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

[2607.22601] The Combined Van-and-Mopeds Routing Problem with Time Windows: Formulation, Variants, and Solution Methods Skip to main content Search Submit Donate Log in Mathematics > Optimization and Control arXiv:2607.22601 (math) [Submitted on 12 Jun 2026 ( v1 ), last revised 25 Aug 2026 (this version, v2)] Title: The Combined Van-and-Mopeds Routing Problem with Time Windows: Formulation, Variants, and Solution Methods Authors: Pedro Lameiras , Alexandre P. Francisco , Adriano Serrano , Cátia Vaz View PDF HTML (experimental) 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. Comments: 35 pages, LaTeX; typos corrected Subjects: Optimization and Control (math.OC) MSC classes: 90C27 ACM classes: G.2.1 Cite as: arXiv:2607.22601 [math.OC] (or arXiv:2607.22601v2 [math.OC] for this version) https://doi.org/10.48550/arXiv.2607.22601 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Alexandre P Francisco [ view email ] [v1] Fri, 12 Jun 2026 14:54:15 UTC (2,363 KB) [v2] Tue, 25 Aug 2026 08:08:13 UTC (2,496 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.OC < prev | next > new | recent | 2026-07 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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