Editing Trip distribution (section)

== Gravity model ==
The gravity model illustrates the macroscopic relationships between places (say homes and workplaces).  It has long been posited that the interaction between two locations declines with increasing (distance, time, and cost) between them, but is positively associated with the amount of activity at each location (Isard, 1956).  In analogy with physics, Reilly (1929) formulated [[Reilly's law of retail gravitation]], and [[J. Q. Stewart]] (1948) formulated definitions of [[demographic gravitation]], force, energy, and potential, now called accessibility (Hansen, 1959).  The [[distance decay]] factor of 1/distance has been updated to a more comprehensive function of generalized cost, which is not necessarily linear - a negative exponential tends to be the preferred form.

The gravity model has been corroborated many times as a basic underlying aggregate relationship (Scott 1988, Cervero 1989, Levinson and Kumar 1995). The rate of decline of the interaction (called alternatively, the impedance or friction factor, or the utility or propensity function) has to be empirically measured, and varies by context.

Limiting the usefulness of the gravity model is its aggregate nature.  Though policy also operates at an aggregate level, more accurate analyses will retain the most detailed level of information as long as possible. While the gravity model is very successful in explaining the choice of a large number of individuals, the choice of any given individual varies greatly from the predicted value. As applied in an urban travel demand context, the disutilities are primarily time, distance, and cost, although discrete choice models with the application of more expansive utility expressions are sometimes used, as is stratification by income or vehicle ownership.

Mathematically, the gravity model often takes the form:

: <math>
T_{ij}  = K_i K_j T_i T_j f(C_{ij} )
</math>

: <math>
\sum_j {T_{ij}  = T_i } ,\sum_i {T_{ij}  = T_j } 
</math>

: <math>
K_i  = \frac{1}
{{\sum_j {K_j T_j f(C_{ij} )} }},K_j  = \frac{1}
{{\sum_i {K_i T_i f(C_{ij} )} }}
</math>

where 
* <math>T_{ij}</math> = Trips between origin ''i'' and destination ''j''
* <math>T_i</math> = Trips originating at ''i''
* <math>T_j</math> = Trips destined for ''j''
* <math>C_{ij}</math> = travel cost between ''i'' and ''j''
* <math>K_i, K_j</math> = balancing factors solved iteratively. See [[Iterative proportional fitting]].
* <math>f</math> = distance decay factor, as in the accessibility model

It is doubly constrained, in the sense that for any ''i'' the total number of trips from ''i'' predicted by the model always (mechanically, for any parameter values) equals the real total number of trips from ''i''. Similarly, the total number of trips to ''j'' predicted by the model equals the real total number of trips to ''j'', for any ''j''.