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Multiplication with Vectors

John has 20...

Question

John has 20,000 $t o i n v e s t i n t h r e e f u n d s$ {F}_{1},{F}_{2} $a n d$ {F}_{3} $. F u n d$ {F}_{1} $i s o f f e r s a r e t u r n o f$ 2{F}_{2} $o f f e r s a r e t u r n o f$ 4{F}_{3} $o f f e r s a r e t u r n o f$ 5 $3000$ in $F_{3}$ and at least twice as much as in $F_{1}$ than in $F_{2}$ . Assuming that the rates hold till the end of the year, what amounts should he invest in each fund in order to maximize the year end return?

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Solution

Let $x$ be the amount invested in $F_{1}$ , $y$ the amount invested in $F_{2}$ and $z$ the amount invested in $F_{1}$ .
$x + y + z = 20, 000$
$z = 20, 000 - (x + y)$
Total return $R$ of all three funds is given by
$R = 2$ % $x + 4$ % $y + 5$ % $z = 0.2 x + 0.04 y + 0.05 (20, 000 - (x + y))$
Simplifies to
$R (x, y) = 1000 - 0.03 y x - 0.01 y$ : This is the return to maximize
Constraints: $x, y \geq 0$ which may be written as $x + y \leq 20, 000$
John invests no more than 3000 $i n$ {F}_{3}$, hence
$z \leq 3000$
Substitute $z$ by $20, 000 - (x + y)$ in the above inequality to obtain
$20, 000 - (x + y) \leq 3000$ which may be written as $x + y \geq 17, 000$
Let us put all the inequalities together to obtain the following system
$⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} x \geq 0 x \geq 0 (x + y) \leq 17000 x + y \leq 20, 000 x \geq 2 y \end{matrix}$
The solution set of the system of inequalities above and the vertices of the feasible solution set obtained are shown.
Vertices
$A$ at intersection of $x + y = 20000$ and $y = 0$ , coordinates of $A : (20000, 0)$
$B$ at intersection of $x + y = 17000$ and $y = 0$ coordinates of $B : (17000, 0)$
$C$ at intersection of $x + y = 17000$ and $x = 2 y$ coordinates of $C : (11333, 5667)$
$D$ is at intersection of $x = 2 y$ and $x + y = 20000$ , coordinates of $D : (13333, 6667)$
Evaluete the return $R (X, y) = 1000 - 0.03 x - 0.01 y$ at each one of the vertices $A (x, y), B (x, y), C (x, y)$ and $D (x, y)$
At $A (20000, 0) : R (20000, 0) = 1000 - 0.03 (20000) - 0.01 (0) = 400$
At $B (17000, 0) : R (17000, 0) = 1000 - 0.03 (17000) - 0.01 (0) = 490$
At $C (11333, 5667) : R (11333, 5667) = 1000 - 0.03 (13333) - 0.01 (5667) = 603$
At $D (13333, 6667) : R (13333, 6667) = 1000 - 0.03 (13333) - 0.01 (6667) = 533$
The return $R$ is maximum at the vertex at $C (11333, 5667)$ where $x = 11333$ and $y = 5667$ and $z = 20, 000 - (x + y) = 3000$
For maximum return, John has to invest 11333 $i n f u n d$ {F}_{1} $,$ $5667$ in fund $F_{2}$ and 3000 $i n f u n d$ {F}_{3}$

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