Journal of Heuristics

, Volume 24, Issue 3, pp 515–550 | Cite as

On a two-phase solution approach for the bi-objective k-dissimilar vehicle routing problem

  • Sandra ZajacEmail author


In the k-dissimilar vehicle routing problem, a set of k dissimilar alternatives for a Capacitated Vehicle Routing Problem (CVRP) has to be determined for a single instance. The tradeoff between minimizing the longest routing and maximizing the minimum dissimilarity between two routings is investigated. Here, spatial dissimilarity is considered. Since short routings tend to be similar to each other, an objective conflict arises. The developed heuristic approach approximates the Pareto-set with respect to this tradeoff. This paper focuses on the generation of a high-quality candidate set of routings from which k routings are extracted with respect to a spatial as well as to an edge-based dissimilarity metric. In particular two algorithmic variants are suggested which differ in generating dissimilar routings. They are further compared to each other as well as to a naive approach. The method is tested on benchmark instances of the CVRP and findings are reported for both metrics. Taking the hypervolume as a quality criterion, it could be shown that the approach provides a good approximation of the Pareto-set for both metrics. An additional comparison to the results of Talarico et al. (Eur J Oper Res 244(1):129–140, 2015) proves its competitive ability.


Bi-objective k-dissimilar vehicle routing problem Pareto-set approximation Candidate set of routings Solution alternatives 


  1. Akgün, V., Erkut, E., Batta, R.: On finding dissimilar paths. Eur. J. Oper. Res. 121(2), 232–246 (2000)CrossRefzbMATHGoogle Scholar
  2. Augerat, P., Belenguer, J., Benavent, E., Corberán, A., Naddef, D., Rinaldi, G. (1998) Computational results with a branch-and-cut code for the capacitated vehicle routing problem. Tech. Rep. RR. 949-M, Université Joseph Fourier, Grenoble, FranceGoogle Scholar
  3. Caramia, M., Giordani, S., Iovanella, A.: On the selection of \(k\) routes in multiobjective hazmat route planning. IMA J. Manag. Math. 21, 239–251 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Christofides, N.: Combinatorial Optimization. John Wiley and Sons Canada Limited, New York (1979)zbMATHGoogle Scholar
  5. 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)CrossRefGoogle Scholar
  6. Croes, G.A.: A method for solving traveling salesman problems. Oper. Res. 6, 791–812 (1958)MathSciNetCrossRefGoogle Scholar
  7. Dantzig, G., Fulkerson, R., Johnson, S.: Solution of a large-scale traveling-salesman problem. Oper. Res. 2, 393–410 (1954)MathSciNetGoogle Scholar
  8. Dell’Olmo, P., Gentili, M., Scozzari, A.: On finding dissimilar Pareto-optimal paths. Eur. J. Oper. Res. 162(1), 70–82 (2005)CrossRefzbMATHGoogle Scholar
  9. Fisher, M.: Optimal solution of vehicle routing problems using minimum \(k\)-trees. Oper. Res. 42(4), 626–642 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Geiger, M.J.: Fast approximation heuristics for multi-objective vehicle routing problems. In: Chio, C., Brabazon, A., Caro, G.A., Ebner, M., Farooq, M., Fink, A., Grahl, J., Greenfield, G., Machado, P., O’Neill, M., Tarantino, E., Urquhart, N. (eds.) Applications of Evolutionary Computation. Lect Notes Comput Sc, vol. 6025, pp. 441–450. Springer, Berlin (2010)CrossRefGoogle Scholar
  11. Johnson, P., Joy, D., Clarke, D.: HIGHWAY 3.01 – An enhancement routing model: Program, description, methodology and revised user’s manual. Tech. rep., Oak Ridge National Laboratories, Washington, D.C. (1992)Google Scholar
  12. Kuby, M., Zhongyi, X., Xiaodong, X.: A minimax method for finding the \(k\) best “differentiated” paths. Geogr. Anal. 29(4), 298–313 (1997)CrossRefGoogle Scholar
  13. Laporte, G.: The vehicle routing problem: An overview of exact and approximate algorithms. Eur. J. Oper. Res. 59(3), 345–358 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Lei, H., Laporte, G., Guo, B.: A generalized variable neighborhood search heuristic for the capacitated vehicle routing problem with stochastic service times. TOP 20(1), 99–118 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Levenshtein, V.: Binary codes capable of correcting deletions, insertions, and reversals. Sov. Phys. Dokl. 10, 707–710 (1966)MathSciNetzbMATHGoogle Scholar
  16. Lin, S., Kernighan, B.W.: An effective heuristic algorithm for the traveling-salesman problem. Oper. Res. 21(2), 498–516 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Løkketangen, A., Oppen, J., Oyola, J., Woodruff, D.L.: An attribute based similarity function for VRP decision support. DMMS 6(2), 65–83 (2012)CrossRefzbMATHGoogle Scholar
  18. Lombard, K., Church, R.L.: The gateway shortest path problem: Generating alternative routes for a corridor location problem. Geogr. Syst. 1, 25–45 (1993)Google Scholar
  19. Martí, R., González Velarde, J.L., Duarte, A.: Heuristics for the bi-objective path dissimilarity problem. Comput. Oper. Res. 36(11), 2905–2912 (2009)CrossRefzbMATHGoogle Scholar
  20. Mladenović, N., Hansen, P.: Variable neighborhood search. Comput. Oper. Res. 24(11), 1097–1100 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Ngueveu, S.U., Prins, C., Wolfler Calvo, R.: A hybrid tabu search for the \(m\)-peripatetic vehicle routing problem. In: Maniezzo, V., Stützle, T., Voß, S. (eds.) Matheuristics, Annals of Information Systems, vol. 10, pp. 253–266. Springer, Boston (2010)Google Scholar
  22. Potvin, J.Y., Rousseau, J.M.: An exchange heuristic for routeing problems with time windows. J. Oper. Res. Soc. 46(12), 1433–1446 (1995)CrossRefzbMATHGoogle Scholar
  23. Prins, C.: A simple and effective evolutionary algorithm for the vehicle routing problem. Comput. Oper. Res. 31(12), 1985–2002 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Savelsbergh, M.W.P.: The vehicle routing problem with time windows: minimizing route duration. INFORMS J. Comput. 4, 146–154 (1992)CrossRefzbMATHGoogle Scholar
  25. Sörensen, K.: Route stability in vehicle routing decisions: a bi-objective approach using metaheuristics. Cent. Europ. J. Oper. Re. 14(2), 193–207 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  26. Suurballe, J.W.: Disjoint paths in a network. Networks 4(2), 125–145 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  27. Taillard, E.: Parallel iterative search methods for vehicle routing problems. Networks 23(8), 661–673 (1993)CrossRefzbMATHGoogle Scholar
  28. Talarico, L., Sörensen, K., Springael, J.: The \(k\)-dissimilar vehicle routing problem. Eur. J. Oper. Res. 244(1), 129–140 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. Toth, P., Vigo, D.: The granular tabu search and its application to the vehicle routing problem. INFORMS J. Comput. 15(4), 333–346 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  30. Uchoa, E., Pecin, D., Pessoa, A., Poggi, M., Vidal, T., Subramanian, A.: New benchmark instances for the capacitated vehicle routing problem. Eur. J. Oper. Res. 257(3), 845–858 (2017)MathSciNetCrossRefGoogle Scholar
  31. Wolfler Calvo, R., Cordone, R.: A heuristic approach to the overnight security service problem. Comput. Oper. Res. 30(9), 1269–1287 (2003)CrossRefzbMATHGoogle Scholar
  32. Yen, J.Y.: Finding the \(k\) shortest loopless paths in a network. Manage. Sci. 17(11), 712–716 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  33. Zajac, S.: The bi-objective \(k\)-dissimilar vehicle routing problem. In: Paias, A., Ruthmair, M., Voß, S. (eds.) Computational Logistics. Lect Notes Comput Sc, vol. 9855, pp. 306–320. Springer International Publishing (2016)Google Scholar
  34. Zitzler, E.: Thiele L (1998) Multiobjective optimization using evolutionary algorithms - a comparative case study. In: Eiben, A.E., Bäck, T., Schoenauer, M., Schwefel, H.P. (eds.) Parallel Problem Solving from Nature - PPSN V. Lect Notes Comput Sc, vol. 1498, pp. 292–301. Springer, Berlin (1998)CrossRefGoogle Scholar
  35. Zitzler, E., Thiele, L.: Multiobjective evolutionary algorithms: a comparative case study and the Strength Pareto approach. IEEE Trans. Evolut. Comput. 3(4), 257–271 (1999)CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Helmut Schmidt University HamburgHamburgGermany

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