Vehicle flow formulation for two-echelon time-constrained vehicle routing problem

Two-echelon routing problems, including variants such as the two-echelon vehicle routing problem (2E-VRP) and the two-echelon location routing problem (2E-LRP), involve assignment and location decisions. However, the two-echelon time-constrained vehicle routing problem (2E-TVRP) that caters to from-linehaul-to-delivery practices does not involve assignment decisions. This routing problem variant for networks with two echelons has not yet attracted enough research interest. Localized or long-distance services suffer from the lack of the assignment decisions between satellites and customers. Therefore, the 2E-TVRP, rather than using assignment decisions, adopts time constraints to decide the routes on each of the two interacting echelons: large-capacity vehicles transport cargoes among satellites on the first echelon, and small-capacity vehicles deliver cargoes from satellites to customers on the second echelon. This study introduces a mixed integer linear programming model for the 2E-TVRP and proposes a heuristic algorithm that incorporates the savings algorithm followed by a variable neighborhood search phase. Illustrative examples are used to test the mathematical formulation and the heuristic and a case study is used to demonstrate that the heuristic can effectively solve realistic-size instances of the 2E-TVRP.