  Let $R$ be a Noetherian local ring. We define the minimal $j$-multiplicity
and almost minimal $j$-multiplicity of an arbitrary $R$-ideal on any finite
$R$-module. For any ideal $I$ with minimal $j$-multiplicity or almost minimal
$j$-multiplicity on a Cohen-Macaulay module $M$, we prove that under some
residual assumptions, the associated graded module ${\rm gr}_I(M)$ is
Cohen-Macaulay or almost Cohen-Macaulay, respectively. Our work generalizes the
results for minimal multiplicity and almost minimal multiplicity obtained by
Sally, Rossi, Valla, Wang, Huckaba, Elias, Corso, Polini, and VazPinto.
