The Combined Van-and-Mopeds Routing Problem with Time Windows: Formulation, Variants, and Solution Methods

Abstract

We introduce and study the Combined Van-and-Mopeds Routing Problem with Time Windows (CVMRPTW), a novel variant of the heterogeneous fleet vehicle routing problem motivated by last-mile parcel delivery in dense European cities. In the CVMRPTW, a single van serves as a mobile base from which on-demand mopeds are deployed to reach customers located in areas inaccessible or inefficient for vans. The primary objective is to minimise the number of mopeds used while guaranteeing that all delivery time windows are met. We propose a Mixed-Integer Linear Programming (MILP) formulation that captures vehicle synchronisation, capacity tracking, and time-window compliance through 22 structured constraint families, and extend it to three operationally motivated variants: the standard model, the active-waiting-van variant, and the common-depot variant. A lexicographic two-stage optimisation framework allows secondary objectives-route duration, combined travel time, and combined distance-to be optimised without sacrificing the primary objective. Three graph-sparsification preprocessing techniques reduce the number of variables and constraints by up to 25%. For instances beyond the practical limits of exact solvers, we develop the Cluster-Based Combined Routing (CBCR) heuristic, which decomposes the problem into a clustering phase, a cluster-level exact solve, and a route-reconstruction phase. Computational experiments on real-world instances derived from OpenStreetMap data for Lisbon and Stuttgart show that the MILP solves instances with up to 48 customers to feasibility within one hour, and that the CBCR heuristic extends tractability to instances with up to 80 customers under favourable conditions.