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Trip Distribution
Modelling
K. Ramachandra Rao
CEL 442: Traffic and Transportation Planning
2
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
3
Transportation Engineering-I
Trip Distribution
4
Urban Transportation Modelling
System
Transportation Planning
Trip Distribution
Four-step model
5
Transportation Engineering-I
Trip Distribution
6
Trip Distribution
Transportation Planning
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
7
Definition and Notation
Transportation Planning
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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Trip Distribution
8
Definition and Notation
Transportation Planning
Trip Distribution
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Growth Factor models- uniform growth
factor
Transportation Planning
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
10
Growth Factor models- average growth
factor
Transportation Planning
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
11
Singly constrained growth factor model -
Fratar model
Transportation Planning
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
12
Fratar model
Transportation Planning
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
k
kik
jij
ij
13
Growth Factor models: advantages
and limitations
Transportation Planning
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
14
Transportation Engineering-I
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
15
Transportation Engineering-I
Trip Distribution
16
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
d
MMKT
1
2
Transportation Planning
Trip Distribution
17
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:
Transportation Planning
n
jij
b
ijj
ij
b
ijj
iij
KCA
KCAPT
1
Trip Distribution
18
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
Transportation Planning
n
jijijj
ijijj
iij
KFA
KFAPT
1
Trip Distribution
19
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
Transportation Planning
Trip Distribution
20
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
Transportation Planning
Trip Distribution
21
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
e
eePT
1
)( )1(
Transportation Planning
Trip Distribution
22
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
Transportation Planning
Trip Distribution
23
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