This study considers the effects of a machine breakdown, inspection, and partial backordering for deteriorating items. Most industries try to reduce facility unavailability by implementing a regular inspection and preventive maintenance since there is a possibility that some machines will breakdown during the production process. Moreover, an emergency purchase policy can be provided for quick response to customer’s backorder. The system also produces imperfect items with different rates before and after the inspection. Rework process and post-sales warranty are launched for the defective items. Unlike previous studies, we applied a fixed-point approach and renewal reward theorem to solve the deteriorating production-inventory model while considering machine breakdown, inspection, and partial backordering. A case example and sensitivity analysis are provided. The sensitivity analysis shows the important parameters that should be considered in designing the inspection plan and the replenishment policy when facility unavailability and imperfect items exist.

The economic production quantity (EPQ) and the economic order quantity (EOQ) models have been widely used in inventory management for vendors and retailers, respectively. However, an optimal solution for the EPQ or the EOQ model is only based on either vendors' or buyers' viewpoint. That is, such an optimal solution is usually beneficial to one of the vendors and buyers, but detrimental to the others. Hence, to establish a long-term cooperative relationship between the vendor and buyer, an integrated inventory model is needed. The purpose of this paper is to investigate a single-vendor single-retailer production-inventory model for products with imperfect quality. It is assumed that a 100% inspection process is conducted to screen out the defectives contained in the received lot and the defectives screened out during the 100% inspection process will be returned to the vendor immediately. By maximizing the expected annual integrated total profit, the optimal number of shipments in a cycle and the optimal size of each shipment are determined. A numerical example is provided to illustrate the proposed model and demonstrate the superiority of the proposed model to a general single supplier-single buyer system not considering lot-splitting shipments.