  We investigate numerically the propagation and the Anderson localization of
plane waves in a one-dimensional lattice chain, where disorder and saturable
nonlinearity are simultaneously present. Using a calculation scheme for solving
the stationary discrete nonlinear Schr\"{o}dinger equation in the fixed input
case, the disorder-averaged logarithmic transmittance and the localization
length are calculated in a numerically precise manner. The localization length
is found to be a nonmonotonic function of the incident wave intensity,
acquiring a minimum value at a certain finite intensity, due to saturation
effects. For low incident intensities where the saturation effect is
ineffective, the enhancement of localization due to Kerr-type nonlinearity
occurs in a way similar to the case without saturation. For sufficiently high
incident intensities, we find that the localization length is an increasing
function of the incident wave intensity, which implies that localization is
suppressed for stronger input intensities, and ultimately approaches a
saturation value. This feature is associated with the fact that the nonlinear
system is reduced to an effectively linear one, when either the incident wave
intensity or the saturation parameter is sufficiently large. We also find that
the nonlinear saturation effect is stronger and more pronounced when the energy
of the incident wave is larger.
