  We study heavy-quarkonia suppression under final-state multiple scatterings
in most central Au~+~Au collisions at RHIC energy. We first calculate the
survival probability of a heavy quarkonium under multiple scattering in
Bjorken's expanding QGP at large $N_c$. Then, we calculate the rapidity
dependence of the nuclear modification factor $R_{AA}$ for heavy-quarkonia
production by considering final-state multiple scatterings in most central
Au~+~Au collisions in a simplified model. In our formula a constant $P_0$ is
also introduced to estimate the possible cold nuclear effects. By fitting the
data for $J/\Psi$ production in most central Au + Au collisions at
$\sqrt{s_{NN}}=200$~GeV at RHIC, we find that the transportation coefficient
$\hat{q}_0\simeq(0.33-0.95)~\mbox{GeV}^2$/fm, and, accordingly, the energy
density at $\tau_0$ is $\epsilon_0\simeq(1.39-5.62)~\mbox{GeV}/\mbox{fm}^3$ in
perturbative thermal QCD. A better understanding of cold nuclear effects is
essential for us to get a more accurate analysis. The small values of the
transportation coefficient $\hat{q}_0$ in our estimate are in sharp contrast
with those obtained by the analysis of high-$p_T$ hadron spectra in Ref. [31].
