Zeitschrift für Operations Research

, Volume 28, Issue 7, pp 193–260 | Cite as

Stochastic scheduling problems I — General strategies

  • R. H. Möhring
  • F. J. Radermacher
  • G. Weiss


The paper contains an introduction to recent developments in the theory of non-preemptive stochastic scheduling problems. The topics covered are: arbitrary joint distributions of activity durations, arbitrary regular measures of performance and arbitrary precedence and resource constraints. The possible instability of the problem is demonstrated and hints are given on stable classes of strategies available, including the combinatorial vs. analytical characterization of such classes. Given this background, the main emphasis of the paper is on the monotonicity behaviour of the model and on the existence of optimal strategies. Existing results are presented and generalized, in particular w.r.t. the cases of lower semicontinuous performance measures or joint duration distributions having a Lebesgue density.

Key words

Analytic behaviour of strategies continuous strategies ES strategies MES strategies machine idleness monotonicity behaviour optimal strategies preselectivity regular measure of performance scheduling problems stability stochastic dynamic optimization stochastically ordered distributions stochastic scheduling 


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  1. Bauer, H.: Wahrscheinlichkeitstheorie. De Gruyter, Berlin 1968.Google Scholar
  2. Bertsekas, D.P., andS.E. Shreve: Stochastic Optimal Control: The Discrete Time Case. Academic Press, New York 1978.Google Scholar
  3. Billingsley, P.: Convergence of Probability Measures. Wiley, New York 1968.Google Scholar
  4. Conway, R.W., W.L. Maxwell, andL.W. Miller: Theory of Scheduling. Addison-Wesley, Reading, MA. 1967.Google Scholar
  5. Dempster, M.A.H.: A stochastic approach to hierarchical planning and scheduling. In: Dempster, M.A.H. et al. (eds.). Deterministic and Stochastic Scheduling. D. Reidel Publishing Company, Dord-Dortrecht 1982, 271–296.Google Scholar
  6. Dempster, M.A.H., J.K. Lenstra, andA.H.G. Rinnooy Kan (eds.): Deterministic and Stochastic Scheduling. D. Reidel Publishing Company, Dordrecht, 1982.Google Scholar
  7. Elmaghraby, S.E.: Activity Networks. Wiley, New York 1977.Google Scholar
  8. Esary, J.D., F. Proschan, andD.W. Walkup: Association of random variables with applications. Annals of Mathematical Statistics, Vol.38, 1967, 1466–1474.Google Scholar
  9. Fishburn, P.C.: Utility Theory for Decision Making. Wiley, New York 1970.Google Scholar
  10. Graham, R.L., E.L. Lawler, J.K. Lenstra, andA.H.G. Rinnooy Kan: Optimization and approximation in deterministic sequencing and scheduling: a survey. Annals of Discrete Math.5, 1979, 287–326.MathSciNetGoogle Scholar
  11. Halmos, P.R.: Measure Theory. Van Nostrand Reinhold, New York 1950.Google Scholar
  12. Hansel, G., andJ.P. Troallic: Measures marginales et theorème de Ford-Fulkerson. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete43, 1978, 245–251.CrossRefGoogle Scholar
  13. Heller, U.: On the shortest overall duration in stochastic project networks. Methods of Oper. Res.42, 1981, 85–104.Google Scholar
  14. Hinderer, K.: Foundation of Non-stationary Dynamic Programming with Discrete Time Parameter. Springer Verlag, Berlin 1970.Google Scholar
  15. —: Grundbegriffe der Wahrscheinlichkeitstheorie. Springer Verlag, Berlin 1972.Google Scholar
  16. Igelmund, G., andF.J. Radermacher: Preselective strategies for the optimization of stochastic project networks under resource constraints. Networks13, 1983a, 1–29.Google Scholar
  17. —: Algorithmic approaches to preselective strategies for stochastic scheduling problems. Networks13, 1983b, 29–48.Google Scholar
  18. Jakobs, K.: Measure and Integral. Academic Press, New York 1978.Google Scholar
  19. Jurecka, W.: Netzwerkplanung im Baubetrieb. Teil 2: Optimierungsverfahren. Bauverlag GmbH, Wiesbaden 1972.Google Scholar
  20. Kadelka, D.: On randomized policies and mixtures of deterministic policies. Methods of Oper. Res.46, 1983, 67–75.Google Scholar
  21. Kamae, T., U. Krengel, andO'Brien: Stochastic inequalities on partially ordered spaces. Ann. Probability5, 1977, 899–912.Google Scholar
  22. Kaerkes, R., R.H. Möhring, W. Oberschelp, F.J. Radermacher, andM.M. Richter: Netzplanoptimierung: Deterministische und stochastische Scheduling-Probleme über geordneten Strukturen. Springer Verlag, to appear in 1985.Google Scholar
  23. Klein Haneveld, W.K.: Distributions with known marginals and duality of mathematical programming with applications to PERT, Preprint 90 (OR-8203). Interfaculateit der Actuariele Wetenschappen en Econometrie, University of Groningen, 1982.Google Scholar
  24. Meilijson, I., andA. Nadas: Convex majorization with an application to the length of critical paths. J. Appl. Prob.16, 1979, 671–677.Google Scholar
  25. Moder, J.J., andC.R. Phillips: Project management with CPM and PERT. Reinhold, New York 1964.Google Scholar
  26. Möhring, R.H.: Scheduling problems with a singular solution, Ann. of Discr. Math.16, 1982, 225–239.Google Scholar
  27. —: Minimizing Costs of resource requirements subject to a fixed completion time in project networks. Oper. Res.32, 1984, 89–102.Google Scholar
  28. Möhring, R.H., andF.J. Radermacher: Introduction to stochastic scheduling problems. In: Neumann, K., Pallaschke, D. (Eds.): Contributions to Operations Research. Proceedings of the Oberwolfach conference on Operations Research. Springer-Verlag, Heidelberg 1985.Google Scholar
  29. Möhring, R.H., F.J. Radermacher, andG. Weiss: Stochastic scheduling problems II: set strategies. (Next volume).Google Scholar
  30. -: Stochastic scheduling problems III: tractable cases. (In preparation).Google Scholar
  31. Neumann, K.: Operations Research Verfahren. Band III, Carl Hanser Verlag, München 1975.Google Scholar
  32. Pinedo, M., andL. Schrage: Stochastic shop scheduling: a survey. In: Dempster, M.A.H. et al. (eds.). Deterministic and Stochastic Scheduling, D. Reidel Publishing Company, Dordrecht, 1982, 181–196.Google Scholar
  33. Radermacher, F.J.: Kapazitätsoptimierung in Netzplänen. Math. Syst. in Econ.40, Anton Hain, Meisenheim 1978.Google Scholar
  34. —: Cost-dependent essential systems of ES-strategies for stochastic scheduling problems. Methods of Oper. Res.42, 1981, 17–31.Google Scholar
  35. —: Optimale Strategien für stochastische Scheduling Probleme. Habilitationsschrift, RWTH Aachen 1981. In: Schriften zur Informatik und Angewandten Mathematik98, RWTH Aachen 1984.Google Scholar
  36. Raiffa, H.: Einführung in die Entscheidungstheorie, Oldenbourg Verlag, München 1973.Google Scholar
  37. Rinnooy Kan, A.H.G.: Machine Scheduling Problems: Classification, Complexity and Computation. Nijhoff, The Hague 1976.Google Scholar
  38. Rival, I.: Assembly lines have no timing anomalies. Preprint, 1984.Google Scholar
  39. Ross, S.M.: Applied Probability Models with Optimization Applications. Holden-Day, San Francisco 1970.Google Scholar
  40. —: Introduction to Probability Models (3d. ed.). Academic Press, New York 1985.Google Scholar
  41. Royden, H.L.: Real Analysis. The Macmillan Company, London 1968.Google Scholar
  42. Schneeweiss, H.: Entscheidungskriterien bei Risiko. Springer Verlag, Berlin 1967.Google Scholar
  43. Seeling, R.: Reihenfolgenprobleme in Netzplänen, Bauwirtschaft, 1972, 1897–1904.Google Scholar
  44. Shogan, A.W.: Bounding distributions for stochastic PERT networks. Networks 7, 1977, 359–381.Google Scholar
  45. Spelde, H.G.: Stochastische Netzpläne und ihre Anwendung im Baubetrieb, Dissertation, RWTH Aachen 1976.Google Scholar
  46. —: Bounds for the distribution function of network variables, Methods of Oper. Res.27, 1977, 113–123.Google Scholar
  47. Stoyan, D.: Comparison methods for queues and other stochastic models. John Wiley, Chichester 1983.Google Scholar
  48. Strassen, V.: The existence of probability measures with given marginals. Ann. Math. Statistics36, 1965, 423–439.Google Scholar
  49. Strauch, R.: Negative dynamic programming. Ann. Math. Statist.37, 1966, 871–890.Google Scholar
  50. Sullivan, R.S., andJ.C. Hayya: A comparison of the method of bounding distributions (MBD) and Monte Carlo simulation for analyzing stochastic acyclic networks. Oper. Res.28, 1980, 614–617.Google Scholar
  51. Wald, A.: Statistical Decisions Functions. Chelsea, New York 1950.Google Scholar
  52. Weber, R.R.: Optimal Organization of Multserver Systems, Ph. D. thesis, Unviersity of Cambridge 1979.Google Scholar
  53. Weber, R.R., P. Varaiya, andJ. Walrand: Scheduling jobs with stochastically ordered processing times on parallel machines to minimize expected flowtime. Memorandum No. UCB/ERL M84/57, Univ. of California 1984.Google Scholar
  54. Weiss, G.: Multiserver stochastic scheduling. In: Dempster, M.A.H. et al. (eds.). Deterministic and Stochastic Scheduling. D. Reidel Publishing Company, Dordrecht 1982a, 157–179.Google Scholar
  55. -: Stochastic bounds on distributions of optimal value functions with applications to PERT networks, flows and reliability. Techn. Report, Lehrstuhl für Informatik IV, Techn. Univ. of Aachen 1982b.Google Scholar
  56. Weiss, G., andM. Pinedo: Scheduling tasks with exponential service times on non-identical processors to minimize various cost functions. J. Appl. Probability17, 1980, 187–202.Google Scholar

Copyright information

© Physica-Verlag 1984

Authors and Affiliations

  • R. H. Möhring
    • 1
  • F. J. Radermacher
    • 2
  • G. Weiss
    • 3
  1. 1.Fachbereich InformatikHochschule HildesheimHildesheim
  2. 2.Lehrstuhl für Informatik und Operations ResearchUniversity of PassauPassau
  3. 3.Department of StatisticsTel-Aviv UniversityRamat-AvivIsrael

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