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# 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

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:

t 'ij = tij E
Where: tij 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
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

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
Multiply each cell in Row 2 by 460/455
Multiply each cell in Row 3 by 400/255
Multiply each cell in Row 4 by 702/570

## 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

t 'ij = tij ( Ei + E j ) / 2

Ei = Pi / pi and E j = Aj / a j
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%

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.

k

Pi Aj
t 'ij = tij . .
pi a j

ik

(A

/ ak )tik

trips

## The Furness Method (Doubly constrained growth factor method)

The model is represented as follows

t 'ij = tij Pi / pi
t ''ij = t 'ij ( Aj /sum of trips attracted to j in the first iteration)
Example of the Fratar Method

Pa
Ab
Ac
=2
=3
=4
=2
pa
ab
ac

## 200 + 400 + 600

t 'ab 200*2*3[
]
200*3 + 400*4 + 600*2

= 423

## 200 + 400 + 600

t 'ac = 400*2*4[
]
200*3 + 400*4 + 600*2
= 1129

## 200 + 400 + 600

]
200*3 + 400*4 + 600*2

= 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
Multiply each cell in row 2 by 460/455
Multiply each cell in row 3 by 400/255
Multiply each cell in row 4 by 702/570

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
Multiply column 2 by 400/464.6
Multiply column 3 by 500/529.5
Multiply column 4 by 802/701.2

Step 4
Continue iterations as in step 2 and step 3 until the origin and destination
factors are within 5% of unity

## 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

## 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 Newtons 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( . Cij)
Where is a measure of peoples 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:

! Pi Pj
Tij = 2
dij

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.

## Tij = !Oi D j f (cij )

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

## 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.

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.

## Tij = !Oi D j f (cij )

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

## 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.
The text passage describes various methods used in trip distribution modeling, including growth factor methods and synthetic models like the gravity model. It provides equations and examples for some of these methods.
no_question

the passage does not contain any explicit mathematical questions or problems to solve. it primarily describes methods and equations used in trip distribution modeling without posing any specific questions or providing answers to such questions.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Sie sind auf Seite 1von 27

# 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

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:

t 'ij = tij E
Where: tij 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
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

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
Multiply each cell in Row 2 by 460/455
Multiply each cell in Row 3 by 400/255
Multiply each cell in Row 4 by 702/570

## 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

t 'ij = tij ( Ei + E j ) / 2

Ei = Pi / pi and E j = Aj / a j
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%

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.

k

Pi Aj
t 'ij = tij . .
pi a j

ik

(A

/ ak )tik

trips

## The Furness Method (Doubly constrained growth factor method)

The model is represented as follows

t 'ij = tij Pi / pi
t ''ij = t 'ij ( Aj /sum of trips attracted to j in the first iteration)
Example of the Fratar Method

Pa
Ab
Ac
=2
=3
=4
=2
pa
ab
ac

## 200 + 400 + 600

t 'ab 200*2*3[
]
200*3 + 400*4 + 600*2

= 423

## 200 + 400 + 600

t 'ac = 400*2*4[
]
200*3 + 400*4 + 600*2
= 1129

## 200 + 400 + 600

]
200*3 + 400*4 + 600*2

= 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
Multiply each cell in row 2 by 460/455
Multiply each cell in row 3 by 400/255
Multiply each cell in row 4 by 702/570

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
Multiply column 2 by 400/464.6
Multiply column 3 by 500/529.5
Multiply column 4 by 802/701.2

Step 4
Continue iterations as in step 2 and step 3 until the origin and destination
factors are within 5% of unity

## 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

## 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 Newtons 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( . Cij)
Where is a measure of peoples 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:

! Pi Pj
Tij = 2
dij

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.

## Tij = !Oi D j f (cij )

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

## 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.

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.

## Tij = !Oi D j f (cij )

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

## 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.
user
The text passage describes various methods used in trip distribution modeling, including growth factor methods and synthetic models like the gravity model. It provides equations and examples for some of these methods.

TYPE: NO_QUESTION

The passage does not contain any explicit mathematical questions or problems to solve. It primarily describes methods and equations used in trip distribution modeling without posing any specific questions or providing answers to such questions.