• Nov 5th 2006, 09:30 PM
pauline112
1.) George Johnson recently inherited a large sum of money; he wants to use a portion of this money to set up a trust fund for his two children. The trust fund has two investment options: (1) a bond fund and (2) a stock fund. The projected returns over the life of the investments are 6% for teh bond fund and 10% for the stock fund. Whatever portion of the inheritance he finally decides to commit to the trust fund, he wants to invest at lesat 30% of that amount in the bond fund. In addition, he wants to select a mix that will enable him to obtain a total return of at least 7.5%.
a. Formulate a linear programming model that can be used to determine the percentage that should be allocated to each of hte possible investment alternatives.
b. Solve the problem using the graphical soluiton procedure.
• Nov 6th 2006, 05:31 AM
CaptainBlack
Quote:

Originally Posted by pauline112
1.) George Johnson recently inherited a large sum of money; he wants to use a portion of this money to set up a trust fund for his two children. The trust fund has two investment options: (1) a bond fund and (2) a stock fund. The projected returns over the life of the investments are 6% for teh bond fund and 10% for the stock fund. Whatever portion of the inheritance he finally decides to commit to the trust fund, he wants to invest at lesat 30% of that amount in the bond fund. In addition, he wants to select a mix that will enable him to obtain a total return of at least 7.5%.
a. Formulate a linear programming model that can be used to determine the percentage that should be allocated to each of hte possible investment alternatives.
b. Solve the problem using the graphical soluiton procedure.

Let b be the percentage invested in the bond func, and s be the percentage
invested in the stock fund.

Then: b+s<=100

(the total investment cannot be more than 100% of the available funds
<= here indicates that there is no requirement that he invest all of the
inheritance)

Also: b>=0.30(s+b)

(b must be at least 30% of the invested funds).

0.06b + 0.10s>=7.5.

These three inequalities together with the requirments that s>=0, b>=0
represent the problem statement, that is:

b+s<=100
b>=0.30(s+b)
0.06b + 0.10s>=7.5
s>=0
b>=0

Which may be rearranged into:

-b-s>=-100
0.7b-0.3s>=0
0.06b + 0.10s>=7.5
s>=0
b>=0

But as these stand they do not constitute a linear program as there is
no objective function specified that is to be maximised (or minimised).
Though a solution to the problem as posed will be the entire feasible region
(if there is one).

(We can probably assume that we are to maximise the return 0.06b + 0.10s
but we have not been asked to).

The attached plot shows the feasible region (shaded) and which vertex of
the region maximises 0.06b + 0.10s.

RonL
• Nov 6th 2006, 08:04 AM
Soroban
Hello, Pauline!

This linear programming problem has a feature
. . that makes it quite unusual . . .

Quote:

George Johnson recently inherited a large sum of money.
He wants to use a portion of this money to set up a trust fund for his two children.
The trust fund has two investment options: (1) a bond fund and (2) a stock fund.
The projected returns over the life of the investments are 6% for the bond fund
and 10% for the stock fund.

Whatever portion of the inheritance he finally decides to commit to the trust fund,
he wants to invest at lesat 30% of that amount in the bond fund.
In addition, he wants a mix with a total return of at least 7.5%.

a. Formulate a linear programming model that can be used to determine the percentage
that should be allocated to the two investment alternatives.

b. Solve the problem using the graphical soluiton procedure.

Let $B$ = percentage invested in the bond fund. . $B \geq 0$
Let $S$ = percentage invested in the stock fund. . $S \geq 0$

So we have: . $B + S \:=\:100$ [1]

We are told that: $B \geq 30$ [2]

The return on the bond fund is: $6B$ dollars.
The return on the stock fund is; $10S$ dollars.
. . The total return is: $6B + 10S$ dollars.

The total return is to be at least 7.5% of the total investment: $7.5(B + S)$

Hence, we have: . $6B + 10S \:\geq \:7.5(B + S)\quad\Rightarrow\quad S \:\geq\:\frac{3}{5}B$ [3]

Graph these statements on a $B\text{-}S$ coordinate system.

Inequaltiy [2]: $B \geq 30$
This is the vertical line $B = 30$ and the region to the right.

Inequality [3]: $S \geq \frac{3}{5}B$
This is the slanted line $S = \frac{3}{5}B$ and the region above it.

So far, we have:
Code:

        S         |      :::::::::::::::::         |      ::::::::::::::::         |      :::::::::::::::         |      ::::::::::::::         |      ::::::::::::*         |      ::::::::*         |      ::::*         |      *         |  *  :       --*-------+-------------- B         |      30

But statement [1] is an equation: $B + S \:=\:100$

There is no "region" . . . the points are on the slanted line.
Code:

        |     100 *      :         | *    :         |  *  :         |    * :         |      oA          *         |      : *    *         |      :  oB         |      *    *         |  *  :      *       --*-------+---------*----         |      30        100

So the "region" is the line segment from $A(30,70)$ to $(62.5,37.5)$

We find that the maximum return is at point $A(30,70)$

Therefore, George should invest 30% in bonds and 70% in stocks.

• Nov 6th 2006, 08:13 AM
CaptainBlack
Quote:

Originally Posted by Soroban
Hello, Pauline!

This linear programming problem has a feature
. . that makes it quite unusual . . .

Let $B$ = percentage invested in the bond fund. . $B \geq 0$
Let $S$ = percentage invested in the stock fund. . $S \geq 0$

So we have: . $B + S \:=\:100$ [1]

We are told that: $B \geq 30$ [2]

This last inequality is not what the condition specifies, it actually says:

"Whatever portion of the inheritance he finally decides to commit to the trust fund, he wants to invest at lesat 30% of that amount in the bond fund"

which is that 0.3*(s+b)<=b

RonL
• Nov 6th 2006, 08:39 AM
Soroban
Hello, Captain!

I believe we're both saying the same thing.

You said: . $B \:\geq \:0.3(B + S)$

Since $B + S \:=\:100$ (they're percentages): . $B \,\geq\,0.3(100) \,=\,30$

• Nov 6th 2006, 09:00 AM
CaptainBlack
Quote:

Originally Posted by Soroban
Hello, Captain!

I believe we're both saying the same thing.

You said: . $B \:\geq \:0.3(B + S)$

Since $B + S \:=\:100$ (they're percentages): . $B \,\geq\,0.3(100) \,=\,30$

But we (however bizarre it may seem) not told that b+s=100:eek:
We just have b+s<=100.

(with one interpretation of what we are to maximise it is true that
at the optimum b+s=100, so with that interpretation the optimum
will be the same, but they are then different problems which happen
to have the same solution:eek: :eek:

Or have I slipped into pedantic mode without noticing?:confused:

RonL
• Sep 20th 2007, 07:46 PM
livingtwo
so which one is right here?
• Sep 20th 2007, 09:53 PM
CaptainBlack
Quote:

Originally Posted by livingtwo
so which one is right here?

Probably neither as the problem is ambiguous, it is not clear if the 7.5%
return is on the investment or the inheritance.

There I believe the main difference between the ywo methods is that
CaptainBlack is calculating for a return of 7.5% on the inheritance while
Soroban is calculating for a return of 7.5% on the investment (as far as
I can tell at this distance in time)

You will note that Soroban says B+S=100, that is he is either assuming all
of the inheritance is invested or that B and S represents the split of what
is invested. CaptainBlack says b+s<=100, that is they are the percentages
of the inheritance invested in each option.

Of course if you had been asked to maximise the monetary return all of the
inheritance would have been invested as the diagram attached to CB's post
shows that the optimum mix would be 70-30 which is the same as Soroban's
optimum.

RonL