Exercise 12.1

Maximise Z = 3x + 4y subject to the constraints : x + y ≤ 4, x ≥ 0, y ≥ 0.

Minimise Z = - 3 x + 4 y subject to x + 2y ≤ 8, 3x + 2y ≤ 12, x ≥ 0, y ≥ 0.

Maximise Z = 5x + 3y subject to 3x + 5y ≤ 15, 5x + 2y ≤ 10, x ≥ 0, y ≥ 0.

Minimise Z = 3x + 5y such that x + 3y ≥ 3, x + y ≥ 2, x, y ≥ 0.

Maximise Z = 3x + 2y subject to x + 2y ≤ 10, 3 x + y ≤ 15, x, y ≥ 0.

Minimise Z = x + 2y subject to 2x + y ≥ 3, x + 2y ≥ 6, x, y ≥ 0. Show that the minimum of Z occurs at more than two points.

Minimise and Maximise Z = 5x + 10 y subject to x + 2y ≤ 120, x + y ≥ 60, x - 2y ≥ 0, x, y ≥ 0.

Minimise and Maximise Z = x + 2y subject to x + 2y ≥ 100, 2x - y ≤ 0, 2x + y ≤ 200; x, y ≥ 0.

Maximise Z = - x + 2y, subject to the constraints: x ≥ 3, x + y ≥ 5, x + 2y ≥ 6, y ≥ 0.

Maximise Z = x + y, subject to x - y ≤ -1, - x + y ≤ 0, x, y ≥ 0.