Hostname: page-component-5c6d5d7d68-wbk2r Total loading time: 0 Render date: 2024-08-11T16:53:11.740Z Has data issue: false hasContentIssue false
  • English
  • Français

A simple proof for optimality of (s, S) policies in infinite-horizon inventory systems

Published online by Cambridge University Press:  14 July 2016

Yu-Sheng Zheng
Yu-Sheng Zheng*
Affiliation:
University of Pennsylvania
*
∗ Postal address: Decision Sciences Department, The Wharton School, University of Pennsylvania, Philadelphia, PA 19104, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

The classical proofs for the existence of a stationary (s, S) inventory policy that minimizes the total discounted or average cost over an infinite horizon are lengthy because they depend heavily on the optimality results for corresponding finite-horizon models. This note presents a simpler alternative. Since optimal stationary (s, S) policies are relatively simple to characterize, it is easy to construct a solution to the optimality equation which is satisfied by an (s, S) policy or an equivalent variant thereof. For the discounted model, the proof characterizes an (s, S) policy that is optimal for all initial inventory positions. This policy can be generated by a simple existing algorithm. For the average-cost model, the optimality proof is completed with some additional arguments, which are simple but novel, to overcome the normal difficulties encountered in models with unbounded one-step expected costs.

Keywords

DISCOUNTED COSTAVERAGE COSTUNBOUNDED ONE-STEP COST
Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 4 , December 1991 , pp. 802 - 810
Copyright
Copyright © Applied Probability Trust 1991 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bell, C. (1970) Improved algorithms for inventory and replacement stocking problems. SIAM J. Appl. Math. 18, 558–566.Google Scholar
[2] Bertsekas, D. (1976) Dynamic Programming and Stochastic Control. Academic Press, New York.Google Scholar
[3] Federgruen, A. and Zipkin, P. (1984) An efficient algorithm for computing optimal (s, S) policies. Operat. Res. 32, 1268–1285.10.1287/opre.32.6.1268Google Scholar
[4] Howard, R. (1960) Dynamic Programming and Markov Processes. Wiley, New York.Google Scholar
[5] Iglehart, D. (1963) Optimality of (s, S) policies in the infinite-horizon dynamic inventory problem. Management Sci. 9, 259–267.10.1287/mnsc.9.2.259Google Scholar
[6] Iglehart, D. (1963) Dynamic programming and stationary analysis of inventory problems. Chapter 1 in Multistage Inventory Models and Techniques, ed. Scarf, H., Gilford, D. and Shelly, M. Stanford University Press.Google Scholar
[7] Johnson, E. (1968) On (s, S) policies. Management Sci. 15, 80–101.10.1287/mnsc.15.1.80Google Scholar
[8] Küenle, C. and Küenle, H. (1977) Durchschnittsoptimale Strategien in markovschen Entscheidungsmodellen bei unbeschränkten Kosten. Math. Operationsforsch. Statist., Ser. Optimization 8, 549–564.10.1080/02331937708842450Google Scholar
[9] Ross, S. (1970) Applied Probability Models with Optimization Applications. Holden-Day, San Francisco.Google Scholar
[10] Ross, S. (1983) Introduction to Stochastic Dynamic Programming. Academic Press, New York.Google Scholar
[11] Scarf, H. (1960) The optimality of (s, S) policies in dynamic inventory problem. In Mathematical Models in Social Sciences, ed. Arrow, K., Karlin, S. and Suppes, P., Stanford University Press.Google Scholar
[12] Veinott, A. F. (1966) On the optimality of (s, S) inventory policies: new conditions and a new proof. SIAM J. Appl. Math. 14, 1067–1083.10.1137/0114086Google Scholar
[13] Veinott, A. F. and Wagner, H. (1965) Computing optimal (s, S) inventory policies. Management Sci. 11, 525–552.10.1287/mnsc.11.5.525Google Scholar
[14] Zheng, Y. S. and Federgruen, A. (1991) Finding optimal (s, S) policies is about as simple as evaluating a single policy. Operat. Res. 39, 654–665.10.1287/opre.39.4.654Google Scholar