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 HoogeveenEmail author
  • Linda van Norden
  • Steef van de Velde


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.


Completion Time Slack Variable Linear Programming Relaxation Total Completion Time Partial Schedule 
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Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  • Han Hoogeveen
    • 1
    Email author
  • 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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