Linear
optimization
Optimisation
linéaire
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© The scientific sentence. 2010
| Mathematics 2: Linear optimization
Examples
Boxes
A large box of volume 1 m3, and
maximum weight of 50 kg, can contain two
kinds of boxes:
Kind A: box of volume 8 dm3 which
weights 300 g and costs 4 $, and
Kind B: box of volume 1 dm3 which
weights 100 g and costs 1 $ .
Wow many boxes of the two different kinds can
contain the large box in order to maximize the pofit?
Solution
Set the unknown:
x is the number of boxes of kind A
y is the number of boxes of kind B
Volume : 8 x + 1 y ≤ 1000
Mass : 300 x + 100 y ≤ 50 000
Or
Volume : 8 x + 1 y ≤ 1000
Mass : 3 x + y ≤ 500
Fonction objective: Z = 4 x + 1 y : maximum
We graph the following equations:
y = - 8 x + 1000
y = - 3 x + 500
| Vertex | Z ($) |
| O(0, 0) | 0 |
| A(0, 500) | 500 |
| B(100, 200) | 600 |
| C(1000/8, 0) = (125,0) | 500 |
To obtain the maximum profit which is 600 $,
we would place within a large box, 100 boxes
of kind A and 200 boxes of kind B.
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