Inventory replenishment control under supply uncertainty

We consider a discrete state, discrete decision epoch inventory replenishment control problem under supply uncertainty. We assume that there is no backlogging, the single period demand d is deterministic, and once an item is placed in inventory, it will not perish. If a units of the product are ordered, then α units are placed into inventory with probability P(α{pipe}a), where ∑_α=0^aP(α{pipe}a) = 1. Let z=d-x, where x is the current inventory level. For the infinite horizon, total discounted cost criterion, we present conditions that guarantee that an optimal replenishment policy δ ^* is such that δ ^*(z)=0 for z≤0, δ ^*(z)≥z≥0, and δ ^*(z)-z is monotonically non-decreasing for z≥0. Such a "staircase" structure has a simple parametric description, which can help to accelerate value iteration and policy iteration.