# Thread: Linear programming with constraints

1. ## Linear programming with constraints

George Johnson would like 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. To reduce the risk resulted from market volatility, he wants to invest at least 30% of the entire amount of trust fund in the bond fund. In addition, he wants to select a portfolio that will enable him to obtain a total return of at least 7.5%.

Formulate a linear program that can be used to determine the percentage allocation to the bond fund and stock fund. The objective of the problem is to maximize the expected total portfolio return.

So far I have:
$\displaystyle z=0.06b+0.1s$ (Maximize expected total portfolio return)

Subject to:

$\displaystyle s\geq 0$
$\displaystyle b\geq 0$
$\displaystyle s+b=1$
$\displaystyle b\geq 0.3$

I'm confused by the part in bold. Is that another constraint? If my goal is to maximize expected return, what does it matter if the person wants at least a 7.5% return? Am I reading something wrong?

2. Originally Posted by downthesun01
George Johnson would like 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. To reduce the risk resulted from market volatility, he wants to invest at least 30% of the entire amount of trust fund in the bond fund. In addition, he wants to select a portfolio that will enable him to obtain a total return of at least 7.5%.

Formulate a linear program that can be used to determine the percentage allocation to the bond fund and stock fund. The objective of the problem is to maximize the expected total portfolio return.

So far I have:
$\displaystyle z=0.06b+0.1s$ (Maximize expected total portfolio return)

Subject to:

$\displaystyle s\geq 0$
$\displaystyle b\geq 0$
$\displaystyle s+b=1$
$\displaystyle b\geq 0.3$

I'm confused by the part in bold. Is that another constraint? If my goal is to maximize expected return, what does it matter if the person wants at least a 7.5% return? Am I reading something wrong?
It is an additional constraint which will either render the problem infeasible or not be tight. So solve without this constraint if the return exceeds 7.5% then you need not worry, if it is less than 7.5% the problem is infeasible (but if this were an exam question you should either include it as a constraint or say why you are ignoring it in the formulation and how it will be treated after solving the LP without it).

CB

3. Thanks. That's what I was thinking. Much appreciated.

4. One more quick thing. Can you see any way that the feasible region would not be triangular in shape? I graphed everything out and got a triangular region bound by:
$\displaystyle s=-0.6b+0.75$
$\displaystyle b\geq 0.3$
$\displaystyle s=1-b$

Before giving us this problem to practice, our professor said that the feasible region would not be triangular in shape for this problem. Either I'm graphing wrong or he was mistaken. Any help would be appreciated.

5. Originally Posted by downthesun01
One more quick thing. Can you see any way that the feasible region would not be triangular in shape? I graphed everything out and got a triangular region bound by:
$\displaystyle s=-0.6b+0.75$
$\displaystyle b\geq 0.3$
$\displaystyle s=1-b$

Before giving us this problem to practice, our professor said that the feasible region would not be triangular in shape for this problem. Either I'm graphing wrong or he was mistaken. Any help would be appreciated.
That looks OK to me, it would only be non-triangular if the return constraint was no more than 7.5%

CB