  Let $X$ be a smooth polarized algebraic surface over the compex number field.
We discuss the invariants obtained from the moduli stacks of semistable sheaves
of arbitrary ranks on $X$. For that purpose, we construct the virtual
fundamental classes of some moduli stacks, and we show the transition formula
of the integrals over the moduli stacks of the $\delta$-stable Bradlow pairs
for the variation of the parameter $\delta$.
  Then, we study the relation among the invariants. In the case $p_g>0$, we
show that the invariants are independent of the choice of a polarization of
$X$. We also show that the invariants can be reduced to the invariants obtained
from the moduli of abelian pairs and the Hilbert schemes. In the case $p_g=0$,
we obtain the weak wall crossing formula and the weak intersection rounding
formula, which describes the dependence of the invariants on the polarization.
