Hostname: page-component-89b8bd64d-z2ts4 Total loading time: 0 Render date: 2026-05-07T09:32:15.380Z Has data issue: false hasContentIssue false
  • English
  • Français

(s, S) Inventory Systems with Random Lead Times: Harris Recurrence and Its Implications in Sensitivity Analysis

Published online by Cambridge University Press:  27 July 2009

Michael C. Fu  and
Jian-Qiang Hu
Michael C. Fu
Affiliation:
College of Business and Management, University of Maryland, College Park, Maryland 20742
Jian-Qiang Hu
Affiliation:
Manufacturing Engineering Department, Boston University, Boston, Massachusetts 02215
Get access Rights & Permissions [Opens in a new window]

Abstract

Most of the previous work on (s, S) inventory systems assumes that lead times for orders are such that orders never cross in time; i.e., the arrival of orders follows the same sequence as the placement of the orders. In this paper we consider more general mechanisms for random lead times. Because the introduction of a general random lead time mechanism makes the system essentially intractable for most performance measures of interest, simulation is a. natural candidate for estimating performance and/or optimizing the system. Two important issues in simulation are the stability and ergodicity of the system. Therefore, we first study some theoretical implications of the mechanism by providing conditions for which the system is stable and Harris ergodic, with the accompanying wide-sense regenerative properties. We then consider the problem of gradient estimation during simulation. Using the technique of perturbation analysis, we derive sample path-based gradient estimates for the finite-horizon average cost per period with respect to the parameters s and S and give a sample path proof of unbiasedness. We then show how stability and ergodicity can be used to simplify the estimators in the limiting infinite-horizon case and to establish strong consistency of the resulting estimators.

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 8 , Issue 3 , July 1994 , pp. 355 - 376
Copyright
Copyright © Cambridge University Press 1994

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.)

Article purchase

Temporarily unavailable

References

1.Asmussen, S. (1987). Applied probability and queues. New York: John Wiley & Sons.Google Scholar
2.Bashyam, S. & Fu, M. (1994). Application of perturbation analysis to a class of periodic review (s, S) inventory systems. Naval Research Logistics 41(1): 47–48.3.0.CO;2-I>CrossRefGoogle Scholar
3.Browne, S. & Zipkin, P. (1991). Inventory models with continuous stochastic demands. Annals of Applied Probability 1(3): 419–435.CrossRefGoogle Scholar
4.Fu, M.C. (1994). Sample path derivatives for (s, S) inventory systems. Operations Research 42(21): 351–364.CrossRefGoogle Scholar
5.Fu, M.C. (1994). Optimization via simulation: A review. Annals of Operations Research (to appear).CrossRefGoogle Scholar
6.Fu, M.C. & Hu, J.Q. (1992). Extensions and generalizations of smoothed perturbation analysis in a generalized semi-Markov process framework. IEEE Transactions on Automatic Control 37: 1483–1500.CrossRefGoogle Scholar
7.Glasserman, P., Hu, J.Q., & Strickland, S.G. (1991). Strongly consistent steady state derivative estimates. Probability in the Engineering and Informational Sciences 5(4): 391–413.CrossRefGoogle Scholar
8.Glasserman, P. & Tayur, S. (1992). Sensitivity analysis for base-stock levels in multi-echelon production-inventory systems. Manuscript.Google Scholar
9.Glasserman, P. & Tayur, S. (1992). The stability of a capacitated, multi-echelon production-inventory systems under a base-stock policy. Manuscript.Google Scholar
10.Gong, W.B. & Ho, Y.C. (1987). Smoothed perturbation analysis of discrete-event dynamic systems. IEEE Transactions on Automatic Control AC-32: 858–867.CrossRefGoogle Scholar
11.Hu, J.Q. & Strickland, S.G. (1990). Strong consistency of sample path derivative estimates. Applied Mathematics Letters 3(4): 55–58.CrossRefGoogle Scholar
12.Sahin, I. (1990). Regenerative inventory systems. New York: Springer-Verlag.CrossRefGoogle Scholar
13.Sigman, K. (1988). Queues as Harris recurrent Markov chains. Queueing Systems: Theory and Applications 3: 179–198.CrossRefGoogle Scholar
14.Sigman, K. (1988). Regeneration in tandem queues with multi-server stations. Journal of Applied Probability 25: 391–403.CrossRefGoogle Scholar
15.Walrand, I. (1988). Introduction to queueing networks. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
16.Whitt, W. (1972). Embedded renewal processes in the GI/G/s queue. Journal of Applied Probability 9: 650–658.CrossRefGoogle Scholar
17.Zheng, Y. & Federgruen, A. (1991). Finding optimal (s, S) policies is about as simple as evaluating a single policy. Operations Research 39: 654–665.CrossRefGoogle Scholar
18.Zipkin, P. (1986). Stochastic leadtimes in continuous-time inventory models. Naval Research Logistics Quarterly 33: 763–774.CrossRefGoogle Scholar