Question

Given the following constraints, find the maximum and minimum values for z. Constraints: 2x−y≤124x+2y≥0x+2y≤6 Optimization Equation: z=2x+5y

Answer 1

max= 16 min= -24

Explanation:

Answer 2

Minimum = 0

Maximum = 15

Explanation:

Given

Optimization Equation:
`z = 2x + 5y`

Constraints:

`2x- y \le 12`

`4x + 2y \ge 0`

`x + 2y \le 6`

`x,y\ge 0`

Required

The maximum and the minimum values of z

To do this, we make use of graphical method.

Plot the constraints on a graph (see attachment)

Get the corner points from the points.

These are the points where
`x,y\ge 0`

So, we have:

`(x_1,y_1) = (0,0)`

`(x_2,y_2) = (0,3)`

`(x_3,y_3) = (6,0)`

Substitute these points in the optimization equation:

`(x_1,y_1) = (0,0)`

`z = 2x + 5y`

`z = 2 * 0 + 5 * 0 = 0`

`(x_2,y_2) = (0,3)`

`z = 2 * 0 + 5 * 3 = 15`

`(x_3,y_3) = (6,0)`

`z = 2 * 6 + 5 * 0 = 12`

So, the values are:

Minimum = 0

Maximum = 15