Submodularity of Joint Replenishment Games

In recent years, many companies have come to realize that they can collaborate in supply chain management to greatly improve their performances. It has been reported that shippers who collaborate and consolidate their small and frequent shipments into full truckloads can reduce the transportation and inventory cost significantly. Inventory pooling is also known to be an effective way to reduce safety stock and increase customer service. By sharing their inventories, the companies can reduce the joint setup cost, which usually incurs jointly for the items in the production, purchasing, and transportation. In particular, the joint replenishment problem (JRP), i.e., the problem to determine the inventory replenishment policy with common setup costs, is a well-established and studied model in the field of supply chain management. However, the willingness to collaborate between different companies often depends on a fair mechanism to allocate the cost or gain, which is considered to be advantageous to all participants. In practice, getting all parties to agree to collaborative commerce has been identified by some as one of the major barriers. We are interested in the question of how the system-wide cost should be allocated among the retailers fairly to encourage cooperation. A proper cost allocation scheme is important particularly when the retailers belong to different firms or are decentralized divisions of an organization. Therefore, cooperative game theory is a natural choice for analyzing the cost allocation issues. A particular type of cooperative game called submodular game automatically guarantees the existence of a fair scheme to allocate the cost (core allocation), and more importantly under this allocation scheme everyone would benefit with more joining the coalition (PMAS). The first major goal of this project is to establish the submodularity of cooperative JRP game.On the other hand, some companies may only want to share the information related to the joint setup cost, and are not willing to reveal their private information. Given the fair cost allocation rule for the joint setup cost (based on the revealed information), the companies usually act on their own to greedily improve their own performances. It is natural to apply non-cooperative game setting under this scenario. However for this model the decision variables and objective functions for each player are in the discrete continuous domain, and therefore we could not use the standard existence theorem of Nash equilibrium. For discrete non-cooperative games, the existence of Nash equilibrium often relies on the submodularity of the game, which ensures that there exists a unique best (worst) equilibrium, and these two equilibriums can be reached simply by the best response mechanism. To establish the submodularity of the non-cooperative JRP game is our second major objective.It also worth noting that, polymatroid optimization, one of the main methodologies which shall be used in this research, has its own impact in optimization theory and applications. In this project, we shall also study the structural results of polymatroid optimization and the applications of it.