Skip to main content

Optimal policy for brownian inventory models with general convex inventory cost

Abstract

We study an inventory system in which products are ordered from outside to meet demands, and the cumulative demand is governed by a Brownian motion. Excessive demand is backlogged. We suppose that the shortage and holding costs associated with the inventory are given by a general convex function. The product ordering from outside incurs a linear ordering cost and a setup fee. There is a constant leadtime when placing an order. The optimal policy is established so as to minimize the discounted cost including the inventory cost and ordering cost.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    Bar-Ilan, A., Sulem, A. Explicit solution of inventory problems with delivery lags. Mathematics of Operations Research, 20: 709–720 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    Bather, J.A. A continuous time inventory model. Journal of Applied Probability, 3: 538–549 (1966)

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    Benkherouf, L., Bensoussan, A. Optimality of an (s, S) policy with compound Poisson and diffusion demands: a quasi-variational inequalities approach. SIAM Journal of Control and Optimization, 48: 756–762 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    Bensoussan, A., Liu, R.H., Sethi, S.P. Optimality of an (s, S) policy with compound Poisson and diffusion demands: a quasi-variational inequalities approach. SIAM Journal of Control and Optimization, 44: 1650–1676 (2005)

    MathSciNet  Article  Google Scholar 

  5. [5]

    Constantinides, G., Richard, S. Existence of optimal simple policies for discounted-cost inventory and cash management in continous time. Operations Research, 26: 620–636 (1978)

    MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    Dai, J.G., Yao, D. Brownian inventory models with convex holding cost: part 1 average-optimal controls. Stochastic Systems, revised, avilable at http://arxiv.org/abs/1110.2831 (2012)

    Google Scholar 

  7. [7]

    Dai, J.G., Yao, D. Brownian inventory models with convex holding cost: part 2 discount-optimal controls. Stochastic Systems, revised, avilable at http://arxiv.org/abs/1110.6572 (2012)

    Google Scholar 

  8. [8]

    Harrison, J.M., Sellke, T.M., Taksar, M.I. Impulse control of Brownian motion. Mathematics of Operations Research, 8: 454–466 (1983)

    MathSciNet  Article  MATH  Google Scholar 

  9. [9]

    Jacod, J., Shiryaev, A.N. Limit Theorems for Stochastic Processes, 2nd ed. Springer, Berlin, 2003

    Book  MATH  Google Scholar 

  10. [10]

    Korn, R. Optimal impulse control when control actions have random consequences. Mathematics of Operations Research, 22: 639–667 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    Ormeci, M., Dai, J.G., Vande Vate, J. Impulse control of Brownian motion: the constrained average cost case. Operations Research, 56: 618–629 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    Richard, S. Optimal impulse control of a diffusion process with both fixed and proportional costs of control. SIAM Journal of Control and Optimization, 15: 79–91 (1977)

    MathSciNet  Article  MATH  Google Scholar 

  13. [13]

    Sulem, A. A solvable one-dimensional model of a diffusion inventory system. Mathematics of Operations Research, 11: 125–133 (1986)

    MathSciNet  Article  MATH  Google Scholar 

  14. [14]

    Wu, J., Chao, X. Optimal control of a Brownian production/inventory system with average cost criterion. preprint (2012)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Da-cheng Yao.

Additional information

Supported by the National Natural Science Foundation of China (No. 11101050).

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Yao, Dc. Optimal policy for brownian inventory models with general convex inventory cost. Acta Math. Appl. Sin. Engl. Ser. 29, 187–200 (2013). https://doi.org/10.1007/s10255-013-0201-y

Download citation

Keywords

  • stochastic inventory model
  • Brownian motion
  • (sS) policy
  • impulse control

2000 MR Subject Classification

  • 60J70
  • 90B05
  • 93E20