lecture 8 - Math 482(Lecture 8 Duality I This lecture...

Math 482 (Lecture 8): Duality I
This lecture covers Section 3.1 of the textbook.
In class exercises
Further discussion of the UIUCbucks example (in above exercise): I came up with some
numbers to illustrate the example:
1=UIUCap, 2=Illin-ice, 3="The482"
max z=2x1+3x2+2.5x3
subject to
1x1+2x2+1.5x3 ≤ 150 (beans)
3x1+1x2+2.0x3 ≤ 300 (milk)
x1,x2,x3 ≥ 0
The corresponding simplex tableau is
x
0
x
1
x
2
x
3
y
1
y
2
0 1 -2 -3 -2.5 0 0
150 0 1 2 1.5 1 0
300 0 3 1 2 0 1
I used the simplex pivot tool to solve it:
x
0
x
1
x
2
x
3
y
1
y
2
270 1 0 0.0000156 0 1.4 0.2
60 0 0 2 1 1.2 -0.4
60 0 1 -1 0 -0.8 0.6
(I rounded some figures).
The final objective function (after turning the problem back to a maximization) is
max 270-0.0000156x2-1.4y1-0.2y2
I CLAIMED that the 1.4 can be interpreted as "how much more money you would make
if you had one more unit of beans and the 0.2 is "how much more money you would
make if you had more more unit of milk". These are called the "shadow prices" .
The theory of duality puts a framework to explain this claim.