5 trip distribution lecture 12-13.unlocked

Trip Distribution

Modelling

K. Ramachandra Rao

CEL 442: Traffic and Transportation Planning

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Outline Introduction

Growth Factor models Fratar model

Stochastic models: Synthetic or Gravity models Calibration of Gravity models

Stochastic models: Intervening and competing opportunity models

Other models Bi and Tri-proportional Approach

Entropy models

Transportation Planning

Trip Distribution

Introduction The task of the trip distribution model is to distribute or

link-up the zonal trip ends, the productions and attractions for each zone as predicted by the trip generation model in order to predict the flow of trips Tij from each production zone to each attraction zone

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

Growth factor method Constant factor method

Average factor method

Fratar method

Furness method

Stochastic methods Gravity model

Opportunity model

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Transportation Engineering-I

Trip Distribution

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Urban Transportation Modelling

System


Trip Distribution

Four-step model

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Trip Distribution

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Trip Distribution


The task of the trip distribution model is to distribute or link-up the zonal trip ends, the productions and attractions for each zone as predicted by the trip generation model in order to predict the flow of trips Tij from each production zone to each attraction zone

Types of models Growth factor models

Stochastic models Gravity models

Intervening opportunities model

Entropy maximizing approach Trip Distribution

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Definition and Notation


Trip pattern in a study area by means of a trip matrix a two dimensional array of cells where rows and columns

represent each of the z zones in the study area

Cells in each row contain trips originating in that zone which have destinations in the corresponding columns

Leading diagonal indicates the corresponding intra-zonal trips

Matrices can be further disaggregated by person type (n) or by mode (k)

The cost element may be considered in terms of distance, time or money units

A generalised cost of travel is the combination of all the main attributes related to the disutility of the journey

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ij

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ij

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DTOT

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Trip Distribution

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Definition and Notation


Trip Distribution

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Growth Factor models- uniform growth

factor


Useful in short term updating trip tables and estimation of through trips or external trips

Let us consider a situation where we have a basic trip matrix t, (from previous studies or estimated from survey data)

We would like to estimate the matrix corresponding to the design year, say 10 yrs into the future

Tij = G*tij for each pair i and j, where G is the ratio of the expanded over the previous number of trips

Trip Distribution

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Growth Factor models- average growth

factor


Tij = [(Gi+Gj)/2 ]*tij for each pair i and j, where

Gi = Ti/ti; Gj = Tj/tj is the ratio of the expanded over the previous number of trips

When the calculated values would not match with the total flows originating or terminating in a zone, then iterative process is used.

Ti (target) Ti (current) Gi = Ti (target)/Ti (current) and Gj = Tj (target)/Tj

(current) and then reuse the equation above till the growth factors approximate to unity

Trip Distribution

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Singly constrained growth factor model -

Fratar model


Fratar model: begins with the base-year interchange data, and does not distinguish between productions and attractions

As there is no distinction between productions and attractions, Tij = Tji the trip generation of each zone is denoted by Ti =Tij for all j

The estimate of target-year trip generation which precedes trip distribution is obtained by Ti (t) = Gi[Ti(b)]; Gi = zonal growth factor for a specific origin or Gj = zonal growth factor for a specific destination

Subsequently the model estimates the target Tij(t), that satisfies the trip balance equation, Ti =Tij

Trip Distribution

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Fratar model


A set of adjustment factors are computed by Ri = Ti(t)/Ti(current), if the adjustment factors are close to unity and trip balance constraint is satisfied the procedure is terminated

Basic Equation:

The expected trip generation of zone I is distributed among all zones so that a specific zone j receives the share according to a zone specific term divided by all the terms competing zones k

Two different values of Tij and Tji would result, but the current value is computed as follows

2

)()()()(

newTnewTcurrentTcurrentT

jiij

jiij

Trip Distribution

)()(

).(tT

RcurrentT

RcurrentTT i

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ij

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Growth Factor models: advantages

and limitations


They preserve the observations as much as consistent with the information available on growth rates

Reasonable for short term planning horizons

Does not take into account changes in transport costs due to improvements in the network, i.e., not sensitive to travel impedance

Breaks down mathematically when new zone is added, after base year, since all base year interchange volumes would be zero using this zone

Trip Distribution

Stochastic/Synthetic models gravity model Based on the presumption that the number of

trips between each pair of zones is proportional to the activities of those zones but inversely proportional to the distance and other resistances among the trips to potential destinations

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

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Trip Distribution

Gravity model

Loose analogy to Newtons law of gravity the attractive force between any two bodies is

directly related to the masses of the bodies and

inversely related to the distance between them

G= gravitational constant

the number of trips between two areas is

directly related to activities in the area

represented by trip generation and inversely

related to the separation between the areas

represented as a function of travel time

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Trip Distribution

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Gravity model

Tij= no. of trips between zones i and j

Pi = no. of trips generated in zone i

K = constant reflecting local conditions which must be empirically determined

Mi,Mj = populations of zones i and j

Dij = distance between zones i and j

Fij = friction factor or travel impedance = cij-b ;exp (-bcij)

Kij = zone-to-zone relationship factor

Aj = measure of attractiveness of zone j

n

jijijj

ijijj

iij

ij

ji

ij

KFA

KFAPT

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MMKT

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2


Trip Distribution

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Gravity model - Calibration Calibration of gravity model involves the determination of

the numerical value of the parameter b that fixes the model to the one that reproduces the base-year observations

The knowledge of the proper value of b fixes the relative relationship between the travel time factor and inter-zonal impedance

Unlike the calibration of a simple linear regression model where the parameters can be solved by a relatively easy minimization of the sum of squared deviations, the calibration of gravity model is accomplished through an iterative procedure:


n

jij

b

ijj

ij

b

ijj

iij

KCA

KCAPT

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Trip Distribution

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Gravity model - Calibration Step 1: The initial value of b is assumed and the trip distribution

equation is used to get Tijs

Step 2: The Tijs computed are compared to those observed during the base year

If the computed volumes are close to the observed volumes, the current value of b is retained

Else, adjustement to b is made the procedure is continued until an acceptable degree of convergence is achieved

Most commonly the friction factor function F is used rather than the parameter c is used in the calibration procedure


n

jijijj

ijijj

iij

KFA

KFAPT

1

Trip Distribution

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Limitations of gravity model

Simplistic nature of impedance and its apparent lack of behavioural basis to explain the destination choice

Dependence on K-factors of adjustment factors

Absence of any variables that reflect the characteristics of the individuals or households who decide which destinations to choose in order to satisfy the needs, destination choice models tend to overcome this problem


Trip Distribution

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Intervening Opportunities model The postulate on which this model is based, from Stouffer,

is

Probability of choice of a particular destination (from a given origin for particular trip purpose) is proportional to the opportunities for trip-purpose satisfaction at the destination at the destination and inversely proportional to all such opportunities that are closer to the origin

The inverse proportionality to the closer opportunities can be interpreted as proportionality to the probability that none of the closer destinations (opportunities) are chosen

The attraction properties of the destination are modelled as opportunities and the impedances are measured in terms of the number of opportunities which are closer


Trip Distribution

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Intervening Opportunities model

This is an attempt to correct the deficiencies of two

previous models Tij= no. of trips between zones i and j

L = Probability of accepting any particular destination/opportunity

Vj,Vj+1 No. of Opportunities passed up to the zones j and j+1, respectively

Vn = the total no. of opportunities

Pi = population in zone i

n

jj

LV

LVLV

iij

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)( )1(


Trip Distribution

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Entropy Maximizing approach Entropy-maximization approach which has been used in

the generation of a wide range of models

gravity model,

shopping models and

location model


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References Meyer, MD and Miller, EJ (2001), Urban Transportation

Planning, McGraw Hill, 2nd Edition

Ortuzar, JD and Willumsen, HCW (2011) Modelling Transport, John Wiley, 4th Edition

Papacostas, CS, and Prevedouros, PD (2001) Transportation Engineering and Planning, Prentice-Hall, 3rd Edition

Khisty, CJ, and Lall, B.K. (2003) Transportation Engineering: An Introduction, Prentice-Hall of India, New Delhi, 3rd Edition

Manheim, ML (1979) Fundamentals of Transportation Systems Analysis, Vol I, The MIT Press, Cambridge

5 trip distribution lecture 12-13.unlocked

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