Altruistic profit allocation rules for joint replenishment with carbon cap-and-trade policy

We study profit allocation rules for joint replenishment among retailers under a carbon cap-and-trade policy. In particular, we consider retailers’ altruistic behavior when designing the profit allocation rules. We show that joint replenishment can increase participants’ total profit and reduce their total amount of carbon emissions. To allocate the total profit from joint replenishment to the retailers, we introduce a joint replenishment game with the carbon cap-and-trade policy. We show this game is convex and propose a profit allocation rule lying in the core of this game. Based on this, we further design an altruistic profit allocation rule by categorizing retailers into efficient ones and non-efficient ones, with the efficient retailers being altruistic who will transfer their surplus carbon allowance to those non-efficient ones. We show that our proposed altruistic profit allocation rule lies in the core of this game. Moreover, we derive results concerning how much carbon allowance, to whom and at what price retailers are willing to transfer in the grand coalition, by considering their altruistic parameters. Our results show that the retailer with the highest altruistic parameter value obtains all the surplus carbon allowance from other retailers, and this particular retailer transfers his surplus carbon allowance to the retailer with the second-highest altruistic parameter value. These results indicate that people do not help other people uniformly; rather, they do so according to the generosity of other people. Based on these results, we derive another altruistic profit allocation rule which also belongs to the core of the game.