Minimize Z = 4x + 6y subject to 2x + 4y ≥ 20 3x + 2y ≤ 24 x, y ≥ 0

Question

Solve the linear programming problem by graphing. Graph the feasible region, list the extreme points and identify the maximum value of Z. You do not have to submit your graph, but please list the equations of the lines that form the feasible region. 

Minimize Z = 4x + 6y 
subject to
2x + 4y ≥ 20
3x + 2y ≤ 24
x, y ≥ 0

Answer 1

The equations are:

x=0 (y-axis)

y=0 (x-axis)

2x+4y=20 or x+2y=10

3x+2y=24

The easiest way to find the minimum value of Z given the constraints is to draw a graph and find out what geometrical figure represents the optimization problem. In this case, the figure is a triangle with vertices at (0,5), (0,12) and (7,1.5) (see below).

Solving the system of equations: 2x+4y=20, 3x+2y=24.

The first equation reduces to x+2y=10, so 2y=10-x, and 3x+10-x=24, 2x=14, x=7, y=1.5. This gives us the vertex (7,1.5). The triangle represents the feasible region defined by the constraints. If we plug the coordinates of the vertices into Z, we get respectively 4x+6y=30, 72, 28+9=37. Clearly 30 is the smallest value, so x=0 and y=5.