Google Scholar 

  • Ignall, E., and L. E. Scharge, “Application of the branch and bound technique to some flow-shop problems,” Operations Research, 13, 400–412 (1965).

    Article  Google Scholar 

  • Kohler, W. H. and K. Steiglitz, “Exact, approximate and guaranteed accuracy algorithms for the flowshop problem n/2/F/F,” Journal ACM, 22, 106–114 (1975).

    Article  Google Scholar 

  • Sayin, S. and S. Karabati, “A bicriteria approach to the two-machine flow shop scheduling problem,” European Journal of Operational Research, 113, 435–449 (1999).

    Article  Google Scholar 

  • Van de Velde, S. L, “Minimizing the sum of the job completion times in the two-machine flow shop by Lagrangian relaxation,” Annals of Operations Research, 26, 257–268 (1990).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Information and Computing Sciences, Utrecht University, P.O. Box 80.089, 3508 TB, Utrecht, The Netherlands

    Han Hoogeveen

  2. Faculty of Electrical Engineering, Mathematics and Computer Science, Department of Software Technology, Delft University of Technology, P.O. Box 5031, 2600 GA, Delft, The Netherlands

    Linda van Norden

  3. Faculty of Business Administration/Rotterdam School of Management, Erasmus University Rotterdam, P.O. Box 1738, 3000 DR, Rotterdam, The Netherlands

    Steef van de Velde

Authors
  1. Han Hoogeveen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Linda van Norden
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Steef van de Velde
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Han Hoogeveen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hoogeveen, H., van Norden, L. & van de Velde, S. Lower bounds for minimizing total completion time in a two-machine flow shop. J Sched 9, 559–568 (2006). https://doi.org/10.1007/s10951-006-8789-x

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10951-006-8789-x

Keywords

Search

Navigation