There are several variants of the simplex method known, which differ by the choice of entering and leaving variables. But neither have I found a reference, which variants are used in which applications, nor have I found an attempt to compare the run-time of those methods. I suspect that a fair share of these variants have remained academic during the past years.
Can anybody provide a reference or an experimental comparison of these methods?
Unfortunately, the vendors of linear programming software have often been unwilling to tell us about the particular tricks that they use in implementing the simplex method, so there aren't many published articles that will give you these kinds of details. Robert Bixby's 2002 paper, Solving Real-World Linear Programs: A Decade and More of Progress is a good place to start your research on this subject. He has given more recent presentations on this topic, but I haven't seen that he's published a follow up paper.
