Multi‐Stage Supply Chain with Production Uncertainty

With supply chains becoming increasingly extended, the uncertainties in the upstream production process can greatly affect the material flows that aim toward meeting the uncertain demands at the downstream. We analyze a two‐location system in which the upstream production facility experiences random capacities and the downstream store faces random demands. Different from the widely used approach that seeks the decomposition of the profit function based on the echelon inventories, our approach builds on the notions of stochastic functions, in particular, the stochastic linearity in midpoint and the directionally concave order. With these notions, we establish the concavity and submodularity of the profit functions in the transformed decision variables. In general, it is optimal to follow a two‐level state‐dependent threshold policy such that an order is issued at a location if and only if the inventory position of that location is below the corresponding threshold. In the special case where the salvage values are linear in the ending inventories, the profit function becomes separable in the inventory positions, and the optimal policy reduces to the echelon base‐stock policy. The effect of the uncertain capacity and demand depends critically on whether the production capacity is limited or ample in relation to the demand. Only when the capacity and the demand do not differ much, the upstream facility carries positive inventory; otherwise, all units produced are shipped immediately toward the downstream. We further extend our analysis to systems with general stochastic production functions and with multiple locations.