Kuliah 3 - Perencanaan Transportasi

8/19/2019 Kuliah 3 - Perencanaan Transportasi

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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

Kuliah 3 - Perencanaan Transportasi

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