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.
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References
Bertsimas, D.: Probabilistic combinatorial optimization problems. PhD Thesis, Massachusetts Institute of Technology, Cambridge, (1988)
Jaillet, P.: Probabilistic traveling salesman problems. PhD Thesis, Massachusetts Institute of Technology, Cambridge, (1985)
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)
Jézéquel, A.: Probabilistic vehicle routing problems. Master’s Thesis, Massachusetts Institute of Technology, Cambridge, (1985)
Bertsimas, D., Jaillet, P., Odoni, A.: A priori optimization. Oper. Res. 38(6), 1019–1033 (1990)
Tillman, F.: The multiple terminal delivery problem with probabilistic demands. Transp. Sci. 3(3), 192–204 (1969)
Bertsimas, D.J.: A vehicle routing problem with stochastic demand. Oper. Res. 40(3), 574–585 (1992)
Laporte, G., Louveaux, F., Mercure, H.: The vehicle routing problem with stochastic travel times. Transp. Sci. 26(3), 161–170 (1992)
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)
Gendreau, M., Laporte, G., Séguin, R.: Stochastic vehicle routing. Eur. J. Oper. Res. 88, 3–12 (1996)
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)
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)
Hjorring, C., Holt, J.: New optimality cuts for a single-vehicle stochastic routing problem. Ann. Oper. Res. 86(0), 569–584 (1999)
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)
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)
Hoos, H., Stützle, T.: Stochastic Local Search: Foundations and Applications. Morgan Kaufmann, San Francisco (2005)
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)
Yang, W., Mathur, K., Ballou, R.H.: Stochastic vehicle routing problem with restocking. Transp. Sci. 34(1), 99–112 (2000)
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)
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)
Secomandi, N., Margot, F.: Reoptimization approaches for the vehicle-routing problem with stochastic demands. Oper. Res. 57(1), 214–230 (2009)
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)
Balaprakash, P.: Estimation-based metaheuristics for stochastic combinatorial optimization: Case studies in stochastic routing problems. PhD Thesis, Université Libre de Bruxelles, Brussels, Belgium (2010)
Stewart Jr, W.R., Golden, B.L.: Stochastic vehicle routing: a comprehensive approach. Eur. J. Oper. Res. 14(4), 371–385 (1983)
Dror, M., Laporte, G., Trudeau, P.: Vehicle routing with stochastic demands: properties and solution frameworks. Transp. Sci. 23(3), 166–176 (1989)
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)
Psaraftis, H.: Dynamic vehicle routing: status and prospects. Ann. Oper. Res. 61(1), 143–164 (1995)
Secomandi, N.: Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands. Comput. Oper. Res. 27(11), 1201–1225 (2000)
Secomandi, N.: A rollout policy for the vehicle routing problem with stochastic demands. Oper. Res. 49(5), 796–802 (2001)
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)
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)
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)
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)
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)
Moscato, P.: On evolution, search, optimization, genetic algorithms and martial arts: towards memetic algorithms. Caltech Concurrent Computation Program Report 826, Caltech, Pasadena, California (1989)
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)
Dorigo, M., Stützle, T.: Ant Colony Optimization. MIT Press, Cambridge (2004)
Dorigo, M., Birattari, M.: Swarm intelligence. Scholarpedia 2(9), 1462 (2007)
Séguin, R.: Problèmes stochastiques de tournées de véhicules. PhD Thesis, Université de Montréal, Montréal, Canada (1994)
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)
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)
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)
Tukey, J.W.: Comparing individual means in the analysis of variance. Biometrics 5(2), 99–114 (1949)
Rubinstein, R.Y.: Simulation and the Monte Carlo Method. Wiley, New York (1981)
Bentley, J.L.: Fast algorithms for geometric traveling salesman problems. ORSA J. Comput. 4(4), 387–411 (1992)
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)
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)
Merz, P., Freisleben, B.: Memetic algorithms for the traveling salesman problem. Complex Syst. 13(4), 297–345 (2001)
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)
Johnson, D.S., McGeoch, L.A., Rego, C., Glover, F.: 8th DIMACS implementation challenge (2001)
Stützle, T.: ACOTSP: A software package of various ant colony optimization algorithms applied to the symmetric traveling salesman problem (2002)
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)
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)
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)
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)
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)
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.
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Balaprakash, P., Birattari, M., Stützle, T. et al. Estimation-based metaheuristics for the single vehicle routing problem with stochastic demands and customers. Comput Optim Appl 61, 463–487 (2015). https://doi.org/10.1007/s10589-014-9719-z
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DOI: https://doi.org/10.1007/s10589-014-9719-z