Abstract
We consider the basic Vehicle Routing Problem (VRP) in which a fleet ofM identical vehicles stationed at a central depot is to be optimally routed to supply customers with known demands subject only to vehicle capacity constraints. In this paper, we present an exact algorithm for solving the VRP that uses lower bounds obtained from a combination of two relaxations of the original problem which are based on the computation ofq-paths andk-shortest paths. A set of reduction tests derived from the computation of these bounds is applied to reduce the size of the problem and to improve the quality of the bounds. The resulting lower bounds are then embedded into a tree-search procedure to solve the problem optimally. Computational results are presented for a number of problems taken from the literature. The results demonstrate the effectiveness of the proposed method in solving problems involving up to about 50 customers and in providing tight lower bounds for problems up to about 150 customers.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Balinski and R. Quandt, On an integer program for a delivery problem, Operations Research 12(1964)300–304.
N. Christofides and A. Mingozzi, Vehicle routing: practical and algorithmic aspects, in:Logistics: Where Ends Have to Meet, ed. C.F.H. van Rijn (Pergamon Press, 1989) pp. 30–48.
N. Christofides, Vehicle routing, in:The Traveling Salesman Problem. A Guided Tour of Combinatorial Optimization, eds. E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan and D.B. Shmoys (Wiley, Chichester, 1985) pp. 431–448.
N. Christofides, A. Mingozzi and P. Toth, Exact algorithms for the vehicle routing problem, based on spanning tree and shortest path relaxations, Mathematical Programming 10(1981)255–280.
N. Christofides, A. Mingozzi and P. Toth, State space relaxation procedures for the computation of bounds to routing problems, Networks 11(1981)145–164.
N. Christofides and S. Eilon, An algorithm for the vehicle dispatching problem, Operational Research Quarterly 20(1969)309–318.
C. Clarke and J.Q. Wright, Scheduling of vehicles from a central depot to a number of delivery points, Operations Research 12(1964)568–581.
G. Cornuéjols and F. Harche, Polyhedral study of the capacitated vehicle routing problem, Management Science Research Report No. 553, Carnegie Mellon University (1989).
S. Eilon, C.D.T. Watson-Gandy and N. Christofides,Distribution Management, Mathematical Modelling and Practical Analysis (Griffin, London, 1971).
M.L. Fisher, Optimal solution of vehicle routing problems using minimumK-trees, Operations Research 42(1994)626–642.
M.L. Fisher and R. Jaikumar, A Generalized Assignment heuristic for vehicle routing, Networks 11(1981).
M. Gendreau, A. Hertz and G. Laporte, A tabu search heuristic for the vehicle routing problem, Report CRT No. 777, University of Montreal (1992), Management Science, to appear.
E. Gillet and L.R. Miller, A heuristic algorithm for the vehicle dispatch problem, Operations Research 22(1974)340–349.
E. Hadjiconstantinou, C.P.A. Weston and N. Christofides, An efficient implementation of an algorithm fork-shortest simple paths, Management School Research Paper, Imperial College (1995).
M. Held, P. Wolfe and H.P. Crowder, Validation of subgradient optimization, Mathematical Programming 6(1974)62–88.
N. Katoh, T. Ibaraki and H. Mine, An Efficient Algorithm fork-shortest simple paths, Networks 12(1982)411–427.
G. Laporte and Y. Nobert, Exact algorithms for the vehicle routing problem, Annals of Discrete Mathematics 31(1987)147–184.
G. Laporte, Y. Nobert and M. Desrochers, Optimal routing under capacity and distance restrictions, Operations Research 33(1985)1050–1073.
S. Lin and L. Kernighan, An effective heuristic algorithm for the travelling salesman problem, Operations Research 21(1973)498–516.
A.P. Lucena, Exact solution approaches for the vehicle routing problem, Ph.D Thesis, Department of Management Science, Imperial College, University of London (1986).
E. Taillard, Parallel iterative search methods for vehicle routing problems, Networks 23(1993)661–673.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hadjiconstantinou, E., Christofides, N. & Mingozzi, A. A new exact algorithm for the vehicle routing problem based onq-paths andk-shortest paths relaxations. Ann Oper Res 61, 21–43 (1995). https://doi.org/10.1007/BF02098280
Issue Date:
DOI: https://doi.org/10.1007/BF02098280