INVENTORY PLANNING IN A DETERMINISTIC ENVIRONMENT: CONCAVE COST SET-UP.

The purpose of this paper is to present an original approach to the dynamic inventory planning problem in discrete time with concave costs, under (1) the general assumption of non-stationary and (2) with any non-negative initial inventory. We show, using the backward dynamic programming equations, that the optimal cost is a piecewise concave function of the initial inventory. This property leads to a first simplification of the backward dynamic programming equations. We then consider the infinite horizon problem. We prove that, under a very general assumption, it is possible to obtain a solution close to the optimal one. Furthermore, we point out that the optimal cost is a piecewise concave function of the initial inventory, as in the case of the finite horizon.