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Strongly Polynomial Algorithms for the High Multiplicity Scheduling Problem

Author

Listed:
  • Dorit S. Hochbaum

    (University of California, Berkeley, California)

  • Ron Shamir

    (Rutgers University, Piscataway, New Jersey)

Abstract

A high multiplicity scheduling problem consists of many jobs which can be partitioned into relatively few groups, where all the jobs within each group are identical. Polynomial, and even strongly polynomial, algorithms for the standard scheduling problem, in which all jobs are assumed to be distinct, become exponential for the corresponding high multiplicity problem. In this paper, we study various high multiplicity problems of scheduling unit-time jobs on a single machine. We provide strongly polynomial algorithms for constructing optimal schedules with respect to several measures of efficiency (completion time, lateness, tardiness, the number of tardy jobs and their weighted counterparts). The algorithms require a number of operations that are polynomial in the number of groups rather than in the total number of jobs. As a by-product, we identify a new family of nxn transportation problems which are solvable in O ( n log n ) time by a simple greedy algorithm.

Suggested Citation

  • Dorit S. Hochbaum & Ron Shamir, 1991. "Strongly Polynomial Algorithms for the High Multiplicity Scheduling Problem," Operations Research, INFORMS, vol. 39(4), pages 648-653, August.
    • Handle: RePEc:inm:oropre:v:39:y:1991:i:4:p:648-653
    DOI: 10.1287/opre.39.4.648
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    Citations

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    Cited by:

    1. Gianfranco Ciaschetti & Lorenzo Corsini & Paolo Detti & Giovanni Giambene, 2007. "Packet scheduling in third-generation mobile systems with UTRA-TDD air interface," Annals of Operations Research, Springer, vol. 150(1), pages 93-114, March.
    2. John J. Clifford & Marc E. Posner, 2000. "High Multiplicity in Earliness-Tardiness Scheduling," Operations Research, INFORMS, vol. 48(5), pages 788-800, October.
    3. N. Brauner & Y. Crama & A. Grigoriev & J. Klundert, 2005. "A Framework for the Complexity of High-Multiplicity Scheduling Problems," Journal of Combinatorial Optimization, Springer, vol. 9(3), pages 313-323, May.
    4. Alexander Grigoriev & Vincent J. Kreuzen & Tim Oosterwijk, 2021. "Cyclic lot-sizing problems with sequencing costs," Journal of Scheduling, Springer, vol. 24(2), pages 123-135, April.
    5. Sterna, Małgorzata, 2021. "Late and early work scheduling: A survey," Omega, Elsevier, vol. 104(C).
    6. Klaus Jansen & Roberto Solis-Oba, 2011. "A Polynomial Time OPT + 1 Algorithm for the Cutting Stock Problem with a Constant Number of Object Lengths," Mathematics of Operations Research, INFORMS, vol. 36(4), pages 743-753, November.
    7. C. T. Ng & T. C. E. Cheng & J. J. Yuan, 2006. "A note on the complexity of the problem of two-agent scheduling on a single machine," Journal of Combinatorial Optimization, Springer, vol. 12(4), pages 387-394, December.
    8. Omri Dover & Dvir Shabtay, 2016. "Single machine scheduling with two competing agents, arbitrary release dates and unit processing times," Annals of Operations Research, Springer, vol. 238(1), pages 145-178, March.
    9. A. Agnetis & S. Smriglio, 2000. "Optimal assignment of high multiplicity flight plans to dispatchers," Naval Research Logistics (NRL), John Wiley & Sons, vol. 47(5), pages 359-376, August.
    10. Christoph Hertrich & Christian Weiß & Heiner Ackermann & Sandy Heydrich & Sven O. Krumke, 2020. "Scheduling a proportionate flow shop of batching machines," Journal of Scheduling, Springer, vol. 23(5), pages 575-593, October.
    11. Sterna, Malgorzata, 2011. "A survey of scheduling problems with late work criteria," Omega, Elsevier, vol. 39(2), pages 120-129, April.
    12. Vergauwen, P.G.M.C. & Busser, K. & Rongen, P. & Verwaijen, R. & Vossen, D.J.L.H., 2001. "Cost accounting and pricing improvement at Helmond Print: using Xeikon digital colour printing equipment: a case study," Research Memorandum 028, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
    13. Selvarajah, Esaignani & Steiner, George, 2006. "Batch scheduling in a two-level supply chain--a focus on the supplier," European Journal of Operational Research, Elsevier, vol. 173(1), pages 226-240, August.
    14. S. Thomas McCormick & Scott R. Smallwood & Frits C. R. Spieksma, 2001. "A Polynomial Algorithm for Multiprocessor Scheduling with Two Job Lengths," Mathematics of Operations Research, INFORMS, vol. 26(1), pages 31-49, February.
    15. Cheng, T.C.E. & Shafransky, Y. & Ng, C.T., 2016. "An alternative approach for proving the NP-hardness of optimization problems," European Journal of Operational Research, Elsevier, vol. 248(1), pages 52-58.
    16. Rubing Chen & Jinjiang Yuan & Yuan Gao, 2019. "The complexity of CO-agent scheduling to minimize the total completion time and total number of tardy jobs," Journal of Scheduling, Springer, vol. 22(5), pages 581-593, October.
    17. Omri Dover & Dvir Shabtay, 2016. "Single machine scheduling with two competing agents, arbitrary release dates and unit processing times," Annals of Operations Research, Springer, vol. 238(1), pages 145-178, March.

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