A Measure Approach for Continuous Inventory Models: Discounted Cost Criterion

This paper develops a new approach to the solution of impulse control problems for continuous inventory models under a discounted cost criterion. The analysis imbeds this stochastic problem in two different infinite-dimensional linear programs, parametrized by the initial inventory level $x_0$, by concentrating on particular functions and capturing the expected (discounted) behavior of the inventory level process and ordering decisions as measures. The first imbedding then naturally leads to the minimization of a nonlinear function representing the cost associated with an $(s,S)$ ordering policy and an optimizing pair determines optimal levels $(s^*,S^*)$. The lower bound arising from this imbedding is tight when $x_0 \geq s^*$ but is a strict lower bound when $x_0 < s^*$. Solving the first linear program determines the value function in the “no order” region and is critical to the formulation of the second linear program. The dual of the second linear program is then solved to provide a tight lower bound...