We consider the case of a two-stage serial supply chain system. This supply chain system involves a single vendor who supplies a single buyer with a single product. The vendor’s production rate is assumed finite. In addition, the demand at the buyer is assumed deterministic. In order to coordinate their replenishment policies and jointly optimize their operational costs, the two supply chain partners fully share their relevant information. For this purpose, we develop an integrated inventory replenishment model assuming linear and fixed backorders costs. Then, we use a hybrid geometric-algebraic method to drive the optimal replenishment policy and the minimum supply chain total cost in a closed form. 1. Introduction Supply chain integration is concerned with functional integration and coordination among the supply chain partners. In an independently managed supply chain, each member in each stage will optimize his own operational costs in a decentralized fashion. Generally, it has been realized that such managerial independence of the supply chain partners may increase the imbalance between demand and supply. Such independence also has been recognized as direct cause of increased costs. This pushed firms towards the full integration of the supply chain resources and the proper coordination of decisions. Researchers report that closer collaboration among the chain partners, increased level of information sharing, and high level of coordination of various decision processes lead to improved customers service and reduced costs. The significant advances in information and communication technologies facilitated the provision and sharing of the business information necessary for efficiency improvement. This, in turn, facilitated the development in the integrated supply chain management. In recent years, numerous articles in supply chain modeling have addressed the issue of inventory coordination. Banerjee [1] introduced the concept of joint economic lot-sizing problem (JELS) for the case of a single vendor and a single purchaser under the assumption of deterministic demand and lot for lot policy. Since then, numerous articles in supply chain modeling focused on the integrated vendor-buyer inventory models and the joint economic lot-sizing problem [2–4]. An extensive review of integrated models which deal with the interaction between a buyer and vendor is presented in [5]. This review classified the literature dealing with the integrated models into four main classes. The first class represents models which deal with joint economic lot sizing policies. The
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