  In representation theory of graded Iwanaga-Gorenstein algebras, tilting
theory of the stable category $\underline{\mathsf{CM}}^{\mathbb{Z}} A$ of
graded Cohen-Macaulay modules plays a prominent role. In this paper we study
the following two central problems of tilting theory of
$\underline{\mathsf{CM}}^{\mathbb{Z}} A$ in the case where $A$ is finite
dimensional: (1) Does $\underline{\mathsf{CM}}^{\mathbb{Z}} A$ have a tilting
object? (2) Does the endomorphism algebras of tilting objects in
$\underline{\mathsf{CM}}^{\mathbb{Z}} A$ have finite global dimension? To the
problem (2) we give the complete answer. We show that the endomorphism algebra
of any tilting object in $\underline{\mathsf{CM}}^{\mathbb{Z}}A$ has finite
global dimension. To the problem (1) we give a partial answer. For this
purpose, first we introduce an invariant $g(A)$ for a finite dimensional graded
algebra $A$. Then, we prove that in the case where $A$ is 1-Iwanaga-Gorenstein,
an inequality for $g(A)$ gives a sufficient condition that a specific
Cohen-Macaulay module $V$ becomes a tilting object in the stable category. As
an application, we study the existence of tilting objects in
$\underline{\mathsf{CM}}^{\mathbb{Z}}\Pi(Q)_w$ where $\Pi(Q)_w$ is the
truncated preprojective algebra of a quiver $Q$ associated to $w\in W_Q$. We
prove that if the underling graph of $Q$ is tree, then
$\underline{\mathsf{CM}}^{\mathbb{Z}}\Pi(Q)_w$ has a tilting object.
