L2 Generalized Representation of a Transportation Network (Transportation and Logistics & Dr. Anna Nagurney)

1. Lecture 2
Generalized Representation of a
Transportation Network
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 2

2. Zurich, Switzerland Railway Network Map
www.vs.inf.ethz.ch
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 2

3. Generalized Representation of a Transportation Network
represented by a mathematical network graph.
1. finite set of nodes
2. set of directed links (arcs, branches, edges).
Examples
In a road network, nodes are where traffic is generated or
attracted to, or intermediate points. Links are the roads.
In an airline network, nodes are the airports, links are the air
routes.
In a freight network, nodes are loading/unloading points and
switching points. Links are made up of tracks.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 2

4. Network 1
Nodes: x1, x2, ... Links: a1, a2, ...
∗ Links have a direction and may be uniquely expressed as follows:
a1 = (x1, x3), a2 = (x3, x2), etc.
O/D pairs w : w1, w2 w1 = (x1, x3), w2 = (x3, x4).
Note: A 2-way street is represented by 2 links going in opposite
directions.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 2

5. Generalized Representation of a Transportation Network
Flows in Transportation Networks
Steady State
Path Flows and Link Flows
Path: A path is a sequence of links connecting an O/D pair
w = (x, y) of nodes. It can be represented by linking all the
distinct links from the origin to the destination.
We exclude all cycles or loops.
Assume the travel demand (rate) is constant in time. Hence the
flows are constant in time.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 2

6. Generalized Representation of a Transportation Network
Notation:
Fp - flow on path p (measured in units/hrs.,users/unit time).
⇒ Path flows are always assumed to be nonnegative.
fa - flow on link a (measured in users/unit time).
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 2

7. US Rail Network
www.mapsofworld.com
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 2

8. Example - Network 2
Nodes: x1, x2, x3 Links: a1, a2, a3
O/D pairs: w1 = (x1, x2), w2 = (x1, x3)
Paths connecting O/D pairs:
Pw1 = {p1}, Pw2 = {p2, p3}, where
p1 = (a1), p2 = (a1, a3), p3 = (a2)
fa1 = Fp1 + Fp2, fa2 = Fp3
fa3 = Fp2
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 2

9. Generalized Representation of a Transportation Network
The general expression relating link flows and path flows:
fa =
p
Fpδap
where
δap=
1, if link a is contained in path p;
0, otherwise.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 2

10. Example - Network 3
Nodes N={x1, x2, x3, x4}
Links L={a1, a2, ..., a8}
O/D pairs W={w1, w2} where
w1=(x1, x2), w2=(x3, x4)
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 2

11. Generalized Representation of a Transportation Network
Paths connecting the O/D pairs
Pw1 = {p1, p2, p3, p4}, where
p1 = (a4), p2 = (a4, a5)
p3 = (a2, a6), p4 = (a1, a3, a6).
Pw2 = {p5, p6, p7, p8}, where
p5 = (a3), p6 = (a5, a7)
p7 = (a8, a2), p8 = (a8, a4, a7).
A link can be contained in 2 (or more) paths connecting an O/D
pair. It can be contained in paths connecting O/D pairs; or both
of the above.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 2

12. ∗ More than 2 path flow patterns may induce the same link flow
pattern, but link flows are ”unique”.
Example
Suppose that we know the link flows. Can we solve for the path
flows uniquely?
Travel demand w1 = (x1, x2) =100/hrs
Travel demand w2 = (x1, x3) =100/hrs.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 2

13. Generalized Representation of a Transportation Network
O/D w1:
path p1 = (a1)
path p2 = (a2)
O/D w2:
path p3 = (a1, a3)
path p4 = (a2, a3)
fa1 = Fp1 + Fp3 ,
fa2 = Fp2 + Fp4 ,
fa3 = Fp3 + Fp4
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 2

14. Generalized Representation of a Transportation Network
Problem:
Given link flows fa’s, find the path flows Fp’s.
Case I:
For O/D pair w1 = (x1, x2), all travelers take path p2 =⇒Fp2 =100,
Fp1 =0.
For O/D pair w2 = (x1, x3), all travelers take path p3 =⇒Fp3 =100,
Fp4 =0.
With this path flow pattern, the induced link flow pattern is:
fa1 = 100, fa2 = 100, fa3 = 100.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 2

15. Generalized Representation of a Transportation Network
Case II:
For O/D pair w1 = (x1, x2), all travelers take path p1 =⇒Fp1 =100,
Fp2 =0.
For O/D pair w2 = (x1, x3), all travelers take path p4 =⇒Fp4 =100,
Fp3 =0.
Here again the induced link flow pattern is:
fa1 = 100, fa2 = 100, fa3 = 100.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 2