Optimal lot sizing and inspection policy for products sold under free-repair warranty

A model is developed to jointly determine the optimal lot size and product inspection policy for a deteriorating production system. In the end of each production of a lot, the state of the system can be detected by inspection. If the process is in-control, then no action is taken and all items in a lot are released for sale with no inspection. If the process is found to be out-of-control, then it is restored to the in-control state and the last K products in a lot are inspected and non-conforming products are reworked before they are released for sale to reduce the post-sale warranty cost. Under this inspection scheme, the optimal lot size and the optimal number of products to be inspected are jointly determined. There exists a unique optimal policy such that the expected total cost per unit time is minimized. Since there is no closed-form expression for the optimal policy, upper bound is derived to facilitate the search process. From the numerical study, we have observed that the model performs significantly better than the traditional EMQ policy.