Multi-categorical set balancing

I have a particular mathematical problem that I would name as multi-categorical set balancing. I don't think this is a new problem but I do not know the correct term for it, therefore I am also looking for the correct terms to describe such problem.

The problem consists in having a list of sets such as the sum of their contents is as similar as possible. We can consider that the pool of sets is infinite. For example we have a pool of sets that look like this: [0, 0, 1], [1, 0, 1], [0, 1, 0]... And if for example we want a list of sets that has 1 of each categories we would choose sets 2 and 3. [1, 0, 1] + [0, 1, 0] = [1, 1, 1]

In my real problem the sets are quite long (+200 categories) and can have values larger than 1, but always integers and always positive. I understand that this problem can have multiple solutions and that for large numbers it might be impossible that the perfect balance is found, therefore I am interested in heuristic methods that find a solution that can fit some restrictions.

What are the appropriate terms for such a problem? There are probably already some algorithms that solve this kind of problem.