Analysis of a decentralized supply chain under partial cooperation

In this article, we analyze a decentralized supply chain consisting of a supplier and two independent retailers. In each order cycle, retailers place their orders at the supplier to minimize inventory-related expected costs at the end of their respective response times. There are two types of lead times involved. At the end of the supplier lead time, retailers are given an opportunity to readjust their initial orders (without changing the total order size), so that both retailers can improve their expected costs at the end of respective retailer lead times (the time it takes for items to be shipped from the supplier to the retailers). Because of the possibility of cooperation at the end of supplier lead time, each retailer will consider the other's order-up-to level in making the ordering decision. Under mild conditions, we prove the existence of a unique Nash equilibrium for the retailer order-up-to levels, and show that they can be obtained by solving a set of newsboy-like equations. We also present computational analysis that provides valuable managerial insight for design and operation of decentralized systems under the possibility of partial cooperation.