  Let $R$ be a standard graded algebra over an infinite field $\mathbb{K}$ and
$M$ a finitely generated $\ZZ$-graded $R$-module. Let $I_1,\ldots I_m$ be
graded ideals of $R$. The functions $r(M/I_1^{a_1}\ldots I_m^{a_m}M)$ and
$r(I_1^{a_1}\ldots I_m^{a_m}M)$ are investigated and their asymptotical
behaviours are given. Here $r(\bullet)$ stands for the reduction number of a
finitely graded module $\bullet$.
