I have a set of $m$ machines that can each produce up to $n$ different widgets, but each machine can only produce a subset of widgets (e.g., machine $1$ can produce widgets $w_1$, $w_3$, and $w_5$, while machine $2$ can produce widgets $w_1$, $w_2$, and $w_4$). I have matrix $A_{m \times n}$ that describes which widgets each machine can produce, a set of row sums $r_1,\dots,r_m$ that represent the maximum widgets each machine can produce, and a set of "target" column sums $c_1,\dots,c_n$ that represent the amount of each widget that I am trying to produce.
Is there a way to calculate if there is a set of weightings $$X_{m,n} =\left\{\begin{array}{lll} x_{1,1} &&+x_{1,3}&&+x_{1,5} & + \dots & = & r_1\\ x_{2,1} &+x_{2,2}&&+x_{2,4} && + \dots & = & r_2\\ &&&\dots &&& = & r_{n} \end{array}\right.$$
where the column sums are $\ge c_1,\dots,c_n$?
From what I have gathered reading other posts, this seems to fit in the realm of contingency tables and transportation polytopes, but I haven't seen anything about solutions to contingency tables with known margins and restrictions imposed by the lower density of $A_{m \times n}$.