is any indecomposable projectiveinjective Amodule a direct summand of tilting module

Yes, any indecomposable projectiveinjective $A$module $P$ is (isomorphic to) a direct summand of every tilting $A$module $T$. Just check that $T \oplus P$ is also a tilting module, and then use the fact that any two tilting $A$modules have the same number of isomorphism classes of idecomposable summands. 


Alternatively, if $P$ is your module and $T$ is a tilting module, you know (from the definition of being tilting) that there is a short exact sequence $0\to P\to T_0\to T_1\to 0$ with $T_0$ and $T_1$ in $\operatorname{add} T$, because $P$ is an indecomposiable projective. But then, since it is also an indecomposable injective, the short exact sequence splits and $P$ is a direct summand of $T_0$. Now what you want follows from KrullSchmidt's theorem. 

