0

(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.

user499701
  • 303
  • This seems incomplete. If the assets have no risk attached to them, you should clearly invest fully in the one with higher yield (including short selling the lower yield one to finance greater investment in the higher yield asset). Absent risks the constraints appear to be entirely random. There could be a cash investment...what are the constraints on that? What are you trying to optimize? One obvious answer is $50%$ goes to high yield, $50%$ to low yield. That optimizes return given that these assets (and $0%$ cash) are the only options. Is that the answer you wanted? – lulu Aug 23 '18 at 12:45
  • @lulu no, I don't think we're meant to know anything about investing for this course. – user499701 Aug 23 '18 at 12:48
  • Ok...but then do you agree with the $50-50$ answer? It passes all your constraints and it clearly optimizes return. No? – lulu Aug 23 '18 at 12:49
  • @lulu I think so but we're not required to solve this. I would get 0 marks for that in an exam. – user499701 Aug 23 '18 at 12:53
  • @user499701 In my view there are some flaws in the solution. For instance: An individual wishes to invest £5000 over the next ... means $x_1+x_2=5000$ and Market research recommends an allocation of at least 25% in A means $x_1\geq 0.25\cdot 5000=1250$ – callculus42 Aug 23 '18 at 13:00
  • I don't understand your inequalities. I see inequalities of the form $p_1≥.25,p_2≤.5$ (using percents instead of pounds). And $2p_1≥p_2$. Where do the $.25,.75$ constraints come from? – lulu Aug 23 '18 at 13:02
  • @callculus I wrote it that way too at first and was told that naturally when we solve this linear program $x_1+x_2$ will equal 5000 if we wanted to maximise the profit function. But I believe strictly from that line $x_1+x_2 \le 5000$ holds. – user499701 Aug 23 '18 at 13:07
  • I think you have omitted critical information. Your constraint, $.25x_1-.75x_2≥0$ is equivalent to $x_1≥3x_2$. (and the one you say is correct is $3x_1≥x_2$). Where does a constraint of this form arise? I don't see anything in your problem statement which suggests this. – lulu Aug 23 '18 at 13:11
  • @user499701 What about the other constraints? 2, $x_1\geq 1250$, $x_1\leq 2500$ for Market research recommends an allocation of at least 25% in A and at most 50% in B – callculus42 Aug 23 '18 at 13:11
  • @lulu I've been trying to follow the given answer to this question and I'm not so sure either. How would you do it in terms of percentages or pounds? – user499701 Aug 23 '18 at 13:11
  • I have told you how I would do it, and it is a complete triviality. As I say, I have to imagine you are leaving off critical information. Where did you get the inequality you claimed? The one that holds that $.25x_1≥.75 x_2$? – lulu Aug 23 '18 at 13:12
  • @lulu this is an official university exam question (supposed to be a straightforward starter question) and the solution is one that is given by my professor. It is annoying me too because either I do not understand it or the solution is incorrect. – user499701 Aug 23 '18 at 13:15
  • Once again: where did you get the inequality $.25x_1-.75x_2≥0$? You wrote that down for some reason, presumably. What was that reason? What about $.5x_1-.5x_2≥0$? What made you think of that? Why did you not use the obvious constraints? $x_1≥.25\times 5000,x_2≤.5\times 5000$? – lulu Aug 23 '18 at 13:16
  • @lulu my first solution was in terms of pounds. $x_1 \ge 1250$ and $x_2 \le 2500$. Then I looked at the solution. Which had written $0.75x_1−0.25x_2≥0$ and $0.5x_1-0.5x_2 \ge 0$ when using the same information. So I figured they wanted us to write it in this way. I tried to think of it as the percentage invested in A must be 25% or more than the rest invested in B (75%). Although this does not entirely make sense to me. – user499701 Aug 23 '18 at 13:24
  • Sorry, that really doesn't make any sense. Clearly you have omitted necessary information. Perhaps you transcribed the problem incorrectly or perhaps it was typed incorrectly on whatever you are looking at or perhaps the official solution refers to an unrelated problem. – lulu Aug 23 '18 at 13:27
  • @lulu I copied and pasted the problem. No information was omitted. – user499701 Aug 23 '18 at 13:28

2 Answers2

2

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

$x_1 \ge 0.25 \times 5000$

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

$\Rightarrow 0.75x_1 \ge 0.25x_2$

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

gandalf61
  • 15,326
0

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.

lulu
  • 70,402