Numerical stability of Simplex method
Yes, there is research on this issue.

The Simplex Method is Not Always Well Behaved, Wlodzimierz Ogryczak

retroLP, An Implementation of the Standard Simplex Method, Gavriel Yarmish and Richard Van Slyke

A Numerically Stable Form of the Simplex Algorithm, Philip E. Gill and Walter Murray

You might also be interested in the revised simplex method.  This method can take advantage of matrix sparsity; it doesn't keep a representation of the entire matrix.
This thesis was of great interest to me: A Comparison of Simplex Method Algorithms.