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

Step 1: Understanding the Concept:
Linear Programming (LP) is a method to achieve the best outcome (such as maximum profit or minimum cost) in a mathematical model whose requirements are represented by linear relationships.
The Corner Point Theorem (or Fundamental Theorem of Linear Programming) states that if a feasible region for an LPP is a convex polygon and an optimal value exists, then it must occur at one of the corner points (vertices) of that region.
If the feasible region is bounded (as specified in this problem), both a maximum and a minimum value are guaranteed to exist at these corner points.
The problem is reduced to checking the value of the objective function \( Z \) at each vertex and selecting the highest result.
Step 2: Key Formula or Approach:
The objective function to evaluate is \( Z = 3x + 4y \).
Substitute the coordinates \( (x, y) \) of each corner point into the objective function.
Step 3: Detailed Explanation:
We are provided with four corner points. We will calculate the value of \( Z \) for each:
1. Evaluate at point \( (0, 0) \):
\[ Z = 3(0) + 4(0) = 0 \]
2. Evaluate at point \( (4, 0) \):
\[ Z = 3(4) + 4(0) = 12 + 0 = 12 \]
3. Evaluate at point \( (2, 3) \):
\[ Z = 3(2) + 4(3) = 6 + 12 = 18 \]
4. Evaluate at point \( (0, 4) \):
\[ Z = 3(0) + 4(4) = 0 + 16 = 16 \]
Now, we compare the set of results: \( \{0, 12, 18, 16\} \).
The highest value among these is 18.
It occurs when \( x = 2 \) and \( y = 3 \).
Step 4: Final Answer:
The maximum value of the objective function \( Z \) is 18.
This matches Option (A).
(Note: Some versions of this memory-based paper highlight Option B incorrectly in summary keys; however, mathematically, 18 is higher than 16, and \( (2, 3) \) is a valid corner point).