Scheduling unit jobs with compatible release dates on parallel machines with nonstationary speeds
Purchase on Springer.com
$29.95 / €24.95 / £19.95*
* Final gross prices may vary according to local VAT.
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.
- A. von Arnim and R. Schrader, The permutahedron of P 4-sparse posets, to appear in Discrete Applied Mathematics.
- 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.
- E. Balas, A linear characterization of permutation vectors, Management Science Research Report 364, Carnegie Mellon University, Pittsburgh, USA, 1975.
- -, On the facial structure of scheduling polyhedra, Mathematical Programming Study 24 (1985), 179–218.
- 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.
- A. Frank and É. Tardos, Generalized polymatroids and submodular flows, Mathematical Programming 42 (1988), 489–563.
- S. Fujishige, Submodular systems and related topics, Mathematical Programming Study 22 (1984), 113–131.
- -, Submodular Functions and Optimization, Annals of Discrete Mathematics, vol. 47, North-Holland, Amsterdam, 1991.
- P. Gaiha and S. K. Gupta, Adjacent vertices on a permutohedron, SIAM Journal of Applied Mathematics 32 (1977), 323–327.
- 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.
- 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.
- J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker, Complexity of machine scheduling problems, Annals of Discrete Mathematics 1 (1977), 343–362.
- 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.
- I. Meilijson and A. Tamir, Minimizing flow time on parallel identical processors with variable unit processing time, Operations Research 32 (1984), 440–448.
- G. L. Nemhauser and L. A. Wolsey, Integer and Combinatorial Optimization, John Wiley & Sons, 1988.
- 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.
- M. Queyranne, Polyhedral approaches to scheduling problems, seminar presented at the CORE, University of Louvain, Belgium, January 26, 1988.
- -, Structure of a simple scheduling polyhedron, Mathematical Programming 58 (1993), 263–285.
- M. Queyranne and A. S. Schulz, Polyhedral approaches to machine scheduling, Preprint 408/1994, Department of Mathematics, Technical University of Berlin, Berlin, Germany, 1994.
- M. Queyranne and Y. Wang, Single-machine scheduling polyhedra with precedence constraints, Mathematics of Operations Research 16 (1991), 1–20.
- -, On the convex hull of feasible solutions to certain combinatorial problems, Operations Research Letters 11 (1992), 1–11.
- R. Rado, An inequality, The Journal of the London Mathematical Society 27 (1952), 1–6.
- 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.
- A. Schrijver, Theory of Linear and Integer Programming, John Wiley & Sons, 1986.
- A. S. Schulz, Polyedrische Charakterisierung von Scheduling Problemen, Diploma Thesis, Department of Mathematics, Technical University of Berlin, Berlin, Germany, 1993.
- W. E. Smith, Various optimizers for single-stage production, Naval Research and Logistics Quarterly 3 (1956), 59–66.
- Scheduling unit jobs with compatible release dates on parallel machines with nonstationary speeds
- Book Title
- Integer Programming and Combinatorial Optimization
- Book Subtitle
- 4th International IPCO Conference Copenhagen, Denmark, May 29–31, 1995 Proceedings
- pp 307-320
- Print ISBN
- Online ISBN
- Series Title
- Lecture Notes in Computer Science
- Series Volume
- Series ISSN
- Springer Berlin Heidelberg
- Copyright Holder
- Additional Links
- Industry Sectors
- eBook Packages
- Author Affiliations
- 1. Faculty of Commerce, University of British Columbia, Main Mall, 2053, Vancouver, B. C., Canada
- 2. Technische Universität Berlin, Fachbereich Mathematik (MA 6-1), Straße des 17. Juni 136, D-10623, Berlin, Germany
To view the rest of this content please follow the download PDF link above.