Name: Mathematical Formulation of LPP: Illustration 2
Uploaded: 2019-07-19
Duration: 8 min 40 s
Description: LPP can be used in many real-life scenarios. Here is one of them. Watch this video to get an idea of how a manufacturing unit plans production using LPP.

Question

Amelie received $$ 120000$ from a trust fund on her $25$ th birthday. She wants to invest it in different funds to maximize the interest she receives.

She can invest in three types of investment:

Type of investment	Interest per annum
Municipal bond	$2 %$
Bank Mutual fund	$3.2 %$
Speculative money market fund	$5 %$

To minimize her risk, she decides to invest only $$ 20000$ in the speculative money market fund. Her tax adviser says she has to invest at least three times as much in the municipal bond as in the bank's mutual fund. Determine the optimum investment amounts for each type of investment.

Accepted Answer

Let $x$ is the amount of money invested in municipal bond and $25$ is the amount invested in bank mutual fund.

Given that she invested only $$ 20000$ in speculative money market.

So, $x + y + 20000 \leq 120000 \Rightarrow x + y \leq 100000$

and given that $x \geq 3 y$ [she has to invest at least three times as much in the municipal bond as in the bank's mutual fund]

So, the constraints are $x + y \leq 100000$ ....(1)

, $x - 3 y \leq 0$ ...(2) $, x \geq 0, y \geq 0$

Let $z$ is total interest returned from all investments

$z = 0.02 x + 0.032 y + 0.05 (20000) = 0.02 x + 0.032 y + 1000$

To maximise $z = 0.02 x + 0.032 y + 1000$

Consider, $x + y = 100000$

at $x = 0 \Rightarrow y = 100000$ , at $y = 0 \Rightarrow x = 100000$

Solving $x + y = 100000$ and $$ 20000$ , we get $x = 75000, y = 25000$

The graph is plotted as follows,

The corner points of feasible region are $A (0, 0), B (100000, 0), C (75000, 25000)$ . The value of $z$ at corner points are

Corner Point $z = 0.02 x + 0.032 y + 1000$

$A (0, 0)$ $0$

$B (100000, 0)$ $3000$

$C (75000, 25000)$ $3300$ Maximum

The maximum value of $z$ is $3300$ at point $(75000, 25000)$ .

Therefore, she should invest $$ 75000$ in municipal bond and $$ 25000$ in bank mutual bond.

Important Questions on Decisions, Decisions (Inequalities)

Learn and Prepare for any exam you want !