 Solution

Form a bipartite graph with one vertex for each row and one for each column, putting an edge $ij$ exactly where $a_{ij}>0$. For a set $S$ of row vertices, let $N(S)$ be its neighbouring column vertices. Since $A$ is a doubly stochastic matrix,The Hall marriage theorem therefore supplies a perfect matching. Its incidence entries define a permutation matrix $P$ supported on the positive entries of $A$, hence on the ones of $B$. Entrywise $P\le B$, so