Multidimensional knapsack STRONGLY NP-complete
In the paper titled "ON MULTIDIMENSIONAL PACKING PROBLEMS" that appeared in SODA and then in SICOMP (http://epubs.siam.org/doi/abs/10.1137/S0097539799356265) we considered several packing problems where items are d-dimensional vectors. We discuss PIPs which are the same as as multidimensional knapsack (we mention this) and their approximability when d is not fixed. You can find hardness results as function of d in that paper.