THE MAXIMUM CAPTURE OR “SPHERE OF INFLUENCE” LOCATION PROBLEM: HOTELLING REVISITED ON A NETWORK*

ABSTRACT. The problem of the ice cream vendor on the beach is extended to the siting of multiple entering servers into the midst of existing servers which are already in place at multiple sites on a network. The network of population nodes, joining arcs, and eligible sites for servers provides the underlying matrix of the problem. The goal of the single firm which is siting the multiple entering servers is the capture of the maximum population from existing servers. This combinatorial problem can be described as a linear integer program for which solutions can easily be derived. Computational experience and insights derived therefrom are reported.     

TESTING SIZE-DENSITY RELATIONSHIPS WITHOUT RATIO-VARIABLES,MULTICOLLINEARITY, LOGARITHMIC TRANSFORMATIONS,OR REGRESSION ANALYSIS*

Using data describing the historical distribution of county seats in the United States, we (1) replicate the traditional ratio-variable test of the size-density law, (2) test a new multivariate, nonratio form of the law, and (3) introduce an alternative method of testing, one which avoids three possible problems in traditional analyses: ratio-variables, multicollinearity, and logarithmic transformation.