A mathematical model to determine the optimal production lot size for a deteriorating production system under an extended product inspection policy is developed.The last-K product inspection policy is considered so that the nonconforming items can be reduced,under which the last K products in a production lot are inspected and the nonconforming items from those inspected are reworked.Consider that the products produced towards the end of a production lot are more likely to be nonconforming,is proposed an extended product inspection policy for a deteriorating production system.That is,in a production lot,product inspections are performed among the middle K_1 items and after inspections,all of the last K_2 products are directly reworked without inspections.Our objective here is the joint optimization of the production lot size and the corresponding extended inspection policy such that the expected total cost per unit time is minimized.Since there is no closed form expression for our optimal policy,the existence for the optimal production inspection policy and an upper bound for the optimal lot size are obtained.Furthermore,an efficient solution procedure is provided to search for the optimal policy.Finally, numerical examples are given to illustrate the proposed model and indicate that the expected total cost per unit time of our product inspection model is less than that of the last-K inspection policy. A mathematical model to determine the optimal production lot size for a deteriorating production system under an extended product inspection policy is developed. The last-K product inspection policy is considered so that the nonconforming items can be reduced, under which the last K products in a production lot are inspected and the nonconforming items from those inspected are reworked. CPCider that the products produced towards the end of a production lot are more likely to be nonconforming, is proposed an extended product inspection policy for a deteriorating production system. What is is, in a production lot, product inspections are performed among the middle K_1 items and after inspections, all of the last k_2 products are directly reworked without inspections.Our objective here is the joint optimization of the production lot size and the corresponding extended inspection policy such that the expected total cost per unit time is minimized.Since there is no closed form expression for our opti mal policy, the existence for the optimal production inspection policy and an upper bound for the optimal lot size are obtained. Further, an efficient solution procedure is provided to search for the optimal policy. Finally, numerical examples are given to illustrate the proposed model and indicate that the expected total cost per unit time of our product inspection model is less than that of the last-K inspection policy.