Question

In a Linear Programming Problem (LPP), if the objective function to maximize is \( Z = 3x + 4y \) and the corner points of the feasible bounded region are \( (0,0), (4,0), (2,3), \) and \( (0,4) \), find the maximum value of \( Z \).

• A. \( 18 \)
• B. \( 16 \)
• C. \( 12 \)
• D. \( 22 \)

Hint:

Always test every single corner point listed in the problem. Even if one coordinate looks large (like \( (0,4) \)), a balanced interior vertex (like \( (2,3) \)) can often generate a higher overall value depending on the objective weights.

Answer 1

The Correct Option is B

Concept: According to the Fundamental Theorem of Linear Programming, the optimal value (maximum or minimum) of a linear objective function across a bounded feasible region always occurs at one of the boundary corner vertices (vertex points).

Step 1: Evaluate the objective function \( Z \) at each corner point. We calculate the value of \( Z = 3x + 4y \) at each of the four given vertices:
• At point \( (0,0) \): \[ Z = 3(0) + 4(0) = 0 \]
• At point \( (4,0) \): \[ Z = 3(4) + 4(0) = 12 \]
• At point \( (2,3) \): \[ Z = 3(2) + 4(3) = 6 + 12 = 18 \]
• At point \( (0,4) \): \[ Z = 3(0) + 4(4) = 16 \]

Step 2: Compare the calculated values to identify the maximum result. Reviewing our list of results: \( \{0, 12, 18, 16\} \). The largest value is 18, which occurs at the coordinate vertex \( (2,3) \).