Description
problems
5.1
5.5
5.9
5.1
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Explanation & Answer
attached is my solution to 5.1, 5.5 and 5.9
Problem 5.1:
a) We maximize P 3x1 2 x2 4 x3 subjected to
x1 x2 2 x3 4
2 x1 3x3 5
2 x1 x2 3x3 7
x1 , x2 , x3 0
The optimal solution to the maximizing problem, by the method of Solver in Excel, is
x1*
5 * 3
, x2 and x3* 0
2
2
Hence, the dual to this linear programming (LP) problem is
Minimize D 4 y1 5 y2 7 y3 subjected to
y1 2 y2 2 y3 3
y1 y3 2
2 y1 3 y2 3 y3 4
y1 , y2 , y3 0
The optimal solution to the dual problem, by the method of Solver in Excel, is
y1* 0, y2*
As demonstrated in Theorem 5.1, we check for
n
m
j 1
i 1
5
2
, and y3*
6
3
n
m
j 1
i 1
c j x*j bi yi* where
c j x*j bi yi*
5
3
5
2
3 2 4 0 4 0 5 7
2
2
6
3
21 21
2
2
Hence, the validity of Theorem 5.1 is checked.
b) We maximize P 5 x1 6 x2 9 x3 8 x4 subjected to
x1 2 x2 3x3 x4 5
x1 x2 2 x3 3x4 3
x1 , x2 , x3 , x4 0
The optimal solution to the maximizing problem, by the method of Solver in Excel, is
x1* 1, x2*...