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MIT 2.098/6.255/15.093
Optimization Methods
Prof. J. Vera, Fall 2007
Homework Assignment 4. Solution Guidelines
The purpose of this assignment is to give you the opportunity to solve a “real” problem using
available software, and to improve the solution procedure. The problem is coded in AMPL and
you may use the student edition of AMPL for this. The pointers to obtain it are in one of the
announcements of the class. I do want, also, that you try some optimization resources available in
the web. You will need to do some extensions to the AMPL code, but they are relatively straight
forward, you can mail us with speciﬁc questions on AMPL.
Problem description:
The problem is the already known network design problem, but in this case it applies to a multi
commodity ﬂow problem. So, Assume we have a network (
N,A
), where the arcs will have to be
actually built to satisfy some give demand between a certain number of origin  destination pairs.
There are
K
of this pairs of nodes.
•
Let
o
k
= (
r
k
,s
k
) denote the origindestination pair
k
,
r
k
∈
N,s
k
∈
N
.
•
For each pair
k
there is a demand of ﬂow,
d
k
which needs to travel from
r
k
to
s
k
.
•
Each arc has a capacity
u
ij
.
•
For each arc there is a unit cost
c
ij
for traveling arc (
i,j
) (we make it independent of the
origindestination pair in this problem).
•
There is a construction cost
G
ij
if we decide to use arc (
i,j
)
∈
A
.
We model this with a variable
x
k
ij
, corresponding to the ﬂow in arc (
i,j
) corresponding to the
origindestination
k
, and with a binary variables
z
ij
which equals 1 if we construct the arc and 0
otherwise. The model is:
min
∑
k
∈
K
∑
(
i,j
)
∈
A
c
k
ij
x
k
ij
+
∑
(
i,j
)
∈
A
G
ij
z
ij
s.t.
∑
j
:(
i,j
)
∈
A
x
k
ij

∑
p
:(
p,i
)
∈
A
x
k
pi
=
b
k
i
,i
∈
N,k
= 1
,...,K
K
∑
k
=1
x
k
ij
≤
u
ij
z
ij
,
∀
(
i,j
)
∈
A
x
ij
≥
0
,z
ij
∈ {
0
,
1
}
,
∀
(
i,j
)
∈
A
where
b
k
i
=
d
k
, i
=
r
k

d
k
, i
=
s
k
0 otherwise
To work with this problem, you have to download the ﬁles “net1v1mult.mod” and “net1v1mult.dat”.
They contain the model and the basic data. To introduce variation into the diﬀerent data instances
you will solve, you will all use diﬀerent sets of data in which basically what changes is the cost infor
mation. To obtain the data which you should use, you have to download the Excel ﬁle “DATA.XLS”,
generate the data, and copy it in the corresponding places in the ﬁle “net1v1mult.dat”, follow the
instructions in the Excel ﬁle.
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 Optimization

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