  In replica exchange Monte Carlo (REM), tuning of the temperature set and the
exchange scheduling are crucial in improving the accuracy and reducing
calculation time. In multi-dimensional simulated tempering, the first order
phase transition is accessible. Therefore it is important to study the tuning
of parameter set and the scheduling of exchanges in the parallel counterpart,
the multi-dimensional REM. We extend Hukushima's constant exchange probability
method to multi-dimensional REM for the parameter set. We further propose a
combined method to use this set and the Bittner-Nussbaumer-Janke's PT tau
algorithm for scheduling. We test the proposed method in two-dimensional spin-1
Blume-Capel model and find that it works efficiently, including the vicinity of
the first order phase transition.
