An easy method to derive the integrated vendor–buyer production–inventory model with backordering using cost-difference rate comparison approach

In growing competitive markets, close cooperative strategies among the vendors and the buyers are essential to cut the joint inventory cost and to have less response time to the supply chain players. Recently, Yang et al. (2007) [2] have presented an integrated two-stage production–inventory model and used the classical differential calculus method to derive the optimal lot size for the vendor and the buyer, and the number of deliveries in supply chain. Moreover, numerous researchers adopted other algebraic methods to derive the optimal solution for inventory models. However, in these papers, the cost comparisons method and arithmetic–geometric mean inequality do not focus on explicitly developing the mathematical expressions for two-stage inventory system with the backorders. In this study, we extend the inventory model proposed by Yang et al. (2007) [2], and consider a three-variable inventory problem and present an alternative approach to determine the optimal replenishment policy for the integrated vendor–buyer inventory system with backordering utilizing the cost-difference rate comparison approach.