Let $R={igoplus}_{alpha{in}Gamma}R_{alpha}$ be a graded integral domain graded by an arbitrary grading torsionless monoid ${Gamma}$, and ? be a semistar operation on R. In this paper we define and study the graded integral domain analogue of ?-Nagata and Kronecker function rings of R with respect to ?. We say that R is a graded Pr$ddot{u}$fer ?-multiplication domain if each nonzero finitely generated homogeneous ideal of R is ?$_f$-invertible. Using ?-Nagata and Kronecker function rings, we give several different equivalent conditions for R to be a graded Pr$ddot{u}$fer ?-multiplication domain. In particular we give new characterizations for a graded integral domain, to be a $P{upsilon}MD$.