Optimal design of acceptance sampling by attributes for sequential tests at consecutive times

Naturally, consumers of products want to have the reliability of the products ensured, whereas producers want to minimize the costs of ensuring reliability. For this adversarial setting, we consider group sequential acceptance sampling by attributes that is conducted every few years. We state the associated producer's stopping problem in a simple tree structure. We give conditions under which an optimal sampling design and minimal expected costs can be calculated using the Bellman principle and provide an algorithm to do so. Finally, we illustrate our approach by applying the optimization to the surveillance of utility meters subject to German regulations. For this, we use realistic and easily adaptable classes of cost functions and predict subsequent observations with a Bayesian analysis using prior knowledge. The prior knowledge is gained from historical observations and assuming the devices to follow a lifetime model. This case study indicates that compared to optimal single sampling plans, optimal sequential sampling can decrease the costs of water meter surveillance by almost 30%.