Skip to main content Accessibility help
×
Home

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

  • Dacheng Yao (a1), Xiuli Chao (a2) and Jingchen Wu (a2)

Abstract

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).

Copyright

Corresponding author

Postal address: Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China. Email address: dachengyao@amss.ac.cn
∗∗ Postal address: Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI 48109-2117, USA. Email address: xchao@umich.edu

Footnotes

Hide All
∗∗∗

Current address: 500 9th Ave N, Seattle, WA 98109, USA. Email address: wjch@umich.edu

Footnotes

References

Hide All
[1] Bather, J. A. (1966). A continuous time inventory model. J. Appl. Prob. 3, 538549.
[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, 620636.
[3] Dai, J. G. and Yao, D. (2013). Brownian inventory models with convex holding cost, Part 1: Average-optimal controls. Stoch. Systems 3, 442499.
[4] Dai, J. G. and Yao, D. (2013). Brownian inventory models with convex holding cost, Part 2: Discount-optimal controls. Stoch. Systems 3, 500573.
[5] Dharmadhikari, S. and Joag-Dev, K. (1988). Unimodality, Convexity, and Applications. Academic Press, Boston, MA.
[6] Harrison, J. M. (1985). Brownian Motion and Stochastic Flow Systems. John Wiley, New York.
[7] Harrison, J. M. and Taylor, A. J. (1978). Optimal control of a Brownian storage system. Stoch. Process. Appl. 6, 179194.
[8] Harrison, J. M., Sellke, T. M. and Taylor, A. J. (1983). Impulse control of Brownian motion. Math. Operat. Res. 8, 454466.
[9] Hax, A. C. and Candea, D. (1984). Production and Inventory Management. Prentice-Hall, Englewood Cliffs, NJ.
[10] Heyman, D. P. and Sobel, M. J. (2004). Stochastic Models in Operations Research, Vol. I, Stochastic Processes and Operating Characteristics. Dover, Mineola, NY.
[11] Ormeci, M., Dai, J. G. and Vande Vate, J. (2008). Impulse control of Brownian motion: the constrained average cost case. Operat. Res. 56, 618629.
[12] Porteus, E. L. (1971). On the optimality of generalized (s, S) policies. Manag. Sci. 17, 411426.
[13] Porteus, E. L. (1972). On the optimality of generalized (s, S) policies under uniform demand densities. Manag. Sci. 18, 644646.
[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, 7991.
[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. 196202.
[16] Sulem, A. (1986). A solvable one-dimensional model of a diffusion inventory system. Math. Operat. Res. 11, 125133.
[17] Wu, J. and Chao, X. (2014). Optimal control of a Brownian production/inventory system with average cost criterion. Math. Operat. Res. 39, 163189.
[18] Zheng, Y.-S. (1992). On properties of stochastic inventory systems. Manag. Sci. 38, 87103.

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed