Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-24T04:43:27.373Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal control policy for a Brownian inventory system with concave ordering cost

Part of: Operations research and management science

Published online by Cambridge University Press:  30 March 2016

Dacheng Yao ,
Xiuli Chao  and
Jingchen Wu
Dacheng Yao*
Affiliation:
Chinese Academy of Sciences
Xiuli Chao*
Affiliation:
University of Michigan
Jingchen Wu
Affiliation:
University of Michigan
*
∗Postal address: Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China. Email address: [email protected]
∗∗Postal address: Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI 48109-2117, USA. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we consider an inventory system with increasing concave ordering cost and average cost optimization criterion. The demand process is modeled as a Brownian motion. Porteus (1971) studied a discrete-time version of this problem and under the strong condition that the demand distribution belongs to the class of densities that are finite convolutions of uniform and/or exponential densities (note that normal density does not belong to this class), an optimal control policy is a generalized (s, S) policy consisting of a sequence of (si, Si). Using a lower bound approach, we show that an optimal control policy for the Brownian inventory model is determined by a single pair (s, S).

Keywords

Brownian inventory system(s, S) policyconcave ordering cost

MSC classification

Primary: 90B05: Inventory, storage, reservoirs
Secondary: 90B30: Production models
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 4 , December 2015 , pp. 909 - 925
Copyright
Copyright © Applied Probability Trust 2015 

Footnotes

∗∗∗

Current address: 500 9th Ave N, Seattle, WA 98109, USA. Email address: [email protected]

References

[1] Bather, J. A. (1966). A continuous time inventory model. J. Appl. Prob. 3, 538–549.CrossRefGoogle Scholar
[2] Constantinides, G. M. and Richard, S. F. (1978). Existence of optimal simple policies for discounted-cost inventory and cash management in continuous time. Operat. Res. 26, 620–636.CrossRefGoogle Scholar
[3] Dai, J. G. and Yao, D. (2013). Brownian inventory models with convex holding cost, Part 1: Average-optimal controls. Stoch. Systems 3, 442–499.Google Scholar
[4] Dai, J. G. and Yao, D. (2013). Brownian inventory models with convex holding cost, Part 2: Discount-optimal controls. Stoch. Systems 3, 500–573.Google Scholar
[5] Dharmadhikari, S. and Joag-Dev, K. (1988). Unimodality, Convexity, and Applications. Academic Press, Boston, MA.Google Scholar
[6] Harrison, J. M. (1985). Brownian Motion and Stochastic Flow Systems. John Wiley, New York.Google Scholar
[7] Harrison, J. M. and Taylor, A. J. (1978). Optimal control of a Brownian storage system. Stoch. Process. Appl. 6, 179–194.Google Scholar
[8] Harrison, J. M., Sellke, T. M. and Taylor, A. J. (1983). Impulse control of Brownian motion. Math. Operat. Res. 8, 454–466.Google Scholar
[9] Hax, A. C. and Candea, D. (1984). Production and Inventory Management. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
[10] Heyman, D. P. and Sobel, M. J. (2004). Stochastic Models in Operations Research, Vol. I, Stochastic Processes and Operating Characteristics. Dover, Mineola, NY.Google Scholar
[11] Ormeci, M., Dai, J. G. and Vande Vate, J. (2008). Impulse control of Brownian motion: the constrained average cost case. Operat. Res. 56, 618–629.Google Scholar
[12] Porteus, E. L. (1971). On the optimality of generalized (s, S) policies. Manag. Sci. 17, 411–426.CrossRefGoogle Scholar
[13] Porteus, E. L. (1972). On the optimality of generalized (s, S) policies under uniform demand densities. Manag. Sci. 18, 644–646.Google Scholar
[14] Richard, S. F. (1977). Optimal impulse control of a diffusion process with both fixed and proportional costs of control. SIAM J. Control Optimization 15, 79–91.Google Scholar
[15] Scarf, H. (1960). The optimality of (s, S) policies in the dynamic inventory problem. In Mathematical Methods in the Social Sciences, 1959, Stanford University Press, pp. 196–202.Google Scholar
[16] Sulem, A. (1986). A solvable one-dimensional model of a diffusion inventory system. Math. Operat. Res. 11, 125–133.CrossRefGoogle Scholar
[17] Wu, J. and Chao, X. (2014). Optimal control of a Brownian production/inventory system with average cost criterion. Math. Operat. Res. 39, 163–189.Google Scholar
[18] Zheng, Y.-S. (1992). On properties of stochastic inventory systems. Manag. Sci. 38, 87–103.CrossRefGoogle Scholar