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

Authors:
Han Hoogeveen;Linda Norden;Steef Velde
Affiliations:
Institute of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands 3508 TB;Faculty of Electrical Engineering, Mathematics and Computer Science, Department of Software Technology, Delft University of Technology, Delft, The Netherlands 2600 GA;Faculty of Business Administration/Rotterdam School of Management, Erasmus University Rotterdam, Rotterdam, The Netherlands 3000 DR
Venue:
Journal of Scheduling
Year:
2006

Citing 3
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Stronger Lagrangian bounds by use of slack variables: applications to machine scheduling problems

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Exact, Approximate, and Guaranteed Accuracy Algorithms for the Flow-Shop Problem n/2/F/ F

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A matheuristic approach for the total completion time two-machines permutation flow shop problem

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An assignment-based lower bound for a class of two-machine flow shop problems

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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.