Set cover with budget on sets I am wondering if this is a studied variant of the Set Cover problem.

We are given a universe $X$, a collection of sets $S = \{S_1, ..., S_m\}$ and integers $c_i$.  We want to cover all elements in $X$ with sets $S'_i \subseteq S_i$ such that $|S'_i| \leq c_i$.  In other words we want to cover the elements with as few sets as possible but we are only allowed to use $c_i$ elements of each set. 

The most similar variant I found is described in "Maximum Coverage Problem with Group Budget Constraints and Applications"

Thank you
Yes, this variant, and in fact a further generalization has been considered in the literature. See the paper below for the problem they call capacitated facility location.

J. Bar-Ilan, G. Kortsarz and D. Peleg, 
Generalized submodular cover problems and applications, 
Theoretical Computer Science, 250:179-200, 2001.