Let $Q$ be a bipartite quiver with vertex set $Q_0$ such that the number of arrows between any source vertex and any sink vertex is constant. Let $β=(β(x))_{x \in Q_0}$ be a dimension vector of $Q$ with positive integer coordinates. Let $rep(Q, β)$ be the representation space of $β$-dimensional representations of $Q$ and $GL(β)$ the base change group acting on $rep(Q, β)$ be simultaneous conjugation. Let $K^β_{\underlineλ}$ be the multiplicity of the irreducible representation of $GL(β)$ of highest weight $\underlineλ$ in the ring of polynomial functions on $rep(Q, β)$. We show that $K^β_{\underlineλ}$ can be expressed as the number of lattice points of a polytope obtained by gluing together two Knutson-Tao hive polytopes. Furthermore, this polytopal description together with Derksen-Weyman's Saturation Theorem for quiver semi-invariants allows us to use Tardos' algorithm to solve the membership problem for the moment cone associated to $(Q,β)$ in strongly polynomial time.