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.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 4
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
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 4
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.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 4
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.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 4
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?
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 4
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:
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 4
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.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 4
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.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 4
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.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 4
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)
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 4
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,
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 4
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
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 4
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 .
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 4
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
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 4
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
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 4
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.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 4
INFORMS Meeting, San Francisco (2005) (with Beckmann, McGuire and Boyce)
For more photos, see http://supernet.som.umass.edu/bmw-informs-2005/bmw50.html
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 4
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
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 4