Advertisement

Annals of Operations Research

, Volume 279, Issue 1–2, pp 1–42 | Cite as

Branch and bound algorithms for solving the multi-commodity capacitated multi-facility Weber problem

  • M. Hakan AkyüzEmail author
  • Temel Öncan
  • İ. Kuban Altınel
Original Research

Abstract

The Multi-commodity Capacitated Multi-facility Weber Problem is concerned with locating I capacitated facilities in the plane in order to satisfy the demands of J customers for K commodities such that the total transportation cost is minimized. This is a multi-commodity extension of the well-known Capacitated Multi-facility Weber Problem and difficult to solve. In this work, we propose two branch-and-bound algorithms for exactly solving this nonconvex optimization problem. One of them considers partitioning of the allocation space while the other one considers partitioning of the location space. We have implemented two lower bounding schemes for both algorithms and tested several branching strategies. The results of an extensive computational study are also included.

Keywords

Facility location–allocation Branch-and-bound algorithm Multi-commodity transportation 

Mathematics Subject Classification

90B06 90B85 90C26 

Notes

Acknowledgements

This research is supported by the Turkish Scientific and Technological Research Council (TÜBİTAK) Research Grant No: 107M462, and Galatasaray University Scientific Research Projects Grant Nos: 07.402.014, 10.402.001 and 10.402.019. The first author acknowledges the partial support of National Graduate Scholarship Program for PhD Students awarded by TÜBİTAK.

References

  1. Akyüz, M. H., Altınel, İ. K., & Öncan, T. (2014). Location and allocation based branch and bound algorithms for the capacitated multi-facility Weber problem. Annals of Operations Research, 222(1), 45–71.CrossRefGoogle Scholar
  2. Akyüz, M. H., Öncan, T., & Altınel, İ. K. (2010). The multi-commodity capacitated multi-facility Weber problem: Heuristics and confidence intervals. IIE Transactions, 42(11), 825–841.CrossRefGoogle Scholar
  3. Akyüz, M. H., Öncan, T., & Altınel, İ. K. (2012a). Efficient approximate solution methods for the multi-commodity capacitated multi-facility Weber problem. Computers and Operations Research, 39(2), 225–237.CrossRefGoogle Scholar
  4. Akyüz, M. H., Öncan, T., & Altınel, İ. K. (2012b). Solving the multi-commodity capacitated multi-facility Weber problem using Lagrangean relaxation. Journal of the Operational Research Society, 63(6), 771–789.CrossRefGoogle Scholar
  5. Akyüz, M. H., Öncan, T., & Altınel, İ. K. (2013). Beam search heuristics for the single and multi-commodity capacitated multi-facility Weber problems. Computers and Operations Research, 40(12), 3056–3068.CrossRefGoogle Scholar
  6. Al-Loughani, L. (1997). Algorithmic approaches for solving the Euclidean distance location-allocation problems. Ph.D. Thesis, Virginia Polytechnic Institute and State University, USA.Google Scholar
  7. Boyacı, B., Altınel, İ. K., & Aras, N. (2013). Approximate solution methods for the capacitated multi-facility Weber problem. IIE Transactions, 45(1), 97–120.CrossRefGoogle Scholar
  8. Brimberg, J., Chen, R., & Chen, D. (1998). Accelerating convergence in the Fermat–Weber location problem. Operations Research, 22(4–5), 151–157.Google Scholar
  9. Brimberg, J., Drezner, Z., Mladenović, N., & Salhi, S. (2014). A new local search for continuous location problems. European Journal of Operational Research, 232(2), 256–265.CrossRefGoogle Scholar
  10. Brimberg, J., Hansen, P., & Mladenović, N. (2006). Decomposition strategies for large-scale continuous location–allocation problems. IMA Journal of Management Mathematics, 17(4), 307–316.CrossRefGoogle Scholar
  11. Brimberg, J., Hansen, P., Mladenović, N., & Salhi, S. (2008). A survey of solution methods for the continuous location–allocation problem. International Journal of Operations Research, 5(1), 1–12.Google Scholar
  12. Brimberg, J., & Love, R. F. (1993). Global convergence of a generalized iterative procedure for the minisum location problem with L\(_{p}\) distances. Operations Research, 41(6), 1153–1163.CrossRefGoogle Scholar
  13. Brimberg, J., Walker, J. H., & Love, R. F. (2007). Estimation of travel distances with the weighted \(l_{p}\) norm: Some empirical results. Journal of Transport Geography, 15(1), 62–72.CrossRefGoogle Scholar
  14. Chen, J., Pan, S., & Ko, C. (2011). A continuation approach for the capacitated multi-facility Weber problem based on nonlinear SOCP reformulation. Journal of Global Optimization, 50(4), 713–728.CrossRefGoogle Scholar
  15. Cooper, L. (1963). Location–allocation problems. Operations Research, 11(3), 331–343.CrossRefGoogle Scholar
  16. Cooper, L. (1972). The transportation-location problem. Operations Research, 20(1), 94–108.CrossRefGoogle Scholar
  17. Drezner, Z. (2007). A general global optimization approach for solving location problems in the plane. Journal of Global Optimization, 37(2), 305–319.CrossRefGoogle Scholar
  18. Drezner, Z., Brimberg, J., Mladenović, N., & Salhi, S. (2015). New local searches for solving the multi-source Weber problem. Annals Operations Research,.  https://doi.org/10.1007/s10479-015-1797-5.CrossRefGoogle Scholar
  19. Drezner, Z., Drezner, T., & Wesolowsky, G. O. (2009). Location with acceleration–deceleration distance. European Journal of Operational Research, 198, 157–164.CrossRefGoogle Scholar
  20. Drezner, Z., & Suzuki, A. (2004). The big triangle small triangle method for the solution of nonconvex facility location problems. Operations Research, 52, 128–135.CrossRefGoogle Scholar
  21. Drezner, Z., Wesolowsky, G. O., & Drezner, T. (2004). The gradual covering problem. Naval Research Logistics, 51, 841–855.CrossRefGoogle Scholar
  22. Durier, R., & Michelot, C. (1985). Geometrical properties of the Fermat–Weber problem. European Journal of Operational Research, 20, 332–343.CrossRefGoogle Scholar
  23. Durier, R., & Michelot, C. (1994). On the set of optimal pointes to the Weber problem: Further results. Transportation Science, 28(2), 141–149.CrossRefGoogle Scholar
  24. Hansen, P., Peeters, D., Richard, D., & Thisse, J. (1985). The minisum and minimax location problems revisited. Operations Research, 33(6), 1251–1265.CrossRefGoogle Scholar
  25. Hansen, P., Peeters, D., & Thisse, J. (1981). On the location of an obnoxious facility. Sistemi Urbani, 3, 299–317.Google Scholar
  26. Hansen, P., Perreur, J., & Thisse, F. (1980). Location theory, dominance and convexity: Some further results. Operations Research, 28, 1241–1250.CrossRefGoogle Scholar
  27. Held, M., Wolfe, P., & Crowder, H. P. (1974). Validation of subgradient optimization. Mathematical Programming, 6, 62–88.CrossRefGoogle Scholar
  28. Klingman, D., Napier, A., & Stutz, J. (1974). NETGEN: A program for generating large scale capacitated assignment, transportation, and minimum cost flow problems. Management Science, 20, 814–821.CrossRefGoogle Scholar
  29. Korovkin, P. P. (1986). Inequalities. Moskow: Mir Publishers.Google Scholar
  30. Luis, M., Salhi, S., & Nagy, G. (2009). Region-rejection based heuristics for the capacitated multi-source Weber problem. Computers and Operations Research, 36(6), 2007–2017.CrossRefGoogle Scholar
  31. Luis, M., Salhi, S., & Nagy, G. (2011). A guided reactive GRASP for the capacitated multi-source Weber problem. Computers and Operations Research, 38(7), 1014–1024.CrossRefGoogle Scholar
  32. Plastria, F. (1992). GBSSS: The generalized big square small square method for planar single-facility location. European Journal of Operational Research, 62, 163–174.CrossRefGoogle Scholar
  33. Plastria, F., & Elosmani, M. (2013). Continuous location of an assembly station. TOP, 22(2), 323–340.CrossRefGoogle Scholar
  34. Rockafellar, R. T. (1970). Convex analysis. New Jersey: Princeton University Press.CrossRefGoogle Scholar
  35. Sherali, H. D., & Adams, W. P. (1999). A reformulation-linearization technique for solving discrete and continuous nonconvex problems. Netherlands: Kluwer Academic Publishers.CrossRefGoogle Scholar
  36. Sherali, H. D., Al-Loughani, I., & Subramanian, S. (2002). Global optimization procedures for the capacitated Euclidean and L\(_{p}\) distance multifacility location-allocation problem. Operations Research, 50(3), 433–448.CrossRefGoogle Scholar
  37. Sherali, H. D., Ramachandran, S., & Kim, S. I. (1994). A localization and reformulation discrete programming approach for the rectilinear distance location-allocation problems. Discrete Applied Mathematics, 49(1–3), 357–378.CrossRefGoogle Scholar
  38. Sherali, H. D., & Tunçbilek, C. H. (1992). A Squared-Euclidean distance location–allocation problem. Naval Research Logistics, 39(4), 447–469.CrossRefGoogle Scholar
  39. Thisse, J., Ward, J. E., & Wendell, R. E. (1984). Some properties of location problems with block and round norms. Operations Research, 32(6), 1309–1327.CrossRefGoogle Scholar
  40. Üster, H., & Love, R. (2000). The convergence of the Weiszfeld algorithm. Computers and Mathematics with Applications, 40, 443–451.CrossRefGoogle Scholar
  41. Ward, J. E., & Wendell, R. E. (1985). Using block norms for location modeling. Operations Research, 33(5), 1074–1090.CrossRefGoogle Scholar
  42. Weiszfeld, E. (1937). Sur le point lequel la somme des distances de n points donné est minimum. Tôhoku Mathematical Journal, 43, 355–386.Google Scholar
  43. Wendell, R. E., & Hurter, A. P. (1973). Location theory, dominance and convexity. Operations Research, 21(1), 314–320.CrossRefGoogle Scholar
  44. Witzgall, C., & Goldman, A. J. (1964). Optimal location of a central facility: Mathematical models and concepts. US Department of Commerce: National Bureau of Standards.Google Scholar
  45. Zainuddin, Z. M., & Salhi, S. (2007). A perturbation-based heuristic for the capacitated multisource Weber problem. European Journal of Operational Research, 179(3), 1194–1207.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • M. Hakan Akyüz
    • 1
    Email author
  • Temel Öncan
    • 1
  • İ. Kuban Altınel
    • 2
  1. 1.Department of Industrial EngineeringGalatasaray UniversityOrtaköy, İstanbulTurkey
  2. 2.Department of Industrial EngineeringBoğaziçi UniversityBebek, İstanbulTurkey

Personalised recommendations