Modeling and solving the two-level generalized median tour problem

In many real-world situations, the beneficiaries of a distribution system may be grouped into clusters, requiring a transportation structure to serve each cluster of beneficiaries efficiently. For instance, in public services, such as health, education, and emergencies, delivery tasks rely on the local authority of each district or cluster. In this context, a two-level distribution system named the Two-Level Generalized Median Tour Problem (TLGMTP) is introduced. The first level addresses product distribution using a specialized vehicle, starting and ending at a depot and visiting some clusters. In this manner, products are delivered to one or more nodes belonging to the visited clusters. The second level comprises smaller vehicles that start their trips from the nodes belonging to the first level and transport the products to one or more nodes located in a non-visited cluster, ensuring that all non-visited clusters of the first level are visited in the second level. Then, the non-visited nodes in each cluster must reach a node in the same cluster to collect their products. In this study, we present, model, and solve the TLGMTP to minimize the total transportation costs. We develop three mathematical formulations and solve them using a branch-and-cut algorithm. Exhaustive computational experiments involving tests and real-world instances are presented to show the efficiency and advantages of the proposed methodology.