Abstract
In the Fermat-Weber problem, the location of a source point in ℝN is sought which minimizes the sum of weighted Euclidean distances to a set of destinations. A classical iterative algorithm known as the Weiszfeld procedure is used to find the optimal location. Kuhn proves global convergence except for a denumerable set of starting points, while Katz provides local convergence results for this algorithm. In this paper, we consider a generalized version of the Fermat-Weber problem, where distances are measured by anl p norm and the parameterp takes on a value in the closed interval [1, 2]. This permits the choice of a continuum of distance measures from rectangular (p=1) to Euclidean (p=2). An extended version of the Weiszfeld procedure is presented and local convergence results obtained for the generalized problem. Linear asymptotic convergence rates are typically observed. However, in special cases where the optimal solution occurs at a singular point of the iteration functions, this rate can vary from sublinear to quadratic. It is also shown that for sufficiently large values ofp exceeding 2, convergence of the Weiszfeld algorithm will not occur in general.
Similar content being viewed by others
References
J. Brimberg, Properties of distance functions and minisum location models, Ph.D. Thesis, McMaster University, Hamilton, Canada (1989).
A.V. Cabot, R.L. Francis and M.A. Stary, A network flow solution to a rectilinear distance facility location problem, AIIE Trans. 2(1970)132–141.
R. Chandrasekaran and A. Tamir, Open questions concerning Weiszfeld's algorithm for the Fermat-Weber location problem, Math. Progr. 44(1989)293–295.
L. Cooper, Location-allocation problems, Oper. Res. 11(1963)37–52.
G. Dahlquist and Å. Björck,Numerical Methods, transl. by N. Anderson (Prentice-Hall, Englewood Cliffs, NJ, 1974).
D.T. Finkbeiner,Elements of Linear Algebra (W.H. Freeman, San Francisco, CA, 1972).
H. Juel and R.F. Love, Fixed point optimality criteria for the location problem with arbitrary norms, J. Oper. Res. Soc. 32(1981)891–897.
I.N. Katz, Local convergence in Fermat's problem, Math. Progr. 6(1974)89–104.
H.W. Kuhn, A note on Fermat's problem, Math. Progr. 4(1973)98–107.
H.W. Kuhn and R.E. Kuenne, An efficient algorithm for the numerical solution of the generalized Weber problem in spatial economics, J. Regional Sci. 4(1962)21–34.
R.F. Love and J.G. Morris, Modelling inter-city road distances by mathematical functions, Oper. Res. Quarterly 23(1972)61–71.
R.F. Love and J.G. Morris, Mathematical models of road travel distances, Manag. Sci. 25(1979)130–139.
R.F. Love, J.G. Morris and G.O. Wesolowsky,Facilities Location: Models and Methods (North-Holland, New York, 1988).
W. Miehle, Link-length minimization in networks, Oper. Res. 6(1958)232–243.
J.G. Morris, Analysis of generalized empirical “distance” function for use in location problems,Int. Symp. on Locational Decisions, Banff, Alberta (1978).
J.G. Morris, Convergence of the Weiszfeld algorithm for Weber problems using a generalized “distance” function, Oper. Res. 29(1981)37–48.
J.G. Morris and W.A. Verdini, Minisuml p distance location problems solved via a perturbed problem and Weiszfeld's algorithm, Oper. Res. 27(1979)1180–1188.
J.M. Ortega and W.C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970).
L.M. Ostresh, On the convergence of a class of iterative methods for solving the Weber location problem, Oper. Res. 26(1978)597–609.
J.-C. Picard and H.D. Ratliff, A cut approach to the rectilinear distance facility location problem, Oper. Res. 26(1978)422–433.
J.-F. Thisse, J.E. Ward and R.E. Wendell, Some properties of location problems with block and round norms, Oper. Res. 32(1984)1309–1327.
E. Weiszfeld, Sur le point lequel la somme des distances den points donnés est minimum, Tohoku Math. J. 43(1937)355–386.
G.O. Wesolowsky and R.F. Love, The optimal location of new facilities using rectangular distances, Oper. Res. 19(1971)124–130.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Brimberg, J., Love, R.F. Local convergence in a generalized Fermat-Weber problem. Ann Oper Res 40, 33–66 (1992). https://doi.org/10.1007/BF02060469
Issue Date:
DOI: https://doi.org/10.1007/BF02060469