urban transport: 4401 - gravity model
DESCRIPTION
Urban transport 4401 notes/homeworkUNSW Concise notes for upcoming transport students in their 4th year of university.This lecture proceeds to talk about the gravity model and its use in transportation models todayTRANSCRIPT
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
Gravity Model Example
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 Example
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
Gravity Model Calculations for Example
Zone j Aj t1j F1j AjF1j AjF1j/Sum(AjF1j) V1j
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
Gravity Model Calculations for 4-Zone Ex.
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?
Gravity Model Calculations for Example
Zone j Aj t1j F1j AjF1j AjF1j/Sum(AjF1j) V1j
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
Gravity Model Calculations for 4-Zone Ex.
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
Gravity Model Calculations for Example
Zone j Aj t1j F1j AjF1j AjF1j/Sum(AjF1j) V1j
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
Gravity Model Example
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