Most recent algorithms for calculating the mixed volume of the support A = (A1, ... , An ) of a polynomial system P(x) = ( p1(x), ... , pn(x)) in Cn by first finding all the fine mixed cells in a fine mixed subdivision of A followed by adding the volumes of all those cells will be presented in this talk. The problem of finding all the fine mixed cells, which plays a crucial role when the polyhedral homotopy method is employed to find all the isolated zeros of P(x), is dealt with by solving a series of Phase I problems in Linear Programming.