Answer to Question #171627 in Operations Research for Nicholas Matheus

Question

A trust fund is planning to invest up to $6000 in two types of bonds.: A and B. Bond A is safer than bond B and carries a dividend of 8% while bond B carries a dividend of 10%. Suppose that the bond’s rule state that no more than $4000 may be invested in bond B, while at least $1500 must be invested in bond A. The amount invested in bond B cannot exceed one half the amount invested in bond A. How much should be invested in each type of bond to maximize the fund’s return?

Answer 1

$x=$ amount invested in A bonds

$y=$ amount invested in B bonds

$$x+y\leq6000$$

$$0\leq y\leq4000$$

$$x\geq1500$$

$$y\leq\dfrac{1}{2}x$$

Our linear optimization problem is:

Maximize $z=0.08x+0.1y$ subject to

$$x+y\leq6000$$

$$0\leq y\leq4000$$

$$x\geq1500$$

$$y\leq\dfrac{1}{2}x$$

$$\def\arraystretch{1.5} \begin{array}{c:c:c} Segment & Equation & z=0.08x+0.1y \\ \hline AB & x=1500, & z=120+0.1y\\ & 0\leq y\leq 750 & \\ \hdashline BC & y=\dfrac{1}{2}x, & z=0.13x \\ & 1500\leq x\leq 4000 & \\ \hdashline CD & y=6000-x, & z=600-0.02x \\ & 4000\leq x\leq 6000 & \\ \hdashline DA & y=0, & z=0.08x \\ & 1500\leq x\leq 6000 & \\ \hdashline \end{array}$$

$$\def\arraystretch{1.5} \begin{array}{c:c} Point & z=0.08x+0.1y \\ \hline A(1500, 0) & z=120\\ \hdashline B(1500, 750) & z=195 \\ \hdashline C(4000, 2000) & z=520 \\ \hdashline D(6000, 0)& z=480 \\ \hdashline \end{array}$$

Because the point $(4000, 2000)$ produces the highest fund’s return we conclude that $4000 should be invested in

How much should be invested in bond A and $2000 should be invested in bond B.