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, it committed a manufacturing fault that requires 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, an inspection takes place that may reveal the fault, if it exists. Based on this information, the company can recall the product anytime it pleases. 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 compute the optimal recall time that minimizes the expected cost of recall.