Properties of Three-Dimensional Median Line Location Models

Springer Science and Business Media LLC - Tập 122 - Trang 71-85 - 2003
Jack Brimberg1, Henrik Juel2, Anita Schöbel3
1Royal Military College of Canada and Groupe d'Études et de Recherche en Analyse des Décisions, Canada
2Informatics and Mathematical Modelling, Richard Petersens Plads, Technical University of Denmark, Lyngby, Denmark
3University of Kaiserslautern Germany

Tóm tắt

We consider the problem of locating a line with respect to some existing facilities in 3-dimensional space, such that the sum of weighted distances between the line and the facilities is minimized. Measuring distance using the l p norm is discussed, along with the special cases of Euclidean and rectangular norms. Heuristic solution procedures for finding a local minimum are outlined.

Tài liệu tham khảo

Brazil, M., tD.H. Lee, J.H. Rubinstein, D.A. Thomas, J.F. Weng, and N.C. Wormald. (2001). “Network Optimisation of Underground Mine Designs.” Submitted to AusIMM Proceedings. Brimberg, J., tH. Juel, and A. Schöbel. (2002). “Linear Facility Location in Three Dimensions – Models and Solution Methods.” Operations Research 50(6), 1050–1057. Brimberg, J. and tR.F. Love. (1993). “Global Convergence of a Generalized Iterative Procedure for the Minisum Location Problem with _p Distances.” Operations Research 41, 1153–1163. Brimberg, J., tR. Chen, and D. Chen. (1998). “Accelerating Convergence in the Fermat–Weber Location Problem.” Operations Research Letters 22, 151–157. Díaz-Báñez, J.M., tJ.A. Mesa, and A. Schöbel. (2002). “Continuous Location of Dimensional Structures.” Report in Wirtschaftsmathematik 79/2002, University of Kaiserslautern. Follert, F. (1995). “Maxmin Location of an Anchored Ray in 3-Space and Related Problems.” In 7th Canadian Conference on Computational Geometry, Quebec. Follert, F., tE. Schömer, J. Sellen, M. Smid, and C. Thiel. (1995). “Computing a Largest Empty Anchored Cylinder and Related Problems.” Technical Report MPI–91–1–001, Max-Planck-Institut für Informatik, Saarbrücken. Houle, M.E. and tG.T. Toussaint. (1988). “Computing the Width of a Set.” IEEE Transactions on Pattern Analysis and Machine Intelligence 10, 760–765. Korneenko, N.M. and tH. Martini. (1993). “Hyperplane Approximation and Related Topics.” In J. Pach (ed.), New Trends in Discrete and Computational Geometry. New York: Springer. Martini, H. and tA. Schöbel. (1998). “Median Hyperplanes in Normed Spaces – A Survey.” Discrete Applied Mathematics 89, 181–195. Martini, H. and tA. Schöbel. (1999). “A Characterization of Smooth Norms.” Geometriae Dedicata, 77 173–183. Minkowski, H. (1967). Gesammelte Abhandlungen. New York: Chelsea. Morris, J.G. and tJ.P. Norback. (1980). “A Simple Approach to Linear Facility Location.” Transportation Science 14, 1–8. Morris, J.G. and tJ.P. Norback. (1983). “Linear Facility Location – Solving Extensions of the Basic Problem.” European Journal of Operational Research 12, 90–94. Norback, J.P. and tJ.G. Morris. (1980). “Fitting Hyperplanes by Minimizing Orthogonal Deviations.” Mathematical Programming 19, 102–105. Schöbel, A. (1998). “Locating Least Distant Lines in the Plane.” European Journal of Operational Research 106(1), 152–159. Schöbel, A. (1999). Locating Lines and Hyperplanes: Theory and Algorithms. Dordrecht: Kluwer. Schömer, E., tJ. Sellen, M. Teichmann, and C. Yap. (1996). “Efficient Algorithms for the Smallest Enclosing Cylinders Problem.” In Proceedings of the 8th Canadian Conference on Computational Geometry. Wesolowsky, G.O. (1975). “Location of the Median Line for Weighted Points.” Environment and Planning A 7, 163–170. Zemel, E. (1984). “An O(n) Algorithm for the Linear Multiple Choice Knapsack Problem and Related Problems.” Information Processing Letters 18, 123–128.