Coordination policy for a two-stage supply chain considering quantity discounts and overlapped delivery with imperfect quality

• This model improves the flaw of occurring shortages during the screening duration.
• This model considers three practical situations to reflex the real-life situation.
• Some properties and theorems are explored to help develop an algorithm.
• A solution procedure free the convexity is published to find the optimal solution.
• Managerial insights are discussed.

Unlike the traditional integrated supplier–buyer coordination model, this research incorporates overlapped delivery and imperfect items into the production–distribution model. This model improves the observable fact that the system might experience shortage during the screening duration and also takes quantity discount into account. This approach has not been discussed in previous integrated supplier–buyer coordination models. The expected annual integrated total cost function is derived and properties and theorems are explored to help develop an algorithm. A solution procedure, free from the convexity associated with an algorithm is established to find the optimal solution. A numerical example is given to illustrate the proposed procedure and algorithm. A sensitivity analysis is made to investigate the effects of five important parameters (the inspect rate, the annual demand, the defective rate, the holding cost, and the receiving cost) on the optimal solution. Managerial insights are also discussed.