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

Abstract

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.