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Journal of Scheduling

, Volume 9, Issue 6, pp 559–568 | Cite as

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

  • Han Hoogeveen
  • Linda van Norden
  • Steef van de Velde
Papers

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.

Keywords

Completion Time Slack Variable Linear Programming Relaxation Total Completion Time Partial Schedule 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 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).CrossRefGoogle Scholar
  2. Cadambi, B. W. and Y. S. Sathe, “Two-machine flowshop scheduling to minimise mean flow time,” Opsearch, 30, 35–41 (1993).Google Scholar
  3. Conway, R. W., W. L. Maxwell, and L. W. Miller, Theory of Scheduling. Addison-Wesley, Reading, MA, (1967).Google Scholar
  4. 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).CrossRefGoogle Scholar
  5. 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).CrossRefGoogle Scholar
  6. 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
  7. 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
  8. 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
  9. Ignall, E., and L. E. Scharge, “Application of the branch and bound technique to some flow-shop problems,” Operations Research, 13, 400–412 (1965).CrossRefGoogle Scholar
  10. 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).CrossRefGoogle Scholar
  11. 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).CrossRefGoogle Scholar
  12. 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

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  • Han Hoogeveen
    • 1
  • Linda van Norden
    • 2
  • Steef van de Velde
    • 3
  1. 1.Institute of Information and Computing SciencesUtrecht UniversityUtrechtThe Netherlands
  2. 2.Faculty of Electrical Engineering, Mathematics and Computer Science, Department of Software TechnologyDelft University of TechnologyDelftThe Netherlands
  3. 3.Faculty of Business Administration/Rotterdam School of ManagementErasmus University RotterdamRotterdamThe Netherlands

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