Trip Distribution Introduction

Trip Generation • The greater the separation between an origin and destination, the less
likely the trip.
• The decay in the willingness to travel function varies by mode.
• The choice of mode influences locational decisions
Trip Distribution • Conventional trip distribution models are estimated for auto trips only
and applied to all modes.
• Justified because more than 80% of all trips are made by cars.
Mode Choice • However, limits ability of model to forecast other modes, important in
recent years.

Route Assignment

Aim to Predict Trips

Trip Distribution
from i to j
• Estimate the number of trips going from zone i to zone j for each
purpose. This requires the travel time (and cost) between zones (Cij) and
the trips produced or attracted to each zone (e.g. Ti, Tj).

Origin \ Destination 1 2 3 Z
1 T11 T12 T13 T1Z
Destination 1 2 3 . . J
2 T21
3 T31 Origin
Z T41 TZZ
1 Cij

2
3

Gravity Model

• Gravity Model - expresses interaction between two places as

a function of the size of the two places and the distance
between them.
– interaction increases with origin (Ti) and destination (Tj) trips.
Illustration
– interaction decreases with distance, of Gravity
– the likelihood of intervening opportunity increases with spatial Model
separation (the cost of travel between zones) (Cij).

http://www.rri.wvu.edu/WebBook/Goetz/slides/figuresII/sld002.htm

1
 Negative Exponential
Opportunities
Form Impedence Function (e-0.08t) Cumulative Opportunities= D*3.14*t2
0.8
45000000

0.7 40000000

35000000
0.6

30000000
0.5

25000000

0.4

20000000

0.3
15000000

0.2
10000000

0.1 5000000

0
0
0 20 40 60 80 100 120 140
0 10 20 30 40 50 60 70 80 90
Time
Time (minutes)

Resulting Trip
Distribution Proportion of Trips
Trip Distribution:
0.16 Impedances
0.14 • This table of impedances requires knowing the congested
travel time on the network (Cij), which itself requires
0.12
knowing demand, and so may require a recursive solution
0.1
method of some kind (i.e. feedback).

0.08
• Interaction between zones is often described by a gravity
0.06
model, in analogy to Newton's Laws of Gravitation. While
f(Cij)=1/Cij2 was used in the past, now a negative exponential
0.04 form f(Cij)=eβCij is often preferred.
0.02

0
0 20 40 60 80 100 120 140
Time

Impedance Function Trip Distribution

(Work Trips) (Work Trips)
TRIP DISTRIBUTION MODEL
WORK TRIPS
Auto Drive Auto Walk to Auto-1 Auto-2 Auto-3+ Walk
to Transit Pax to Transit 0.400

Transit
a TIME 0.05 -0.11 -0.08 -0.08 -0.07 -0.06 -0.14 0.350 to Transit
(2.3) (-3.8) (-7.9) (-17.2) (-16.4) (-10.6) (-11.6) to Transit
Transit
0.300 Auto-1
b TIME0.5 - 0.642 0.265 - - - -
Relative Probability per Opportunity

Auto-2
- (2.1) (2.3) - - - - Auto-3+
0.250
c TIME 2
-0.00106 - - - - - -
(-4.6) - - - - - - 0.200

d CONST -2.92 -2.90 -1.91 -0.97 -1.03 -1.31 -0.58

0.150

r-squared 0.87 0.88 0.98 0.94 0.94 0.87 0.94

0.100

0.050
0.5 2
f (Cijm ) = e at+ bt + ct + d
0.000
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
TIME (in 5 minute cohorts)

2
 Trip Patterns Are Like
Coin Tossing
Coins
• Number of ways 4 coins can be tossed and come zone 1 2 3
up all heads is 1. (HHHH) zone
1 2 1 1
• Number of ways they can be tossed and come up 2 2 0 2 1

heads, 2 tails is 6. (HHTT, HTHT, HTTH,

7!
THTH, THHT, TTHH) w(Tij ) = = 1260
n!
2!1!1!0!2!1!
w= n

∏n! i
i =1

• 1=4!/4!0!, 6=4!/2!2! €

But We Have Additional

Most Likely Outcome
Information
• Column and Row Totals. Consider just the T!
max w(Tij ) =
row totals: ∏ Tij!
ij
• 4!/2!1!1! = 12
subjectto
• 3!/0!1!2! = 3
• 12 x 3 = 36 << 1260
∑T ij = Ti
j

∑T ij = Tj
i

T = ∑∑Tij = ∑Ti = ∑Tj

j i i j

One More Constraint Lagrangians

T!      
+ ∑ λi  Ti − ∑Tij  + ∑ λj  Tj − ∑Tij  + β C − ∑∑Tij cij 
∑∑T c ij ij =C ∏ Tij!
ij
i  j  j  i   i j 
i j

• C is quantity of resources, cij is travel cost. • Transform the constraints so that instead of
€ X=Y, X-Y=0
• Put transformed constraints into objective
€ function (multiplied by a Langrangian
multiplier (lambda or beta).

3
 Transform with
Evaluate the Maximums
Natural Logs
ln N!≈ N ln N − N ∂T
= − lnTij − λi − λj − βcij = 0
∂Tij
∂ ln N!
≈ ln N lnTij = − λi − λj − βcij
∂N
€ −λi −λj −βc ij
Tij = e
€ € €
€

Substituting Back Which Gives

−λ
∑e −λi −λj −βc ij
=0 e j e −λi
j
= Ai ; = Bj
−λi −λj −βc ij
Ti Tj
∑e =0
−βc ij
i
Tj Tij = Ai BjTiTj e
−λj Ti
e = −λj −βc ij
;e −λi = −λi −βc ij
∑e ∑e €
€ j i

Gravity Model is Most Entropy Maximization

Probable Distribution Typically Trips can be represented as a function of
productions (Ti), attractions (Tj) and Costs/Times (C) such as
of Trips Tij = f(Tj, Ti, Cij)

• Negative exponential is most likely distance Tij = K iK j TiT j f (Ci j )

decay formulation. Note this form is where :
downward sloping, so it is like a demand βC ij
curve. f (Ci j ) = e
• But we don’t know C, so we need to solve 1 1
Ki = Kj =
for Beta and search for A and B ∑K Tj j f (Cij ) ∑ K T f (C
i i ij )
• Beta dictates how sensitive travel is to cost. €
Solve iteratively for Ki and Kj
€ €

4
 Objective of Mode
Mode Choice
Choice
Trip Generation • AGGREGATE: Estimate the number of trips from each zone to each
zone by purpose that take mode m.
• DISAGGREGATE: Estimate the probability that a particular trip
(purpose, time, zone-zone) by a specific individual will take mode m.
Trip Distribution • Typically forecasters use a “discrete choice” model, that predicts distinct
(or discrete or qualitative) choices (bus vs. car) rather than continuous
ones (3.4 trips vs. 3.6).
Mode Choice • Logit is the most popular version of mode choice model.

Route Assignment

The Logit Model Disaggregate Choice Model

U mij • Pm - probability of • Disaggregate Choice Model - include variables other than travel time in
e determining the probability of making a trip.
Pm = U mij
taking mode m • The logit model and gravity model have been shown by Wilson to be of

∑e • Umij - Utility of mode

m between OD pair ij
essentially the same form.
• The application of these models differ
– the gravity model uses impedance by travel time, perhaps stratified
m
for an individual (or a by socioeconomic variables, in determining the probability of trip
s.t. representative traveler)
making,
– a discrete choice approach brings those variables inside the utility or
impedance function. Discrete choice models require more
• Umij = f(Cij,…)
∑P m =1 information to estimate and more computational time.

Relationship of Logit Typical Model Structure

and Gravity MODE CHOICE

• The functional relationship between the modern gravity

model (negative exponential form) and the logit mode
choice model are very similar, enabling simultaneous choice
AU1 AU2 AU3+ WALK
models to be easily developed. WCT ADT APT
BIKE

• The key difference is that the gravity model is typically • (WCT) walk connected transit • Alternative Structures include Nests
much more aggregate. • (ADT) auto connect transit (driver • Advantages: Nests Allow Model to
alone/park and ride) Capture Relationships between
• (APT) auto connect transit (auto modes, skirt IIA property.
passenger/kiss and ride)
• Disadvantages: Increased
• (AU1) auto driver (no passenger)
computation, Marginal
• (AU2) auto 2 occupants Improvement in Estimation
• (AU3+) auto 3+ occupants
• (WK/BK) walk/bike

5
 Independence of Probability vs.
Irrelevant Alternatives Difference in Weights
• Property of Logit (but not all Discrete Choice models) Probability

• If you add a mode, it will draw from present modes in proportion to 1

their existing shares. 0.9

• Example: Suppose a mode were removed. Where would those travelers 0.8

go. IIA says they will go to other modes in same proportion that other 0.7

travelers are currently using them. However, if we eliminated Kiss and 0.6

Ride, a disproportionate number may use Park and Ride or carpool. 0.5
Nesting allows us to reduce this problem. However, there is an issue of
0.4
the proper Nest.
0.3
• Other alternatives include more complex models (e.g. Mixed Logit)
which are more difficult to estimate. 0.2

0.1

0
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
W1-W2

Utility and Probability Transformation

 P 
log i  = v(xi )
1− Pi  PA
= e vA
• Probability of choice i increases as utility (v) 1− PA
increases PA = e vA − PA e vA
• Observed utility = mean utility + random (error)
term PA (1+ e vA ) = e v A
€ • u(x) = v(x) + e(x)
e vA
• If e(x) is normal, we have a probit model, if e(x) is PA =
Weibull distributed, we have a logit model. 1+ e vA

Utility Expression (e.g.) Estimation

N
vA = β0 + β1( cA − cT ) + β2 ( tA − tT ) + β3 I + β4 N ∏ f ( y x ,θ )
L* = n n
n=1

∂L
=0
€ • Utility of auto depends on cost of auto, cost
€
∂θˆN
of transit, auto travel time, transit travel time, N
income, and number of travelers ln L* = ∑ ln f ( yn xn ,θ )
n=1

€
€

6
 Two Choices (0, 1) Likelihood Function
• Either choose X or not-X. f(1,1,1,0,1)

0.09
Likelihood Function for the Sample {1,1,1,0,1}

• Let probability of X be γ, P(not-X)=1-γ 0.08

• Suppose there are 5 observations {1,1,1,0,1} 0.07

0.06

• We want to find the maximum likelihood 0.05

estimator γ. 0.04

• If γ is 0, P of drawing sample is 0
0.03

0.02

• If γ is 0.1, Probability is f(1)f(1)f(1)f(0)f(1) = 0.01

0.1*0.1*0.1*0.9*0.1 = 0.00009 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Bi-nomial Logit
Maximum Log-Likelihood
Likelihood
N n
L* = ∏ P (1− P )Yi 1−Y i ∂l
n=1
i
n
i = ∑ Yi − Pˆi = 0
∂β i =1
( )
l = ln L = ∑[Yi ln Pi + (1− Yi ) ln(1− Pi )]
*

i =1
• Set partial derivatives to zero
€ e v (xauto )
Pauto =
1+ e v (xauto )
n €
€ [
l = ln L* = ∑ Yi v(xauto )− ln(1+ e v (xauto ) )
i =1
]
€

Route Assignment Route Assignment

• A uto A s s ign m ent: How travelers choose a route between A and B.
Trip Generation
• Volu me Delay Function (VDF) (or Link Performance Function
( L P F )): As the traffic flow on a link increases, the travel time on that link
Trip Distribution increases. The cost that a driver imposes on others is called the marginal cost.
However, when making decisions, a driver only faces his own cost (the average
cost) and ignores any costs imposed on others (the marginal cost).
• Average Cost = Total Cost/ Q
Mode Choice • Marginal Cost = d Total Cost / dQ

Route Assignment

7
 Volume-Delay Function Wardrop’s Principles
Link Travel Time
Volume Delay Function
• Each link has a link • User Equilibrium • System optimal
3 – Each user acts to minimize
volume delay function his/her own cost, subject to – Each user acts to
2.5
(link performance every other doing the same – minimize the total
Travel times are equal on all travel time on the
2
function) relating travel used routes and lower than on
time (Cost) on that link any unused route. system.
– What people choose is efficient
1.5
and total flow on the for them which however need
1 link. This is analogous not be efficient for the network.

0.5
to an average cost
curve used in
0
0 500 1000
Flow per Lane
1500 2000 2500 economics.

Conservation of Flow Example

An important factor in road assignment factor is the • Solve for the flows on Links 1 and 2
in the Simple Network just shown if
conservation of flow. This means that the number of the volume delay function on link 1:
vehicles entering the intersection equals the number of – C1 = 5 + 2* Q1
• and the function on link 2:
vehicles exiting the intersection for a given period of time.
– C2 = 10+Q2.
(except for sources and sinks) • Time (Cost) is equal on all used Link 1
routes so Origin Destination
– C1=C2 Qo=1000 Qd=1000
Similarly, the number of vehicles entering the back of the link Link 2
– And Q1+Q2=1000
equals the number exiting the front (over a long period of – 5+2*(1000- Q2 )=10+ Q2
time). – 1995=3 Q2
– Q2 =665; Q1 =335

Problem: User
Problem (continued)
Equilibrium
• Example. Given a flow of six (6) units from origin “o” to destination “r”. • A. What is the flow and travel time on each link
Flow on each route ab is designated with Qab in the Time Function.
Apply Wardrop's Network Equilibrium Principle (Users Equalize (complete the table below) for Network A?
Travel Times on all used routes)

Link Time Flow Time

Network A Function
p r o-p 5 *Qop
p-r 25 + Qpr
o-q 20 + 2*Qoq
o q
q-r 5 *Qqr

8
 Problem (Solution)
These four links are really 2 links O-P-R and O-Q-R, because by conservation of flow
Example 2 (Continued)
Qop = Qpr and Qoq=Qqr.
By Wardrop's Equilibrium Principle, The travel time (cost) on each used route must By the conservation of flow principle
be equal. So 13/7Qoqr = 41/7
Qoqr+Qopr = 6 Qoqr = 41/13 = 3.15
Copr = Coqr. Qopr = 6 - Qoqr
OR Qopr=2.84
By substitution
25+ 6*Qopr= 20+7*Qoqr Check
Qoqr = 5/7 + 6/7 (6 - Qoqr) = 41/7 -6/7
Qoqr 42.01=25+6(2.84) ? 20+7(3.15) = 42.05
5+ 6*Qopr= 7*Qoqr
Check (within rounding error)
Qoqr = 5/7 + 6/7 Qopr
Link Time Flow Time
Function
Link Time
o-p 5 *Qop 2.84 14.2
Function
p-r 25 + Qpr 2.84 27.84
o-p-r 25+6 *Qopr
o-q 20 + 2*Qoq 3.15 26.3
o-q-r 20 + 7*Qoqr
q-r 5 *Qqr 3.15 15.75

Example 2b: System

Questions
Optimal
• What is the system optimal assignment for the previous example
– Conservation of Flow: Qoqr+Qopr = 6 • Questions?
– Total Delay = Qopr(25+ 6*Qopr) + Qoqr(20+7*Qoqr)
– 25 Qopr + 6 Qopr^2 + (6-Qopr)( 62 - 7Qopr)
– 25 Qopr + 6 Qopr^2 + 372 - 62 Qopr - 42 Qopr + 7 Qopr^2
– 13 Qopr^2 - 79 Qopr + 372 • Please turn in minute paper at end of class.
TWO SOLUTION METHODS:
“Solver”, “Analytic”
• Analytic
– Min Total Delay
– dDelay/dQ = 26 Qopr - 79 = 0
– Qopr = 79/26 = 3.04
– Qoqr = 6-Qopr=2.96