In this paper, we investigated the boundedness of multilinear fractional strong maximal operator MR,α associated with rectangles or related to more general basis with multiple weights A ( p,q),R . In the rectangles setting, we first gave an end-point estimate of MR,α, which not only extended the famous linear result of Jessen, Marcinkiewicz and Zygmund, but also extended the multilinear result of Grafakos, Liu, Pérez and Torres (α = 0) to the case 0 < α < mn. Then, in one weight case, we gave several equivalent characterizations between MR,α and A ( p,q),R , by applying a different approach from what we have used before. Moreover, a sufficient condition for the two weighted norm inequality of MR,α 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 ( p,q),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.Date: December 28, 2015.