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

Alain Bensoussan · R. H. Liu · SURESH P. SETHI

111 citations0/0 files storedSIAM Journal on Control and Optimization

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Supply Chain and Inventory Management Stochastic processes and financial applications Advanced Queuing Theory Analysis

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 examplefor the general case.

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References

1993Inventory Control in a Fluctuating Demand Environment 1978Existence of Optimal Simple Policies for Discounted-Cost Inventory and Cash Management in Continuous Time 1986Stochastic leadtimes in continuous‐time inventory models 1966A continuous time inventory model 1978(s, S) Policies Under Continuous Review and Discrete Compound Poisson Demand

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