This paper investigates the effects of investment and inspection policies on an integrated production–inventory model involving defective items and upstream advance-cash-credit payment provided by the supplier. In this model, retailers offer customers a downstream credit period. Furthermore, the defective rate of the item can be improved through capital co-investment by the supplier and retailer. The objective of this study was to determine the optimal shipping quantity, order quantity, and investment alternatives for maximizing the supply chain's joint total profit per unit time. An algorithm was developed to obtain the optimal solution for the proposed problem. Several numerical examples are used to demonstrate the proposed model and analyze the effects of parameters changes on the optimal solutions. Finally, management implications for relevant decision makers are obtained from the numerical examples.

Abstract. This study investigated a production-inventory model with defective items under a two-part trade credit, where the agreement of conditionally freight concession is considered in an integrated supply chain. It is assumed here that the retailer conducts the inspection process before selling incoming items. All the defective items are discovered, stored, and sold as a single batch to a secondary market at a decreased price. Furthermore, shortages are allowed and completely backlogged for the retailer. The purpose of this study is to determine the optimal number of shipments per production cycle for the supplier and the optimal length of time when there is no inventory shortage and replenishment cycle for the retailer, such that the total pro t function has a maximum value. In theoretical analysis, the existence and uniqueness of the optimal solutions are shown, and an algorithm is developed to nd the optimal solutions. Furthermore, numerical examples are presented to demonstrate the solution procedures, and a sensitivity analysis of the optimal solutions regarding all parameters is also carried out.