About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 624398, 15 pages
http://dx.doi.org/10.1155/2013/624398
Research Article

A Heuristic Algorithm for Constrained Multi-Source Location Problem with Closest Distance under Gauge: The Variational Inequality Approach

Department of Mathematics, College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received 1 August 2013; Accepted 27 August 2013

Academic Editor: Abdellah Bnouhachem

Copyright © 2013 Jian-Lin Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. Weber, UBer Den Standort Der Industrien, 1. Teil: Reine Theorie Des Standortes, Mohr Siebeck, Tübingen, Germany, 1909.
  2. E. Weiszfeld, “Sur le point pour lequel la somme des distances de n points donnes est minimum,” Tohoku Mathematical Journal, vol. 43, pp. 355–386, 1937.
  3. R. F. Love, J. G. Morris, and G. O. Wesolowsky, Facilities Location: Models and Methods,, vol. 7, North-Holland Publishing, Amsterdam, Netherlands, 1988. View at MathSciNet
  4. F. Plastria, “Continuous location problems: research, results and questions,” in Facility Location: A Survey of Applications and Methods, Z. Drezner, Ed., pp. 225–260, Springer, New York, NY, USA, 1995.
  5. C. D. Bennett and A. Mirakhor, “Optimal facility location with respect to several regions,” Journal of Regional Science, vol. 14, no. 1, pp. 131–136, 1974.
  6. J. Brimberg and G. O. Wesolowsky, “Minisum location with closest Euclidean distances,” Annals of Operations Research, vol. 111, pp. 151–165, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. Z. Drezner and G. O. Weslowsky, “Optimal location of a facility relative to area demands,” Naval Research Logistics Quarterly, vol. 27, no. 2, pp. 199–206, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. J. Jiang and X. Yuan, “A Barzilai-Borwein-based heuristic algorithm for locating multiple facilities with regional demand,” Computational Optimization and Applications, vol. 51, no. 3, pp. 1275–1295, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. A. Suzuki and Z. Drezner, “The p-center location problem in an area,” Location Science, vol. 4, no. 1-2, pp. 69–82, 1996. View at Scopus
  10. G. O. Wesolowsky and R. F. Love, “Location of facilities with rectangular distances among point and area destinations,” Naval Research Logistics Quarterly, vol. 18, pp. 83–90, 1971.
  11. E. Carrizosa, M. Muñoz-Márquez, and J. Puerto, “The weber problem with regional demand,” European Journal of Operational Research, vol. 104, no. 2, pp. 358–365, 1998. View at Scopus
  12. R. E. Stone, “Some average distance results,” Transportation Science, vol. 25, no. 1, pp. 83–91, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  13. J. Puerto and A. M. Rodríguez-Chía, “On the structure of the solution set for the single facility location problem with average distances,” Mathematical Programming, vol. 128, no. 1-2, pp. 373–401, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  14. Z. Drezner and G. O. Wesolowsky, “Location models with groups of demand points,” INFOR Journal, vol. 38, no. 4, pp. 359–372, 2000. View at Scopus
  15. O. Berman, Z. Drezner, and G. O. Wesolowsky, “Location of facilities on a network with groups of demand points,” IIE Transactions, vol. 33, no. 8, pp. 637–648, 2001. View at Publisher · View at Google Scholar · View at Scopus
  16. J. Brimberg and G. O. Wesolowsky, “Note: facility location with closest rectangular distances,” Naval Research Logistics, vol. 47, no. 1, pp. 77–84, 2000. View at MathSciNet
  17. S. Nickel, J. Puerto, and A. M. Rodriguez-Chia, “An approach to location models involving sets as existing facilities,” Mathematics of Operations Research, vol. 28, no. 4, pp. 693–715, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. R. F. Love and J. G. Morris, “Mathematical models of road travel distances,” Management Science, vol. 25, no. 2, pp. 130–139, 1979. View at Scopus
  19. J. E. Ward and R. E. Wendell, “A new norm for measuring distance which yields linear location problems,” Operations Research, vol. 28, pp. 836–844, 1980.
  20. J. E. Ward and R. E. Wendell, “Using block norms for location modeling,” Operations Research, vol. 33, no. 5, pp. 1074–1090, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. C. Witzgall, “Optimal location of a central facility, mathematical models and concepts,” Report 8388, National Bureau of Standards, Washington, DC, USA, 1964.
  22. C. Michelot and O. Lefebvre, “A primal-dual algorithm for the Fermat-Weber problem involving mixed gauges,” Mathematical Programming, vol. 39, no. 3, pp. 319–335, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. R. Durier, “On Pareto optima, the Fermat-Weber problem, and polyhedral gauges,” Mathematical Programming, vol. 47, no. 1, pp. 65–79, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. E. Carrizosa, E. Conde, M. Munoz-Márquez, and J. Puerto, “Simpson points in planar problems with locational constraints. The polyhedral-gauge case,” Mathematics of Operations Research, vol. 22, no. 2, pp. 291–300, 1997. View at Scopus
  25. S. Nickel, “Restricted center problems under polyhedral gauges,” European Journal of Operational Research, vol. 104, no. 2, pp. 343–357, 1998. View at Scopus
  26. M. Cera, J. A. Mesa, F. A. Ortega, and F. Plastria, “Locating a central hunter on the plane,” Journal of Optimization Theory and Applications, vol. 136, no. 2, pp. 155–166, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. F. Plastria, “Asymmetric distances, semidirected networks and majority in Fermat-Weber problems,” Annals of Operations Research, vol. 167, pp. 121–155, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. I. Norman Katz and S. R. Vogl, “A Weiszfeld algorithm for the solution of an asymmetric extension of the generalized Fermat location problem,” Computers & Mathematics with Applications, vol. 59, no. 1, pp. 399–410, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. H. Minkowski, Theorie der Konvexen Körper, Gesammelte Abhandlungen, Teubner, Berlin, 1911.
  30. Z. Drezner, Facility Location: A Survey of Applications and Methods, Springer, New York, NY, USA, 1995. View at MathSciNet
  31. A. Ghosh and G. Rushton, Spatial Analysis and Location-Allocation Models, Van Nostrand Reinhold, New York, NY, USA, 1987.
  32. L. Cooper, “Solutions of generalized location equilibrium models,” Journal of Regional Science, vol. 7, pp. 1–18, 1967.
  33. N. Megiddo and K. J. Supowit, “On the complexity of some common geometric location problems,” SIAM Journal on Computing, vol. 13, no. 1, pp. 182–196, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. L. Cooper, “Heuristic methods for location-allocation problems,” SIAM Review, vol. 6, pp. 37–53, 1964. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. J.-L. Jiang and X.-M. Yuan, “A heuristic algorithm for constrained multi-source Weber problem: the variational inequality approach,” European Journal of Operational Research, vol. 187, no. 2, pp. 357–370, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. J.-l. Jiang, K. Cheng, C.-C. Wang, and L.-p. Wang, “Accelerating the convergence in the single-source and multi-source Weber problems,” Applied Mathematics and Computation, vol. 218, no. 12, pp. 6814–6824, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. Y. Levin and A. Ben-Israel, “A heuristic method for large-scale multi-facility location problems,” Computers & Operations Research, vol. 31, no. 2, pp. 257–272, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. K. Fukunaga, Introduction to Statistical Pattern Recognition, Academic Press, Boston, Mass, USA, 1990. View at MathSciNet
  39. F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, NY, USA, 2004.
  40. M. C. Ferris and C. Kanzow, “Engineering and economic applications of complementarity problems,” SIAM Review, vol. 39, no. 4, pp. 669–713, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. K. Addi, B. Brogliato, and D. Goeleven, “A qualitative mathematical analysis of a class of linear variational inequalities via semi-complementarity problems: applications in electronics,” Mathematical Programming, vol. 126, no. 1, pp. 31–67, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  42. A. Bnouhachem, H. Benazza, and M. Khalfaoui, “An inexact alternating direction method for solving a class of structured variational inequalities,” Applied Mathematics and Computation, vol. 219, no. 14, pp. 7837–7846, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  43. A. Barbagallo and P. Mauro, “Time-dependent variational inequality for an oligopolistic market equilibrium problem with production and demand excesses,” Abstract and Applied Analysis, vol. 2012, Article ID 651975, 35 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  44. J. Gwinner, “Three-field modelling of nonlinear nonsmooth boundary value problems and stability of differential mixed variational inequalities,” Abstract and Applied Analysis, vol. 2013, Article ID 108043, 10 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  45. Y.-B. Zhao and J.-Y. Yuan, “An alternative theorem for generalized variational inequalities and solvability of nonlinear quasi-PxM2a;-complementarity problems,” Applied Mathematics and Computation, vol. 109, no. 2-3, pp. 167–182, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  46. H. Uzawa, “Iterative methods for concave programming,” in Studies in Linear and Nonlinear Programming, K. J. Arrow, L. Hurwicz, and H. Uzawa, Eds., pp. 154–165, Stanford University Press, Stanford, Calif, USA, 1958.
  47. B. S. He, “A new method for a class of linear variational inequalities,” Mathematical Programming, vol. 66, no. 2, pp. 137–144, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  48. B. S. He, “A modified projection and contraction method for a class of linear complementarity problems,” Journal of Computational Mathematics, vol. 14, no. 1, pp. 54–63, 1996. View at Zentralblatt MATH · View at MathSciNet
  49. N. Xiu, C. Wang, and J. Zhang, “Convergence properties of projection and contraction methods for variational inequality problems,” Applied Mathematics and Optimization, vol. 43, no. 2, pp. 147–168, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  50. B. C. Eaves, “On the basic theorem of complementarity,” Mathematical Programming, vol. 1, no. 1, pp. 68–75, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet