  The generalized quantum group $\mathcal{U}(\epsilon)$ of type $A$ is an
affine analogue of quantum group associated to a general linear Lie
superalgebra $\mathfrak{gl}_{M|N}$. We prove that there exists a unique $R$
matrix on tensor product of fundamental type representations of
$\mathcal{U}(\epsilon)$ for arbitrary parameter sequence $\epsilon$
corresponding to a non-conjugate Borel subalgebra of $\mathfrak{gl}_{M|N}$. We
give an explicit description of its spectral decomposition, and then as an
application, construct a family of finite-dimensional irreducible
$\mathcal{U}(\epsilon)$-modules which have subspaces isomorphic to the
Kirillov-Reshetikhin modules of usual affine type $A_{M-1}^{(1)}$ or
$A_{N-1}^{(1)}$.
