Question

(b) Minimize \( Z = x + 2y \) under the following constraints: \[ 2x + y \geq 3, \quad x + 2y \geq 6, \quad x \geq 0, \quad y \geq 0. \]

Hint:

Linear programming problems are solved by evaluating the objective function at the vertices of the feasible region.

Answer 1

Plot the constraints on a graph to form the feasible region. The vertices of the feasible region are determined by solving the intersection points: \[ 2x + y = 3 \quad \text{and} \quad x + 2y = 6. \] Substitute the vertices into \( Z = x + 2y \): \[ Z(0, 3) = 0 + 2(3) = 6, \quad Z(1, 2) = 1 + 2(2) = 5, \quad Z(3, 0) = 3 + 2(0) = 3. \] The minimum value of \( Z \) is \( 3 \).