Dynamic Coordination In A Supply Chain With Production Capacity Uncertainty

With the trend of globalization, supply chains are increasingly extended and uncertainties associated with material flow along the supply chain have become a major challenge in the planning process. In this study, we examine the effect of supply uncertainty on dynamically managing a two-stage supply chain under both centralized and decentralized controls. Specifically, the downstream faces demand uncertainty, while the upstream faces supply uncertainty in the sense that the output at this stage is a stochastic function of its input. In each period, the downstream places a replenishment order to the upstream aiming at meeting the demand in the coming period, and the upstream, in turn, places a production order to generate additional output. Local leftover inventories are carried over to the next period and the unmet demand at the downstream is backordered. Even under centralized control, this is a challenging, nonconcave optimization problem due to the presence of supply uncertainty. Applying the concept of directional concave ordering (Shaked and Shanthikumar 2007) and extending the notion of stochastic linearity (Feng and Shanthikumar 2016), we develop a concave transformation of the problem for a general class of stochastic supply functions, which allows us to analyze the dynamic problem and characterize the optimal policy. In general, it is optimal for each stage to follow a threshold policy so that a production (replenishment) order is issued at the upstream (downstream) if and only if the local inventory level is below a threshold. In the special case when the upstream supply uncertainty comes from a random production capacity, the optimal threshold policy reduces to the conventional echelon base stock policy (Clark and Scarf 1960). The optimal base stock levels increase when the downstream demand becomes stochastically larger or when the upstream capacity becomes stochastically smaller.