Let G = (X, Y,E(G)) be a bipartite graph with vertex set V (G) = X [ Y and edge set E(G) and let g and f be two non-negative integer-valued functions defined on V (G) such that g(x)≤f(x) for each x ∈ V (G). A (g, f)-factor of G is a spanning subgraph F of G such that g(x)≤dF (x)≤f(x) for each x ∈ V (F); a (g, f)-factorization of G is a partition of E(G) into edge-disjoint (g, f)-factors. In this paper it is proved that every bipartite (mg+ m− 1,mf− m+ 1)-graph has (g, f)-factorizations randomly k-orthogonal to any given subgraph with km edges if k ≤ g(x) for any x ∈ V (G) and has a (g, f)-factorization k-orthogonal to any given subgraph with km edges if k-1≤g(x) for any x ∈ V (G) and that every bipartite (mg,mf)-graph has a (g, f)-factorization orthogonal to any given m-star if 0≤g(x)≤f(x) for any x ∈ V (G). Furthermore, it is shown that there are polynomial algorithms for finding the desired factorizations and the results in this paper are best possible.