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ISYE3133C  Engineering optimization
Final exam  Practice questions
Instructions:
•
These are sample questions to practice for the ﬁnal exam.
•
The ﬁnal exam will consist of a subset of these questions, so you should really know how to do all of them.
•
Remember that you may use anything to solve these questions or to check that your answer is correct, but
you won’t be able to resort to anything during the ﬁnal, so keep that in mind.
•
Show ALL your work.
1
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View Full Document Question 1
Solve the following LP using only graphical solution methods:
min
2
x
1
+4
x
2
+3
x
3
+4
x
4
s.t.
x
1
+2
x
2
+
x
4
≥
4
x
2
+
x
3
+
x
4
≥
3
x
i
≥
0
,
∀
i
= 1
,...,
4
Question 2
Consider the following LP in Standard Form:
max

4
x
1
+
x
2

6
x
3
s.t.

x
1
+
x
2
+4
x
4
+
x
5
= 2
x
1
+
x
3
+3
x
4
+
x
6
= 6
2
x
1

x
2

2
x
3
+7
x
4
= 3
x
i
≥
0
,
∀
i
= 1
,...,
6
For basis
B
=
{
1
,
2
,
3
}
a) What is the associated primal solution?
b) What is the associated dual solution?
c) What are the reduced costs of the nonbasic variables?
d) Is the basis primal/dual feasible? Is it optimal?
Question 3
Consider the following LP:
max
cx
s.t.
Ax
≤
b
x
≥
0
where
c
is an
n
dimensional vector,
A
is an
m
×
n
matrix,
b
is an
m
dimensional vector and
x
is an
n
dimensional
vector of variables.
a) Write down the dual of this LP
b) Suppose that we have a dual feasible solution
y
∈
R
m
with objective function value equal to 20. What can
we say about the optimal objective function of the primal?
c) Suppose that the dual is unbounded, what can we say about the primal?
d) Suppose that the optimal objective function of the dual is
14
, what can we say about the optimal objective
function of the primal?
e) Suppose that the dual is infeasible, what can we say about the primal?
f) Suppose we have the optimal solution to the dual
y
*
. Describe in detail how you would obtain the optimal
solution to the primal.
Question 4
Consider the following LP:
max 2
x
1
+3
x
2

4
x
3
+
x
4
s.t.
x
1
+
x
3
+3
x
4
≥
3
3
x
1
+
x
2
+6
x
3

x
4
≤
5
x
1

4
x
3
+2
x
4
≥
4
x
i
≥
0
,
∀
i
= 1
,...,
4
a) Write the dual of the LP.
b) Write the complementary slackness conditions for the LP.
c) Given that the optimal solution for the dual is
(0
,

0
.
5
,
0
.
25)
, what is the optimal solution for the primal?
Question 5
A company produces six products in the following fashion: Each unit of raw material purchased
and processed yields four units of product 1, two units of product 2 and one unit of product 3. Up to 1,200 units
of product 1 can be sold, and up to 300 units of product 2 can be sold. Each unit of product 1 can be sold or
processed further. Each unit of product 1 that is processed yields a unit of product 4. Demand for products 3 and
4 is unlimited. Each unit of product 2 can be sold or processed further. Each unit of product 2 that is processed
further yields 0.8 units of product 5 and 0.3 units of product 6. Up to 1,000 units of product 5 can be sold, and
up to 800 units of product 6 can be sold. Up to 3,000 units of raw material can be purchased at
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This test prep was uploaded on 04/18/2008 for the course ISYE 3133 taught by Professor Juanpablovielma during the Spring '08 term at Georgia Institute of Technology.
 Spring '08
 JuanPabloVielma
 Optimization

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