An integrated vendor–buyer inventory model with backorder price discount and effective investment to reduce ordering cost

The single-vendor single-buyer integrated production inventory system has been an object of study for a long time, but little is known about the effect of investing in reducing ordering cost on the integrated inventory models with backorder price discount and variable lead time. The purpose of this article is to investigate in the continuous review model with backorder price discount and variable lead time to effectively increase investment and to reduce the joint expected annual total cost. The integrated strategy discussed here is one in which the buyer orders a quantity, then the vendor produces n times order quantity in each production cycle, in order to reduce setup cost. In addition, the buyer offers backorder price discounts to the customers that may motivate the customers’ desire for backorders, and buyer ordering cost can be reduced through effective investment. An integrated inventory model is established to find the optimal solutions of order quantity, ordering cost, backorder price discount, lead time, and the number of shipments from the vendor to the buyer in one production run, so that the joint expected annual total cost incurred has the minimum value. Furthermore, numerical examples are used to demonstrate the benefits of the model.