Why is my linear programming model wrong?

Question

(Investment problem) An individual wishes to invest £5000 over the next yearin two types of investments: Investment A yields 5% and investment B yields 8%. Market research recommends an allocation of at least 25% in A and at most 50% in B.Moreover, investment in A should be at least half the investment in B. How should the fund be allocated to the two investments? Model this problem as a linear programming problem.

My linear program is:

Let $x_1$ and $x_2$ be the amounts invested in $A$ and $B$ respectively.

$maximise$ $0.05x_1+0.08x_2$

$s.t.$
$x_1+x_2 \le 5000$

$0.25x_1-0.75x_2 \ge 0$

$0.5x_1-0.5x_2 \ge 0$

$2x_1-x_2 \ge 0$

$x_1, x_2 \ge 0$

However the solution given has:

$0.75x_1−0.25x_2≥0$ instead of $0.25x_1-0.75x_2 \ge 0$

What am I not seeing here? I can't seem to wrap my head around this.

Answer 1

An allocation of at least $25\%$ of $\$5000$ in $A$ gives

$x_1 \ge 0.25 \times 5000$

$\Rightarrow x_1 \ge 0.25(x_1+x_2)$

$\Rightarrow 0.75x_1 \ge 0.25x_2$

$\Rightarrow 0.75x_1 - 0.25x_2 \ge 0$

Answer 2

This problem appears to be missing critical information. As stated, there's no explaining either the inequalities the OP uses (or omits) or the supposedly official answer.

Letting $p_A,p_B$ be the percent allocations to the two assets, all we are told is that we wish to optimize $$.05p_A+.08p_B$$ constrained by

$$p_A+p_B≤1\quad p_A≥.25\quad 0≤p_B≤.5\quad 2p_A≥p_B$$

(Note: I am making up the $p_B≥0$ constraint, though I assume it is intended. It's not important...the max is independent of that constraint.).

Of course, it is trivial to solve this (you just get $p_A=p_B=.5$) but I assume that this is far from the intended problem.