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

Authors:
Alain Bensoussan;R. H. Liu;Suresh P. Sethi
Affiliations:
-;-;-
Venue:
SIAM Journal on Control and Optimization
Year:
2005

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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.