Carbon emissions in a multi-echelon production-inventory model with lead time constraints

We develop a deterministic optimization model that incorporates carbon emissions in a multi-echelon production-inventory model with lead time constraints. We impose that each customer order must be delivered within the due date fixed by the customer. The quantity that cannot be delivered on time is a lost sale. We consider a multi-echelon supply chain with different external suppliers, different manufacturing facilities, and different distribution centers. We adopt a general inventory policy. Indeed, we do not impose any constraints on the stock level that must be kept for each product in each facility in each period and on the procurement order quantities in the different facilities. Carbon emissions are associated with the decisions of manufacturing of intermediate and final products, ordering (transportation) from external and internal suppliers, and inventory positioning of the different products in the different stages of the supply chain. We first deal with the case of carbon emissions tax and then turn to the case of carbon emissions cap. We use the model to provide a series of insights that would be of interest for firms and policy makers. Such insights would be difficult to obtain with classical production-inventory models. For instance, the integration of lead times permits to show how the amount of carbon emissions is non-monotone with the variation of customer lead time and orders frequency. Also, the consideration of a general inventory policy permits to show how some particular policies (such as the base stock and the fixed order quantity) leads to increasing emissions. In addition, we capitalize on the multi-echelon aspect of our model in order to study the effect of individual emissions caps on each facility with comparison to a global cap on the entire supply chain. For instance, we demonstrate that individual caps can achieve significant lower emissions but can paradoxically lead to increasing the per unit emissions. We also show how a share of emissions can improve per unit emissions without deteriorating total emissions.