Question

A person wants to invest 75,000 in options A and B, which yield returns of 8% and 9% respectively. He plans to invest at least 15,000 in Plan A, 25,000 in Plan B, and keep Plan A ≤ Plan B. Formulate the LPP to maximize the return.

• A. maximize Z=0.08x+0.09y,
  x≥15000,
  y≥25000,
  x+y≤75000,
  x≤y, x,y≥0
• B. maximize Z=0.08x+0.09y,
  x≥15000,
  y≥25000,
  x+y≤75000,
  x≥y, x,y≥0
• C. maximize Z=0.08x+0.09y,
  x≥15000,
  y≥25000,
  x+y≤75000,
  x≥y, x,y≥0
• D. maximize Z=0.08x+0.09y,
  x≥15000,
  y≥25000,
  x+y≤75000,
  x≤y, x,y≥0

Hint:

When formulating a Linear Programming Problem (LPP), it’s essential to carefully define all the constraints that impact the decision variables. Each constraint reflects a real-world limitation or requirement that must be adhered to when solving the problem. In this case, we have constraints on individual investments, total investment, and the relationship between the investments in Plan A and Plan B. By accurately interpreting and translating these constraints into mathematical inequalities, you can form an effective LPP that ensures the investment strategy is optimal and feasible. Additionally, always ensure that the final solution respects these constraints for a practical solution.

Answer 1

The Correct Option is D

The Linear Programming Problem (LPP) for maximizing the return must satisfy the given constraints:

\(x \geq 15,000\): At least Rs.15,000 is invested in Plan A.

\(y \geq 25,000\): At least Rs.25,000 is invested in Plan B.

\(x + y \leq 75,000\): The total investment does not exceed Rs.75,000.

\(x \leq y\): The investment in Plan A does not exceed the investment in Plan B.

\(x, y \geq 0\): Investments cannot be negative.

Thus, the correct representation of the LPP is option (4).

Answer 2

The Linear Programming Problem (LPP) for maximizing the return must satisfy the given constraints:

Step 1: First constraint - Investment in Plan A:

The first constraint is related to the amount of money that must be invested in Plan A. The problem specifies that the investment in Plan A must be at least Rs. 15,000. This condition ensures that there is a minimum amount allocated to Plan A, which is essential for the formulation of the investment strategy. This is represented as: \[ x \geq 15,000 \] where \( x \) denotes the amount invested in Plan A.

Step 2: Second constraint - Investment in Plan B:

The second constraint dictates that the investment in Plan B must be at least Rs. 25,000. This ensures that enough funds are allocated to Plan B to meet the investment goals. It represents a lower bound for the investment in Plan B, ensuring that Plan B is sufficiently funded. This condition is represented by: \[ y \geq 25,000 \] where \( y \) denotes the amount invested in Plan B.

Step 3: Third constraint - Total investment constraint:

The total investment in both plans combined should not exceed Rs. 75,000. This constraint ensures that the total funds available for investment are used optimally and do not exceed the budget. This upper bound ensures that the total allocation is controlled and limited by the available resources. This constraint is represented by: \[ x + y \leq 75,000 \] where \( x \) and \( y \) represent the amounts invested in Plans A and B, respectively.

Step 4: Fourth constraint - Relationship between investments in Plans A and B:

This constraint ensures that the investment in Plan A does not exceed the investment in Plan B. This condition might be imposed due to specific strategic or financial goals that require a larger investment in Plan B. It ensures that Plan B is more heavily invested than Plan A, which might reflect different risk profiles or returns associated with each plan. This condition is expressed as: \[ x \leq y \] where \( x \) and \( y \) are the investments in Plans A and B, respectively.

Step 5: Fifth constraint - Non-negativity constraint:

The non-negativity constraint ensures that the investments in both plans cannot be negative. It is a basic but essential requirement that ensures that the investment values are realistic and feasible in a financial context. This constraint is represented by: \[ x, y \geq 0 \] where \( x \) and \( y \) represent the investments in Plans A and B, respectively.

Conclusion: In summary, the correct representation of the LPP involves these five key constraints. The LPP aims to maximize returns while adhering to these restrictions, ensuring a balanced and sustainable investment strategy.

Thus, the correct matching of the constraints and objective is option (4).