## Spatial Interaction Models

Spatial interaction (gravity) models have been employed by urban planners and transportation engineers since the 1960s. Models of this type have their roots in the laws of physics which quantify the attraction of one entity to another based on their relative weights. In the case of modeling land use, a set of discrete zones is defined for the study area and the cost to travel the distance between each pair of zonal midpoints is calculated. The weight of a zone represents its relative attractiveness, such as urban area population density, office density of an employment center, or area of retail space in a store, while the distance factor represents the cost impedance of traveling between zones. Distance can be quantified as Euclidean or least cost path measures, accounting for different road types, speed limits, and congestion. Gravity models have been used in the analysis of optimal commercial facility siting and migration patterns, among other applications. They generally follow the form of eqn [3], where the interaction between zone i and zone j is proportional to the product of the relative attraction of each zone divided by the distance between the zones multiplied by a constant.

The gravity model is represented by

where Iij - the interaction between zones i and j, Pi and Pj - the weight of zones i and j, Dij - distance between zones i and j, and k - an empirically determined constant whose value adjusts the ratio to actual conditions.

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