optimal production and ordering strategies with defective items and allowable shortage under two-part trade credit

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 the integration supply chain. we assume the inspection process is conducted by the retailer before selling incoming items. all the defective items are discovered, stored and then 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 wherein there is no inventory shortage and replenishment cycle for the retailer such that the total profit function has a maximum value. in theoretical analysis, the existence and uniqueness of the optimal solutions are shown and an algorithm is developed to find the optimal solutions. furthermore, numerical examples are presented to demonstrate the solution procedures and a sensitivity analysis of the optimal solutions regarding all parameters are also carried out.