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Part III: Travel Demand Modeling

11 Second Step of Four Step Modeling (Trip Distribution)

Chapter Overview

This chapter describes the second step, trip distribution, of the four-step travel demand modeling (FSM). This step focuses on the procedure that distributes the trips after trip generation has been modeled, meaning after trips generated from or attracted to each zone in the study area are understood. The input for this step of FSM is the output from the previous step discussed in Chapter 10, trip generation plus the interzonal transportation costs introduced in this chapter. Based on the concepts of the gravity model, the trip flows between pairs of zones can be calculated as an origin-to-destination (O-D) matrix. The essential concepts and techniques, such as growth factors and calibration methods, for this step are also discussed in this chapter.

Learning Objectives

Student Learning Outcomes

• Explain trip distribution and how to relate it to the first step (trip generation) results.
• Summarize the factors that determine the level of attractiveness of zones in a travel demand model.
• Summarize and compare different methods for trip distribution estimation within FSM.
• Complete the trip distribution step by balancing total trip productions and attractions after the trip distribution step.

Introduction

This chapter delves into trip distribution, which is the second step of the Four-Step Model (FSM). After generating trip productions and attractions (P-A trips) by zone in the first step, the next task is to compute the number of trips between each pair of zones, referred to as trip distribution. These outputs are commonly known as Origin-Destination pairs (O-D pairs or Tij, as discussed in Chapter 9), indicating the number of trips between Zone i (origin) and Zone j (destination) (Levine, 2010). Essentially, trip distribution transforms the outcomes of the first FSM step into a comprehensive matrix detailing origins and destinations in Traffic Analysis Zones (TAZs). It also considers travel impedance factors, such as travel time or cost, for each O-D pair. Figure 11.1 illustrates the input (P-A trips) and outputs (trip tables) of this step in the model, highlighting the role of impedance functions initially introduced in Chapter 3 and discussed further in this chapter.

Recall from Chapter 10, that each step of the FSM answers a question specific to the step but central to model determining travel demand in a study area.  For the trip distribution step, the main question is “What portion of trips produced in or attracted to a zone would go to each of the other zones?” There are several methods typically used to estimate trip distribution. While growth factor models and the intervening opportunities model are used , the gravity model is the most common.

There are a few foundational components to consider prior to calculating trip distribution. It is important to note that these components are independent of the FSM framework, or the methods used for trip distribution estimation. However, they serve as inputs for estimating trip distribution. As previously mentioned, trip distribution constitutes the second step of the FSM, where trip productions are allocated to all other zones. The outcomes yield a matrix that displays the number of both intrazonal and interzonal trips in a single table (Lincoln MPO, 2011).

The attractiveness of a zone is influenced by several factors (Cesario, 1973):

• Uniqueness : This factor indicates how unique a service or employment center is and thus attracts more trips regardless of distance.
• Distance: The spatial separation, distance, between two zones plays an impedance role, meaning that the further the two zones are from each other, the fewer trips will be distributed between them.
• Closeness to other services: The assumption is that proximity to other desirable services will result in more trips to that zone within an urban area.
• Urban or rural area : The assumption is that the attraction rate for a zone differs based on its urban or rural classification, while controlling for other factors.

In addition to the destination’s attractiveness factors, the origin’ has an emissivity factor, which is usually represented by population, employment, or income (Cesario, 1973). With a general understanding of the factors affecting trip distribution from origin and destination, we can now proceed with an introduction to methodology.

Gravity Model

As we discussed, the gravity model is the most common method used to estimate trip distribution. Gravity models are easy to understand and very accurate, and they can also accommodate varied factors such as population, employment, socio-demographics, and transportation systems. Almost all U.S. Departments of Transportation (DOTs) use gravity models. In contrast, the Growth Factor Model, discussed in a subsequent section, requires additional data about trip distribution in the base year and an estimate of the number of future trips in each zone, which is only sometimes available (Meyer, 2016).

It is important to remember that the Gravity Model is built based on the number of trips made between two zones. The number of trips is directly linked to the total number of attractions in the destination zone and inversely proportional to a function of cost, which may be represented by travel time or trip cost (Council, 2006). The formula gets its name from Newton’s Law of Gravity , which states that the attractiveness between two bodies is related to their mass (positive attraction) and the distance between them (negative attraction) (Verlinde, 2011). In transportation modeling, the two main factors are trip production and attraction, along with the time duration of travel or the cost of travel. While using the gravity model is simple, determining the optimal value for the impedance factor can be difficult. This value is highly dependent on the context and can vary by circumstances.

Equation below shows the fundamental equation of trip distribution:

Trips between TAZ1 and TAZ2=Trips prodduced in TAZ1*(Attractiveness of TAZ2 /Attractiveness of all TAZs  

As equation (1) shows, the total trips between zones are equal to the products of the trips produced in a zone, a ratio of the attractiveness of the destination zone, and the total attractiveness of all zones. We can represent the gravity model in several different ways. Remodifying equation the original equation, the gravity model can be rewritten as:

Trips ij =Productions i *(Attractions j *FF ij *k ij /∑Attractions j *FF ij *k ij )   

Where Trips ij is the number of trips between zone i and zone j , Prouctionsi is trip production in zone i , Attractionsj is total trips attracted to zone j , FF ij is the friction factor (travel impedance) between i and j , and K ij are the socio-economic factors of zones i and j . These values will be elaborated later in this chapter.

From the above equations, the mathematical format of gravity model can be seen in equation below:

T ij = number of trips that are produced in zone i and attracted to zone j

P i = total number of trips produced in zone i

A j = number of trips attracted to zone j

F ij = a value which is an inverse function of travel time

K ij = socioeconomic adjustment factor for interchange ij

Recall that the Pi and Aj values are determined through the trip generation process (refer to Chapter 10), and the sum of all productions and attractions should be equal (PE, 2017). Numerous studies confirm that people value travel time differently based on the purpose of the trip (like work trips vs. recreational trips) (Hansen, 1962; Allen, 1984; Thill & Kim, 2005). Therefore, it is rational to compute the gravity model for each trip purpose using different impedance factors (Meyer, 2016).

Impedance Factor

The impedance factor (aka friction factor) is a value that varies for different trip purposes because, with the FSM model, the assumption is that travel behavior depends on trip purpose. Impedance captures the spatial separation between two zones, represented as travel time or cost.

Friction factors (FF) can be estimated using different measure, as follows:

• A simple measure of friction is the travel time between the zones.
• Another method is adopting an exponential formula with the 1/exp (m × Tij) friction factor, where m is the average travel time calculated using empirical data.
• Gamma distribution uses scaling factors to estimate distribution (Cambridge Systematics, 2010; Meyer, 2016).

The impedance factor reflects the difficulty of traveling between two zones. The friction factor is higher when accessibility between two zones is easy and is zero if no individual is willing to travel between two zones.

There is also a calibration step in the friction factor estimation process. For calibration, trip generation and attraction values are distributed between O-D pairs using the gravity model. Next, the number of trips is compared with a particular amount of time to the results of the O-D survey (observed data). If the numbers do not match, calibration adjusts for the friction factor. When using travel time as the measure of impedance, the relationship between the friction factor and time in the is represented as t-1, t–2, e– t (Ashford & Covault, 1969).

The friction factor is estimated for the entire analysis area. However, such an assumption is limiting because travel costs or time has different implications for different households. For example, a toll on a specific highway may result in disparate use. The cost burden or friction factor may be too high for low-income individuals or households, meaning a factor greater than zero, but negligible for higher-income households. In this case the friction factor for higher-income commuters is zero.

The friction factors for different trip purposes can also be specified. The figure below (Figure 11.2) shows the function of the friction factor appropriate to the time and for different trip purposes. As the figure shows, there is a direct relationship between friction factor and travel time. According to this figure, for each trip purpose, there is a perception about the length or impedance of trip. Beyond certain length, friction factor approaches zero, meaning a high level of disutility of the trip and this threshold is different for each trip purpose. In very general terms, a friction factor Fij that is an inverse function of travel impedance Wij is used in trip distribution to plug in the travelers’ willingness to travel between zone i and zone j .

In very general terms, a friction factor F ij that is an inverse function of travel impedance W ij is used in trip distribution to plug in the travelers’ willingness to travel between zone i and zone j .

Travel demand modeling is influenced by various socio-economic factors that affect travel behavior and demand for different purposes. Chapter 10 highlights the most significant factors in travel demand modeling: income, auto ownership, availability of multimodal transportation systems, age, and job type (Pan et al., 2020). Therefore, the K-Factor method was developed and plugged into the gravity model to represent variation in socio-economic factors and adjust interzonal trips accordingly. For example, a blue-collar employee working in a low-income suburb may exhibit different travel behaviors (in terms of mode choice and frequency of travel) compared to a white-collar employee working in the central city with a higher income. The K-factor is determined and plugged into the gravity formula to accommodate such differences.

Recall the classic land use models presented in Chapter 4. Based on the proximity to employment centers or the central business districts, different neighborhoods offer housing and job options tailored to individuals in different income brackets. For example, the earnings of employees in chain restaurants significantly contrast with those working in Central Business District (CBD) headquarters. Prevailing land use policies and the typical American development pattern heighten this disparity. Consequently, these groups are likely to inhabit geographically distant areas in a country like the United States. Furthermore, people of varying income levels or social statuses may respond differently to travel impediments, such as travel time or cost.

Calibration of K values is determined by comparing the estimated results and observed data for the base year (Tawfik & Rakha, 2012). The K numeric value will be above one (>1) if the socio-economic factors contribute to more travel and less than one (< 1) if otherwise (Meyer, 2016). Figure 11.3 shows the mean number of trips for different age groups (K-factor) and various trip purposes. Accordingly, calculating friction factors and K-factors for different purposes and socio-economic groups yields a better fit to the data.

11.2.3 Example 1

Consider a small area with three zones (TAZs). Table 11.1 shows the trip generation results for each zone, and Table 11.2 shows the travel time for each pair of zones. The friction factor is also given in this example as a function of travel time in Table 11.3. The intrazonal travel time for zone 1 is larger than that of most other inter-zone times because of the geographical characteristics of the zone and lack of access within the area. Using this information, please calculate the number of trips for each pair of zones.

For the calculation of trip distribution between the three zones, the trip generation and attraction table from step one of the FSM model is the input data, and then the gravity model is used for calculation. Tables 11.1, 11.2, and 11.3 represent the trip generated and attracted for each zone, travel time between each pair of zones, and friction factor derived from the travel time.

Now with this information, we can start the calculation process. First, we have to estimate the attractiveness of each zone using the equation (1)

For example, for zone 1 we have:

Attractiveness1= 210*26=5460

Attractiveness2= 210*35=7350

Attractiveness3= 350*35=12250

Now, we use the pivotal formula of the gravity model (equation 2). Accordingly, we have (K-factor set to 1):

The result of the calculation is summarized in Table 11.4:

However, our calculations’ results do not match the already existing and observed data. The mentioned mismatch is why calibration and balancing of the matrix are needed. In other words, we must perform more than one iteration of the model to generate more accurate results. For performing a double or triple iteration, we use a formula discussed at the end of this chapter (example adapted from: Garber & Hoel, 2018).

Growth Factor Model

After successfully calibrating and validating the data we have estimated, we can also apply the gravity model to forecast future travel behavior or travel patterns in our study area. Future trip distributions can be predicted by using the change in land-use data, socioeconomic data, or any other changes in the whole system. We can calculate trip distribution from the O-D table for either base or forecasting year when the friction factor and K-factor data are unavailable or unsatisfactorily calibrated. Depending on historical trends and data, growth factor models are limited if an observed O-D table is unavailable. Similar to the trip generation step, growth factor models cannot incorporate updated travel time as the change in travel time between zones can highly affect travel patterns (Qsim, 2016).

Fratar method

One of the most common mathematical formulas of the growth factor model is the Fratar method, shown in the following equation. Through his method, the future distribution of trips from one zone is equal to the present distribution multiplied by the growth factor of the destination zone between now and the forecasting year (Heanue & Pyers, 1966). The formula to calculate future trip values is shown in equation below:

T ij =number of trips estimated from zone  to zone t i  =present trip generation in zone G x =growth factor of zone T i  =future trip generation in zone t ix =number of trips between zone  and other zones t ij =present trips between zone  and zone G j =growth factor of zone

The following section will discuss an example illustrating the application of the Fratar method.

The case study area of this example consists of four TAZs. Table 11.5 shows the current trip distributions. Assuming the growth rate for each TAZ is shown in Table 11.6, the next step is to calculate the number of trips between each two TAZs in the future year.

To solve this problem, apply the Fratar Method using the required two estimates for each pair. These estimates should be averaged; the resulting value will be the final T ij . Based on the formula, calculations are as follows:

Based on the calculations, the first iteration of the method will yield the following table:

To estimate future trip rates between zones, use the Fratar formula, as shown in Table 11.7. However, there is a problem with the estimated total number of trips generated in each zone not being equal to the actual trip generation. Therefore, a second iteration is necessary. In this second iteration, we use the new O-D matrix as the input to calculate new growth ratios. The trip generation is estimated to occur in the next five years based on the preceding calculation. As an exercise, you can conduct as many iterations as needed to bring the estimated and actual trip generations into alignment.

In a hypothetical area, we are interested in determining the number of trips attracted by three different shopping malls at various distances from a university campus that generates about 2,000 trips per day. In Figure 11.4, the hypothetical area, the number of trips generated by the campus, and the total number of trips attracted for each zone are presented:

•  socioeconomic adj. Factor K=1.0
•  Calibration factor C=2.0

As the first step, we need to calculate the friction factor for each pair of zones based on travel time (t). Given is the following formula with which we calculate friction factor:

Next, using the friction factor, we use the gravity model to calculate the relative attractiveness of each zone. In Table 11.8 , you can see how calculations are being carried out for each zone.

Next, with having relative attractiveness of each zone (or probability of attracting trips), we plug in the trip generation rate for the campus (6,000) to finally estimate the number of trips attracted from the campus to each zone. Figure 11.5 shows the final results.

Model Calibration and Validation

Model validation is an integral part of all simulation and modeling procedures. One of the most essential steps in FSM modeling is developing a procedure to calibrate its final outputs (predictions) with actual and observed data. To do this, model parameters are adjusted so that the observed data and estimations have fewer mismatches (Meyer, 2016). After such adjustments, the model with calibrated parameters can help in simulation and future scenario analyses.

After completing the trip distribution step, it is important to compare model calibration and adjustment results in each category (i.e., by trip purpose) with recorded real-world trips from the O-D survey.

If the two values are not identical, model parameters, like FF or K-factors, are reassigned and re-run the gravity model. The process continues until the observed data and estimations are very close (ratio between 0.9 and 1.1).

The following example shows the process of trip distribution step with calibration.

11.4.1 Example 4

This example demonstrates the calibration process. The first step is to identify the model’s inputs, which are the outcomes of the trip generation process. The following tables show the results obtained from surveys and actual trip data, as well as the travel time between each pair of zones (represented by the friction factor (FF)) and the socioeconomic conditions (Tables 11.10, 11.11, and 11.12 ). The column with the heading “A’ “in Table 11.11 represents observed generated trips.

here are several formulas, such as negative exponential or inverse power function, that can be used to calculate friction factors from impeding factors like travel cost or time, as discussed in previous sections. To estimate the number of trips between each pair of zones, we use the gravity formula and input the necessary data. Table 11.14 shows the results of trip distribution for each pair of zones. However, the total number of trips attracted from our calculations is different from observed data.

Now, by looking to the last table we can see that the total number of trips produced is exactly matching to the results of the trip generation table. But the total attractions and actual data have a mismatch. In the next step, we apply the calibration methods in order to make our final results more accurate.

In the first iteration of calibration, we have to generate a value called column factor, which is the result of dividing actual data attraction by estimated attractions. Then we apply this number for each pair in the same column. In Table 11.15, we can observe that the sum of attractions is now the same as the actual data, but the sum of generation amounts is now different from actual data generation. In this step, we perform another iteration, the same as the first iteration but instead of column factor, we plug in row factor value, which is the result of dividing actual data trip generation by estimated generation.

A third iteration is needed because the sum of attraction is still different from the actual data, and we must generate another column factor. The results are shown in Table 11.17.

Based on the results of the third iteration results, we see the attractions are now accurate, and trip generations have very insignificant differences with actual data. At this point, we can stop the calibration. However, the procedure can continue to calibrate results to decrease the difference as much as possible. In other words, the sensitivity of the calibration, the threshold for the row and column factors, can be adjusted by the modeler.

In this chapter, we demonstrated the procedure, application and other details of the second step of FSM modeling framework. Using the concept of gravity-based accessibility, we saw how the production and attraction table can be transformed into a trip distribution matrix. By using simple numerical examples, we showed how different methods can be applied to calculate trips between pair of zones. Assumption of homogenous behavior, assumption of static and sequential behavior, aggregation biases, and less emphasis on lots of social and physical barriers. Dynamic modeling (concurrent mode and destination choice), micro-simulation, agent-based models or newer methods such machine learning have made several enhancement to the traditional model. Collection of real-time data as well as increase in computational capacity has opened such prospects in travel demand modeling and trip distribution studies.

• Emissivity is a quantity that represents the trip production rate of a neighborhood, similar to attractiveness for trip attraction.
• Intra zonal trips are those trips that both ends of the trip is in the same zone.
• Interzonal Trips are those trips where one end of the trip is in a different zone.
• Uniqueness is a quantity defined for a TAZ that indicates how unique that zone or trip attraction center is.
• Gamma distribution is a probability distribution that is used for converting travel times into impedance functions

Blue-collar employee is a worker who usually performs manual and low-skill duties for their work.

White-collar employee is a worker who is high-skill and performs professional, or administrative work.

• Zonal Emissivity refers to a quantity that represents the trip making rates for that zone. Factors affecting this feature can be population, employment, income level, vehicle ownership, etc.

Key Takeaways

In this chapter, we covered:

• What trip distribution is and the factors that determine attractiveness of zones for travel demand.
• Different modeling frameworks appropriate for trip distribution and their mathematical formulation.
• What advantages and disadvantages of different methods and assumptions in trip distribution are.
• How to perform a trip distribution manually using simplified transportation networks.

Prep/quiz/assessments

• What factors affect the attractiveness of the zones in trip distribution, and what input data is needed to measure such attractiveness?
• What are the advantages and disadvantages of the three trip distribution methods (gravity model, intervening opportunities, and Fratar model)?
• What are the friction factor and K-factor in trip distribution, and how do they help to calibrate model results?
• How should we balance trip attraction and production after performing trip distribution? Explain.

Allen, B. (1984). Trip distribution using composite impedance. Transportation Research Record , 944 , 118–127.

Seggerman, KE. (2010). Increasing the integration of TDM into the land use and development process. Fairfax County (Virginia) Department of Transportation, May. Department of Transportation.

Cesario, F. J. (1973). A generalized trip distribution model. Regional Science Journal , 13 (2), 1973-08

Council, A. T. (2006). National guidelines for transport system management in Australia 2006 .  Australia Transportation Council. https://www.atap.gov.au/sites/default/files/National_Guidelines_Volume_1.pdf

Garber, N. J., & Hoel, L. A. (2018).  Traffic and highway engineering . Cengage Learning.

Hansen, W. G. (1962). Evaluation of gravity model trip distribution procedures . Highway Research Board Bulletin, 347 . https://onlinepubs.trb.org/Onlinepubs/hrbbulletin/347/347-007.pdf

Ned Levine (2015).  CrimeStat : A spatial statistics program for the analysis of crime incident locations (v 4.02). Ned Levine & Associates, Houston, Texas, and the National Institute of Justice, Washington, D.C. August.

Lima & Associates. (2011). Lincoln travel demand model . Lincoln Metropolitan Planning Organization. (2011). https://www.lincoln.ne.gov/files/sharedassets/public/planning/mpo/projects-amp-reports/tdm11.pdf

Meyer, M. D. (2016). Transportation planning handbook . John Wiley & Sons.

NHI. (2005). Introduction to Urban Travel Demand Forecasting . In National Highway Administration (Ed.), Introduction to Urban Travel Demand Forecasting. American University. . National Highway Institute : Search for Courses (dot.gov)

Pan, Q., Jin, Z., & Liu, X. (2020). Measuring the effects of job competition and matching on employment accessibility. Transportation Research Part D: Transport and Environment , 87 , 102535. https://doi.org/10.1016/j.trd.2020.102535

PE Lindeburg, M. R. (2017). PPI FE civil review – A comprehensive FE civil review manual (First edition). PPI, a Kaplan Company.

Qasim, G. (2015). Travel demand modeling: AL-Amarah city as a case study . [Unpublished Doctoral dissertation , the Engineering College University of Baghdad]

Tawfik, A. M., & Rakha, H. A. (2012). Human aspects of route choice behavior: Incorporating perceptions, learning trends, latent classes, and personality traits in the modeling of driver heterogeneity in route choice behavior . Virginia Tech Transportation Institute . Blacksburg, Virginia   https://vtechworks.lib.vt.edu/handle/10919/55070

Thill, J.-C., & Kim, M. (2005). Trip making, induced travel demand, and accessibility. Journal of Geographical Systems , 7 (2), 229–248. https://doi.org/10.1007/s10109-005-0158-3

Verlinde, E. (2011). On the origin of gravity and the laws of Newton. Journal of High Energy Physics , 2011 (4), 1–27. https://link.springer.com/content/pdf/10.1007/JHEP04(2011)029.pdf

Transportation Land-Use Modeling & Policy Copyright © by Mavs Open Press. All Rights Reserved.

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3.5: Destination Choice

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• David Levinson et al.
• Associate Professor (Engineering) via Wikipedia

Everything is related to everything else, but near things are more related than distant things. - Waldo Tobler's 'First Law of Geography’

Destination Choice (or trip distribution or zonal interchange analysis ), is the second component (after Trip Generation, but before Mode Choice and Route Choice) in the traditional four-step transportation forecasting model. This step matches tripmakers’ origins and destinations to develop a “trip table”, a matrix that displays the number of trips going from each origin to each destination. Historically, trip distribution has been the least developed component of the transportation planning model.

Table: Illustrative Trip Table

Where: \(T_{ij}\) = Trips from origin i to destination j .

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, which follow the same structure.

Fratar Models

The simplest trip distribution models (Fratar or Growth models) simply extrapolate a base year trip table to the future based on growth,

\[T_{ij,y+1}=g*T_{ij,y}\]

• \(T_{ij,y}\) - Trips from \(i\) to \(j\) in year \(y\)
• \(g\) - growth factor

Fratar Model takes no account of changing spatial accessibility due to increased supply or changes in travel patterns and congestion.

Gravity Model

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 auto ownership.

Mathematically, the gravity model often takes the form:

\[T_{ij}=r_is_jT_{O,i}T_{D,j}F(C_{ij})\]

\[\displaystyle \sum{j}T_{ij}=T_{O,i}, \displaystyle \sum{i} T_{ij} = T_{D,j}\]

\[r_i = (\displaystyle \sum{j} s_jT_{D,j}f(C_{ij}))^{-1}\]

\[s_j=(\displaystyle \sum{i} r_iT_{O,i}f(C_{ij}))^{-1}\]

• \(r_i,s_j\) = balancing factors solved iteratively.
• \(f\) = impedance or distance decay factor

Balancing a matrix

Balancing a matrix can be done using what is called the Furness Method , summarized and generalized below.

1. Assess Data, you have \(T_{O,i}\),\(T_{D,j}\), \(C_{ij}\)

2. Compute \(f(C_{ij})\), e.g.

\[f(C_{ij})=C_{ij}^{-2}\]

\[f(C_{ij})=e^{\beta C_{ij}}\]

3. Iterate to Balance Matrix

(a) Multiply Trips from Zone \(i(T_i)\) by Trips to Zone \(j(T_j)\) by Impedance in Cell \(ij(f(C_{ij})\) for all \(ij\)

(b) Sum Row Totals \(T'_{O,i}\), Sum Column Totals \(T'_{D,j}\)

(c) Multiply Rows by \(N{O,i}=T_{O,i}/T'_{O,i}\)

(d) Sum Row Totals \(T'_{O,i}\), Sum Column Totals \(T'_{D,j}\)

(e) Compare \(T_{O,i}\) and \(T'_{O,i}\), \(T_{D,j}\) \(T'_{D,j}\) if within tolerance stop, Otherwise go to (f)

(f) Multiply Columns by \(N_{D,j}=T_{D,j}/T'_{D,j}\)

(g) Sum Row Totals \(T'_{O,i}\), Sum Column Totals \(T'_{D,j}\)

(h) Compare \(T_{O,i}\) and \(T'_{O,i}\), \(T_{D,j}\) and \(T'_{D,j}\) if within tolerance stop, Otherwise go to (b)

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 (as quoted in Florian et al. (1975) ) and others, and later formal mathematical programming techniques were established by Evans (1976).

Feedback and time budgets

A key point in analyzing feedback is the finding in earlier research by Levinson and Kumar (1994) 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 by Barnes and Davis (2000).

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.

In terms of time budgets:

• 1440 Minutes in a Day
• Time Spent Traveling: ~ 100 minutes + or -
• Time Spent Traveling Home to Work: 20 – 30 minutes + or -

Research has found that auto commuting times have remained largely stable over the past forty years, despite significant changes in transportation networks, congestion, household income, land use pattern, family structure, and labor force participation. The stability of travel times and distribution curves gives a good basis for the application of trip distribution models for relatively long term forecasting.

Example 1: Solving for impedance

You are given the travel times between zones, compute the impedance matrix \(f(C_{ij})\), assuming \(f(C_{ij})=C_{ij}^{-2}\).

Compute impedances (\(f(C_{ij})\))

Impedance Matrix ( \(f(C_{ij})\) )

You are given the travel times between zones, trips originating at each zone (zone1 =15, zone 2=15) trips destined for each zone (zone 1=10, zone 2 = 20) and asked to use the classic gravity model \(f(C_{ij})=C_{ij}^{-2}\)

(a) Compute impedances (\(f(C_{ij})\))

(b) Find the trip table

Balancing Iteration 0 (Set-up)

Balancing Iteration 1 ( \(T_{ij,iteration1}=T_{O,i}T_{D,j}f(C_{ij})\) )

Balancing Iteration 2 ( \(T_{ij,iteration2}=T_{ij,iteration1}*N_{O,i,iteration1\) )

Balancing Iteration 3 ( ​​​​\(T_{ij,iteration3}=T_{ij,iteration2}*N_{D,j,iteration2\) )

Balancing Iteration 4 ( ​​​​​\(T_{ij,iteration4}=T_{ij,iteration3}*N_{O,i,iteration3\) )

Balancing Iteration 16 ( \(T_{ij,iteration16}=T_{ij,iteration15}*N_{D,i,iteration5\) )

So while the matrix is not strictly balanced, it is very close, well within a 1% threshold, after 16 iterations. The threshold refers to the proximity of the normalizing factor to 1.0.

Additional Questions

1. Identify five independent variables that you believe affect trip generation. Pose hypotheses about how each variable affects number of trips generated.

2. Identify three different types of trip distribution models. Which one includes the most information? Which one is most common?

3. You are given the following situation: The towns of Saint Cloud and Minneapolis, separated by 110 km as the crow flies, are to be connected by a railroad, a freeway, and a rural highway. Answer the following questions related to this problem

Trip Generation and Distribution

Your planners have estimated the following models for the AM Peak Hour

\[T_{O,i}=500+0.5*{HH}_i\]

\[T_{D,j}=250+0.5*{OFFEMP_j}+0.355*{OTHEMP_j}+0.094*{RETEMP_j}\]

Where: \(T_{O,i}=\) Person Trips Originating in Zone i

\(T_{D,j}=\)Person Trips Destined for Zone j

\(HH_i=\) Number of Households in Zone i

\({OFFEMP}_j=\)Office Employees in Zone j

\({OTHEMP}_j=\)Other Employees in Zone j

\({RETEMP}_j\)Retail Employees in Zone j

Your are also given the following data

The travel time between zones (in minutes) is given by the following matrix:

(a) (10) What are the number of AM peak hour person trips originating in and destined for Saint Cloud and Minneapolis.

(b) (10) Assuming the origins are more accurate, normalize the number of destination trips for Saint Cloud and Minneapolis.

(c) (10) Assume a gravity model where the impedance \(f(C_{ij})=C_{ij}^{-2}\). Estimate the proportion of trips that go from Saint Cloud to Minneapolis. Solve your matrix within 5 percent of a balanced solution.

• What are the different kinds of models for trip distribution? How do they differ?
• What factors do conventional Trip Distribution models neglect?
• In a day, how many minutes are spent traveling for the average person, traveling to work? Why might this be stable, not stable?
• Do travelers have a travel time tolerance or a travel time budget?
• What affects travel impedance?
• If impedance increases, will willingness to travel increase or decrease? Is a person more likely to travel to a closer place or a farther place?
• Why does willingness to travel have a negative exponential form?
• Briefly describe the gravity model? How might the gravity model be extended to depend on more than just size (opportunities) and distance (impedance)?
• Why is balancing a matrix an iterative procedure?
• What importance does impedance play in balancing iterations
• Give two functions of impedance (f(Cij)) used in gravity models.
• How do you calculate impedance between zones?
• What is congested travel time?
• Conventional trip distribution models are estimated for which mode?
• How is trip distribution applied?
• What can a trip distribution curve tell you about maximum and average trip times and willingness to take trips?
• What factors are multiplied to give a resulting trip distribution curve
• \(r_i\) - Calibration parameter for Origins
• \(s_j\) - Calibration parameter for Destinations

• View Record

TRID the TRIS and ITRD database

A COMPARATIVE EVALUATION OF TRIP DISTRIBUTION PROCEDURES

THE RESULTS OF A RESEARCH PROJECT DESIGNED TO EVALUATE ON A COMMON BASIS THE FRATAR, GRAVITY, INTERVENING OPPORTUNITIES AND COMPETING OPPORTUNITIES TRIP DISTRIBUTION PROCEDURES ARE REPORTED. EACH OF THE PROCEDURES WAS CALIBRATED USING THE 1948 WASHINGTON, D. C., O-D SURVEY TRAVEL DATA AS A BASE. PROJECTIONS WERE MADE TO 1955 USING THE PROCEDURES RECOMMENDED BY THE PRINCIPAL DEVELOPERS OF THE TECHNIQUES. THE 1955 PROJECTIONS WERE THEN COMPREHENSIVELY TESTED AGAINST THE 1955 WASHINGTON, D. C., O-D SURVEY TRAVEL DATA. EACH PROCEDURE IS EVALUATED FOR TRAVEL PATTERN SIMULATION ABILITY AS WELL AS THE FORECASTING STABILITY OF THE PARAMETERS. VARIOUS METHODS EVALUATE THE ACCURACY OF THE MODELS INCLUDING TRIP LENGTH FREQUENCY DUPLICATION, SCREENLINE CHECKS, SPECIFIC MOVEMENT CHECKS AND OVERALL STATISTICAL EVALUATIONS OF THE ESTIMATED MOVEMENTS. THESE TESTS ARE PERFORMED FOR EACH TECHNIQUE AND COMPARISONS OF THE RELATIVE ACCURACIES ARE ALSO MADE. APPROPRIATE CHANGES IN THE CALIBRATION PROCEDURES ARE RECOMMENDED. /AUTHOR/

• Record URL: http://onlinepubs.trb.org/Onlinepubs/hrr/1966/114/114-003.pdf
• Distribution, posting, or copying of this PDF is strictly prohibited without written permission of the Transportation Research Board of the National Academy of Sciences. Unless otherwise indicated, all materials in this PDF are copyrighted by the National Academy of Sciences. Copyright © National Academy of Sciences. All rights reserved.
• Heanue, Kevin E
• Pyers, Clyde E
• Cleveland, Donald E
• Vogt, Robert S
• Brokke, G E
• Howe, Robert T
• Publication Date: 1966
• Media Type: Print
• Features: Figures; References; Tables;
• Pagination: pp 20-50
• Monograph Title: Origin and destination: Methods and Evaluation
• Highway Research Record
• Issue Number: 114
• Publisher: Highway Research Board

Subject/Index Terms

• TRT Terms: Alternatives analysis ; Calibration ; Evaluation ; Forecasting ; Gravity models ; Length ; Methodology ; Origin and destination ; Simulation ; Stability (Mechanics) ; Statistical analysis ; Traffic assignment ; Travel ; Travel patterns
• Uncontrolled Terms: Comparative analysis ; Fratar method ; Trip
• Geographic Terms: Washington (District of Columbia)
• Old TRIS Terms: Competing opportunity ; Intervening opportunity ; Screen line
• Subject Areas: Data and Information Technology; Highways; Operations and Traffic Management;

Filing Info

• Accession Number: 00227481
• Record Type: Publication
• Files: TRIS, TRB
• Created Date: Aug 15 2004 2:36AM

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Using Fratar

Fratar distribution is the process of modifying a matrix of values based upon a set of production and attraction factors for each of the zones in the matrix. The process is a relatively simple iterative one:

In the first iteration, each row in the matrix is factored according to its production factor. At the end of the iteration, the row totals will match the target row values, but the column totals will most likely not match their targets.

In the second iteration each column in the modified matrix is factored according its attraction factor. At then end of the iteration, the column totals will match the target column values, but the row totals may not match their targets.

This process continues for some number of iterations; the row and column totals should converge towards the target totals. When the criteria for convergence is met, the process is complete.

A complete convergence (target row and column totals obtained for all zones) can be obtained only if the target grand control totals for rows and columns are the same. The program makes adjustments to guarantee that the target grand totals do match. It is possible that the user input values and specifications can preclude obtaining matching totals. In such cases, the program will fatally terminate.

This section discusses:

Specifying target values

Controlling target totals

Convergence — Iteration control

Multiple purposes

There are several typical ways in which the control totals can be specified: direct values, growth factors (explicit and implicit), or some combination of both. All specifications are via the SETPA control statements. An array of production values (Ps) and an array of attraction values (As) are maintained for each purpose. To simplify this description, the term "value" will be used to mean either direct values or growth factors. There must be a set of production values and attraction values for each matrix to be factored. They are input to the program via P, A, PGF, and AGF expressions. If direct values are to be input, the P and A expressions are used. If growth factors are to be input, the PGF and AGF expressions are used. Direct values and growth factors can be mixed for a purpose, but a complete understanding of the SETPA statement is necessary.

Each of the keyword expressions is computed for an array of values for all zones. P[1] = ZI.1.HBWP2000 would cause the program to simulate the expression:

Similarly, A[]=, PGF[]=, and AGF[]= expressions are computed for corresponding arrays. To provide the capability of mixing P and PGF for a purpose, the SETPA statement may include the basic INCLUDE and EXCLUDE filter specifications. If either, or both, of these filters are specified on a SETPA statement, they apply to all expressions on that statement. To specify P and PGF for the same purpose, separate SETPA statements are used; each would have its own zonal filter set. If the sets overlap, the latest SETPA values replace any prior values. If the final value for a P or A is 0, the program revises it to a growth factor of 1.0.

In this example, the values for zones 1-500 would be the direct values obtained directly from the ZI.1.HBWP2000 array, and the values for zones 501-550 would be the growth factors obtained from the ZI.2.EXTW array (divided by 2).

In most cases the values will be obtained from ZDATI (zonal data) files, or LOOKUP functions, but that is not an absolute requirement. Standard numerical expressions (J being the only viable variable that could be included) are used to compute the values. Sometimes, it is desirable to input specific values.

A special feature of these expressions is that if the result is less than zero, it is not stored. After all SETPA P,A,PGF, and AGF expressions are processed, the program performs a zonal (I) loop, obtaining the matrix values for each purpose. The matrices are obtained by solving the SETPA MW[]= expressions. Again, the INCLUDE and EXCLUDE filters are employed, but care must be exercised, if they are specified. The MW expressions are array notation, but applied for each I zone. Therefore the filters will apply to both the I and J zones.

After processing the input matrix, the target totals for any growth factor values can be fully determined (value = gf * input). Next, the program adjusts the values to insure that the P and A totals match for each purpose. There are several options for adjustment; they are specified by the use of the CONTROL keywords on the SETPA statement. There may be a CONTROL specification for each purpose, and if the CONTROL for any purpose is specified more than one time, the latest value prevails. If no CONTROL is specified it defaults to PA. The valid values for CONTROL are: P, A, PA, PL, AL, and PAL.

The meanings are:

P - The P totals control; all values in the A array will be factored so that the A totals will match the P totals.

A - The A totals control; all values in the P array will be factored so that the P totals will match the A totals.

PA - All values in both the P and A arrays will be factored so that their totals will match the average of the initial totals.

Sometimes only certain zones are to be modified, and the remainder of the zones are to be kept constant. The program keeps track of the zones that are eligible for modification by noting which zones have target values that differ from the input value by more than one. If the letter "L" is appended to any of the CONTROL values, it indicates that the modifications are Limited to only the zones that have change. Use of the this feature can, in some cases, lead to a situation where a matrix grand total can not be properly computed. If that is the case, the program will fatally terminate.

PL - The P totals control. The changed zones in the A array will be factored so that the final A total will match the P total.

AL - The A totals control. The changed zones in the P array will be factored so that the final P total will match the A total.

PAL - The values in P array for zones that have P changes, and the values in the A array for zones that have A changes will be factored in such a manner that the final P and A totals match the average of the initial P and A totals.

It is impossible to modify any cell, column, or row of the input matrix that has zero to begin with. If a target value is specified for a zone that initially had no total, a warning message is issued. Traditionally, some modelers would scale a matrix by a value (usually 10, or 100), and then fill in all empty cells with one. This is not necessarily a good, or bad, solution. But, because of the potential problems associated with this approach, zero accountability is not included in this program directly. If the scaling scheme is to be applied, a prior application of the Matrix program can be used to scale and fill in a matrix in any desired manner. It could also be achieved by setting the SETPA MW expression to: max(1,mi.n.n*10).

A concern is when to stop the iterating process; there are several ways to control it. The user can specify a maximum number of iterations, so that no matter how the convergence is progressing, the process will not exceed that number. After each iteration, the program computes an RMS error value based upon the integer differences between the computed and target row or column totals. After odd iterations, column total differences are checked, and after even iterations, row differences are checked. If the RMSE value is less than the MAXRMSE parameter value, the solution is achieved.

It is believed that this process will eventually reach convergence. But if, due to some unforeseen conditions, the RMSE value begins to oscillate, the program detects the oscillation, and terminates the process at the minimum RMSE. If there are multiple matrices being factored, they may reach optimum solutions at different times. If this happens, the "solved" matrices are held steady, and the others continue to be processed.

A small example of this process:

As dictated by row factoring, the row totals are correct. But, the column totals do not quite match the target. Another iteration is performed, and the results appear as:

The column totals are now on target, but the row totals are not quite on target.

This process goes on, back and forth, until either the RMSE drops to the MAXRMSE level, or the number of iterations reaches the MAXITERS value. In this example, the final solution is reached after 5 iterations (MAXRMSE=0.01 and MAXITERS=10).

All values are shown to the nearest integer and thus may not total exactly. Internally, the values are carried with more precision.

Mulitple Purposes

The number of purposes is determined by the highest P, A, PGF, AGF, or MW index found on any SETPA control statement. The program assumes that there will be purposes from one, monotonically, through that highest index. (FRATAR allows up to 20 trip purposes.) The distribution is performed prior to entering the main Matrix program I-loop. When the main I-loop is processed, MW[1] through MW[highest purpose] are initialized with the final matrices from the Fratar distribution. After the factoring process is complete, a standard Matrix program I-loop is performed.

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Modelling Trip Distribution Using the Gravity Model and Fratar's Method

2021, Mathematical Modelling of Engineering Problems

Trip Distribution is a difficult and significant model in the urban transportation planning process. This paper creates and assesses a satisfactory model of the trip distribution stage for the Nasiriyah city by using two models, Gravity and Fratar methods. A large sample was used for developing the model. The research methodology depends on discussing the theoretical fundamentals of the various methods for estimating the trips distribution and examining the suitability of these fundamentals for the conditions of the selected study area. Two different models had been used, namely; Frater and Gravity model. These models were calibrated using real data. The study tests the accuracy of the models, including overall statistical assessments of the predicted movements. Finally, the study recommended to use Fratar Method. These results had been confirmed to the literature that, if the area is a homogenous growth, the best model is the growth factor (Fratar&#39;s method) and if the area is e...

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In this study, travel demand in Tehran, the capital of Iran, is modeled in GIS using 3191 census blocks of the city. In the proposed model, the census blocks are considered as traffic analysis zones used for simulating travel origins and destinations in Tehran. Proper demographic data needed for modeling the travel demand are gathered and linked to the census blocks. After data preparation, trip production and attraction are calculated for each block using multiple regression methods. Different types of travel (i.e. for different purposes such as work, study, shopping etc.) cannot be explained by one model only. Therefore, several models are developed for each type of travel considering appropriate parameters. In the next stage, distribution of trips is modeled according to the calculated trip production and attraction, and the distance between the census blocks. The shortest path distance along the street network is considered as the distance between blocks. Suitable friction (impedance) functions and separate gravity-based models are developed for each travel type, in order to distribute estimated trips. Finally, the origin-destination matrix is achieved as the output of trip distribution. Using this matrix, daily trips between various locations are estimated. Having good knowledge about travel demand and travel spatial patterns can be very helpful for urban transportation planners and decision makers to effectively improve and develop urban transportation system.

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IJIRAE:: AM Publications, India

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Trip Generation and Trip Distribution

Fundamentals of trip generation.

• Every trip has two ends, and we need to know where both of them are. Because land use can be divided into two broad category (residential and non-residential) we have models that are household based and non-household based.
• Trip generation is thought of as a function of the social and economic attributes of households
• The first part is determining how many trips originate in a zone and the second part is how many trips are destined for a zone aka “productions” and attractions Production and attractions differ from origins and destinations. Trips are produced by households even when they are returning home (that is, when the household is a destination).

Actvities for Trip Generation

• Trips are categorized by purposes, the activity undertaken at a destination location.
• Major activities are home, work, shop, school, eating out, socializing, recreating, and serving passengers (picking up and dropping off).
• There are numerous other activities that people engage on a less than daily or even weekly basis, such as going to the doctor, banking, etc. Often less frequent categories are dropped and lumped into the catchall “Other”.

Specifying models

The number of trips originating from or destined to a purpose in a zone are described by trip rates (a cross-classification by age or demographics is often used) or equations.

$$T_h = f(housing \ units, household \ size, age, income, accessibility, vehicle \ ownership)$$

$$T_w = f(jobs(area \ of \space \ by \ type, occupancy \ rate) )$$

Shopping Trips

$$T_s = (number \ of \ retail workers, type \ of \ retail, area, location, competition)$$

Estimating Models

To estimate trip generation at the home end, a cross-classification model can be used. This is basically constructing a table where the rows and columns have different attributes, and each cell in the table shows a predicted number of trips, this is generally derived directly from data.

In the example cross-classification model: The dependent variable is trips per person. The independent variables are dwelling type (single or multiple family), household size (1, 2, 3, 4, or 5+ persons per household), and person age.

Non-home-end

• The trip generation rates for both “work” and “other” trip ends can be developed using Ordinary Least Squares (OLS) regression relating trips to employment by type and population characteristics.
• The variables used in estimating trip rates for the work-end are Employment in Offices ($E_{off}$) , Retail($E_{ret}$) , and Other($E_{oth}$)

A typical form of the equation can be expressed as:

$$T_{D,i} = a_1 E_{off,i} + a_2 E_{oth,i} + a_3 E_{ret,i}$$

$T_{D,i}$ - Person trips attracted per worker in Zone k $E_{off,i}$ - office employment in the ith zone $E_{oth,i}$ - other employment in the ith zone $E_{ret,i}$ - retail employment in the ith zone $a_1,a_2,a_3$ - model coefficients

Normalization

• For each trip purpose (e.g. home to work trips), the number of trips originating at home must equal the number of trips destined for work. Two distinct models may give two results.
• There are several techniques for dealing with this problem. One can either assume one model is correct and adjust the other, or split the difference.
• It is necessary to ensure that the total number of trip origins equals the total number of trip destinations, since each trip interchange by definition must have two trip ends.

The rates developed for the home end are assumed to be most accurate,

The basic equation for normalization:

Sample Problems

Planners have estimated the following models for the AM Peak Hour

$$T_{O,i}=1.5∗H_{i}$$ $$T_{D,j} =(1.5∗E_{off,j})+(1∗E_{oth,j})+(0.5∗E_{ret,j})$$ Where:

$T_{O,i}$ = Person Trips Originating in Zone i

$T_{D,j}$ = Person Trips Destined for Zone j

$H_{i}$ = Number of Households in Zone i

You are also given the following data

A. What are the number of person trips originating in and destined for each city?

B. Normalize the number of person trips so that the number of person trip origins = the number of person trip destinations. Assume the model for person trip origins is more accurate.

Solution to Trip Generation Problem Part A

Solution to trip generation problem part b.

Modelers have estimated that the number of trips leaving Rivertown ( TO ) is a function of the number of households (H) and the number of jobs (J), and the number of trips arriving in Marcytown ( TD ) is also a function of the number of households and number of jobs.

$$T_{O}=1{H}+0.1{J};R^2=0.9$$ $$T_{D}=0.1{H}+1{J};R^2=0.5$$ Assuming all trips originate in Rivertown and are destined for Marcytown and:

Rivertown: 30000 H, 5000 J

Marcytown: 6000 H, 29000 J

Determine the number of trips originating in Rivertown and the number destined for Marcytown according to the model.

Which number of origins or destinations is more accurate? Why?

$$T_{Rivertown} =T_{O} ; T_{O}= 1(30000) + 0.1(5000) = 30500 trips$$

$$T_{MarcyTown}=T_D ; T_{D}= 0.1(6000) + 1(29000) = 29600 trips$$

Origins($T_{Rivertown}$) because of the goodness of fit measure of the Statistical model ($R^2=0.9$).

Trip Distribution

Everything is related to everything else, but near things are more related than distant things. - Waldo Tobler’s ‘First Law of Geography’

Destination Choice (or trip distribution or zonal interchange analysis), is the second component (after Trip Generation, but before Mode Choice and Route Choice) in the traditional four-step transportation forecasting model. This step matches tripmakers’ origins and destinations to develop a “trip table”, a matrix that displays the number of trips going from each origin to each destination. Historically, trip distribution has been the least developed component of the transportation planning model.

General Form

$$T_{ij} = T_{i} * P(T_{j})$$

Where: $T_{ij}$ = Trips from origin i to destination j.

$T_{i}$ = total trips originating at zone i

$P(T_{j})$ = probability measure that trips will be attracted to zone j

• Singly Constrained $$\sum_{i} T_{ij}= D_{i} \ or \sum_{j} T_{ij}= O_{i}$$
• Doubly Constrained $$\sum_{i} T_{ij}= D_{i} \ and \sum_{j} T_{ij}= O_{i}$$

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, which follow the same structure.

Friction Factor Model

• Friction factors express the effect that travel that travel time has on the number of trips traveling between two zones.

Exponential

$$f(C_{ij}) = e ^ {-c(c_{ij})} \ c >0$$

Inverse Power

$$f(C_{ij}) = c_{ij} ^ {-b} \ b >0$$

$$f(C_{ij}) = a \ X \ c_{ij} ^ {-b} \ X\ e ^ {-c(c_{ij})} \ a >0, \ b >0, \ c >0$$

Friction factors were developed using a gamma function to estimate the friction factors and application of the trip distribution model to identify the best-fit for the average trip length and trip length frequency distributions.

Fratar Model

The simplest trip distribution models (Fratar or Growth models) simply extrapolate a base year trip table to the future based on growth,

$$T_{ij,y+1} = g * T_{ij,y}$$

• $T_{ij,y}$ - Trips from i to j in year y
• g - growth factor

Fratar Model takes no account of changing spatial accessibility due to increased supply or changes in travel patterns and congestion.

Gravity Model

The Gravity Model assumes that the number of trips between two zones is

• directly proportional to the trips produced and attracted to both zones, and
• inversely proportional to the travel time between the zones. The distance decay factor of ${ distance^{-1}}$ 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 is much like Newton’s theory of gravity. The gravity model assumes that the trips produced at an origin and attracted to a destination are directly proportional to the total trip productions at the origin and the total attractions at the destination. 

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 auto ownership.

Mathematically, the gravity model often takes the form:

$$T_{ij}=r_is_jT_{O,i}T_{D,j}F(C_{ij})$$

$$\displaystyle \sum{j}T_{ij}=T_{O,i}, \displaystyle \sum{i} T_{ij} = T_{D,j}$$

$$r_i = (\displaystyle \sum{j} s_jT_{D,j}f(C_{ij}))^{-1}$$

$$s_j=(\displaystyle \sum{i} r_iT_{O,i}f(C_{ij}))^{-1}$$

$T_{ij}$ = Trips between origin i and destination j

$T_{O,i}$ = Trips originating at i

$T_{D,j}$ = Trips destined for j

$C_{ij}$ = travel cost between i and j

$r_i,s_j$ = balancing factors solved iteratively.

$f$ = impedance or distance decay factor

It is doubly constrained so that Trips from i to j equal number of origins and destinations.

Balancing Matrix

Balancing a matrix can be done using what is called the Furness Method, summarized and generalized below.

• Assess Data, you have $T_{O,i} , T_{D,j} , C_{ij}$
• Compute f(Cij) , e.g.

$$f(C_{ij})=C_{ij}^{-2}$$

$$f(C_{ij})=e^{\beta C_{ij}}$$

• Iterate to Balance Matrix

(a) Multiply Trips from Zone i ($T_i$) by Trips to Zone j($T_j$) by Impedance in Cell ij($f(Cij)$ for all ij

(b) Sum Row Totals $T′ {O,i}$ , Sum Column Totals $T′ {D,j}$

(c) Multiply Rows by $N_{O,i} =T_{O,i}/T′_{O,i}$

(d) Sum Row Totals $T′ {O,i}$ , Sum Column Totals $T′ {D,j}$

(e) Compare $T_{O,i}$ and $T′ {O,i}$ , $T {D,j}$, $T′_{D,j}$ if within tolerance stop, Otherwise go to (f)

(f) Multiply Columns by $N_{D,j}=T_{D,j}/T′_{D,j}$

(g) Sum Row Totals $T′ {O,i}$ , Sum Column Totals $T′ {D,j}$

(h) Compare $T_{O,i}$ and $T′ {O,i}$ , $T {D,j}$ and $T′_{D,j}$ if within tolerance stop, Otherwise go to (b)

You are given the travel times between zones, compute the impedance matrix $f(C_{ij})$ , assuming $f(C_{ij})=C_{ij}^{-2}$

Compute impedances ( $f(C_{ij})$ )

You are given the travel times between zones, trips originating at each zone (zone1 =15, zone 2=15) trips destined for each zone (zone 1=10, zone 2 = 20) and asked to use the classic gravity model $f(C_{ij})=C_{ij}^{-2}$

Travel Time OD Matrix

(a) Compute impedances ( $f(C_{ij})$ )

Impedance Matrix

(b) Find the trip table

Balancing Iteration 0

Balancing iteration 1, balancing iteration 2, balancing iteration 3, balancing iteration 4, balancing iteration 16.

So while the matrix is not strictly balanced, it is very close, well within a 1% threshold, after 16 iterations. The threshold refers to the proximity of the normalizing factor to 1.0.

Comparison of Trip Distribution Models

Last updated on Apr 2, 2022

Highway Traffic Analysis and Design pp 38–55 Cite as

Trip distribution

• R. J. Salter 2  

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Trip distribution is another of the major aspects of the transportation simulation process and although generation, distribution and assignment are often discussed separately, it is important to realise that if human behaviour is to be effectively simulated then these three processes must be conceived as an interrelated whole.

• Gravity Model
• Factor Method
• Traffic Analysis
• Transportation Study

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T. J. Fratar, Vehicular trip distributions by successive approximations, Traff. Q. , 8 (1954), 53–64

Google Scholar  

K. P. Furness, Time function iteration, Traff. Engng Control , 7 (1965), 458–60

Ministry of Transport, Land Use/Transport Studies for Smaller Towns, Memorandum to Divisional Road Engineers and Principal Regional Planners , London (1965)

J. C. Tanner, Factors affecting the amount of travel, DSIR Road Research Technical Paper No. 51, HMSO, London (1961)

S. A. Stouffer, Intervening opportunities—a theory relating mobility and distance, Am. Soc. Rev. , 5 (1940), 347–56

Article   Google Scholar  

A. R. Tomazinia and G. Wickstrom, A new method of trip distribution in an urban area, Highw. Res. Bd Bull. , 374 (1962), 254–7

Chicago Area Transportation Study, Final Report , 11 (1960)

H. C. Lawson and J. A. Dearinger, A comparison of four work trip distribution models, Proc. Am. Soc. Civ. Engrs , 93 (November 1967), 1–25

F. R. Ruiter, Discussion on R. W. Whitaker and K. E. West. The intervening opportunities model: a theoretical consideration, Highw. Res. Rec . 250, Highway Research Board (1968)

K. E. Heanue and C. E. Pyers, A comparative evaluation of trip distribution procedures, Highw. Res. Rec . 114, Highway Research Board (1966)

W. R. Blunden, The Land-Use/Transport System , Pergamon, Oxford (1971)

Greater London Council, Movement in London , County Hall, London (1969)

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Salter, R.J. (1989). Trip distribution. In: Highway Traffic Analysis and Design. Palgrave, London. https://doi.org/10.1007/978-1-349-20014-6_7

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