  In this paper, we consider $\mathbb{Z}^{r}-$graded modules on the
$\mathrm{Cl}(X)$ $-$graded Cox ring $\mathbb{C}[x_{1},\dotsc,x_{r}]$ of a
smooth complete toric variety $X$. Using the theory of Klyachko filtrations in
the reflexive case, we construct a collection of lattice polytopes codifying
the multigraded Hilbert function of the module. We apply this approach to
reflexive $\mathbb{Z}^{s+r+2}$-graded modules over non-standard bigraded
polynomial rings $\mathbb{C}[x_{0},\dotsc,x_{s},y_{0},\dotsc,y_{r}]$. In this
case, we give sharp bounds for the multigraded regularity index of their
multigraded Hilbert function, and a method to compute their Hilbert polynomial.
