Transportation - gravity model

1. Advanced
Transportation Engineering
The Four-Step Model:
II. Trip Distribution

2. The
Gravity
Model

3. The Gravity Model
The bigger the friction factor,
the more trips that are encouraged
Tij = Qij =
Pi Aj FijKij
ΣAzFizKi
j
Fij = 1 / Wc
ij
(Productions)(Attractions)(Friction Factor)
Sum of the (Attractions x Friction Factors) of the Zones
=
= Pipij
& ln F = - c ln W

4. Some of the Variables
Tij = Qij = Trips Volume between i & j
Fij =1/Wc
ij = Friction Factor
Wij = Generalized Cost (including travel time, cost)
c = Calibration Constant
pij = Probability that trip i will be attracted to zone j
kij = Socioeconomic Adjustment Factor

5. To Apply the Gravity Model
What we need…
1. Productions, {Pi}
2. Attractions, {Aj}
3. Skim Tables {Wij)
 Target-Year Interzonal Impedances

6. Gravity Model Example 8.2
 Given:
 Target-year Productions, {Pi}
 Relative Attractiveness of Zones, {Aj}
 Skim Table, {Wij}
 Calibration Factor, c = 2.0
 Socioeconomic Adjustment Factor, K = 1.0
 Find:
 Trip Interchanges, {Qij}

7. Attractions vs. Attractiveness
 The number of attractions to a particular zone
depends upon the zone’s attractiveness
 As compared to the attractiveness of all the other
competing zones and
 The distance between them
 Two zones with identical attractiveness may
have a different number of attractions due to one’s
remote location
 Thus, substituting attractions for attractiveness
can lead to incorrect results

8. Σ(AzFizKij)
Calculate Friction Factors, {Fij}
TAZ Productions
1 1500
2 0
3 2600
4 0
Σ 4100
TAZ "Attractiveness"
1 0
2 3
3 2
4 5
Σ 10
TAZ 1 2 3 4
1 5 10 15 20
2 10 5 10 15
3 15 10 5 10
4 20 15 10 5
Calibration Factor
c = 2.0
Socioeconomic Adj. Factor
K = 1.0
TAZ 1 2 3 4
1 0.0400 0.0100 0.0044 0.0025
2 0.0100 0.0400 0.0100 0.0044
3 0.0044 0.0100 0.0400 0.0100
4 0.0025 0.0044 0.0100 0.0400
TAZ 1 2 3 4 Σ
1 0.0000 0.0300 0.0089 0.0125 0.0514
2 0.0000 0.1200 0.0200 0.0222 0.1622
3 0.0000 0.0300 0.0800 0.0500 0.1600
4 0.0000 0.0133 0.0200 0.2000 0.2333
TAZ 1 2 3 4
1 0.0000 0.5838 0.1730 0.2432
2 0.0000 0.7397 0.1233 0.1370
3 0.0000 0.1875 0.5000 0.3125
4 0.0000 0.0571 0.0857 0.8571
TAZ 1 2 3 4 Σ
1 0 876 259 365 1500
2 0 0 0 0 0
3 0 488 1300 813 2600
4 0 0 0 0 0
Σ 0 1363 1559 1177 4100
Target-Year Inter-zonal
Impedances, {Wij}
F11=
1
52 = 0.04
AjFijKij=A4F34K34= (5)(0.01)(1.0)
= 0.05
Given…
Find Denominator of Gravity Model Equation {AjFijKij}
Find Probability that Trip i will be attracted to Zone j, {pij}
Find Trip Interchanges, {Qij}
pij =
AjFijKij
0.16
=
0.05
= 0.3125
Qij = Pipij = (2600)(0.3125) = 813
Fij =
1
Wc
ij
=

9. Keep in mind that the socioeconomic
factor, K, can be a matrix of value
rather than just one value
TAZ 1 2 3 4
1 1.4 1.2 1.7 1.9
2 1.2 1.1 1.1 1.4
3 1.7 1.1 1.5 1.3
4 1.9 1.4 1.3 1.6

10. Calibration of
the Gravity Model
 When we talk about calibrating the
gravity model, we are referring to
determining the numerical value c
 The reason we do this is to fix the
relationship between the travel-time
factor and the inter-zonal impedance

11. Calibration of
the Gravity Model
 Calibration is an iterative process
 We first assume a value of c and then use:
Qij = Pi
Aj Fij
Σ(Ax Fix)
[ ]
Qij = Tij = Trips Volume between i & j
Fij =1 / Wc
ij = Friction Factor
Wij = Generalized Cost (including travel time, cost)
c = Calibration Constant

12. Calibration of
the Gravity Model
 The results are then compared with the
observed values during the base year
 If the values are sufficiently close, keep c
 The results are expressed in terms of the
appropriate equation relating F and W with c
 If not, then adjust c and redo the procedure

13. Trip-Length Frequency Distribution
 Compares the observed and computed
Qij values

14. Gravity Model
Calibration Example
Given:
Zone 2
Zone 3
Zone 4
Zone 5
Zone 1
5 TAZ City
TAZ Productions
1 500
2 1000
3 0
4 0
5 0
Σ 1500
TAZ "Attractiveness"
1 0
2 0
3 2
4 3
5 5
Σ 10
TAZ 3 4 5
1 5 10 15
2 10 5 15
Target-Year Inter-zonal Impedances, {Wij}
5
5
10
10
15
15
Find: c and Kij to fit the base data
TAZ 3 4 5
1 300 150 50
2 180 600 220
Base-Year Trip Interchange Volumes, {tij}

15. Relating F & W
 Starting with:
 Take the natural log of both sides
 Now c is the slope of a straight
line relating ln F and ln W
Fij =
1
Wc
ij
ln F = - c ln W
ln
F
ln W

16. Trip-Length Frequency Distribution
 Group zone pairs by max Wij
 Sum Interchange Volumes for each set of zone pairs
 Find f = (Σtix) / (Σtij)
TAZ 3 4 5
1 5 10 15
2 10 5 15
Target-Year Inter-zonal Impedances, {Wij}
TAZ 3 4 5
1 300 150 50
2 180 600 220
Base-Year Trip Interchange Volumes, {tij}
W Σtij f
5 300 600 900 0.60
10 150 180 330 0.22
15 50 220 270 0.18
Σ 1500 1.00
tij Values
Zone Pairs
13, 24
14, 23
15, 25

17. Let’s assume c = 2.0
TAZ 3 4 5
1 5 10 15
2 10 5 15
Target-Year Inter-zonal
Impedances, {Wij} Base-Year Trip Interchange Volumes, {tij}
First Iteration
TAZ 3 4 5 Σ
1 300 150 50 500
2 180 600 220 1000
Friction Factor, {Fij} with c = 2.0
TAZ 3 4 5
1 0.040 0.010 0.004
2 0.010 0.040 0.004
Fij =
1
Wc
ij
= F13=
1
52 = 0.04

18. c = 2.0
First Iteration
Aj Fij
TAZ 3 4 5 Σ
1 0.605 0.227 0.168 1.000
2 0.123 0.740 0.137 1.000
TAZ 3 4 5 Σ
1 0.080 0.030 0.022 0.132
2 0.020 0.120 0.022 0.162
Aj Fij
Σ(Ax Fix)
[ ]
TAZ "Attractiveness"
1 0
2 0
3 2
4 3
5 5
Σ 10
Friction Factor, {Fij} with c = 2.0
TAZ 3 4 5
1 0.040 0.010 0.004
2 0.010 0.040 0.004

19. c = 2.0
First Iteration
TAZ 3 4 5 Σ
1 0.605 0.227 0.168 1.000
2 0.123 0.740 0.137 1.000
TAZ 3 4 5 Σ
1 303 113 84 500
2 123 740 137 1000
W Σtij f
5 303 740 1042.2 0.69
10 113 123 236.73 0.16
15 84 137 221.02 0.15
Σ
1 5 0 0 1 . 0 0
15, 25
Zone Pairs tij Values
13, 24
14, 23
Aj Fij
Σ(Ax Fix)
[ ]
Qij = Pi
Aj Fij
Σ(Ax Fix)
[ ]
TAZ Productions
1 500
2 1000
3 0
4 0
5 0
Σ 1500

20. Trip-Length FrequencyDistribution
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
5 10 15
Base Year Iteration 2 (c=1.5)

21. Let’s assume c = 1.5
TAZ 3 4 5
1 5 10 15
2 10 5 15
Target-Year Inter-zonal
Impedances, {Wij} Base-Year Trip Interchange Volumes, {tij}
Second Iteration
TAZ 3 4 5 Σ
1 300 150 50 500
2 180 600 220 1000
Friction Factor, {Fij} with c = 1.5
Fij =
1
Wc
ij
= F13=
1
51.5 = 0.089
TAZ 3 4 5
1 0.089 0.032 0.017
2 0.032 0.089 0.017

22. c = 1.5
Second Iteration
Aj Fij
Aj Fij
Σ(Ax Fix)
[ ]
TAZ "Attractiveness"
1 0
2 0
3 2
4 3
5 5
Σ 10
Friction Factor, {Fij} with c = 1.5
TAZ 3 4 5
1 0.089 0.032 0.017
2 0.032 0.089 0.017
TAZ 3 4 5 Σ
1 0.179 0.095 0.086 0.360
2 0.063 0.268 0.086 0.418
TAZ 3 4 5 Σ
1 0.497 0.264 0.239 1.000
2 0.151 0.642 0.206 1.000

23. c = 1.5
Second Iteration
Aj Fij
Σ(Ax Fix)
[ ]
Qij = Pi
Aj Fij
Σ(Ax Fix)
[ ]
TAZ Productions
1 500
2 1000
3 0
4 0
5 0
Σ 1500
TAZ 3 4 5 Σ
1 0.497 0.264 0.239 1.000
2 0.151 0.642 0.206 1.000
TAZ 3 4 5 Σ
1 249 132 120 500
2 151 642 206 1000
W Σtij f
5 249 642 891.06 0.59
10 132 151 283.26 0.19
15 120 206 325.67 0.22
Σ
1 5 0 0 1 . 0 0
Zone Pairs tij Values
13, 24
14, 23
15, 25

24. Trip-Length FrequencyDistribution
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
5 10 15
Base Year Iteration 1 (c=2.0) Iteration 2 (c=1.5)

25. K Factors
 Even after calibration, there will typically still be
discrepancies between the observed & calculated data
 To “fine-tune” the model, some employ socioeconomic
adjustment factors, also known as K-Factors
 The intent is to capture special local conditions between some
zonal pairs such as the need to cross a river
Kij = Rij
1-Xi
1 - XiRij
Rij = ratio of observed to calculated Qij (or Tij)
Xi = ratio of the base-year Qij to Pi (total productions of zone i)

26. K Factor Example
Rij =
Observed Qij
Calculated Qij
= R13 =
300
249
= 1.20
Xi =
Base-Year Qij
Pi
= X1 =
300
500
= 0.60
Kij = Rij
1-Xi
1 - XiRij
= K13 = 1.20
1-0.6
1–(0.6)(1.2)
= 1.71

27. The Problem with K-Factors
 Although K-Factors may improve the model
in the base year, they assume that these
special conditions will carry over to future
years and scenarios
 This limits model sensitivity and undermines the
model’s ability to predict future travel behavior
 The need for K-factors often is a symptom
of other model problems.
 Additionally, the use of K-factors makes it more
difficult to figure out the real problems

28. Limitations of
the Gravity Model
 Too much of a reliance on K-Factors in
calibration
 External trips and intrazonal trips cause
difficulties
 The skim table impedance factors are often too
simplistic to be realistic
 Typically based solely upon vehicle travel times
 At most, this might include tolls and parking costs
 Almost always fails to take into account how things
such as good transit and walkable neighborhoods affect
trip distribution
 No obvious connection to behavioral decision-making