Transportation Planning Module

Modelling Trip Distribution- Lecture 6

Dr. Chro H. Ahmed

Ph.D in Rail Transit System Planning and Design
University of Sulaimani- College of Engineering
 6.1 The Role of Trip Distribution:
▪ Trip distribution is a process by which the trips
generated in one zone are allocated
to other zones in the study area. These trips may
be within the study area (internal-internal) or
between the study area and areas outside the
study area (internal-external).

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 6.1 The Role of Trip Distribution:
▪ For example, if the trip generation analysis results
in an estimate of 200 HBW trips in zone 10, then
the trip distribution analysis would determine
how many of these trips would be made between
zone 10 and all the other internal zones.
In addition, the trip distribution process
considers internal-external trips (or vice versa)
where one end of the trip is within the study area
and the other end is outside the study area.
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 6.1 The Role of Trip Distribution:
▪ The trip interchanges can be represented by mean of a trip matrix (Tij) which
gives the number of trips generated by zone i that are attracted to zone j.

▪ Trip generation essentially determines the row and column totals for the
forecast year.

σ𝑗 𝑇𝑖𝑗 = 𝑃𝑖 = number of trips generated by zone i

σ𝑖 𝑇𝑖𝑗 = 𝐴𝑖 = number of trips attracted to zone j

Trip distribution ‘fills in’ the body of the trip matrix.

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 Trip Matrix

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 6.2 Methods for Trip Distribution Modelling:
▪ Trip interchanges increase with decrease impedance (distance, travel time,
travel cost) between zones.

▪ Trip interchanges increase with increased zone “attractiveness” (square

footage of retail or population).

There are several basic methods for trip distribution, among these are:

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 6.2 Methods for Trip Distribution Modelling:
1. Growth factor models – scale up observed base year trip matrix

2. Gravity/entropy maximization models- aggregate (zonal-level) models

3. Destination choice models- disaggregate (individual or household level) models

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 6.3 Growth Factor Models
Growth factor models work on the basis of scaling up an observed base year trip
matrix by applying a set of factors which account for zonal growth:

the general form of these models is:

𝑇𝑖𝑗 = 𝑇𝑖𝑗0 × 𝐸𝑖𝑗

Where:

𝑇𝑖𝑗 is the number of trips between zone I and zone j in some future (forecast year),

𝑇𝑖𝑗0 is the number of trips between i and j in the base year, and

𝐸𝑖𝑗 is the growth factor.

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 6.3 Growth Factor Models
There are different types of growth factor models, distinguished by different
methods of calculating 𝐸𝑖𝑗 .

1. Uniform Growth Factor

𝐸𝑖𝑗 = 𝐸

Where: 𝐸 = 𝑇Τ𝑇 0 and T is the total number of trips in the future year, and 𝑇 0
is the total number of trips in the base year.

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 6.3 Growth Factor Models
2. Average Growth Factor
𝑝
𝐸𝑖𝑗 = 𝐸𝑖 + 𝐸𝑗𝑎 Τ2
𝑝
Where: 𝐸𝑖 = 𝑃𝑖 Τσ𝑗 𝑇𝑖𝑗0 𝑎𝑛𝑑 𝐸𝑗𝑎 = 𝐴𝑗 Τσ𝑗 𝑇𝑖𝑗0

A large number of other similar growth factor methods also exist.

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 7.3 Growth Factor Models
Example: Basic Data
Zone Total Future trip Gens

1 2 3

Zone 1 20 90 150 260 400

2 40 30 40 110 500

3 90 100 10 200 300

Total 150 220 200 570

Future trip Atts

400 400 400

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 Uniform Growth Factor

G= 1200/570=2.105263
Zone Total Future trip Gens

1 2 3

Zone 1 42 189 316 547 400

2 84 63 84 232 500

3 189 211 21 421 300

Total 316 463 421

Future trip Atts

400 400 400
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 Average Growth Factor

(400/260+400/150)/2= (400/220+400/260)/2= (400/200+400/260)/2=

2.1 1.68 1.77 2.1 1.68 1.77

G= 3.61 3.18 3.27 (400/150+500/110)/2=

3.61
(400/220+500/110)/2=
3.18
(400/200+500/110)/2=
3.27
2.08 1.66 1.75 (400/150+300/200)/2= (400/220+300/200)/2= (400/200+300/200)/2=
2.08 1.66 1.75

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 Average Growth Factor

G= 2.1 1.68 1.77

3.61 3.18 3.27
2.08 1.66 1.75

Zone Total Future trip Gens

1 2 3
Zone 1 42 151 265 458 400
2 144 95 131 371 500
3 188 166 18 371 300
Total 374 412 414

Future trip Atts

400 400 400 Dr. Chro H. Ahmed
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 6.3 Growth Factor Models
3. Doubly Constrained Growth Factors
An interesting problem is generated when 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 τi and τj . The application of an ‘average’ growth
factor, say Fij = 0.5 (τi + τj ) is only a poor compromise as none of the two targets or
trip-end constraints would be satisfied.

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 6.3 Growth Factor Models
3. Doubly Constrained Growth Factors
Historically, several iterative methods have been proposed to derive an estimated
trip matrix that fulfills both sets of trip-end constraints or the two sets of growth
factors, which are essentially equivalent.

All of these methods entail the computation of a series of intermediate correction

coefficients, subsequently applied to cell entries in each respective row or column.
Following the application of these corrections, the totals for each column are
computed and compared with the target values. If substantial differences are
identified, new correction coefficients are calculated and applied as needed.
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 6.3 Growth Factor Models
3. Doubly Constrained Growth Factors
In transport these methods are known by their authors as Fratar in the US and
Furness elsewhere. For example Furness (1965) introduced ‘balancing factors’ Ai and
Bj as follows:
Tij = tij · τi · τj · Ai · Bj
or incorporating the growth rates into new variables ai and bj:
Tij = tij · ai · bj
with ai = τi Ai and bj = τj Bj.
The factors ai and bj (or Ai and Bj) must be calculated so that the following constraints
are satisfied. Dr. Chro H. Ahmed
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 6.3 Growth Factor Models
3. Doubly Constrained Growth Factors

This means that the sum of the trips in a row should equal the total number of trips
emanating from that zone; the sum of the trips in a column should correspond to the
number of trips attracted to that zone.

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 6.3 Growth Factor Models
3. Doubly Constrained Growth Factors
This is achieved in an iterative process as follows:

1. set all bj = 1.0 and solve for ai; in this context, ‘solve for ai’ means find the
correction factors ai that satisfy the trip generation constraints;

2. with the latest ai solve for bj, e.g. satisfy the trip attraction constraints;

3. keeping the bj’s fixed, solve for ai and repeat steps (2) and (3) until the changes
are sufficiently small.

This method produces solutions within 3 to 5% of the target values in a few iterations
when certain conditions are met.
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 6.3 Growth Factor Models
3. Doubly Constrained Growth Factors
If reliable information is available to estimate both Oi and Dj then the model must
satisfy both conditions; in this case the model is said to be doubly constrained. In
some cases there will be information only about one of these constraints, for
example to estimate all the Oi’s, and therefore the model will be said to be singly
constrained. Thus a model can be origin or production constrained if the Oi’s, are
available, or destination or attraction constrained if the Dj’s are at hand.

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 Example 1: The base year trip matrix for a study area consisting of three zones is
given below.

The productions from the zone 1,2 and 3 for the horizon year is expected to grow
to 98, 106, and 122 respectively. The attractions from these zones are expected to
increase to 102, 118, 106 respectively. Compute the trip matrix for the horizon
year using doubly constrained growth factor model using Furness method.
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 Solution:
The sum of the attractions in the horizon year, i.e.

Oi = 98+106+122 = 326.

The sum of the productions in the horizon year, i.e.

Dj = 102+118+106 = 326.

They both are found to be equal. Therefore we can proceed. The first step is to fix bj
= 1, and find balancing factor ai. ai = Oi/oi, then find Tij = ai X tij . So

▪ a1 = 98/78 = 1.26

▪ a2 = 106/92 = 1.15

▪ a3 = 122/82 = 1.49 Dr. Chro H. Ahmed

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 Solution:
Further T11 = t11 x a1 = 20 x 1.26 = 25.2.

Similarly T12 = t12 x a2 = 36 x 1.15 = 41.4. etc. Multiplying a1 with the first row of the
matrix, a2 with the second row and so on, matrix obtained is as shown below.

Also 𝑑𝑗1 = 25.2 + 41.4 + 32.78 = 99.38

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 Solution:

▪ In the second step, find bj = Dj/𝑑𝑗1 and Tij = tij x bj . For example b1 = 102/99.38 =
1.03, b2 = 118/125.26 = 0.94 etc.,

▪ T11 = t11 x b1 = 25.2 x 1.03 = 25.96 etc. Also 𝑂𝑖1 = 25.96 + 35.53 + 36.69 = 98.18.
The matrix is as shown below:

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 Solution:

Therefore error can be computed as ; Error =σ 𝑂𝑖 − 𝑂𝑖1 + σ 𝐷𝑗 − 𝑑𝑗

𝑬𝒓𝒓𝒐𝒓 = 𝟗𝟖. 𝟏𝟖 − 𝟗𝟖 + 𝟏𝟎𝟓. 𝟗𝟑 − 𝟏𝟎𝟔 + 𝟏𝟐𝟏. 𝟔𝟕 − 𝟏𝟐𝟐 + ȁ𝟏𝟎𝟐. 𝟑𝟔 −

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 Example 2: the following table represents a doubly constrained growth factor
problem:

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 Solution: The solution to this problem, after three iterations on rows and columns
(three sets of corrections for all rows and three for all columns), is shown in the
following table:

Note that this estimated matrix is within 1% of meeting the target trip ends, more
than enough accuracy for this problem. Dr. Chro H. Ahmed
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 6.4 Advantages and Disadvantages of Growth Factor Models

6.4.1 Advantages :

1. Growth-factor methods are simple to understand and make direct use of

observed trip matrices and forecasts of trip-end growth.

2. They preserve the observations as much as is consistent with the information

available on growth rates.

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 6.4 Advantages and Disadvantages of Growth Factor Models

6.4.2 Disadvantages :

1. they are probably only reasonable for short-term planning horizons or when
changes in transport costs are not to be expected.

2. They require an observed (sampled) trip matrix; this is an expensive data item.

3. The methods are heavily dependent on the accuracy of the base-year trip matrix.

4. Any error in the base-year may well be amplified by the application of successive
correction factors.

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 6.4 Advantages and Disadvantages of Growth Factor Models

6.4.2 Disadvantages :

6. If portions of the base-year matrix are unobserved, they will continue to remain
unobserved in the forecasts. Consequently, these methods cannot be employed
to populate unobserved cells in partially observed trip matrices..

7. These methods do not consider changes in transport costs resulting from

improvements (or new congestion) in the network. As a result, they have limited
applicability in the analysis of policy options involving new modes, new links,
pricing policies, and new zones.

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 6.5 Synthetic or Gravity Models
Originally generated from an anology with Newton’s gravitational law.
Tij = α Pi Pj / (dij)2
(Casey, 1955)
Tij = Trips from zone i to zone j
α= Proportionality factor
Pi = Population of zone i
Pj = Population of zone j
dij = Distance between zone i and zone j

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 6.5 Synthetic or Gravity Models

The model was further generalized by assuming that the effect of distance or
“separation” could be modelled better by a decreasing function (deterrence
function).

Tij = αOi Dj f (cij)

where f (cij) is a generalized function of the travel costs with one or more parameters
for calibration. This function often receives the name of ‘deterrence function’
because it represents the disincentive to travel as distance (time) or cost increases.
Popular versions for this function are:

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 6.5 Synthetic or Gravity Models

f (cij) = exp(−βcij) Exponential function

f (cij) = cij−n Power function

f (cij) = cnij exp(−βcij) Combined function

Generalized cost of travel

Cij = a1 tij + a2 Fij + a3 Pj + δ

tij : Travel time
Fij : Fare (Travel cost)
Pj : Parking cost
δ : Modal penalty that represents all other attributes not included in the
generalized cost (e.g., comfort, safety, convenience)
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 6.5 Synthetic or Gravity Models

As in the growth factor model, here also we have singly and doubly constrained models.
The expression Tij = AiOiBjDjf(cij ) is the classical version of the doubly constrained
model. Singly constrained versions can be produced by making one set of balancing
factors Ai or Bj equal to one. Therefore we can treat singly constrained model as a
special case which can be derived from doubly constrained models. Hence we will limit

our discussion to doubly constrained models. As seen earlier, the model has the
functional form, Tij = AiOiBjDjf(cij )

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 6.5 Synthetic or Gravity Models
෍ 𝑇𝑖𝑗 = ෍ AiOiBjDjf(cij )
𝑖 𝑖

But

σ𝑖 𝑇𝑖𝑗 = Dj Therefore,

Dj = BjDj σ𝑖 AiOif(cij )

From this we can find the balancing factor Bj as

Bj =1/ σ𝒊 AiOif(cij ) Bj depends on Ai which can be found out by the following

equation:

Ai =1 / σ𝒋 BjDjf(cij )

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 6.5 Synthetic or Gravity Models

We can see that both Ai and Bj are interdependent. Therefore, through some
iteration procedure similar to that of Furness method, the problem can be solved.
The procedure is discussed below:

1. Set Bj = 1, find Ai using equation Ai =1 / σ𝒋 BjDjf(cij )

2. Find Bj using equation Bj =1/ σ𝒊 AiOif(cij )

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 6.5 Synthetic or Gravity Models

3. Compute the error as E =σ 𝑂𝑖 − 𝑂𝑖1 + σ 𝐷𝑗 − 𝐷𝑗1 where Oi corresponds to

the actual productions from zone i and 𝑂𝑖1 is the calculated productions from that

zone. Similarly Dj are the actual attractions from the zone j and 𝐷𝑗1 are the
calculated attractions from that zone.

4. Again set Bj = 1 and find Ai, also find Bj . Repeat these steps until the
convergence is achieved

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 6.6 Application of Gravity Models

Please refer to the “Traffic and Highway Engineering –Part 3-Transportation

Planning” for more details and solved examples on Gravity model.

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 6.7 Gravity Models Summary

Advantages:

▪ Has a sounder theoretical and statistical basis than growth factor methods

▪ Can model with non-marginal changes

▪ Does not require complete O-D survey

Disadvantages:

▪ Calibration of deterrence parameter can be difficult

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 THANKS!

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