Linear programming

Question

Find the minimum value of:

C=4x+3y

Subject to the following constraints:

x ≥ 0

y ≥ 0

2x+3y ≥ 6

3x -2y ≤ 9

x+5y ≤ 20

Answer 1

Hi Kalil, here's the graphical way to solve it.

1. Draw a cartesian plane (X-Y)
2. Draw each constraint and point in the direction the constraint allows values
3. x ≥ 0 : draw a line at x=0 and draw arrows pointing up (positive)
4. y ≥ 0 : draw a line at y=0 and draw arrows pointing right (positive)
5. 2x+3y ≥ 6 : convert this inequality to a line (y=mx+b) and draw the line. This would be y = -2x/3 + 2. Draw an arrow pointing up and to the right (positive x and y).
6. Do the same as c to 3x -2y ≤ 9 and x+5y ≤ 20. Note that these are upper limits, which means that when you draw the lines on your graph, draw the arrows pointing down and to the left (negative x and y).
7. Identify all of the vertices of the shape you now have drawn. It helps to shade in the region that is entirely constrained within the constrain lines you have drawn in steps a-d.
8. For each vertex (or corner), plug in the coordinates into the original equation C=4x+3y. Take the minimum value of C.

Hope that helps!