  The sign problem is a notorious problem, which occurs in Monte Carlo
simulations of a system with the partition function whose integrand is not real
positive. The basic idea of the factorization method applied on such a system
is to control some observables in order to determine and sample efficiently the
region of configuration space which gives important contribution to the
partition function. We argue that it is crucial to choose appropriately the set
of the observables to be controlled in order for the method to work
successfully in a general system. This is demonstrated by an explicit example,
in which it turns out to be necessary to control more than one observables.
Extrapolation to large system size is possible due to the nice scaling
properties of the factorized functions, and known results obtained by an
analytic method are shown to be consistently reproduced.
