Optimal Inventory Policy with Subadditive Ordering Costs and Stochastic Demands

Abstract

We consider a discrete review, single product, dynamic inventory model. With the exception of the ordering cost function, we make the usual assumptions (see [8]). The distinguishing feature of our model is that we merely assume that the ordering cost function for each period is non-decreasing and subadditive. Denoting the smallest minimizer of the expected costs for period i by $\bar y_i $, we establish that when x is the inventory level entering period i, the optimal stock level $y_i (x)$ after ordering in period i is x if $x\geqq \bar y_i $. Moreover, if the ordering cost (apart from a linear term) is eventually constant, then we can show that $y_i ( \cdot )$ is lower semicontinuous and satisfies $y_i (x) = S_i $ for $x < s_i $ and $y_i (x)\leqq S_i $ for $s_i \leqq x < \bar y_i $ where $s_i \leqq \bar y_i \leqq S_i $. Also, simple conditions are given which imply (the planning horizon theorem) $S_i = \bar y_i $. Upon assuming that the ordering cost is the multiple set-up cost, a partial characterizatio...