Question

Find the maximum value of the linear objective optimization function \[ Z = 4x + y \] evaluated over a feasible region bounded by the corner vertices: \[ (0,0), \ (3,0), \ (2,3), \ \text{and} \ (0,4). \]

• A. \( 12 \)
• B. \( 4 \)
• C. \( 11 \)
• D. \( 16 \)

Hint:

For Linear Programming Problems, always evaluate the objective function at all corner points of the feasible region. The optimal value always occurs at a vertex.

Answer 1

The Correct Option is A

Concept: In Linear Programming Problems (LPP), the maximum or minimum value of an objective function occurs at one of the corner points (vertices) of the feasible region.

Step 1: Identify the objective function.
The objective function is: \[ Z = 4x + y \] The feasible region has the corner points: \[ (0,0), \ (3,0), \ (2,3), \ (0,4) \]

Step 2: Evaluate the objective function at each corner point.
\[ \begin{aligned} Z(0,0) &= 4(0) + 0 = 0 [0.2cm] Z(3,0) &= 4(3) + 0 = 12 [0.2cm] Z(2,3) &= 4(2) + 3 = 8 + 3 = 11 [0.2cm] Z(0,4) &= 4(0) + 4 = 4 \end{aligned} \]

Step 3: Determine the maximum value.
The obtained values are: \[ 0,\ 12,\ 11,\ 4 \] The maximum among these is: \[ 12 \] Hence, the maximum value of the objective function is: \[ \boxed{12} \] Therefore, the correct answer is: \[ \boxed{\text{(A)}} \]