Lecture 4 The Basic Problem - Anna NagurneyLecture 4 The Basic Problem Dr. Anna Nagurney John F....

Lecture 4The Basic Problem

Dr. Anna Nagurney

John F. Smith Memorial ProfessorIsenberg School of Management

University of MassachusettsAmherst, Massachusetts 01003

c©2009

Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 4

The Basic Problem

The Basic Problem

Traffic Assignment or Traffic Equilibrium Problemwith Fixed Demands

Statement

Given the travel demands dw associated with every O/D pair andthe user’s travel cost ca(fa) for every link a, determine the pathflows Fp for all paths, and the link flows fa for all links.

Hence, given the demands dw , and travel costs, ca(fa), for all a,

compute the path flows Fp and the link flows fa.


The Basic Problem

We know that Path and Link flows must be feasible, i.e.

dw =∑p∈Pw

Fp

for all O/D pairs w .

(satisfies demand)

Also, Fp ≥ 0, for all paths p.

A load pattern f = fa ; a ∈ L, that is induced by a feasible flowpattern F via the equation:

fa =∑p

Fpδap,

for all a ∈ L is called feasible.Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 4

A Behavioral Assumption for Choosing Paths

A Noncooperative Game (Nash Equilibrium)

John F. Nashwww.search.tvnz.co.nz

Assume that a rational user will make a unilateral decision to change his path,if assuming that all other users are fixed, he can decrease his travel cost bychanging his path.

If no users have any incentive to change paths, the traffic assignment is said tobe

USER OPTIMIZED

or in

EQUILIBRIUM


Before giving a mathematical definition of an equilibrium state, wewill look at some examples.

Network Example

w = (x , y), dw = 30, ca(fa) = gafa + ha, wherega1 = 3, ha1 = 30ga2 = 2, ha2 = 20

Claim: The following is the user-optimized traffic pattern. Notethat here links & paths, hence link & path flows coincide.

Fp1 = 10, Fp2 = 20Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 4

The Basic Problem

With the computed travel costs along paths p1, p2:

ca1 = Cp1 = 60ca2 = Cp2 = 60.

∗ Observe that the costs on the paths are equal.

Assume that the flow δ > 0 switches from p1 to p2. Then, the newuser cost along path p2 (or link a2) would be:

ca2( ´fa2) = 2 (20 + δ) + 20

= 60 + 2δ > ca2(fa2) = 60.

So not cost-wise to have a switch from path p2 to p1.


The Basic Problem

Suppose we also want to evaluate a switch from p2 to p1. Then,the new user cost along path p1 (or link a1) would be:

ca1( ´fa1) = 3 (10 + δ) + 30

= 60 + 3δ > ca2(fa2) = 60.

Also, not cost-wise!

∗ Hence, the pattern (10,20) satisfies the definition and it isuser-optimized.


Suppose that we add another link, a3, to the network:

ca3(fa3) = fa3 + 80

Is the original flow pattern still U-O?

Why or why not?


The Basic Problem

Definition of U-O Equilibrium

We will now define when f is a U-O pattern.

To do this we must compare f with other flow patterns f ′ whichwe will construct as follows:

Consider any path r such that Fr > 0 (a used path) and choose anumber δ, such that 0 ≤ δ ≤ Fr .

Also, consider a path q that connects the same O/D pair of nodesas r and constructs F ′ from F as follows:


The Basic Problem

New flow pattern F ′:

F ′r = Fr − δ

F ′q = Fq + δ

F ′p = Fp, for all other paths.

Observe that F ′ is feasible.


The Basic Problem

Definition

We say that a U-O pattern F has been reached if for every pair ofpaths q, r with Fr > 0, (and w fixed), for every possible δ > 0, ifthe following inequalities hold:

Cr (f ) ≤ Cq(f ′) true for q, r ∈ Pw & for all w .

COMMENT

This is a difficult test. Moreover, we can’t compute theequilibrium via this method.


The Basic Problem

Wardrop’s Principle (1952)

Necessary and sufficient conditionsfor a User-Optimized Pattern (U-O)

A feasible flow pattern F and its induced link load pattern f is U-Oif and only if, for every O/D pair w there exists a numbering of thepaths p1, ..., ps , ps+1, ..., pm that connect w such that:

Cp1(f ) = Cp2(f ) = ... = Cps (f ) = vw ≤ Cps+1(f ) ≤ ...Cpm(f )

and

Fpr > 0, for pr ; r = 1, ..., s.

Fpr = 0, for pr ; r = s + 1, ...,m.


Computation of the U-O equilibrium for the following specialnetwork.

• O/D pair w = (x , y)

• m-disjoint paths

• fixed travel demand dw

• travel cost: ca(fa) = gafa + ha, Cp(f ) = ca(fa)


The Basic Problem

Suppose that we know the critical s. Then the U-O conditionstake the form:

ga1fa1 + ha1 = ga2fa2 + ha2 = ... = gas fas + has = vw

≤ gas+1fas+1 + has+1 ≤ ... ≤ gam fam + ham (I)

far = Fpr > 0; r = 1, ..., s,

far = Fpr = 0; r = s + 1, ...,m,


The Basic Problem

Observe that (I) can be rewritten as:

vw =fa1+

ha1ga1

1ga1

= ... =fas +

hasgas

1gas

≤fas+1+

has+1gas+1

1gas+1

≤ ... ≤fam+

hamgam

1gam

A fact from linear algebra:

x1y1

= x2y2

= ... =

∑i

xi∑i

yi

By using this fact from linear algebra:

vw =

S∑i=1

fai +s∑

i=1

hai

gai

s∑i=1

1

gai

=

dw+

s∑i=1

hai

gai

s∑i=1

1

gai


The Basic Problem

How do we compute the critical s?Suppose that we have ordered the hai ’s.

far = vw−hargar

, r = 1, ..., s

far = 0, r = s + 1, ...,m

We know that

vw−hargar

> 0; r = 1, ..., s

vw > har ⇒ vw > har > has

vw − har ≤ 0, for r = s + 1, ...,m⇒ vw ≤ has+1 .


The Basic Problem

Hence,

(∗) ha1 ≤ ha2 ≤ ... ≤ has−1 ≤ has ≤ vw ≤ has+1 ≤ ... ≤ ham .

Since

vw =

dw+

s∑i=1

hai

gai

s∑i=1

1

gai

Define

v rw =

dw+

r∑i=1

hai

gai

r∑i=1

1

gai


The Basic Problem

The Exact Equilibration Algorithm (U-O)

• Sort the hai ’s in non-decreasing order and relabel accordingly.

Set r = 1 and ham+i =∞.

• Compute

v rw =

dw+

r∑i=1

hai

gai

r∑i=1

1

gai


The Basic Problem

The Exact Equilibration Algorithm (U-O) (continued)

• CheckIf

har < v rw ≤ har+1 ,

then STOP.

Set the critical s = r ;

Fpr = v rw−hargar

; r = 1, ..., s

Fpr = 0; r = s + 1, ...,m.

Else set r = r + 1 and goto Step 2.


INFORMS Meeting, San Francisco (2005) (with Beckmann, McGuire and Boyce)

For more photos, see http://supernet.som.umass.edu/bmw-informs-2005/bmw50.html


http://supernet.som.umass.edu/bmw-informs-2005/bmw50.html

References

⇒ Dafermos S. and Sparrow F. (1969) The Traffic AssignmentProblem for a General Network. Journal of Research of the NationalBureau of Standards 73B:91-118

⇒ Boyce D., Mahmassani H., and Nagurney A. (2005) A Retrospectiveon Beckmann, McGuire and Winsten’s Studies in the Economics ofTransportation. Papers in Regional Science, 84 :85-103http://supernet.som.umass.edu/articles/jrs.pdf


http://supernet.som.umass.edu/articles/jrs.pdf

Lecture 4 The Basic Problem - Anna NagurneyLecture 4 The Basic Problem Dr. Anna Nagurney John F....

Documents

Transcript of Lecture 4 The Basic Problem - Anna NagurneyLecture 4 The Basic Problem Dr. Anna Nagurney John F....