Question

The corner points of the feasible region determined by the system of linear constraints are \( (0, 0), (0, 40), (20, 40), (60, 20) \) and \( (60, 0) \). If the objective function of an LPP is \( Z = 4x + 3y \), then the maximum value is :

• A. \( 200 \)
• B. \( 300 \)
• C. \( 240 \)
• D. \( 120 \)

Hint:

To avoid calculation errors, organize your vertex evaluations inside a small table mapping points directly to calculated values.

Answer 1

The Correct Option is B

Concept: According to the Corner Point Theorem in Linear Programming, the optimal value (maximum or minimum) of an objective function always occurs at one of the vertices (corner points) of the bounded feasible region. We calculate the value of \( Z \) at each vertex to find the largest one.

Step 1: Evaluate \( Z = 4x + 3y \) at each corner point systematically. Let's compute the value of the objective function for all five points:
• At point \( (0, 0) \): \[ Z = 4(0) + 3(0) = 0 + 0 = 0 \]
• At point \( (0, 40) \): \[ Z = 4(0) + 3(40) = 0 + 120 = 120 \]
• At point \( (20, 40) \): \[ Z = 4(20) + 3(40) = 80 + 120 = 200 \]
• At point \( (60, 20) \): \[ Z = 4(60) + 3(20) = 240 + 60 = 300 \]
• At point \( (60, 0) \): \[ Z = 4(60) + 3(0) = 240 + 0 = 240 \]

Step 2: Identify the maximum value. Comparing all the calculated objective values: \[ 0, \, 120, \, 200, \, 300, \, 240 \] The maximum value among these options is \( 300 \), which occurs at the coordinate point vertex \( (60, 20) \).