A person wants to invest at least ₹20,000 in plan A and \₹30,000 in plan B. The return rates are 9% and 10% respectively. He wants the total investment to be ₹80,000 and investment in A should not exceed investment in B. Which of the following is the correct LPP model (maximize return $ Z $)?
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In LPP, the objective function represents the goal of the problem (maximizing profit, return, etc.), and the constraints represent the limitations or conditions that need to be satisfied.
Let \( x \) represent the investment in plan A and \( y \) represent the investment in plan B.
The problem conditions are as follows:
1. Return rates are 9% for plan A and 10% for plan B. The total return \( Z \) is given by:
\[
Z = 0.09x + 0.1y
\]
This function calculates the aggregate return from both investment plans.
2. The total investment must be a minimum of ₹80,000, leading to the constraint:
\[
x + y \geq 80000
\]
3. Investment in plan A cannot exceed investment in plan B, formulated as:
\[
x \leq y
\]
4. Minimum investments required are ₹20,000 for plan A and ₹30,000 for plan B, resulting in the constraints:
\[
x \geq 20000 \quad \text{and} \quad y \geq 30000
\]
The Linear Programming Problem (LPP) to maximize the return \( Z \) is therefore:
\[
\text{Maximize } Z = 0.09x + 0.1y
\]
Subject to the following constraints:
\[
x + y \geq 80000, \quad x \leq y, \quad x \geq 20000, \quad y \geq 30000
\]
The correct option is (A).