Cost and emission allocation for joint replenishment systems subject to carbon constraints

We consider joint replenishment among multiple retailers subject to various types of carbon constraints. In particular, we consider four types of carbon constraints: carbon tax, strict carbon cap, carbon cap-and-offset, and carbon cap-and-price. Under each carbon policy, we first identify the optimal joint ordering policy. Then, we study the cost allocation problem in the framework of cooperative game theory. For the joint replenishment game with carbon tax, we show that a simple proportional rule belongs to the core of the corresponding game. For the joint replenishment games with strict carbon cap, carbon cap-and-offset, and cap-and-price wherein the penalty rate is larger than the reward rate, we show that the cost allocation rules based on the Lagrangian multipliers belong to the core of the corresponding games. Interestingly, for the joint replenishment game with cap-and-price wherein the penalty rate is less than the reward rate, we show that the corresponding joint replenishment game may not satisfy subadditivity and may have an empty core. This implies that joint replenishment is not feasible in this situation. In addition, we further discuss carbon emission allocation game under the strict carbon cap policy. We show that the carbon emission game does not necessarily satisfy subadditivity, and therefore, we propose an algorithm to generate an optimal coalition structure.

Automated summary

This article explores joint replenishment among multiple retailers under different carbon constraints. By considering various carbon policies, optimal ordering strategies are determined, and the problem of cost allocation is studied. It is found that in the joint replenishment game with a carbon tax, a simple proportional rule belongs to the core. In the joint replenishment games with strict carbon cap, carbon cap-and-offset, and carbon cap-and-price, where the penalty rate is higher than the reward rate, cost allocation rules based on the Lagrangian multipliers are part of the core. Interestingly, in the joint replenishment game with carbon cap-and-price and a lower penalty rate, subadditivity may not be satisfied, and the core may be empty. This suggests that joint replenishment is not feasible in this situation. Additionally, the carbon emission allocation game under the strict carbon cap policy is further discussed, and an algorithm is proposed to generate an optimal coalition structure.