L10 The Extended Model (Transportation and Logistics & Dr. Anna Nagurney)

1. Lecture 10
The Extended Model
Dr. Anna Nagurney
John F. Smith Memorial Professor
Isenberg School of Management
University of Massachusetts
Amherst, Massachusetts 01003
c 2009
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 10

2. The Extended Model
The Extended Model
Recall that the standard model assumes that the user cost on each
link is a function solely of the flow on that link.
A more general model is the extended model in which the cost on
a link can depend upon the entire vector of link flows. Hence, for
the extended model, we have that
ca = ca(f ), ∀a ∈ L,
where f denotes the vector of link flows such that
f = (fa, fb, ..., fn).
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 10

3. The Extended Model
Hence, in the extended model we have that the user link cost
functions are of the form:
ca = ca(f ), ∀a ∈ L.
The user travel cost on path f is, thus,
Cp = Cp(f ) =
a∈L
ca(f )δap
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 10

4. Roads in Atlanta, GA
Google Maps
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 10

5. The Extended Model
Simplest Example
User’s cost is linear:
ca(f ) =
b∈L
gabfb + ha, ∀a ∈ L.
(We expect, however, that the main effect is from gaafa.)
Total link cost can then be expressed as:
ˆca(f ) =
b∈L
gabfb × fa + hafa
In particular, ca may be independent of some of the flows.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 10

6. The Extended Model
Example
Two way street:
a = (x, y), ¯a = (y, x)
ca(f ) = gaafa + ga¯af¯a + ha
ˆca(f ) = gaaf 2
a + ga¯af¯afa + hafa
c¯a(f ) = g¯a¯af¯a + g¯aafa + h¯a
ˆc¯a(f ) = g¯a¯af 2
¯a + g¯aafaf¯a + h¯af¯a
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 10

7. The Extended Model
U-O Equilibrium Conditions
(Statement is the same as previous one, but user costs on the
paths require now more computation.)
For every O/D pair w, there exists an ordering of the paths
connecting w, such that:
Cp1 (f ) = ... = Cps (f ) = vw ≤ Cps+1 (f ) ≤ ... ≤ Cpm (f )
F∗
pr
> 0; r = 1, ..., s.
F∗
pr
= 0; r = s + 1, ..., m.
where
Cp(f ) =
a∈L
ca(f )δap
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 10

8. The Extended Model
Hence, there exists an optimization reformulation of the
user-optimized conditions if the Jacobian of the user link
travel cost functions is symmetric.
If the Jacobian matrix is positive-definite, then we are guaranteed
that there is a unique link flow pattern.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 10

9. The Extended Model
Associated Network
We pose the problem, given a transportation network
T = (G, D, C) where,
G: graph of nodes & links
D: demands
C: cost structure
Is it possible to construct an associated network T∗ = (G, D, C∗)
C∗: total link costs
so that
Cp(f ) = ˆC∗
p (f ) - marginal total costs, ∀ paths p.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 10

10. The Extended Model
This is true if the Jacobian matrix
Λ =



∂ca
∂fa
∂ca
∂fb
... ∂ca
∂fn
...
...
...
...
∂cn
∂fa
∂cn
∂fb
... ∂cn
∂fn



is symmetric.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 10

11. The Extended Model
The symmetry condition means that
∂ca
∂fb
= ∂cb
∂fa
, for all links a, b.
Moreover, if the Jacobian Matrix of the link travel cost functions
is positive definite, we are guaranteed a unique link flow pattern.
How to verify if the Jacobian matrix is positive definite?
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 10

12. The Extended Model
One way, is if it is strictly diagonally dominant, then it is positive
definite, i.e. if
aii >
j=i
|aij |
for each row of the matrix
A =



a11 a12 · · · a1n
...
...
...
...
an1 an2 · · · ann



Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 10

13. Cargo Containers at Aqaba Port, Jordan
healingiraq.blogspot.com
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 10

14. The Extended Model
S-O Optimality Conditions
(Kuhn-Tucker Conditions)
For every w ∈ W :
ˆCp1
(f ) = · · · = ˆCps
(f ) = λw ≤ ˆCps +1
(f ) ≤ · · · ≤ ˆCpm
(f )
Fpr > 0; r = 1, · · · , s
Fpr = 0; r = s + 1, · · · , m,
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 10

15. The Extended Model
where
by the Chain Rule
ˆCp(f ) = ∂S
∂Fp
=
b∈L
∂S
∂fb
×
∂fb
∂Fp
=
b∈L
∂S
∂fb
× δbp =
b∈L
∂(
a
ˆca(f ))
∂fb
× δbp =
a b
∂ˆca
∂fb
δbp
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 10

16. References
⇒ Dafermos S. (1971) An extended traffic assignment model
with applications to two-way traffic, Transportation Science,
5: pp. 366-389
⇒ Dafermos S. and Sparrow F. (1969) The traffic assignment
problem for a general network, Journal of Research of the
National Bureau of Standards, 73B: pp. 91-118
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 10

17. Additional Reading
For more advanced formulations and associated theory, see
Professor Nagurney’s Fulbright Network Economics lectures.
http://supernet.som.umass.edu/austria lectures/fulmain.html
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 10