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Stochastic scheduling problems I — General strategies

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Abstract

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.

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References

  • Bauer, H.: Wahrscheinlichkeitstheorie. De Gruyter, Berlin 1968.

    Google Scholar 

  • Bertsekas, D.P., andS.E. Shreve: Stochastic Optimal Control: The Discrete Time Case. Academic Press, New York 1978.

    Google Scholar 

  • Billingsley, P.: Convergence of Probability Measures. Wiley, New York 1968.

    Google Scholar 

  • Conway, R.W., W.L. Maxwell, andL.W. Miller: Theory of Scheduling. Addison-Wesley, Reading, MA. 1967.

    Google Scholar 

  • 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 

  • 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 

  • Elmaghraby, S.E.: Activity Networks. Wiley, New York 1977.

    Google Scholar 

  • 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 

  • Fishburn, P.C.: Utility Theory for Decision Making. Wiley, New York 1970.

    Google Scholar 

  • 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.

    MathSciNet  Google Scholar 

  • Halmos, P.R.: Measure Theory. Van Nostrand Reinhold, New York 1950.

    Google Scholar 

  • Hansel, G., andJ.P. Troallic: Measures marginales et theorème de Ford-Fulkerson. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete43, 1978, 245–251.

    Article  Google Scholar 

  • Heller, U.: On the shortest overall duration in stochastic project networks. Methods of Oper. Res.42, 1981, 85–104.

    Google Scholar 

  • Hinderer, K.: Foundation of Non-stationary Dynamic Programming with Discrete Time Parameter. Springer Verlag, Berlin 1970.

    Google Scholar 

  • —: Grundbegriffe der Wahrscheinlichkeitstheorie. Springer Verlag, Berlin 1972.

    Google Scholar 

  • Igelmund, G., andF.J. Radermacher: Preselective strategies for the optimization of stochastic project networks under resource constraints. Networks13, 1983a, 1–29.

    Google Scholar 

  • —: Algorithmic approaches to preselective strategies for stochastic scheduling problems. Networks13, 1983b, 29–48.

    Google Scholar 

  • Jakobs, K.: Measure and Integral. Academic Press, New York 1978.

    Google Scholar 

  • Jurecka, W.: Netzwerkplanung im Baubetrieb. Teil 2: Optimierungsverfahren. Bauverlag GmbH, Wiesbaden 1972.

    Google Scholar 

  • Kadelka, D.: On randomized policies and mixtures of deterministic policies. Methods of Oper. Res.46, 1983, 67–75.

    Google Scholar 

  • Kamae, T., U. Krengel, andO'Brien: Stochastic inequalities on partially ordered spaces. Ann. Probability5, 1977, 899–912.

    Google Scholar 

  • 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.

  • 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.

  • Meilijson, I., andA. Nadas: Convex majorization with an application to the length of critical paths. J. Appl. Prob.16, 1979, 671–677.

    Google Scholar 

  • Moder, J.J., andC.R. Phillips: Project management with CPM and PERT. Reinhold, New York 1964.

    Google Scholar 

  • Möhring, R.H.: Scheduling problems with a singular solution, Ann. of Discr. Math.16, 1982, 225–239.

    Google Scholar 

  • —: Minimizing Costs of resource requirements subject to a fixed completion time in project networks. Oper. Res.32, 1984, 89–102.

    Google Scholar 

  • 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 

  • Möhring, R.H., F.J. Radermacher, andG. Weiss: Stochastic scheduling problems II: set strategies. (Next volume).

  • -: Stochastic scheduling problems III: tractable cases. (In preparation).

  • Neumann, K.: Operations Research Verfahren. Band III, Carl Hanser Verlag, München 1975.

    Google Scholar 

  • 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 

  • Radermacher, F.J.: Kapazitätsoptimierung in Netzplänen. Math. Syst. in Econ.40, Anton Hain, Meisenheim 1978.

    Google Scholar 

  • —: Cost-dependent essential systems of ES-strategies for stochastic scheduling problems. Methods of Oper. Res.42, 1981, 17–31.

    Google Scholar 

  • —: Optimale Strategien für stochastische Scheduling Probleme. Habilitationsschrift, RWTH Aachen 1981. In: Schriften zur Informatik und Angewandten Mathematik98, RWTH Aachen 1984.

    Google Scholar 

  • Raiffa, H.: Einführung in die Entscheidungstheorie, Oldenbourg Verlag, München 1973.

    Google Scholar 

  • Rinnooy Kan, A.H.G.: Machine Scheduling Problems: Classification, Complexity and Computation. Nijhoff, The Hague 1976.

    Google Scholar 

  • Rival, I.: Assembly lines have no timing anomalies. Preprint, 1984.

  • Ross, S.M.: Applied Probability Models with Optimization Applications. Holden-Day, San Francisco 1970.

    Google Scholar 

  • —: Introduction to Probability Models (3d. ed.). Academic Press, New York 1985.

    Google Scholar 

  • Royden, H.L.: Real Analysis. The Macmillan Company, London 1968.

    Google Scholar 

  • Schneeweiss, H.: Entscheidungskriterien bei Risiko. Springer Verlag, Berlin 1967.

    Google Scholar 

  • Seeling, R.: Reihenfolgenprobleme in Netzplänen, Bauwirtschaft, 1972, 1897–1904.

  • Shogan, A.W.: Bounding distributions for stochastic PERT networks. Networks 7, 1977, 359–381.

    Google Scholar 

  • Spelde, H.G.: Stochastische Netzpläne und ihre Anwendung im Baubetrieb, Dissertation, RWTH Aachen 1976.

    Google Scholar 

  • —: Bounds for the distribution function of network variables, Methods of Oper. Res.27, 1977, 113–123.

    Google Scholar 

  • Stoyan, D.: Comparison methods for queues and other stochastic models. John Wiley, Chichester 1983.

    Google Scholar 

  • Strassen, V.: The existence of probability measures with given marginals. Ann. Math. Statistics36, 1965, 423–439.

    Google Scholar 

  • Strauch, R.: Negative dynamic programming. Ann. Math. Statist.37, 1966, 871–890.

    Google Scholar 

  • 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 

  • Wald, A.: Statistical Decisions Functions. Chelsea, New York 1950.

    Google Scholar 

  • Weber, R.R.: Optimal Organization of Multserver Systems, Ph. D. thesis, Unviersity of Cambridge 1979.

  • 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.

  • 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 

  • -: 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.

  • 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 

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Authors and Affiliations

  1. Fachbereich Informatik, Hochschule Hildesheim, D-3200, Hildesheim

    R. H. Möhring

  2. Lehrstuhl für Informatik und Operations Research, University of Passau, D-8390, Passau

    F. J. Radermacher

  3. Department of Statistics, Tel-Aviv University, Ramat-Aviv, Israel

    G. Weiss

Authors
  1. R. H. Möhring
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  2. F. J. Radermacher
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  3. G. Weiss
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Möhring, R.H., Radermacher, F.J. & Weiss, G. Stochastic scheduling problems I — General strategies. Zeitschrift für Operations Research 28, 193–260 (1984). https://doi.org/10.1007/BF01919323

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