A pension fund manager decides to invest a total of at most $40 million in U.S. Treasury bonds paying 6% annual interest and in mutual funds paying 8% annual interest. He plans to invest at least...
A pension fund manager decides to invest a total of at most $40 million in U.S. Treasury bonds paying 6% annual interest and in mutual funds paying 8% annual interest. He plans to invest at least $5 million in bonds and at least $20 million in mutual funds. Bonds have an initial fee of $100 per million dollars, while the fee for mutual funds is $200 per million. The fund manager is allowed to spend no more than $7000 on fees. How much should be invested in each to maximize annual interest? What is the maximum interest?
Such problems are called linear programming ones. We have a two-dimensional domain formed by straight lines, and a linear function to maximize on this domain. To solve such a problem it is desirable to draw the domain and a line where the function to maximize has some value (all such lines are parallel).
Denote a number of millions invested in bonds as `x` and in funds as `y.` Then the domain is formed by the inequalities
`100x+200ylt=7000,` or `x+2ylt=70.`
They form a quadrilateral domain, please look at the picture given by the attached link. The function to maximize is
The line drawn in orange is `6x+8y=350.` The parallel line `6x+8y=A,` touching the domain and with the greatest `A` is the desired one.
It is evident from the picture that the point `(10,30)` is the nearest to the orange line. This means $10 million should be invested in U.S. Treasury bonds and $30 million in mutual funds. The annual interest will be `10,000(6*10+8*30)=` 3,000,000($).