Stochastic Dynamic Inventory Problem Under Explicit Inbound Transportation Cost and Capacity

We study a practical generalization of the classical stochastic dynamic inventory problem where privately owned trucks with limited cargo capacity are used to transport the replenishment quantity. The resulting replenishment cost function also includes the traditional fixed setup cost, and it is known as a multiple setup cost structure, which leads to complicated cost-to-go functions in the problem of interest. We introduce the concepts of non-(Δ, C)-decreasing and $non\text{-}(\text{Δ},C{)}_{\mathbb{N}}^K\text{-}decreasing$, and we develop a sophisticated replenishment policy called the $(Q,\overrightarrow{s},\overrightarrow{S})$ policy. We examine the sufficient conditions under which the new policy is optimal. Our results offer a detailed characterization of optimal policies and generalize the existing theory on the concepts of K-convexity and non-K-decreasing. In doing so, we are able to investigate a traditional stochastic inventory problem which has remained open in the literature for more than four decades.

The online appendix is available at https://doi.org/10.1287/opre.2017.1625.