Answer to Question #289627 in Operations Research for seth

Question

A trust fund is planning to invest up to P600,000 in

two type of bonds: A and B. Bond A carries a

dividend of 25% and B, 30%. Suppose the fund rules

state that no more than P400,000 may be invested in

bond B, while at least P150,000 must be 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 in thousands

$y=$ amount invested in B bonds in thousands

$$x+y\leq600$$ $$0\leq y\leq400$$ $$x\geq150$$

Our linear optimization problem is:

Maximize $z=0.25x+0.30y$ subject to

$$x+y\leq600$$ $$0\leq y\leq400$$ $$x\geq150$$

$$\def\arraystretch{1.5} \begin{array}{c:c:c} Segment & Equation & z=0.25x+0.30y \\ \hline AB & y=400, & z=0.25x+120\\ & 150\leq x\leq 200 & \\ \hdashline BC & y=600-x, & z=180-0.05x\\ & 200\leq x\leq 600 & \\ \hdashline DC & y=0, & z=0.25x \\ & 200\leq x\leq 600 & \\ \hdashline DA & x=150, & z=37.5+0.30y \\ & 0\leq y\leq 400 & \\ \hdashline \end{array}$$

$$\def\arraystretch{1.5} \begin{array}{c:c} Point & z=0.25x+0.30y \\ \hline A(150, 400) & z=157.5\\ \hdashline B(200,400) & z=170 \\ \hdashline C(600, 0) & z=150 \\ \hdashline D(150, 0)& z=37.5 \\ \hdashline \end{array}$$

Because the point $(200, 400)$ produces the highest fund’s return we conclude that P200,000 should be invested in bond A and P400,000 should be invested in bond B to maximize the fund’s return.