I have a video game(Factorio, if you have heard of it) that I like to play that involves many different kinds of ratios in order to optimize the production of items that require other items. Normally I can solve for the ratios, but this one is a little more complicated. Here is the problem:
I have 3 different kinds of factories in my setup. For simplicity, we will call them X, Y, and Z. We will also have 3 different products we will be discussing, which we will call H, L, and G.
Factory X takes no input(it does, but it is a fourth product that can be supplied unlimited from an external source, so shouldn't be needed to solve the problem), and produces all three outputs. It produces 55.575 of H, 100.035 of L, and 122.265 of G.
Factory Y takes H as an input, and has L as an output. It consumes 131 of H and produces 127.725 of L.
Factory Z takes L as an input, and has G as an output. Is consumes 98.25 of L and produces 65.5 of G.
The problem to solve is to find the ratio of X:Y:Z(where X, Y, and Z are all greater than zero) that will avoid any excess product of H or L, leaving only G as the net output.
I am pretty sure that this problem falls in the realm of Linear Algebra(which I admittingly have only a basic knowledge of), something to do with matrices and/or simultaneous equations. Trying to solve for X, Y, and Z, though, would require 3 equations, and I have sadly only been able to come up with either 2, or a third at the cost of accidentally creating a fourth unknown.
Here is what I have tried thus far, in case I was on the right track:
First attempt:
0 = H = 55.575X - 131Y + 0Z
0 = L = 100.035X + 127.725L - 98.25Z
G = 122.265X + 0Y + 65.5Z
(Doesn't work due to creating fourth unknown, G)
Second attempt:
Y = (55.575X)/131
Z = (100.035X + 127.725Y)/98.25
(Doesn't work due to only having 2 equations)
I would love to see the solution to this problem, and see what exactly I am missing here. Thank you in advance for any help!
EDIT:
From another source, I had a few people solving the problem. I am led to believe that there are too many unknowns to solve for a perfect ratio, and also that perhaps there may not actually be a completely perfect ratio. That said, with a ratio of 1000000000000 : 424236641221 : 1569675572520, there is only a 0.000000000088752720331681537543048644709316% error from whatever the perfect ratio would be, which in incredibly close. I will follow up with the person's calculations to obtain said ratio.
EDIT:
Turns out that it is a repeating decimal answer, so the above was not in simplest form. In its simplest form, the answer is 52400:22230:82251
EDIT:
Corrected type on second equation, changing 100.032 to 100.035
Best Answer
Let's go back to your first attempt. We have the system of equations:
\begin{align} 0 &= 55.575X - 131Y + 0Z \\ 0 &= 100.032X + 127.725Y - 98.25Z \\ G &= 122.265X + 0Y + 65.5Z \end{align}
Now we have four unknowns, but remember we're just trying to find a ratio between $X$, $Y$, and $Z$. This means we can fix any of them at $1$ and find the ratio from there. Let $X=1$, and we will find.
\begin{align} 0 &= 55.575 - 131Y\\ 0 &= 100.032 + 127.725Y - 98.25Z \\ G &= 122.265+65.5Z \end{align}
We can solve this system now, I used WolframAlpha to do so and got $$G=\frac{900307}{4000}, X = 1, Y = \frac{2223}{5240}, Z = \frac{411247}{262000}$$
Now we wish to find integer results for $Y$ and $Z$ so we multiply and find $$262000: 111150: 411247$$ as the ratio $X:Y:Z$.
I've now checked my work by plugging it back into the original equations and find an output of $G = 5.89701\cdot 10^7$.
Happy engineering!