Modeling and Solving the Capacitated Vehicle Routing Problem on Trees

Part of the Operations Research/Computer Science Interfaces book series (ORCS, volume 43)


Capacitated vehicle routing problems (CVRPs) form the core of logistics planning and are hence of great practical and theoretical interest. This chapter considers the CVRP on trees (TCVRP), a problem that often naturally arises in railway, river, and rural road networks. Our objective is to build high-quality models that exploit the tree structure of the problem that can also be easily implemented within the framework of a modeling language (a feature desired by practitioners) like AMPL, GAMS, or OPL. We present two new integer programming models for the TCVRP that explicitly take advantage of the tree structure of the graph. The two models are implemented using the AMPL model building language, and compared along several metrics—computation time, quality of the linear programming relaxation, and scalability—to examine their relative strengths.

Key words

Capacitated vehicle routing on trees integer linear programming formulations high-level modeling languages 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    C. Basnet, L.R. Foulds, and J.M. Wilson. Heuristics for vehicle routing on tree-like networks.Journal of the Operational Research Society, 50:627–635, 1999.CrossRefGoogle Scholar
  2. 2.
    I. Berger, J.-M. Bourjolly, and G. Laporte. Branch-and-bound algorithms for the multi-product assembly line balancing problem.European Journal of Operational Research, 58:215–222, 1992.CrossRefGoogle Scholar
  3. 3.
    G. Clarke and J. Wright. Scheduling of vehicles from a central depot to a number of delivery points.Operations Research, 12:568–581, 1964.Google Scholar
  4. 4.
    E. G. Coffman, M. R. Garey, and D. S. Johnson. Approximation algorithms for bin packing – an updated survey. In G. Ausiello, M. Lucertini, and P. Serafini, editors,Algorithms design for computer system design, pages 49–106. Springer-Verlag, New York, 1984.Google Scholar
  5. 5.
    R. Fourer, D. M. Gay, and B. W. Kernighan.AMPL: A Modeling Language for Mathematical Programming. The Scientific Press, 1993.Google Scholar
  6. 6.
    ILOG. Cplex 9.0 reference manual, 2003.Google Scholar
  7. 7.
    J. Kallrath, editor.Modeling Languages in Mathematical Optimization, volume 88 ofApplied Optimization. Springer Publishing Company, 2004.Google Scholar
  8. 8.
    M. Labb´e, G. Laporte, and H. Mercure. Capacitated vehicle routing on trees.Operations Research, 39:616–622, 1991.CrossRefGoogle Scholar
  9. 9.
    P. Mbaraga, A. Langevin, and G. Laporte. Two exact algorithms for the vehicle routing problem on trees.Naval Research Logistics, 46:75–89, 1999.CrossRefGoogle Scholar
  10. 10.
    G. L. Nemhauser and L. A. Wolsey.Integer and Combinatorial Optimization. John Wiley and Sons, New York, 1988.Google Scholar
  11. 11.
    P. Toth and D. Vigo, editors.The Vehicle Routing Problem, volume 9 of SIAM Monographs on Discrete Mathematics and Applications.SIAM, Philadelphia, PA, 2002.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of Industrial Engineering and Operations ResearchUniversity of CaliforniaBerkeley
  2. 2.The Robert H. Smith School of Business and Institute for Systems ResearchUniversity of MarylandCollege Park

Personalised recommendations