  The goals of this paper are of two aspects. Firstly, we introduce the notion
of generalized numerical Kodaira dimension with multiplier ideal sheaf and
establish the subadditivity inequalities in terms of this notion, which can be
used to give an analytic proof of O. Fujino's result on the subadditivity of
the log Kodaira dimensions. Secondly, motivated by Zhou-Zhu's subadditivity of
generalized Kodaira dimensions, we adopt another definition of generalized
Kodaira dimension with multiplier ideal sheaf and show they are equal by using
Okounkov bodies. As one application, we show that the superadditivity part in
Zhou-Zhu's setting also holds true. As another application, we give an
alternative proof of Zhou-Zhu's subadditivity formula, in the case when the
singular metric $h_L$ has analytic singularities, by using generalized Iitaka
fibrations.
