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IE521 Advanced Optimization
HW4
Due December 28 (8:00 AM)
1. (20 points) Consider the following LP.
max
6
x
1
+ 10
x
2
+ 9
x
3
+ 20
x
4
s.t.
4
x
1
+ 9
x
2
+ 7
x
3
+ 10
x
4
≤
600
x
1
+
x
2
+ 3
x
3
+ 40
x
4
≤
400
3
x
1
+ 4
x
2
+ 2
x
3
+
x
4
≤
500
x
1
,x
2
,x
3
,x
4
≥
0
Optimal solution to this LP is
z
= 2800
/
3
,x
1
= 400
/
3
,x
4
= 20
/
3
,x
2
=
x
3
= 0 and
s
1
=
s
2
=
0
,s
3
= 280
/
3.
(a) Write the dual of the LP.
(b) Using complimentary slackness and the given optimal primal solution, ﬁnd the optimal
dual solution.
2. (20 points) Consider the following LP:
max
5
x
1
+ 3
x
2
+
x
3
s.t.
2
x
1
+
x
2
+
x
3
≤
6
x
1
+ 2
x
2
+
x
3
≤
7
x
1
,x
2
,x
3
≥
0
Graphically solve the dual of this LP. Then use complementary slackness to solve this maxi
mization problem.
3. (20 points) Zales Jewelers uses rubies and sapphires to produce two types of rings. A type 1
ring requires 2 rubies, 3 sapphires, and 1 hour of jewelers labor. A type 2 ring requires 3 rubies,
2 sapphires, and 2 hours of jewelers labor. Each type 1 ring sells for $400; type 2 sells for $500.
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 Fall '11
 ZelihaAkça
 Optimization

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