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International Conference on Integer Programming and Combinatorial Optimization

IPCO 1995: Integer Programming and Combinatorial Optimization pp 307–320Cite as

Scheduling unit jobs with compatible release dates on parallel machines with nonstationary speeds

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 920))

Abstract

We consider the problem of nonpreemptively scheduling a set of jobs with identical processing requirements (unit jobs) on parallel machines with nonstationary speeds. In addition to the case of uniform machines, this allows for such predictable effects as operator learning and tool wear and tear, as well as such planned activities as machine upgrades, maintenance and the preassignment of other operations, all of which may affect the available processing speed of the machine at different points in time. We also allow release dates that satisfy a certain compatibility property. We show that the convex hull of feasible completion time vectors is a supermodular polyhedron. For nonidentical but compatible release dates, the supermodular function defining this polyhedron is the Dilworth truncation of a (non supermodular) function defined in a natural way from the release dates. This supermodularity result implies that the total weighted flow time can be minimized by a greedy algorithm. Supermodular polyhedra thus provide a general framework for several unit job, parallel machine scheduling problems and for their solution methods.

This work was done in part during mutual visits of the authors to each other's institution. This research was supported in part by a research grant of the Natural Sciences and Engineering Research Council (NSERC) of Canada to the first author, and by the graduate school “Algorithmische Diskrete Mathematik”. The latter is supported by the Deutsche Forschungsgemeinschaft, grant We 1265-1.

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References

  1. A. von Arnim and R. Schrader, The permutahedron of P 4-sparse posets, to appear in Discrete Applied Mathematics.

    Google Scholar 

  2. A. von Arnim and A. S. Schulz, Facets of the generalized permutahedron of a poset, Preprint 386/1994, Department of Mathematics, Technical University of Berlin, Berlin, Germany, 1994, to appear in Discrete Applied Mathematics.

    Google Scholar 

  3. E. Balas, A linear characterization of permutation vectors, Management Science Research Report 364, Carnegie Mellon University, Pittsburgh, USA, 1975.

    Google Scholar 

  4. -, On the facial structure of scheduling polyhedra, Mathematical Programming Study 24 (1985), 179–218.

    Google Scholar 

  5. J. L. Bruno, E. G. Coffman Jr., and R. Sethi, Scheduling independent tasks to reduce mean finishing time, Communications of the ACM 17 (1974), 382–387.

    Google Scholar 

  6. A. Frank and É. Tardos, Generalized polymatroids and submodular flows, Mathematical Programming 42 (1988), 489–563.

    Google Scholar 

  7. S. Fujishige, Submodular systems and related topics, Mathematical Programming Study 22 (1984), 113–131.

    Google Scholar 

  8. -, Submodular Functions and Optimization, Annals of Discrete Mathematics, vol. 47, North-Holland, Amsterdam, 1991.

    Google Scholar 

  9. P. Gaiha and S. K. Gupta, Adjacent vertices on a permutohedron, SIAM Journal of Applied Mathematics 32 (1977), 323–327.

    Google Scholar 

  10. J.-B. Lasserre and M. Queyranne, Generic scheduling polyhedra and a new mixed-integer formulation for single-machine scheduling, Integer Programming and Combinatorial Optimization (E. Balas, G. Cornuéjols, and R. Kannan, eds.), Carnegie Mellon University, 1992, Proceedings of the 2nd IPCO Conference, pp. 136–149.

    Google Scholar 

  11. E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, and D. B. Shmoys, Sequencing and scheduling: Algorithms and complexity, Logistics of Production and Inventory (S. C. Graves, A. H. G. Rinnooy Kan, and P. H. Zipkin, eds.), Handbooks in Operations Research and Management Science, vol. 4, North-Holland, Amsterdam, The Netherlands, 1993, pp. 445–522.

    Google Scholar 

  12. J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker, Complexity of machine scheduling problems, Annals of Discrete Mathematics 1 (1977), 343–362.

    Google Scholar 

  13. L. Lovász, Submodular functions and convexity, Mathematical Programming: The State of the Art — Bonn 1982 (A. Bachem, M. Grötschel, and B. Korte, eds.), Springer, 1983, pp. 235–257.

    Google Scholar 

  14. I. Meilijson and A. Tamir, Minimizing flow time on parallel identical processors with variable unit processing time, Operations Research 32 (1984), 440–448.

    Google Scholar 

  15. G. L. Nemhauser and L. A. Wolsey, Integer and Combinatorial Optimization, John Wiley & Sons, 1988.

    Google Scholar 

  16. W. R. Pulleyblank, Polyhedral combinatorics, Optimization (G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, eds.), Handbooks in Operations Research and Management Science, vol. 1, North-Holland, Amsterdam, The Netherlands, 1989, pp. 371–446.

    Google Scholar 

  17. M. Queyranne, Polyhedral approaches to scheduling problems, seminar presented at the CORE, University of Louvain, Belgium, January 26, 1988.

    Google Scholar 

  18. -, Structure of a simple scheduling polyhedron, Mathematical Programming 58 (1993), 263–285.

    Google Scholar 

  19. M. Queyranne and A. S. Schulz, Polyhedral approaches to machine scheduling, Preprint 408/1994, Department of Mathematics, Technical University of Berlin, Berlin, Germany, 1994.

    Google Scholar 

  20. M. Queyranne and Y. Wang, Single-machine scheduling polyhedra with precedence constraints, Mathematics of Operations Research 16 (1991), 1–20.

    Google Scholar 

  21. -, On the convex hull of feasible solutions to certain combinatorial problems, Operations Research Letters 11 (1992), 1–11.

    Google Scholar 

  22. R. Rado, An inequality, The Journal of the London Mathematical Society 27 (1952), 1–6.

    Google Scholar 

  23. P. H. Schoute, Analytical treatment of the polytopes regularly derived from the regular polytopes, Johannes Müller, Amsterdam, The Netherlands, 1911, Verhandelingen der Koninklijke Akademie van Wetenschappen XI.3.

    Google Scholar 

  24. A. Schrijver, Theory of Linear and Integer Programming, John Wiley & Sons, 1986.

    Google Scholar 

  25. A. S. Schulz, Polyedrische Charakterisierung von Scheduling Problemen, Diploma Thesis, Department of Mathematics, Technical University of Berlin, Berlin, Germany, 1993.

    Google Scholar 

  26. W. E. Smith, Various optimizers for single-stage production, Naval Research and Logistics Quarterly 3 (1956), 59–66.

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Commerce, University of British Columbia, Main Mall, 2053, Vancouver, B. C., Canada

    Maurice Queyranne

  2. Technische Universität Berlin, Fachbereich Mathematik (MA 6-1), Straße des 17. Juni 136, D-10623, Berlin, Germany

    Andreas S. Schulz

Authors
  1. Maurice Queyranne
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  2. Andreas S. Schulz
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Egon Balas Jens Clausen

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© 1995 Springer-Verlag Berlin Heidelberg

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Queyranne, M., Schulz, A.S. (1995). Scheduling unit jobs with compatible release dates on parallel machines with nonstationary speeds. In: Balas, E., Clausen, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 1995. Lecture Notes in Computer Science, vol 920. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-59408-6_60

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  • DOI: https://doi.org/10.1007/3-540-59408-6_60

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-59408-6

  • Online ISBN: 978-3-540-49245-0

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