Exact solution of location–routing problems with heterogeneous fleet and weight-based carbon emissions

This paper investigates a location–routing problem with heterogeneous fleet and weight-based carbon emissions, aiming to minimize total operating costs, including depot and vehicle fixed costs and carbon emissions. The problem is formulated using an extended formulation, from which a relaxation is derived. The relaxation is further strengthened with additional cuts. For this problem, we devise an exact algorithm that combines a branch-price-and-cut algorithm with state-of-the-art technologies. The exact algorithm relies on a novel bidirectional labeling algorithm; it merges labels through cost compensation, leading to an effective solution. We conduct comprehensive computational experiments to validate the proposed algorithm, evaluating key factors related to carbon emissions. Our findings confirm the algorithm's efficacy, highlighting its potential as a useful tool for addressing the location routing problem in heterogeneous fleet scenarios.