Editing Trip distribution (section)

== Mathematics ==

At this point in the transportation planning process, the information for zonal interchange analysis is organized in an origin-destination table.  On the left is listed trips produced in each zone.  Along the top are listed the zones, and for each zone we list its attraction.  The table is ''n'' x ''n'', where ''n'' = the number of zones.

Each cell in our table is to contain the number of trips from zone ''i'' to zone ''j''.  We do not have these within-cell numbers yet, although we have the row and column totals.  With data organized this way, our task is to fill in the cells for tables headed ''t''&nbsp;=&nbsp;1 through say ''t''&nbsp;=&nbsp;''n''.

Actually, from home interview travel survey data and attraction analysis we have the cell information for ''t''&nbsp;=&nbsp;1.  The data are a sample, so we generalize the sample to the universe.  The techniques used for zonal interchange analysis explore the empirical rule that fits the ''t'' = 1 data.  That rule is then used to generate cell data for ''t'' = 2, ''t'' = 3, ''t'' = 4, etc., to ''t'' = ''n''.

The first technique developed to model zonal interchange involves a model such as this:

: <math>
T_{ij}  = T_i\frac{{A_j f\left( {C_{ij} } \right)K_{ij} }}
{{\sum_{j' = 1}^n {A_{j'} f\left( {C_{ij'} } \right)K_{ij'} } }}
</math>

where:
* <math>T_{ij}</math> : trips from i to j.
* <math>T_i</math> : trips from i, as per our generation analysis
* <math>A_j</math> : trips attracted to  j, according to generation analysis
* <math>f(C_{ij})</math> : [[travel cost friction]] factor, say = <math>C_{ij}^b</math>
* <math>K_{ij}</math> : Calibration parameter
 
Zone ''i'' generates ''T''<sub>&nbsp;''i''</sub> trips; how many will go to zone ''j''?  That depends on the attractiveness of ''j'' compared to the attractiveness of all places;  attractiveness is tempered by the distance a zone is from zone ''i''. We compute the fraction comparing ''j'' to all places and multiply ''T''<sub>&nbsp;;''i''</sub> by it.

The rule is often of a gravity form:

: <math>
T_{ij}  = a\frac{{P_i P_j }}
{{C_{ij}^b }}
</math>

where:
* <math>P_i; P_j</math> : populations of  ''i''  and  ''j''
* <math>a; b</math> :  parameters

But in the zonal interchange mode, we use numbers related to trip origins (''T''<sub>&nbsp;;''i''</sub>) and trip destinations (''T''<sub>&nbsp;;''j''</sub>) rather than populations.

There are many model forms because we may use weights and special calibration parameters, e.g., one could write say:

: <math>
T_{ij}  = a\frac{{T_i^c T_j^d }}
{{C_{ij}^b }}
</math>

or

: <math>
T_{ij}  = \frac{{cT_i dT_j }}
{{C_{ij}^b }}
</math>

where:
* ''a, b, c, d'' are parameters
* <math>C_{ij}</math> : travel cost (e.g. distance, money, time)
* <math>T_j</math> : inbound trips, destinations
* <math>T_i</math> : outbound trips, origin