Supply Chain Network Design with Nonlinear Emissions and Cap on Carbon Footprint

Abstract

We consider a green supply chain network design problem with one manufacturing facility, one product, multiple distribution centers, and multiple retailers, and where carbon footprint is capped. We account for cost as well as emissions from storage at the distribution centres and transportation. The latter is done either using electric trucks or diesel trucks. While emissions for electric trucks is linearly dependent on load, emission for diesel trucks is hard to capture. To remedy this, we consider linear, concave, and convex functions of the load transported. We optimize the location of distribution centers, the allocation of demand, and the transportation modes among diesel and electric trucks. We present a nonlinear mixedinteger model, propose a Lagrangian relaxation to decompose the model by echelon in order to isolate the nonlinearity in one of the subproblems, and devise a Lagrangian heuristic based on the solution of the other subproblem. Through a case study, we evaluate the impact of various emission functions and levels of emission restrictions on the structure of the supply chain, as well as the trade-offs between the utilization of diesel trucks and electric trucks. Our findings reveal that the type of emission function significantly influences the use of diesel versus electric trucks and the distribution of cost components. With a concave function, diesel truck usage slightly increases as the cap on per unit emission increases when electric trucks are cost-efficient and can rise by up to 30% when diesel trucks are the cheaper option. In contrast, a convex function shifts the main cost contribution to transportation emissions, with diesel truck usage decreasing as the emission coefficient increases. The linear function shows mixed trends, aligning with either the concave or convex cases under different conditions. Moreover, extending electric truck range when the emission function is convex heavily favors their utilization in most trips, while under concave emission functions, still diesel trucks are favored over electric ones as the limit on emission loosens.