Optimal decision rules for product recalls

We consider a company that has just manufactured and sold a number of copies of a product. It is known that, with a small probability, the company committed a manufacturing fault that will require a recall. The company is able to observe the expiration times of the sold items whose distribution depends on whether the fault is present or absent. At the expiration of each item, a public inspection takes place that may reveal the fault, if it exists. Based on this information, the company can recall the product at any moment. On recall, each customer is paid back the price of the product. If the company is not able to recall before an inspection reveals the fault, it pays a fine per item sold which is assumed to be much larger than the price of the product. We first compute the optimal recall time that minimizes the expected cost of recall. Next we derive and solve a stationary limit recall problem and show that the original problem converges to the limit problem as the number of items initially sold increases to 1. Finally, we propose two extensions of the original problem and compute the optimal recall times for these. In the first extension, the expired items are inspected only if they expire earlier than expected; in the second extension, the company is able to conduct internal/private inspections on the expired items. In adition to optimality proofs, we provide numerical examples and simulation results for all of the three models.