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Decompositions, Network Flows, and a Precedence Constrained Single-Machine Scheduling Problem

Author

Listed:
  • François Margot

    (GSIA, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, Pennsylvania 15213)

  • Maurice Queyranne

    (Faculty of Commerce, University of British Columbia, Vancouver, British Columbia, Canada)

  • Yaoguang Wang

    (PeopleSoft, Inc., Pleasanton, California 94566)

Abstract

We present an in-depth theoretical, algorithmic, and computational study of a linear programming (LP) relaxation to the precedence constrained single-machine scheduling problem 1|prec|(Sigma) j w j C j to minimize a weighted sum of job completion times. On the theoretical side, we study the structure of tight parallel inequalities in the LP relaxation and show that every permutation schedule that is consistent with Sidney's decomposition has total cost no more than twice the optimum. On the algorithmic side, we provide a parametric extension to Sidney's decomposition and show that a finest decomposition can be obtained by essentially solving a parametric minimum-cut problem. Finally, we report results obtained by an algorithm based on these developments on randomly generated instances with up to 2,000 jobs.

Suggested Citation

  • François Margot & Maurice Queyranne & Yaoguang Wang, 2003. "Decompositions, Network Flows, and a Precedence Constrained Single-Machine Scheduling Problem," Operations Research, INFORMS, vol. 51(6), pages 981-992, December.
    • Handle: RePEc:inm:oropre:v:51:y:2003:i:6:p:981-992
    DOI: 10.1287/opre.51.6.981.24912
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    References listed on IDEAS

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    1. Jeffrey B. Sidney, 1975. "Decomposition Algorithms for Single-Machine Sequencing with Precedence Relations and Deferral Costs," Operations Research, INFORMS, vol. 23(2), pages 283-298, April.
    2. Maurice Queyranne & Yaoguang Wang, 1991. "Single-Machine Scheduling Polyhedra with Precedence Constraints," Mathematics of Operations Research, INFORMS, vol. 16(1), pages 1-20, February.
    3. Leslie A. Hall & Andreas S. Schulz & David B. Shmoys & Joel Wein, 1997. "Scheduling to Minimize Average Completion Time: Off-Line and On-Line Approximation Algorithms," Mathematics of Operations Research, INFORMS, vol. 22(3), pages 513-544, August.
    4. C. N. Potts, 1985. "A Lagrangean Based Branch and Bound Algorithm for Single Machine Sequencing with Precedence Constraints to Minimize Total Weighted Completion Time," Management Science, INFORMS, vol. 31(10), pages 1300-1311, October.
    5. Jean-Claude Picard, 1976. "Maximal Closure of a Graph and Applications to Combinatorial Problems," Management Science, INFORMS, vol. 22(11), pages 1268-1272, July.
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    Cited by:

    1. Rostami, Salim & Creemers, Stefan & Leus, Roel, 2019. "Precedence theorems and dynamic programming for the single-machine weighted tardiness problem," European Journal of Operational Research, Elsevier, vol. 272(1), pages 43-49.
    2. Felix Happach & Lisa Hellerstein & Thomas Lidbetter, 2022. "A General Framework for Approximating Min Sum Ordering Problems," INFORMS Journal on Computing, INFORMS, vol. 34(3), pages 1437-1452, May.
    3. José R. Correa & Martin Skutella & José Verschae, 2012. "The Power of Preemption on Unrelated Machines and Applications to Scheduling Orders," Mathematics of Operations Research, INFORMS, vol. 37(2), pages 379-398, May.
    4. Robbert Fokkink & Thomas Lidbetter & László A. Végh, 2019. "On Submodular Search and Machine Scheduling," Management Science, INFORMS, vol. 44(4), pages 1431-1449, November.
    5. Tanaka, Shunji & Sato, Shun, 2013. "An exact algorithm for the precedence-constrained single-machine scheduling problem," European Journal of Operational Research, Elsevier, vol. 229(2), pages 345-352.
    6. José R. Correa & Andreas S. Schulz, 2005. "Single-Machine Scheduling with Precedence Constraints," Mathematics of Operations Research, INFORMS, vol. 30(4), pages 1005-1021, November.
    7. Andreas S. Schulz & Nelson A. Uhan, 2011. "Near-Optimal Solutions and Large Integrality Gaps for Almost All Instances of Single-Machine Precedence-Constrained Scheduling," Mathematics of Operations Research, INFORMS, vol. 36(1), pages 14-23, February.
    8. Christoph Ambühl & Monaldo Mastrolilli & Nikolaus Mutsanas & Ola Svensson, 2011. "On the Approximability of Single-Machine Scheduling with Precedence Constraints," Mathematics of Operations Research, INFORMS, vol. 36(4), pages 653-669, November.

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