Suppose that $b_1,\dots,b_m \in \mathbb Z^n$ are not linearly independent over $\mathbb Z$ (otherwise the problem is trivial). Given another element $w\in \mathbb Z^n$ is there a way to determine (algorithmically) if $w$ can be written as a linear combination of $b_1,\dots,b_m$ with positive integer coefficients?
Edit: the $b_i$ under my consideration have exactly one negative component, and all the other components are either 1 or 0, and the sum of the components is 0. I say this, because in this special case there may be a specific solution.