I have $15$ equations in $80$ unknowns $x_i$. Each unknown is either $0$ or $1$. How do you find a solution to this? If there are multiple solutions, I want to see a couple. If one doesnt exist, how to find a best fit?
The system is of the following form
$$\sum_i a_{ij}x_i=R_j$$
So the $a_{ij}$ and $R_j$ are given strictly positive numbers, where $j$ goes from $1$ to $15$ and $i$ goes from $1$ to $80$.
As you probably know, this is called a $0-1$ linear integer programming problem.
Unfortunately, it is NP-hard so you can't do much better than exhaustive trials, and $2^{80}$ is out of reach.
If the $R_j$ aren't so large compared to the $a_{ij}$ (positive), the the number of ones will be bounded and the number of combinations will be lower. Maybe in your case other heuristics apply.