History
Over the years, modelers have used several different formulations of trip distribution. The first was the Fratar or Growth model (which did not differentiate trips by purpose). 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. (Simple Growth factor model, Furness Model and Detroit model are models developed at the same time period) The next models developed were the gravity model and the intervening opportunities model. The most widely used formulation is still the gravity model. While studying traffic inMathematics
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'' = 1 through say ''t'' = ''n''. Actually, from home interview travel survey data and attraction analysis we have the cell information for ''t'' = 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: : where: * : trips from i to j. * : trips from i, as per our generation analysis * : trips attracted to j, according to generation analysis * : travel cost friction factor, say = * : Calibration parameter Zone ''i'' generates ''T'' ''i'' 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'' ;''i'' by it. The rule is often of a gravity form: : where: * : populations of ''i'' and ''j'' * : parameters But in the zonal interchange mode, we use numbers related to trip origins (''T'' ;''i'') and trip destinations (''T'' ;''j'') rather than populations. There are many model forms because we may use weights and special calibration parameters, e.g., one could write say: : or : where: * ''a, b, c, d'' are parameters * : travel cost (e.g. distance, money, time) * : inbound trips, destinations * : outbound trips, originGravity 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). TheEntropy analysis
Wilson (1970) gives another way to think about zonal interchange problem. This section treats Wilson’s methodology to give a grasp of central ideas. To start, consider some trips where there are seven people in origin zones commuting to seven jobs in destination zones. One configuration of such trips will be: : where 0! = 1. That configuration can appear in 1,260 ways. We have calculated the number of ways that configuration of trips might have occurred, and to explain the calculation, let’s recall those coin tossing experiments talked about so much in elementary statistics. The number of ways a two-sided coin can come up is , where n is the number of times we toss the coin. If we toss the coin once, it can come up heads or tails, . If we toss it twice, it can come up HH, HT, TH, or TT, four ways, and . To ask the specific question about, say, four coins coming up all heads, we calculate . Two heads and two tails would be . 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. However, the notion of most likely state comes not from this thinking; it comes from statistical mechanics, a field well known to Wilson and not so well known to transportation planners. The result from statistical mechanics is that a descending series is most likely. Think about the way the energy from lights in the classroom is affecting the air in the classroom. If the effect resulted in an ascending series, many of the atoms and molecules would be affected a lot and a few would be affected a little. The descending series would have many affected not at all or not much and only a few affected very much. We could take a given level of energy and compute excitation levels in ascending and descending series. Using the formula above, we would compute the ways particular series could occur, and we would conclude that descending series dominate. That is more-or-less Boltzmann's Law, : That is, the particles at any particular excitation level ''j'' will be a negative exponential function of the particles in the ground state, , the excitation level, , and a parameter , which is a function of the (average) energy available to the particles in the system. 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. For the same travel pattern in the O-D matrix used before, we would have row and column totals, i.e.: Consider the way the four folks might travel, 4!/(2!1!1!) = 12; consider three folks, 3!/(0!2!1!) = 3. All travel can be combined in 12×3 = 36 ways. The possible configuration of trips is, thus, seen to be much constrained by the column and row totals. 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 is the travel cost from ''i'' to ''j''. The discussion thus far contains the central ideas in Wilson’s work, but we are not yet to the place where the reader will recognize the model as it is formulated by Wilson. First, writing the function to be maximized using Lagrangian multipliers, we have: : where and are the Lagrange multipliers, having an energy sense. Second, it is convenient to maximize the natural log (ln) rather than , for then we may useIssues
Congestion
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). Haney (1972) suggests internal assumptions about travel time used to develop demand should be consistent with the output travel times of the route assignment of that demand. 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 and others, and later formal mathematical programming techniques were established by Suzanne Evans.Stability of travel times
A key point in analyzing feedback is the finding in earlier research 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 CitiesBarnes, G. and Davis, G. 2000. ''Understanding Urban Travel Demand: Problems, Solutions, and the Role of Forecasting'', University of Minnesota Center for Transportation Studies: Transportation and Regional Growth Study The stability of travel times and distribution curves over the past three decades gives a good basis for the application of aggregate trip distribution models for relatively long term forecasting. This is not to suggest that there exists a constant travel time budget.See also
*Footnotes
References
* Allen, B. 1984 Trip Distribution Using Composite Impedance Transportation Research Record 944 pp. 118–127 * Ben-Akiva M. and Lerman S. 1985 Discrete Choice Analysis, MIT Press, Cambridge MA * Boyce, D., Lupa, M. and Zhang, Y.F. 1994 Introducing “Feedback” into the Four-Step Travel Forecasting Procedure vs. the Equilibrium Solution of a Combined Model presented at 73rd Annual Meeting of Transportation Research Board * Haney, D. 1972 Consistency in Transportation Demand and Evaluation Models, Highway Research Record 392, pp. 13–25 1972 * Hansen, W. G. 1959. How accessibility shapes land use. Journal of the American Institute of Planners, 25(2), 73–76. * Heanue, Kevin E. and Pyers, Clyde E. 1966. A Comparative Evaluation of Trip Distribution Procedures, * Levinson, D. and Kumar A. 1995. A Multi-modal Trip Distribution Model. Transportation Research Record #1466: 124–131. * Portland MPO Report to Federal Transit Administration on Transit Modeling * Reilly, W.J. (1929) “Methods for the Study of Retail Relationships” University of Texas Bulletin No 2944, Nov. 1929. * Reilly, W.J., 1931. The Law of Retail Gravitation, New York. * Ruiter, E. 1967 Improvements in Understanding, Calibrating, and Applying the Opportunity Model Highway Research Record No. 165 pp. 1–21 * Stewart, J.Q. (1948) “Demographic Gravitation: Evidence and Application” Sociometry Vol. XI Feb.–May 1948 pp 31–58. * Stewart, J.Q., 1947. Empirical Mathematical Rules Concerning the Distribution and Equilibrium of Population, Geographical Review, Vol 37, 461–486. *Stewart, J.Q., 1950. Potential of Population and its Relationship to Marketing. In: Theory in Marketing, R. Cox and W. Alderson (Eds) ( Richard D. Irwin, Inc., Homewood, Illinois). * Stewart, J.Q., 1950. The Development of Social Physics, American Journal of Physics, Vol 18, 239–253 * Voorhees, Alan M., 1956, "A General Theory of Traffic Movement," 1955 Proceedings, Institute of Traffic Engineers, New Haven, Connecticut. * Whitaker, R. and K. West 1968 The Intervening Opportunities Model: A Theoretical Consideration Highway Research Record 250 pp. 1–7 * Wilson, A.G. A Statistical Theory of Spatial Distribution Models Transportation Research, Volume 1, pp. 253–269 1967 * Wohl, M. 1963 Demand, Cost, Price and Capacity Relationships Applied to Travel Forecasting. Highway Research Record 38:40–54 * Zipf, G. K., 1946. The Hypothesis on the Intercity Movement of Persons. American Sociological Review, vol. 11, Oct * Zipf, G. K., 1949. Human Behaviour and the Principle of Least Effort. Massachusetts {{Transportation-planning Transportation planning