Decision Processes in Continuous Time

  • Martin J. Beckmann
Part of the Ökonometrie und Unternehmensforschung book series (ÖKONOMETRIE, volume 9)


Consider a system whose output rate decreases with time until corrective action (maintenance, replacement) is taken. Let

t denote the time elapsed since the last action,

a(t) the output rate



Dynamic Program Continuous Time Inventory Level Preventive Maintenance Terminal Condition 
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References and Selected Reading to Part Four

  1. Barlow, R. E., F. Proschan, and L. C. Huraen: Mathematical Theory of Reliability. London: Wiley 1965.Google Scholar
  2. Barlow, R. E., and L. Hunter: Optimum Preventive Maintenance Policies. OR 8, 1, 90–100 (1960).CrossRefGoogle Scholar
  3. Beckenbach, E. F. (ed.): Modern Mathematics for the Engineer. 2nd series. New York: McGraw Hill 1961.Google Scholar
  4. Beckmann, M., and R. Muth: An Inventory Policy for a Case of Lagged Delivery. MS 2, 2, 145–155 (1956).Google Scholar
  5. Beckmann, M.: An Inventory Model for Arbitrary Interval and Quantity Distributions of Demand. MS 8, 1, 35–57 (1961).Google Scholar
  6. Bellman, R. E.: Equipment Replacement Policy. SIAM 3, 133–136 (1955).Google Scholar
  7. Bellman, R. E.: Dynamic Programming Ch. XII, 245–282. Princeton: Princeton University Press 1957.Google Scholar
  8. Bellman, R. E.: Some Applications of the Theory of Dynamic Programming. A Review. OR 2, 3, 275–288 (1954).CrossRefGoogle Scholar
  9. Bellman, R. E., I. L. Glicksberg, and O. A. Gross: Some Aspects of the Mathematical Theory of Control Processes. Santa Monica: RAND Corp. R-313, 1958.Google Scholar
  10. Bellman, R. E., and St. E. Dreyfus: Applied Dynamic Programming. Ch. V, 180–206. Princeton: Princeton University Press 1962.Google Scholar
  11. Bellman, R. E.: (ed.): Mathematical Optimization Techniques. Berkeley and Los Angeles: University of California Press 1963.Google Scholar
  12. Bellman, R. E., and R. Kalaba: Dynamic Programming and Modern Control Theory. New York: Academic Press 1965.Google Scholar
  13. Cartaino, T. F., and S. E. Dreyfus: Application of Dynamic Programming to the Airplane Minimum Time-to-climb Problem. Aeronautical Engineering Review 16, 6, 74–77 (1957).Google Scholar
  14. Descamps, R.: Remplacement et Programmation Dynamique. Métra 4, 2, 181–199 (1965).Google Scholar
  15. Dreyfus, S. E.: A Generalized Equipment Replacement Study. SIAM 8, 3, 425–435 (1960).Google Scholar
  16. Dreyfus, S. E.: Dynamic Programming and the Calculus of Variations. New York–London: Academic Press 1965.Google Scholar
  17. Feller, W.: An Introduction to Probability Theory and its Applications. II, p. 178. New York: Wiley 1966.Google Scholar
  18. Jorgenson, D. W., Mccall and Radner: Optimal Replacement Polity. Amsterdam: North Holland Publ. 1967.Google Scholar
  19. Maffei, R. B.: Planning Advertising Expenditures by Dynamic Programming Methods. Management Technology 1, 2, 94–100 (1960).Google Scholar
  20. Meij, J. L.: Depreciation and Replacement Policy. Amsterdam: North Holland Publishing Comp. 1961.Google Scholar
  21. Pontryagin, L. S. et al.: The Mathematical Theory of Optimal Processes. New York: Wiley 1962.Google Scholar
  22. Radner, R., and D. W. Jorgenson: Opportunistic Replacement of a Single Part in the Presence of Several Monitored Parts. RAND Corp., RM–3057–PR, Nov. 1962.Google Scholar
  23. Weiss, G. H.: On the Theory of Replacement of Machinery with a Random Failure Time, NRLQ 3, 279–293 (1956).CrossRefGoogle Scholar
  24. White, D. J.: Optimal Revision Periods, Journal Math. Anal. Appl. 4, 353–365 (1962).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 1968

Authors and Affiliations

  • Martin J. Beckmann
    • 1
    • 2
  1. 1.Institut für Ökonometrie und UnternehmensforschungUniversität BonnGermany
  2. 2.Brown UniversityProvidenceUSA

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