Question

A linear program is given by Maximize z = 60x1 + 60x2 + 36x3 Subject to 12x1 + 12x2 + 0x3 ≤ 72 30x1 + 18x2 + 18x3 ≤ 180 18x1 + 30x2 + 18x3 ≤ 180 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0 For this linear program the maximum value of the objective function is z max= ? and it

A linear program is given by
Maximize z = 60x1 + 60x2 + 36x3
Subject to
12x1 + 12x2 + 0x3 ≤ 72
30x1 + 18x2 + 18x3 ≤ 180
18x1 + 30x2 + 18x3 ≤ 180
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0
For this linear program the maximum value of the objective function is
z max= ?
and it achieves this value at the point
(x1, x2, x3) = ?

Answer 1

See below.

Explanation:

Linear programming problems can be solved using standard algorithms such as those found in matlab or mathematica. Those algorithms accept data in a standardized format. In mathematica the linear programming must be submitted within the format

`min z = c cdot x`

subjected to

`A cdot x ge b`
`x ge 0`

In the present case we have

`z = max c cdot x` with

`c = (60,60,36)`
`x = (x_1,x_2,x_3)`

and

`A = ((12, 12, 0),(30, 18, 18),(18, 30, 18))`
`b = ((72), (180), (180))`

with

`A cdot x le b`

now

`{(min c cdot x),(A cdot x ge b),(x ge 0):}equiv{(max c cdot x),(A cdot x le b),(x ge 0):}`

so submitting we obtain the answer

`z = 432` at `x = (3,3,2)`