Economic design of offline inspections for a batch production process

We consider a batch production process that can be either stable or unstable, in which inspection is performed offline after production of the batch is completed. The quality of a batch can be estimated with a desired level of certainty by inspecting only a sample of its units. In order to minimise the expected total cost per batch, which includes the costs of inspection, of false acceptance and of false rejection, we propose an economic inspection plan in which only a fraction of the batches, rather than each batch, is inspected. We prove that the expected total cost is a strictly quasiconvex function of the inspection interval. We establish necessary and sufficient conditions for the optimal inspection interval to be finite and propose an efficient algorithm to obtain its value. We demonstrate for the case of a single-sampling plan where the proposed economic approach outperforms the common procedure of inspecting every batch, and the proposed algorithm is very efficient.