Mathematical Programming

, Volume 58, Issue 1–3, pp 263–285 | Cite as

Structure of a simple scheduling polyhedron

  • Maurice Queyranne
Article

Abstract

In a one-machine nonpreemptive scheduling problem, the feasible schedules may be defined by the vector of the corresponding job completion times. For given positive processing times, the associated simple scheduling polyhedronP is the convex hull of these feasible completion time vectors. The main result of this paper is a complete description of the minimal linear system definingP. We also give a complete, combinatorial description of the face lattice ofP, and a simple, O(n logn) separation algorithm. This algorithm has potential usefulness in cutting plane type algorithms for more difficult scheduling problems.

Key words

Scheduling polyhedron polyhedra scheduling 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Bachem and M. Grötschel, “Characterizations of adjacency of faces of polyhedra,”Mathematical Programming Study 14 (1981) 1–22.Google Scholar
  2. A. Bachem and M. Grötschel, “New aspects of polyhedral theory,” in: B. Korte, ed.,Modern Applied Mathematics — Optimization and Operations Research (North-Holland, Amsterdam, 1982) pp. 51–106.Google Scholar
  3. E. Balas, “On the facial structure of scheduling polyhedra,”Mathematical Programming Study 24 (1985) 179–218.Google Scholar
  4. S.P. Bansal, “Single machine scheduling to minimize weighted sum of completion times with secondary criterion — a branch and bound approach,”European Journal of Operational Research 5 (1980) 177–181.Google Scholar
  5. M. Barbut, ed.,Ordres Totaux Finis (Mouton, Paris, 1971).Google Scholar
  6. M. Ben-Or, “Lower bounds for algebraic computation trees,”Proceedings 15th ACM Annual Symposium on Theory of Computing (May 1983) 80–86.Google Scholar
  7. J.P. Benzécri, “Sur l'analyse des préférences,” in: M. Barbut, ed. (1971).Google Scholar
  8. M. Dyer and L.A. Wolsey, “Formulating the single machine sequencing problem with release dates as a mixed integer program,”Discrete Applied Mathematics 26 (1990) 255–270.Google Scholar
  9. J. Edmonds, “Submodular functions, matroids and certain polyhedra,” in: R. Guy, ed.,Combinatorial Structures and Their Applications (Gordon and Breach, New York, 1970) pp. 68–87.Google Scholar
  10. R.E. Fox, “OPT—an answer for America,”Inventories and Production, Nov.–Dec. 1982, 10–19.Google Scholar
  11. S. Fujishige, “Submodular systems and related topics,”Mathematical Programming Study 22 (1984) 113–131.Google Scholar
  12. G. Grätzer,General Lattice Theory (Academic Press, New York, 1978).Google Scholar
  13. M. Grötschel, L. Lovász and A. Schrijver, “The ellipsoid method and its consequences in combinatorial optimization,”Combinatorica 1 (1981) 169–197.Google Scholar
  14. B. Grünbaum,Convex Polytopes (Wiley, London, 1967).Google Scholar
  15. G. Th. Guilbaud and P. Rosenstiehl, “Analyse algébrique d'un scrutin,” in: M. Barbut, ed. (1971).Google Scholar
  16. K. Jordan,Calculus of Finite Differences (Chelsea, New York, 1947).Google Scholar
  17. G. Kreweras, “Représentation polyédrique des préordres complets finis,” in: M. Barbut, ed. (1971).Google Scholar
  18. E.L. Lawler, “Recent results in the theory of machine scheduling,” in: A. Bachem, M. Grötschel and B. Korte, eds.,Mathematical Programming: The State of the Art — Bonn 1982 (Springer, Berlin, 1983) pp. 202–234.Google Scholar
  19. E.L. Lawler, J.K. Lenstra and A.H.G. Rinnooy Kan, “Recent developments in deterministic sequencing and scheduling: a survey,” in: M.A.H. Dempster, J.K. Lenstra and A.H.G. Rinnooy Kan, eds.,Deterministic and Stochastic Scheduling (Reidel, Dordrecht, 1982) pp. 35–73.Google Scholar
  20. E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan and D.B. Shmoys, “Sequencing and scheduling: Algorithms and complexity,” Report BS-8909, Centre for Mathematics and Computer Science (Amsterdam, 1989).Google Scholar
  21. J.K. Lenstra and A.H.G. Rinnooy Kan, “New directions in scheduling theory,”Operations Research Letters 2 (1984) 255–259.Google Scholar
  22. J.K. Lenstra and A.H.G. Rinnooy Kan, “Scheduling theory since 1981: An annotated bibliography,” Report BW 188, Centre for Mathematics and Computer Science (Amsterdam, 1983).Google Scholar
  23. L. Lovász, “Submodular functions and convexity,” in: A. Bachem, M. Grötschel and B. Korte, eds.,Mathematical Programming: The State of the Art — Bonn 1982 (Springer, Berlin, 1983) pp. 235–257.Google Scholar
  24. J.-C. Picard and M. Queyranne, “Selected applications of minimum cuts in networks,”INFOR 20 (1982) 394–422.Google Scholar
  25. M.E. Posner, “Minimizing weighted completion times with deadlines,”Operations Research 33 (1985) 562–574.Google Scholar
  26. W.R. Pulleyblank, “Polyhedral Combinatorics,” in: A. Bachem, M. Grötschel and B. Korte, eds.,Mathematical Programming: The State of the Art — Bonn 1982 (Springer, Berlin, 1983) pp. 101–123.Google Scholar
  27. F.P. Preparata and M.I. Shamos,Computational Geometry: An Introduction (Springer, New York, 1985).Google Scholar
  28. M. Queyranne, “Structure of a nonpreemptive one-machine scheduling polyhedron,” TIMS/ORSA Joint National Meeting, New Orleans, May 1987.Google Scholar
  29. M. Queyranne and Y. Wang, “Single-machine scheduling polyhedra with precedence constraints,”Mathematics of Operations Research 16 (1991) 1–20.Google Scholar
  30. R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).Google Scholar
  31. W.E. Smith, “Various optimizers for single-stage production,”Naval Research and Logistics Quarterly 3 (1956) 59–66.Google Scholar
  32. J. Stoer and C. Witzgall,Convexity and Optimization in Finite Dimensions I (Springer, Berlin, 1970).Google Scholar
  33. P. Tseng, private communication (July 1986).Google Scholar
  34. A.C. Yao and R.L. Rivest, “On the polyhedral decision problem,”SIAM Journal on Computing 9 (1980) 343–347.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1993

Authors and Affiliations

  • Maurice Queyranne
    • 1
  1. 1.University of British ColumbiaVancouverCanada

Personalised recommendations