"Spatial nodes" in discrete location problems

Discrete location models often assume an underlying network where demands originate at point nodes. To apply these models to planar regions with continuously distributed demand, the region is usually partitioned into "zones" and the demand from each zone is assumed to originate at a point, usually the zone centroid. Thus, the point node in the underlying network represents a spatial zone with a finite area. This paper examines the effect of approximating these "spatial nodes" by point nodes. In some problem scenarios, the approximation does not affect the solution. However, especially when the locational criterion includes the consideration of intra-zonal travel cost variances (e.g. travel time variance) and demands may originate anywhere within zones of nonzero area, point nodes do not give an accurate evaluation of the performance of a locational design. To illustrate the application of the concept of spatial nodes, a model is formulated for locating p (fire-fighting) units in a region having continuously distributed demand with the objective of minimizing a nonlinear function of arrival times of the first and second closest units to any (fire) incident. A heuristic "site-substitution" procedure is presented that solves the formulated model.