Pricing and ordering decisions in a supply chain with imperfect quality items and inspection under buyback of defective items

In this article, the economic production and inventory model in a three-layer supply chain including one distributor, one manufacturer and one retailer for a single-product and general demand functions under three scenarios is developed. We assume that during the production process, both healthy and defective items are generated. As the first scenario, we develop the first model, in which the defective items are not reworked and all considered as scrape, while in the second model, we assume that the defective items are reworked and are sold as perfect item. In the second scenario, we assume that defective item can be sold with lower price than the selling price. Moreover, raw materials with imperfect quality are sent back from a distributor to outside supplier under a lower price. Determining the order quantity of the distributor and the selling prices of the distributor and the manufacturer as well as the retailer was the goal of this article such that the total profit of each member is maximised. In order to solve the models, the Stackelberg approach is employed between the members, and the concavity of the profit functions is proved using several theorems. Then, closed form solutions are derived for the decision variables and a solution algorithm is proposed to determine the optimal solutions. Finally, a numerical example is presented to illustrate the applicability of the model.