  In this paper, a double-pivot simplex method is proposed. Two upper bounds of
iteration numbers are derived. Applying one of the bounds to some special
linear programming (LP) problems, such as LP with a totally unimodular matrix
and Markov Decision Problem (MDP) with a fixed discount rate, indicates that
the double-pivot simplex method solves these problems in a strongly polynomial
time. A variant of Klee-Minty cube is used to show that the estimated bounds of
the iteration numbers are very tight. Numerical test on three variants of
Klee-Minty cubes is performed for the problems with sizes as big as $200$
constraints and $400$ variables. Dantzig's simplex method cannot handle
Klee-Minty cube problem with $200$ constraints because it needs about $2^{200}
\approx 10^{60}$ iterations. But the proposed algorithm performs extremely good
for all three variants.
