Optimal policies for a deterministic continuous-time inventory model with several suppliers

This paper is concerned with determining the optimal inventory policy for an infinitehorizon deterministic continuous-time continuous-state inventory model, where, in the absence of intervention, changes in inventory level are governed by a differential evolution equation. The decision maker has the option of ordering from several suppliers, each of which entails differing ordering and purchasing costs. The objective is to select the supplier and the size of the order that minimizes the discounted cost over an infinite planning horizon. The optimal policy is formulated as the solution of a quasi-variational inequality. It is shown that there are three possibilities regarding its solvability: it has a unique solution that corresponds to an (s, S) policy; it does not admit a solution corresponding to an (s, S) policy but does have a unique solution that corresponds to a generalized (s, S) policy; or, it does not admit a solution corresponding to an (s, S) policy or a generalized (s, S) policy. A necessary and sufficient condition for each possibility is obtained. Examples illustrate their occurrence.