  We introduce and study the quantum toroidal algebra
$\mathcal{E}_{m|n}(q_1,q_2,q_3)$ associated with the superalgebra
$\mathfrak{gl}_{m|n}$ with $m\neq n$, where the parameters satisfy
$q_1q_2q_3=1$. We give an evaluation map. The evaluation map is a surjective
homomorphism of algebras $\mathcal{E}_{m|n}(q_1,q_2,q_3) \to
\widetilde{U}_q\,\widehat{\mathfrak{gl}}_{m|n}$ to the quantum affine algebra
associated with the superalgebra $\mathfrak{gl}_{m|n}$ at level $c$ completed
with respect to the homogeneous grading, where $q_2=q^2$ and $q_3^{m-n}=c^2$.
We also give a bosonic realization of level one
$\mathcal{E}_{m|n}(q_1,q_2,q_3)$-modules.
