F R E I G H T T R A V E L D E M A N D M O D E L I N G

C I V L 7 9 0 9 / 8 9 8 9

D E P A R T M E N T O F C I V I L E N G I N E E R I N G

U N I V E R S I T Y O F M E M P H I S

1

Lecture-11: Freight Assignment

11/21/2014

Outline2

1. Traffic Assignment-General concept

2. Application in Freight

3. Operational freight traffic assignment

4. Case Studies

Overview3

The procedure used to obtain expected traffic volume on the network is known as trip assignment.

Assignment Definition4

Given

A graph representation of the urban transportation network

The associated link performance functions, and

An origin-destination matrix

Find

the flow and the associated travel time on each of the network links.

This problem is known as that of traffic assignment as the objective is to assign the O-D matrix onto the network.

Link Performance Functions

5

Mathematical Relationship Between Traffic Flow and Travel Time

Traffic Flow Traffic Flow

Linear Relationship Non-Linear Relationship

Ro

ute

Tra

vel

Tim

e

Cap

acit

y

Free-Flow

Travel Time

Link Performance Functions6

A steady-state link performance function is a positive, increasing, and convex curve.

Typical link performance functions do not consider queued vehicles in the traffic stream

Link Performance Functions7

0

0.2

0.4

0.6

0.8

1

1.2

0 500 1,000 1,500 2,000 2,500

Volume in Vehicles/hour

Tra

vel ti

me o

n lin

k

C

Vtt 10

V=volume, C=capacity, t0=free flow travel time

0

0.2

0.4

0.6

0.8

1

1.2

0 500 1,000 1,500 2,000 2,500

Volume in Vehicles/hour

Tra

vel ti

me o

n lin

k

0.4,15.0

Assignment Methods8

All-or-nothing (AON)

User equilibrium (UE)

System optimum (SO)

Stochastic user equilibrium (SUE)

Multiple user class assignment

Stochastic multiple user class assignment

Classification of Traffic Assignment9

User ClassCapacity Restraint

Stochastic Effects Included

No Yes

Single user class

No All-or-nothingDial's Stochastic

Yes UE SUE

Multiple user class

NoA-O-N with multiple user classes

Dial's Stochastic

Yes UE multiple user classes SUE

All or Nothing Assignment10

Simplest assignment methods

Does not consider any congestion effects

Absence of congestion suggests that link costs are fixed

All drivers are assigned to one route and no driver is assigned to other (less attractive routes)

All or Nothing Assignment11

min z ( )

,

0 , ,

n n

a a

a

rs

k rs

k

rs

k

y t y

subject to

g q r s

g k r s

minimizing the total travel time over a network with fixed travel time

the auxiliary flow variable for path k connecting O-D pair r-s

Definition of Equilibria12

To solve the traffic assignment problem, it is required that the rule by which motorists choose a route be specified.

It is reasonable to assume that every motorist will try to minimize his or her own travel time when traveling form origin to destination.

A stable condition is reached only when no traveler can improve his/her travel time by unilaterally changing routes.

Equilibrium13

UE definition implies that

motorists have full information (choice set and travel times),

motorists consistently make the correct route choice decision

all motorists are identical in their behavior

These assumptions can be partially relaxed in the context of route choice under information provision.

distinction between the travel time that individuals perceiveand the actual travel time

This definition characterizes the stochastic-user-equilibrium (SUE) condition.

Wardrop’s Equillibrium14

Criterion-1

The journey times on all routes actually uses are equal and less than those which would be experienced by a single vehicle on any unused route

Criterion-2

User equilibrium conditions traffic arranges itself in such a way that no individual trip maker can reduce his/her path costs by switching routes.

A simple example of UE15

O D

Link 1

Link 2

t

x

x2

t2(x2) t1(x1)

x1

qOD = x1 +x2

Operational UE16

Operational definition of UE:

For each O-D pair, at user equilibrium, the travel time on all used paths is

equal, and (also) less than or equal to the travel time that would be

experienced by a single vehicle on any unused path.

Path 1

Cost

Path 2

Cost

Trips on Path 1 Trips on Path 2

Example UE17

User Equilibrium is reached when no traveler can improve his

travel time by unilaterally changing routes.

100

(30 min)

(40 min)

(65 min)

100

(40 min)

110

Formulating the Assignment Problem18

NOTATIONS

Network G (N,A)

N is set of consecutively numbered nodes

A is a set of consecutively numbered arcs (links)

R denote the set of origin centroids (which are the nodes at which flows are generated)

S denote the set of destination centroids (which are the nodes at which flows terminate)

qrs is the trip rate between origin “r” and destination “s” during the period of analysis

xa and ta, represent the flow and travel time, respectively, on link a

Formulating the Assignment Problem19

( )

a a a

rs

k

t t x

f

NOTATIONS

: where . represents the relationship between flow and travel time for link at a

: represents flow on path connecting origin and destination such that path k Krsk r s

rs

kc : is travel time on path is the sum of the travel time on the links comprising this path.k

, ,

where

1 if link is part of path connecting O-D pair

0 otherwise

rsrs

k a rs

a k

rs

k

c t k K r R s S

k r s

Using the same indicator variable, the link flow can be expressed as a function of the path flow, that is

,

rs rs

a k a k

a r s

x f

User Equilibrium Formulation20

0

,

min z(x) = ( )

subject to

,

0 , ,

ax

a

a

rs

k rs

k

rs

k

rs rs

a k a k

a r s

t d

f q r s

f k r s

x f a

flow conservation constraint

Non negativity constraint

definitional constraints𝛿𝑎,𝑘𝑟𝑠 =

1 𝑖𝑓 𝑙𝑖𝑛𝑘 𝑎 𝑖𝑠 𝑜𝑛 𝑝𝑎𝑡ℎ 𝑘 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑜 − 𝑑 𝑝𝑎𝑖𝑟 𝑟𝑠0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

Significance of User Equilibrium 21

Significance:

Reasonable assumption for representation of human behavior

In order to asses the network performance for given demands UE conditions are assumed

Limitations:

Assumption that each user minimizes travel time implies each user has perfect information on all conditions and routes

Individuals are assumed to behave identically

UE Note22

Demand for travel depends on the activity pattern, and hence not uniform over time and space.

However transportation planners analyze networks only for certain periods of the day – morning peaks, evening peaks etc. depending on objective of analysis

=> O-D flows are considered constant for such analysis (steady-state) -> static assignment

Flow is present simultaneously on all links of a path (static conditions)

System Optimal Assignment23

min z(x) = ( )

subject to

,

0 , ,

a a a

a

rs

k rs

k

rs

k

x t x

f q r s

f k r s

flow conservation constraint

Non negativity constraint

Total system travel time

0

,

min z(x) = ( )

subject to

,

0 , ,

ax

a

a

rs

k rs

k

rs

k

rs rs

a k a k

a r s

t d

f q r s

f k r s

x f a

𝛿𝑎,𝑘𝑟𝑠 =

1 𝑖𝑓 𝑙𝑖𝑛𝑘 𝑎 𝑖𝑠 𝑜𝑛 𝑝𝑎𝑡ℎ 𝑘 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑜 − 𝑑 𝑝𝑎𝑖𝑟 𝑟𝑠0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

Definitional constraint

SO Properties24

The SO formulation is subject to the same set of constraints as the UE problem and differs only in its objective function

The SO flow pattern does not generally represent an equilibrium solution in congested networks

Consequently, the SO flow pattern is not an appropriate descriptive model of actual user behavior

Significance of System Optimal 25

1. In many transportation system analysis problem it is useful to know the best performance possible for the network and OD demand

2.This is useful for control action (pricing, tolling) as well as to compare alternative solution strategies

3.Solution procedures for SO are virtually identical to those for UE

Solving UE26

Solving UE27

Substitute x2= 12-x1

Differentiate w.r.t x1 and equate to zero

x1= 5.8, x2 = 6.2.

Solving SO28

Let us consider the same example

For SO

Solving SO29

Substitute x2= 12-x1

Differentiate the equation and set it to zero

x1 = 5.3,x2= 6.7

Comparison of Methods30

Type t1 t2 x1 x2 UE Z(x) SO Z(x)

AON 10.00 15.00 12.00 0.00 336.00 552.00

UE 27.40 27.40 5.80 6.20 239.90 328.80

SO 30.10 25.60 5.30 6.70 240.53 327.55

Stochastic Methods31

Emphasize the variability in driver perception of cost

Need to consider second best routes

No perfect information about network characteristics

Different travel costs perception

Eliminates “zero volume” links

Requires large number of iterations and hence a longer run time

See more in Modeling Transport by Ortuzar and Williumsen, Chapter 10.

Multiclass traffic assignment32

Older approach

Pce conversion

Preload the usual class first

The

Modern approach

Each user class is assigned simultaneously.

See a worked out example in Nagurney (2000)

Available in TransCAD, Cube, VISUM

Nagurney, A. (2000). A Multiclass, Multicriteria Traffic Network Equilibrium Model. Mathematical and Computer Modelling 32 (2000) 393-411 .

Model Validation33

Truck counts

By vehicle class

By facility type

By time of day

Screen lines

Cordon lines

Develop RMSE, R2 or other goodness-of-fit measures

Screenlines Example34

Share of counts per screenline

> 75%

Share of counts per screenline

50% - 75%

< 50%

Model Validation Example35

All vehicles all day

R² = 0.7826

0

20,000

40,000

60,000

80,000

100,000

120,000

140,000

0 20,000 40,000 60,000 80,000 100,000 120,000 140,000

Sim

ula

tio

n

Count

VMT based comparison36

0

10

20

30

40

50

60

Inte

rsta

te

Fre

eway

s/Ex

pre

ssw

ays

Oth

er P

rin

cip

alA

rte

rial

Min

or

Art

eri

al

Co

llect

or

Loca

l

Dai

ly V

MT

Mill

ion

s

Observed Modeled

Volume class comparison37

0%

20%

40%

60%

80%

100%

120%

140%

<5,000 5,000-10,000

10,000-20,000

20,000-30,000

30,000-40,000

40,000-50,000

50,000-60,000

60,000-70,000

70,000-80,000

80,000-90,000

90,000-100,000

>100,000

%R

MS

E

Volume Class

State of Practice38

Source: Freight demand modeling:

Tools for public sector decision making

QRFM based

State of Practice (sample only)39

FAF AON (older versions) Freight only SUE (TransCAD based)

Florida Multiclass UE

Indiana QRFM

Iowa AON

Maryland Multiclass UE

Oregon Multiclass SUE

Ohio Multi UE

Wisconsin Multiclass UE