Word Problem Linear progrmming

Question

Mr. Lothbrok has $10,000 to invest in two types of mutual funds, Coleman High-yield Fund, and
Coleman Equity Fund. The High-yield fund gives an annual yield of 6%, while the Equity fund earns
10%. Mr. Lothbrok would like to invest at least $5000 in the High-yield fund and at least $2000 in the
Equity fund. How much money should he invest in each to maximize his annual yield, and what is the
maximum yield?

Answer 1

It's true that you can solve this as a linear programming problem, so let's do that.

 

Let x = the amount invested in the High-yield fund, and let y = the amount invested in the Equity fund.

 

The objective function would be the amount of yield: 0.06x + 0.10y

 

The total amount available to invest is 10000, so that gives us one constraint:  x + y <= 10000

 

The minimums he wants to invest in each fund gives us two more constraints:

 

x >= 5000

y >= 2000

 

It's often helpful to draw the problem on graph paper.  Use units of $1000 per space.  Draw the lines for each of the three constraints and shade the regions that satisfy the inequalities. You will see that the feasible region is a triangle bordered by x = 5000 on the left, by y = 2000 on the bottom, and by x + y = 10000 to the upper right.

 

The following points are the vertices of the triangle:  (5000, 2000), (8000, 2000), and (5000, 5000).  The objective function will be a maximum at one of these points.  So let's evaluate the objective function at each:

 

(5000, 2000):  0.06x + 0.10y = 0.06 * 5000 + 0.10 * 2000 = 300 + 200 = 500

(8000, 2000):  0.06x + 0.10y = 0.06 * 8000 + 0.10 * 2000 = 480 + 200 = 680

(5000, 5000):  0.06x + 0.10y = 0.06 * 5000 + 0.10 * 5000 = 300 + 500 = 800

 

The maximum occurs at (5000, 5000), so the amount he should invest is 5000 in each fund.  The return will be 800, for an 8% total yield.

 

 

BUT, it's also possible to solve this problem logically, without linear programming.

 

First of all, you know that he should invest all $10,000.  Any money he invests will earn something, but any money he doesn't invest will earn nothing.  You also know that the Equity fund gives a higher return, so he should invest as much money in that fund as he can, subject to the minimum investments he wants to make in each fund.  Since he wants to invest at least $5,000 in the High-yield fund, he will only be able to invest $5,000 in the Equity fund.  That gives the same result as the linear programming approach.