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Problem Set 7
IE406 Introduction to Mathematical Programming
Dr. Ralphs
Due October 31, 2007
1. Bertsimas 5.12
2. Bertsimas 5.13
3. Consider the following linear programming problem and its optimal ﬁnal tableau shown below.
min

2
x
1

x
2
+
x
3
s.t.
x
1
+ 2
x
2
+
x
3
≤
12

x
1
+
x
2

2
x
3
≤
3
x
1
, x
2
, x
3
≥
0
Final tableau:
x
1
x
2
x
3
x
4
x
5
0
3
3
2
0
24
1
2
1
1
0
12
0
3

1
1
1
15
(a) Determine the optimal dual solution by examining the tableau.
(b) Determine the range of values of the right hand side of the ﬁrst constraint for which the
basis shown above remains optimal.
(c) Suppose that after obtaining the optimal solution depicted in the ﬁnal tableau above, it
was revealed that the following set of constraints were left out and must also be satisﬁed:
2
x
1
+ 3
x
2
≤
20
x
1

x
2
+
x
3
≤
11
2
x
1

3
x
3
≤
23
Use constraint generation to obtain an optimal solution after augmenting the original
LP with these three new constraints. (Hint
: This only requires a few calculations.)
4. The output of a paper mill consists of standard rolls 110 inches (110”) wide, which are cut
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This note was uploaded on 08/06/2008 for the course IE 406 taught by Professor Ralphs during the Fall '08 term at Lehigh University .
 Fall '08
 Ralphs

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