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A new ILP-based refinement heuristic for Vehicle Routing Problems

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Abstract

In this paper we address the Distance-Constrained Capacitated Vehicle Routing Problem (DCVRP), where k minimum-cost routes through a central depot have to be constructed so as to cover all customers while satisfying, for each route, both a capacity and a total-distance-travelled limit.

Our starting point is the following refinement procedure proposed in 1981 by Sarvanov and Doroshko for the pure Travelling Salesman Problem (TSP): given a starting tour, (a) remove all the nodes in even position, thus leaving an equal number of ``empty holes'' in the tour; (b) optimally re-assign the removed nodes to the empty holes through the efficient solution of a min-sum assignment (weighted bipartite matching) problem. We first extend the Sarvanov-Doroshko method to DCVRP, and then generalize it. Our generalization involves a procedure to generate a large number of new sequences through the extracted nodes, as well as a more sophisticated ILP model for the reallocation of some of these sequences. An important feature of our method is that it does not rely on any specialized ILP code, as any general-purpose ILP solver can be used to solve the reallocation model.

We report computational results on a large set of capacitated VRP instances from the literature (with symmetric/asymmetric costs and with/without distance constraints), along with an analysis of the performance of the new method and of its features. Interestingly, in 13 cases the new method was able to improve the best-know solution available from the literature.

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References

  1. Ahuja, R.K., Ergun, O., Orlin, J.B., Punnen, A.B.: A survey of very large-scale neighborhood search techniques. Discrete Applied Math. 123, 75–102 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  2. Balas, E., Simonetti, N.: Linear time dynamic programming algorithms for new classes of Restricted TSPs. Informs J. Comput. 13, 56–75 (2001)

    Article  MathSciNet  Google Scholar 

  3. Baldacci, R., Hadjiconstantinou, E.A., Mingozzi, A.: An exact algorithm for the capacitated vehicle routing problem based on a two-commodity network flow formulation. Oper. Res. 2004 (forthcoming)

  4. Burkard, R.E., Deineko, V.G.: Polynomially solvable cases of the traveling salesman problem and a new exponential neighborhood. Comput. 54, 191–211 (1995)

    MATH  MathSciNet  Google Scholar 

  5. Cordeau, J.-F., Gendreau, M., Hertz, A., Laporte, G., Sormany, J.-S.: New heuristics for the vehicle routing problem. In: A. Langevin, D. Riopel (eds.). Logistics Systems: Design and Optimization, Kluwer, Boston, 2005 forthcoming

  6. Christofides, N., Eilon, S.: Expected distances in distribution problems. Oper. Res. Quarterly 20 (4), 437–443 (1969)

    Google Scholar 

  7. Christofides, N., Mingozzi, A., Toth, P.: The vehicle routing problem. Combinatorial Optimization (Wiley, Chichester), 1979

  8. Deineko, V.G., Woeginger, G.J.: A study of exponential neighborhoods for the travelling salesman problem and for the quadratic assignment problem. Math. Programming 87, 519–542 (2000)

    MATH  MathSciNet  Google Scholar 

  9. Dell'Amico, M., Fischetti, M., Toth, P.: Heuristic algorithms for the multiple depot vehicle scheduling problem. Management Sci. 39, 115–125 (1993)

    MATH  Google Scholar 

  10. Ergun, O.: New neighborhood search algorithms based on exponentially large neighborhoods. Operations Research Center Thesis, MIT, 2001

  11. Ergun, O., Orlin, J.B.: Two dynamic programming methodologies in very large scale neighborhood search applied to the Traveling Salesman Problem. Working Paper 4463-03, MIT Sloan School of Management, 2003

  12. Ergun, O., Orlin, J.B., Steele-Feldman, A.: A computational study on a large-scale neighborhood search algorithm for the vehicle routing problem with capacity and distance constraints. Working paper, School of Ind. and Sys. Engr. Georgia Institute of Technology, Atlanta, 2002

  13. Ergun, O., Orlin, J.B., Steele-Feldman, A.: Creating very large scale neighborhoods out of smaller ones by compounding moves: a study on the Vehicle Routing Problem. Working Paper 4393-02, MIT Sloan School of Management, 2002

  14. Firla, R.T., Spille, B., Weismantel, R.: Exponentialirreducible neighborhoods for combinatorial optimization problems. Math. Meth. Oper. Res. 56, 29–44 (2002)

    MATH  MathSciNet  Google Scholar 

  15. Fischetti, M., Lodi, A., Martello, S., Toth, P.: A polyhedral approach to simplified crew scheduling and vehicle scheduling problems. Management Sci. 47 (6), 833–850 (2001)

    Google Scholar 

  16. Fischetti, M., Toth, P., Vigo, D.: A branch and bound algorithm for the capacitated vehicle routing problem on directed graphs. Oper. Res. 42, 846–859 (1994)

    MATH  MathSciNet  Google Scholar 

  17. Fisher, M.: Vehicle routing. In: M.O. Ball, T.L. Magnanti, C.L. Monma, G.L. Nemhauser (eds.). Network Routing, Handbooks in OR & MS Vol. 8, Elsevier, Amsterdam, 1995

  18. Fisher, M.L., Jaikumar, R.: A generalized assignment heuristic for vehicle routing. Networks 11, 109–124 (1981)

    MathSciNet  Google Scholar 

  19. Fukasawa, R., Poggi de Aragão, M., Reis, M., Uchoa, E: Robust branch-and-cut-and-price for the capacitated vehicle routing problem, Relatórios de Pesquisa em Engenharia de Produção, RPEP 3 (8), Universidade Federal Fluminense, 2003

  20. Gillett, B.E., Miller, L.R.: A heuristic algorithm for the vehicle dispatch problem. Oper. Res. 22, 340–349 (1974)

    Article  MATH  Google Scholar 

  21. Gendreau, M., Hertz, A., Laporte, G.: A Tabu Search Heuristic for the VRP, Technical Reoprt CRT-777, 1991

  22. Gendreau, M., Hertz, A., Laporte, G.: A tabu search heuristic for the vehicle routing problem. Management Sci. 40, 1276–1290 (1994)

    MATH  Google Scholar 

  23. Gendreau, M., Laporte, G., Potvin, J.-Y.: Metaheuristics for the capacitated VRP. In: P. Toth, D. Vigo (eds.). The Vehicle Routing Problem. SIAM Monographs on Discrete Mathematics and Applications, 2002

  24. Golden, B.L., Wasil, E.A., Kelly, J.P., Chao, I-M.: Metaheuristics in Vehicle Routing. In: T.G. Crainic, G. Laporte (eds.). Fleet Management and Logictics. Kluwer, Boston, 1998 pp. 33–56

  25. Glover, F., Punnen, A.P.: The traveling salesman problem: New solvable cases and linkages with the development of approximation algorithms. J. Oper. Res. Soc. 48, 502–510 (1997)

    Article  MATH  Google Scholar 

  26. Gutin, G.M.: On an approach to solving the traveling salesman problem, Proc. The USSR Conference on System Research (Moscow, USSR), 1984 pp. 184–185 (in Russian)

  27. Gutin, G.M.: Exponential neighborhood local search for the traveling salesman problem. Comput. Oper. Res. 26, 313–320 (1999)

    MATH  MathSciNet  Google Scholar 

  28. Gutin, G., Yeo, A., Zverovitch, A.: Exponential neighborhoods and domination analysis for the TSP. In: G. Gutin, A. Punnen (eds.). The Traveling Salesman Problem and its Variations. Kluwer, 2002 pp. 223–256

  29. ILOG Cplex 8.1: user's manual and reference manual, ILOG, S.A., http://www.ilog.com/, 2003

  30. ILOG Concert Technology 1.2: User's manual and reference manual, ILOG, S.A., http://www.ilog.com/, 2003

  31. Johnson, D.S., McGeoch, L.A.: Experimental analysis of heuristics for the STSP. In: G. Gutin, A.P. Punnen (eds.). The Traveling Salesman Problem and Its Variations Kluwer Academic Publishers, 9, pp. 369–487, 2002

  32. Laporte, G., Semet, F.: Classical heuristics for the capacitated VRP. In: P. Toth, D. Vigo (eds.). The Vehicle Routing Problem SIAM Monographs on Discrete Mathematics and Applications, Chap. 5, 2002

  33. Li, F., Golden, B.L., Wasil, E.A.: Very large-scale vehicle routing: New test problems, algorithms, and results. Comput. Oper. Res. 2004 (forthcoming)

  34. Lysgaard, J., Letchford, A.N., Eglese, R.W.: A new branch-and-cut algorithm for the capacitated vehicle routing problem. Math. Prog. 100 (2), 423–442 2004

    MathSciNet  Google Scholar 

  35. Mester, D., Braysy, O.: Active guided evolution strategies for the large scale vehicle routing problems with time windows. Comput. Oper. Res. 2004 (forthcoming)

  36. Punnen, A.P.: The Traveling Salesman Problem: new polynomial approximation and domination analysis. J. Information Optim. Sci. 22, 191–206 (2001)

    MATH  MathSciNet  Google Scholar 

  37. Ralphs, T.: Branch and Cut and Price Applications: Vehicle Routing, http://branchandcut.org/VRP/

  38. Rego, C., Glover, F.: Local search and metaheuristcs. In: G. Gutin, A. Punnen (eds.). The Traveling Salesman Problem and its Variations, Kluwer, 2002 pp. 309–368

  39. Reimann, M., Doerner, K., Hartl, R.F.: D-Ants: Savings based ants divide and conquer the vehicle routing problem. Comput. Oper. Res. 2004 (forthcoming)

  40. Sarvanov, V.I., Doroshko, N.N.: The approximate solution of the travelling salesman problem by a local algorithm with scanning neighborhoods of factorial cardinality in cubic time, (in Russian), Software: Algorithms and Programs 31, Mathematical Institute of the Belorussian Academy of Sciences, Minsk, 1981, pp. 11–13

  41. Taillard, E.: Eric Taillard's Page, Vehicle Routing Instances, http://ina.eivd.ch/collaborateurs/etd/problemes.dir/vrp.dir/vrp.html.

  42. Tarantilis, C.-D., Kiranoudis, C.T.: Bone Route: An adaptive memory-based method for effective fleet management. Ann. Oper. Res. 115, 227–241 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  43. Toth, P., Vigo, D.: An overview of vehicle routing problems. In: P. Toth, D. Vigo (eds.). The Vehicle Routing Problem, SIAM Monographs on Discrete Mathematics and Applications, 2002

  44. Vigo, D.: VRPLIB: A Vehicle Routing Problem LIBrary, http://www.or.deis.unibo.it/research_pages/OR instances/VRPLIB/VRPLIB. html

  45. Wenger, K.M.: Generic cut generation methods for routing problems. Ph.D Dissertation, Institute of Computer Science, University of Heidelberg, 2003

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Authors and Affiliations

  1. DEI, University of Padova, Via Gradenigo 6/A, 35131, Padova, Italy

    Roberto De Franceschi & Matteo Fischetti

  2. DEIS, University of Bologna, Viale Risorgimento 2, 40136, Bologna, Italy

    Paolo Toth

Authors
  1. Roberto De Franceschi
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  2. Matteo Fischetti
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  3. Paolo Toth
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Correspondence to Paolo Toth.

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Work supported by M.I.U.R. and by C.N.R., Italy.

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Franceschi, R., Fischetti, M. & Toth, P. A new ILP-based refinement heuristic for Vehicle Routing Problems. Math. Program. 105, 471–499 (2006). https://doi.org/10.1007/s10107-005-0662-8

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