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...

Friday, December 12, 2014

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...

Hello!

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 $\displaystyle{x}$ and in funds as $\displaystyle{y}.$ Then the domain is formed by...

Hello!

Denote a number of millions invested in bonds as $\displaystyle{x}$ and in funds as $\displaystyle{y}.$ Then the domain is formed by the inequalities

$\displaystyle{x}+{y}\leq{40}$
$\displaystyle{x}\geq{5}$
$\displaystyle{y}\geq{20}$
$\displaystyle{100}{x}+{200}{y}\leq{7000},$ or $\displaystyle{x}+{2}{y}\leq{70}.$

They form a quadrilateral domain, please look at the picture given by the attached link. The function to maximize is

$\displaystyle{x}\cdot{1},{000},{000}\cdot{6}%+{y}\cdot{1},{000},{000}\cdot{8}%=$

$\displaystyle={60},{000}{x}+{80},{000}{y}={10},{000}{\left({6}{x}+{8}{y}\right)}.$

The line drawn in orange is $\displaystyle{6}{x}+{8}{y}={350}.$ The parallel line $\displaystyle{6}{x}+{8}{y}={A},$ touching the domain and with the greatest $\displaystyle{A}$ is the desired one.

It is evident from the picture that the point $\displaystyle{\left({10},{30}\right)}$ 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 $\displaystyle{10},{000}{\left({6}\cdot{10}+{8}\cdot{30}\right)}=$ 3,000,000($).

Notes

Friday, December 12, 2014