# Optimal way to pack items with multidimensional weight such that the number of items is minimized?

I am given a set of items S = {a1,a2,a3,...,an}. Each item has a corresponding M dimensional bit vector indicating the properties of that item. For example, if item x has corresponding vector: {0, 1, 0}, then item x does not have property 1, does have property 2, and does not have property 3.

I am also given a M dimensional "goal" vector indicating the number of items I need that match each property. For example, if my goal vector is: {1, 3, 2}, then I need 1 item with property 1, 3 items with property 2, and 2 items with property 3.

My question is, how can I select a subset of items from the set S such that the constraints listed in the goal vector are met exactly and so that the number of items selected from S is minimized? What is the name of this problem, and what is an efficient algorithm for solving it? Also, what algorithm would I use for the opposite scenario where I would like to maximize (instead of minimize) the number of items used?

Could this be a variation on the knapsack problem or the bin packing problem?