An integrated inventory model with immediate return for defective items under unequal-sized shipments

The purpose of this paper is to investigate a single-vendor single-retailer production-inventory model with immediate return for imperfect items under an unequal-sized policy. It is assumed that the batch quantity will be received by the retailer in n shipments. The first shipment is of small size followed by (n-1) equal-sized shipments of size just equal to the first shipment size times the ratio of the production rate to the demand rate. Moreover, 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. First, the expected annual integrated total profit is constructed. Next, by maximizing the expected annual integrated total profit, the optimal number of shipments in a cycle and the optimal sizes of the shipments in a cycle are determined. Finally, a numerical example is provided to illustrate the proposed model and demonstrate the usefulness of the proposed model when compared with the integrated model under equal-sized policy.