Optimality of an (s, S) Policy with Compound Poisson and Diffusion Demands: A Quasi-Variational Inequalities Approach

SIAM Journal on Control & Optimization, Vol. 44, No. 5, pp. 1650-1676, 2006

28 Pages Posted: 26 Feb 2008 Last revised: 14 Apr 2008

See all articles by Alain Bensoussan

Alain Bensoussan

University of Texas at Dallas - Naveen Jindal School of Management; Hong Kong Polytechnic University - Faculty of Business; Ajou University

R. H. Liu

University of Dayton - Department of Mathematics

Suresh Sethi

University of Texas at Dallas - Naveen Jindal School of Management

Abstract

We prove that an (s, S)policy is optimal in a continuous-review stochastic inventory model with a fixed ordering cost when the demand is a mixture of (i) a diffusion process and a compound Poisson process with exponentially distributed jump sizes, and (ii) a constant demand and a compound Poisson process. The proof uses the theory of impulse control. The Bellman equation of dynamic programming for such a problem reduces to a set of quasi-variational inequalities (QVI). An analytical study of the QVI leads to showing the existence of an optimal policy as well as the optimality of an (s, S) policy. Finally, the combination of a diffusion and a general compound Poisson demand is not completely solved. We explain the difficulties and what remains open. We also provide a numerical example for the general case.

Keywords: (s, S) policy, compound Poisson process, diffusion process, economic order quantity model, impulse control, quasi-variational inequalities, stochastic inventory model, EOQ Model

JEL Classification: C61, M11

Suggested Citation

Bensoussan, Alain and Liu, R. H. and Sethi, Suresh, Optimality of an (s, S) Policy with Compound Poisson and Diffusion Demands: A Quasi-Variational Inequalities Approach. SIAM Journal on Control & Optimization, Vol. 44 Issue 5, 1650-1676, 2006. Available at SSRN: https://ssrn.com/abstract=1097452

Alain Bensoussan (Contact Author)

University of Texas at Dallas - Naveen Jindal School of Management ( email )

800 West Campbell Rd
SM 30
Richardson, TX 75080-3021
United States
9728836117 (Phone)

HOME PAGE: http://www.utdallas.edu/~axb046100/

Hong Kong Polytechnic University - Faculty of Business

Dept SEEM
Systems Engr * Engr Mgmt
Hong Kong, Hong Kong
China

Ajou University ( email )

Ajou
France

R. H. Liu

University of Dayton - Department of Mathematics ( email )

300 College Park
Dayton, OH 45469
United States
(937) 229-1021 (Phone)
(937) 229-2566 (Fax)

HOME PAGE: http://academic.udayton.edu/ruihualiu/

Suresh Sethi

University of Texas at Dallas - Naveen Jindal School of Management ( email )

800 W. Campbell Road, SM30
Richardson, TX 75080-3021
United States

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