The commodity constrained split delivery vehicle routing problem considering carbon emission: Formulations and a branch-and-cut method

This paper introduces the Green Commodity constrained Split Delivery Vehicle Routing Problem (GC-SDVRP), which involves designing efficient and environmentally friendly delivery routes that reduce the CO${}_2$ emissions associated with transporting multiple commodities. In this problem, different commodities demanded by a customer can be delivered by one or more vehicles, if beneficial, which poses additional modeling and solution challenges. We propose a relaxed formulation that provides a lower bound on the optimal value of the GC-SDVRP, and adapt two other formulations from the literature to address this problem. Additionally, we develop a branch-and-cut (BC) method based on two of these formulations, and introduce a procedure for deriving feasible solutions to the GC-SDVRP from solutions obtained with the relaxed formulations. The results of computational experiments performed on benchmark instances indicate the superior performance of the BC method based on the proposed formulation. Furthermore, they show that, contrary to the traditional objective of minimizing distance, the GC-SDVRP is significantly easier to solve to optimality and can reduce CO${}_2$ emissions by 2.59% compared to the problem that minimizes total travel distance. Our investigation also reveals that increasing vehicle capacity improves solution quality in the GC-SDVRP, while split delivery can enable further reductions in CO${}_2$ emissions. Finally, although increasing the number of commodities imposes challenges in solving the problem, the possibility of split delivery mitigates its impact on the value of the final solution, indicating that an increase in the number of commodities does not necessarily result in higher CO${}_2$ emissions.