THE WEBER PROBLEM: FREQUENCY OF DIFFERENT SOLUTION TYPES AND EXTENSION TO REPULSIVE FORCES AND DYNAMIC PROCESSES*

ABSTRACT. The frequency of occurrence of the different types of solutions to the Weber problem is studied. These solutions are: a location at an attraction point due to a dominant force, to incompatible angles, or to concavity; a location at infinity; a location inside the polygon; and a location outside the polygon. Situations involving both attraction and repulsion points are examined in the triangle and in the more-than-three-sided polygon context, and methods for solving the corresponding problems are compared. A trigonometric solution is proposed for the triangle case involving one repulsion and two attraction points. The variation in the frequency of a location at an attraction point when the number of attraction points increases while the number of repulsion points remains the same is observed as well. Implications of the results are studied for the analysis of dynamic location processes.     

Further Understanding Spatial Inertia: A Reply to Plastria and Rosing

Our position with respect to Plastria and Rosing's comments isqualified. Some of their comments are irrelevant because they attack a general statement that is not included in Tellier and Vertefeuille's paper (1995). Portions of their comments on the mathematical framework are relevant and have some implication for the validity of the mathematical relations proposed. A new analysis of the local behavior of the Weberian optimum is presented and its probabilistic interpretation is emphasized. On the basis of this analysis and ofempirical observation, we maintain that Tellier and Vertefeuille's paper was generically correct with regard to spatial inertia.     

UNDERSTANDING SPATIAL INERTIA: CENTER OF GRAVITY,POPULATION DENSITIES,THE WEBER PROBLEM,AND GRAVITY POTENTIAL*

ABSTRACT. In this paper we attempt to clarify the theoretical links between the concepts of “center of gravity” and “point of maximum population density” which describe the present, and the concepts of “minimum of the comprehensive Weber problem” and “maximum comprehensive gravity potential” which guide the future. Critical values of the characteristic parameters of the relevant functions are estimated. Implications for the understanding of spatial inertia are discussed.     

EQUILATERAL TRIANGLE VS. HEXAGON: A MOST SIMPLE DEMONSTRATION

ABSTRACT. This note presents a most simple proof of the superiority of hexagons over equilateral triangles of the same size in terms of minimization of the average distance between consumers and facilities. It also stresses that, under the same assumptions, facilities are closer to each other in a triangular grid than in an hexagonal grid and that this fact should not be neglected.