1. CE 254 Transportation Engineering
The Four-Step Model: II. Trip Distribution
Wes Marshall, P.E University of Connecticut February 2008

2. The Basic Transportation Model…
Study Area Zones
Attributes of Zones
Socioeconomic Data
Land Use Data
“Cost” of Travel btw. Zones
The Road Network
Inputs
Traffic Volume by Road Link
Mode Splits
Emissions
Outputs

3. What’s in the Black Box?
The Four-Step Model

4. The Four-Step Modeling Process…
Trip Generation
Trip Distribution
Mode Choice
Trip Assignment
WHY?

5. 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

6. Nevertheless, most agencies still use the good ol’ four-step model
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

7. Overview of the Four-Step Model

8. } } Gravity Model TRIP GENERATION Input:
- No. of Housing Units - Office, Industrial SF
- HH Size
- Income
- No. of Cars
Household Socioeconomic Data
Land Use Data
Examples of HH socioeconomic data
Survey Data }
Examples of land use data
Iterative Process
Process:
Model residential trip productions and non-residential trip attractions w/ - Regression Models
- Trip-Rate Analysis
- Cross-Classification Models
Output:
- i.e. columns of trip productions and trip attractions
Trip Ends by purpose
TRIP DISTRIBUTION
Input:
Trip Ends by purpose
Process:
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
Output:
- A matrix of trips between each TAZ… also called a “trip table”
Trip Interchanges
MODE CHOICE
Input:
Trip Interchanges
Output:
Trip Table by Mode
TRIP ASSIGNMENT
Input:
Trip Table by Mode
Output:
Daily Link Traffic Volumes
- i.e. traffic flows on network, ridership on transit lines

9. Trip Interchance Model
MODE CHOICE
Iterative Process
Input:
Trip Interchanges
Process:
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
Output:
Trip Table by Mode
TRIP ASSIGNMENT
Input:
Trip Table by Mode
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
Output:
- i.e. traffic flows on network, ridership on transit lines
Daily Link Traffic Volumes

10. 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

11. 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

12. Preparing for a Four-Step Model
Before jumping into trip generation, we first have to set up our project…
Define study area and boundaries
Establish the transportation network
Create the zones

13. 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

14. Establish the Network
Roads are represented by a series of links & nodes
Links are defined by speed and capacity
Turns are allowed at nodes
Node
Link

15. 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

16. 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

17. 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

18. Trip Generation

19. 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

20. 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

21. Trip Generation Input: Socioeconomic Data Land Use Data Output:1 2 3 4 5 6 8 7
Output:
Trip Ends by purpose (i.e. work)
in columns of productions & attractions

22. Trip Generation Trip Distribution
The question is… how do we allocate all the productions among all the attractions?
Trip Matrix or Trip Table
Zone 1

23. Trip Distribution

24. 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

25. 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

26. 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

27. Basic Criteria for TD
Criteria for allocating all the productions among all the attractions
Cost of trip
Travel Time
Actual Costs
Attractiveness
Quantity of Opportunity
Desirability of Opportunity

28. How to Distribute the Trips?
Growth Factor Models
Gravity Model

29. 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

30. 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?
Trip Matrix, t (2008)
Trip Matrix, T (2018)

31. Some of the More Popular Growth Factor Models
Uniform Growth Factor
Singly-Constrained Growth Factor
Average Factor
Detroit Factor
Fratar Method

32. Uniform Growth Factor Model

33. Uniform Growth Factor i = I = Production Zone j = J = Attraction Zone
Tij = τ tij for each pair i and j
Tij = Future Trip Matrix
tij = Base-year Trip Matrix
τ = General Growth Rate

34. Uniform Growth Factor If we assume τ = 1.2, then… Tij = τ tij
Tij = τ tij for each pair i and j
Tij = Future Trip Matrix
tij = Base-year Trip Matrix
τ = General Growth Rate
Uniform Growth Factor
If we assume τ = 1.2, then…
Trip Matrix, t (2008)
Tij = τ tij
= (1.2)(5)
= 6
Trip Matrix, T (2018)

35. Uniform Growth Factor
The Uniform Growth Factor is typically used for 1 or 2 year horizons
However, assuming that trips grow at a standard uniform rate is a fundamentally flawed concept

36. Singly-Constrained Growth Factor Model

37. 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…
attractions, j
zones
productions, i

38. 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)

39. Singly-Constrained Growth Factor Method
Tij = tij (Target Pi) / (Σj)
= 5 (400 / 355)
= 5.6

40. 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)

41. 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…

42. Average Growth Factor Model

43. Average Growth Factor Function
Fik + Fjk
g ( )= 2
Fik Fjk
g ( )=
F.k
F = Growth Factor
= Ratio of Target Trips to
Previous Iteration Trips
k = Iteration Number
F.k=
ΣTi
Σtik-1 =
Σ(Future Trips)
Σ(Current Iteration Trips)

44. Previous Iteration Trip End
The Basic Steps…
Collect Inputs
Matrix of Existing Trips, {tij}
Vector of Future Trips Ends, {Ti}
Compute Growth Factor for each zone
Compute Inter-zonal Flows
Compute Trips Ends
If tik = Ti for each zone i, then stop… otherwise, go back to “Step 1”
>
ΣTi
Target Trip End
Fik= =
Σtik-1
Previous Iteration Trip End
>
tijk = tijk-1 [g(Fik, Fjk, …)] for each ij pair
tik = Σtijk for each zone i

45. Growth Factor Models: Average Factor Example
1.330= 2

48. Fratar Method Growth Factor Model

49. Fratar Method tjik-1 Fjk tijk=Ti Replace Step 2 with… Σtizk-1Fzk
Tijk=Tjik=
tijk + tjik 2
Balance Matrix with…

50. Growth Factor Models: Fratar Method ExampleTi
Future Trips
115
Fi(1) = = =
= 2.09
ti(0)
Current Trips 55
Σtizk-1Fzk = (t110)(F11)+(t120)(F21)+(t130)(F31)+(t140)(F41)
=(0)(2.09) + (25)(1.30) + (10)(1.59) +(20)(1.46)
= 78
Σtizk-1Fzk = (t210)(F11)+(t220)(F21)+(t230)(F31)+(t240)(F41)
=(25)(2.09) + (0)(1.30) + (60)(1.59) +(30)(1.46)
= 191

51. Growth Factor Models: Fratar Method Example
t431 = (t340)(T4)(F31)/Σ[(t4z0)(Fz1)]
= (15)(95)(1.59) / (105)
= 22
t121 = (t210)(T1)(F21)/Σ[(t1z0)(Fz1)]
= (25)(115)(1.30) / (78)
= 48

52. Now onto Iteration 2… Balance Flows = (tij1 + tji1) / 2
= ( ) / 2
= 23
Now onto Iteration 2…

53. Iteration 2…
Balance Flows…
& Compute Values

54. Iteration 3…
Balance Flows…
& Compute Values

55. 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

56. The Gravity Model

57. The Inspiration for 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
M1 M2
F = k r2

58. 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

59. Some of the Variables Tij = Trips between i & j i = production zone
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
Tij = Trips between i & j
i = production zone
j = attraction zone
Tij = f (Pi, Aj, Cij)
Cij = Generalized cost of trip from i to j

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

61. 2 Types of Gravity Models
Parametric
Fits equation to curve
Non-parametric
Uses look-up table for bars
% Trips
Time (min)

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

63. Gravity Model Example 8.2 Given: Find: 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}

64. 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

65. 1 Fij = Wcij Find Trip Interchanges, {Qij}
Find Denominator of Gravity Model Equation {AjFijKij}
Calculate Friction Factors, {Fij}
Find Probability that Trip i will be attracted to Zone j, {pij}
Given…
Given…
Target-Year Inter-zonal Impedances, {Wij}
Calibration Factor c = Socioeconomic Adj. Factor K = 1.0
Calculate Friction Factors, {Fij} 1 1
F11= 1 52 =
0.04
Fij = =
Fij =
Wcij
Wcij
Find Denominator of Gravity Model Equation {AjFijKij}
AjFijKij=A4F34K34 = (5)(0.01)(1.0) = 0.05
Find Probability that Trip i will be attracted to Zone j, {pij}
AjFijKij
0.05
pij = = =
Σ(AzFizKij)
0.16
Find Trip Interchanges, {Qij}
Qij = Pipij = (2600)(0.3125) = 813

66. Keep in mind that the socioeconomic factor, K, can be a matrix of value rather than just one value

67. 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

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

69. 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

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

71. Gravity Model Calibration Example
Given:
Zone 2
Zone 3
Zone 4
Zone 5
Zone 1 10 5 10 5
Target-Year Inter-zonal Impedances, {Wij} 15 15
Base-Year Trip Interchange Volumes, {tij}
5 TAZ City
Find: c and Kij to fit the base data

72. Relating F & W 1 Fij = Wcij ln F = - c ln 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 1
Fij =
Wcij
ln F = - c ln W
ln F
ln W

73. Trip-Length Frequency Distribution
Target-Year Inter-zonal Impedances, {Wij}
Base-Year Trip Interchange Volumes, {tij}
Group zone pairs by max Wij
Sum Interchange Volumes for each set of zone pairs
Find f = (Σtix) / (Σtij)

74. First Iteration First Iteration 1 = F13= = 0.04 52
Let’s assume c = 2.0
Target-Year Inter-zonal Impedances, {Wij}
Base-Year Trip Interchange Volumes, {tij}
Friction Factor, {Fij} with c = 2.0 1
= F13=
= 0.04 52

75. Friction Factor, {Fij} with c = 2.0
First Iteration
First Iteration
c = 2.0
Friction Factor, {Fij} with c = 2.0
Aj Fij

76. First Iteration
First Iteration
c = 2.0

78. First Iteration Second Iteration 1 = F13= = 0.089 51.5
Let’s assume c = 1.5
Target-Year Inter-zonal Impedances, {Wij}
Base-Year Trip Interchange Volumes, {tij}
Friction Factor, {Fij} with c = 1.5 1
= F13=
= 0.089
51.5

79. Friction Factor, {Fij} with c = 1.5
First Iteration
Second Iteration
c = 1.5
Friction Factor, {Fij} with c = 1.5
Aj Fij

80. First Iteration
Second Iteration
c = 1.5

82. K Factors 1-Xi Kij = Rij 1 - XiRij
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
1-Xi
Kij = Rij
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)

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

84. 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

85. 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

86. 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
White Collar Jobs
Zone 3
Income=20000
Zone 2
Zone 1
Zone 4
Income=75000
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