Cost Allocation for Joint Replenishment Models

Abstract

We consider the one-warehouse multiple retailer inventory model with a submodular joint setup cost function. The objective of this model is to determine an inventory replenishment policy that minimizes the long-run average system cost over an infinite time horizon. Although the optimal policy for this problem is still unknown, a class of easy-to-implement power-of-two policies are 98% effective. This paper focuses on how the cost, under an optimal power-of-two policy, should be allocated to the retailers. This question generates an interesting cooperative game. We prove that this cooperative game has a non-empty core. The key to our result is a strong duality theorem for the one-warehouse multiple retailer problem under power-of-two policies.

Collaborative replenishment in the presence of intermediaries

“…However, Anily and Haviv (2007) prove that if replenishment policies follow the powers-of-two (Jackson et al, 1985) structure, so that downstream players are not forced to synchronize all of their orders, the collaborative replenishment is always beneficial and the total cost is submodular on the set of players. Zhang (2009) extends this result to situations where players are allowed to have a joint inventory stocking point and obtains similar results. Van den Heuvel et al (2007) introduce and investigate the class of economic lot-sizing games wherein players face periodic, yet deterministic, demand and have the option to place joint orders.…”

“…As we prove, the class of submodular CRI situations contains situations wherein the replenishment cost components from intermediaries and manufacturers are themselves submodular. Single-source instances of such joint replenishment models are extensively studied in Meca et al (2004), Anily and Haviv (2007), Zhang (2009), Van den Heuvel et al (2007), Hartman et al (2000), andÖzen et al (2011) among others. Therefore, the second type of results presented in this paper holds for multi-product-multi-source extensions of aforementioned models.…”

Pricing and timing of consolidated deliveries in the presence of an express alternative: Financial and environmental analysis

“…Dror and Hartman (2007), Anily and Haviv (2007), and Zhang (2009) In our research, however, we assume that the upstream rm (the service provider) dictates which transport alternatives that are available on the market. Based on our observations, we believe that this captures several transport markets well.…”

Survey of cooperative inventory games and extensions

“…Not addressed in Roundy's work is the allocation of the optimal POT policy cost among the different items. Anily and Haviv (2007), and Zhang (2009) studied this topic. The problem is described below.…”