Optimizing carbon emissions in green logistics for time-dependent routing

This paper considers a green vehicle routing problem termed the time-dependent green vehicle routing problem with time windows (TDGVRPTW). The TDGVRPTW is an extension of the green vehicle routing problem with time windows in green logistics. It considers time-dependent vehicle speed and aims to minimize carbon emissions. Since the travel times and carbon emissions between locations depend on the departure time from the starting location, optimizing carbon emissions requires determining the optimal departure times from the depot and customers. This paper presents a branch-price-and-cut (BPC) algorithm to solve the TDGVRPTW. The problem is formulated based on a set-partitioning model. To solve the pricing problem associated with the set-partitioning model, we first define backward non-dominated time-dependent arcs and optimal timed routes. We also introduce three types of routes, forward, backward, and patchwork optimal timed routes, and analyze their characteristics. Then, we introduce a method to adjust the times in backward and patchwork optimal timed routes from the latest to the earliest. We prove that this method guarantees correctness and effectively reduces complexity. We propose a bidirectional labeling algorithm to generate routes. The pricing problem employs state-of-the-art techniques, including limited memory subset row cuts (lm-SRCs), ng-route relaxation, and bucket graphs to enhance the algorithm's efficiency. The BPC algorithm is tested on a set of instances derived from benchmark cases in the literature. Its various components and pricing strategies are evaluated, and its performance is compared with other algorithms from the literature. The results confirm the effectiveness of the proposed algorithm and its newly designed components in solving the TDGVRPTW.