Abstract
The Vehicle Routing Problem with Time windows (VRPTW) is an extension of the capacity constrained Vehicle Routing Problem (VRP). The VRPTW is NP-Complete and instances with 100 customers or more are very hard to solve optimally. We represent the VRPTW as a multi-objective problem and present a genetic algorithm solution using the Pareto ranking technique. We use a direct interpretation of the VRPTW as a multi-objective problem, in which the two objective dimensions are number of vehicles and total cost (distance). An advantage of this approach is that it is unnecessary to derive weights for a weighted sum scoring formula. This prevents the introduction of solution bias towards either of the problem dimensions. We argue that the VRPTW is most naturally viewed as a multi-objective problem, in which both vehicles and cost are of equal value, depending on the needs of the user. A result of our research is that the multi-objective optimization genetic algorithm returns a set of solutions that fairly consider both of these dimensions. Our approach is quite effective, as it provides solutions competitive with the best known in the literature, as well as new solutions that are not biased toward the number of vehicles. A set of well-known benchmark data are used to compare the effectiveness of the proposed method for solving the VRPTW.
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References
J. Desrosier, Y. Dumas, M.M. Solomon, and F. Soumis, “Time constraint routing and scheduling,” in Handbooks in Operations Research and Management Science, Vol. 8, Network Routing, edited by M.O. Ball, T/L Magnanti, C.L Monma, G.L. Nemhauser (eds.). Elsevier Science Publishers: Amsterdam, pp. 35–139, 1995.
J.F. Cordeau, G. Desaulniers, J. Desrosiers, M.M. Solomon, and F. Soumis, “The VRP with time windows,” to appear in the Vehicle Routing Problem, Chapter 7, edited by P. Toth and D. Vigo, SIAM Monographs on Discrete Mathematics and Applications, 2001.
O. Braysy and M. Gendreau, “Vehicle routing problem with time windows, Part 1: Route construction and local search algorithms,” SINTEF Applied Mathematics Report, Department of Optimization, Norway, 2001.
O. Braysy and M. Gendreau, “Vehicle routing problem with time windows, Part II: Metaheuristics,” SINTEF Applied Mathematics Report, Department of Optimization, Norway, 2001.
M.R. Garey and D.S. Johnson, Computers and Intractability, A Guide to The Theory of NP-Completeness, W. H. Freeman and Company, 1979.
J.K. Lenstra and A.H.G. Rinnooy Kan, “Complexity of vehicle routing problem with time windows,” Networks, vol. 11, pp. 221–227, 1981.
N. Kohl. “Exact methods for time constrained routing and related scheduling problems,” PhD Thesis, Department of Mathematical Modeling, Technical University of Denmark, 1995.
L.M. Gambardella, E. Taillard, and G. Agazzi, “MACS-VRPTW: A multiple ant colony system for vehicle routing problems with time windows,” in edited by David Corne, Marco Dorigo, and Fred Glover, New Ideas in Optimization,, McGraw-Hill: London, 1999, pp. 63–76.
D.E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison Wesley, 1989.
K.Q. Zhu. “A diversity-controlling adaptive genetic algorithm for the vehicle routing problem with time windows,” in Proceedings of the 15th IEEE International Conference on Tools for Artificial Intelligence (ICTAI 2003), 2003, pp. 176–183.
B. Ombuki, M. Nakamura, and O. Maeda, “A hybrid search based on genetic algorithms and tabu search for vehicle routing,” in 6th IASTED Intl. Conf. On Artificial Intelligence and Soft Computing (ASC 2002), edited by A.B. Banff, H Leung, ACTA Press, pp. 176–181, July 2002.
H. Wee Kit, J. Chin, and A. Lim, “A hybrid genetic algorithm for the vehicle routing problem,” International Journal on Artificial Intelligence Tools (to appear).
J.Y. Potvin and S. Bengio, “The vehicle routing problem with time windows—Part II: Genetic search,” INFORMS Journal of Computing, vol. 8, pp. 165–172, 1996.
O. Braysy, “A new genetic algorithms for vehicle routing problem with time windows based on hybridization of a genetic algorithm and route construction heuristics,” in Proceedings of the University of Vaasa, Research Papers, p. 227, 1999.
K.Q. Zhu, “A new genetic algorithm for VRPTW,” in IC-AI 2000, Las Vegas, USA.
J. Homberger and H. Gehring. “Two evolutionary meta-heuristics for the vehicle routing problem with time windows,” INFORMS Journal on Computing, vol. 37, no. 3, pp. 297–318, 1999.
K.C. Tan, L.L. Hay, and O. Ke. “A hybrid genetic algorithm for solving vehicle routing problems with time window constraints,” Asia-Pacific Journal of Operational Research, vol. 18, no. 1, pp. 121–130, 2001.
S. Thangiah. “Vehicle routing with time windows using genetic algorithms,” in Applications Handbook of Genetic Algorithms: New Frontiers, Vol. II, CRC Press, Boca Raton, 1995, pp. 253–277.
W.C. Chiang and Russell, “Simulated annealing metaheuristic for the vehicle routing problem with time windows,” Annals of Operations Research, vol. 63, pp. 3–27, 1996.
L.M. Rousseau, M. Gendreau, and G. Peasant. “Using constraint-based operators to solve the vehicle routing problem with time windows,” Journal of Heuristics (forthcoming).
Y. Rochat and E.D. Taillard. “Probabilistic diversification and intensification in local search for vehicle routing,” Journal of Heuristics, vol. 1, pp. 147–167, 1995
W.C. Chiang and Russell. “A reactive tabu search metaheuristic for the vehicle routing problem with time windows,” INFORMS Journal on Computing, vol. 9, pp. 417–430, 1997.
E.D. Taillard, P. Badeau, M. Gendreau, F. Gueritin, and J.-Y Potvi, “A tabu search heuristic for the vehicle routing problem with soft time windows,” Transportation Science, vol. 31, pp. 170–186, 1997.
P. Badeau, M. Gendreau, F. Guertin, J.-Y. Potvin, and E. D. Taillard, “A parallel tabu search heuristic for the vehicle routing problem with time windows,” Transportation Research-C, vol. 5, pp. 109–122, 1997.
J.F. Cordeau, G. Larporte, and A. Mercier, “A unified tabu search heuristic for vehicle routing problems with time windows,” Journal of the Operational Research Society, vol. 52, pp. 928–936, 2001.
B. De Backer, V. Furnon, P. Shaw, P. Kilby, and P. Prosser, “Solving vehicle routing problems using constraint programming and metaheuristics,” Journal of Heuristics, vol. 6, pp. 501–523, 2000.
P. Shaw, “Using constraint programming and local search methods to solve vehicle routing problems,” in Proceedings of the Fourth International Conference on Principles and Practice of Constraint Programming (CP'98), edited by M. Maher and J.-F. Puget, Springer-Verlag, pp. 417–431, 1998.
H. Gehring and J. Homberger, “Parallelization of a two-phased metaheuristic for routing problems with time windows,” Asia- Pacific Journal of Operational Research, vol. 18, pp. 35–47, 2001.
J.Y. Potvin and J.M. Rousseau, “A parallel route building algorithm for the vehicle routing and scheduling problem with time windows,” European Journal of Operational Research, vol. 66, pp. 331–340, 1993.
W.C. Chiang and R. Russell, “Hybrid heuristics for the vehicle routing problem with time windows,” Transportation Science, vol. 29, no. 2, 1995.
S.R. Thangiah, “A hybrid genetic algorithms, simulated annealing and tabu search heuristic for vehicle routing problems with time windows,” in Practical Handbook of Genetic Algorithms, Volume III: Complex Structures, edited by L. Chambers, CRC Press, pp. 347–381, 1999.
K. C. Tan, L. H. Lee, Q. L. Zhu, and K. Ou. “Heuristic methods for vehicle routing problem with time windows,” Artificial Intelligent in Engineering, pp. 281–295, 2001.
M. Dorigo and L. M. Gambardella, “Ant Colony System: A cooperative learning approach to the traveling salesman problem,” IEEE Transactions on Evolutionary Computation, vol. 1, pp. 53–66, 1997.
C.A. Coello Coello, D.A. Van Veldhuizen, and G.B. Lamont, Evolutionary Algorithms for Solving Multi-Objective Problems, Kluwer Academic Publishers, 2002.
C. M. Fonseca and P. J. Fleming. “An overview of evolutionary algorithms in multiobjective optimization,” Evolutionary Computation, vol. 3, no. 1, pp. 1–16, 1995.
D. A. Van Veldhuizen and G. B. Lamont, “Multiobjective evolutionary algorithms: Analyzing the state-of-the-art,” Evolutionary Computation, vol. 8, no. 2, pp. 125–147, 2000.
K. Deb, Multi-objective Optimization using Evolutionary Algorithms, John Wiley, 2001.
E. Zitzler and L. Thiele, “Multiobjective evolutionary algorithms: A comparative case study and the strength Pareto approach,” IEEE Transactions on Evolutionary Computation, vol. 3, no. 4, pp. 257–271, 1999.
J. Knowles and D. Corne, “The Pareto archived evolution strategy: A new baseline algorithm for Pareto multiobjective optimisation: NSGA-II,” in Proceedings CEC, 1999, pp. 98–105.
K. Deb, S. Agrawa, A. Pratap, and T. Meyarivan, “A fast elitist non-dominated sorting genetic algorithm for multi-objective optimization: NSGA-II,” in Proceedings PPSN VI, Springer-Verlag, 2000, pp. 849–858.
B. J. Ross and H. Zhu, “Procedural texture evolution using multi-objective optimization,” New Generation Computing, vol. 22, no. 3, pp. 271–293, 2004.
Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs (3rd ed.), Springer-Verlag: New York, 1998.
M.M. Solomon. “Algorithms for the vehicle routing and scheduling problems with time window constraints,” Operations Research, vol. 35, no. 2, pp. 254–265, 1987.
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Ombuki, B., Ross, B.J. & Hanshar, F. Multi-Objective Genetic Algorithms for Vehicle Routing Problem with Time Windows. Appl Intell 24, 17–30 (2006). https://doi.org/10.1007/s10489-006-6926-z
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DOI: https://doi.org/10.1007/s10489-006-6926-z