Trip Distribution 1 - Growth-Factor Methods and Gravity Models

1. Introduction

Trip distribution: Allocation of total number of trips, produced and attracted by each zone (trip generation), to particular destinations, i.e. their distribution over space.

Trip production and attraction provide an idea of the level of trip making in a study area,

• Trip production and attraction are seldom enough for modelling and decision making (i.e., transportation planning).

• What is needed is a better idea/model/approach of trip making pattern, from where to where do trips take place and travel modes chosen.

Trip pattern can be formulated in at least two different ways:

• Trip matrix or trip table which stores trips made from an Origin to a Destination during a particular time period, and it is also called an Origin-Destination (O-D) trip matrix.

  • The O-D trip matrix may be further disaggregated by person type and purpose or perhaps the activity undertaken at each end of the trip.

• Production-Attraction (P-A) matrix or table which considers the factors that generate and attract trips, with Home generally being treated as the producing end, and Work, Shop etc. as the attracting end.

  • By necessity, a P-A matrix will cover a longer time span, (usually a day) than an O-D matrix.

1.1 Definitions and Notation

A trip distribution model aims to estimate the number of trips (\(T_{ij}, i,j=1,2,\cdots,z\)) from origin zone \(i\) to destination zone \(j\) based on the outputs of a trip generation model (\(O_{i}, D_{j}\)) subject to the necessary balance constraints/equations.

Customary to represent trip pattern in a study area by means of a trip matrix.

• A two-dimensional array of cells where rows and columns represent each of the \(z\) zones in the study area (including external zones).

  • \(i,j = 1, 2, \cdots, z\) are the indices of traffic zone, where \(i\) represents the origin zone and \(j\) represents the destination zone

  • \(T_{ij}\) is number of trips from origin \(i\) to destination \(j\)

  • \([T_{ij}]_{z \times z}\) or \(\mathbf{T}\) is the total array

  • Lower case letters, \(t_{ij}\), \(o_i\) and \(d_j\) indicate observations from a sample or from earlier study.

  • Capital case letters represent our targets, or the values we are trying to model for the \(z\) corresponding modelling period.

• The matrix can be further disaggregated, for example, by person type (\(n\)) and/or by travel mode (\(k\)).

  • trips from zones \(i\) to \(j\) by travel mode \(k\) and person type \(n\).

  • the total number of trips originating at zone \(i\) by travel model \(k\) and person type \(n\).

  • \(T_{ij}^n = \sum_k T_{ij}^{nk}\)

  • \(T = \sum_{ij} T_{ij}\)

  • \(t = \sum_{ij} t_{ij}\)

In cases where it is of interest to distinguish the proportion of trips using a particular travel mode and travel cost between two points:

• \(p_{ij}^{k}\) is the proportion of trips from \(i\) to \(j\) by mode \(k\);

• \(c_{ij}^{k}\) is the cost of travelling between \(i\) to \(j\) by mode \(k\).

1.2 Two Balance Equations

The sum of the trips in a row of O-D trip matrix should equal the total number of trips emanating from that zone and the sum of the trips in a column should correspond to the number of trips attracted to that zone:

\[\begin{cases} \displaystyle O_i = \sum_{j=1}^z T_{ij} & i = 1, 2, \cdots, z \qquad (1) \\ \displaystyle D_j = \sum_{i=1}^z T_{ij} & j = 1, 2, \cdots, z \qquad (2) \end{cases} \quad \Rightarrow \quad T = \sum_{i=1}^z O_i = \sum_{j=1}^z D_j = \sum_{j=1}^z \sum_{i=1}^z T_{ij} \]

• Doubly constrained: If reliable information is available to estimate both \(O_i\) and \(D_j\), trip distribution models must satisfy both balance equations (1) and (2).

• Singly constrained: If there is only information about one of these two balance equations (1) and (2). For example we have all \(O_i\).

  • A model can be origin or production constrained if the \(O_i\)'s are available, or destination or attraction constrained if the \(D_j\)'s are at hand.

1.3 Generalized Travel Cost

Cost element may be considered in terms of distance, time or money units.

• Convenient to use a measure combining all the main attributes related to the disutility/impedance of a trip, i.e. generalized travel cost.

• Typically a linear function of the attributes of a trip weighted by coefficients which attempt to formulate their relative importance as perceived by travelers.

\[C_{ij} = a_1 t^v_{ij} + a_2 t^w_{ij} + a_3 t^t_{ij} + a_4 t^n_{ij} + a_5 F_{ij} + a_6 \phi_j + \delta \]

• Where: \(t^v_{ij}\) is the in-vehicle travel time between \(i\) and \(j\);

  \(t^w_{ij}\) is the walking time to and from stops (stations) or from parking area/lot;

  \(t^t_{ij}\) is the waiting time at stops (or time spent searching for a parking space);

  \(t^n_{ij}\) is the interchange (transfer) time, if any;

  \(F_{ij}\) is a monetary charge: the fare charged to travel between \(i\) and \(j\) or the cost of using the car for that journey, including any tolls or congestion charges (note that car operating costs are often not well perceived and that electronic means of payment tend to blur somehow the link between use and payment);

  \(\phi_j\) is a terminal (typically parking) cost associated with the journey from \(i\) to \(j\);

  \(\delta\) is a modal penalty, a parameter representing all other attributes not included in the generalised measure so far, e.g. safety, comfort and convenience;

  \(a_1, a_2, \cdots, a_6\) are weights attached to each element of cost; they have dimensions appropriate for conversion of all attributes to common units, e.g. money or time.

• If generalized travel cost is measured in money units (i.e., \(a_5 = 1\)) then \(a_1\) = value of time (more precisely value of in-vehicle time) as its units are money/time.

• In that case, \(a_2\) and \(a_3\) would be the values of walking and waiting time respectively (In many practical studies they are taken to be two or three times the expected value of \(a_1\).)

2. Growth-Factor Methods

Consider a situation where we have a base year O-D trip matrix \([t_{ij}]_{z \times z}\), perhaps obtained from a previous study or estimated from recent survey data. We aim to forecast a horizon-year O-D trip matrix \([T_{ij}]_{z \times z}\) that should fulfil one or two balance equations.

2.1 Uniform Growth Factor Models

If the only information available is about a general growth rate \(\tau\) for the entire study area, we can assume that it will apply to each cell in the O-D trip matrix:

\[T_{ij} = \tau \cdot t_{ij}, \quad \forall \ i,j = 1,2,\cdots,z \]

2.2 Singly Constrained Growth-Factor Methods

Consider the situation where information is available on the expected growth in trips originating in each zone. In this case we can apply the origin-specific growth factors (\(\tau_i\)) to corresponding rows in O-D trip matrix, namely:

\[\begin{array}{ll} T_{ij} = \tau_i \cdot t_{ij}, &\forall i,j=1,2,\cdots,Z \\ \sum_{j} T_{ij} = O_i, & \forall i=1,2,\cdots,Z \end{array} \]

where origin-specific growth factors \(\tau_i\) :

\[\tau_i = \dfrac{O_i}{o_i} = \dfrac{O_i}{\sum_{j}t_{ij}}, \qquad \forall \ i = 1, 2, \cdots, z \]

If information is available for trips attracted to each zone, the destination-specific growth factors (\(\Gamma_j\)) applied to columns.

\[\begin{array}{ll} T_{ij} = \Gamma_j \cdot t_{ij}, &\forall i,j=1,2,\cdots,Z \\ \sum_{i} T_{ij} = D_j, &\forall j=1,2,\cdots,Z \end{array} \]

where destination-specific growth factors \(\Gamma_j\) :

\[\Gamma_j = \dfrac{D_j}{d_j} = \dfrac{D_j}{\sum_{i}t_{ij}}, \qquad \forall \ j = 1, 2, \cdots, z \]

2.3 Doubly-Constrained Growth Factor Models

• Information is available on the future number of trips originating and terminating in each zone.

  • This implies different growth rates for trips in and out of each zone and consequently having two sets of growth factors for each zone, say \(\tau_i\) and \(\Gamma_j\).

  • The application of an "average" growth factor, say \(T_{ij} = 0.5 (\tau_i+\Gamma_j)\), is a poor compromise as none of the two balance equations would be satisfied.

Doubly-constrained growth factor model

Given \(O_i \ (i = 1,2,\cdots,z)\) and \(D_j\ (j = 1,2,\cdots,z)\), find balancing factors \(A_i \ (i = 1,2,\cdots,z)\) and \(B_j \ (j = 1,2,\cdots,z)\) that fulfill the following three conditions:

\[\begin{array}{ll} T_{ij} = \tau_i \cdot \Gamma_j \cdot A_i \cdot B_j \cdot t_{ij} , & \forall \ i,j=1,2,\cdots,z \\ \sum_{j} T_{ij} = O_i, & \forall i=1,2,\cdots,z \\ \sum_{i} T_{ij} = D_i, & \forall j=1,2,\cdots,z \\ \end{array} \]

The doubly-constrained growth factor model can be equivalently formulated below by setting \(a_i = \tau_{i} \cdot A_i\) and \(b_j = \Gamma_j \cdot B_j\).

\[\begin{array}{ll} T_{ij} = a_i \cdot b_j \cdot t_{ij}, & \forall \ i,j=1,2,\cdots,z \\ \sum_{j} T_{ij} = O_i, & \forall i=1,2,\cdots,z \\ \sum_{i} T_{ij} = D_i, & \forall j=1,2,\cdots,z \\ \end{array} \]

Bi-proportional problem: Find \(a_i\) and \(b_j\) fulfilling the above three conditions.

2.4 Advantages and Limitations

Advantage:

Growth-factor methods are simple to understand and make direct use of observed trip matrices and forecasts of trip-end growth. They preserve the observations as much as is consistent with the information available on growth rates.

Limitations:

• Advantage is also their limitation as they are probably only reasonable for short-term planning horizons or when changes in travel costs are not to be expected.

• Require a base-year O-D trip matrix \([t_{ij}]_{z \times z}\) (this is an expensive data item in practice).

• Heavily dependent on the accuracy of the base-year trip matrix.

• Do not account for changes in travel cost due to transport network improvement and new transport policies.

• They are of limited use in the analysis of policy options involving new travel modes, new links, pricing policies and new zones.

3. Gravity Models

3.1 Basic Gravity Model

• Start from assumptions about group trip making behaviour and the way this is influenced by external factors such as total trip ends and distance travelled.

• A gravity model originally generated from an analogy with Newton's gravitational law.

• A gravity model estimates trips for each cell in the matrix without directly using the observed trip pattern (i.e., base-year O-D matrix \([t_{ij}]_{z \times z}\)).

– It is sometimes called a synthetic model as opposed to growth factor model.

\[T_{ij} = \frac{\alpha P_i P_j}{d^2} \]

• where: \(P_i\) and \(P_j\) are the populations of zones \(i\) and \(j\),

  \(d\) is the travel impedance such as distance and generalized travel cost between zones \(i\) and \(j\),

  and \(\alpha\) is a proportional factor with unit \(\text{trips} \cdot \text{distance}^2 / \text{population}^2\)

3.2 Basic Gravity Model

The conceptual gravity model can be further generalized by assuming that the effect of distance or 'separation' could be modelled better by a decreasing function, to be specified, of the distance or travel cost/time between the zones.

\[T_{ij} = \alpha \cdot O_i \cdot D_j \cdot f(c_{ij}) \]

• where \(f(c_{ij})\) is a generalised function of the travel costs with one or more parameters for calibration. It is known as deterrence function because it describes the disincentive to travel as distance (time or cost) increases.

Popular versions for the deterrence function \(f(c_{ij})\) are:

\[\begin{array}{ll} f(c_{ij}) = \exp \, (-\beta \, c_{ij}) & \text{exponential function} \\ f(c_{ij}) = c_{ij}^{-n} & \text{power function} \\ f(c_{ij}) = c_{ij}^n \, \exp \, (-\beta \, c_{ij}) & \text{combined function} \end{array} \]

where \(n\) and \(\beta\) are parameters require to be calibrated.

A more general version of deterrence function (step-wise function)

\[f(c_{ij}) = \sum_{k=1}^{K} F_{k} \delta_{ijk} \]

• where \(F_{k}\) is the mean value for cost bin \(k \ (k=1,2,\cdots,K)\) (called separation factor),

  and \(\delta_{ijk}\) is equal to 1 if travel (cost or time) from zones \(i\) to \(j\) falls in the range \(k\); otherwise, equal to 0.

3.3 Singly or Doubly Constrained Gravity Models

Replace \(\alpha\) in the basic gravity model to the below form similar to the growth factor model:

\[T_{ij} = A_i \cdot O_i \cdot B_j \cdot D_j \cdot f(c_{ij}) \]

Subsume \(O_i\) and \(D_j\) into these factors \(a_i\) and \(b_j\) and rewrite the model as:

\[T_{ij} = a_i \cdot b_j \cdot f(c_{ij}) \]

where \(a_i=A_i \cdot O_i\) and \(b_j = B_j \cdot D_j\)

Singly-Constrained gravity models

• Either origin or destination constrained is satisfied.

• It can be produced by making one set of balancing factors \(A_i\) or \(B_j\) equal to one.

• For an origin-constrained gravity model ( i.e., \(B_j = 1.0, \forall \ j\) ), and adopting:

  \[\begin{array}{ll} T_{ij} = A_i \cdot O_i \cdot D_j \cdot f(c_{ij}), & \forall i,j = 1,2,\cdots,z \\ \sum_{j} T_{ij} = O_i, & \forall i=1,2,\cdots,z \end{array} \]

  we can derive \(A_i\) is:

  \[A_i = \frac{1}{\sum_j D_j f(c_{ij})} \\ \]

• For a destination-constrained gravity model ( i.e., \(A_i = 1.0, \forall \ i\) )

  \[\begin{array}{ll} T_{ij} = O_i \cdot B_j \cdot D_j \cdot f(c_{ij}), & \forall i,j = 1,2,\cdots,z \\ \sum_{i} T_{ij} = D_i, & \forall i=1,2,\cdots,z \end{array} \]

  we can derive \(B_j\) is:

  \[B_j = \frac{1}{\sum_i O_i f(c_{ij})} \\ \]

Doubly-Constrained Gravity Model

\[\begin{array}{ll} T_{ij} = A_i \cdot O_i \cdot B_j \cdot D_j \cdot f(c_{ij}), & \forall \ i,j = 1, 2, \cdots z \\ \sum_j T_{ij} = O_i, & \forall \ i = 1, 2, \cdots z \\ \sum_i T_{ij} = D_j, & \forall \ j = 1, 2, \cdots z \end{array} \]

we can derive \(A_i\) and \(B_j\) are:

\[\begin{array}{ll} A_i = \dfrac{1}{\sum_j B_j D_j f(c_{ij})}, & \forall \ i = 1, 2, \cdots z \\ B_j = \dfrac{1}{\sum_i A_i O_i f(c_{ij})}, & \forall \ j = 1, 2, \cdots z \\ \end{array} \]

By using the factors \(a_i=A_i \cdot O_i\) and \(b_j = B_j \cdot D_j\), the double-constrained gravity model can be reformulated as below:

\[\begin{array}{ll} T_{ij} = a_i \cdot b_j \cdot f(c_{ij}), & \forall \ i,j = 1, 2, \cdots z \\ \sum_j T_{ij} = O_i, & \forall \ i = 1, 2, \cdots z \\ \sum_i T_{ij} = D_j, & \forall \ j = 1, 2, \cdots z \end{array} \]

where \(\big[ f(c_{ij}) \big]_{z \times z}\) likes the base-year O-D matrix \(\big[ t_{ij} \big]_{z \times z}\).

4. Calibration of Gravity Models

Before using a gravity distribution model it is necessary to calibrate it.

• This makes sure that its parameters are such that the model comes as close as possible to reproducing the base-year trip pattern.

• In the case of calibration one is conditioned by the functional form and the number of parameters of the chosen model, i.e. the deterrence function and all coefficients.

The validation task is different.

• We want to make sure the model is appropriate for the decisions likely to be tested with it.

• It may be that the gravity model is not a sufficiently good representation of reality for the purpose of examining a particular set of decisions.

• Validation tasks depend on the nature of the policies and projects to be assessed.

• A general strategy for validating a model would then be to check whether it can reproduce a known state of the system with sufficient accuracy.

4.1 Trip length distribution (TLD)

Trip length (or cost) distribution (TLD)

• Suppose all the trips with sample origin-destination (e.g., from zone \(i\) to zone \(j\)) share the same trip length (or cost)

• Observed/estimated trip length (or cost) distribution (TLD)

• Observed/estimated TLD can be used to calibrate the parameters of deference function.

• TLD can be also estimated from a survey.

TLD can be modelled by step-wise deterrence function:

\[f(c_{ij}) = \sum_{k=1}^{K} F_{k} \delta_{ijk} \]

• where \(c_{ij}\) the trip length (or cost) from zone \(i\) to zone \(j\)

  \(F_{k}\) is the mean value for cost bin \(k \ (k=1,2,\cdots,K)\) (called separation factor),

  \(S_k\) is frequency number of trips for each bin, that is, trip length distribution,

  and \(\delta_{ijk}\) is equal to 1 if travel (cost or time) from zones \(i\) to \(j\) falls in the range \(k\); otherwise, equal to 0.

Then we have the following model:

\[\begin{array}{ll} T_{i j k}= a_{i} \cdot b_{j} \cdot F_{k} \cdot \delta_{i j k} & i=1,\cdots, z \ ; j=1, \cdots, z \ ; k = 1, \cdots K \\ \sum_{j} \sum_{k} T_{i j k}=O_{i}, & i=1, \cdots, z \\ \sum_{i} \sum_{k} T_{i j k}=D_{j}, & j=1, \cdots, z \\ \sum_{k} \delta_{i j k}=1, & i=1, \cdots, z \ ; j=1, \cdots, z \\ \sum_{i} \sum_{j} T_{i j k} = S_{k}, & k=1, \cdots, K \end{array} \]

Each O-D pair should fall in only one cost bin, namely, \(\sum_{k=1}^{K}\left(F_{k} \cdot \delta_{i j k}\right) \in\left\{F_{1}, F_{2}, \cdots, F_{k}\right\}\).

Hence, we have:

\[T_{i j} = \sum_{k=1}^{K} T_{i j k} = a_{i} \cdot b_{j} \cdot \sum_{k=1}^{K}\left(F_{k} \cdot \delta_{i j k}\right) \]

5. Partial Matrix Techniques

\[\begin{array}{c|ccc|c} & 1 & 2 & 3 & o_i \\ \hline 1 & & b & c & \\ 2 & d & e & f & \\ 3 & g & & i & \\ \hline d_j \\ \end{array} \quad \overset{\text{adjusted to}}{\longrightarrow} \quad \begin{array}{c|ccc|c} &1&2&3& O_i \\ \hline 1&&&& P \\ 2&&&& Q \\ 3&&&& R \\ \hline D_j & S & T & U \\ \end{array} \quad \overset{\text{result}}{\longrightarrow} \quad \begin{array}{c|ccc|c} &1&2&3& O_i \\ \hline 1&A&B&C \\ 2&D& E & F \\ 3&G&F&I \\ \hline D_j \\ \end{array} \]

Approach

Suppose the distribution model is \(T_{ij} = a_i \cdot b_j\)

• Step 1: Use the bi-proportional algorithm based on the current data to estimate factors \(a_i\) and \(b_j\)

  \[\begin{array}{ccc|c} & b & c & P \\ d & e & f & Q \\ g & & i & R \\ \hline S & T & U \\ \end{array} \quad \rightarrow \quad \begin{array}{ccc|c} &&& a_1 \\ &&& a_2 \\ &&& a_3 \\ \hline b_1 & b_2 & b_3 \\ \end{array} \]

• Step 2: The missing data (i.e., \(A\) and \(H\)) are estimated by \(T_{ij} a_i \cdot b_j\).

  And then re-estimate factors \(a_i\) and \(b_j\) using the adjusted trip generation and attraction (i.e., \(P-A\), \(R-H\), \(S-A\) and \(T-H\) where \(A = a_{1} \cdot b_{1}\) and \(H = a_{3} \cdot b_{2}\))

\[\begin{array}{ccc|l} & b & c & P - A \\ d & e & f & Q \\ g & & i & R -H \\ \hline S-A & T-H & U \\ \end{array} \quad \rightarrow \quad \begin{array}{ccc|c} &&& \hat{a}_1 \\ &&& \hat{a}_2 \\ &&& \hat{a}_3 \\ \hline \hat{b}_1 & \hat{b}_2 & \hat{b}_3 \\ \end{array} \quad \rightarrow \quad \begin{array}{ccc} A & B & C \\ D & E & F \\ G & F & I \\ \\ \end{array} \]

• Step 3: The missing data (i.e., \(A\) and \(H\)) are re-estimated by \(T_{ij} = f(\hat{a}_i, \hat{b}_j)\), where \(\hat{a}_i\) and \(\hat{b}_j\) are the re-estimated values of \(a_i\) and \(b_j\) in Step 2.

6. Practical Considerations

6.1 Treatment of external zones

It may be quite reasonable to postulate the suitability of a synthetic trip distribution model in a study area, in particular for internal-to-internal trips.

• However, a significant proportion of the trips may have at least one end outside the area. The suitability of a model which depends on trip distance or cost, a variable essentially undefined for external trips, is thus debatable.

• Common practice in such cases is to take these trips outside the synthetic modelling process:

  • Roadside interviews are undertaken on cordon points at the entrance/ exit to the study area.

  • The resulting matrix of external-external (E − E) and external-internal (E − I) trips is then updated and forecasted using growth factor models, in particular those of Furness (i.e., bi-proportional algorithms).

6.2 Intra-zonal Trips

Given the limitations of any zoning system, the cost values given to centroid connectors are a very crude but necessary approximation to those experienced in reality. The idea of an intra-zonal trip cost is then poorly represented by these centroid connector costs.

It is actually preferable to remove intra-zonal trips from the synthetic modelling process and to forecast those using even simpler approaches. This typically assumes that intra-zonal trips are a fixed proportion of the trip ends calculated by the trip generation models.

6.3 Journey Purposes

Different models are normally used for different trip purposes and/or person types.

• Typically, the journey to work will be modelled using a doubly constrained gravity model

• Almost all other purposes will be modelled using singly constrained models. This is because it is often difficult to estimate trip attractions accurately for shopping, recreational and social trips and therefore proxies for trip attractiveness are used: retail floor space, recreational areas, population.

Some trip purposes may be more sensitive to cost and therefore deserve the use of different values for the deterrence function.

6.4 K Factors

Most individual decisions on residential location and/or choice of employment incorporate many other factors; therefore, the gravity model could only model destination choice at an aggregate level if the importance of these other factors were much reduced on aggregation. However, there are always aggregate effects that do not conform to a simple gravity model.

Introduction of an additional set of parameters \(K_{ij}\) to the gravity model to improve model calibration:

\[T_{ij} = K_{ij} \cdot A_i \cdot O_i \cdot B_{j} \cdot D_{j} \cdot \exp(-\beta \, c_{ij}) \]

The best advice that can be given in respect of K factors is: try to avoid them. If a study area has a small number of zone pairs (say, less than 5% of the total) with a special trip making association which is likely to remain in the future, then the use of a few \(K\) factors might be justified, sparingly and cautiously. But the use of a model with a full set of \(K\) factors cannot be justified.