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Optimal Control Policies for Stochastic Inventory Systems with Endogenous Supply

Published online by Cambridge University Press:  27 July 2009

A. Federgruen  and
Y.-S. Zheng
A. Federgruen
Affiliation:
Graduate School of Business, Columbia University, New York, New York 10027
Y.-S. Zheng
Affiliation:
Department of Decision Sciences, Wharton School, University of PennsylvaniaPhiladelphia, Pennsylvania 19104
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Abstract

We consider an inventory system with compound Poisson demands replenished by discrete production of units on a single-server facility. This facility may start a vacation at any production completion epoch; at the completion of a vacation the inventory level is inspected to decide whether or not to resume production. Unit production and vacation times are independent and identically distributed with general distributions. Under an (s, S) policy, production is terminated when the inventory level reaches a critical level S, and production is resumed when the inventory level, upon completion of a vacation, is at or below a prespecified value s. In this paper we prove that an (s, S) policy is optimal among all possible policies. We also derive a highly efficient algorithm for the determination and evaluation of an optimal (s, S) policy.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 7 , Issue 2 , April 1993 , pp. 257 - 272
Copyright
Copyright © Cambridge University Press 1993

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References

1.Altiok, T. (1989). (R, r) production/inventory systems. Operations Research 37(2): 266–276.CrossRefGoogle Scholar
2.Arrow, K., Harris, T., & Marschak, J. (1951). Optimal inventory policy. Econometrica 19: 250–272.CrossRefGoogle Scholar
3.Boxma, O. (1989). Workloads and waiting times in single-server systems with multiple customer classes. Queueing Systems, Theory and Applications 5: 185–214.CrossRefGoogle Scholar
4.Chaudhry, M. & Templeton, J. (1983). A first course in bulk queues. New York: John Wiley Interscience.Google Scholar
5.Doshi, B.T. (1986). Queueing systems with vacations – A survey. Queueing Systems 1: 29–66.CrossRefGoogle Scholar
6.Federgruen, A. & So, K. (1989). Optimal time to repair a broken server. Advances in Applied Probability 21:376–397.CrossRefGoogle Scholar
7.Federgruen, A. & So, K. (1991). Optimality of threshold policies in single server queueing systems with server vacations. Advances in Applied Probability 23: 388–405.CrossRefGoogle Scholar
8.Federgruen, A. & Zheng, Y. (1992). Characterization and efficient computation of optimal policies for general endogenously supplied inventory systems, (unabridged version of this paper). Working Paper. Graduate School of Business, Columbia University, New York, 10027.Google Scholar
9.Federgruen, A. & Zipkin, P. (1984). An efficient algorithm for computing optimal (s, S) policies. Operations Research 34: 1268–1285.CrossRefGoogle Scholar
10.Fuhrmann, S. & Cooper, R. (1985). Stochastic decompositions in the M/G/1 queue with server vacations. Operations Research 33: 1117–1129.CrossRefGoogle Scholar
11.Gavish, B. & Graves, S.C. (1980). A one product production/inventory problem under continuous review policy. Operations Research 28: 1228–1236.CrossRefGoogle Scholar
12.Gavish, B. & Graves, S.C. (1981). Production/inventory systems with a stochastic production rate under a continuous review policy. Computers and Operations Research 8(3): 169–183.CrossRefGoogle Scholar
13.Heyman, D.P. & Sobel, M.J. (1982). Stochastic models in operations research, Vol. I. New York: McGraw-Hill.Google Scholar
14.Iglehart, D. (1963). Dynamic programming and stationary analysis inventory problems. In Scarf, H., Guilford, D., & Shelly, M. (eds.), Multi-stage inventory models and techniques. Stanford, CA: Stanford University Press, Ch. 1.Google Scholar
15.Johnson, E. (1968). On (s, S) policies. Management Science 15: 80–101.CrossRefGoogle Scholar
16.Kuenle, C. & Kuenle, H. (1977). Durchschnittsoptimale Strategien in Markovschen Entscheidungsmodellen bei Unbeschraenkten Kosten. Mathematische Operations forschung and Statistik Serie Optimisation 8: 549–564.CrossRefGoogle Scholar
17.Lee, H. & Nahmias, S. (1993). Single product, single location models. In Graves, S., Kan, A. Rinnooy, & Zipkin, P. (eds.), Handbook in operations research and management science. Vol. 4: Logistics of production inventory. Amsterdam: North Holland, Ch. 2.Google Scholar
18.Scarf, H. (1960). The optimality of (s, S) policies in the dynamic inventory problem. In Arrow, K., Karlin, S., & Suppes, P. (eds.), Mathematical methods in the social sciences. Stanford, CA: Stanford University Press, pp. 196–202.Google Scholar
19.Sobel, M.J. (1969). Optimal average-cost policy for a queue with start-up and shut-down costs. Operations Research 17: 145–162.CrossRefGoogle Scholar
20.Srinivasan, M. & Lee, H. (1991). The (s, S) policy for random review production/inventory systems with compound Poisson demands and arbitrary processing times. Management Science 37: 813–833.CrossRefGoogle Scholar
21.Takagi, H. (1987). Queueing analysis of vacation models, Part I: M/G/1; Part II: M/G/1 with vacations. Technical Report TR87-O032. IBM Tokyo Research Laboratory, Tokyo, Japan.Google Scholar
22.Tijms, H.C. (1980). An algorithm for denumerable state semi-Markov decision problems with application to controlled production and queueing systems. In Hartley, R. et al. (eds.), Recent developments in Markov decision theory. New York: Academic Press, pp. 143–179.Google Scholar
23.Tijms, H.C. (1986). Stochastic modelling and analysis. Chichester, England: John Wiley.Google Scholar
24.Veinott, A. (1966). On the optimally of (s, S) inventory policies: New condition and a proof. SIAM Journal on Applied Mathematics 14: 1067–1083.CrossRefGoogle Scholar
25.Veinott, A. & Wagner, H. (1965). Computing optimal (s, S) inventory policies. Management Science 11: 525–552.CrossRefGoogle Scholar
26.Weber, R. & Stidham, S. (1987). Optimal control of service rates in networks of queues. Advances in Applied Probability 19: 202–218.CrossRefGoogle Scholar
27.Wolff, R.W. (1989). Stochastic modeling and the theory of queues. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
28.Zheng, Y. (1991). A simple proof for the optimality of (s, S) policies for infinite horizon inventory problems. Journal of Applied Probability 28: 802–810.CrossRefGoogle Scholar
29.Zheng, Y. & Federgruen, A. (1991). Finding optimal (s, S) policies is about as simple as evaluating a single policy. Operations Research 39: 654–666.CrossRefGoogle Scholar
30.Zipkin, P. (1989). Lecture notes in inventory theory. Graduate School of Business, Columbia University, New York.Google Scholar