We provide a generalization of pseudo-Frobenius numbers of numerical semigroups to the context of the simplicial affine semigroups. In this way, we characterize the Cohen-Macaulay type of the simplicial affine semigroup ring $\mathbb{K}[S]$. We define the type of $S$, ${\rm{type}}(S)$, in terms of some Apery sets of $S$ and show that it coincides with the Cohen-Macaulay type of the semigroup ring, when $\mathbb{K}[S]$ is Cohen-Macaulay. If $\mathbb{K}[S]$ is a $d$-dimensional Cohen-Macaulay ring of embedding dimension at most $d+2$, then ${\rm{type}}(S)\leq 2$. Otherwise, ${\rm{type}}(S)$ might be arbitrary large and it has no upper bound in terms of the embedding dimension.

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