Wes Marshall, P.E. University of Connecticut February 2008 CE 254 Transportation Engineering The...

Wes Marshall, P.E. University of ConnecticutFebruary 2008

CE 254Transportation Engineering

The Four-Step Model:

II. Trip Distribution

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

What’s in the Black Box?

The Four-Step Model

The Four-Step Modeling Process…

I. Trip Generation

II. Trip Distribution

III. Mode Choice

IV. Trip Assignment

WHY?

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

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

Overview of the Four-Step Model

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:

- 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:

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

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

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

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

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

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

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

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

Trip Generation

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

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

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

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

Trip Distribution

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

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

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

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

How to Distribute the Trips?

I. Growth Factor Models

II. Gravity Model

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

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 ? ? ? ?

Some of the More Popular Growth Factor Models

Uniform Growth Factor Singly-Constrained Growth Factor Average Factor Detroit Factor Fratar Method

Uniform Growth Factor Model

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


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


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

Singly-Constrained Growth Factor Model

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


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


Tij = tij (Target Pi) / (Σj) = 5 (400 / 355)= 5.6


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


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


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

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…

Average Growth Factor Model

Average Growth Factor Function

F = Growth Factor = Ratio of Target Trips to

Previous Iteration Trips k = Iteration Number

g ( )=Fi

k + Fjk

2

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

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


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

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


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

Fratar Method Growth Factor Model

Fratar Method

tijk=Ti

tjik-1 Fj

k

Σtizk-1Fz

kReplace Step 2 with…

Tijk=Tji

k=tij

k + tjik

2Balance Matrix with…

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

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

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


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…

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


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

Iteration 3…

Balance Flows…


1 2 3 4

1 0 44 29 422 43 0 79 293 29 81 0 254 42 29 24 0

& Compute Values

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

The Gravity Model

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

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

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

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

2 Types of Gravity Models

1. Parametric Fits equation to curve

2. Non-parametric Uses look-up table for bars

Time (min)

% T

rips

To Apply the Gravity Model

What we need…

1. Productions, {Pi}

2. Attractions, {Aj}

3. Skim Tables {Wij) Target-Year Interzonal Impedances

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}

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

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

=

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

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


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


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

Trip-Length Frequency Distribution

Compares the observed and computed Qij values

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


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}

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


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

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

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)[ ]


1 02 03 24 35 5Σ 10


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


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)

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


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

c = 1.5

Second Iteration

Aj Fij

Aj Fij

Σ(Ax Fix)[ ]


1 02 03 24 35 5Σ 10


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

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


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)

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)

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

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

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

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

Wes Marshall, P.E. University of Connecticut February 2008 CE 254 Transportation Engineering The...

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