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

Answer:

Minimum = 0

Maximum = 15

Step-by-step 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