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A memetic NSGA-II for the bi-objective mixed capacitated general routing problem

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Abstract

The mixed capacitated general routing problem (MCGRP) is concerned with the determination of the optimal vehicle routes to service a set of customers located at nodes and along edges or arcs on a mixed weighted graph representing a complete transportation network. Although MCGRP generalizes many other routing problems and yields better models for several practical problems such as newspaper delivery and urban waste collection, this is still an underinvestigated problem. Furthermore, most of the studies have focused on the optimization of just one objective, that is, cost minimization. Keeping in mind the requirement of industries nowadays, MCGRP has been addressed in this paper to concurrently optimize two crucial objectives, namely, minimization of routing cost and route imbalance. To solve this bi-objective form of MCGRP, a multi-objective evolutionary algorithm (MOEA), coined as Memetic NSGA-II, has been designed. It is a hybrid of non-dominated sorting genetic algorithm-II (NSGA-II), a dominance based local search procedure (DBLSP), and a clone management principle (CMP). The DBLSP and CMP have been incorporated into the framework of NSGA-II with a view to empowering its capability to converge at/or near the true Pareto front and boosting diversity among the trade-off solutions, respectively. In addition, the algorithm also contains a set of three well-known crossover operators (X-set) that are employed to explore different parts of the search space. The algorithm was tested on a standard benchmark of twenty three standard MCGRP instances of varying complexity. The computational experiments verify the effectiveness of Memetic NSGA-II and also show the energetic effects of using DBLSP, CMP and X-set together while finding the set of potentially Pareto optimal solutions.

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Notes

  1. In reports from computational experiments on the MCGRP, the convention is to use the numerical value of the total traversal costs for a solution.

References

  • Alba, E., Dorronsoro, B.: Solving the vehicle routing problem by using cellular genetic algorithms. Evolutionary Computation in Combinatorial, pp. 11–20. Springer, Berlin (2004)

    Chapter  Google Scholar 

  • Bach, L., Hasle, G., Wøhlk, S.: A lower bound for the node, edge, and arc routing problem. Comput. Oper. Res. 40(4), 943–952 (2013)

    Article  MathSciNet  Google Scholar 

  • Baños, R., Ortega, J., Gil, C., Máquez, A.L., de Toro, F.: A hybrid meta-heuristic for multi-objective vehicle routing problems with time windows. Comput. Ind. Eng. 65(2), 286–296 (2013)

    Article  Google Scholar 

  • Basseur, M., Seynhaeve, F., Talbi, E.: Path relinking in pareto multi-objective genetic algorithms. In: Coello, C., Aguirre, A., Zitzler, E. (eds.) Evolutionary Multi-criterion Optimization, vol. 3410, pp. 120–134. Springer, Berlin (2005)

    Chapter  Google Scholar 

  • Beasley, J.E.: Route first-cluster seconds methods for vehicle routing. J. Manag. Sci. 11(4), 403–408 (1983)

    Google Scholar 

  • Beichl, I., Sullivan, F.: The metropolis algorithm. Comput. Sci. Eng. 2(1), 65–69 (2000)

    Article  Google Scholar 

  • Berger, J., Barkaoui, M.: A new hybrid genetic algorithm for the capacitated vehicle routing problem. J. Oper. Res. Soc. 54(12), 1254–1262 (2003)

    Article  MATH  Google Scholar 

  • Bosco, A., Laganà, D., Musmanno, R., Vocaturo, F.: Modeling and solving the mixed capacitated general routing problem. Optim. Lett. 7(7), 1451–1469 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  • Bramel, J., Coffman, E.G., Shor, P.W., Simchi-Levi, D.: Probabilistic analysis of the capacitated vehicle routing problem with unsplit demands. Oper. Res. 40(6), 1095–1106 (1991)

    Article  MathSciNet  Google Scholar 

  • Christofides, N., Mingozzi, A., Toth, P., Sandi, C. (eds.): Chapter 11. Combinatorial Optimization. John Willey, Chichester (1979)

  • Clarke, G., Wright, J.W.: Scheduling of vehicles from a central depot to a number of delivery points. Oper. Res. 12(4), 568–581 (1964)

    Article  Google Scholar 

  • Cordeau, J.-F., Laporte, G., Mercier, A.: A unified tabu search heuristic for vehicle routing problems with time windows. J. Oper. Res. Soc. 52(8), 928–936 (2001)

    Article  MATH  Google Scholar 

  • Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6, 182–197 (2002)

    Article  Google Scholar 

  • Fisher, M.L., Greenfield, A.J., Jaikumar, R.: A computrized vehicle routing application. Interfaces 12(4), 42–52 (1982)

    Article  Google Scholar 

  • Gendreau, M., Hertz, A., Laporte, G.: A tabu search heuristic for the vehicle routing problem. Manag. Sci. 40(10), 1276–1290 (1994)

    Article  MATH  Google Scholar 

  • Gillett, B., Miller, L.: A heuristic for the vehicle dispatching problem. Oper. Res. 22, 340–349 (1974)

    Article  MATH  Google Scholar 

  • Goldberg, D.E., Robert Lingle, J.: Alleles loci and the travelling salesman problem. In: Proceedings of the 1st International Conference on Genetic Algorithms and their applications pp.154–159. (1985)

  • Golden, B.L., Wasil, E.A., Kelly, J.P., Chao, I.M.: Metaheuristics in Vehicle Routing. Springer, Kluwer, Boston (1998)

    Google Scholar 

  • Gutiérrez, J.C.A., Soler, D., Hervás, A.: The capacitated general routing problem on mixed graphs. Revita Invest. Oper. 22(5), 15–26 (2002)

    Google Scholar 

  • Hasle, G.: Routing applications in newspaper delivery. Report A23753, SINTEF, Oslo, Norway. ISBN: 978-82-14-05310-4 (2012)

  • He, R., Xu, W., Sun, J., Zu, B.: Balanced k-means algorithm for partitioning areas in large-scale vehicle routing problem. In: IEEE Third International Symposium on Intelligent Information Technology Application, Vol. 3 pp. 87–90 (2009)

  • Jozefowiez, N., Semet, F., Talbi, E.G.: Enhancements of NSGA II and its application to the vehicle routing problem with route balancing. In: Proceedings of the 7th international conference on Artificial Evolution pp. 131–142. (2006)

  • Jozefowiez, N., Semet, F., Talbi, E.G.: Target aiming pareto search and its application to the vehicle routing problem with route balancing. J. Heuristics 13(5), 455–469 (2007)

    Article  Google Scholar 

  • Jozefowiez, N., Semet, F., Talbi, E.G.: Multi-objective vehicle routing problems. Eur. J. Oper. Res. 189(2), 293–309 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  • Jozefowiez, N., Semet, F., Talbi, E.G.: An evolutionary algorithm for the vehicle routing problem with route balancing. Eur. J. Oper. Res. 195(3), 761–769 (2009)

    Article  MATH  Google Scholar 

  • Kim, B.-I., Kim, S., Sahoo, S.: Waste collection vehicle routing problem with time windows. Comput. Oper. Res. 33(12), 3624–3642 (2006)

    Article  MATH  Google Scholar 

  • Kokubugata, H., Moriyama, A., Kawashima, H.: A practical solution using simulated annealing for general routing problems with nodes, edges, and arcs. In: Stuetzle, T., Birattari, M., Hoos, H.H. (eds.) Proceedings of the International conference on Engineering stochastic local search algorithms: designing, implementing and analyzing effective heuristics, volume 4638 of Lecture Notes in Computer Science, pp. 136–149. Springer, Berlin, Heidelberg (2007)

    Google Scholar 

  • Lacomme, P., Prins, C., Sevaux, M.: A genetic algorithm for a bi-objective capacitated arc routing problem. Comput. Oper. Res. 33(12), 3473–3493 (2006)

    Article  MATH  Google Scholar 

  • Murata, T., Itai, R.: Multi-objective vehicle routing problems using two-fold EMO algorithms to enhance solution similarity on non-dominated solutions. In: Proceedings of the Third international conference on Evolutionary Multi-Criterion Optimization, pp. 885–896. Springer, Berlin, Heidelberg (2005)

  • Nagata, Y., Bräysy, O.: Edge assembly-based memetic algorithm for the capacitated vehicle routing problem. Networks 54(4), 205–215 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  • Najera, A.G., Bullinaria, J.A.: An improved multi-objective evolutionary algorithm for the vehicle routing problem with time windows. Comput. Oper. Res. 38(1), 287–300 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  • Oliver, I.M., Smith, D.J., Holland, J.R.C.: A study of permutation crossover operators on the travellng salesman problem. In: Proceedings of 2nd International Conference on Genetic Algorithms and Their Application pp. 224–230. (1987)

  • Osman, I.H.: Metastrategy simulated annealing and tabu search algorithms for the vehicle routing problem. Ann. Oper. Res. 41(4), 421–451 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  • Pandit, R., Muralidharan, B.: A capacitated general routing problem on mixed networks. Comput. Oper. Res. 22(5), 465–478 (1995)

    Article  MATH  Google Scholar 

  • Pasia, J.M., Derner, K.F., Hartl, R.F., Reimann, M.: A population-based local search for solving a bi-objective vehicle routing. In: European conference on Evolutionary computation in combinatorial optimization pp. 166–175 (2007)

  • Pasia, J.M., Derner, K.F., Hartl, R.F., Reimann, M.: Solving a bi-objective vehicle rouitng problem by pareto ant colony optimization. In: Proceedings of Engineering Stochastic Local Search Algorithms pp. 187–191. Springer, Brussels, Belgium (2007)

  • Pisinger, D., Ropke, S.: An adaptive large neighborhood search heuristic for the pick up and delivery problem with time windows. Transp. Sci. 40(4), 455–472 (2006)

    Article  Google Scholar 

  • Prins, C.: A simple and effective evolutionary algorithm for the vehicle routing problem. Comput. Oper. Res. 31(12), 1985–2002 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • Prins, C., Bouchenoua, S.: A memetic algorithm solving the VRP, the CARP and GENERAL routing problems with nodes, edges and arcs. In: Hart, W.E., Krasnogor, N., Smith, J.E. (eds.) Recent Advances in Memetic Algorithms. Studies in Fuzziness and Soft Computing, vol. 166, pp. 65–85. Springer, Berlin, Heidelberg (2004)

    Chapter  Google Scholar 

  • Reimann, M., Doerner, K., Hartl, R.F.: D-ants: Savings based ants divide and conquer the vehicle routing problem. Comput. Oper. Res. 31(4), 563–591 (2004)

    Article  MATH  Google Scholar 

  • Rochat, Y., Taillard, E.D.: Probabilistic diversification and intensification in local search for vehicle routing. J. Heuristics 1, 147–167 (1995)

    Article  MATH  Google Scholar 

  • Solomon, M.M.: Algorithms for the vehicle routing and scheduling problems with time window constraints. Oper. Res. 35(2), 254–265 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  • Tan, K.C., Chew, Y.H., Lee, L.H.: A hybrid multi-objective evolutionary algorithm for solving truck and trailer vehicle routing problems. Eur. J. Oper. Res. 172(3), 855–885 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • Toth, P., Vigo, D.: The granular tabu search and its application to the vehicle-routing problem. INFORMS J. Comput. 15(4), 333–346 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Whitley, L.D., Starkweather, T., Fuquay, D.: Scheduling problems and traveling salesman: the genetic edge recombination operator. In: Proceedings of the 3rd International Conference on Genetic Algorithms, pp. 133–140. George Mason University, Fairfax, Virginia, USA (1989)

    Google Scholar 

  • Xu, H., Fan, W., Wei, T., Yu, L.: An or-opt NSGA-II algorithm for multi-objective vehicle routing problem with time windows. IEEE International Conference on Automation Science and Engineering, pp. 309–314. Key Bridge, Marriott, Washington DC, USA (2008)

    Google Scholar 

  • Zitzler, E., Laumanns, M., Thiele, L.: Evolutionary Methods for Design, Optimisation, and Control, chapter SPEA2: Improving the strength pareto evolutionary algorithm for multiobjective optimization pp. 19–26, CIMNE, Barcelona, Spain (2002)

  • Zitzler, E., Thiele, L.: Multiobjective evolutionary algorithms: a comparative case study and the strength pareto approach. IEEE Trans. Evol. Comput. 3(4), 257–271 (1999)

    Article  Google Scholar 

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Authors and Affiliations

  1. Università degli Studi Roma Tre, Dipartimento di Ingegneria via della vasca navale 79, 00146, Roma, Italy

    Santosh Kumar Mandal & Dario Pacciarelli

  2. Molde University College, Specialized University in Logistics, 6411, Molde, Norway

    Arne Løkketangen

  3. Applied Mathematics, SINTEF, Oslo, Norway

    Geir Hasle

Authors
  1. Santosh Kumar Mandal
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  2. Dario Pacciarelli
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  3. Arne Løkketangen
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  4. Geir Hasle
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Corresponding author

Correspondence to Dario Pacciarelli.

Additional information

This paper is dedicated to the memory of Prof. Arne Løkketangen who passed away unexpectedly on 10th June 2013, just before this work was completed.

Appendices

Appendix 1

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Appendix 2

See Figs. 4, 5, 6, 7, 8, 9, 10, and 11.

Fig. 10
figure 10

Pareto front of CBMix19 without DBLSP

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Fig. 11
figure 11

Pareto front of CBMix19 without DBLSP

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Mandal, S.K., Pacciarelli, D., Løkketangen, A. et al. A memetic NSGA-II for the bi-objective mixed capacitated general routing problem. J Heuristics 21, 359–390 (2015). https://doi.org/10.1007/s10732-015-9280-7

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