Optimal way to pack items with multidimensional weight such that the number of items is minimized?
This problem is NP-hard, so you can't expect a general algorithm that is efficient and always correct.  In particular, in the special case where the goal vector has only 1's and 0's in it, it becomes an instance of the exact cover problem.
Your problem can be viewed as an instance of the multidimensional subset sum problem.
If you have to solve it in practice, I would suggest formulating it as an instance of integer linear programming and using an off-the-shelf ILP solver.