The Weber Problem: Solution and Interpretation*

Abstract

In location theory, the so-called "Weber Problem" has been especially cherished by a large number of authors. It has been generalized, extended, reformulated, criticized, but never forgotten. I intend to present in this paper a trigonometric solution to the triangle problem. An attempt also will be made to extend it to the general polygon case. In the second part, the concept of the "ideal weight" will be modified in order to extend the scope of Weberian theory beyond the sole consideration of transportation costs.' This will be based upon a demonstration that the Weberian analogy between the equilibrium of physical forces and the locational equilibrium was justified and can be preserved in cases where forces vary over distance. Finally, we shall discuss certain fundamental characteristics of the Weberian approach.Let us recall the basic assumptions and formulation of the problem. Weber considers a uniform plain where it is always possible to go directly from one point to any other at a given rate. In this two-dimensional space, the market site, the location of "localized"z resources, and the spatial dis-"The author would like to express his thanks to Dr. John B. Parr for his comments and he1 in drafting the final version of this aper; he would also l i e to thank Dr. Bruce Allen, br. Benjamin Stevens, Dr. Julian kolpert, Anthony Mumphrey, and Michel Boiyert for their assistance.The "ideal weight" at a given localized resource site can be expressed as the product of the technical coefficient corresponding to this resource multiplied by the relevant transport rate. More generally, the "ideal weight" is the magnitude of the force oriented toward a given site. In this paper, we shall use the term "force" in preference to "ideal weight."* Weber makes the distinction between "localized" materials and ubiquities, the former being located only at particular sites in two-dimensional space.Luc-Normand Tellier is a graduate student in regional science at the University of Pennsylvania and is associated with the Centre de Recherches Urbaines et Rdgiowles of the Znstitut National de la Recherche Scientifique, Montreal, Quebec.

Majority Voting and the Location of Salutary Public Facilities

“…Then, the minimum point can be found by defining the components of equation (5) but it coincides with the Weber point only in the special case of an equilateral triangle. Further contrasts between the behavior of P and the Weber point are indicated by comparison with the results obtained by Tellier (1972). To compare the properties of P with that of the centroid, which minimizes the sum AP2 + BP2 + CP', note that the centroid divides AU in the ratio of 2:1, and similarly for BV and CW.…”

“…Let ( P I , P2), (al, a2), (bl, b2), and (cl, c2) be the coordinates of points P, A, B , and C respectively. Then, the minimum point can be found by defining the components of equation (5) Tellier (1972). To compare the properties of P with that of the centroid, which minimizes the sum AP2 + BP2 + CP', note that the centroid divides AU in the ratio of 2:1, and similarly for BV and CW.…”

Spatial optimization for land acquisition problems: A review of models, solution methods, and GIS support

“…A well‐known continuous‐space optimization approach is the Weber problem (Drezner, Klamroth, Schobel, & Wesolowsky, ; Tellier, ; Wesolowsky, ), involving the identification of a point (e.g. a factory) to minimize the total transportation costs to a finite number of demand locations (e.g.…”

“…For cases with three demand locations, the problem is sometimes referred to as the Fermat problem (Wesolowsky, ) and can be solved using a geometrical approach to the triangle problem (Simpson, ). For cases of more than three demand points, geometrical approaches become infeasible and the problem is commonly solved by an iterative method called the Weiszfeld algorithm (Church & Murray, ; Drezner et al., ; Tellier, ).…”

A context-based geoprocessing framework for optimizing meetup location of multiple moving objects along road networks

“…The goal is to find the location that 4 minimizes the sum of the weighted transportation costs. Direct numerical solutions have been developed (Weiszfeld 1937, Tellier 1972. The work by Vardi and Zhang (2001) showed that a modified version of Weiszfeld's algorithm for solving the Weber problem converged monotonically to a unique solution.…”