Maximize P = 3x1 + 2x2 subject to the constraints below. Illustrate the feasible region and indicate the point or points that maximize P.
2x1 + x2 ≤ 8 x1 + 2x2 ≤ 10 x1, x2 ≥ 0
2x1 + x2 = 8 x1 + 2x2 = 10
x1 = 2
x2 = 4
2x1 + x2 = 8 x1 + 2x2 = 10
x1 = 2
x2 = 4
2x1 + x2 ≤ 8 (The red line)
| x1 | x2 |
| 0 | 8 |
| 2 | 4 |
| 4 | 0 |
x1 + 2x2 ≤ 10 (The blue line)
| x1 | x2 |
| 0 | 5 |
| 2 | 4 |
| 10 | 0 |
The feasible region is the shaded area below and to the left of the the two lines.
The objective function will have its maximum value at one of the vertices of the feasible region: (0,0), (4,0), (2,4), (0,5). You must evaluate the objective function at each of these four points in order to determine the maximum value.
| x1 | x2 | P = 3x1 + 2x2 |
| 0 | 0 | 0 |
| 4 | 0 | 12 |
| 2 | 4 | 14 |
| 0 | 5 | 10 |
The maximum value of the objective function is 14 at the point (2,4).
The evaluation of the objective function can also be done using matrices:
