Trip distribution
Work trip distribution is the way that travel demand models understand how people take jobs.
There are trip distribution models for other (non-work) activities such as the choice of location for grocery shopping, which follow the same structure.
This structure extrapolated a base year trip table to the future based on growth, but took no account of changing spatial accessibility due to increased supply or changes in travel patterns and congestion.
This formula has been instrumental in the design of numerous transportation and public works projects around the world.
The Fratar model was shown to have weakness in areas experiencing land use changes.
Some theoretical problems with the intervening opportunities model were discussed by Whitaker and West (1968) concerning its inability to account for all trips generated in a zone which makes it more difficult to calibrate, although techniques for dealing with the limitations have been developed by Ruiter (1967).
With the development of logit and other discrete choice techniques, new, demographically disaggregate approaches to travel demand were attempted.
The application of these models differs in concept in that the gravity model uses impedance by travel time, perhaps stratified by socioeconomic variables, in determining the probability of trip making, while a discrete choice approach brings those variables inside the utility or impedance function.
Because of computational intensity, these formulations tended to aggregate traffic zones into larger districts or rings in estimation.
Allen (1984) used utilities from a logit based mode choice model in determining composite impedance for trip distribution.
At this point in the transportation planning process, the information for zonal interchange analysis is organized in an origin-destination table.
The techniques used for zonal interchange analysis explore the empirical rule that fits the t = 1 data.
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.
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: where 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.
We are solving the equation: An important point is that as n gets larger, our distribution gets more and more peaked, and it is more and more reasonable to think of a most likely state.
The two paragraphs above have to do with ensemble methods of calculation developed by Gibbs, a topic well beyond the reach of these notes.
Returning to the O-D matrix, note that we have not used as much information as we would have from an O and D survey and from our earlier work on trip generation.
We put this point together with the earlier work with our matrix and the notion of most likely state to say that we want to subject to where: and this is the problem that we have solved above.
Wilson adds another consideration; he constrains the system to the amount of energy available (i.e., money), and we have the additional constraint, where C is the quantity of resources available and
back into our constraint equations, we have: and, taking the constant multiples outside of the summation sign Let we have which says that the most probable distribution of trips has a gravity model form,
One of the key drawbacks to the application of many early models was the inability to take account of congested travel time on the road network in determining the probability of making a trip between two locations.
Although Wohl noted as early as 1963 research into the feedback mechanism or the “interdependencies among assigned or distributed volume, travel time (or travel ‘resistance’) and route or system capacity”, this work has yet to be widely adopted with rigorous tests of convergence, or with a so-called “equilibrium” or “combined” solution (Boyce et al. 1994).
While small methodological inconsistencies are necessarily a problem for estimating base year conditions, forecasting becomes even more tenuous without an understanding of the feedback between supply and demand.
Initially heuristic methods were developed by Irwin and Von Cube [2] and others, and later formal mathematical programming techniques were established by Suzanne Evans.
[3] A key point in analyzing feedback is the finding in earlier research[4] that commuting times have remained stable over the past thirty years in the Washington Metropolitan Region, despite significant changes in household income, land use pattern, family structure, and labor force participation.
Similar results have been found in the Twin Cities[5] The stability of travel times and distribution curves over the past three decades[when?]
gives a good basis for the application of aggregate trip distribution models for relatively long term forecasting.
