An Integer Linear Programming Local Search for Capacitated Vehicle Routing Problems
- 8.5k Downloads
In this chapter we address the classical Vehicle Routing Problem (VRP), where (at most) k minimum-cost routes through a central depot are constructed to cover all customers while satisfying, for each route, both a capacity and a total-distance-traveled limit. We present a Local Search algorithm for VRP, based on the exploration of an exponential neighborhood by solving an Integer Linear Programming (ILP) problem. Our starting point is the following refinement heuristic procedure proposed by De Franceschi et al.: given an initial solution to be possibly improved, (a) select several customers from the current solution, and build the restricted solution obtained from the current one by extracting (i.e., short-cutting) the selected customers; (b) reallocate the extracted customers to the restricted solution by solving an ILP problem, in the attempt of finding a new improved solution. We present a generalization of the neighborhood proposed in this method, and investigate the Column Generation Problem associated with the Linear Programming (LP) relaxation of the ILP formulation corresponding to the neighborhood. In particular, we propose a two-phase approach for the neighborhood exploration, which first reduces the neighborhood size through a simple heuristic criterion, and then explores the reduced neighborhood by solving the corresponding ILP formulation through the (heuristic) solution of the Column Generation Problem associated with its LP relaxation. We report computational results on capacitated VRP instances from the literature (with/without distance constraints), which are usually used as benchmark instances for the considered problem. In several cases, the proposed algorithm is able to find the new best-known solution in the literature.
Unable to display preview. Download preview PDF.
- 1.P. Augerat, J.M. Belenguer, E. Benavent, A. Corberán, D. Naddef, and G. Rinaldi. Computational results with a branch and cut code for the capacitated vehicle routing problem. Techinal Report RR 949-M, Université Joseph Fourier, Grenoble, 1995.Google Scholar
- 4.J-F. Cordeau, G. Laporte, M. W. P. Savelsbergh, and D. Vigo. Vehicle routing, inHandbooks in Operations Research and Management Science, Vol. 14 (C. Barnhart and G. Laporte eds.). North–Holland, Amsterdam, 367–428 , 2007.Google Scholar
- 5.J-F. Cordeau, M. Gendreau, A. Hertz, G. Laporte, and J-S. Sormany. New heuristics for the vehicle routing problem, inLogistics Systems: Design and Optimization (A. Langevin and D. Riopel eds.). Springer–Verlag, New York, 279–297 , 2005.Google Scholar
- 6.N. Christofides, A. Mingozzi, and P. Toth. The vehicle routing problem, inCombinatorial Optimization (N. Christofides, A. Mingozzi, P. Toth and C. Sansi eds.). Wiley, Chichester, 315–338 , 1979.Google Scholar
- 9.M. Gendreau, A. Hertz, and G. Laporte. A tabu search heuristic for the VRP. Technical Report CRT-777, 1991.Google Scholar
- 11.B.L. Golden, E.A. Wasil, J.P. Kelly, and I-M. Chao. Metaheuristics in vehicle routing, inFleet Management and Logistics (T.G. Crainic and G. Laporte eds.). Kluwer Academic, Boston, 33–56 , 1998.Google Scholar
- 12.G.M. Gutin. On an approach to solving the traveling salesman problem (in Russian).Proceedings of the USSR Conference on System Research (Moscow, USSR), 184–185 , 1984.Google Scholar
- 14.ILOG Cplex 8.1: User’s Manual and Reference Manual, ILOG, S.A., tt http://www.ilog.com, 2003.Google Scholar
- 15.ILOG Cplex 10.0: User’s Manual and Reference Manual, ILOG, S.A., tt http://www.ilog.com, 2006.Google Scholar
- 17.F. Li, B.L. Golden, and E.A. Wasil. Very large-scale vehicle routing: new test problems, algorithms, and results.Computers and Operations Research, 32:1165-1179, 2005.Google Scholar
- 19.D. Mester and O. Bräysy. Active guided evolution strategies for large scale vehicle routing problems. Working paper, University of Haifa, Israel, 2004.Google Scholar
- 25.C. Rego and C. Roucairol. A parallel tabu search algorithm using ejection chains for the vehicle routing problem, inMeta-Heuristics: Theory and Applications (I.H. Osman and J.P. Kelly eds.). Kluwer, Boston, MA, 661–675 , 1996.Google Scholar
- 28.V.I. Sarvanov and N.N. Doroshko. The approximate solution of the travelling salesman problem by a local algorithm with scanning neighborhoods of factorial cardinality in cubic time (in Russian), inSoftware: Algorithms and Programs, 31. Mathematical Institute of the Byelorussian Academy of Sciences, Minsk, 11–13 , 1981.Google Scholar
- 29.E.D. Taillard. Eric Taillard’s Page, Vehicle Routing Instances, http://mistic.heig-vd.ch/taillard/problemes.dir/vrp.dir/vrp.html..
- 32.P. Toth and D. Vigo. An overview of vehicle routing problems, inThe Vehicle Routing Problem (P. Toth and D. Vigo eds.). SIAM Monographs on Discrete Mathematics and Applications, 2002.Google Scholar
- 34.P. Toth and D. Vigo.The Vehicle Routing Problem (P. Toth and D. Vigo eds.). SIAM Monographs on Discrete Mathematics and Applications, 2002.Google Scholar
- 35.D. Vigo. VRPLIB: A Vehicle Routing Problem LIBrary, tt http://www.or.deis.unibo.it/research.html.Google Scholar
- 37.J. Xu and J.P. Kelly. A network flow-based tabu search heuristic for the vehicle routing problem.Transportation Science, 30:379–393, 1996.Google Scholar