So, I posted on stack overflow looking for a reasonably fast algorithm to solve systems of linear Diophantine inequalities and was pointed to this article by Cheng-Zhi Gao and Yu-Lin Dong. The problem is, they give the algorithm on pages 350-351, but part of step (3) appears to be missing.

My question for math overflow is, therefore, whether anyone knows either of another such algorithm or has an idea as to what the missing part of Gao and Dong's algorithm is.