  Let $\mathbb{N} \mathcal{A}$ be the monoid generated by $\mathcal{A} =
{\mathbf{a}_1, ..., \mathbf{a}_n} \subseteq \mathbb{Z}^d.$ We introduce the
homogeneous catenary degree of $\mathbb{N} \mathcal{A}$ as the smallest $N \in
\mathbb N$ with the following property: for each $\mathbf{a} \in \mathbb{N}
\mathcal{A}$ and any two factorizations $\mathbf{u}, \mathbf{v}$ of
$\mathbf{a}$, there exists factorizations $\mathbf{u} = \mathbf{w}_1, ...,
\mathbf{w}_t = \mathbf{v} $ of $\mathbf{a}$ such that, for every $k,
\mathrm{d}(\mathbf{w}_k, \mathbf{w}_{k+1}) \leq N,$ where $\mathrm{d}$ is the
usual distance between factorizations, and the length of $\mathbf{w}_k,
|\mathbf{w}_k|,$ is less than or equal to $\max{|\mathbf{u}|, |\mathbf{v}|}.$
We prove that the homogeneous catenary degree of $\mathbb{N} \mathcal{A}$
improves the monotone catenary degree as upper bound for the ordinary catenary
degree, and we show that it can be effectively computed. We also prove that for
half-factorial monoids, the tame degree and the $\omega$-primality coincide,
and that all possible catenary degrees of the elements of an affine semigroup
of this kind occur as the catenary degree of one of its Betti elements.
