CE 303 - Transportation EngineeringTransportation Planning

Lecture 3 - Trip Distribution

Having determined trip generations Qi (productions Pi and attractions Aj) the next step is the estimation of target year interchange trip volumes Qij between all pairs of zones. This process is called the trip distribution. It is illustrated by Fig 3.1.

Fig. 3.1 Process of estimation of trip distribution.

The rule of trip distribution ensures that all trip attracting zones js compete with each other to attract trips produced by each zone i. Most trips will be attracted by the zone j with highest attractiveness aj and lowest travel impedance Wij.

Travel impedance Wij between zone i producing a trip and zone j attracting the trip is associated with the level of difficulty of travel, travel time, out of pocket travel cost and the like between the two zones. Wij

has a definite effect on the choice of the attracted zone for a trip produced in zone i. In other words choice of j depends on Wij in addition to aj. Therefore, highest attractiveness aj and lowest impedance Wij are controlling factors in trip interchanges between two zones. Trip distribution is estimated using one of the two common mathematical models. They are the Gravity model and the Fratar model.

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3.1 The Gravity model

The Gravity model is based on Newton’s law of gravitation,

(3.1)

Application of this law to trip distribution takes the form,

(3.2)

k and c are parameters of the model.

The model can be calibrated using usual mathematical modeling techniques. According to the model, the interchange volume Qij between a trip producing zone i and a trip attracting zone j is directly proportional to the magnitude of trip productions Pi of zone i and trip attractiveness aj of zone j and is inversely proportional to a function of the impedance Wij between the two zones.

k can be eliminated by applying a trip production balance constraint,

(3.3)

The constraint states that the total trips Pi produced by zone i is equal to the sum of overall shares of Pi

trips distributed to the zones x. (x є n) n is the total number of zones.

(3.4)

Solving for k we have,

(3.5)

which is the expression for k that ensures the trip balance constraint given by Eq. 3.3 is satisfied. Now, substituting for k we have,

(3.6)

is the proportion of the trips produced by zone i that is attracted by zone j in competition with

all trip attracting zones x.

The gravity model is often written as,

(3.7)

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where is known as the travel time factor.

Finally, a set of interzonal socioeconomic adjustment factors Kij are introduced during calibration to incorporate the effects of a limited number of independent variables not included in the model. The resulting model is,

(3.8)

Where pij is the probability that a trip generated by zone i will be attracted by zone j.

If Aj is the target year trip attractions of zone j, the following equation results,

(3.9)

Example: The target year trip productions and relative attractiveness of a four zone city have been estimated as follows,

Zone, i Trip Productions, Pi Attractiveness, Ai

1 1500 02 0 33 2600 24 0 5

The calibration of the Gravity model for this city estimated the parameter c to be 2.0 and all socioeconomic adjustment factors to be equal to unity. Apply the Gravity model to estimate all target year interchange volumes Qij and to estimate the total target year attractions of each zone. The target year interzonal impedances Wij were found to be as follows,

Zone iInterzonal impedance Wij

Zone j1 2 3 4

1 5 10 15 202 10 5 10 153 15 10 5 104 20 15 10 5

AnswerNote: The trip generation data indicate that there are three types of zones in this city. Zone 1 is purely residential (no attractiveness). Zones 2 and 4 are purely non residential (no productions)., while zone 3 is a mixed land use zone.(production and attractiveness). The diagonal elements of impedance matrix represent intrazonal impedances, i.e. the impedance associated with trips that begin and end within each zone.Calculate distribution of trip productions (by zones 1 and 3). The calculation of interchange volumes using

for zones i=1 and i=3 are as follows in table form,

For i =1, P1 = 1500.j aj F1j aj F1jK1j P1j Q1j3

K1j

1 0 0.0400 1. 0 0.0000 0.000 02 3 0.0100 1. 0 0.0300 0.584 8753 2 0.0044 1. 0 0.0089 0.173 2604 5 0.0025 1. 0 0.0125 0.243 365

Total 0.0514 1.000 P1=1500

For i =3, P3 = 2600.j aj F3j K3j aj F3jK3j P3j Q3j

1 0 0.0044 1. 0 0.0000 0.000 02 3 0.0100 1. 0 0.0300 0.188 4883 2 0.0400 1. 0 0.0800 0.500 13004 5 0.0100 1. 0 0.0500 0.312 812

Total 0.1600 1.000 P3=2600

Now calculate the total target year trip attractions of the non residential zones (j=2, j=3, j=4) applying

A2 = 875+488 = 1363A3 = 260+1300 = 1560A4 = 365+812 = 1177

The following trip table gives the summary of the solution,

Zone iInterzonal trips Qij

Zone j1 2 3 4 Total

1 0 875 260 365 15002 0 0 0 0 03 0 488 1300 812 26004 0 0 0 0 0

Total 0 1363 1560 1177 4100

Note: It is possible that, trips produced by the mixed land use zone 3 are attracted by the non residential sector of the same zone. The other diagonal elements in the trip matrix are zero.

Example: You are a planning consultant to a property development company that is considering the construction of a major shopping centre in a city. At present, the city consists of three residential zones and the central business district (CBD), where all shopping activity is concentrated. Your client can acquire land for the proposed shopping centre at any location shown to him by you. The client is interested in your prediction of the shopping trips the centre will attract if built to compete with the CBD. The following data have been made available to you,

1. Daily shopping trip production per person

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Household income level (X3)

Household size (X1

persons/household)

Personal car ownership per household(X2)

0 1 2

I≤ 2 0.2 0.3 0.4 3 0.1 0.2 0.3≥ 4 0.1 0.2 0.3

II≤ 2 0.3 0.4 0.5 3 0.2 0.2 0.4≥ 4 0.2 0.2 0.5

2. The relative shopping attractiveness of commercial zones have been found to be given by the following multiple regression equation A = 5*Xa + 3*Xb.

Xa = area of shopping floor space provided in hectares Xb = available parking area in hectares.

3. Land use and socioeconomic projections (commercial zones)

ZoneBase year Target year

Xa Xb Xa Xb

4 (CBD) 3.0 2.0 3.0 2.55 (Shopping centre) 0.0 0.0 2.0 3.0

4. Land use and socioeconomic projections (residential zones)

Zone X1 X2 X3Number of households

Base year Target year

1

2 0 I 300 5002 1 I 300 4003 1 I 200 3002 2 II 0 50

2

2 1 I 400 5002 1 II 300 2003 2 I 200 3003 0 I 100 400

3

1 1 II 200 2002 2 II 300 4003 2 II 400 3004 2 II 200 400

5. Gravity model parameters

(a) , where W = interzonal impedance in minutes

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Figure showing network and interzonal impedances

(b) Interzonal socioeconomic adjustment factors (Kij)

Zone iZone j

4 (CBD) 5 (Proposed shopping centre)1 1.0 0.92 0.9 1.23 1.0 1.0

Using the information 1 to 5 calculate all target year interchange volumes and the target patronage (trips attracted) of the two commercial zones.

AnswerApply the calibrated trip generation models and the available land use and socioeconomic projections to find the target year productions and relative attractiveness of the five zones. The shopping trip production model is a disaggregated cross classification model. Considering the units of the calibrated production rate, the contribution of each household type to the total zonal production is the product of (number of households)*(household size)*(trips per person). Therefore for the trip producing zones 1, 2 and 3 the trip productions are

Zone 1 Zone 2 Zone 3500 x 2 x 0.2 = 200 500 x 2x 0.3 = 300 200 x 1x 0.4 = 80400 x 2 x 0.3 = 240 200 x 2x 0.4 = 160 400 x 2x 0.5 = 400300 x 3 x 0.2 = 180 300 x 3x 0.3 = 270 300 x 3x 0.4 = 360 50 x 2 x 0.5 = 50 400 x 3x 0.1 = 120 400 x 4x 0.5 = 800

P1 = 670 P2 = 850 P3 = 1640

The target year attractiveness of the two competing commercial zones is calculated via the calibrated trip attractiveness equations as follows,

A4 = 5 x 3 + 3 x 2.5 = 22.5A5 = 5 x 2 + 3 x 3.0 = 19.0

The target year interchange volumes are computed using the Gravity model with the given c = 1 and the given Kij factors. The following trip matrix results,

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Zone iZone j

Pi4 (CBD) 5 (Shopping centre)1 166 504 6702 400 450 8503 1354 286 1640Aj 1920 1240 3160

The trip matrix shows that 1240 out of the 3160 daily shopping trips (39% of the total) will be attracted by the proposed shopping centre.

3.2 Fratar model

Fratar model is a growth factor (Gi) model. It begins with base year interchange flow (Qi(b)). It considers only the interzonal trips, i.e. trips produced and/or attracted outside the boundary of each zone are considered. It also considers no direction, meaning Qij = Qji. The estimated target year trip generation as,

Qi(t) = Gi * Qi(b) (3.10)

The model estimates the target year trip distribution Qij(t) which satisfies the trip balance equation,

(3.11)

Zone iGrowth

factor (Gi)1 22 23 34 1

(b) Estimated growth factors

(a) Base year trip interchange dataFig. 3.1 An example of Fratar model inputs

3.2.1 Fratar Model procedure

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The Fratar model procedure is an iterative process. It requires successive approximation of target year trip generation Qi (current). Target year trip interchange volumes are computed based on anticipated growth of the two zones at either end of each interchange. The implied estimated target year trip generated is compiled according to,

(3.12)

Then the target year trip generation is tested against current year trip generation for convergence.

(3.13)

At convergence, Ri = 1. If not calculate new interzonal flow,

(3.14)

Eq. 3.14 means that the expected trip generation of zone i is distributed among all zones so that a specific zone j receives its share according to a zone specific term divided by the sum of these terms for all “competing” zones x. The target year trip generation from zone i allotted to interchange i-j resulting from the Eq. 3.14 will not necessarily be the same as the target year trip generation from zone j allotted to interchange j-i. i.e. Qij ≠ Qji.But Fratar model employs only one interzonal volume estimate Qij = Qji. Therefore, the two values should be averaged. Thus,

(3.15)

These values are used to calculate the new adjustment factors Ri. Check whether Ri values are close to unity. If they are not close unity Repeat the procedure to calculate Qi(current) until Ri = 1 or ≈1.

3.2.2 Limitations of Fratar model1. Fratar model break down mathematically when a new zone is built (introduced) after the base year.2. Convergence to the target year generation totals is not always possible.3. The model is not sensitive to impedance Wij.

Example: Consider the base year trip distribution of a simple four zone system given below,

Zone iZone j

Qi(b) Gi1 2 3 41 0 20 30 15 65 22 20 0 10 40 70 23 30 10 0 35 75 34 15 40 35 0 90 1

Qj(b) 65 70 75 90

Find the target year trip distribution.Answer

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Use the trip balance equation to calculate the base year trip generation for the four zones.

Zone i Base year trip generation Qi(b) Target year trip generation Qi(t)= Qi(b)*Gi

1 65 1302 70 1403 75 2254 90 90

For the first iteration equate the adjustment factors Rj to the growth factors and the current interchange flows to the base year interchange volumes.

To apply Eq 3.14, we need,

Zone iQij(current)

Zone j1 2 3 4 Ri Qi(t)

1 0 20 30 15 2 1302 20 0 10 40 2 1403 30 10 0 35 3 2254 15 40 35 0 1 90

From which we can have,

Zone iQij(current)*Rj

Zone j1 2 3 4 ∑Qij(current)*Rj

1 0 40 90 15 1452 40 0 30 40 1103 60 20 0 35 1154 30 80 105 0 215

Zone iQij(new)Zone j

1 2 3 41 0 36 81 132 51 0 38 513 117 39 0 684 13 33 44 0

Applying Eq 3.15, we have,

Zone i

Qij(current)Zone j

1 2 3 4

1 00 44 99 13 1562 44 00 39 42 1243 99 39 00 57 1944 13 42 57 00 112

Qj(current) 156 124 194 112

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Now, apply Eq. 3.13, to calculate the target year trip generation of each zone using the

results above.

Zone i Qi(t) Qi(current) Ri

1 130 156 0.832 140 124 1.133 225 194 1.164 90 112 0.80

The factors show that the current trip generations of zones 1 and 4 are over estimated and the current trip generations of Zones 2 and 3 are under estimated. For better estimates it is necessary repeat the process using the latest adjustment factors Ris and the latest trip matrix as the current trip interchange volumes.

The steps are,

1. Applying Eq 3.14, we have,

Zone iQij(new)Zone j

1 2 3 41 0 36 86 82 44 0 54 413 108 57 0 594 8 35 47 0

2. Applying Eq 3.15,

and applying Eq. 3.13, we have,

Zone i

Qij(current)Zone j

1 2 3 4 Qj(current) Ri

1 0 40 97 8 156 0.902 40 0 56 38 124 1.043 97 56 0 53 194 1.094 8 38 53 0 112 0.91

Qj(current)

We can see that the Ri values are getting closer to unity. If we do one more iteration we can have R1 = 0.97, R2 = 0.98, R3 = 1.04, R4 = 1.00.

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