Practical/heuristic algorithm for multi set-cover
You'd have to tweak the limits (in particular max_level may be too low), but for at least some "real" problems this is within the bounds of Knuth's algorithm M.

See also


The documentation of algorithm D, which describes the basic file format; the documentation of algorithm M just explains the changes made to generalise it;
the latest preprint of fascicle 5c of The Art of Computer Programming, which at some point will get as far as explaining the algorithm, but already has a number of exercises that will give you an idea of the problems that Knuth has solved with it.