Communication Networks
Telematics and Computer Networks Group
WS 10/11
Prof. Dr.-Ing. Günter Schäfer
Tutorial 1
Precondition: ODYSSEUS
Please download the Java software Odysseus from the lecture’s website. You can run
it after extraction by executing odysseusXX.bat (Windows) or odysseusXX.sh
(Linux) where XX is ether 32 or 64 (bit), depending on your system.
Excercise 1
Transformation of constraints and matrix representation
Given the following optimization problem:
Objective
maximize f (h, i , j ) = h + 2i + j
Constraints
3h + i ≤ 10
i + j = 12
j ≥5
h, i , j ≥ 0
a) Transform all constraints excepting h, i , j ≥ 0 to equalities, i.e. the constraints
will look like the second one afterwards.
Hint: You have to introduce slack variables.
b) Construct the matrix A and the vector b needed for the matrix representation
A · x = b of the constraints where x = (h, i, j , ∗, ∗)T . Please replace * with
the names of your slack variables. Furthermore, note down the vector c of the
objective f (x ) = c T · x .
Hint: You will need the constraints of part (a)
c) Run Odysseus, choose the file ”Exercise1.otask” from the folder ”Tutorial 1”
and click on the tab Optimization (Matrix). Enter your matrix A and the vectors b, c and x . If the dimensions of your vectors and matrix are correct, all
fields with white background should be filled out. Select ”=” as the relation
symbol and max as optimization goal. Hit the Optimize button and note down
UB1en.tex @ November 30, 2010 : 15:29
the results.
Hint: You don’t have to give the constraint x ≥ 0 explicitly.
d) Transform the original constraints to inequalities, so that all relation symbols
become ”≤”. Again, create the matrix A and the vectors b needed for the
problem’s matrix representation A · x ≤ b as well as the objective vector c, this
time with x = (h, i , j )T . Optimize the problem using Odysseus (don’t forget to
change the relation symbol) and compare the results with (a).
Hint: The second condition has to be split into two inequalities.
Excercise 2
Simple design problem and variants
You are supposed to design a network for a small company, that wants to interconnect
its six locations. This network will have to satisfy a demand of 1000 kbps between
location 1 and 6 (demand h1 ) as well as a demand of 400 kbps between the locations 2
and 5 (demand h2 ). Four potential paths (candidate paths) have already been defined
and the costs for each link have been calculated. The candidate paths per demand
are numbered consecutively starting with 1.
Edge labels
3
EdgeID (Cost in e/kbps)
4
4 (1)
Node labels
6 (2)
1 (1)
s1 = 1
1000
[s*/t* = ] NodeID
[Demand]
t1 = 6
2 (2)
1000
5 (1)
7 (1)
Demands
h1 = 1000 kbps (from s1 to t1 )
h2 = 400 kbps (from s2 to t2 )
Candidate paths
s2 = 2
400
3 (2)
t2 = 5
Demand h1
1-5-7
1-4-6
400
Demand h2
3
2-5
Figure 1: Network with two demands.
a) Find an optimal linkage (regarding the costs), that satisfies the requirements
given above. Run Odysseus, select ”Exercise2a.otask” from the folder ”Tutorial 1” and enter your optimization problem in matrix representation in the tab
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Optimization (Matrix).
Hint: Look at the example from the script under the topic Dimensioning Problems and apply the transformations from exercise 1.
b) There has been an error in the calculation of the link costs. The costs for
the link between 4 and 6 are actually 1e/kbps and the link from 5 to 6 costs
2e/kbps. What effects will these changes have to your planed network? Hit the
button Choose task... in Odysseus and select the file ”Exercise2b.otask” from
the folder ”Tutorial 1”.
c) You were informed that there is already a 500 kbps link between the locations
2 and 3 and a 100 kbps link between 5 and 6. Find a solution that takes the
existing links into account. Again, hit the Choose task... button and select the
file ”Exercise2c.otask” from the ”Tutorial 1” folder. Formulate the problem as
a general optimization problem in sum representation (see script) to make it
reusable for further projects. Switch to the Optimization (Sum) tab in Odysseus
and enter your problem.
Bonus: Evaluate the networks fault tolerance.
d) Because of a report from the accounting department they will only provide a
3000e budget for your project, you have to adapt your network design. Again,
formulate the problem in general form to make it reusable. What consequences
has the limited budget for your project? Will you have to negotiate with the
managing board about your budget? Hit the Choose task... button and select
”Exercise2d.otask” from the folder ”Tutorial 1”.
Attention: The linkcapacities should be integer values!
Hint: Consider the problem, that you may not be able to satisfy all demands
100%, but a factor r ≤ 1 could help. How would you have to optimize this
factor?
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