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PERENCANAAN TRANSPORTASI
Lecture 3 : Trip Distribution
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Trip Distribution
• Trip generation: Trip productions & attractions in eachTraffic Analysis Zone (TAZ). (how many trips start ineach TAZ and how many trips end in each TAZ)
• Trip distribution links productions to attractions foreach TAZ pair (models convert Ps and As into trip tablesrepresenting movements between TAZs)
• Trip distribution models account for land-use &transportation system/spatial separation of TAZs
• Different models used for trip distribution (noregressions)
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Trip Distribution Function
HBW Trip Generation HBW Trip
Distribution
(balanced)
PHBW: 300
AHBW
:700
PHBW: 500
AHBW
: 250PHBW: 550
AHBW: 400
PHBW: 300
AHBW:700
PHBW: 500
AHBW: 250
PHBW: 550
AHBW: 400
100
300
150
200200400
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Behavioral Foundations?
• Assumptions about group trip-makingbehavior
– Number of productions
– Number of attractions
– Distance traveled and other travel costs
• Trip making is…
– Directly related to production – Directly related to attraction
– Inversely related to travel cost impedance
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The Gravity Model (GM)
• Derives its name and basic premise from
Newton’s Law of Gravity
• Newton’s Gravitational Law states that “……“the force of attraction between two bodies is
directly proportional to the mass of the two
bodies and inversely proportional to the
square of the distance between the twobodies”
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The Gravity Model (GM)
• Gravity model used in transportation planning
predicts that relative number of trips made
between two TAZs is a direct function of the
Ps & As in the two TAZs and inversely related
to the separation of the two TAZs
• TAZs with greater numbers of Ps and As
exchange more trips; TAZs that are fartherfrom each other exchange fewer trips
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Mathematical representation
T ij = number of trips from zone i to zone j
Pi = trip production in TAZ i
A j = trip attraction in TAZ j
f ij = Friction between TAZs i & j (rep. spatial separation between 2 TAZs)
k ij = Optional socioeconomic or trip distribution adjustment factor for
interchange between the two TAZs
m
n
ijij j
ijij j
iij
k f A
k f A P T
1
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What does it mean?
• GM states that the trips produced in TAZ i –> Pi
• Will be distributed to each other TAZ j –> T ij
• According to the relative attractiveness of each TAZ j –>
• A j / ∑A j
• And the relative accessibility of each TAZ j –> f ij / ∑f ij
• And the relative socioeconomics of each TAZ j –> k ij / ∑k ij
• And that’s it!
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Mathematical Expressions :
Gravity Models• First and simplest form (1955)
T ij = Pi P j d ij 2
T is number of trips between two zones i and j
P is population
d is distance
is proportionality factorUsed to explain shopping trips
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Generalizations of the First Gravity
ModelT ij = Oi D j d ij
n
O is productions
D is attractionsn is exponent that must be calibrated
T ij = Oi D j f(cij )
f(cij) is generalized function of travel costs
It is called friction factor, deterrence function, orimpedance – captures disincentive to travel
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Gravity Model
• Distance is replaced by impedance, i.e., thedifficulty in traveling between two zones
• Fn of travel time, travel distance, cost
• Generalized cost function:C ij = a1 t ij + a2 F ij + a3 Pij
Cij = Cost of traveling between i and jtij= Travel times (in-vehicle, out-of-vehicle)
Fij= Out-of-pocket costsPij= Terminal or parking costsai = Parameters
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“Deterrence” Functions
• Exponential
f(c ij ) =
exp(-c*c ij )
• Power
f(c ij ) = c ij - b
• Combined
f(c ij ) =
c ij - b exp(-c*c ij ) 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1 2 3 4 5 6 7 8 9 10
D e t e r a n c e
Cost
0.1
2
0.5 & 0.1
Exponential
Combined
Power
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Trip Distribution: Process
A. GIVEN: Impedance matrix (Auto/Transit travel times)
To
From 1 2 3 4 5
__________________________________
1 5.00 7.60 13.98 16.57 18.052 9.42 5.00 9.98 12.56 14.05
3 17.64 15.15 5.00 11.55 14.25
4 21.68 15.86 13.94 5.00 13.16
5 22.18 17.50 18.21 13.16 5.00
___________________________________B. and EITHER an impedance function
f(C ij ) = a C ij -b . e (-c(cij)) where a > 0 and c >= 0
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Process
Calibration of gravity model—NCHRP 365, 1995
Trip purpose a b c
___________________________________________________
HBW 28507 0.020 0.123HBO 139173 1.285 0.094
NHB 219113 1.332 0.010
___________________________________________________
For example: f(C ij ) = 28507 * C ij (-0.020) * e (-0.123 (cij))
OR a friction factor lookup table
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ProcessC. Obtain “accessibility” for the region
To
From 1 2 3 4 5
____________________________________________________
1 14923.85 10743.81 4845.38 3513.04 2920.64
2 8553.85 14923.85 ...
3 …
4
5 ____________________________________________________
D. AND use predicted P & A (from trip generation)
= 28507 * 5 (-0.020) * exp (-0.123 (5))
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Process
D.AND use predicted P & A (from trip generation)
Zone AVINC AVHHS AVCAR Pi Aj
____________________________________________________1 22.9 1.9 0.7 7331 58119
2 36.4 1.7 0.8 24666 15455
3 42.6 34451 7848
4 14933 7259
5 30284 22984 ____________________________________________________
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Process
E. OUTPUT trip table (flows) from the gravity model (origin-
constrained)
To
From 1 2 3 4 5 Pi ____________________________________________________
1 5462 1046 239 161 423 7331
2 13009 6034 1638 1098 2888 24666
3 ... 34451
4 149335 30284
____________________________________________________
Aj 58119 15455 7848 7259 22984 111665
= 7331 [(58119*14923)/ (58119*14923)+(15455*10743.81)+…)]
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Process
E. OUTPUT trip table (flows) from the gravity model
To
From 1 2 3 4 5 Pi
____________________________________________________1 6107.66 732.60 366.74 73.96 50.04 7331
2 16211.14 4712.48 2797.77 563.54 381.07 24666
3 ... 34451
4 14933
5 30284 ____________________________________________________
Aj 58119 15455 7848 7259 22984 111665
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Origin-Constrained Example
T iJ = Oi D J f(ciJ ) Σ J D J f(ciJ ) O3 = 602. Cij and D j are given.
cij 2-3 4-5 6-7 8-10 11-15 16-20 21-25 26-30 31-40
f(cij) 87 45 29 18 10 6 4 3 2
T32 = 602 * 15399 26473 = 350Zone Attraction c3j f(c3j) D j f(c3j) T3j
Fr To D j (Min)
3 1 1080 20 6 6480 147
3 2 531 7 29 15399 350
3 3 76 5 45 3420 78
3 4 47 10 18 846 19
3 5 82 25 4 328 8
SUM 26473 602
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Two Contemporary Models
T ij = Pi [A j f(cij )] [Σ j A j f(cij )]
T ij = Ai [P j f(cij )] [Σ i Pi f(cij )]
First is origin-constrained: total number of trips predicted
to originate in zone i equals productions Oi.
Second is destination-constrained: total number of tripspredicted to end in zone j equals attractions D j.
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Two Contemporary Models
T ij = Oi D j f(cij ) k ij Σ j D j f(cij ) k ij
T ij = Oi D j f(cij ) k ij Σ i Oi f(cij ) k ij
kij is adjustment factor – “k factor” – that helps with
model calibration. K factors may be set to 1.
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Homework
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