A Markov Additive Process is a bi-variate Markov process $(ξ,J)=\big((ξ_t,J_t),t\geq0\big)$ which should be thought of as a multi-type Lévy process: the second component $J$ is a Markov chain on a finite space $\{1,\ldots,K\}$, and the first component $ξ$ behaves locally as a Lévy process, with local dynamics depending on $J$. In the subordinator-like case where $ξ$ is nondecreasing, we establish several results concerning the moments of $ξ$ and of its exponential functional $I_ξ=\int_{0}^{\infty} e^{-ξ_t}\mathrm dt,$ extending the work of Carmona et al., and Bertoin and Yor. We then apply these results to the study of multi-type self-similar fragmentation processes: these are self-similar analogues of Bertoin's homogeneous multi-type fragmentation processes Notably, we encode the genealogy of the process in a tree, and under some Malthusian hypotheses, compute its Hausdorff dimension in a generalisation of our previous work.