A person wants to invest an amount of ₹ 75,000. He has two options A and B yielding 8% and 9% return respectively on the invested amount. He plans to invest at least ₹15,000 in Plan A and at least ₹25,000 in Plan B. Also he wants that his investment in Plan A is less than or equal to his investment in Plan B. Which of the following options describes the given LPP to maximize the return (where x and y are investments in Plan A and Plan B respectively)?

Question

Prepp Team · Accepted Answer

Linear Programming Problem Formulation for Investment Planning

Let's break down the problem step by step to formulate the Linear Programming Problem (LPP) that maximizes the return on investment.

Defining Variables

First, we need to define the variables representing the amounts invested in each plan. The question specifies this:

Let $x$ be the amount invested in Plan A.
Let $y$ be the amount invested in Plan B.

Identifying the Objective Function

The goal is to maximize the total return. The return from Plan A is 8% of $x$, which is $0.08x$. The return from Plan B is 9% of $y$, which is $0.09y$. The total return, let's call it $Z$, is the sum of returns from both plans.

Objective Function: Maximize $ Z = 0.08x + 0.09y $

Establishing Constraints

The problem describes several conditions or restrictions on the investment amounts. These translate into constraints for the LPP.

Minimum Investment in Plan A: The person plans to invest at least ₹15,000 in Plan A.
Constraint: $ x \geq 15000 $
Minimum Investment in Plan B: The person plans to invest at least ₹25,000 in Plan B.
Constraint: $ y \geq 25000 $
Total Investment Amount: The person wants to invest an amount of ₹75,000. While "an amount of" often implies an exact total, LPP problems can sometimes use inequalities based on context or common interpretations. The provided formulation indicates the total investment must be less than or equal to ₹75,000.
Constraint: $ x + y \leq 75000 $
Relationship between Investments in Plan A and Plan B: The investment in Plan A must be less than or equal to the investment in Plan B.
Constraint: $ x \leq y $
Non-negativity Constraints: Investment amounts cannot be negative. While the minimum investment constraints ($x \geq 15000$ and $y \geq 25000$) already ensure $x$ and $y$ are non-negative, it is standard practice in LPP to explicitly state the non-negativity constraints.
Constraints: $ x \geq 0, y \geq 0 $

Summarizing the LPP

Combining the objective function and all the constraints, the complete LPP formulation to maximize the return is:

Maximize $ Z = 0.08x + 0.09y $

Subject to:

$ x \geq 15000 $
$ y \geq 25000 $
$ x + y \leq 75000 $
$ x \leq y $
$ x \geq 0, y \geq 0 $

Comparing with Options

Let's compare our derived LPP formulation with the given options:

Formulation Component	Our Derived LPP	Option 1	Option 2	Option 3	Option 4
Objective Function	Maximize $ Z = 0.08x + 0.09y $	Maximize $ Z = 0.08x + 0.09y $	Maximize $ Z = 0.08x + 0.09y $	Maximize $ Z = 0.08x + 0.09y $	Maximize $ Z = 0.08x + 0.09y $
Constraint 1 ($x$)	$ x \geq 15000 $	$ x \geq 15000 $	$ x \geq 15000 $	$ x \geq 15000 $	$ x \geq 15000 $
Constraint 2 ($y$)	$ y \geq 25000 $	$ y \geq 25000 $	$ y \geq 25000 $	$ y \geq 25000 $	$ y \geq 25000 $
Constraint 3 (Total)	$ x + y \leq 75000 $	$ x + y = 75000 $	$ x + y \geq 75000 $	$ x + y \leq 75000 $	$ x + y \leq 75000 $
Constraint 4 ($x$ vs $y$)	$ x \leq y $	$ x \leq y $	$ x \leq y $	$ x \geq y $	$ x \leq y $
Constraint 5 (Non-negativity)	$ x \geq 0, y \geq 0 $	$ x \geq 0, y \geq 0 $	$ x \geq 0, y \geq 0 $	$ x \geq 0, y \geq 0 $	$ x \geq 0, y \geq 0 $

Comparing the formulations, Option 4 exactly matches our derived LPP formulation.

Revision Table: Key LPP Terms

Term	Explanation	Example from this problem
Decision Variables	The quantities you need to decide on (what you can control).	$x$ (investment in Plan A), $y$ (investment in Plan B)
Objective Function	The function you want to maximize or minimize.	Maximize $ Z = 0.08x + 0.09y $ (Total return)
Constraints	Limitations or restrictions on the decision variables.	$x \geq 15000$, $y \geq 25000$, $x + y \leq 75000$, $x \leq y$
Non-negativity Constraints	Requirement that decision variables must be non-negative.	$x \geq 0$, $y \geq 0$
Feasible Region	The set of all points ($x, y$) that satisfy all constraints.	The area on a graph where all inequalities are true.

Additional Information: Solving an LPP

Once the LPP is formulated, it can be solved using various methods to find the optimal values of $x$ and $y$ that maximize the objective function within the feasible region defined by the constraints.

Graphical Method: For LPPs with two variables ($x$ and $y$), we can plot the constraints as lines on a graph. The feasible region is the area that satisfies all inequalities. The optimal solution usually occurs at one of the corner points (vertices) of the feasible region.
Simplex Method: For LPPs with more than two variables or a large number of constraints, the simplex method is a more systematic algebraic approach.

To solve this specific LPP graphically, you would plot the lines $x = 15000$, $y = 25000$, $x + y = 75000$, and $x = y$. Then, you would identify the region that satisfies $x \geq 15000$, $y \geq 25000$, $x + y \leq 75000$, and $x \leq y$. The corner points of this feasible region would be evaluated in the objective function $Z = 0.08x + 0.09y$ to find the maximum return.

Prepp Team · Answer

Maximize $ Z = 0.08x + 0.09y $ $ x \geq 15000 $ $ y \geq 25000 $ $ x + y = 75000 $ $ x \leq y $ $ x, y \geq 0 $

Prepp Team · Answer

Maximize $ Z = 0.08x + 0.09y $ $ x \geq 15000 $ $ y \geq 25000 $ $ x + y \geq 75000 $ $ x \leq y $ $ x, y \geq 0 $

Prepp Team · Answer

Maximize $ Z = 0.08x + 0.09y $ $ x \geq 15000 $ $ y \geq 25000 $ $ x + y \leq 75000 $ $ x \geq y $ $ x, y \geq 0 $

Formulation Component	Our Derived LPP	Option 1	Option 2	Option 3	Option 4
Objective Function	Maximize \( Z = 0.08x + 0.09y \)	Maximize \( Z = 0.08x + 0.09y \)	Maximize \( Z = 0.08x + 0.09y \)	Maximize \( Z = 0.08x + 0.09y \)	Maximize \( Z = 0.08x + 0.09y \)
Constraint 1 (\(x\))	\( x \geq 15000 \)	\( x \geq 15000 \)	\( x \geq 15000 \)	\( x \geq 15000 \)	\( x \geq 15000 \)
Constraint 2 (\(y\))	\( y \geq 25000 \)	\( y \geq 25000 \)	\( y \geq 25000 \)	\( y \geq 25000 \)	\( y \geq 25000 \)
Constraint 3 (Total)	\( x + y \leq 75000 \)	\( x + y = 75000 \)	\( x + y \geq 75000 \)	\( x + y \leq 75000 \)	\( x + y \leq 75000 \)
Constraint 4 (\(x\) vs \(y\))	\( x \leq y \)	\( x \leq y \)	\( x \leq y \)	\( x \geq y \)	\( x \leq y \)
Constraint 5 (Non-negativity)	\( x \geq 0, y \geq 0 \)	\( x \geq 0, y \geq 0 \)	\( x \geq 0, y \geq 0 \)	\( x \geq 0, y \geq 0 \)	\( x \geq 0, y \geq 0 \)