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On the (S – 1, S) Stock Model for Renewal Demand Processes: Poisson's poison

Published online by Cambridge University Press:  27 July 2009

Marcel A. J. Smith  and
Rommert Dekker
Marcel A. J. Smith
Affiliation:
Erasmus University Rotterdam, P.O. Box 1738, 300O DR Rotterdam, The Netherlands
Rommert Dekker
Affiliation:
Erasmus University Rotterdam, P.O. Box 1738, 300O DR Rotterdam, The Netherlands
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Abstract

In the standard (S – 1, S) stock model, demand follows a Poisson process. It has appeared to many stock analysts that this model causes an abundance of stock in reality. In case demand is caused by failure or is derived from another process, demand typically does not follow a Poisson process. In this paper, we discuss the (S – 1, S) stock model where demand follows a renewal process and the lead time is deterministic. Moreover, we will extend this to compound renewal demand and multi-echelon inventory systems. Our goal is to show the severe influence of taking the Poisson process for granted.

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 11 , Issue 3 , July 1997 , pp. 375 - 386
Copyright
Copyright © Cambridge University Press 1997

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References

1.Axsäter, S. (1993). Continuous review policies for multi-level inventory systems with stochastic demand. In Graves, S.C., Kan, A.H.G. Rinnooy, & Zipkin, P.H. (eds.), Logistics of production and inventory. Amsterdam: North-Holland, pp. 175–198.CrossRefGoogle Scholar
2.Barbour, A.D., Hoist, L., & Janson, S. (1992). Poisson approximation. Oxford Studies in Probability 2. Oxford: Clarendon Press.CrossRefGoogle Scholar
3.Cox, D.R. & Miller, H.D. (1965). The theory of stochastic processes. London: Chapman and Hall.Google Scholar
4.Graves, S.C. (1985). A multi-echelon inventory model for a repairable item with one-for-one replenishment. Management Science 31: 1247–1256.CrossRefGoogle Scholar
5.Harrison, P.G. (1990). Laplace transform inversion and passage-time distributions in Markov processes. Journal of Applied Probability 27: 74–87.CrossRefGoogle Scholar
6.Higa, I., Feyerherm, A.M., & Machado, A.L. (1975). Waiting time in an (S – 1, S) inventory system. Operations Research 23: 674–680.CrossRefGoogle Scholar
7.Iseger, P.W. den., Smith, M.A.J., & Dekker, R. (1996). Computing compound distributions faster! Research Report 9627, Econometric Institute, Erasmus University, Rotterdam.Google Scholar
8.Johnson, M.A. & Taaffe, M.R. (1989). Matching moments to phase distributions: Mixtures of Erlang distributions of common order. Stochastic Models 5: 711–743.CrossRefGoogle Scholar
9.Kalpakam, S. & Sapna, K.P. (1996). A lost sales (S – 1, S) perishable inventory system with renewal demand. Naval Research Logistics 43: 129–142.3.0.CO;2-D>CrossRefGoogle Scholar
10.Khintchine, A.Y. (1969). Mathematical methods in the theory of queueing. In Griffin's Statistical Monographs and Courses, Vol. 7. London: Griffin.Google Scholar
11.Lee, H.L. (1987). A multi-echelon inventory model for repairable items with emergence lateral transshipments. Management Science 33: 1302–1316.CrossRefGoogle Scholar
12.Muckstadt, J.A. (1973). A model for a multi-item, multi-echelon, multi identure inventory system. Management Science 20: 472–481.CrossRefGoogle Scholar
13.Nahmias, S. (1981). Managing repairable item inventory systems: A review. In Schwarz, L.B. (ed.), Multi-level production/inventory control systems. Studies in Management Science 16. Amsterdam: North-Holland, pp. 253–277.Google Scholar
14.Rooij, P.J. de. (1995). Stock control and spare parts. Master's thesis, Econometric Institute, Erasmus University, Rotterdam.Google Scholar
15.Schaeffer, M.K. (1983). A multi-item maintenance center inventory model for low-demand repairable items. Management Science 29: 1062–1068.CrossRefGoogle Scholar
16.Sherbrooke, C.C. (1968). METRIC: A multi-echelon technique for recoverable item control. Operations Research 16: 122–141.CrossRefGoogle Scholar
17.Sherbrooke, C.C. (1975). Waiting time in an (S – 1, S) inventory system–Constant service time case. Operations Research 23: 819–820.CrossRefGoogle Scholar
18.Sherbrooke, C.C. (1986). VARI-METRIC: Improved approximation for multi-indenture, multiechelon availability models. Operations Research 34: 311–319.CrossRefGoogle Scholar
19.Sherbrooke, C.C. & Feeney, G.J. (1966). The (S – 1, S) inventory policy under compound Poisson demand. Management Science 12: 391–411.Google Scholar
20.Tijms, H.C. (1986). Stochastic modelling and analyses: A computational approach. Chichester: John Wiley & Sons.Google Scholar