Logic-Based Decomposition Methods for the Travelling Purchaser Problem

  • Kyle E. C. Booth
  • Tony T. Tran
  • J. Christopher Beck
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9676)


We present novel branch-and-check and logic-based Benders decomposition techniques for the Travelling Purchaser Problem, an important optimization problem with applications in vehicle routing, logistics, and warehouse management. Our master problem determines a set of markets and directed travel arcs that satisfy product purchase constraints with relaxed travel costs. Our subproblem identifies subtours within this master assignment and produces a set of generalized subtour elimination cuts. We show that the proposed technique demonstrates strong performance on the asymmetric problem variants, finding optimal solutions to previously unsolved instances, while performing competitively on a number of symmetric problem classes. Furthermore, our model is implemented unchanged for the four problem variants whereas other state-of-the-art approaches are variant-specific.


Travelling Salesman Problem Master Problem Valid Inequality Uncapacitated Facility Location Problem Subtour Elimination Constraint 
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  1. 1.
    Achterberg, T.: SCIP: solving constraint integer programs. Math. Program. Comput. 1(1), 1–41 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Applegate, D., Bixby, R., Cook, W., Chvátal, V.: On the solution of traveling salesman problems. Rheinische Friedrich-Wilhelms-Universität Bonn (1998)Google Scholar
  3. 3.
    Balas, E., Toth, P.: Branch and bound methods for the traveling salesman problem. Technical report MSRR-488, DTIC Document (1983)Google Scholar
  4. 4.
    Beck, J.C.: Checking-up on branch-and-check. In: Cohen, D. (ed.) CP 2010. LNCS, vol. 6308, pp. 84–98. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  5. 5.
    Bontoux, B., Feillet, D.: Ant colony optimization for the traveling purchaser problem. Comput. Oper. Res. 35(2), 628–637 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Burt, C.N., Lipovetzky, N., Pearce, A.R., Stuckey, P.J.: Approximate uni-directional benders decomposition. In: Proceedings of PlanSOpt-15 Workshop on Planning, Search and Optimization AAAI-15 (2015)Google Scholar
  7. 7.
    Cambazard, H., Penz, B.: A constraint programming approach for the traveling purchaser problem. In: Milano, M. (ed.) CP 2012. LNCS, vol. 7514, pp. 735–749. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  8. 8.
    Desrochers, M., Laporte, G.: Improvements and extensions to the Miller-Tucker-Zemlin subtour elimination constraints. Oper. Res. Lett. 10(1), 27–36 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Erlenkotter, D.: A dual-based procedure for uncapacitated facility location. Oper. Res. 26(6), 992–1009 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Flood, M.M.: The traveling-salesman problem. Oper. Res. 4(1), 61–75 (1956)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Geoffrion, A.M.: Generalized benders decomposition. J. Optim. Theor. Appl. 10(4), 237–260 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Goerler, A., Schulte, F., Voß, S.: An application of late acceptance hill-climbing to the traveling purchaser problem. In: Pacino, D., Voß, S., Jensen, R.M. (eds.) ICCL 2013. LNCS, vol. 8197, pp. 173–183. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  13. 13.
    Goldbarg, M.C., Bagi, L.B., Goldbarg, E.F.G.: Transgenetic algorithm for the traveling purchaser problem. Eur. J. Oper. Res. 199(1), 36–45 (2009)CrossRefzbMATHGoogle Scholar
  14. 14.
    Golden, B.L., Levy, L., Vohra, R.: The orienteering problem. Naval Res. Logistics (NRL) 34(3), 307–318 (1987)CrossRefzbMATHGoogle Scholar
  15. 15.
    Hooker, J.N., Ottosson, G.: Logic-based benders decomposition. Math. Program. 96(1), 33–60 (2003)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Laporte, G.: Generalized subtour elimination constraints and connectivity constraints. J. Oper. Res. Soc. 37, 509–514 (1986)CrossRefzbMATHGoogle Scholar
  17. 17.
    Laporte, G.: The traveling salesman problem: an overview of exact and approximate algorithms. Eur. J. Oper. Res. 59(2), 231–247 (1992)CrossRefzbMATHGoogle Scholar
  18. 18.
    Laporte, G., Nobert, Y.: A cutting planes algorithm for the m-salesmen problem. J. Oper. Res. Soc. 31, 1017–1023 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Laporte, G., Riera-Ledesma, J., Salazar-González, J.-J.: A branch-and-cut algorithm for the undirected traveling purchaser problem. Oper. Res. 51(6), 940–951 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Miliotis, P.: Integer programming approaches to the travelling salesman problem. Math. Program. 10(1), 367–378 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Miller, C.E., Tucker, A.W., Zemlin, R.A.: Integer programming formulation of traveling salesman problems. J. ACM (JACM) 7(4), 326–329 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Padberg, M., Rinaldi, G.: A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems. SIAM Rev. 33(1), 60–100 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ramesh, T.: Traveling purchaser problem. Opsearch 18(1–3), 78–91 (1981)zbMATHGoogle Scholar
  24. 24.
    Riera-Ledesma, J., Salazar-González, J.-J.: Solving the asymmetric traveling purchaser problem. Ann. Oper. Res. 144(1), 83–97 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Singh, K.N., van Oudheusden, D.L.: A branch and bound algorithm for the traveling purchaser problem. Eur. J. Oper. Res. 97(3), 571–579 (1997)CrossRefzbMATHGoogle Scholar
  26. 26.
    Thorsteinsson, E.S.: Branch-and-check: a hybrid framework integrating mixed integer programming and constraint logic programming. In: Walsh, T. (ed.) CP 2001. LNCS, vol. 2239, pp. 16–30. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  27. 27.
    Tran, T.T., Araujo, A., Beck, J.C.: Decomposition methods for the parallel machine scheduling problem with setups. INFORMS J. Comput. 28(1), 83–95 (2016)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Kyle E. C. Booth
    • 1
  • Tony T. Tran
    • 1
  • J. Christopher Beck
    • 1
  1. 1.Department of Mechanical and Industrial EngineeringUniversity of TorontoTorontoCanada

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