Transportation Systems Analysis
Modeling
CEE 3604
Introduction to Transportation
Engineering
Dr. A.A. Trani
Virginia Tech
November 2009
Organization
• Discuss all four steps in transportation systems
planning and modeling
• Discuss urban applications of the transportation
systems modeling approach
• If you want to know more about this topic take a
senior class called: Transportation Planning (CEE
4624)
Why do We Need a Transportation
Systems Planning and Modeling?
• Because transportation engineers plan, design and
construct facilities
• Because predicting how people travel is more
difficult than predicting a “nuclear reaction at the
molecular level” (true statement from Los Alamos
Physicists)
• Keeping up with demand is difficult in constrained
budget environments
The Basic Idea and Few Steps
Trip Generation
Trip Distribution
Predicts trips from
zone to zone
Distributes trips
between zones
Mode Split/Choice
Splits trips among
various modes of
travel
Traffic Assignment
Assigns trips among
various transport
networks
Reston
Population = 60,000
Household Income = $55,000
Car Ownership = 2.1 (per house)
Transportation
Planning Idea
Washington DC
Population = 230,000
Household Income = $45,000
Car Ownership = 1.3 (per house)
Road Network
Fairfax
Population = 120,000
Household Income = $70,000
Car Ownership = 2.3 (per house)
Centroids
Reston
Interzone trips = 230,000 person-trips
Intrazone trips = 70,000 person-trips
How Many Trips?
Washington DC
Interzone trips = 400,000 person-trips
Intrazone trips = 130,000 person-trips
Road Network
Fairfax
Interzone trips = 360,000 person-trips
Intrazone trips = 100,000 person-trips
Basic Definitions
• Intrazone trips – trips that stay within the zone where
the person making the trips starts its journey
– A trip to a shopping center
– A trip to drop children to school
• Interzone trips – trips that extend beyond the zone
where the person starts its journey
– Commuting trip to work
– Commuting trip to airport, train station to make a longdistance trip
• The definition of a zone in our context is a subarea of
interest in our study with similar socio-economic
characteristics or perhaps physical boundaries
What Drives the Number of Trips?
• Number of persons per household
• Number of cars per household
• Income levels
• Road infrastructure density (lane-km or road per
square kilometer)
• Many others
Back to General Transportation
Planning Method
Trip Generation
Trip Distribution
Mode Split/Choice
Traffic Assignment
Trip Generation
• Use of cross classification tables
• Provides a snapshot of potential trips per household
• Obtained through surveys
• Socio-economic parameters dictate trips
Persons per household
1
2
3
4 or more
Vehicles per household
0
1
1.02
1.9
2.12
3.25
2.15
3.75
3.96
5.02
2 or more
Trip Rate Table for Urban Areas
(units are trips per household per day)
2.1
3.7
3.9
6.54
Sample Surveys Done in the US
• National Household Travel Survey (NHTS)
• American Travel Survey (ATS)
http://nhts.ornl.gov/
http://www.bts.gov/publications/1995_american_travel_survey/
Trip Generation Output
• A trip matrix of trip Attractions (Aj) and trip
Productions (Pi)
• The matrix predicts all trips produced and attracted
to and from every zone
• Trip attractions depend on variables like
employment, retail floor space, etc.
Zone
Reston
Fairfax
DC
Productions
230000
360000
400000
Attractions
200000
200000
590000
Attraction and Production Table for Sample Area
(units are trip-persons per day)
Techniques to Perform Trip
Generation Models
• Cross classification trip rate tables for trip productions
Persons per household
1
2
3
4 or more
Vehicles per household
0
1
1.02
1.9
2.12
3.25
2.15
3.75
3.96
5.02
2 or more
2.1
3.7
3.9
6.54
• Regression analysis for trip attractions
Trip attractions = A + B * (employment)
where: A and B are regressions constants to be
obtained using statistical regression techniques such
as the least-squares method
Back to General Transportation
Planning Method
Trip Generation
Trip Distribution
Mode Split/Choice
Traffic Assignment
Trip Distribution
• Answers the question:
• Where do the trips generated go?
Reston
Distance = 20 km
Distance = 10 km
Washington DC
Fairfax
Trip Distribution
• Methods
• Gravity Model (just like the attraction between planets!)
• Growth factor models (Fratar Models)
Reston
Productions = 230,000
Attractions = 200,000
Distance = 20 km
Distance = 10 km
Fairfax
Washington DC
Productions = 360,000
Attractions = 200,000
Productions = 400,000
Attractions = 590,000
Gravity Model Formulation
Tij = Pi Aj Fij / S (Aj Fij)
where
Pi = Productions at zone I
Aj = Attractions at zone j
Fij = Impedance of travel between I and j
Reston
Productions = 230,000
Attractions = 200,000
Distance = 20 km
Distance = 10 km
Fairfax
Washington DC
Productions = 360,000
Attractions = 200,000
Productions = 400,000
Attractions = 590,000
What is the Impedance (Fij)?
• A common term to state that there is resistance to
travel between two zones
• The resistance is proportional to the travel time
between the zones (time ij)
Fij = Cij exp(-alpha) or
Reston
Cij = travel time
Distance = 20 km
Travel time = 1 hour
Distance = 10 km
Travel time = 30 minutes
Washington DC
Output of Trip Distribution
• A trip interchange matrix (Tij)
• How many trips go from zone I to zone j
Origin Zone
Reston
Fairfax
DC
Reston
T reston-reston
T fairfax-reston
T dc-reston
Destination Zone
Fairfax
T reston-fairfax
T fairfax-fairfax
T dc-fairfax
DC
T reston-dc
T fairfax-dc
T dc-dc
Back to General Transportation
Planning Method
Trip Generation
Trip Distribution
Mode Split/Choice
Traffic Assignment
Trip Mode Split
• Estimates the number of trips made taking a specific
mode of transportation
• For the sample area, travelers will have choices of
mode:
– Bus
– Auto
– Rapid transit
– Walk
– Bicycle
Mode Split or Mode Choice
• Out-of-pocket costs (Cost ij via mode k) is important
• Travel time (time ij via mode k) is important
How many trips by auto?
How many by transit?
Reston
Travel time (transit) = 1 hour
Travel cost (transit) = $1.50
Travel time (auto) = 45 minutes
Travel cost (auto) = $5.00 (includes parking)
Washington DC
Mode Split Formulation
Um = Utility of travel using mode m
Zmj = travel characteristics (time and cost)
Bm = Mode specific constant
aj = Model parameter (from calibration)
e
= stochastic term with zero mean
Calculating Probabilities of
Travel by a given Mode (Logit Model)
• W. McFadden (Nobel Price winner 30 years ago)
developed a fundamental model called Logit Model
to predict people’s choice in economic terms
• Basis for today’s logit models used in
transportation
Pm =
Um
e
åe
Um
m
Pm = probability that mode
m is selected
M = index over all modes
included in the choice set
Example of Mode Split Equation
• A mode split has been calibrated using the maximum likelihood
technique (an advanced statistical method)
• The following equation has been obtained as follows:
Um = bm - 0.25C - 0.02T
where: C is the out-of-pocket cost ($), T is the travel time (minutes)
and the values of the mode specific constants (betas) are:
Transit = 0.30
Auto = 2.2
Back to the Original Problem
How many trips by auto?
How many by transit?
Reston
Travel time (transit) = 1 hour
Travel cost (transit) = $1.50
Travel time (auto) = 45 minutes
Travel cost (auto) = $5.0 (includes parking)
Washington DC
Calculation of Utilities (Um)
u = bm - 0.25C - 0.02T
Uauto = 2.2 - 0.25 (5) - 0.02 (45) = 0.05
Utransit = 0.3 - 0.25 (1.5) - 0.02 (60) = -1.275
Reston
Travel time (transit) = 60 minutes
Travel cost (transit) = $1.50
Travel time (auto) = 45 minutes
Travel cost (auto) = $5.00 (includes parking)
Washington DC
Estimate Probabilities of
Travel by Mode m
Uauto = 2.2 - 0.25 (5) - 0.02 (45) = 0.05
Utransit = 0.3 - 0.25 (1.5) - 0.02 (60) = -1.275
eU auto
e 0.05
Pauto =
= 0.05 -1.275 = 0.79
Um
åe e + e
m
Ptransit
eU transit
e-1.275
=
= 0.05 -1.275 = 0.21
Um
åe e + e
m
Interpretation of Results
• The probability that a traveler from Reston to DC
uses auto is 79%
• The probability that a traveler from Reston to DC
uses transit is 21%
• Why is this important?
– Because as a transportation engineer you have to
plan how many lanes of highway should you
provide between Reston and DC
– You also need to figure out how many transit
vehicles will be needed and how often they
should be scheduled
Sensitivity of Logit Model Results
Interpretation of Results
• If the auto cost is $1.00 the model predicts a
ridership of 9% for the bus (compared to 21%)
– This is a bargain in using the auto mode
– the bust still captures a small fraction of the riders
• If the auto cost is $20.00 the model predicts a
ridership of 9% for the auto mode
– This provides incentives for riders to take the bus
– The cost of auto is quite high and forces many
decision makers to “walk away” from auto mode
Back to General Transportation
Planning Method
Trip Generation
Trip Distribution
Mode Split/Choice
Traffic Assignment
Traffic Assignment (Final Step in
Transportation Systems Planning)
Reston
What routes are selected
by travelers?
Route 1
Washington DC
Route 3
Route 2
Link ij
Road Network
Fairfax
How do Travelers select Routes?
• Consideration of travel time and congestion in
transportation links
• Travelers tend to take routes that minimize travel
time
• After a long period of time traveling a network, a
traveler selects routes that reach equilibrium for that
traveler
– For example, if two routes are feasible to take me from
an origin (say Reston) to a destination (say DC), I will
select these routes in a way that gains in travel time
are not possible once we load the network
Travel Time vs Demand
Route 1
Travel Time
Route 2
Route 1
Total
Route 2
t
Demand
V1
Traffic Volume
V2
VT
Calculation of Travel Times
• Use any of the known traffic flow models
• For example:
• Greenshield’s model
Speed
Flow
Travel Time
uf
u = uf - k
kk
uf 2
q = uf k - k
kk
d
t=
u
Other Ways to Find Travel Times
on Highway Links
• Use of empirical data is useful in finding travel times
if the model is suspected not follow Greenshield or
Greenberg models
Other Ways to Find Travel Times
• Use of empirical data is useful in finding travel times
if the model is suspected not to follow Greenshield or
Greenberg models
Computational Example
(Two-Zone Network)
qf
Reston
qa
Freeway
(2 lanes per side)
6000 person-trips/hr
Arterial
Road (3 lanes per side)
Washington DC
Find qa and qf (volumes on arterial and freeway,
respectively)
Sample Problem (Traffic Assignment)
• Two zones are linked by a simple highway network
with network characteristics as shown:
• Freeway
– vf_freeway = 110;
% free flow speed in kilometers per hour
– kj_freeway = 75;
% jamming density in vehicles per km-lane
– d_freeway = 30;
% length of freeway (km)
– N_freeway = 2;
% number of lanes per side
• Arterial road
– vf_arterial = 90;
% free flow speed in kilometers per hour
– kj_arterial = 80;
% jamminf density in vehicles per km-lane
– d_arterial = 33;
% length of arterial (km)
– N_arterial = 3;
% number of lanes on arterial road
Problem
• Assign traffic so that volumes on the freeway and the
arterial road reach equilibrium assignment
• Equilibrium means: if a travelers switches from a link
to another one, there is no gain in travel time
• In other words, assign volumes so that travel times
on the freeway and the arterial are the same
Solution: Use Traffic Assignment
Simulator (traffic_assignment.m)
• Simple Matlab script to ease computations
• Uses Greenshield’s traffic flow model to estimate
travel time
• Inputs:
– Trips between zones (person trips)
– Vehicle occupancy (passengers per vehicle)
• Outputs:
–
Freeway Speed (km/hr)
–
Freeway Travel Time (minutes)
– Freeway Volume per lane (veh/hr)
– Total Freeway Volume(veh/hr)
–
Freeway Capacity (veh/hr)
– Freeway Number of Lanes (lanes)
Running traffic_assignment.m
• The program requires that you enter the percent of
the trips to be assigned to each link
• Try the following parameters: 6000 person-trips,
vehicle occupancy = 1.2 persons/veh and 60% of
trips assigned to the freeway
–
Freeway Speed (km/hr) 83.7228
–
Freeway Travel Time (minutes) 21.4995
–
Freeway Volume per lane (veh/hr) 1500
–
Total Freeway Volume(veh/hr) 3000
–
Freeway Capacity (veh/hr) 4125
–
–
Arterial Speed (km/hr) 80.7071
–
Arterial Travel Time (minutes) 24.5331
–
Arterial Volume per lane (veh/hr) 666.6667
–
Total Arterial Volume(veh/hr) 2000
–
Arterial Capacity (veh/hr) 5400
Note:
travel times
are not in
equilibrium
Running traffic_assignment.m
• Assign more traffic to the freeway to balance the
travel times
• Try the following parameters: 6000 person-trips,
vehicle occupancy = 1.2 persons/veh and 70.7% of
trips assigned to the freeway
–
Freeway Speed (km/hr) 75.8006
–
Freeway Travel Time (minutes) 23.7465
–
Freeway Volume per lane (veh/hr) 1767.5
–
Total Freeway Volume(veh/hr) 3535
–
Freeway Capacity (veh/hr) 4125
–
–
Arterial Speed (km/hr) 83.4139
–
Arterial Travel Time (minutes) 23.7371
–
Arterial Volume per lane (veh/hr) 488.3333
–
Total Arterial Volume(veh/hr) 1465
–
Arterial Capacity (veh/hr) 5400
Note:
travel times
are in
equilibrium
System is
In user-equilibrium
Applications to Intercity Travel
• Intercity travelers are faced with similar decisions as
urban travelers
• Mode choices are based on attributes of the mode:
– Travel time
– Travel cost
– Route convenience
– Trip purpose, etc.
• Describe the study done for NASA in the period
2001-2006
• Small Aircraft Transportation System (SATS)
On-demand (Air Taxi) Air Transportation
Assumptions
•
Assumptions:
– SATS aircraft is very light jet vehicle
• High mission reliability
• High perceived level of safety
• 350 knots cruise speed
• All-weather (pressurized)
– SATS aircraft cost (VT Eclipse 500 PW610F model)
• Baseline cost $1.50 per seat-mile
• 60% load factor
• 2 professional pilots
– SATS airport set (3,364 public airports, paved runways > 3kft, all
weather equipped)
– SATS access and egress times driven by airport set selected
Assumptions (continuation)
– Commercial airline service network (year 2000 419 airports in the continental U.S.)
– Commercial air fares based on 2000 Department
of Transportation data (12 million fares)
– No constraints in pilot production and aircraft
production
– No constraints in ATC/ATM capacity
Mode Choice (Modal Split)
Auto
Air Taxi (SATS)
Commercial Aviation
Route1
Route2... Route n
Include Airport Choice
Multi-route Mode Split/Choice Model
Utility function = Um = Bm +
Pm =
Um
e
åe
Um
m
S aj z
mj
+e
Probability of selecting
mode m
Auto Travel Time Estimation
Airport-to-Airport Travel Times
450 commercial airports
2001 Official Airline Guide
Airline Network Structure
Detailed Trip Analysis
Air Taxi (SATS) Travel Time Map
Cost of Service (Air Modes)
2.5
Business Class
Notional SATS Jet
Fare ($/seat-mile)
2
1.5
1
0.5
0
0
200
400
600
800
1000
One-way Distance (statute miles)
• Airline fares from 12 million fares (DB1B DOT data)
• SATS cost (Virginia Tech projections)
1200
1400
Traffic Assignment (Which Route?)
• Aircraft vs auto trajectories
Airway Route
I-95 Route
A1-A Route
Market Share Screen
Market Share By Segment
SATS Very Light Jet
$1.50 per seat-mile
SATS Trip Demand in NE Corridor
(Using 2000 Census Socio-Economic Data)
SATS Very Light Jet
$1.50 per seat-mile
3,416 airports
SATS Demand at Airports
(Using 2000 Census Socio-Economic Data)
SATS Very Light Jet
$3.50 per seat-mile
3,416 airports
Fastest Travel Times by Mode
(from Grafton Co., NH)
SATS Single-Engine Aircraft
200 knots cruise speed
700 mile range
Model Output
Automobile
Low Income (<$25K)
Medium Income ($25-50K)
Medium Income ($50-100K)
Market Share (%)
High Income (> $100K)
Airline
SATS
$1.50/seat-mile
Distance (statute miles)
National-level Demand Statistics
by Distance (SATS @ $1.50 per seat-mile)
Person-trips
Auto Mode
Airline Mode
SATS Mode
One-way Distance (statute miles)
Market Share of SATS
(Business Trips Only)
7,000,000
2.7% Market Share
6,000,000
SATS Person-Trips
5,000,000
4,000,000
3364 Airports
724 Airports
3,000,000
1.0% Market Share
2,000,000
0.5% Market Share
1,000,000
0
1.00
1.50
2.00
2.50
3.00
3.50
4.00
Cost ($/seat-mile)
Annual Intercity Business Trips in the U.S. (all modes) = 225 million
Nation-wide Mobility (Hours Saved)
60,000,000
(Total Hours Traveled Without SATS –
Total Hours Traveled With SATS)
Hours Saved Using SATS
50,000,000
40,000,000
3364 Airports
724 Airports
30,000,000
20,000,000
10,000,000
0
1.00
1.50
2.00
2.50
Cost ($/seat-mile)
3.00
3.50
4.00
Fuel Used by SATS
Fuel Used by SATS Operations (kg)
Fuel Used by Airlines = 44,000,000,000 kg.
1,000,000,000
900,000,000
800,000,000
700,000,000
600,000,000
3364 Airports
724 Airports
500,000,000
400,000,000
300,000,000
200,000,000
100,000,000
0
1.00
1.50
2.00
2.50
3.00
Cost ($/seat-mile)
3.50
4.00
Increased Adoption of SATS With
Increased Household Income
Frequency Plot of Total Travel
Time Savings
SATS Cost = $1.50 per seat-mile
3091 counties in the US
1800
1611
1400
1200
1000
800
528
260
222
1
0,
00
0
25
0,
00
0
20
0,
00
0
10
0,
00
00
Total Hours Saved per Year
3
3
0
23
14
15
20
,0
,0
00
41
30
00
,0
20
00
,0
10
0
00
5,
0
00
3,
0
00
2,
0
00
1,
0
100
12
50
200
00
253
,0
400
40
600
0
Frequency (No. of Counties)
1600
Average Speed Gains (by Trip)
SATS Cost = $1.50 per seat-mile
Average Speed Gains per Trip (Miles per Hour)
0.0 to 0.1
0.9 to 1.0
2.0 to 3.0
0.5 to 0.6
1.5 to 1.7
4.0 to 5.0
Total Travel Time Savings
SATS Very Light Jet
$1.50 per seat-mile
Per Capita Travel Time Savings
SATS Very Light Jet
$1.50 per seat-mile