Put $V= \mathbb{C}^3$. Let $D \subset V$ be an isolated singularity and
$\mu: \tilde{V} \rightarrow V$ be a resolution of singularities whose exceptional locus $E$ has normal crossing support. We can define $c_j \in \mathbb{Z}$ such that $K_{\tilde{V}} + \tilde{D}　= \mu^* (K_V +D)+ \sum c_j E_j $ where $\tilde{D}$ is the strict transform of $D$ and $E = \bigcup E_j$ is the irreducible decomposition.

Question Is there $\mu$ such that $c_j \le 0$ for all $j$?

Put $V= \mathbb{C}^3$. Let $D \subset V$ be an isolated singularity and
$\mu: \tilde{V} \rightarrow V$ be a log resolution of the pair $(V,D)$ whose exceptional locus $E$ and the strict transform $\tilde{D}$ satisfies that $\tilde{D} \cup E$ has a normal crossing support. We can define $c_j \in \mathbb{Z}$ such that $K_{\tilde{V}} + \tilde{D}　= \mu^* (K_V +D)+ \sum c_j E_j $ where $E = \bigcup E_j$ is the irreducible decomposition.

Question Is there $\mu$ such that $c_j \le 0$ for all $j$?