Effective birationality for sub-pairs with real coefficients

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 $|-mK_X-sum_ilceil mb_irceil B_i|$ and $|-mK_X-sum_ilfloor mb_irfloor B_i|$ define birational maps for any $mge m_0$ provided that $B_i$ are pseudo-effective Weil divisors, $b_iinGamma$, and $-(K_X+sum_ib_iB_i)$ is big. When $Gammasubset[0,1]$ 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 $epsilon>0$.