Skip to main content
SpringerLink
Log in

Location and allocation based branch and bound algorithms for the capacitated multi-facility Weber problem

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

Given the locations of J customers, their demands and I capacitated facilities, the Capacitated Multi-facility Weber Problem (CMWP) is concerned with locating I facilities in the plane to satisfy the demand of J customers with the minimum total transportation cost which is proportional to the distance between them. We propose two types of branch and bound algorithms for the ℓ r distance CMWP with 1≤r<∞. One of them is an allocation space based branch and bound algorithm for which a new branching variable selection strategy and new lower bounding procedures have been proposed. The other one is new and partitions the location space. Based on extensive computational experiments we can say that the proposed algorithms are promising and perform better than the existing ones.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Algorithm 1
Algorithm 2

Similar content being viewed by others

References

  • Akyüz, M. H., Öncan, T., & Altınel, İ. K. (2012). Efficient approximate solution methods for the multi-commodity capacitated multi-facility Weber problem. Computers & Operations Research, 39, 225–237.

    Article  Google Scholar 

  • Al-Loughani, L. (1997). Algorithmic approaches for solving the Euclidean distance location-allocation problems. PhD thesis, Virginia Polytechnic Institute and State University, Virginia, USA.

  • Aras, N., Yumuşak, S., & Altınel, İ. K. (2006). Solving the capacitated multi-facility Weber problem by simulated annealing, threshold accepting and genetic algorithms. In K. F. Doerner, M. Gendreau, P. Greistorfer, W. J. Gutjahr, R. F. Hartl & M. Reimann (Eds.), Metaheuristics—progress in complex systems optimization (pp. 91–112). New York: Springer.

    Google Scholar 

  • Aras, N., Altınel, İ. K., & Orbay, M. (2007). New heuristic methods for the capacitated multi-facility Weber problem. Naval Research Logistics, 54, 21–32.

    Article  Google Scholar 

  • Aras, N., Orbay, M., & Altınel, İ. K. (2008). Efficient heuristics for the rectilinear distance capacitated multi-facility Weber problem. Journal of the Operations Research Society, 59, 64–79.

    Article  Google Scholar 

  • Bazaraa, M. S., Sherali, H. D., & Shetty, C. M. (1993). Nonlinear programming: theory and algorithms (2nd ed.). New Jersey: Wiley.

    Google Scholar 

  • Boyacı, B. (2009). Solving the capacitated multi-facility Weber problem approximately. MS thesis, Boğaziçi University, İstanbul, Türkiye.

  • 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, 1153–1163.

    Article  Google Scholar 

  • Brimberg, J., Chen, R., & Chen, D. (1998). Accelerating convergence in the Fermat–Weber location problem. Operations Research Letters, 22, 151–157.

    Article  Google Scholar 

  • Cooper, L. (1972). The transportation-location problem. Operations Research, 20, 94–108.

    Article  Google Scholar 

  • Drezner, Z., & Suzuki, A. (2004). The big triangle small triangle method for the solution of nonconvex facility location problems. Operations Research, 52, 128–135.

    Article  Google Scholar 

  • Drezner, Z., Wesolowsky, G. O., & Drezner, T. (2004). The gradual covering problem. Naval Research Logistics, 51, 841–855.

    Article  Google Scholar 

  • Drezner, Z., Wesolowsky, G. O., & Drezner, T. (2007). A general global optimization approach for solving location problems in the plane. Journal of Global Optimization, 37, 305–319.

    Article  Google Scholar 

  • Drezner, Z., Drezner, T., & Wesolowsky, G. O. (2009). Location with acceleration–deceleration distance. European Journal of Operational Research, 198, 157–164.

    Article  Google Scholar 

  • Frenk, J. B. G., Melo, M. T., & Zhang, S. (1994). A Weiszfeld method for a generalized l p distance minisum location model in continuous space. European Journal of Operational Research, 2, 111–117.

    Google Scholar 

  • Geoffrion, A. M. (1997). Integer programming by implicit enumeration and Balas’ method. SIAM Review, 9, 178–190.

    Article  Google Scholar 

  • Hansen, P., Perreur, J., & Thisse, F. (1980). Location theory, dominance and convexity: some further results. Operations Research, 28, 1241–1250.

    Article  Google Scholar 

  • Hansen, P., Peeters, D., & Thisse, F. (1981). On the location of an obnoxious facility. Sistemi Urbani, 3, 299–317.

    Google Scholar 

  • Held, M., Wolfe, P., & Crowder, H. P. (1974). Validation of subgradient optimization. Mathematical Programming, 6, 62–88.

    Article  Google Scholar 

  • Idrissi, H., Loridan, P., & Michelot, C. (1988). Approximation of solutions for location problems. Journal of Optimization Theory and Applications, 56, 127–143.

    Article  Google Scholar 

  • Korovkin, P. P. (1986). Inequalities. Moscow: Mir.

    Google Scholar 

  • Luis, M., Salhi, S., & Nagy, G. (2009). Region-rejection based heuristics for the capacitated multi-source Weber problem. Computers & Operations Research, 36, 2007–2017.

    Article  Google Scholar 

  • Luis, M., Salhi, S., & Nagy, G. (2011). A guided reactive GRASP for the capacitated multi-source Weber problem. Computers & Operations Research, 38, 1014–1024.

    Article  Google Scholar 

  • Martello, S., & Toth, P. (1990). Knapsack problems: algorithms and computer implementations. New York: Wiley.

    Google Scholar 

  • Plastria, F. (1992). GBSSS—the generalized big square small square method for planar single-facility location. European Journal of Operational Research, 62, 163–174.

    Article  Google Scholar 

  • Selim, S. (1979). Biconvex programming and deterministic and stochastic location allocation problems. PhD thesis, Georgia Institute of Technology, Atlanta, USA.

  • Sherali, H. D., & Adams, W. P. (1999). A reformulation-linearization technique for solving discrete and continuous nonconvex problems. Dordrecht: Kluwer Academic.

    Book  Google Scholar 

  • Sherali, H. D., & Nordai, F. L. (1988). Np-hard, capacitated, balanced p-median problems on a chain graph with a continuum of link demands. Mathematics of Operations Research, 13, 32–49.

    Article  Google Scholar 

  • Sherali, H. D., & Tunçbilek, C. H. (1992). A squared-Euclidean distance location–allocation problem. Naval Research Logistics, 39, 447–469.

    Article  Google Scholar 

  • Sherali, H. D., Ramachandran, S., & Kim, S. (1994). A localization and reformulation discrete programming approach for the rectilinear distance location–allocation problems. Discrete Applied Mathematics, 49, 357–378.

    Article  Google Scholar 

  • 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, 443–448.

    Google Scholar 

  • Thisse, J. F., Ward, J. E., & Wendell, R. E. (1984). Some properties of location problems with block and round norms. Operations Research, 32, 1309–1327.

    Article  Google Scholar 

  • Ward, J. E., & Wendell, R. E. (1985). Using block norms for location modeling. Operations Research, 33, 1074–1090.

    Article  Google Scholar 

  • Weiszfeld, E. (1937). Sur le point lequel la somme des distances de n points donné est minimum. Tôhoku Mathematics Journal, 43, 355–386.

    Google Scholar 

  • Weiszfeld, E., & Plastria, F. (2009). On the point for which the sum of the distances to n given points is minimum. Annals of Operations Research, 167, 7–41.

    Article  Google Scholar 

  • Wendell, R. E., & Hurter, A. P. (1973). Location theory, dominance and convexity. Operations Research, 21, 314–320.

    Article  Google Scholar 

  • Witzgall, C. (1964). Optimal location of central facility: mathematical models and concepts (Report No. 8388). National Bureau of Standards (NBS).

  • Zainuddin, Z. M., & Salhi, S. (2007). A perturbation-based heuristic for the capacitated multisource Weber problem. European Journal of Operational Research, 179, 1194–1207.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Industrial Engineering Ortaköy, Galatasaray University, İstanbul, 34357, Turkey

    M. Hakan Akyüz & Temel Öncan

  2. Department of Industrial Engineering Bebek, Boğaziçi University, İstanbul, 34342, Turkey

    İ. Kuban Altınel

Authors
  1. M. Hakan Akyüz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. İ. Kuban Altınel
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Temel Öncan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Hakan Akyüz.

Additional information

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.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Akyüz, M.H., Altınel, İ.K. & Öncan, T. Location and allocation based branch and bound algorithms for the capacitated multi-facility Weber problem. Ann Oper Res 222, 45–71 (2014). https://doi.org/10.1007/s10479-012-1221-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-012-1221-3

Keywords

Search

Navigation