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Computational Optimization and Applications

, Volume 61, Issue 2, pp 463–487 | Cite as

Estimation-based metaheuristics for the single vehicle routing problem with stochastic demands and customers

  • Prasanna Balaprakash
  • Mauro Birattari
  • Thomas Stützle
  • Marco Dorigo
Article

Abstract

The vehicle routing problem with stochastic demands and customers (VRPSDC) requires finding the optimal route for a capacitated vehicle that delivers goods to a set of customers, where each customer has a fixed probability of requiring being visited and a stochastic demand. For large instances, the evaluation of the cost function is a primary bottleneck when searching for high quality solutions within a limited computation time. We tackle this issue by using an empirical estimation approach. Moreover, we adopt a recently developed state-of-the-art iterative improvement algorithm for the closely related probabilistic traveling salesman problem. We integrate these two components into several metaheuristics and we show that they outperform substantially the current best algorithm for this problem.

Keywords

Metaheuristics Empirical estimation Vehicle routing with stochastic demands and customers 

Notes

Acknowledgments

This research has been supported by “E-SWARM – Engineering Swarm Intelligence Systems”, an European Research Council Advanced Grant awarded to Marco Dorigo (Grant Number 246939). The authors acknowledge support from the Fonds de la Recherche Scientifique, F.R.S.-FNRS of the French Community of Belgium.

References

  1. 1.
    Bertsimas, D.: Probabilistic combinatorial optimization problems. PhD Thesis, Massachusetts Institute of Technology, Cambridge, (1988)Google Scholar
  2. 2.
    Jaillet, P.: Probabilistic traveling salesman problems. PhD Thesis, Massachusetts Institute of Technology, Cambridge, (1985)Google Scholar
  3. 3.
    Jaillet, P.: A priori solution of a travelling salesman problem in which a random subset of the customers are visited. Oper. Res. 36(6), 929–936 (1988)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Jézéquel, A.: Probabilistic vehicle routing problems. Master’s Thesis, Massachusetts Institute of Technology, Cambridge, (1985)Google Scholar
  5. 5.
    Bertsimas, D., Jaillet, P., Odoni, A.: A priori optimization. Oper. Res. 38(6), 1019–1033 (1990)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Tillman, F.: The multiple terminal delivery problem with probabilistic demands. Transp. Sci. 3(3), 192–204 (1969)CrossRefGoogle Scholar
  7. 7.
    Bertsimas, D.J.: A vehicle routing problem with stochastic demand. Oper. Res. 40(3), 574–585 (1992)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Laporte, G., Louveaux, F., Mercure, H.: The vehicle routing problem with stochastic travel times. Transp. Sci. 26(3), 161–170 (1992)CrossRefMATHGoogle Scholar
  9. 9.
    Jaillet, P.: Stochastic routing problems. In: Andreatta, G., Mason, F., Serafini, P. (eds.) Advanced School on Stochastics in Combinatorial Optimization, pp. 192–213. World Scientific, Singapore (1987)Google Scholar
  10. 10.
    Gendreau, M., Laporte, G., Séguin, R.: Stochastic vehicle routing. Eur. J. Oper. Res. 88, 3–12 (1996)CrossRefMATHGoogle Scholar
  11. 11.
    Laporte, G., Louveaux, F.V., Mercure, H.: Models and exact solutions for a class of stochastic location-routing problems. Eur. J. Oper. Res. 39(1), 71–78 (1989)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Gendreau, M., Laporte, G., Séguin, R.: An exact algorithm for the vehicle routing problem with stochastic demands and customers. Transp. Sci. 29(2), 143–155 (1995)CrossRefMATHGoogle Scholar
  13. 13.
    Hjorring, C., Holt, J.: New optimality cuts for a single-vehicle stochastic routing problem. Ann. Oper. Res. 86(0), 569–584 (1999)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Laporte, G., Louveaux, F., Van Hamme, L.: An integer L-shaped algorithm for the capacitated vehicle routing problem with stochastic demands. Oper. Res. 50(3), 415–423 (2002)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Rei, W., Gendreau, M., Soriano, P.: Local branching cuts for the 0–1 integer L-shaped algorithm. Technical Report CIRRELT-2007-23, CIRRELT, Montréal, Canada (2007)Google Scholar
  16. 16.
    Hoos, H., Stützle, T.: Stochastic Local Search: Foundations and Applications. Morgan Kaufmann, San Francisco (2005)Google Scholar
  17. 17.
    Gendreau, M., Laporte, G., Séguin, R.: A tabu search algorithm for the vehicle routing problem with stochastic demands and customers. Oper. Res. 44(3), 469–477 (1996)CrossRefMATHGoogle Scholar
  18. 18.
    Yang, W., Mathur, K., Ballou, R.H.: Stochastic vehicle routing problem with restocking. Transp. Sci. 34(1), 99–112 (2000)CrossRefMATHGoogle Scholar
  19. 19.
    Chepuri, K., Homem-de-Mello, T.: Solving the vehicle routing problem with stochastic demands using the cross-entropy method. Ann. Oper. Res. 134(1), 153–181 (2005)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Bianchi, L., Birattari, M., Chiarandini, M., Manfrin, M., Mastrolilli, M., Paquete, L., Rossi-Doria, O., Schiavinotto, T.: Hybrid metaheuristics for the vehicle routing problem with stochastic demands. J. Math. Model. Algorithms 5(1), 91–110 (2006)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Secomandi, N., Margot, F.: Reoptimization approaches for the vehicle-routing problem with stochastic demands. Oper. Res. 57(1), 214–230 (2009)CrossRefMATHGoogle Scholar
  22. 22.
    Rei, W., Gendreau, M., Soriano, P.: A hybrid Monte Carlo local branching algorithm for the single vehicle routing problem with stochastic demands. Transp. Sci. 44(1), 136–146 (2010)CrossRefGoogle Scholar
  23. 23.
    Balaprakash, P.: Estimation-based metaheuristics for stochastic combinatorial optimization: Case studies in stochastic routing problems. PhD Thesis, Université Libre de Bruxelles, Brussels, Belgium (2010)Google Scholar
  24. 24.
    Stewart Jr, W.R., Golden, B.L.: Stochastic vehicle routing: a comprehensive approach. Eur. J. Oper. Res. 14(4), 371–385 (1983)CrossRefMATHGoogle Scholar
  25. 25.
    Dror, M., Laporte, G., Trudeau, P.: Vehicle routing with stochastic demands: properties and solution frameworks. Transp. Sci. 23(3), 166–176 (1989)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Dror, M.: Modeling vehicle routing with uncertain demands as a stochastic program: properties of the corresponding solution. Eur. J. Oper. Res. 64(3), 432–441 (1993)CrossRefMATHGoogle Scholar
  27. 27.
    Psaraftis, H.: Dynamic vehicle routing: status and prospects. Ann. Oper. Res. 61(1), 143–164 (1995)CrossRefMATHGoogle Scholar
  28. 28.
    Secomandi, N.: Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands. Comput. Oper. Res. 27(11), 1201–1225 (2000)CrossRefMATHGoogle Scholar
  29. 29.
    Secomandi, N.: A rollout policy for the vehicle routing problem with stochastic demands. Oper. Res. 49(5), 796–802 (2001)CrossRefMATHGoogle Scholar
  30. 30.
    Birattari, M., Balaprakash, P., Stützle, T., Dorigo, M.: Estimation-based local search for stochastic combinatorial optimization using delta evaluations: a case study in the probabilistic traveling salesman problem. INFORMS J. Comput. 20(4), 644–658 (2008)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Balaprakash, P., Birattari, M., Stützle, T., Dorigo, M.: Adaptive sample size and importance sampling in estimation-based local search for the probabilistic traveling salesman problem. Eur. J. Oper. Res. 199(1), 98–110 (2009)CrossRefMATHGoogle Scholar
  32. 32.
    Balaprakash, P., Birattari, M., Stützle, T., Yuan, Z., Dorigo, M.: Estimation-based ant colony optimization and local search for the probabilistic traveling salesman problem. Swarm Intell 3(3), 223–242 (2009)CrossRefGoogle Scholar
  33. 33.
    Balaprakash, P., Birattari, M., Stützle, T., Dorigo, M.: Estimation-based metaheuristics for the probabilistic traveling salesman problem. Comput. Oper. Res. 37(11), 1939–1951 (2010)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Lourenço, H.R., Martin, O., Stützle, T.: Iterated local search. In: Glover, F., Kochenberger, G. (eds.) Handbook of Metaheuristics. International Series in Operations Research and Management Science, vol. 57, pp. 321–353. Kluwar Academic Publishers, Norwell (2002)Google Scholar
  35. 35.
    Moscato, P.: On evolution, search, optimization, genetic algorithms and martial arts: towards memetic algorithms. Caltech Concurrent Computation Program Report 826, Caltech, Pasadena, California (1989)Google Scholar
  36. 36.
    Moscato, P.: Memetic algorithms: a short introduction. In: Corne, D., Dorigo, M., Glover, F. (eds.) New Ideas in Optimization, pp. 219–234. McGraw Hill, London (1999)Google Scholar
  37. 37.
    Dorigo, M., Stützle, T.: Ant Colony Optimization. MIT Press, Cambridge (2004)CrossRefMATHGoogle Scholar
  38. 38.
    Dorigo, M., Birattari, M.: Swarm intelligence. Scholarpedia 2(9), 1462 (2007)CrossRefGoogle Scholar
  39. 39.
    Séguin, R.: Problèmes stochastiques de tournées de véhicules. PhD Thesis, Université de Montréal, Montréal, Canada (1994)Google Scholar
  40. 40.
    Bowler, N.E., Fink, T.M.A., Ball, R.C.: Characterization of the probabilistic traveling salesman problem. Phys. Rev. E 68(3), 036703–036710 (2003)CrossRefGoogle Scholar
  41. 41.
    Gutjahr, W.J.: A converging ACO algorithm for stochastic combinatorial optimization. In: Albrecht, A., Steinhofl, K. (eds.) Stochastic Algorithms: Foundations and Applications. LNCS, vol. 2827, pp. 10–25. Springer, Berlin (2003)CrossRefGoogle Scholar
  42. 42.
    Gutjahr, W.J.: S-ACO: an ant based approach to combinatorial optimization under uncertainty. In: Dorigo, M., Birattari, M., Blum, C., Gambardella, L.M. (eds.) Ant Colony Optimization and Swarm Intelligence, 5th International Workshop, ANTS 2004. LNCS, vol. 3172, pp. 238–249. Springer, Berlin (2004)Google Scholar
  43. 43.
    Tukey, J.W.: Comparing individual means in the analysis of variance. Biometrics 5(2), 99–114 (1949)CrossRefMathSciNetGoogle Scholar
  44. 44.
    Rubinstein, R.Y.: Simulation and the Monte Carlo Method. Wiley, New York (1981)CrossRefMATHGoogle Scholar
  45. 45.
    Bentley, J.L.: Fast algorithms for geometric traveling salesman problems. ORSA J. Comput. 4(4), 387–411 (1992)CrossRefMATHMathSciNetGoogle Scholar
  46. 46.
    Bianchi, L., Knowles, J., Bowler, N.: Local search for the probabilistic traveling salesman problem: correction to the 2-p-opt and 1-shift algorithms. Eur. J. Oper. Res. 162, 206–219 (2005)CrossRefMATHMathSciNetGoogle Scholar
  47. 47.
    Bianchi, L., Campbell, A.: Extension of the 2-p-opt and 1-shift algorithms to the heterogeneous probabilistic traveling salesman problem. Eur. J. Oper. Res. 176(1), 131–144 (2007)CrossRefMATHMathSciNetGoogle Scholar
  48. 48.
    Merz, P., Freisleben, B.: Memetic algorithms for the traveling salesman problem. Complex Syst. 13(4), 297–345 (2001)MATHMathSciNetGoogle Scholar
  49. 49.
    Dorigo, M., Gambardella, L.M.: Ant colony system: a cooperative learning approach to the traveling salesman problem. IEEE Trans. Evol. Comput. 1(1), 53–66 (1997)CrossRefGoogle Scholar
  50. 50.
    Johnson, D.S., McGeoch, L.A., Rego, C., Glover, F.: 8th DIMACS implementation challenge (2001)Google Scholar
  51. 51.
    Stützle, T.: ACOTSP: A software package of various ant colony optimization algorithms applied to the symmetric traveling salesman problem (2002)Google Scholar
  52. 52.
    Penky, J.F., Miller, D.L.: A staged primal-dual algorithm for finding a minimum cost perfect two-matching in an undirected graph. ORSA J. Comput. 6(1), 68–81 (1994)CrossRefGoogle Scholar
  53. 53.
    Johnson, D.S., McGeoch, L.A.: The travelling salesman problem: a case study in local optimization. In: Aarts, E.H.L., Lenstra, J.K. (eds.) Local Search in Combinatorial Optimization, pp. 215–310. Wiley, Chichester (1997)Google Scholar
  54. 54.
    Balaprakash, P., Birattari, M., Stützle, T., Dorigo, M.: Estimation-based metaheuristics for the the vehicle routing problem with stochastic demands and customers. IRIDIA Supplementary page (2011)Google Scholar
  55. 55.
    Balaprakash, P., Birattari, M., Stützle, T.: Improvement strategies for the F-Race algorithm: sampling design and iterative refinement. In: Bartz-Beielstein, T., Blesa, M.J., Blum, C., Naujoks, B., Roli, A., Rudolph, G., Sampels, M. (eds.) Hybrid Metaheuristics. LNCS, vol. 4771, pp. 113–127. Springer, Berlin (2007)CrossRefGoogle Scholar
  56. 56.
    Birattari, M., Yuan, Z., Balaprakash, P., Stützle, T.: F-Race and iterated F-Race: an overview. In: Bartz-Beielstein, T., Chiarandini, M., Paquete, L., Preuss, M. (eds.) Empirical Methods for the Analysis of Optimization Algorithms. Springer, New York (2010)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Prasanna Balaprakash
    • 1
  • Mauro Birattari
    • 2
  • Thomas Stützle
    • 2
  • Marco Dorigo
    • 2
  1. 1.Mathematics and Computer Science DivisionArgonne National LaboratoryArgonneUSA
  2. 2.IRIDIA, CoDEUniversité Libre de BruxellesBrusselsBelgium

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