-ppt4 trip distribution

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Trip Distribution Models

1.GrowthFactor/Fratar Method

A simple method to distribute trips in a studyarea. Assumptions of the modela. the distribution of future trips from a given

origin zone is proportional to the present tripdistribution

b. this future distribution is modified by thegrowth factor of the zone to which these trips areattached


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The Fratar formula can be written as

Example: Fratar Method

An origin zone i with 20 base-year trips going to zones a, b, and c numbering4, 6, and 10, respectively, has growth rates of 2, 3, 4, and 5 for i, a, b, and c,respectively. Determine the future trips from i to a, b, and c in the future year.

!

4

6

10

20Given:


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The Gravity Model

The most widely used trip distributionmodel The model states that the number of tripsbetween two zones is directly proportionalto the number of trip attractions generatedby the zone of destination and inverselyproportional to a function of time of travel

between the two zones.


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The gravity model is expressed as

Singly Constrained model when information isavailable about the expected growth tripsoriginating in each zone only or the other way, tripsattracted to each zone only Doubly Constrained model when information isavailable on the future number of trips originatingand terminating in each zone.

Single Constrained vs. Doubly Constrained model


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For a doubly constrained gravity model, the adjustedattraction factors are computed according to the formula

Gravity Model Example


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Solution:

Iteration 1 :


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Calibrating a Gravity Model

Calibrating of a gravity model is accomplished bydeveloping friction factors and developingsocioeconomic adjustment factors

Friction factors reflect the effect travel time ofimpedance has on trip making

A trial-and-error adjustment process is generally

adopted

One other way is to use the factors from a paststudy in a similar urban area


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Three items are used as input to the gravity model forcalibration:

1.Production-attraction trip table for eachpurpose

2.Travel times for all zone pairs, includingintrazonal times

3. Initial friction factors for each increment oftravel time

The calibration process involves adjusting the friction factor

parameter until the planner is satisfied that the modeladequately reproduces the trip distribution as represented bythe input trip table from the survey data such as the trip-time frequency distribution and the average trip time.

The Calibration Process1. Use the gravity model to distribute trips based on initialinputs.

2. Total trip attractions at all zones j, as calculated by themodel, are compared to those obtained from the inputobserved trip table.

3. If this comparison shows significant differences, theattraction Aj is adjusted for each zone, where a differenceis observed.

4. The model is rerun until the calculated and observedattractions are reasonably balanced.

5. The models trip table and the input travel time table canbe used for two comparisons: the trip-time frequencydistribution and the average trip time. If there aresignificant differences, the process begins again.


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Figure 11-7 shows the results of four iterations comparing travel-time frequency.


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An example of smoothed values of F factors in Figure 11-9. In general, values of F decreases as travel time increases, and may take the

form F varies as t-1, t-2, or e-t.

Figure 11-9 Smoothed Adjusted Factors, Calibration 2

A more general term used for representing travel time (or a

measure of separation between zones) is impedance, and therelationship between a set of impedance (W) and frictionfactors (F) can be written as:

Example:A gravity model was calibrated with the following results:

"#! $ " " % & ' ( )) )*

! ! +,-* +,./ +,.* +,.) +,)/

Using the f as the dependent variable, calculate parameterA and c of the equation.


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Solution:

The equation can be written as

ln F = ln A c ln W

% )+-/ )+0/ .+,( .+&, .+0)

-+-* -+*& -+'/ -+(' -+/'

These figures yield the following values of A = .07 and c = .48.

Hence, F = 0.07/W0.48

1 23

4 ,+//*0&/

54 ,+//)*)'

#64#

54 ,+/(('((

##

,+,.',.

2$ *

2

) ,+.-0-'/ ,+.-0-'/ -*,+*/)/ ,+,,,---

#4 - ,+,,.,-) ,+,,,'00

& ,+.-/&

! !

! .+0-.)( ,+,*)/& *. +',.- )+*),* .+(/0&( .+*''(/ .+(/0&( .+*''(/

7 ,+&') ,+,.&'.) )(+0.&) ,+,,,--- ,+*-/-* ,+-(.'* ,+*-/-* ,+-(.'*

Since, lnA = -2.73218,

A = e^(-2.73218)A = .065

and

c = - (-.461)

c = 0.461


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Seatwork (40 pts)

1. A four zone city has the following productions and attractions

The travel time matrix is

Apply the gravity models to distribute the trips (K ij = 1). Perform threeiterations.

2. An approximately 2-mile-square city has a central business district (CBD) of

mile square with a uniform population density of 100 persons/acre. The densityform the edge of the CBD to the outer edge of the city on all four sides decreases

uniformly to 5 persons/acre. Sketch the population density pattern and determinethe total population (1 sq mi = 640 acres).

END

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