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A Double Band Control Policy of a Brownian Perishable Inventory System

Published online by Cambridge University Press:  27 July 2009

David Perry
David Perry
Affiliation:
Department of Statistics, The University of Haifa, Haifa, Israel, 31905
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Abstract

The blood bank system is a typical example of a perishable inventory system. The commodity arrival and customer demand processes are stochastic. However, the stored items have a constant lifetime. In this study, we introduce a diffusion approximation to this system. The stock level is represented by the amount of items arriving during the age of the oldest item; it is assumed to fluctuate as an alternating two-sided regulated Brownian motion between barriers 0 and 1. Hittings of level 0 are outdatings and hittings of level 1 are unsatisfied demands. Also, there are two predetermined switchover levels, a and b, with 0 ≤ a < b ≤ 1. Whenever the stock level process upcrosses level b, the controller generates a switch in the drift from γ = γ0 to γ = γ1, while downcrossings of level a generate switches from γ1 to γ0. A useful martingale is introduced for analyzing the stationary law of the controlled process as well as the total expected discounted cost.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 11 , Issue 3 , July 1997 , pp. 361 - 373
Copyright
Copyright © Cambridge University Press 1997

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References

1.Bather, J. (1966). A continuous time inventory model. Journal of Applied Probability 3: 538–549.CrossRefGoogle Scholar
2.Bather, J. (1968). A diffusion model for the control of a dam. Journal of Applied Probability 5: 55–71.CrossRefGoogle Scholar
3.Chung, K.L. & Williams, R.J. (1990). Introduction to stochastic integral, 2nd ed.Boston: Birkhauser.CrossRefGoogle Scholar
4.Cohen, J.W. (1982). The single server queue, 2nd ed.Amsterdam: North-Holland.Google Scholar
5.Foschini, G.J. (1977). On heavy traffic diffusion analysis and dynamic routing in packet switched networks. In Chandy, K.M. and Reiser, M. (eds.), Computer performance. Amsterdam: North-Holland, pp. 419–514.Google Scholar
6.Harrison, J.M. (1985). Brownian motion and stochastic flow system. New York: Wiley.Google Scholar
7.Kaspi, H. & Perry, D. (1983). Inventory system of perishable commodities. Advances in Applied Probability 15: 674–685.CrossRefGoogle Scholar
8.Kella, O. & Whitt, W. (1992). A useful martingale for stochastic storage process with Lévy input. Journal of Applied Probability 29: 396–403.CrossRefGoogle Scholar
9.Nahmias, S. (1982). Perishable inventory theory: A review. Operations Research 30: 680–708.CrossRefGoogle ScholarPubMed
10.Newell, G.F. (1979). Approximate Behavior of Tandem Queues, Lecture Notes in Economics and Mathematical Systems, 171. New York: Springer-Verlag.CrossRefGoogle Scholar
11.Newell, G.F. (1982). Application of queueing theory. London: Chapman & Hall.CrossRefGoogle Scholar
12.Perry, D. & Asmussen, S. (1995). Rejection rules in the M/G/l type queue. Queueing Systems 19: 105–130.CrossRefGoogle Scholar
13.Perry, D. & Bar-Lev, S.K. (1989). A control of Brownian storage system with two switchover drifts. Stochastic Analysis and Its Application 7: 103–115.CrossRefGoogle Scholar
14.Perry, D. & Posner, M.J.M. (1990). Control of input and demand rates in inventory systems of perishable commodities. Naval Research Logistics 37: 85–97.3.0.CO;2-F>CrossRefGoogle Scholar
15.Rath, J. (1975). Controlled queues in heavy traffic. Advances in Applied Probability 7: 656–671.CrossRefGoogle Scholar
16.Rath, J. (1977). The optimal policy for a controlled Brownian motion process. SIAM Journal on Applied Mathematics 32: 115–125.CrossRefGoogle Scholar
17.Taylor, H.M. (1975). A stopped Brownian motion formula. The Annals of Probability 3: 234–246.CrossRefGoogle Scholar
18.Whitt, W. (1974). Heavy traffic limit theorems for queues: A survey. In Clarke, A.B. (ed.), Mathematical Methods in Queueing Theory, Lecture Notes in Economics and Mathematical Systems, 98. New York: Springer-Verlag, pp. 307–350.CrossRefGoogle Scholar
19.Zuckerman, D. (1977). Two stage output procedure of a finite dam. Journal of Applied Probability 14: 421–425.CrossRefGoogle Scholar
20.Zuckerman, D. (1979). N-stage output procedure of a finite dam. Zeitschrift für Operations Research 12: 179–187.Google Scholar