- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Annual Issues
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 624398, 15 pages
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.
- A. Weber, UBer Den Standort Der Industrien, 1. Teil: Reine Theorie Des Standortes, Mohr Siebeck, Tübingen, Germany, 1909.
- 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.
- R. F. Love, J. G. Morris, and G. O. Wesolowsky, Facilities Location: Models and Methods,, vol. 7, North-Holland Publishing, Amsterdam, Netherlands, 1988.
- 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.
- 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.
- J. Brimberg and G. O. Wesolowsky, “Minisum location with closest Euclidean distances,” Annals of Operations Research, vol. 111, pp. 151–165, 2002.
- 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.
- 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.
- A. Suzuki and Z. Drezner, “The p-center location problem in an area,” Location Science, vol. 4, no. 1-2, pp. 69–82, 1996.
- 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.
- 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.
- R. E. Stone, “Some average distance results,” Transportation Science, vol. 25, no. 1, pp. 83–91, 1991.
- 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.
- Z. Drezner and G. O. Wesolowsky, “Location models with groups of demand points,” INFOR Journal, vol. 38, no. 4, pp. 359–372, 2000.
- 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.
- J. Brimberg and G. O. Wesolowsky, “Note: facility location with closest rectangular distances,” Naval Research Logistics, vol. 47, no. 1, pp. 77–84, 2000.
- 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.
- R. F. Love and J. G. Morris, “Mathematical models of road travel distances,” Management Science, vol. 25, no. 2, pp. 130–139, 1979.
- 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.
- J. E. Ward and R. E. Wendell, “Using block norms for location modeling,” Operations Research, vol. 33, no. 5, pp. 1074–1090, 1985.
- C. Witzgall, “Optimal location of a central facility, mathematical models and concepts,” Report 8388, National Bureau of Standards, Washington, DC, USA, 1964.
- 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.
- R. Durier, “On Pareto optima, the Fermat-Weber problem, and polyhedral gauges,” Mathematical Programming, vol. 47, no. 1, pp. 65–79, 1990.
- 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.
- S. Nickel, “Restricted center problems under polyhedral gauges,” European Journal of Operational Research, vol. 104, no. 2, pp. 343–357, 1998.
- 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.
- F. Plastria, “Asymmetric distances, semidirected networks and majority in Fermat-Weber problems,” Annals of Operations Research, vol. 167, pp. 121–155, 2009.
- 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.
- H. Minkowski, Theorie der Konvexen Körper, Gesammelte Abhandlungen, Teubner, Berlin, 1911.
- Z. Drezner, Facility Location: A Survey of Applications and Methods, Springer, New York, NY, USA, 1995.
- A. Ghosh and G. Rushton, Spatial Analysis and Location-Allocation Models, Van Nostrand Reinhold, New York, NY, USA, 1987.
- L. Cooper, “Solutions of generalized location equilibrium models,” Journal of Regional Science, vol. 7, pp. 1–18, 1967.
- 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.
- L. Cooper, “Heuristic methods for location-allocation problems,” SIAM Review, vol. 6, pp. 37–53, 1964.
- 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.
- 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.
- 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.
- K. Fukunaga, Introduction to Statistical Pattern Recognition, Academic Press, Boston, Mass, USA, 1990.
- F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, NY, USA, 2004.
- M. C. Ferris and C. Kanzow, “Engineering and economic applications of complementarity problems,” SIAM Review, vol. 39, no. 4, pp. 669–713, 1997.
- 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.
- 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.
- 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.
- 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.
- Y.-B. Zhao and J.-Y. Yuan, “An alternative theorem for generalized variational inequalities and solvability of nonlinear quasi--complementarity problems,” Applied Mathematics and Computation, vol. 109, no. 2-3, pp. 167–182, 2000.
- 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.
- B. S. He, “A new method for a class of linear variational inequalities,” Mathematical Programming, vol. 66, no. 2, pp. 137–144, 1994.
- 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.
- 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.
- B. C. Eaves, “On the basic theorem of complementarity,” Mathematical Programming, vol. 1, no. 1, pp. 68–75, 1971.