Question

If the vertices of the finite feasible solution region are \( (0, 6), (3, 3), (9, 9), (0, 12) \), then the maximum value of the objective function \( z = 6x + 12y \) is __________.

Hint:

To find the maximum or minimum value of the objective function in a linear programming problem, evaluate the objective function at each vertex of the feasible region. The maximum or minimum value will occur at one of the vertices.

Answer 1

Step 1: The objective function is \( z = 6x + 12y \). 

Step 2: To find the maximum value of the objective function, we evaluate \( z \) at each of the given vertices: - At \( (0, 6) \): \[ z = 6(0) + 12(6) = 0 + 72 = 72. \] - At \( (3, 3) \): \[ z = 6(3) + 12(3) = 18 + 36 = 54. \] - At \( (9, 9) \): \[ z = 6(9) + 12(9) = 54 + 108 = 162. \] - At \( (0, 12) \): \[ z = 6(0) + 12(12) = 0 + 144 = 144. \] 

Step 3: The maximum value of \( z \) is 162, which occurs at the vertex \( (9, 9) \). Thus, the maximum value of the objective function is \( 162 \).