1

Static and Static and Dynamic Dynamic NetworksNetworks

Patrizia Daniele Patrizia Daniele Department of Mathematics and Department of Mathematics and

Computer SciencesComputer SciencesUniversity of Catania, ITALYUniversity of Catania, ITALY

Visiting Scholar at DEAS (Harvard Visiting Scholar at DEAS (Harvard University) University)

Spring Semester 2006Spring Semester 2006

2

Edward Elgar PublishingEdward Elgar Publishing August 2006.August 2006.

3

AcknowledgementsAcknowledgements1 Introduction1 Introduction 2 The Traffic Equilibrium Problem 2 The Traffic Equilibrium Problem

2.1 Static Case 2.1 Static Case 2.2 Dynamic Case2.2 Dynamic Case2.3 Existence Theorems 2.3 Existence Theorems 2.4 Additional Constraints2.4 Additional Constraints2.5 Calculation of the Solution2.5 Calculation of the Solution2.6 Delay and Elastic Model2.6 Delay and Elastic Model2.7 Sources and Remarks 2.7 Sources and Remarks

3 Evolutionary Spatial Price 3 Evolutionary Spatial Price EquilibriumEquilibrium3.1 The Price Formulation3.1 The Price Formulation3.2 The Quantity Formulation3.2 The Quantity Formulation3.3 Economic Model for Demand-3.3 Economic Model for Demand-Supply MarketsSupply Markets3.4 Sources and Remarks3.4 Sources and Remarks

4 The Evolutionary Financial 4 The Evolutionary Financial ModelModel4.1 Quadratic Utility Function4.1 Quadratic Utility Function4.2 General Utility Function4.2 General Utility Function4.3 Policy Intervention4.3 Policy Intervention4.4 A Numerical Financial 4.4 A Numerical Financial ExampleExample4.5 Sources and Remarks 4.5 Sources and Remarks

5 Projected Dynamical Systems 5 Projected Dynamical Systems 5.1 Finite-Dimensional PDS 5.1 Finite-Dimensional PDS 5.2 Infinite-Dimensional PDS5.2 Infinite-Dimensional PDS5.3 Common Formulation5.3 Common Formulation5.4 General Uniqueness Results5.4 General Uniqueness Results5.5 The Relation Between the 5.5 The Relation Between the Two TimeframesTwo Timeframes5.6 Computational Procedure5.6 Computational Procedure5.7 Numerical Dynamic Traffic 5.7 Numerical Dynamic Traffic ExamplesExamples5.8 Sources and Remarks5.8 Sources and Remarks

A Definitions and PropertiesA Definitions and PropertiesB Weak ConvergenceB Weak ConvergenceC Generalized DerivativesC Generalized DerivativesD Variational InequalitiesD Variational InequalitiesE Quasi-Variational InequalitiesE Quasi-Variational InequalitiesF Infinite Dimensional DualityF Infinite Dimensional DualityBibliographyBibliographyIndexIndex

4

Wardrop J.G. (1952), “Some theoretical aspects of road traffic research”, Proceedings of the Institute of Civil Engineers, Part II, pp. 325-378.

Beckmann M.J., McGuire C.B. and Winsten C.B. (1956), Studies in the Economics of Transportation, Yale University Press, New Haven, Connecticut.

Dafermos S.C. and Sparrow F.T. (1969), “The traffic assignment problem for a general nerwork”, Journal of Research of the National Bureau of Standards 73B, 91-118.

Smith M.J. (1979), “The Existence, Uniqueness and Stability of Traffic Equilibrium”, Transportation Research, 138, 1979, 295-304.

Dafermos S. (1980), “Traffic equilibria and variational inequalities”, Transportation Science 114, 42-54.

Kinderlehrer D. and Stampacchia G. (1980), An Introduction to Variational Inequalities and Their Applications, Academic Press, New York.

5

Static Traffic NetworkStatic Traffic Network

Links: , : , 1, 2, , : 1, 2, ,i j iA P P i j p a i n

1 2Nodes: , , , pN P P P

O/D Pairs: : 1,2, ,jW w j l N N

path connecting 1, ,j r jR w j l

Assumptions: 1, , and j j l m l

6

ExampleExample

P4

P1

P3

P2

7

1Path Flow Vector: , , mmF F F R

1, ,

1

Set of all paths: l

j r r mj

R

1, ,1, ,

Link-Path Incidence Matrix:

1 if

0 if

i nirr m

i rir

i r

a R

a R

1Link Flow Vector: , , nnf f f R

8

1

Connection between and : m

i ir rr

f F

f F f F

1

Path Cost Vector:

, , mmC F C F C F R

1

Link Cost Vector:

, , nnc f c f c f R

9

Travel Demand: 0, 1, ,j j l

1 1

Connection between and :

'

n n

r ir i ir ii i

c f C F

C F c f c F

C F c F

1, ,1, ,

O/D Pair-Path Incidence Matrix:

1 if

0 if

j ljrr m

r j

jrr j

R

R

10

Wardrop’s Principle (1952):Wardrop’s Principle (1952):

Set of Feasible Flows: :mK F F R

1

Traffic Conservation Law:

1, ,m

jr r jr

F j l F

is an equilibrium , , ,

if ( ) ( ) 0

j j

r s r

H K w W r s

C H C H H

11

Theorem (Smith, 1979):Theorem (Smith, 1979):

Existence TheoremExistence Theorem

Theorem 1:Theorem 1:

1

is an equilibrium solves the Variational Inequality

, 0 0m

r r rr

H K H

C H F H F K C H F H

convex, bounded, closed, nonempty

: continuous

: , 0

n

n

K

C K

H K C H F H F K

R

R

12

Existence and Uniqueness TheoremsExistence and Uniqueness Theorems

Theorem 2:Theorem 2:

Theorem 3:Theorem 3:

convex, bounded, closed, nonempty

: contin andstrictly monotoneuous

! : , 0

n

n

K

C K

H K C H F H F K

R

R

2

convex, closed, nonempty; : s.t.

, , 0

,

! : , 0

nn

nn

n nK C K

C x B y x y x y

C x C y M x y M

H K C H F H F K

RR

RR

R R

13

Direct MethodDirect Method

1 1

1, , 0, 1, ,m m

jr r j j j jr rr r l

F j l F F j l

11

, , : 0, 1, , ; , 1, ,m

m ll m r jr r j

r l

K F F F F r l m F j l

R

: , 0 :mH K C H F H F K F F R

Initial VI:Initial VI:

New feasible set:New feasible set:

New VI:New VI:

1 1

: 0,m l

r jr j r rr l j

H K C H C H F H F K

14

0H

H K

: , 0H K H F H F K

Step 1:Step 1:

Step 2 (First Type Face):Step 2 (First Type Face):

1

1

0

: 0, 1, , , ; 0;

, 1, ,

ss

s m lr s

m

jr r jr lr s

K K F

F F r l m r s F

F j l

R

15

New VI: New VI:

Step 3 (Second Type Face):Step 3 (Second Type Face):

0and 0

s s

ss

s s

HH

H K

: , 0s s s s s s s sH K H F H F K

1

1 111

1 and m

jj r r j

r l

K K j l F

16

1 1

1

11 1

0and 0

j j

jl

j j

HH

H K

1 1

1 1

1 1

11

11 1

: 0, ;

, 1, , , ;

j j m lr

m m

jr r j j r r jr l r lr l r l

K F F r l

F j l j j F

R

New VI:New VI:

Step 4: Step 4: mixture of steps 2 and 3mixture of steps 2 and 3

1 1 1 1 1 1 1 1: , 0,j j j j j j j jH K H F H F K

17

Numerical ExampleNumerical Example

P4

P1

P3

P2

1 2 1 2

1 3 1 3

3 2 3 2

3 4 3 4

4 2 4 2

1 2

1 2

0 199

,

2

2

1

2

, 100

P P P P

P P P P

P P P P

P P P P

P P P P

W P P

c f

c f

c f

c f

c f

P P

1 1 2,R P P

2 1 3 3 2, ,R P P P P

3 1 3 3 4 4 2, , ,R P P P P P P

18

1 1

2 2 3

3 2 3

2

2 2 1

3 2 2

C F F

C F F F

C F F F

31 2 3 1 2 3 1 2 3

3 1 2

21 2 1 2 1 2

, , : , , 0, 100

100

, : , 0, 100

K F F F F F F F F F F

F F F

K F F F F F F F

R

R

1 3 1 1 2 3 2 2

1 1 1 2 2 2

0

0

C H C H F H C H C H F H

H F H H F H F K

VI:VI:

19

1

1 1

1

1 1

11

2 2

3

1 2 1

1

2 2

3

3 5040 11

401 20 if 0 195 :

11195

11if 195 199 :

0 100 where 100

201

40 199

and 040

j

j j

j

j j

HH

H H

H KH

K F F F

H

HH H

H KH

20

Dynamic Traffic NetworkDynamic Traffic Network

Ran B. and Boyce D.E. (1996), Ran B. and Boyce D.E. (1996), Modeling Dynamic Modeling Dynamic Transportation Networks: an Intelligent System Transportation Networks: an Intelligent System Oriented ApproachOriented Approach, Second Edition, Springer-Verlag, , Second Edition, Springer-Verlag, Berlin, Germany.Berlin, Germany.

Daniele P., Maugeri A., Oettli W. (1999), Daniele P., Maugeri A., Oettli W. (1999), “Time-“Time-Dependent Variational Inequalities”Dependent Variational Inequalities”, , Journal of Journal of Optimization Theory and ApplicationsOptimization Theory and Applications, , 104104, 543-555., 543-555.

Raciti F. (2001), Raciti F. (2001), “Equilibrium in Time-Dependent Traffic “Equilibrium in Time-Dependent Traffic Networks with Delay”Networks with Delay”, in , in Equilibrium Problems: Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Nonsmooth Optimization and Variational Inequality ModelsModels, F. Giannessi, A. Maugeri and P.M. Pardalos , F. Giannessi, A. Maugeri and P.M. Pardalos ((edseds)), Kluwer Academic Publishers, The Netherlands, , Kluwer Academic Publishers, The Netherlands, 247-253.247-253.

Scrimali L. (2004), Scrimali L. (2004), “Quasi-Variational Inequalities in “Quasi-Variational Inequalities in Transportation Networks”Transportation Networks”, , M3AS: Mathematical Models M3AS: Mathematical Models and Methods in Applied Sciencesand Methods in Applied Sciences, , 1414, 1541-1560., 1541-1560.

21

Generalized Wardrop’s Principle:Generalized Wardrop’s Principle:

0,TT

is a dynamic equilibrium

, , and a.e. in

if

or

q s

q q s s

H K

w W q s w

C H t C H t

H t t H t t

T

Set of Feasible Flows:

: , a.e. inK F t F t t F t t L T

, , 1pL p RRL T

22

is a dynamic equilibrium

, 0

, 0

H K

C H F H F K

C H t F t H t dt

T

*, , , ,G F G t F t dt G F T L L

Canonical Bilinear form on Canonical Bilinear form on LL* * xx LL:

Theorem (Variational Formulation)Theorem (Variational Formulation)

23

Preliminary DefinitionsPreliminary Definitions

EE real topological vector space, real topological vector space, K K E E convex, convex, C: K → EC: K → E**: :

CC hemicontinuoushemicontinuous if if upper semicontinuous on upper semicontinuous on KK;;

CC hemicontinuoushemicontinuous along line segmentsalong line segments if if

upper semicontinuous on the upper semicontinuous on the line segment [line segment [xx, , yy];];

CC pseudomonotonepseudomonotone if if

, ,y K C y

, ,x y K xyC ,

, ,x y K

, 0 , 0.y x C x y x C y

24

Existence TheoremsExistence Theorems

Theorem 1 (Daniele, Maugeri and Oettli, 1999)Theorem 1 (Daniele, Maugeri and Oettli, 1999)

XX real topological vector space, real topological vector space, KK XX nonempty and nonempty and convex; convex;

FF:: K→ X K→ X*:*:

1.1.

2.2. FF pseudomonotone and hemicontinuous along line pseudomonotone and hemicontinuous along line segments.segments.

Then, Then,

nonempty and compact, compact and convex:

\ : , 0;

A K B K

x K A y B F x y x

: , 0x A F x y x y K

25

Theorem 2 (Daniele, Maugeri and Oettli, Theorem 2 (Daniele, Maugeri and Oettli, 1999)1999)

XX real topological vector space, real topological vector space, KK XX nonempty nonempty and convex; and convex;

FF:: K→ X K→ X*:*:

1.1.

2.2. FF hemicontinuous. hemicontinuous.

Then, Then,

nonempty and compact, compact and convex:

\ : , 0;

A K B K

x K A y B F x y x

: , 0x A F x y x y K

26

Corollary (Daniele, Maugeri and Oettli, Corollary (Daniele, Maugeri and Oettli, 1999)1999)

KK L L and and CC:: K→ K→ LL **. . Each of the following Each of the following conditions is sufficient for the existence of conditions is sufficient for the existence of the solution to the EVI:the solution to the EVI:

1.1. CC strongly hemicontinuous on strongly hemicontinuous on KK and and AA K K nonempty and compact and nonempty and compact and BB K K compact:compact:

2.2. CC weakly hemicontinuous on weakly hemicontinuous on K;K;

3.3. CC pseudomonotone and hemicontinuous pseudomonotone and hemicontinuous along line segments.along line segments.

\ , : , 0;H K A F B C H F H

27

Spatial Market NetworksSpatial Market NetworksNagurney A. (1993), Nagurney A. (1993), Network Economics - A Network Economics - A Variational Inequality ApproachVariational Inequality Approach, Kluwer Academic , Kluwer Academic Publishers, Dordrecht, The Netherlands.Publishers, Dordrecht, The Netherlands.

Nagurney A. and Zhang D. (1996), Nagurney A. and Zhang D. (1996), “Stability of Spatial “Stability of Spatial Price Equilibrium Modeled as a Projected Dynamical Price Equilibrium Modeled as a Projected Dynamical System”System”, , Journal of Economic Dynamics and ControlJournal of Economic Dynamics and Control, , 2020, , 43-63.43-63.

Daniele P. (2001), Daniele P. (2001), “Variational Inequalities for Static “Variational Inequalities for Static Equilibrium Market: Lagrangean Function and Duality”Equilibrium Market: Lagrangean Function and Duality”, , in in Equilibrium Problems: Nonsmooth Optimization Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Modelsand Variational Inequality Models, F. Giannessi, A. , F. Giannessi, A. Maugeri and P. Pardalos Maugeri and P. Pardalos ((edseds)), Kluwer Academic , Kluwer Academic Publishers, Dordrecht, The Netherlands, 43-58.Publishers, Dordrecht, The Netherlands, 43-58.

Daniele P. (2003), Daniele P. (2003), “Evolutionary Variational Inequalities “Evolutionary Variational Inequalities and Economic Models for Demand-Supply Markets”and Economic Models for Demand-Supply Markets”, , M3AS: Mathematical Models and Methods in Applied M3AS: Mathematical Models and Methods in Applied SciencesSciences, , 44, no. 13, 471-489., no. 13, 471-489.

28

Economic ModelEconomic Model

P1 PnP2

Qm

Q2Q1

Supply Markets

Demand Markets

…

……

…

29

g g RRnn: total supply vector: total supply vector p p RRnn: total supply price vector: total supply price vector

f f RRmm: total demand vector: total demand vector q q RRmm: total demand price vector: total demand price vector

x x RRnmnm: flow vector: flow vector c c RRnmnm: flow cost vector: flow cost vector

s s RRnn: supply excess vector: supply excess vector ττ RRmm: demand excess vector: demand excess vector

Static CaseStatic Case

30

1 1

1, , and 1, ,m n

i ij i j ij jj i

g x s i n f x j m

1 2 31 1 1 1

, 0, 0,n m n m

i j iji j i j

K p q x K K K

Set of feasible vectors:Set of feasible vectors:

Given functions:Given functions:

1 2 3: , : , :

, ,

n m nmg K f K c K

g g p f f q c c x

R R R

31

Definition (Equilibrium Conditions)Definition (Equilibrium Conditions)

, , equilibrium

0 ; 0 1, , ;

0 ; 0 0 1, , ;

if 0

if 0 1,..., ; 1,...,

if

i i i i i i

j j j j j j

j ij

iji ij j

ijj ij

u p q x

s p p p p s i n

q q q q j m

q x

p c q x x i n j m

q x x

32

Theorem (Variational Formulation)Theorem (Variational Formulation)

* * * * * *

* * * * * *

1 1 1 1

* * * *

1 1

, , equilibrium , 0

that is:

0

n m m n

i ij i i j ij j ji j j i

n m

i ij j ij iji j

u p q x K v u u u u K

g p x p p f q x q q

p c x q x x u K

33

Dynamic CaseDynamic Case g(t)g(t):[:[0,T0,T] → ] → LL22([([0,T0,T]], , RRnn)): total supply function: total supply function p(t)p(t):[:[0,T0,T] → ] → LL22([([0,T0,T]], , RRnn)): supply price function: supply price function

f (t)f (t):[:[0,T0,T] → ] → LL22([([0,T0,T]], , RRmm)): total demand function: total demand function q(t)q(t):[:[0,T0,T]] → → LL22([([0,T0,T]], , RRmm)): demand price function: demand price function

x(t)x(t):[:[0,T0,T]] → → LL22([([0,T0,T]], , RRnmnm)): flow function: flow function c(t)c(t):[:[0,T0,T]] → → LL22([([0,T0,T]], , RRnmnm)): unit cost function: unit cost function

s(t)s(t):[:[0,T0,T]] → → LL22([([0,T0,T]], , RRnn)): supply excess function: supply excess function ττ(t)(t):[:[0,T0,T]] → → LL22([([0,T0,T]], , RRmm)): demand excess function: demand excess function

34

1

1

1, , and

1, ,

m

i ij ij

n

j ij ji

g p t x t s t i n

f q t x t t j m

1 2 3

1 1 1 1

, , ,n m n m

iji j iji ji j i j

K K K K

p t p t q t q t x t x t

Set of feasible vectors:Set of feasible vectors:

Given functions:Given functions:

21

22

23

: 0, , ,

: 0, , ,

: 0, ,

n

m

nm

g p t K L T

f q t K L T

c x t K L T

R

R

R

35

Definition (Equilibrium Conditions)Definition (Equilibrium Conditions)

, , dynamic equilibrium

1, , , 1, , and a.e. in 0, :

0 ; 0;

0 ; 0;

if

if

if

i i i i iii

j j j j j jj

ijj ij

ijiji ij j

ijj ij

u t p t q t x t K

i n j m T

s t p t p t p t p t p t s t

t q t q t q t q q t

q t x t x t

p t c t q t x t x x t

q t x t x t

36

Theorem (Variational Formulation)Theorem (Variational Formulation)

* * * *

* *

* * *

01 1

* * *

1 1

* * * *

1 1

, , dynamic equilibrium

, 0 , ,

that is:

0

n mT

i ij i ii j

m n

j ij j jj i

n m

i ij j ij iji j

u t p t q t x t K

v u u u u t p t q t x t K

g p t x t p t p t

f q t x t q t q t

p t c x t q t x t x t dt

u t p t

, ,q t x t K

37

Theorem (Existence)Theorem (Existence)KK LL andand vv:: K→ K→ LL*. *. Each of the following conditions Each of the following conditions

is sufficient for the existence of the solution to the is sufficient for the existence of the solution to the EVI:EVI:

1.1.

2.2.

3.3.

1 1 2 2 3 3

1 1 2 2 3 3

1 1 1 2 2 3 3 2 1 2 3

1 2 1

strongly hemicontinuous and , ,

compact and , , convex and compact:

\ \ \ , :

, 0;

pseudomonotone and hemicontinuous

along line segments;

v u A K A K A K

B K B K B K

u K A K A K A u B B B

v u u u

v u

v

weakly hemicontinuous.u

38

General Formulation of the set General Formulation of the set KK for traffic network problems, for traffic network problems,

spatial price equilibrium spatial price equilibrium problems, financial equilibrium problems, financial equilibrium

problemsproblems

2

1

0, , : ( ) ( ) ( ) a.e.in 0, ;

( ) ( ) a.e. in 0 , 0,1 , 1,..., , 1,...,

, , 0, , : given functions

q

q

ji i j jii

p q

u t L T t u t t T

u t t ,T i q j l

L T

K R

R

39

Discretization ProcedureDiscretization Procedure

1

0 1 0 1

1

,

Partitions of 0, :

, , , ,0

max : 1, , , lim 0.

0, , 0, , : ,

1, ,

n n

j jn n

N Nn n n n n n n

j jn n n n n

n

m m mn jt t

n

T

t t t t t t T

t t j N

P T v L T v v

j N

R R R

40

1 1

2

1,

2

2

1:

0, , : , a.e. in 0, ,

, , 0

, ,

0, , , 0, ,

jn

j j jn n n

t

n j jt t tn n

m

m m

v v s dst t

F t L T F t T

F t

C t F t A t F t B t

A t L T B t L T

K R

R R

41

1

1

0

1

,

,

:

, 0,

jnn

jn

jn

jn

T

Nt

tj

nj

t n n nj j jt

nj

A t H t B t F t H t dt

A t H t B t F t H t dt

H t

A t H t B t F t H t dt

F t

K

K

42

1 11 1

1

1

Finite-dimensional Variational Inequality:

: , 0,

where

1 1; .

, piecewise constant

approximation to th

j jn n

j jn n

n

n n n n n n nj m j j j j j j m

t tn nj jj j j jt t

n n n n

Nj j n

n n n jj

H A H B F H F

A A t dt B B t dtt t t t

H t t t H

K K

e solution of EVI

43

TheoremTheorem positive definite a.e. in 0, the set

(weakly) compact, with feasible cluster points solutions

to the EVI.

n nA t T U H

2

, , 1

, 1

0, , : ( ) ( ) ( ) a.e.in 0, ;

, 0, ( ) a.e. in 0

: : a.e.in , ;

a.e. in ,

m

n mj n j nn j j j j j

j nj j j

F t L T t F t t T

t t F t t ,T

F F t t

F t t

K R

K R

44

LemmaLemma nn j

j K K K

0

Initial problem:

, , 0T

H t t

C t H t F t H t dt F t t

K

K

TheoremTheorem

1

1

positive definitea.e. in 0, ,

has (weakly) cluster points solutions to the initial problem.

nNj j n

n n n jj

A t T H t t t H

45

0

0

strictly monotone

0,T , uniquesolution to the EVI:

, 0, 0,

: solving the initial value problem:

, , , ,

0,

q

F

u t C

F u t v t u t t T

x x F x

x x

K

R

R K K

K

Continuity assumptionContinuity assumption

Projected Dynamical System Projected Dynamical System (PDS)(PDS)

46

Dupuis P. and Nagurney A. (1993), Dupuis P. and Nagurney A. (1993), “Dynamical Systems “Dynamical Systems and Variational Inequalities”and Variational Inequalities”, , Annals of Operations Annals of Operations ResearchResearch, , 4444, 9-42., 9-42.

Nagurney A. and Zhang D. (1996), Nagurney A. and Zhang D. (1996), Projected Dynamical Projected Dynamical Systems and Variational Inequalities with Systems and Variational Inequalities with ApplicationsApplications, Kluwer Academic Publishers, Boston, , Kluwer Academic Publishers, Boston, Massachusetts.Massachusetts.

Cojocaru M.G. (2002), Cojocaru M.G. (2002), “Projected Dynamical Systems on “Projected Dynamical Systems on Hilbert Spaces”Hilbert Spaces”, PhD Thesis, Queen’s University, Canada., PhD Thesis, Queen’s University, Canada.

Cojocaru M.G. and Jonker L.B. (2004), Cojocaru M.G. and Jonker L.B. (2004), “Existence of “Existence of Solutions to Projected Differential Equations in Hilbert Solutions to Projected Differential Equations in Hilbert Spaces”Spaces”, , Proceedings of the American Mathematical Proceedings of the American Mathematical SocietySociety, , 132132, 183-193., 183-193.

Cojocaru M.G., Daniele P. and Nagurney A. (2005), Cojocaru M.G., Daniele P. and Nagurney A. (2005), ““Projected Dynamical Systems and Evolutionary Projected Dynamical Systems and Evolutionary ((Time-Time-DependentDependent)) Variational Inequalities Via Hilbert Spaces Variational Inequalities Via Hilbert Spaces with Applications”with Applications”, , Journal of Optimization Theory and Journal of Optimization Theory and ApplicationsApplications, , 2727, no. 3, 1-15., no. 3, 1-15.

47

•Discretization the time interval [0,Discretization the time interval [0,TT];];•Sequence of PDS;Sequence of PDS;•Calculation of the critical points of the PDS;Calculation of the critical points of the PDS;•Interpolation of the sequence of critical points;Interpolation of the sequence of critical points;•Approximation of the equilibrium curve.Approximation of the equilibrium curve.

New Discretization New Discretization ProcedureProcedure

48

Traffic NetworkTraffic Network

2 4 2 2 20 1 1 2j j n

n n n n n

D

O

1

2

2 21 2

1 2

32 0

, , ,20 1

1

0,2 , :0 ; 0 3;

a.e.in 0,2

F ttA t F t B t

F t

F L F t t F t

F t F t t

K R

49

21 2 1 2

1 1 1 2 2 2

2 1

1 1 1

, 0

2 1 2 1:0 , 0 3,

2 1 32 1 0

2

2 1

2 1 3 2 12 1

2

n n n n n n nj j j j j j j

n n n n n nj j j j j j

n n n n n n n nj j j j j j j j

n nj j

n nj j j

A H B F H F

j jF F F F F

n n

jH F H H F H F

n

jF F

n

j jH H F

n n

K

K R

K

1 0n njH

50

1

2 22

1

2 1 1if 1 1

2 2 6

4 2 1if 1

2 3 2 1 2 2 6

10 if 1 1

2 2 6

4 2 1 2 2 1 1if 1

2 3 2 1 2 2 6

2 21 ,

nj

nj

nn

n jj

j nj

nH

j n nj

n j

nj

Hn j n j n

jn n j

H t j j Hn n

51

Direct MethodDirect Method

1 1 1 2 2 2

2 21

2 1 2

22 1 1 1

1 1 1 1

32 1 0

2

0,2 , : 0 ;

0 3; , a.e. in 0, 2

0,2 , :0

13 2 0

2

t H t F t H t H t F t H t

F t F t L F t t

F t F F t t

F t t F t F t L F t t

t H t t F t H t F t

K R

K R

K

52

1

2 2

1 2 1 2

6if 0 1

2

2 1 6if 1 2

2 6 2

60 if 0 1

2

2 4 1 6if 1 2

2 6 2

, ,n n

t t

H tt

tt

t

H tt t

tt

H t H t H t H t

53

54

Traffic NetworkTraffic Network

2 21 2

1 2

30,2 , : 0 ( ) ,0 a.e.in 0,2 ;

2

a.e. in 0 2

F t L F t t F t t

F t F t t ,

K R

0

D

1 1

2 2

Cost functions:

1

2

C H t H t

C H t H t

55

0

2 2 2 2

1 2 1 2

0

1 0 2 0 1 2

0 0 0

Vector field:

: 0,2 , 0,2 ,

, 1, 2 .

: 0, ,8 sequence of PDS defined by :4

, 1, 2 on

30, 0,

2t

F L L

F H t F H t H t H t

kt k

F H t H t H t H t

t t x y t

R R

K

56

tt00 HH11((tt00)) HH22((tt00))

00 00 00

1/41/4 1/41/4 00

1/21/2 1/21/2 00

3/43/4 3/43/4 00

11 11 00

5/45/4 9/89/8 1/81/8

3/23/2 5/45/4 1/41/4

7/47/4 11/811/8 3/83/8

22 3/23/2 1/21/2

1

2

Explicit formulas:

if 0 1

1if 1 2

2

0 if 0 1

1if 1 2

2

t tH t t

t

tH t t

t

0

0

21 0 2 0

1 0 2 0 1 0 2 0

Unique equilibrium at :

, :

, ,t

t

H t H t

F H t H t N H t H t

K

R

57