  In this paper, we investigated the boundedness of multilinear fractional
strong maximal operator $\mathcal{M}_{\mathcal{R},\alpha}$ associated with
rectangles or related to more general basis with multiple weights
$A_{(\vec{p},q),\mathcal{R}}$. In the rectangles setting, we first gave an
end-point estimate of $\mathcal{M}_{\mathcal{R},\alpha}$, which not only
extended the famous linear result of Jessen, Marcinkiewicz and Zygmund, but
also extended the multilinear result of Grafakos, Liu, P\'{e}rez and Torres
($\alpha=0$) to the case $0<\alpha<mn.$ Then, in one weight case, we gave
several equivalent characterizations between $\mathcal{M}_{\mathcal{R},\alpha}$
and $A_{(\vec{p},q),\mathcal{R}}$, by applying a different approach from what
we have used before. Moreover, a sufficient condition for the two weighted norm
inequality of $\mathcal{M}_{\mathcal{R},\alpha}$ was presented and a version of
vector-valued two weighted inequality for the strong maximal operator was
established when $m=1$. In the general basis setting, we further studied the
properties of the multiple weights $A_{(\vec{p},q),\mathcal{R}}$ conditions,
including the equivalent characterizations and monotonic properties, which
essentially extended one's previous understanding. Finally, a survey on
multiple strong Muckenhoupt weights was given, which demonstrates the
properties of multiple weights related to rectangles systematically.
