L9 Modeling Extensions (Transportation and Logistics & Dr. Anna Nagurney)

1. Lecture 9
Modeling Extensions
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 9

2. Highway Traffic
www.env-ne.org
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 9

3. Modeling Extensions
Modeling Extensions
Standard Model ca = ca(fa)
The more general model to handle 2-way traffic & intersections:
Extended Model ca = ca(f )
Multimodal Model to handle multimodal networks.
c1
a = c1
a (f 1
a , f 2
a , · · · , f k
a )
...
ck
a = ck
a (f 1
a , f 2
a , · · · , f k
a )
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 9

4. Modeling Extensions
Extended Model
ca = ca(f ) : f = fa : a ∈ L
Travel cost on path p:
Cp = Cp(f ) =
a∈L
ca(f )δap
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 9

5. Modeling Extensions
Simplest Example
User’s cost:
ca(f ) =
b∈L
gabfb + ha
(expect main effect to be from gaafa)
Total link cost:
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 9

6. Modeling Extensions
User Equilibrium with Elastic Demand
Until now all the models that we have considered assumed a fixed
demand associated with the O/D pairs.
In many situations, however, the demands may be influenced by the
level of service on the network.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 9

7. Modeling Extensions
For example, as congestion increases, motorists may decide to use
a different mode of travel (e.g., subway) or forego some trips
altogether.
To handle this situation, we introduce an elastic demand function.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 9

8. Increasing Travel Demand, India
blogs.taz.de
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 9

9. Modeling Extensions
The Elastic Demand Model
For each O/D pair w, we have now a demand function:
dw = dw (λw ),
where λw denotes the disutility of traveling between O/D pair w.
We also assume that dw admits an inverse
λw = λw (dw ).
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 9

10. Modeling Extensions
Then the conservation of flow equations are given by:
dw =
p∈Pw
Fp, for all w
and
fa =
p
Fpδap for all a
We also assume (for the time being) that
ca = ca(fa), for all a.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 9

11. Modeling Extensions
U-O (Equilibrium) Conditions
A feasible path flow and demand pattern (F∗, d∗) is said to be
U-O if and only if for each O/D pair w, and each p ∈ Pw :
Cp(f ∗)
= λw (d∗
w ), if F∗
p > 0
≥ λw (d∗
w ), if F∗
p = 0
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 9

12. Modeling Extensions
Equivalent Minimization Formulation
Minimize
a
fa
a
ca(x)dx −
w
dw
0
λw (y)dy
subject to:
fa =
p
Fpδap, for all a
dw =
p∈Pw
Fp, for all w
Fp ≥ 0, for all p.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 9

13. Modeling Extensions
Although there exist algorithms to compute the equilibrium in the
elastic demand problem, the elastic demand problem can be
transformed into a fixed demand problem, for which efficient
algorithm exists.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 9

14. Modeling Extensions
Solution by Network Representation
Excess Demand Reformulation
For each O/D pair w, we construct a path connecting O/D pair w
consisting of a single link which we denote by w. Given an upper
bound ¯dw > dw (λw ), for every w, we construct associated
functions on the links thus:
cw = λw (¯dw − fw )
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 9

15. Fw denotes the flow on a new disjoint path w.
Example
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 9

16. References
⇒ Gartner N. (1980), Optimal Traffic Assignment with Elastic Demands: A
Review; Part II: Algorithmic Approaches, Transportation Science, 14: pp.
192-208.
⇒ Nagurney A. and Dong J. (2002), Supernetworks: Decision-Making for
the Information Age, Edward Elgar Publishing
http://supernet.som.umass.edu/bookser/supbook.html
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 9