lecture 3 - fratar algorithm

Trip Distribution

Trip Distribution

Two known sets of trip ends are connected together to form a tip matrix between origins and destinations.

In the literature there are two basic methods by which this connection can be achieved.

1.  Growth factor method Constant factor method Average factor method Fratar method Furness method

2.  Synthetic Methods

Gravity model

Opportunity model

Growth Factor Methods

Assumptions:

•  The trip making pattern will remain the same in the future as it is in the base year

•  The volume will increase according to the growth in the generating and attracting zones

Uniform Growth (constant) factor method

Assumes:

•  That growth in all zones will increase in an uniformed manner

•  That the existing traffic pattern will be the same for the future but that the volume will change

•  That the growth which is expected to take place in the survey area will have an equal effect on all trips made in the area

The equation used in this method is as follows:

Where: t’ij is the future number of trips from zone i to zone j

tij is the present number of trips from zone i to zone j

E is the constant factor derived by dividing the number of trip ends in the future year by the base year

Disadvantages of this method

1.  Tends to overestimate the trips between densely populated zones which probably have little further development potential

2.  Tends to underestimate the future trips between underdeveloped zones which could be extensively populated in the future

'ij ijt t E=

Example of Uniform growth factor method

Given: 4x4 matrix = demand in the base year

Uniform growth rate = 20%

Method:

Each cell in the matrix will be multiplied by 1.2 (to get a 20% increase).

Singly constrained growth factor methods

Used where information is available on the expected growth in trips originating from zones.

Consider the following trip matrix

Method

Multiply each cell in Row 1 by 400/355




Average factor method

This method tries to take into account the varying rates of growth in trip making in different zones.

The equation is as follows:

Where and

Pi = future trip production of zone i

pi = present trip production of zone i

Aj = future trip attractions of zone j

aj = present trip attractions of zone j

Note:

•  After the first iteration, trip productions will not match trip attractions

•  Futher iterations will be necessary to achieve accuracy of 1-5%

' ( ) / 2ij ij i jt t E E= +

/i i iE P p= /j j jE A a=

Disadvantages of this method

1.  This method suffers the same disadvantages as the constant growth factor method

2.  If many iterations need to be preformed then the accuracy of the resulting trips matrix may be questioned.

The Fratar method

•  This procedure must be interated by summing the total number of future trips

•  As before iterations are continued until accuracy is achieved

' . .( / )

k

ikji

ij ij ki j

k k ik

tAPt tp a A a t

=∑

∑

The Furness Method (Doubly constrained growth factor method)

The model is represented as follows

sum of trips attracted to j in the first iteration)

Example of the Fratar Method

' /ij ij i it t P p=

'' ' ( /ij ij jt t A=

2 3 4 2Pa Ab Ac Adpa ab ac ad

= = = =

200 400 600' 200*2*3[ ]200*3 400*4 600*2abt

+ +

+ +423=

200 400 600' 400*2*4[ ]200*3 400*4 600*2act

+ +=

+ +

1129=

200 400 600' 600*2*2[ ]200*3 400*4 600*2adt

+ +=

+ +

847=

There are four steps to the calculation

1.  Total the outgoing trips for each zone and multiply by the zonal growth factor to obtain the predicted origin out going totals

2.  Multiply the lines in the matrix by the appropriate growth factor

3.  Total the incoming trips into each zone and divide by the predicted incoming totals to obtain the destination factors

4.  Repeat the iteration process until the origin or destination factor being calculated is sufficiently close to unity (within 5%)

Examples of the Furness Method

Example 1:

Consider the following trip matrix with the target origins and destinations for the future year included.

Base year trip matrix for doubly constrained problem.

Step 1 is already done, sum totals for each row.

Step 2

Multiply each cell in row 1 by 400/355




The table above shows the trip matrix after application of ratio of target to existing totals to the rows

Step 3

Multiply column 1 by 260/257.7




Step 4

Continue iterations as in step 2 and step 3 until the origin and destination factors are within 5% of unity

Comments on Growth Factor Methods

•  The accuracy is dependent upon the accuracy achieved in defining the growth factor used.

•  Growth factor process is not related to the factors which influence trip makers.

Synthetic Models

•  Allow for the inclusion of travel cost

•  Try to include the causes influencing present day travel patterns

•  Assume that these underlying causes will remain the same in the future

Trip distribution in the NTA model

The trip distribution model (TDM) is used to determine the pattern of trips between sets of trip generators and trip attractions.

This is a sub-model of the full NTA model.

The function of the model is to determine to what zones the trips are generated at any particular origin will travel. As the model has 666 zones, the matrix of trip patterns is 666 x 666.

The TAGM model also uses trip distributions from the base year. These are based upon observed travel patterns from the GDA household and education surveys and from the POWCAR dataset. The travel patterns represented by the base year matrices are also calibrated in the GDA transport model to ensure that when assigned to the transport networks, the model outputs closely match the observed network characteristics. This process - known as base year model calibration.

The calibration of the base year model also generates base year travel costs that can (optionally) be fed back into the TDM to impact on travel patterns.

Therefore the TDM mode: - creates an all day forecast year trip generations and attractions from the TAGM - Base year trip distribution matrices for the am-peak and off-peak periods.

The TDM uses a combination of two methods to derive forecast year travel patterns in the form of 666 x 666 trip matrices for each modelled time period:

Factoring: Where well established trip patterns exist in the base year, this pattern is retained and simply factored up to the new forecast year trip generations and attractions.

Sectoring: The GDA Transport Model has the facility to aggregate trip patterns using a 75 strategic zone system (called sectors). In the case of a green field site development in a zone or where there is insufficient data in the base year to determine the pattern of trips, the base year trip pattern of the 75 zone sector containing the green field zone is used to give the equivalent forecast year trip pattern.

In addition to using Factoring and Sectoring, the TDM can also (optionally) use the travel costs that are output during the route choice / trip assignment stage of the model to influence travel patterns in the forecast year. If this option is chosen, the TDM uses a form of “gravity model” to determine travel patterns for green field zones that have development in the forecast year, or for zones where there is a major change in population or travel costs between the base and forecast years.

The Gravity Distribution is based on the Newton’s gravitational formula, and in modelling trip distribution takes the form:

Tij = Oi . Dj . f(Cij)

Where Tij is the number of trips between origin O and destination D. Oi is the total trips generated at origin O and Dj is the number of trips attracted to destination D. f(Cij) is called the deterrence function based on the cost of travel between O and D.

The TDM uses a log-normal form of the deterrence function as follows:

f(Cij) = EXP(λ . Cij)

Where λ is a measure of people’s sensitivity to travel costs.

The outputs from the TDM are in the form of trip matrices (666 x 666) for each of the six journey purposes divided by car available and car not available persons (i.e. twelve trip matrices). These trip matrices are the essential inputs into the next three stages of the model described in the next section.

The Gravity Distribution Model Distribution models of a different kind have been developed to assist in the forecasting future trip patterns when important changes in the network take place.

These models make assumptions about group trip making behaviour and the way this is influenced by external factors such as total trip ends and distance travelled. The most widely used of these models is the gravity model.

This model estimates trips for each cell in the matrix without directly using the observed trip pattern.

The gravity model takes the following functional form:

Tij =!PiPjdij2

Where :Pi and Pj = populations of the towns O-D pair dij = distance between i and j ! = proportionality factor

This approach was seen to be too simplistic, and improvements have been introduced. One such improvement was the assumption that the effect of distance (or separation) could be modelled better using a decreasing function.

This function is often called a deterrence function.

Properties of the Gravity Model 1. It provides a more rigorous way of specifying the mathematical properties of the resulting model 2. The use of a mathematical programming framework also facilities the use of a standard tool-kit of solution methods and the analysis of the efficiency of alternative algorithms 3. The theoretical framework used to generate the model also assists in providing an improved interpretation of the solutions generated by it 4. Just because the gravity model can be generated in a number of ways does not make it correct. The appropriateness of the model depends upon the acceptability of the assumptions required for its generation and interpretation.

Tij =!OiDj f (cij )where: f (cij ) is a generalised function of the travel cost with one or more parameters for calibration.

lecture 3 - fratar algorithm

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