assignment models - didatticaweb

Prof. Ing. Antonio Comi Teoria dei Sistemi di Trasporto 1 + 2 (9 CFU)

Teoria dei Sistemi di Trasporto 1 + 2 (6/9 CFU)

A. A. 2020 - 2021

Assignment models

(elements)

prof. ing. Antonio Comi

Department of Enterprise Engineering

University of Rome Tor Vergata

Prof. Ing. Antonio Comi Teoria dei Sistemi di Trasporto 1 + 2 (9 CFU)Prof. Ing. Antonio Comi Teoria dei Sistemi di Trasporto 1 + 2 (9 CFU)

Transportation system model structure

2

TRANSPORTATION SYSTEM MODEL

TRANSPORT

FACILITIES AND

SERVICES

MODELS OF ACTIVITY

LOCATION AND

LEVEL

ACTIVITY

SYSTEM

Level of Service attributes

(times, costs)

ASSIGNMENT

MODELS

DEMAND MODEL

O/D MATRICES

Performance functions

SUPPLY MODEL

Transportation

networks

Flows

Evaluation


Assignment models

3

They simulate the interaction of demand and supply on a

transportation network allowing:

➢ calculation of user flows

➢ calculation of performance measures for each supply element

(network link)

resulting from origin–destination (O-D) demand flows, path choice

behaviour, and the mutual interactions between supply and demand.

Assignment model outputs are inputs required for design and/or

evaluation of transportation projects.


Classification criteria

The system state simulated through assignment models depends on

assumptions about user behaviour (demand functions, path choice,

available information) and the approach used for representing supply–

demand interactions.

✓ Supply-demand interactions➢ Equilibrium assignment

➢ Process assignment

✓ Cost functions➢ Congested networks

➢ Not-Congested networks

✓ Path choice behaviour➢ Deterministic model

➢ Stochastic model

4


Classification [1/3]

Demand-supply interaction

The user equilibrium (UE) represents equilibrium configurations of

the system, that is, configurations in which demand, path, and link

flows are consistent with the costs that they produce in the network.

From a mathematical point of view, equilibrium assignment can be

defined as the problem of finding a flow vector that reproduces itself

based on the correspondence defined by the supply and demand

models (fixed point models, or else with variational inequality or

optimization models).

5




The alternative approach for representing supply–demand

interactions leads to between-period (or day-to-day) dynamic process

(DP) assignment models.

It is assumed that the system evolves over time

✓ through possibly different feasible states,

✓ as a result of changes in the number of users undertaking trips, path choices,

supply performance, and so on.

One of the mechanisms that drives the changes from one state to

another is the dependency between flows and costs.

6




In a given reference period, the system state – defined by the demand,

path, and link flows and the corresponding costs – may be internally

inconsistent, and this may cause a change towards a different state in

the following reference periods.

Dynamic process assignment models explicitly simulate the evolution

of the system state based on the mechanisms underlying path choice

and information acquisition, which in turn determine user choices in

successive reference periods.

By analogy, equilibrium assignment could be termed within-day

static assignment.

7


Assignment models classification

8

DNL = Deterministic Network Loading

AoN = All or Nothing

SNL = Stochastic Network Loading

DUE = Deterministic User Equilibrium

SUE = Stochastic User Equilibrium

PATH CHIOCE MODEL

DETERMINISTIC STOCHASTIC

NOT CONGESTED

NETWORKDNL (AoN) SNL

CONGESTEDEQUILIBRIUM DUE SUE

NETWORK DYNAMIC

PROCESSDDP SDP

DDP = Deterministic Dynamic Process

SDP = Stochastic Dynamic Process



9

ASSIGNMENT MODELS Path choice model

Deterministic Stochastic

c = cost DNL (AoN) SNL

c = c(f) equilibrium DUE SUE







Assignment with inelastic and elastic

demandAssignment models with inelastic/elastic demand assume that O-D

demand flows are independent/dependent of changes in network

costs that may occur as a result of congestion.

Elastic demand

demand flows vary with congestion costs; demand flows are therefore a function

of path costs resulting from congestion, as well as of activity system attributes.

Depending on the modelling context, demand might be assumed variable in

certain choice dimensions only. For example, it might be assumed that the total

O-D matrix is cost-independent (meaning that frequency and destination choices

are not influenced by cost variations), but that mode choice is affected by the

relative costs of the available modes; in this way, multimode assignment models

are obtained. Obviously, from a practical viewpoint, demand elasticity is relevant

only for congested networks where costs depend on flows.

10


Schematic representation of assignment

models

11


Fields of Application

✓ Assignment models as estimators of the present state (monitoring)

of the transportation systemThe assignment model is applied to estimate other quantities that would be too.

✓Assignment models for simulating the effects of modifications to

the transportation system.

Assignment models are used to estimate the changes in relevant network

variables due to changes in supply and/or demand.

✓Assignment models for the estimation (or update) of travel demand.

Assignment models are seeing increasing application for the estimation of O-D

demand flows and/or for the calibration of demand models.

12


Definitions and assumptionsRelationship between link costs and path costs

13

Let:

o be a origin centroid node;

d be a destination centroid node;

od be an origin–destination pair;

Iod be the set of paths for O-D pair od; each path k is uniquely

associated with one and only one O-D pair od;

A be the overall link–path incidence matrix;

c be the link cost vector, with entries cl;

C be the overall path cost vector, with entries Cod (the path costs

for users of O-D pair od);

C = AT c


Definitions and assumptions

101101)4,3(5

010010)4,2(4

001001)3,2(3

000100)3,1(2

000011)2,1(1

654321

=A

14

+

+

+

++

=

===

=

34

24

3423

3413

2412

342312

5

4

3

2

1

6

5

4

3

2

1

10000

01000

10100

10010

01001

10101

cc

cc

cc

cc

ccc

c

c

c

c

c

cAC

C

C

C

C

C

C

C TADD

G (N,L)

N (1, 2, 3,4)

L (1, 2),(1, 3),(2, 3),(2, 4),(3, 4)

Origins1, 2, 3 Destination4

6

Relationship between link costs and path costs C = AT c

1

2

3

4

GRAPHPATHS

3

4

4

2

3

41

2

3

41

2

451

2

42

3

1 43

1

2

3

4

5

=

=

1

2

3

1

2

10000

01000

10100

10010

01001

10101

1

2

4

2

4

6cAC TC = AT c



15

For not congested networks:

c = cost

For congested networks:

c = c(f)

C = AT c(f)

Let:

f be the link flow vector, with entries, with entries fl;

F be the overall vector of path flows, consisting of the vectors of

path flows Fod related to each pair od;

will be:

f = A F

0,0

0,5

1,0

1,5

2,0

2,5

3,0

180 540 900 1260 1620 1980 2340 2700

flusso

co

sto

g=2

g=3

g=4



101101)4,3(5

010010)4,2(4

001001)3,2(3

000100)3,1(2

000011)2,1(1

654321

=A

16

=

=

800

1321

179

867

117

16

101101

010010

001001

000100

000011

1861

1439

195

867

133

FAf

G (N,L)

N (1, 2, 3, 4)

L (1, 2),(1, 3),(2, 3),(2, 4),(3, 4)

Origins1, 2, 3 Destination 4

1

2

3

GRAPHPATHS

1

2

3

4

5

Relationship between link flows and path flows f = A F

6

4

3

4

4

2

3

41

2

3

41

2

451

2

42

3

1 43


Path choice model

17

d whose components are the demand values dod for each O-D pair.

Path choice behaviour

Random Utility Model

p[k/od] = Prob[Vk - Vj j - k j Є Iod] od,k

Deterministic utility model Stochastic utility model

0=ε 0ε


Path choice model

18

pod = pod(Cod) od

P = P(C)

pod be the vector of path choice probabilities for users of O-D pair

od;

P the matrix of path choice probabilities for users of O-D pair

od, with a column for each O-D pair od and a row for each

path k;

the generic entry is p[k/od] if exist path k between od, 0

otherwise.


1-4 2-4 3-4

1 0.016 0 0

2 0.117 0 0

3 0.867 0 0

4 0 0.119 0

5 0 0.881 0

6 0 0 1.000

Paths

Path choice matrix

19

( )( )

od,k

od,nn

exp Cp k / od

exp C

1

−=

−

=

14 24 34

0.0160.119

= 0.117 ; p = ; = 1.0000.881

0.867

p p

O-D

HP: stochastic path choice model.

6

4

2

4

2

1

=

C

P


Path flow

20

Fk = dod p[k/od]

F = P (C) d

where

Fk = flow on path k

dod = demand flow on od pair

p[k/od] = path k (from o to d) choice probability

F = path flow vector

P = matrix of path choice probabilities

C = path cost vector

d = demand vector

A = link-path incidence matrix

c = link cost vector


Assignment model

21

Fk = dod p[k/od]

F = P (C) d

f = A F = A P (C) d

F = path flow vector;

P = matrix of path choice probabilities;

C = path cost vector;

d = demand vector;

f = flow cost vector;

A = link-path incidence matrix;

c = link cost vector.


Uncongested network assignment models

22


SNL assignment modelsPath flows

23

( )( )

1α Cα

Cαp

nodn

kodkod =

−

−=

= ,

,,

exp

exp

1

2

4

2

4

6

C

000.1p ;881.0

119.0p ;

867.0

117.0

016.0

p 342414 =

=

=

Percorsi

Coppie O-D 1-4 2-4 3-4

1 0.016 0 0

2 0.117 0 0

3 0.867 0 0

4 0 0.119 0

5 0 0.881 0

6 0 0 1.000

=

=

800

1500

1000

000.100

0881.00

0119.00

00867.0

00117.0

00016.0

800

1321

179

867

117

16dPF

HP: stochastic path choice model

HP: constant link costs

( )( )

od,k

od,nn

exp Cp k / od

exp C

1

−=

−

=


SNL assignment modelsPath flows

24

fNL = fNL(c; d) = A P(AT c) d c

16133 1 1 0 0 0 0

117867 0 0 1 0 0 0

867195 1 0 0 1 0 0

1791438 0 1 0 0 1 0

13211862 1 0 1 1 0 1

800

=

f = A · F


SNL assignment models

25

d1-4 = 1000

16

1616

3

2

1 4

2

1 4

117 117

3

1 4

867 867

d2-4 = 1500

1

3

2

4179

179

2

4

1321

d3-4 = 800

3

2

1 4

800

Link flows

3

2

1 4

1862867

195

1438133


DNL (AoN) assignment modelsPath flow

26

1 ;1

0 ;

1

0

0

1

2

4

2

4

6

342414 =

=

=

= pppC

1-4 2-4 3-4

1 0 0 0

2 0 0 0

3 1 0 0

4 0 0 0

5 0 1 0

6 0 0 1

PathsO-D

=

=

800

1500

1000

100

010

000

001

000

000

800

1500

0

1000

0

0dPF

HP: deterministic path choice model

HP: constant link costs


DNL (AoN) assignment modelsLink flows

27

fNL = fNL(c; d) = A P(AT c) d c

=

800

1500

0

1000

0

0

101101

010010

001001

000100

000011

1800

1500

0

1000

0

f = A · F


DNL (AoN) assignment modelsPath flow

28

d1-4 = 1000

0

00

3

2

1 4

2

1 4

0 0

3

1 4

1000 1000

d2-4 = 1500

1

3

2

40

0

2

4

1500

d3-4 = 800

3

2

1 4

800

Link flows

3

2

1 4

18001000

0

15000



29

ASSIGNMENT MODELS Path choice

Deterministic Stochastic

c = cost DNL (AoN) SNL

c = c(f) equilibrium DUE SUE







Assignment models classificationEquilibrium

30


Assignment models classificationEquilibrium

31

c = c (f)

f = A F

c = c (A F)

circular dependence

between flows and

costs

F* = P [AT c (A F*)] d

f* = A P [AT c (f*)] d

The state, or configuration, of a transport network in a specific

reference time (for example in the morning rush hour of a day),

can be defined by the path flows vector F, from this it is

possible going back to the link flows vector f and then to the

link costs and path costs vector.


Equilibrium problem: notionsExample

32

od

1

2

veh100

hodd =

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100f1

c1

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100f2

c2

( ) ffc 111 5.030 +=

( ) ffc 222 5.050 +=


Equilibrium problem: notionsExample

33

55107590

500805030100

22

11

22

11

=→==→=

−−−−−−−−−−−=→=

=+=→=

C fC f

C fCf

6530

6570

22

11

=→=

=→=

Cf

Cf

c

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

c1

c2

f1

40 20 06080100

f2

70

30

c

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