Lot Size Decisions for Vendor-Buyer System with Quantity Discount, Partial Backorder, and Stochastic Demand

This paper presents production-inventory model for two-echelon system consisting of single vendor and single buyer. The proposed model contributes to the current inventory literature by incorporating quantity discount scheme into stochastic vendor-buyer model. Almost all vendor-buyer inventory models have discussed this scheme in single-echelon system and deterministic demand situation. Here, we assume that the demand of the buyer is normally distributed and the unmet demand is considered to be partially backordered. In addition, the lead time is variable and consists of production time and nonproductive time. The quantity discount is developed by using all-units quantity discounts. Finally, an iterative procedure is proposed to obtain all decision variables and numerical examples are provided to show the application of the proposed procedure. 1. Introduction Supply chain is the sequence of business processes and activities from suppliers through customers that provide products, services, and information to achieve customer satisfaction, that is, a chain that can quickly respond to customer’s requirement. Integration of different entities in the supply chain is an important way to gain competitive advantage and customer satisfaction [1]. In recent years, research dealing with inventory management in supply chain system has attracted considerable attention from many scholars. Goyal [2] is among the first authors who studies integrated inventory model for single-vendor single-buyer system. He introduces a model for situation in which vendor produces a lot based on an infinite production rate and transfers it to the buyer by a lot-for-lot policy. He shows that making inventory decisions jointly among vendor and buyer can result in a substantial cost reduction compared to individual decisions. The framework proposed by Goyal [2] has encouraged many researchers to present various types of integrated vendor-buyer system. Banerjee [3] relaxes the assumption of lot-for-lot policy and infinite production rate and proposes a model where the vendor produces a batch at finite production rate and then delivers it equally to the buyer. Goyal [4] also relaxes the lot-for-lot assumption and introduces a more general lot sizing model. He argues that producing a batch which is made up of equal shipments generally produced lower total cost, but the whole batch must be completed before the first shipment is made. A number of researchers, including Goyal [5], Hill [6], Hill [7], Goyal and Nebebe [8], Hoque and Goyal [9], Hill and Omar [10], and Zhou and Wang [11] develop a