Abstract
This paper introduces skewed norms, i.e. norms perturbed by a linear function, which are useful for modelling asymmetric distance measures. The Fermat-Weber problem with mixed skewed norms is then considered. Using subdifferential calculus we derive exact conditions for a destination point to be optimal, thereby correcting and completing some recent work on asymmetric distance location problems. Finally the classical dominance theorem is generalized to Fermat-Weber problems with a fixed skewed norm.
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References
R. Chen, An improved method for the solution of the problem of location on an inclined plane, RAIRO, Recherche Operationelle/Operations Research 25(1991)45–53.
Z. Drezner and G.O. Wesolowsky, The asymmetric distance location problem, Transp. Sci. 23(1989)201–207.
R. Durier and C. Michelot, Geometrical properties of the Fermat-Weber problem, Europ. J. Oper. Res. 20(1985)332–343.
R.L. Francis and J.A. White,Facility Layout and Location (Prentice-Hall, Englewood Cliffs, NJ, 1974).
M.J. Hodgson, R.T. Wong and J. Honsaker, Thep-centroid problem on an inclined plane, Oper. Res. 35(1987)221–233.
H. Juel and R.F. Love, Fixed point optimality criteria for the Weber problem with arbitrary norms, J. Oper. Res. Soc. 32(1981)891–897.
H.W. Kuhn and E. Kuenne, An efficient algorithm for the numerical solution of the generalised Weber problem in spatial economics, J. Regional Sci. 4(1962)21–33.
R.F. Love and J.G. Morris, Modelling inter-city road distances by mathematical functions, Oper. Res. Quarterly 23(1972)61–71.
H. Minkowski,Theorie der konvexen körper, Gesammelte Abhandlungen, Vol. 2 (Teubner, Berlin, 1911).
J.G. Morris, Convergence of the Weiszfeld algorithm for Weber problems using a generalized “distance” function, Oper. Res. 29(1981)37–48.
F. Plastria, A note on “Fixed point optimality criteria for the location problem with arbitrary norms”, J. Oper. Res. Soc. 34(1983)164–165.
R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).
J.E. Ward and R.E. Wendell, Using block norms for location modelling, Oper. Res. 33(1985)1074–1090.
A. Weber, Über den Standort der Industrien, Tübingen [transl.: C.J. Friedrich,Alfred Weber's Theory of the Location of Industries (Chicago University Press, Chicago, 1929)].
E. Weiszfeld, Sur le point pour lequel la somme des distances den points donnés est minimum, Tohoku Math. J. 43(1937)355–386.
C. Witzgall, Optimal location of a central facility, mathematical models and concepts, Report 8388, US Department of Commerce, National Bureau of Standards (1964).
C. Witzgall, On convex metrics, J. Res. NBS, B: Math. and Math. Phys. 69B(1965)175–177.
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Plastria, F. On destination optimality in asymmetric distance Fermat-Weber problems. Ann Oper Res 40, 355–369 (1992). https://doi.org/10.1007/BF02060487
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DOI: https://doi.org/10.1007/BF02060487