Sequential inspection in a two-stage system

We study a quality control problem in a two-stage tandem system. We assume that at each stage units are processed in batches, and the defect rates are random variables with known distributions. Final products are supplied to customers under warranties or service contracts, with penalty costs associated. Our focus is on coordinating the inspection procedures at the two stages. Using a stochastic dynamic programming formulation, we show that the optimal policy at stage 1 is characterized by a sequence of thresholds, and at stage 2, by a priority structure, as well as a threshold structure, The key to optimality is a so-called K-submodularity property, which is a strengthening of the usual notion of submodularity