Buying ingredients for a cake from many different stores and need to minimise the number of stores - algorithm name?
If it was not for multiples the problem would be Set Cover.
With multiples, a natural name would be Multiset Cover though I cannot say I have encountered this name before.

Obviously, Multiset Cover has Set Cover as a special case.
Hence (the decision variant of) it is NP-hard and you will not find a truly efficient algorithm.
On the other hand, Multiset Cover can be seen as a special case of integer linear programming in the same way that cover problems generally can.