Homework 5
Due:
March 3, 2009
Exercise 4, page 155
Write the dual for each of the following primal problems:
(a)
maximize
z
=

5
x
1
+ 2
x
2
subject to

x
1
+
x
2
≤ 
2
2
x
1
+ 3
x
2
≤
5
x
1
, x
2
≥
0
.
(b)
minimize
z
= 6
x
1
+ 3
x
2
subject to
6
x
1

3
x
2
+
x
3
≥
2
3
x
1
+ 4
x
2
+
x
3
≥
5
x
1
, x
2
, x
3
≥
0
.
(c)
maximize
z
=
x
1
+
x
2
subject to
2
x
1
+
x
2
= 5
3
x
1

x
2
= 6
x
1
, x
2
unrestricted
.
Exercise 5, page 163
:
Consider the following LP:
maximize
z
= 2
x
1
+ 4
x
2
+ 4
x
3

3
x
4
subject to
x
1
+
x
2
+
x
3
= 4
x
1
+ 4
x
2
+
x
4
= 8
x
1
, x
2
, x
3
, x
4
≥
0
.
Using
x
3
and
x
4
as starting variables, the optimal tableau is given as
Basic
x
1
x
2
x
3
x
4
Solution
z
2
0
0
3
16
x
3
0
.
75
0
1

0
.
25
2
x
2
0
.
25
1
0
0
.
25
2
Write the associated dual problem and determine its optimal solution in two ways.
1
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Exercise 3(a), Page 167
:
Consider the following LP model:
maximize
z
= 3
x
1
+ 2
x
2
+ 5
x
3
subject to
x
1
+ 2
x
2
+
x
3
+
x
4
= 30
3
x
1
+ 2
x
3
+
x
5
= 60
x
1
+ 4
x
2
+
x
6
= 20
x
1
, x
2
, x
3
, x
4
, x
5
, x
6
≥
0
.
Check the optimality and the feasibility of the following basic solution:
Basic variables = (
x
4
, x
3
, x
6
), and Inverse =
1

1
2
0
0
1
2
0
0
0
1
.
Exercise 6, page 168
:
Consider the following LP model:
maximize
z
= 5
x
1
+ 2
x
2
+ 3
x
3
subject to
x
1
+ 5
x
2
+ 2
x
3
≤
b
1
x
1

5
x
2

6
x
3
≤
b
2
x
1
, x
2
, x
3
≥
0
.
The following optimal tableau corresponds to specific values of
b
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 Spring '09
 Nedich
 Operations Research, Optimization, BMW Sports Activity Series, X1, BMW X5

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