Abstract
The Fermat–Weber problem is considered with distance defined by a quasimetric, an asymmetric and possibly nondefinite generalisation of a metric. In such a situation a distinction has to be made between sources and destinations. We show how the classical result of optimality at a destination or a source with majority weight, valid in a metric space, may be generalized to certain quasimetric spaces. The notion of majority has however to be strengthened to supermajority, defined by way of a measure of the asymmetry of the distance, which should be finite. This extended majority theorem applies to most asymmetric distance measures previously studied in literature, since these have finite asymmetry measure.
Perhaps the most important application of quasimetrics arises in semidirected networks, which may contain edges of different (possibly zero) length according to direction, or directed edges. Distance in a semidirected network does not necessarily have finite asymmetry measure. But it is shown that an adapted majority result holds nevertheless in this important context, thanks to an extension of the classical node-optimality result to semidirected networks with constraints.
Finally the majority theorem is further extended to Fermat–Weber problems with mixed asymmetric distances.
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References
Albert, G. E. (1940). A note on quasimetric spaces. Bulletin of the American Mathematical Society, 46, 219.
Bourbaki, N. (1943). Topologie générale. Paris: Herman.
Blumenthal, L. M. (1970). Theory and applications of distance geometry. Bronx: Chelseta Publ. Co.
Buchanan, D. J., & Wesolowsky, G. O. (1993). Locating a noxious facility with respect to several polygonal regions using asymmetric distances. Computers and Operations Research, 20, 151–165.
Carrizosa, E., & Rodriguez-Chia, A. M. (1997). Weber problems with alternative transportation systems. European Journal of Operational Research, 97, 87–93.
Carrizosa, E., Conde, E., Munõz, M., & Puerto, J. (1997). Simpson points in planar problems with locational constraints. The polyhedral gauge case. Mathematics of Operations Research, 22, 297–300.
Cech, E. (1966). Topological spaces. London: Interscience.
Cera, M., & Ortega, F. A. (2002). Locating the median hunter among n mobile prey on the plane. International Journal of Industrial Engineering, 9, 6–15.
Cera, M., Mesa, J. A., Ortega, F. A., & Plastria, F. (2008). Locating a central hunter on the plane. Journal of Optimization Theory and Applications, 136(2), 155–166.
Chen, R. (1991). An improved method for the solution of the problem of location on an inclined plane. RAIRO Operations Research, 25, 45–53.
Deza, M. M., & Laurent, M. (1997). Geometry of cuts and metrics. Berlin: Springer.
Dijkstra, E. W. (1959). A note on two problems in connexion with graphs. Numerische Mathematik, 1, 269–271.
Drezner, Z., & Wesolowsky, G. O. (1989). The asymmetric distance location problem. Transportation Science, 23(3), 201–207.
Durier, R. R. (1990). On Pareto optima, the Fermat–Weber problem and polyhedral gauges. Mathematical Programming, 47, 65–79.
Durier, R., & Michelot, C. (1985). Geometrical properties of the Fermat–Weber problem. European Journal of Operational Research, 20(3), 332–343.
Fliege, J. (1994). Some new coincidence conditions in minisum multifacility location problems with mixed gauges. Studies in Locational Analysis, 7, 49–60.
Fliege, J. (1997). Nondifferentiability detection and dimensionality reduction in minisum multifacility location problems. Journal of Optimization Theory and Applications, 94(2), 365–380.
Fliege, J. (1998). A note on ‘On Pareto optima, the Fermat–Weber problem, and polyhedral gauges’. Mathematical Programming, 84(2), 435–438.
Fliege, J. (2000). Solving convex location problems with gauges in polynomial time. Studies in Locational Analysis, 14, 153–172.
Hakimi, S. L. (1964). Optimal location of switching centers and the absolute centers and medians of a graph. Operations Research, 12, 450–459.
Hiriart-Urruty, J. B., & Lemaréchal, C. (2001). Fundamentals of convex analysis. Berlin: Springer.
Hinojosa, Y., & Puerto, J. (2003). Single facility location problems with unbounded unit balls. Mathematical Methods in Operations Research, 58, 87–104.
Hodgson, M. J., Wong, R. T., & Honsaker, J. (1987). The p-centroid problem on an inclined plane. Operations Research, 35(2), 221–233.
Idrissi, H., Lefebvre, O., & Michelot, C. (1988). A primal dual algorithm for a constrained Fermat–Weber problem involving mixed gauges. Recherche Operationnelle / Operations Research, 22, 313–330.
Idrissi, H., Lefebvre, O., & Michelot, C. (1989). Duality for constrained multifacility location problems with mixed norms and applications. Annals of Operations Research, 18, 71–92.
Kaufman, L., & Rousseeuw, P. (1990). Finding groups in data. New York: Wiley.
Kelley, J. (1975). General topology. New York: Springer.
Labbe, M., Peeters, D., & Thisse, J.-F. (1995). Location on networks. In M. Ball et al. (Eds.), Handbooks in OR & MS : Vol. 8. Network routing (pp. 551–624). Amsterdam: North-Holland.
Levy, J. (1967). An extended theorem for location on a network. Operational Research Quarterly, 18, 433–442.
Lindenbaum, A. (1926). Contributions à l’étude de l’espace métrique. Fundamenta Mathematica, 8, 209–222.
Love, R. F., Morris, J. G., & Wesolowski, G. O. (1988). Facility location: Models and methods. New York: Norh Holland.
Michelot, C., & Lefebvre, O. (1987). A primal-dual algorithm for the Fermat–Weber problem involving mixed gauges. Mathematical Programming, 39, 319–335.
Minieka, E. (1979). The chinese postman problem for mixed networks. Management Science, 25, 643–648.
Minkowski, H. (1911). Theorie der konvexen Körper, Gesammelte Abhandlungen (Vol. 11). Berlin: Teubner.
Mirchandani, P. B. (1975). Analysis of stochastic networks in emergency service systems, IRP-TR-15-75, Operations Research Center, Massachusetts Institute of Technology, Cambridge, MA, USA.
Nickel, S. (1998). Restricted center problems under polyhedral gauges. European Journal of Operational Research, 104, 343–357.
Nickel, S., Puerto, J., & Rodriguez-Chía (2003). An approach to location models involving sets as existing facilities. Mathematics of Operations Research, 28, 693–715.
Papini, P., & Puerto, J. (2005). Location problems with different norms for different points. Jouranl of Optimization Theory and Applications, 125, 673–695.
Plastria, F. (1984). Localization in single facility location. European Journal of Operational Research, 18, 215–219.
Plastria, F. (1987). Solving general continuous single facility location problems by cutting planes. European Journal of Operational Research, 29, 329–332.
Plastria, F. (1992a). GBSSS, the generalized big square small square method for planar single facility location. European Journal of Operational Research, 62, 163–174.
Plastria, F. (1992b). On destination optimality in asymmetric distance Fermat–Weber problems. Annals of Operations Research, 40, 355–369.
Plastria, F. (1992c). A majority theorem for Fermat–Weber problems in quasimetric spaces with applications to semidirected networks. In J. Moreno (Ed.), Proceedings of the sixth meeting of the EURO working group on locational analysis (pp. 153–165). Tenerife, Spain: Puerto de la Cruz.
Plastria, F. (1994). Fully geometric solutions to some planar minimax location problems. Studies in Locational Analysis, 7, 171–183.
Ribeiro, H. (1943). Sur les espaces a métrique faible. Portugalia Mathematicae, 4, 21–40.
Rinow, W. (1961). Die innere Geometrie der metrischen Räume. Berlin: Springer (in German).
Rockafellar, T. (1970). Convex analysis. Princeton: Princeton University Press.
Ward, J. E., & Wendell, R. E. (1985). Using block norms for location modeling. Operations Research, 33, 1074–1090.
Wendell, R. E., & Hurter, A. P. Jr. (1973). Optimal locations on a network. Transportation Science, 7, 18–33.
Witzgall, C. (1964). Optimal location of a central facility, mathematical models and concepts. Report 8388, National Bureau of Standards, Washington DC, USA.
Witzgall, C. (1965). On convex metrics. Journal of Research of the National Bureau of Standards – B. Mathematics and Mathematical Physics, 69B(3), 175–177.
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Plastria, F. Asymmetric distances, semidirected networks and majority in Fermat–Weber problems. Ann Oper Res 167, 121–155 (2009). https://doi.org/10.1007/s10479-008-0351-0
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DOI: https://doi.org/10.1007/s10479-008-0351-0