Retailers are often faced with a trade-off between setup costs, which arise when placing an order, and the associated inventory holding costs. In operations literature, something that has received a considerable amount of attention is the joint replenishment problem (JRP), which is a certain class of retailer inventory problems that arise under various circumstances. For example, it is common for multiple retailers to place a joint order from a single supplier to save ordering costs. Additionally, a group of retailers may also decide to share the same transport resources to reduce expenses.

Under these circumstances, there are two types of replenishment costs: a fixed major setup cost that is charged whenever an order is placed, and a retailer-specific minor setup cost for each retailer if they decide to replenish the order. In addition, a retailer also incurs a holding cost per item, over a period of time.

To address this issue, researchers Simai He, Jay Sethuraman, Xuan Wang, Jiawei Zhang aim to determine whether there is an inventory policy that maximises the long-term average total cost for an entire ordering system. To do so, they factored in power-of-two (POT) policies, “in which the replenishment interval of each retailer is an integer power-of-two times a base planning period”. With an optimal POT policy, the researches aimed to determine the best way of allocating total cost among retailers so that every retailer benefits from cooperation.

Different from the majority of the existing literature, the authors looked into the cost sharing problem of joint replenishment from a non-cooperative game-theoretic perspective. The researchers explain, “our non-cooperative approach in fact also has an underlying cooperative element, where the retailers agree upfront on how to share the fixed major setup cost when they place a joint order”. In this case, each retailer is allowed to freely choose their replenishment interval according to their needs.

The cost sharing problem—in which the costs are partitioned among different members—was considered in the researchers’ non-cooperative approach. For the allocation problem, they prosed the following rules: retailers pay their own holding cost, and whenever an order is placed, all ordering retailers pay their minor setup cost and pay an equal share of the major setup cost incurred.

“With an appropriate choice of the base planning period, the price of stability is equal to 1 when the retailers do not differ too much in the ratios of their minor setup cost to holding cost parameters”, the authors explain. Their results show that a simple cost allocation rule can be used as an effective mechanism to induce cost-efficient outcomes that are consistent with the strategic behaviour of others. They also proposed a simple allocation rule, where retailers pay an equal share of the major setup cost incurred during an order – avoiding any instances of “free riders” who do not contribute to the major setup cost.

Finally, the authors add, “it is of interest to investigate allocation-schemes other than the equal-division rule to share the major setup cost that might guarantee a more efficient outcome in the decentralized system…one may consider a more realistic setting in which the demand faced by retailers is uncertain and nonstationary”.