# Multistart Branch and Bound for Large Asymmetric Distance-Constrained Vehicle Routing Problem

• Samira Almoustafa
• Said Hanafi
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 31)

## Abstract

In this chapter we revise and modify an old branch-and-bound method for solving the asymmetric distance-constrained vehicle routing problem suggested by Laporte et al. in 1987. It is based on reformulating distance-constrained vehicle routing problem into a travelling salesman problem and use of assignment problem as a lower bounding procedure. In addition, our algorithm uses the best-first strategy and new tolerance-based branching rules. Since our method was fast but memory consuming, it could stop before optimality is proven. Therefore we introduce the randomness, in case of ties, in choosing the node of the search tree. If an optimal solution is not found, we restart our procedure. In that way we get multistart branch-and-bound method. As far as we know instances we solved exactly (up to 1,000 customers) are much larger than instances considered for other VRP models from the recent literature. So, despite its simplicity, this proposed algorithm is capable of solving the largest instances ever solved in the literature. Moreover, this approach is general and may be used in solving other types of vehicle routing problems.

## References

1. 1.
Almoustafa, S., Goldengorin,B., Tso, M., Mladenović, N.: Two new exact methods for asymmetric distance-constrained vehicle routing problem. Proceedings of SYM-OP-IS. Belgrade, pp. 297–300 (2009)Google Scholar
2. 2.
Balas, E., Toth, P.: Branch and bound methods. In: Lawer, et al. (eds.) The Traveling Salesman Problem, pp. 361–401. Wiley, Chichester (1985)Google Scholar
3. 3.
Baldacci, R., Mingozzi, A.: An unified exact method for solving different classes of vehicle routing problems. Math. Program. Ser. A 120(2), 347–380 (2009)
4. 4.
Baldacci, R., Toth, P., Vigo, D.: Recent advances in vehicle routing exact algorithms. 4OR 5(4), 269–298 (2007)Google Scholar
5. 5.
Baldacci, R., Mingozzi, A., Roberti, R.: Recent exact algorithms for solving the vehicle routing problem under capacity and time window constraints (invited review). Eur. J. Oper. Res. doi:10.1016/j.ejor.2011.07.037, 218(1), 1–6 (2011, in press)Google Scholar
6. 6.
Christofides, N., Mingozzi, A., Toth, P.: State space relaxation procedures for the computation of bounds to routing problems. Networks 11(2), 145–164 (1981)
7. 7.
Clarke, G., Wright, J. V.: Scheduling of vehicles from a central depot to a number of delivery points. Oper. Res. 12(4), 568–581 (1964)
8. 8.
Dantzig, G.B., Fulkerson, D.R., Johnson, S.M.: Solution of a large-scale traveling salesman problem. Oper. Res. 2, 393–410 (1954)
9. 9.
Goldengorin, B., Jager, G., Molitor, P.: Tolerances applied in combinatorial optimization. J. Comp. Sci. 2(9), 716–734 (2006)
10. 10.
Haimovich, M., Rinnooy Kan, A.H.G., Stougie, L.: Analysis of heuristic routing problems. In: Golden, et al. (eds.) Vehicle Routing: Methods and Studies, pp. 47–61. North Holland, Amsterdam (1988)Google Scholar
11. 11.
Hansen, P., Mladenović, N., Moreno Pé, J.A.: Variable neighbourhood search: methods and applications. Ann. Oper. Res. 175(1), 367–407 (2010)Google Scholar
12. 12.
Jonker, R., Volgenant, A.: Improving the hungarian assignment algorithm. Oper. Res. Lett. 5(4), 171–175 (1986)
13. 13.
Kara, I.: Two indexed polynomial size formulations for vehicle routing problems. Technical Report (2008/01). Baskent University, Ankara/Turkey (2008)Google Scholar
14. 14.
Koltai, T., Terlaky, T.: The difference between the managerial and mathematical interpretation of sensitivity analysis results in linear programming. Int. J. Prod. Econ. 65(3), 257–274 (2000)
15. 15.
Kuhn, H.W.: The Hungarian method for the assignment problem. Naval Res. Logist. Q. 2, 83–97 (1955)
16. 16.
Laporte, G.: The vehicle routing problem: An overview of exact and approximate algorithms. Eur. J. Oper. Res. 59(3), 345–358 (1992)
17. 17.
Laporte, G.: What you should know about the vehicle routing problem. Naval Res. Logist. 54(8), 811–819 (2007)
18. 18.
Laporte, G., Nobert, Y.: Exact algorithms for the vehicle routing problem. Ann. Discrete Math. 31, 147–184 (1987)
19. 19.
Laporte, G., Nobert, Y., Desrochers, M.: Optimal routing under capacity and distance restractions. Oper. Res. 33(5), 1050–1073 (1985)
20. 20.
Laporte, G., Nobert, Y., Taillefer, S.: A branch and bound algorithm for the asymmetrical distance-constrained vehicle routing problem. Math. Model. 9(12), 857–868 (1987)
21. 21.
Lawer, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B.: The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley-Interscience series in Discrete Mathematics. Chichester, Wiley (1985)Google Scholar
22. 22.
Lenstra, J.K., Rinnooy Kan, A.H.G.: Some simple applications of the traveling salesman problem. Oper. Res. Q. 26(4), 717–734 (1975)Google Scholar
23. 23.
Letchford. A. N., Salazar-Gonźalez, J.J.: Projection results for vehicle routing. Math. Program. Ser. B. 105, 251–274 (2006)Google Scholar
24. 24.
Lin, C., Wen, U.: Sensitivity analysis of the optimal assignment. Discrete Optim. 149(1), 35–46 (2003)
25. 25.
Mladenović, N., Hansen, P.: Variable neighbourhood search. Comp. Oper. Res. 24(11), 1097–1100 (1997)
26. 26.
Nemhauser, G.L., Wolsey, L.A.: Integer and combinatorial optimization. Discrete Math. Optim. Wiley, New York (1988)
27. 27.
Paschos, V.Th.: An overview on polynomial approximation of NP-hard problems. Yugoslav J. Oper. Res. 19(1), 3–40 (2009)
28. 28.
Pessoa, A., Poggi de Aragão, M., Uchoa, E.: Robust branch-cut-and-price algorithms for vehicle routing problems. In: Golden, B., et al. (eds.) The Vehicle Routing Problem Latest Advances and New Challenges. Operations Research/Computer Science Interfaces Series, Springer, New York, vol. 43, Part II, pp. 297–325 (2008)Google Scholar
29. 29.
Toth, P., Vigo, D.: The Vehicle Routing Problem. SIAM Monographs on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics, Philadelphia (2002)
30. 30.
Turkensteen, M., Ghosh, D., Goldengorin, B., Sierksma, G.: Tolerance-based branch and bound algorithms for the ATSP. Eur. J. Oper. Res. 189, 775–788 (2008)
31. 31.
Volgenant, A.: An addendum on sensitivity analysis of the optimal assignment. Eur. J. Oper. Res. 169(1), 338–339 (2006)

## Authors and Affiliations

• Samira Almoustafa
• 1
Email author
• Said Hanafi
• 2