  For $\epsilon$-lc Fano type varieties $X$ of dimension $d$ and a given finite
set $\Gamma$, we show that there exists a positive integer $m_0$ which only
depends on $\epsilon,d$ and $\Gamma$, such that both $|-mK_X-\sum_i\lceil
mb_i\rceil B_i|$ and $|-mK_X-\sum_i\lfloor mb_i\rfloor B_i|$ define birational
maps for any $m\ge m_0$ provided that $B_i$ are pseudo-effective Weil divisors,
$b_i\in\Gamma$, and $-(K_X+\sum_ib_iB_i)$ is big. When $\Gamma\subset[0,1]$
satisfies the DCC but is not finite, we construct an example to show that the
effective birationality may fail even if $X$ is fixed, $B_i$ are fixed prime
divisors, and $(X,B)$ is $\epsilon'$-lc for some $\epsilon'>0$.
