Spring 2010
CS530 – Analysis of Algorithms
Homework 6
H
OMEWORK
6,
DUE
M
ARCH
17
You must prove your answer to every question.
Problems with a
(
*
)
in place of a score may be a little too advanced, or
too challenging to most students, so I do not assign a score to them. But I
will still note if you solve them.
Problem 1
(29.56)
.
(25pts) Solve the following linear programming prob
lem, by the simplex method.
(This is the politician’s problem we started
with.)
minimize
x
1
+
x
2
+
x
3
+
x
4
subject to

2
x
1
+
8
x
2
+
10
x
4
50,
5
x
1
+
2
x
2
100,
3
x
1

5
x
2
+
10
x
3

2
x
4
25,
x
1
,
x
2
,
x
3
,
x
4
0.
Solution.
The auxiliary variable for the first phase (to find a feasible solu
tion) is called
x
5
. We muliply by

1 all the inequalities and the objective
function, in order to get a standard form. I present the solution as a list of
tableaux. The variable numbers appear at the head of their columns for non
basic variables (following the number of the column) and at the head of their
rows for basic variables (folowing the number of the row). The last two lines
are the value (with reversed sign) and the coefficients of the two objective
functions: the original one, and the function

x
5
.
3 times 5
1 1
2 2
3 3
4 4
5 5
1 6
50
2
8
0
10
1
2 7
100
5
2
0
0
1
3 8
25
3
5
10
2
1
0
1
1
1
1
0
0
0
0
0
0
1
Bringing
x
5
into the basis:
3 times 5
1
1
2
2
3
3
4
4
5
7
1
6
50.000
7.000
6.000
0.000
10.000
1.000
2
5
100.000
5.000
2.000
0.000
0.000
1.000
3
8
75.000
2.000
7.000
10.000
2.000
1.000
0.000
1.000
1.000
1.000
1.000
0.000
100.000
5.000
2.000
0.000
0.000
1.000
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Spring 2010
CS530 – Analysis of Algorithms
Homework 6
Optimizing the objective function found in the last line.
3 times 5
1
6
2
2
3
3
4
4
5
7
1
1
7.143
0.143
0.857
0.000
1.429
0.143
2
5
64.286
0.714
6.286
0.000
7.143
0.286
3
8
60.714
0.286
8.714
10.000
4.857
0.714
7.143
0.143
1.857
1.000
2.429
0.143
64.286
0.714
6.286
0.000
7.143
0.286
3 times 5
1
6
2
8
3
3
4
4
5
7
1
1
13.114
0.115
0.098
0.983
0.951
0.213
2
5
20.489
0.508
0.721
7.214
3.639
0.229
3
2
6.967
0.033
0.115
1.148
0.557
0.082
20.081
0.082
0.213
3.131
1.394
0.295
20.489
0.508
0.721
7.214
3.639
0.229
3 times 5
1
6
2
8
3
5
4
4
5
7
1
1
15.906
0.046
0.000
0.136
0.455
0.182
2
3
2.840
0.070
0.100
0.139
0.504
0.032
3
2
10.228
0.114
0.000
0.159
1.136
0.046
28.974
0.138
0.100
0.434
0.185
0.196
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 Spring '09
 Linear Programming, Algorithms, Optimization, Standard form, Analysis of algorithms, objective function

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