wes marshall, p.e. university of connecticut february 2008 ce 254 transportation engineering the...
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Wes Marshall, P.E. University of ConnecticutFebruary 2008
CE 254Transportation Engineering
The Four-Step Model:
II. Trip Distribution
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The Basic Transportation Model…1) Study Area Zones 2) Attributes of Zones
Socioeconomic Data Land Use Data “Cost” of Travel btw. Zones
3) The Road Network
Traffic Volume by Road Link Mode Splits Emissions
Inpu
tsO
utpu
ts
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What’s in the Black Box?
The Four-Step Model
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The Four-Step Modeling Process…
I. Trip Generation
II. Trip Distribution
III. Mode Choice
IV. Trip Assignment
WHY?
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The Four-Step Model
The main reason we use the four-step model is: To predict roadway traffic volumes
& traffic problems such as congestion and pollution emissions
In turn, we typically use the models to compare
several transportation alternatives
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The Four-Step Model
Originally developed in the 1950s with the interstate highway movement Since the 1950s, researchers have developed a
multitude of advanced modeling techniques
Nevertheless, most agencies still use the good ol’ four-step model
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Overview of the Four-Step Model
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Model residential trip productions and non-residential trip attractions w/ - Regression Models- Trip-Rate Analysis- Cross-Classification Models
- A matrix of trips between each TAZ… also called a “trip table”
- i.e. columns of trip productions and trip attractions
- No. of Housing Units- Office, Industrial SF
- HH Size- Income- No. of Cars
Iter
ativ
e P
roce
ss
Land Use Data
Input: Household Socioeconomic Data }Examples of HH socioeconomic data
}Examples of land use data
Output: Trip Ends by purpose
Input: Trip Ends by purpose
Output: Trip Interchanges
TRIP GENERATION
TRIP DISTRIBUTION
Process:
Survey Data
- Growth Factor Models- Not as accurate as Gravity Model- Used for external trips or short-term planning
- Gravity Model- Used for regional or long-term planning
Process:
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- i.e. traffic flows on network, ridership on transit lines
Iter
ativ
e P
roce
ssInput: Trip Interchanges
Output: Trip Table by Mode
Input: Trip Table by Mode
Output: Daily Link Traffic Volumes
MODE CHOICE
TRIP ASSIGNMENT
Finds trip interchanges between i & j for each mode- Function of Trip Maker, Journey, and Transport Facility
- Trip End Model- Mode plays role in trip ends- Typically used for small cities with little traffic and little transit- No accounting for the role that policy decisions play in mode choice
- Trip Interchance Model- Use when LOS is important, transit is a true choice, highways are congested, and parking is
limited
Process:
Allocate trips to links between nodes i & j- Function of Path to Destination and Minimum Cost (time & money)
- Identify Attractive Routes via Tree Building- Shortest Path Algorithm or Dijkstra’s Algorithm
- Assign Portions of Matrix to Routes / Tree- User Equilibrium, Heuristic Methods, Stochastic Effects w/ Logit
- Search for Convergence
Process:
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Some General Problems with the Conventional Methodology
Huge focus on vehicular traffic A transit component is typical in better models
Typically forecasts huge increases in traffic Leads to engineers building bigger roads
to accommodate “forecast” traffic Which leads to induced traffic and congestion…
right back where we started when we needed the bigger roads in the first place
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Some General Problems with the Conventional Methodology
Pedestrians and bicyclists are rarely included Level of geography is difficult
for non-motorized modes Network scale is insignificant Input variables are too limited
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Preparing for a Four-Step Model
Before jumping into trip generation, we first have to set up our project…
1) Define study area and boundaries
2) Establish the transportation network
3) Create the zones
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Defining the Study Area
3 Basic Types Regional
Statewide or a large metro area Used to predict larger patterns of traffic distribution,
growth, and emissions Corridor
Major facility such as a freeway, arterial, or transit line Used to evaluate traffic
Site or Project Proposed development or small scale change
(i.e. intersection improvement) Used to evaluate traffic impact
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Roads are represented by a series of links & nodes
Links are defined by speed and capacity Turns are allowed at nodes
LinkNode
Establish the Network
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Establish the Network
Typically only main roads and intersections are included Even collector roads are often excluded
This practice is becoming less common as the processing power of computers has increased
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Creating Zones
Create Traffic Analysis Zones (TAZ) Uniform land use Bounded by major roads Typically small in size (about the
size of a few neighborhood blocks) The State of Connecticut model has
~2,000 zones that cover 5,500 square miles and over 3.4 million people
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Creating Zones
All modeled trips begin in a zone and are destined for a zone Zones are usually large enough that most
pedestrian and bicycle trips start and end in the same zone (and thus not modeled)
Also, the typical data we collect about zones in terms of population and employment information is not enough to predict levels of walking and biking
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Trip Generation
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Trip Generation
Using socioeconomic data, we try to estimate how many trips are “produced” by each TAZ For example, we might use linear regression to
estimate that a 2-person, 2-car household with a total income of $90,000 makes 2 home-based work trips per day
Using land use data, we estimate how many trips are “attracted” to each TAZ For example, an 3,000 SF office might bring in
12 work trips per day
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Trip Generation
The process considers the total number of trips Thus, walking and biking trips have not
been officially excluded (although most models ignore them completely)
The trips are generated by trip purpose such as work or shopping Recreational or discretionary trips are
difficult to include
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Trip Generation
1
23
4
5
6
87
TAZ Productions
1 122 193 354 45 56 107 138 22
TAZ Attractions
1 92 123 44 385 456 67 48 2
Input:
Output:
Socioeconomic DataLand Use Data
Trip Ends by purpose (i.e. work) in columns of productions & attractions
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Trip Generation Trip Distribution
The question is… how do we allocate all the productions among all the attractions?
TAZ Productions
1 122 193 354 45 56 107 138 22
TAZ Attractions
1 92 123 44 385 456 67 48 2
Zone 2
Zone 3
Zone 4
Zone 5
Zone 6
Zone 7
Zone 8
Zone 1
TAZ 1 2 3 4 5 6 7 8
12345678
Trip Matrixor
Trip Table
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Trip Distribution
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Trip Distribution
We link production or origin zones to attraction destination zones
A trip matrix is produced
The cells within the trip matrix are the “trip interchanges” between zones
Zone 1
TAZ 1 2 3 4 5 6 7 8
12345678
Trip Matrix
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Trip Interchanges
Decrease with distance between zones In addition to the distance between zones,
total trip “cost” can include things such as tolls and parking costs
Increase with zone “attractiveness” Typically includes square footage of retail
or office space but can get much more complicated
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Trip Distribution
Similar to Trip Generation, all the modes are still lumped together by purpose (i.e. work, shopping) This creates a problem for non-vehicular
trips because distance affects these trips very differently
Additionally, many walking and biking trips are intra-zonal & difficult to model
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Basic Criteria for TD
Criteria for allocating all the productions among all the attractions Cost of trip
Travel TimeActual Costs
AttractivenessQuantity of OpportunityDesirability of Opportunity
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How to Distribute the Trips?
I. Growth Factor Models
II. Gravity Model
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Growth Factor Models
Growth Factor Models assume that we already have a basic trip matrix
Usually obtained from a previous study or recent survey data
TAZ 1 2 3 4
1 5 50 100 2002 50 5 100 3003 50 100 5 1004 100 200 250 20
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Growth Factor Models
The goal is then to estimate the matrix at some point in the future For example, what would the trip matrix
look like in 10 years time?
TAZ 1 2 3 4
1 5 50 100 2002 50 5 100 3003 50 100 5 1004 100 200 250 20
Trip Matrix, t (2008)
Trip Matrix, T (2018)
TAZ 1 2 3 4
1 ? ? ? ?2 ? ? ? ?3 ? ? ? ?4 ? ? ? ?
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Some of the More Popular Growth Factor Models
Uniform Growth Factor Singly-Constrained Growth Factor Average Factor Detroit Factor Fratar Method
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Uniform Growth Factor Model
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Uniform Growth Factor
Tij = τ tij for each pair i and jTij = Future Trip Matrix tij = Base-year Trip Matrix τ = General Growth Rate
i = I = Production Zone j = J = Attraction Zone
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Uniform Growth Factor
TAZ 1 2 3 4
1 5 50 100 2002 50 5 100 3003 50 100 5 1004 100 200 250 20
TAZ 1 2 3 4
1 6 60 120 2402 60 6 120 3603 60 120 6 1204 120 240 300 24
Trip Matrix, t (2008)
Trip Matrix, T (2018)
If we assume τ = 1.2, then…
Tij = τ tij
= (1.2)(5)= 6
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Uniform Growth Factor
The Uniform Growth Factor is typically used for 1 or 2 year horizons
However, assuming that trips growat a standard uniform rate is a fundamentally flawed concept
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Singly-Constrained Growth Factor Model
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Singly-Constrained Growth Factor Method
Similar to the Uniform Growth Factor Method but constrained in one direction For example, let’s start with our base
matrix, t…
TAZ 1 2 3 4
1 5 50 100 2002 50 5 100 3003 50 100 5 1004 100 200 250 20
attractions, j
prod
uctio
ns, i
zones
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Singly-Constrained Growth Factor Method
Instead of one uniform growth factor, assume that we have estimated how many more or less trips will start from our origins…
Now all we have to do is multiply each row by the ratio of (Target Pi) / (Σj)
TAZ 1 2 3 4 Σj Target Pi
1 5 50 100 200 355 4002 50 5 100 300 455 4603 50 100 5 100 255 4004 100 200 250 20 570 702
Σi 205 355 455 620 1635 1962
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Singly-Constrained Growth Factor Method
Tij = tij (Target Pi) / (Σj) = 5 (400 / 355)= 5.6
TAZ 1 2 3 4 Σj Target Pi
1 5.6 56.3 112.7 225.4 400 4002 50.5 5.1 101.1 303.3 460 4603 78.4 156.9 7.8 156.9 400 4004 123.2 246.3 307.9 24.6 702 702
Σi 257.7 464.6 529.5 710.2 1962 1962
TAZ 1 2 3 4 Σj Target Pi
1 5 50 100 200 355 4002 50 5 100 300 455 4603 50 100 5 100 255 4004 100 200 250 20 570 702
Σi 205 355 455 620 1635 1962
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Singly-Constrained Growth Factor Method
Can also perform the singly-constrained growth factor method for a destination specific future trip table By multiplying each column
by the ratio of (Target Aj) / (Σi)TAZ 1 2 3 4 Σj
1 5 50 100 200 3552 50 5 100 300 4553 50 100 5 100 2554 100 200 250 20 570
Σi 205 355 455 620 1635
Target Aj 180 406 380 740 1706
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Overview of the Singly-Constrained Growth Factor Methodology
One of the simplest trip distribution techniques Used with existing trip table & future trip ends Typically, we balance flows after processing
This means that the total number of productions equals the total number of attractions (or in terms of origins & destinations)
Tij = Tji
But there are more advanced growth factor models…
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Average Growth Factor Model
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Average Growth Factor Function
F = Growth Factor = Ratio of Target Trips to
Previous Iteration Trips k = Iteration Number
g ( )=Fi
k + Fjk
2
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The Basic Steps…1) Collect Inputs
Matrix of Existing Trips, {tij} Vector of Future Trips Ends, {Ti}
2) Compute Growth Factor for each zone
3) Compute Inter-zonal Flows
4) Compute Trips Ends
5) If tik = Ti for each zone i, then stop…
otherwise, go back to “Step 1”
Fik=
ΣTi
Σtik-1 =
Target Trip End
Previous Iteration Trip End
tijk = tij
k-1 [g(Fik, Fj
k, …)] for each ij pair
tik = Σtij
k for each zone i
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Growth Factor Models: Average Factor Example
Iteration 1: Fij(1) = [Fi(1) + Fj(1)] / 21 2 3 4
1 1.546 1.606 1.285 1.1292 1.466 1.526 1.205 1.0493 1.711 1.771 1.450 1.2944 1.591 1.651 1.330 1.174
1.330=1.41+1.25
2
g ( )=Fi
k + Fjk
2g ( )=
Fik + Fj
k
2
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Iteration 2: Fij(2) = [Fi(2) + Fj(2)] / 21 2 3 4
1 1.099 1.119 1.051 0.9752 1.059 1.079 1.011 0.9353 1.049 1.069 1.001 0.9254 0.994 1.014 0.946 0.870
tij(2) = tij(1) * Fij(2)
1 2 3 4 ti(2) Ti Fi(3) = Ti / ti(2)
1 0.00 107.78 371.57 628.61 1108 1200 1.082 77.65 0.00 499.73 434.55 1012 1050 1.043 220.83 115.43 0.00 56.26 393 380 0.974 324.31 443.43 94.41 0.00 862 770 0.89
tj(2) 623 667 966 1119
Tj 670 730 950 995
Fj(2) = Tj / tj(1) 1.08 1.10 0.98 0.89
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tij(2) = tij(1) * Fij(2)
1 2 3 4 ti(2) Ti Fi(3) = Ti / ti(2)
1 0.00 107.78 371.57 628.61 1108 1200 1.082 77.65 0.00 499.73 434.55 1012 1050 1.043 220.83 115.43 0.00 56.26 393 380 0.974 324.31 443.43 94.41 0.00 862 770 0.89
tj(2) 623 667 966 1119
Tj 670 730 950 995
Fj(2) = Tj / tj(1) 1.08 1.10 0.98 0.89
Iteration 3: Fij(3) = [Fi(3) + Fj(3)] / 21 2 3 4
1 1.078 1.088 1.032 0.9842 1.058 1.068 1.012 0.9643 1.023 1.033 0.977 0.9294 0.983 0.993 0.937 0.889
tij(3) = tij(2) * Fij(3)
1 2 3 4 ti(3) Ti Fi(4) = Ti / ti(3)
1 0.00 117.21 383.41 618.82 1119 1200 1.072 82.15 0.00 505.66 419.09 1007 1050 1.043 225.89 119.19 0.00 52.29 397 380 0.964 318.76 440.11 88.45 0.00 847 770 0.91
tj(3) 627 677 978 1090
Tj 670 730 950 995
Fj(4) = Tj / tj(3) 1.07 1.08 0.97 0.91
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Fratar Method Growth Factor Model
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Fratar Method
tijk=Ti
tjik-1 Fj
k
Σtizk-1Fz
kReplace Step 2 with…
Tijk=Tji
k=tij
k + tjik
2Balance Matrix with…
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Growth Factor Models: Fratar Method Example
Fi(1) =Ti
ti(0)Future TripsCurrent Trips=
Σtizk-1Fz
k = (t110)(F1
1)+(t120)(F2
1)+(t130)(F3
1)+(t140)(F4
1) =(0)(2.09) + (25)(1.30) + (10)(1.59) +(20)(1.46) = 78
=11555 = 2.09
Σtizk-1Fz
k = (t210)(F1
1)+(t220)(F2
1)+(t230)(F3
1)+(t240)(F4
1) =(25)(2.09) + (0)(1.30) + (60)(1.59) +(30)(1.46) = 191
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Growth Factor Models: Fratar Method Example
Iteration 1: tij(1) = tij(0) Ti Fj(1) / Σ[tiz(0) Fz(1)]
1 2 3 4
1 0 48 24 432 41 0 75 343 23 87 0 244 38 35 22 0
t431 = (t34
0)(T4)(F31)/Σ[(t4z
0)(Fz1)]
= (15)(95)(1.59) / (105) = 22
t121 = (t21
0)(T1)(F21)/Σ[(t1z
0)(Fz1)]
= (25)(115)(1.30) / (78) = 48
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Balance Flows with [tij(1) + tji(1)] / 2
1 2 3 4 ti(1) Ti Fi(2) Σtiz(1) Fz(2)
1 0 45 23 41 109 115 1.06 1062 45 0 81 35 160 150 0.94 1673 23 81 0 23 127 135 1.06 1234 41 35 23 0 99 95 0.96 100
Iteration 1: tij(1) = tij(0) Ti Fj(1) / Σ[tiz(0) Fz(1)]
1 2 3 4
1 0 48 24 432 41 0 75 343 23 87 0 244 38 35 22 0
Balance Flows = (tij1 + tji
1) / 2 = (22 + 24) / 2 = 23
Now onto Iteration 2…
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Balance Flows with [tij(1) + tji(1)] / 2
1 2 3 4 ti(1) Ti Fi(2) Σtiz(1) Fz(2)
1 0 45 23 41 109 115 1.06 1062 45 0 81 35 160 150 0.94 1673 23 81 0 23 127 135 1.06 1234 41 35 23 0 99 95 0.96 100
Iteration 2: tij(2) = tij(1) Ti Fj(2) / Σ[tiz(1) Fz(2)]
1 2 3 4
1 0 46 27 422 43 0 77 303 27 83 0 244 41 31 23 0
Iteration 2…
Balance Flows… & Compute Values
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Iteration 3…
Balance Flows…
Iteration 3: tij(3) = tij(2) Ti Fj(3) / Σ[tiz(2) Fz(3)]
1 2 3 4
1 0 44 29 422 43 0 79 293 29 81 0 254 42 29 24 0
& Compute Values
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Limitations of the Fratar Model
Breaks down mathematically with a new zone Convergence to the target year not always possible The model does not reflect travel times or cost of
travel between zones Thus, this model as well as the other growth factor
models are only used for External trips through the zones or Short-term horizon years
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The Gravity Model
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The big idea behind the gravity model is Newton’s law of gravitation…
The force of attraction between 2 bodies is directly proportional to the product of masses between the two bodies and inversely proportional to the square of the distance
The Inspiration for the Gravity Model
F = kM1 M2
r2
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The Inspiration for the Gravity Model
In terms of transportation planning and trip distribution: The zones correspond to the objects The attributes of the zones in terms of the
relative proportion of productions and attractions represent the mass of the objects
The distance between the zones is captured by the distance between the objects
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Some of the Variables
Tij = Qij = Trips Volume between i & j
Fij =1/Wcij = 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
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The Gravity Model
The bigger the friction factor, the more trips that are encouraged
Tij = Qij =Pi Aj FijKij
ΣAzFizKij
Fij = 1 / Wcij
(Productions)(Attractions)(Friction Factor)Sum of the (Attractions x Friction Factors) of the Zones=
= Pipij
& ln F = - c ln W
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2 Types of Gravity Models
1. Parametric Fits equation to curve
2. Non-parametric Uses look-up table for bars
Time (min)
% T
rips
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To Apply the Gravity Model
What we need…
1. Productions, {Pi}
2. Attractions, {Aj}
3. Skim Tables {Wij) Target-Year Interzonal Impedances
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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}
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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
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Σ(AzFizKij)
Calculate Friction Factors, {Fij}
TAZ Productions
1 15002 03 26004 0Σ 4100
TAZ "Attractiveness"
1 02 33 24 5Σ 10
TAZ 1 2 3 4
1 5 10 15 202 10 5 10 153 15 10 5 104 20 15 10 5
Calibration Factorc = 2.0
Socioeconomic Adj. FactorK = 1.0
TAZ 1 2 3 4
1 0.0400 0.0100 0.0044 0.00252 0.0100 0.0400 0.0100 0.00443 0.0044 0.0100 0.0400 0.01004 0.0025 0.0044 0.0100 0.0400
TAZ 1 2 3 4 Σ
1 0.0000 0.0300 0.0089 0.0125 0.05142 0.0000 0.1200 0.0200 0.0222 0.16223 0.0000 0.0300 0.0800 0.0500 0.16004 0.0000 0.0133 0.0200 0.2000 0.2333
TAZ 1 2 3 4
1 0.0000 0.5838 0.1730 0.24322 0.0000 0.7397 0.1233 0.13703 0.0000 0.1875 0.5000 0.31254 0.0000 0.0571 0.0857 0.8571
TAZ 1 2 3 4 Σ
1 0 876 259 365 15002 0 0 0 0 03 0 488 1300 813 26004 0 0 0 0 0Σ 0 1363 1559 1177 4100
Target-Year Inter-zonal Impedances, {Wij}
F11=152 = 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
Wcij
=
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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 1.92 1.2 1.1 1.1 1.43 1.7 1.1 1.5 1.34 1.9 1.4 1.3 1.6
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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
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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
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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
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Trip-Length Frequency Distribution
Compares the observed and computed Qij values
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Gravity ModelCalibration Example
Given:
Zone 2
Zone 3
Zone 4
Zone 5
Zone 1
5 TAZ City
TAZ Productions
1 5002 10003 04 05 0Σ 1500
TAZ "Attractiveness"
1 02 03 24 35 5Σ 10
TAZ 3 4 5
1 5 10 152 10 5 15
Target-Year Inter-zonal Impedances, {Wij}
5
510
10
1515
Find: c and Kij to fit the base data
TAZ 3 4 5
1 300 150 502 180 600 220
Base-Year Trip Interchange Volumes, {tij}
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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
Wcij
ln F = - c ln W
ln F
ln W
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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 10 152 10 5 15
Target-Year Inter-zonal Impedances, {Wij}TAZ 3 4 5
1 300 150 502 180 600 220
Base-Year Trip Interchange Volumes, {tij}
W Σtij f
5 300 600 900 0.6010 150 180 330 0.2215 50 220 270 0.18Σ 1500 1.00
tij ValuesZone Pairs
13, 2414, 2315, 25
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Let’s assume c = 2.0
TAZ 3 4 5
1 5 10 152 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 5002 180 600 220 1000
Friction Factor, {Fij} with c = 2.0
Fij =1
Wcij
= F13=1
52 = 0.04
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c = 2.0
First Iteration
Aj Fij
TAZ 3 4 5 Σ
1 0.605 0.227 0.168 1.0002 0.123 0.740 0.137 1.000
TAZ 3 4 5 Σ
1 0.080 0.030 0.022 0.1322 0.020 0.120 0.022 0.162
Aj Fij
Σ(Ax Fix)[ ]
TAZ "Attractiveness"
1 02 03 24 35 5Σ 10
Friction Factor, {Fij} with c = 2.0
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c = 2.0
First Iteration
TAZ 3 4 5 Σ
1 0.605 0.227 0.168 1.0002 0.123 0.740 0.137 1.000
TAZ 3 4 5 Σ
1 303 113 84 5002 123 740 137 1000
W Σtij f
5 303 740 1042.2 0.6910 113 123 236.73 0.1615 84 137 221.02 0.15Σ 1500 1.00
15, 25
Zone Pairs tij Values
13, 2414, 23
Aj Fij
Σ(Ax Fix)[ ]
Qij = PiAj Fij
Σ(Ax Fix)[ ]
TAZ Productions
1 5002 10003 04 05 0Σ 1500
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Trip-Length Frequency Distribution
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)
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Let’s assume c = 1.5
TAZ 3 4 5
1 5 10 152 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 5002 180 600 220 1000
Friction Factor, {Fij} with c = 1.5
Fij =1
Wcij
= F13=1
51.5 = 0.089
TAZ 3 4 5
1 0.089 0.032 0.017
2 0.032 0.089 0.017
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c = 1.5
Second Iteration
Aj Fij
Aj Fij
Σ(Ax Fix)[ ]
TAZ "Attractiveness"
1 02 03 24 35 5Σ 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 0.3602 0.063 0.268 0.086 0.418
TAZ 3 4 5 Σ
1 0.497 0.264 0.239 1.0002 0.151 0.642 0.206 1.000
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c = 1.5
Second IterationAj Fij
Σ(Ax Fix)[ ]
Qij = PiAj Fij
Σ(Ax Fix)[ ]
TAZ Productions
1 5002 10003 04 05 0Σ 1500
TAZ 3 4 5 Σ
1 0.497 0.264 0.239 1.0002 0.151 0.642 0.206 1.000
TAZ 3 4 5 Σ
1 249 132 120 5002 151 642 206 1000
W Σtij f
5 249 642 891.06 0.5910 132 151 283.26 0.1915 120 206 325.67 0.22Σ 1500 1.00
Zone Pairs tij Values
13, 2414, 2315, 25
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Trip-Length Frequency Distribution
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)
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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)
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K Factor Example
Rij =Observed Qij
Calculated Qij= R13 =
300249 = 1.20
Xi =Base-Year Qij
Pi= X1 =
300500 = 0.60
Kij = Rij1-Xi
1 - XiRij= K13 = 1.20
1-0.61–(0.6)(1.2)
= 1.71
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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
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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
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Limitations of the Gravity Model
The model fails to reflect the characteristics of the individuals or households who decide which destinations to choose in order to satisfy their activity needs
Zone 2
Zone 3
Zone 4Zone 1
Income=75000
Income=20000
White Collar Jobs
Blue Collar Jobs
The gravity model does not take this type of situation into account without using K-Factors… which leads back into another whole set of problems