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Journal of Global Optimization

, Volume 71, Issue 4, pp 871–889 | Cite as

Global optimization algorithm for capacitated multi-facility continuous location-allocation problems

  • Cristiana L. Lara
  • Francisco Trespalacios
  • Ignacio E. Grossmann
Article

Abstract

In this paper we propose a nonlinear Generalized Disjunctive Programming model to optimize the 2-dimensional continuous location and allocation of the potential facilities based on their maximum capacity and the given coordinates of the suppliers and customers. The model belongs to the class of Capacitated Multi-facility Weber Problem. We propose a bilevel decomposition algorithm that iteratively solves a discretized MILP version of the model, and its nonconvex NLP for a fixed selection of discrete variables. Based on the bounding properties of the subproblems, \(\epsilon \)-convergence is proved for this algorithm. We apply the proposed method to random instances varying from 2 suppliers and 2 customers to 40 suppliers and 40 customers, from one type of facility to 3 different types, and from 2 to 32 potential facilities. The results show that the algorithm is more effective at finding global optimal solutions than general purpose global optimization solvers tested.

Keywords

Location-allocation problem Weber problem Nonconvex optimization Generalized disjunctive programming Mixed-integer nonlinear programming 

Notes

Acknowledgements

The first and third authors gratefully acknowledge financial support from the Center for Advanced Process Decision-making at Carnegie Mellon University, and CAPES Foundation—Ministry of Education of Brazil (Scholarship no 13241-13-3).

References

  1. 1.
    Akyüz, M.H., Öncan, T., Altınel, İ.K.: Solving the multi-commodity capacitated multi-facility Weber problem using lagrangean relaxation and a subgradient-like algorithm. J. Oper. Res. Soc. 63(6), 771–789 (2012).  https://doi.org/10.1057/jors.2011.81 CrossRefGoogle Scholar
  2. 2.
    Al-Loughani, I.M.: Algorithmic approaches for solving the Euclidean distance location and location-allocation problems. Ph.D. thesis, Department of Industrial and Systems Engineering, Virginia Polytechnic Institute and State University (1997)Google Scholar
  3. 3.
    Aras, N., Altnel, K., Orbay, M.: New heuristic methods for the capacitated multi-facility Weber problem. Nav. Res. Logist. (NRL) 54(1), 21–32 (2007).  https://doi.org/10.1002/nav.20176 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brimberg, J., Hansen, P., Mladonovic, N., Salhi, S.: A survey of solution methods for the continuous location allocation problem. Int. J. Oper. Res. 5(1), 1–12 (2008)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chen, J.S., Pan, S., Ko, C.H.: A continuation approach for the capacitated multi-facility Weber problem based on nonlinear SOCP reformulation. J. Glob. Optim. 50(4), 713–728 (2011).  https://doi.org/10.1007/s10898-010-9632-7 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cooper, L.: The transportation-location problem. Oper. Res. 20(1), 94–108 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cooper, L.: The fixed charge problem I: a new heuristic method. Comput. Math. Appl. 1(1), 89–95 (1975).  https://doi.org/10.1016/0898-1221(75)90010-3 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cooper, L.: An efficient heuristic algorithm for the transportation-location problem. J. Reg. Sci. 16(3), 309 (1976)CrossRefGoogle Scholar
  9. 9.
    Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002).  https://doi.org/10.1007/s101070100263 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gamrath, G., Fischer, T., Gally, T., Gleixner, A.M., Hendel, G., Koch, T., Maher, S.J., Miltenberger, M., Müller, B., Pfetsch, M.E., Puchert, C., Rehfeldt, D., Schenker, S., Schwarz, R., Serrano, F., Shinano, Y., Vigerske, S., Weninger, D., Winkler, M., Witt, J.T., Witzig, J.: The SCIP optimization suite 3.2. Tech. Rep. 15-60, ZIB, Takustr.7, 14195 Berlin (2016)Google Scholar
  11. 11.
    IBM: IBM ILOG CPLEX Optimization Studio CPLEX User’s Manual. Tech. rep. (2015)Google Scholar
  12. 12.
    Khun, H.W., Kuenne, R.E.: An efficient algorithm for the commercial solution of the generallized Weber problem in spatial economics. J. Reg. Sci. 4(2), 21–33 (1962)CrossRefGoogle Scholar
  13. 13.
    Lara, C.L., Grossmann, I.E.: Global optimization for a continuous location-allocation model for centralized and distributed manufacturing. In: 26th European Symposium on Computed Aided Process Engineering (ESCAPE). Portoroz, Slovenia (2016)Google Scholar
  14. 14.
    Luis, M., Salhi, S., Nagy, G.: A constructive method and a guided hybrid GRASP for the capacitated multi-source Weber problem in the presence of fixed cost. J. Algorithms Comput. Technol. 9(2), 215–232 (2015).  https://doi.org/10.1260/1748-3018.9.2.215 MathSciNetCrossRefGoogle Scholar
  15. 15.
    McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part I – convex underestimating problems. Math. Program. 10(1), 147–175 (1976).  https://doi.org/10.1007/BF01580665 CrossRefzbMATHGoogle Scholar
  16. 16.
    Misener, R., Floudas, C.A.: ANTIGONE: algorithms for continuous / integer global optimization of nonlinear equations. J. Glob. Optim. 59(2–3), 503–526 (2014).  https://doi.org/10.1007/s10898-014-0166-2 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Sherali, A.D., Shetty, C.M.: The rectilinear distance location-allocation problem. A I I E Trans. 9(2), 136–143 (1977).  https://doi.org/10.1080/05695557708975135 MathSciNetGoogle Scholar
  18. 18.
    Sherali, H.D., Al-Loughani, I., Subramanian, S.: Global optimization procedures for the capacitated euclidean and \(\ell _p\) distance multifacility location-allocation problems. Oper. Res. 50(3), 433–448 (2002).  https://doi.org/10.1287/opre.50.3.433.7739 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sherali, H.D., Nordai, F.L.: NP-hard, capacitated, balanced p-median problems on a chain graph with a continuum of link demands. Math. Oper. Res. 13(1), 32–49 (1988).  https://doi.org/10.1287/moor.13.1.32 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sherali, H.D., Tuncbilek, C.H.: A squared-euclidean distance location-allocation problem. Nav. Res. Logist. (NRL) 39(4), 447–469 (1992).  https://doi.org/10.1002/1520-6750(199206)39:4%3c447::AID-NAV3220390403%3e3.0.CO;2-O MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103(2), 225–249 (2005).  https://doi.org/10.1007/s10107-005-0581-8 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Trespalacios, F., Grossmann, I.E.: Review of mixed-integer nonlinear and generalized disjunctive programming methods. Chem. Ing. Tech. 86(7), 991–1012 (2014).  https://doi.org/10.1002/cite.201400037 CrossRefGoogle Scholar
  23. 23.
    Weber, A., Friedrich, C.J.: Theory of the Location of Industries. University of Chicago Press, Chicago (1929)Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Chemical EngineeringCarnegie Mellon UniversityPittsburghUSA
  2. 2.Corporate Strategic ResearchExxonMobil Research and Engineering CompanyAnnandaleUSA

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