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The Capacitated Arc Routing Problem: Valid Inequalities and Facets

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Abstract

In this paper we study the polyhedron associated with the Capacitated Arc Routing Problem (CARP) where a maximum number K of vehicles is available. We show that a subset of the facets of the CARP polyhedron depends only on the demands of the required edges and they can be derived from the study of the Generalized Assignment Problem (GAP). The conditions for a larger class of valid inequalities to define facets of the CARP polyhedron still depend on the properties of the GAP polyhedron. We introduce the special case of the CARP where all the required edges have unit demand (CARPUD) to avoid the number problem represented by the GAP. This allows us to make a polyhedral study in which the conditions for the inequalities to be facet inducing are easily verifiable. We give necessary and sufficient conditions for a variety of inequalities, which are valid for CARP, to be facet inducing for CARPUD.

The resulting partial description of the polyhedron has been used to develop a cutting plane algorithm for the Capacitated Arc Routing Problem. The lower bound provided by this algorithm outperformed all the existing lower bounds for the CARP on a set of 34 instances taken from the literature.

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References

  1. A. Assad, W. Pearn, and B.L. Golden, "The capacitated chinese postman problem: Lower bounds and solvable cases," American J. of Mathematical and Man. Sci., vol. 7, pp. 63-88, 1987.

    Google Scholar 

  2. A. Assad and B.L. Golden, "Arc routing methods and applications," Handbook on Operations Research and Management Science, North-Holland, vol. 6, pp. 375-483, 1995.

    Google Scholar 

  3. E. Balas and E. Zemel, "Facets of the knapsack polytope from minimal covers," SIAM Journal on Applied Mathematics, vol. 34, pp. 119-148, 1978.

    Google Scholar 

  4. J.M. Belenguer, "El poliedro del problema de rutas por arcos con capacidades," Ph.D. Thesis, Universitat de Valencia, 1990.

  5. J.M. Belenguer and E. Benavent, “Polyhedral results on the capacitated arc routing problem,” Dep. Estadística e Inv. Op., T.R. 1-92, Univ. de Valencia, 1992.

  6. J.M. Belenguer, E. Benavent, and F. Cognata, "A tabu heuristic for the capacitated arc routing problem," 1996, in preparation.

  7. E. Beltrami and L. Bodin, "Networks and vehicle routing for municipal waste collection," Networks, vol. 4, pp. 65-94, 1974.

    Google Scholar 

  8. E. Benavent, V. Campos, A. Corberan, and E. Mota, "The capacitated arc routing problem. A heuristic algorithm," Questiió, vol. 14, nos.1-3, pp. 107-122, 1990.

    Google Scholar 

  9. E. Benavent, V. Campos, A. Corberan, and E. Mota, "The capacitated arc routing problem. Lower bounds," Networks, vol. 22, pp. 669-690, 1992.

    Google Scholar 

  10. L.D. Bodin and S.J. Kursh, "A detailed description of a computer system for the routing and scheduling of street sweepers," Computers and Oper. Res., vol. 6, pp. 181-198, 1979.

    Google Scholar 

  11. N. Christofides, “The optimum traversal of a graph,” OMEGA, vol. 1, pp. 719-732, 1973.

    Google Scholar 

  12. CPLEX Version 2.0, CPLEX Optimization Inc., 1992.

  13. G. Cornuejols and F. Harche, "Polyhedral study of the capacitated vehicle routing problem," Mathematical Programming, vol. 60, pp. 21-52, 1993.

    Google Scholar 

  14. R.W. Eglese and L.Y.O. Li, "Efficient routing for winter gritting," Journal of the Operational Research Society, vol. 43, pp. 1031-1034, 1992.

    Google Scholar 

  15. H.A. Eiselt, M. Gendreau, and G. Laporte, "Arc routing problems, Part 1: The Chinese postman problem," Operations Research, vol. 43, pp. 231-242, 1995.

    Google Scholar 

  16. H.A. Eiselt, M. Gendreau, and G. Laporte, "Arc routing problems, Part 2: The rural postman problem," Operations Research, vol. 43, pp. 399-414, 1995.

    Google Scholar 

  17. M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, 1979.

  18. B.L. Golden and R. Wong, "Capacitated arc routing problems," Networks, vol. 11, pp. 305-315, 1981.

    Google Scholar 

  19. B.L. Golden, J.S. DeArmon, and E.K. Baker, "Computational experiments with algorithms for a class of routing problems," Computers and Oper. Res., vol. 10, pp. 47-59, 1983.

    Google Scholar 

  20. E.S. Gottlieb and M.R. Rao, "The generalized assignment problem: Valid inequalities and facets," Mathematical Programming, vol. 46, pp. 31-52, 1990.

    Google Scholar 

  21. E.S. Gottlieb and M.R. Rao, "(1, k)-Configuration facets for the generalized assignment problem," Mathematical Programming, vol. 46, pp. 53-60, 1990.

    Google Scholar 

  22. F. Harche and G. Rinaldi, "The capacity inequalities for the capacitated vehicle routing problem," 1993, in preparation.

  23. G.L. Nemhauser and L.A. Wolsey, Integer and Combinatorial Optimization, Wiley: New York, 1988.

    Google Scholar 

  24. M.W. Padberg, "(1, k)-Configurations and facets for packing problems," Mathematical Programming, vol. 18, pp. 94-99, 1980.

    Google Scholar 

  25. M.W. Padberg and M.R. Rao, "Odd minimum cut-sets and b-matchings," Mathematics of Oper. Res., vol. 7, pp. 67-80, 1982.

    Google Scholar 

  26. M.W. Padberg and G. Rinaldi, "Optimization of a 532-city symmetric traveling salesman problem by branch-and-cut," Operations Research Letters, vol. 6, pp. 1-7, 1987.

    Google Scholar 

  27. W.L. Pearn, "New lower bounds for the capacitated arc routing problems," Networks, vol. 18, pp. 181-191, 1988.

    Google Scholar 

  28. W.L. Pearn, "Approximate solutions for the capacitated arc routing problem," Computers and Oper. Res., vol. 1, no.16, pp. 589-600, 1989.

    Google Scholar 

  29. W.L. Pearn, "Augment-insert algorithms for the capacitated arc routing problem," Computers and Oper. Res., no. 18, pp. 189-198, 1991.

    Google Scholar 

  30. W.L. Pearn, A.A. Assad, and B. L. Golden "Transforming arc routing into node routing problems," Computers and Oper. Res., no. 14, pp. 285-288, 1987.

    Google Scholar 

  31. H. Stern and M. Dror, "Routing electric meter readers," Computers and Oper. Res., vol. 6, pp. 209-223, 1979.

    Google Scholar 

  32. Z. Win, "Contributions to routing problems," Ph.D. Thesis, University of Augsburg, 1988.

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Authors and Affiliations

  1. Departamento de Estadística e Investigación Operativa, Universitat de València, Dr. Moliner 50, 46100, Burjassot (Valencia), Spain

    J.M. Belenguer & E. Benavent

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  1. J.M. Belenguer
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  2. E. Benavent
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Belenguer, J., Benavent, E. The Capacitated Arc Routing Problem: Valid Inequalities and Facets. Computational Optimization and Applications 10, 165–187 (1998). https://doi.org/10.1023/A:1018316919294

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