  The existence of a tilting or silting object is an important feature for an
algebraic triangulated category since it gives an equivalence with the derived
category of a ring. By applying tilting theory, we study Cohen-Macaulay
representations of $\mathbb N$-graded Artin-Schelter Gorenstein algebras
$A=\bigoplus_{i\in \mathbb N}A_i$, where do not assume that $A_0$ is a field.
This is a large class of noncommutative Gorenstein rings containing Gorenstein
orders. In this paper, we concentrate on the case where $A$ has dimension one.
Under the assumptions that $A$ is ring-indecomposable and $A_0$ has finite
global dimension, we show that the stable category
$\operatorname{\underline{\mathsf{CM}}}_0^{\mathbb Z}A$ always admits an
(explicitly constructed) silting object. We also show that
$\operatorname{\underline{\mathsf{CM}}}_0^{\mathbb Z}A$ admits a tilting object
if and only if either $A$ is Artin-Schelter regular or the average Gorenstein
parameter $p^A_\mathrm{av} \in {\mathbb Q}$ of $A$ is non-positive. These
results are far-reaching generalizations of the results of Buchweitz, Iyama,
and Yamaura. We give two different proofs of the second result; one is based on
Orlov-type semiorthogonal decompositions, and the other is based on a more
direct calculation. We apply our results to a Gorenstein tiled order $A$ to
prove that $\operatorname{\underline{\mathsf{CM}}}^{\mathbb Z}A$ is equivalent
to the derived category of the incidence algebra of an (explicitly constructed)
poset. We also apply our results and Koszul duality to prove that the derived
category $\mathsf{D}^\mathrm{b}(\operatorname{\mathsf{qgr}} A)$ of a smooth
noncommutative projective quadric hypersurface $\operatorname{\mathsf{qgr}} A$
admits an (explicitly constructed) tilting object, which contains the tilting
object of $\operatorname{\underline{\mathsf{CM}}}^{\mathbb Z}A$ due to Smith
and Van den Bergh as a direct summand.
