L11 The Multimodal Model (Transportation and Logistics & Dr. Anna Nagurney)

1. Lecture 11
The Multimodal 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 11

2. Multimodal Transportation
www.burohappold.com
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 11

3. The Multimodal Model
The Multimodal Model
There are k different modes of transportation using the network:
(1, 2, ..., k)
Assumptions
• Each mode has its own cost function.
• Each mode contributes to its own and other modes’ costs in
an individual way.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 11

4. The Multimodal Model
Multimodal Transportation Networks
Topological characteristics of networks remain the same as the
single-modal.
Transportation characteristics now change: The O/D travel
demands, path flows, link flows, and path costs now change and
become k dimensional vectors.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 11

5. The Multimodal Model
Travel Costs
ci
a : user or personal travel cost of mode i on link a
Assumptions
c1
a = c1
a (f 1
a , · · · , f k
a )
...
...
ck
a = ck
a (f 1
a , · · · , f k
a )
- extension of the standard model
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 11

6. The Multimodal Model
Travel cost on path p for mode i:
Ci
p = Ci
p(f ) =
a
ci
a(f 1
a , ..., f k
a )δap
Total link cost of mode i on link a:
ˆci
a = ci
a(f 1
a , ..., f k
a ) × f i
a = ˆci
a(f 1
a , ..., f k
a )
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 11

7. The Multimodal Model
Example of user cost functions
Linear model
ci
a(f 1
a , ..., f k
a ) =
j
gij
a f j
a + hi
a
c1
a = g11
a f 1
a + g12
a f 2
a + ... + h1
a
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 11

8. Example: 2 modes
c1
a = 10f 1
a + 5f 2
a + 10
c2
a = 5f 2
a + 4f 1
a + 5
c1
b = 6f 1
b + 4f 2
b + 10
c2
b = 3f 2
b + 2f 1
b + 15
d1
xy = 10 d2
xy = 20
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 11

9. The Multimodal Model
Conservation of Flow Equations
di
w =
p∈Pw
Fi
p, for all O/D pairs w & modes i.
F is feasible if Fi
p ≥ 0, for all i and p, and the conservation of flow
equation above holds.
f i
a = flow or load on link a induced by mode i.
f i
a =
p
Fi
p δap
f = (f 1
a , ..., f k
a ) is feasible when these equations are satisfied by a
feasible path flow pattern.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 11

10. The Multimodal Model
Travel Demands
With every O/D pair w, we associate a vector travel demand
dw = (d1
w , d2
w , ..., dk
w ),
where di
w is the travel demand associated with O/D pair w and
mode i.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 11

11. The Multimodal Model
Flows
Fp is now the flow through path p, where Fp is the vector with
components Fp = (F1
p , F2
p , ..., Fk
p ),
and
Fi
p is the flow on path p by mode i.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 11

12. The Multimodal Model
User-Optimized or Equilibrium Conditions
For each mode i, and every O/D pair w, f is an equilibrium if and
only if:
Ci
p1
(f ) = · · · = Ci
ps
(f ) = vi
w ≤ Ci
ps+1
(f ) ≤ · · · ≤ Ci
pm
(f )
Fi
pr
∗ > 0; r = 1, · · · , s.
Fi
pr
∗ = 0; r = s + 1, · · · , m.
where Ci
p(f ) =
a
ci
a(f 1
a , · · · , f k
a ) δap
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 11

13. The Multimodal Model
The System-Optimized Problem
Minimize S(f ) =
i a
ˆci
a(f 1
a , · · · , f k
a )
subject to:
di
w =
p∈Pw
Fi
p, for all i and w
f i
a =
p∈Pw
Fi
p δap, for all i and a
Fi
p ≥ 0, for all i and p.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 11

14. The Multimodal Model
Optimality Conditions for S-O Problem
For each mode i and O/D pair w, f is S-O if and only if:
ˆC
i
p1
(f ) = · · · = ˆC
i
ps
(f ) = λi
w ≤ ˆC
i
ps +1
(f ) ≤ · · · ≤ ˆC
i
pm
(f )
Fi
pr
> 0; r = 1, · · · , s
Fi
pr
= 0; r = s + 1, · · · , m ,
where
ˆC
i
p(f ) = ∂S
∂Fi
p
= by chain rule
b j
∂ˆcj
b
∂f i
b
δbp
ˆC
i
p(f ) : marginal costs
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 11

15. Modes of Transportation
Minnesota Department of Transportation
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 11

16. The Multimodal Model
Reduction of Multimodal Networks into a Single Mode Network
Suppose we have a multimodal network. we can construct an
equivalent single-modal (but extended cost) network as follows.
The network will have a new topology, new travel demands, etc.
We do this by making multiple copies of the network, one copy for
each mode of transportation.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 11

17. Example
(multimodal network)
d1
xy = 5, d2
xy = 10
c1
a = 10f 1
a + 5f 2
a + 10 c2
a = 5f 2
a + 3f 1
a + 6
c1
b = 5f 1
b + 4f 2
b + 3 c2
b = 6f 2
b + 3f 1
b + 10
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 11

18. Transformation into a single-modal (extended cost) problem:
O/D pair w1 = (x1, y1) O/D pair w2 = (x2, y2)
ca1 = 10fa1 + 5fa2 + 10 ca2 = 5fa2 + 3f a1 + 6
cb1 = 5fb1 + 4fb2 + 3 cb2 = 6fb2 + 3fb1 + 10
dx1y1 = 5 dx2y2 = 10
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 11

19. Important Notice
We always denote the U-O, equivalently,
transportation network equilibrium solution in
either link or path flows with a ”*”.
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 11

20. References
⇒ Dafermos SC (1972) The traffic assignment problem for multi-class user
transportation networks.Transportation Science 6: 73-78
Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 11

21. 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 11