Urban Transport: 4401 - Gravity Model

CVEN 4401/9405

Trip Distribution: Gravity Model Introduction

Dr. Lauren Gardner

Disadvantages of Growth Factor Models Advantages

Simple

No LOS information needed

Disadvantages

May break down mathematically when a new zone is added (e.g. housing development is built after base year)

Convergence to the target-year generation totals is not always possible

The model is not sensitive to impedance (No project/policy effect)

No congestion impact

Generally only used when Gravity model, or more sophisticated models are unavailable

Gravity Model Analogue to Newton’s Gravity Model:

“The force of attraction between two bodies is directly proportional

to the product of the masses of the two bodies and inversely

proportional to the square of the distance between them”

Has been applied to many situations involving human interaction

Volume of long distance phone calls

When applied to trip distribution the functional form is:

2

21

r

mmkF

c

ij

ji

ijW

APkV

Gravity Model

Function states the interchange volume between a trip-

producing zone I and a trip-attracting zone J is:

Directly proportional to the magnitude of the trip productions

of the zone I and trip attractions of zone J

Inversely proportional to a function of the impedance, Wijc

between the zones.

Independent variable: Vij

Dependent variables: productions, attractions, impedences

Parameters: k and c

Need to be calibrated using base year data

Gravity Model Assumptions

A trip produced in zone i is more likely to be attracted to an attraction zone that has a higher number of attractions

A trip produced in zone i is more likely to be attracted to an attraction zone that is closer to zone i (when Fij is a function of travel time)

Fij = 1/Wijc

Note: Circled part would not be affected if all attraction terms were multiplied by a constant factor

j

ijj

ijj

iijFA

FAPV Relative attractiveness of zone j

compared to all other zones

Friction Factor Friction factor (or travel time factor)

Fij is a measure of the impedance from i to j

Friction Factor Parameters: a and b are estimated through calibration

Fij is often estimated as a function of only travel time

Does changing the value of a change the results?

What behaviour does a larger b value mean?

b

ij

ijat

F1

Socioeconomic Adjustment Factors

Kij incorporates the effects not captured by the limited

number of independent variables in the model

Model Parameter k is estimated through calibration

iji

j

ijijj

ijijj

iij pPKFA

KFAPV

Relative attractiveness of zone j

compared to all other zones

Gravity Model Example 1

Cities A B C D

Productions 4724 901 193 108

Attractions 4909 774 174 69

Transport planners want to know how many people from town B go shopping in

each region

Production and Attractions for Shopping Trips

O/D A B C D

A 7 35 45 40

B 35 5 20 12

C 45 20 3 8

D 40 12 8 2

Average Travel Times for Region in minutes

Gravity Model Example

Use the travel demand model:

First need to calculate friction factors:

Where a=1000; b=2

F21 = 1000/352 = 0.816

F22 = 1000/52 = 40

F23 = 1000/202 = 2.5

F24 = 1000/122 = 6.944

b

ijij atF

j

ijj

ijj

iijFA

FAPV

TT from B to: A (1) B (2) C (3) D (4)

35 5 20 12


Now Calculate the trip interchanges, Vij using the P-A table

F21 = 0.816; F22 = 40; F23 = 2.5; F24 = 6.944

If we decrease a from 1000 to 100 will that impact the resulting Vij values?

Cities A (1) B (2) C (3) D (4)

Productions 4724 901 193 108

Attractions 4909 774 174 69

101)944.6*69()5.2*174()40*774()816.0*4909(

816.0*490990121

244233222211

2112

2

211221

V

FAFAFAFA

FAP

FA

FAPV

j

jj


Gravity Model Calculations for Example

Zone j Aj t2j F2j AjF2j AjF2j/Sum(AjF2j) V2j

A (1) 4909 35 0.816 4007.3 0.112 101

B (2) 774 5 40 30960 0.863 777

C (3) 174 20 2.5 435 0.012 11

D (4) 69 12 6.994 479.2 0.013 12

5926 35881.5 1 901

j

ijj

ijj

iijFA

FAPV

If we increase b from -2 to -1 will that impact the resulting Vij values?

Friction Factor Parameters

The function could overestimate or underestimate the number of short trips

If the results from the previous example overestimate the number of trips that stay in region B (TT=5), while underestimating the number of trips between B and A (TT=35) should be b parameter be increased or decreased?

What form of friction factor should be used? Need to calibrate

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20 25

Fri

cti

on

Fac

tor

Travel Time

b=-0.5

b=-1

b=-2

Friction Factor Parameters Consider the following example when short trips are not the most common

trip length in the travel data analysed

How many of the 100 trips produced in zone 1 will find destinations in each zone: 1,2,3,4?

If travel time doesn’t matter where will all the trips go?

If travel time is the most important factor where will all the trips go?

Zone 1 Zone 2

Zone 4Zone 3

P1 = 100

A1 = 50A2 = 200

A3 = 75 A4 = 675

j TT1j

1 2

2 5

3 10

4 15

Gravity Model Calculations for 4-Zone Ex.

Assume Fij = 500*tij-2

What is the problem with these results?

V(11) > A1: Can not have more trip interchanges than attractions

Potential Causes:

P and A predictions are incorrect

Friction Factor form is incorrect

Overestimating short trips and underestimating long trips



A (1) 50 2 125 6250 0.515 52

B (2) 200 5 20 4000 0.330 33

C (3) 75 10 5 375 0.031 3

D (4) 675 15 2.222 1500 0.124 12

1000 12125 1 901


Change b=-1; so Fij = 500*tij-1

What should the friction factor form be? What is the shape of the expected trip length distribution for trips made my

automobile in a given region?

Are 5 mile trips more common than 3 mile trips?

Are short trips (a few blocks) more common than 1-2 mile trips?



A (1) 50 2 250 12500 0.213 21

B (2) 200 5 100 20000 0.340 34

C (3) 75 10 50 3750 0.064 6

D (4) 675 15 33.33 22500 0.383 38

1000 58750 1 100

Trip Length Distribution Curves It is well known that very short trips are not normally the

most frequent vehicle trips

0

0.5

1

1.5

2

2.5

0 5 10 15 20 25 30

Fri

cti

on

Fac

tor

Travel Time

t^(-2)

t^(b)*exp(ct)

atijb

atijbect(ij)

Tanner Function

Monotonically decreasing


Use Tanner function: Change b=2, c=-0.5; so Fij = 500*tij2 e-0.5t

Most trips now go to the second closest zone

Will the Vij values change if a is increased or decreased?

Sometimes intrazonal trips, Tii require a separate models



A (1) 50 2 735.76 36787.94 0.119 12

B (2) 200 5 1026.06 205212.50 0.664 66

C (3) 75 10 336.90 25267.30 0.082 8

D (4) 675 15 62.22 41999.84 0.136 14

1000 309267.58

6

1 100


Cities A B C D

Productions 793 1143 5803 6583

Attractions 2527 3627 1757 1497

Production and Attractions for Shopping Trips

ISSUE:

∑P ≠ ∑A (i.e. 14322 ≠ 9408)

SOLUTION:

Planners have more confidence in their Production predictions so multiply all Aj values by

(∑P/∑A)

Cities A B C D

Productions 793 1143 5803 6583

Attractions 3847 5521 2675 2279

Balanced Production and Attractions for Shopping Trips

Now ∑P = ∑A = 14,322

Gravity Model Example: Fill the Blanks

TT A B C D FF(ij) A B C D

A 2.4 11 8.2 6.6 A 17.36

B 11 2.1 7.4 5.5 B

C 8.2 7.4 3.4 9.9 C 8.65

D 6.6 5.5 9.9 2.3 D 2.30

A*FF A B C D Sum Vij A B C D Tot P

A 0 A 793

B 0 7534 B 1143

C 0 C 1831 5803

D 18253 0 29813 D 6583

Tot A 4014 7521 1190 1598 14322

Cities A B C D

Productions 793 1143 5803 6583

Attractions 3847 5521 2675 2279

Balanced Production and Attractions for Shopping Trips

Travel Times Compute FF=100/t^2

Compute A*FF Compute Vij

Gravity Model ExampleTT A B C D FF(ij) A B C D

A 2.4 11 8.2 6.6 A 17.36 0.83 1.49 2.30

B 11 2.1 7.4 5.5 B 0.83 22.68 1.83 3.31

C 8.2 7.4 3.4 9.9 C 1.49 1.83 8.65 1.02

D 6.6 5.5 9.9 2.3 D 2.30 3.31 1.02 18.90

A*FF A B C D Sum Vij A B C D Tot P

A 0 4563 3978 5232 13773 A 0 263 229 301 793

B 3179 0 4884 7534 15597 B 233 0 358 552 1143

C 5721 10083 0 2325 18129 C 1831 3227 0 744 5803

D 8831 18253 2729 0 29813 D 1950 4030 603 0 6583

Tot A 4014 7521 1190 1598 14322

Cities A B C D

Productions 793 1143 5803 6583

Attractions 3847 5521 2675 2279ISSUE: Change in Attraction of zones 2-4

Need better calibrated friction factors

How to Adjust Friction Factors

1. friction factor exponent is trip lengths are not being

represented correctly

2. Use a different friction factor equation, in case the actual

trip length distribution is not monotonically decreasing

(e.g. Tanner function)

3. Look for other variables besides travel time which better

explain how travellers choose between multiple

destinations

Urban Transport: 4401 - Gravity Model

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