I have the following problem:
Let $a \in \mathbb{Z}^{n}$ and $b \in \mathbb{Z}^{k+1}$ with known integer values and $b(k+1)=1$.
Note: If it makes it easier, let $k<n$
I need to find a matrix $X$ with integer values such that $a=Xb$ and it satisfies following restrictions $$\forall i,j\quad X(i,j) \geq 0 , \forall j<k+1:\sum_{i} X(i,j) \leq n_{j}$$ with each $n_{j}$ also a known integer (so the sum of each column is restricted). Furthermore I want to minimize the sum of the last column.
Note that I reduced a problem at work to this statement and don't recall ever studying such a problem during my times at university (I haven't done much math since then). I also didn't find anything interesting on the internet. Is this even possible to solve?