Approximation for vector bin packing
Create an instance $y_1,\ldots,y_n$ of bin packing by taking $y_i = \max(x_i^1,\ldots,x_i^d)$, and solve it using the given approximation algorithm. The solution is feasible for the vector bin packing problem. On the other hand, given a solution for the vector bin packing problem, you can construct a solution for the "max" bin packing problem using at most a factor $d$ more bins by splitting each bin into $d$ bins according to the largest coordinate.