Question

solve the following linear programming problem using graphical methods
Minimize z= 2x+8y
Subject to x-y > -7
3x+2y > 24
x > 0
y > 0
find the minimum z value and name the points

Answer 1

Task:

Solve the following linear programming problem using graphical methods

Minimize $z = 2x + 8y$

Subject to x-y > -7

$3x + 2y > 24$

x>0

y>0

Find the minimum z value and name the points

Solution.

1. Depict straight lines $x - y = -7$ and $3x + 2y = 24$

$x - y = -7 \Leftrightarrow y = x + 7$ . Put $x = 0$ and $x = -7$ . $x - y = -7$ passes through the points $(0,7)$ and $(7,0)$ .

$3x + 2y = 24$ passes through the points $(8,0)$ and $(0,12)$ .

Conditions $x - y > -7$ , $3x + 2y > 24$ , $x > 0$ , $y > 0$ correspond to part of the plane, highlighted in grey.

2. $z = 2x + 8y \Leftrightarrow 4y = 1/2 * z - x \Leftrightarrow y = 1/8 * z - 1/4 * x$ .

All lines $z = 2x + 8y$ will have the same slope and are moved up or down depending on the value of $z$ .

For example, $z = 0$ , $x + 4y = 0$ or $z = 24$ , $x + 4y = 12$ .

3. As we can see, in order to satisfy the conditions and for minimize $z = 2x + 8y$ , this line should cross with the gray part of the plane and pass through the point (8,0).

Our sought-for line is highlighted in red.

Put in equation $z = 2x + 8y$ $x = 8$ and $y = 0$ we get $z = 16$ . Minimum value of $z$ is 16, but it is not achieved, since $x > 0$ and $3x + 2y > 24$ . So, $z > 16$ .

Answer:

$z > 16$ . Minimum value of $z$ is 16, but it is not achieved.