  We prove that the class of positive operators from $L_\infty (\mu)$ to $Y$
has the Bishop-Phelps-Bollob\'as property for any positive measure $\mu$,
whenever $Y$ is a uniformly monotone Banach lattice with a weak unit. The same
result also holds for the pair $(c_0, Y)$ for any uniformly monotone Banach
lattice $Y.$ Further we show that these results are optimal in case that $Y$ is
strictly monotone.
