Linear Algebra/Objective Function & Constraints

Question

Consider the problem:

“LUX Investments has three different possibilities. The first investment will pay 3% after the first year, the second 3.5%, and the third 6%. LUX has decided to invest $400,000 in these investments for one year. Due to risks, they have decided that the most they will invest in any one of these investments is $250,000. They have also made the decisions that the amount invested at 3% will not be less than the total of the other two and that the amount invested at 6% should not exceed the amount invested at 3.5%. How much should they invest in each in order to satisfy their conditions and maximize their expected interest on these investments?”

(a) Find the objective function.

(b) Find the list of constraints.

Answer 1

Nathan

Well suppose they invest x in the first, and y in the second, then automatically they invest 400000 - x - y in the third

After first year interest will be 0.03 x + 0.035 y + 0.06 (400000 - x - y). That's the objective function .

Constraints :

x, y, 400000 - x - y all < 250,000

x >= y + 400000 - x - y

400000 - x - y <= y

x,y, 400000 - x - y >= 0 (obviously).

So 8 constraints.

Draw the 8 constraint lines. For all the vertices of the region enclosed evaluate the objective function and take the vertex where that is maximised. From that you know x, y and you can calculate the third.

Mike