  We consider generalized Orlicz-Morrey spaces
$M_{\Phi,\varphi}(\mathbb{R}^{n})$ including their weak versions
$WM_{\Phi,\varphi}(\mathbb{R}^{n})$. We find the sufficient conditions on the
pairs $(\varphi_{1},\varphi_{2})$ and $(\Phi, \Psi)$ which ensures the
boundedness of the fractional maximal operator $M_{\alpha}$ from
$M_{\Phi,\varphi_1}(\mathbb{R}^{n})$ to $M_{\Psi,\varphi_2}(\mathbb{R}^{n})$
and from $M_{\Phi,\varphi_1}(\mathbb{R}^{n})$ to
$WM_{\Psi,\varphi_2}(\mathbb{R}^{n})$. As applications of those results, the
boundedness of the commutators of the fractional maximal operator
$M_{b,\alpha}$ with $b \in BMO(\mathbb{R}^{n})$ on the spaces
$M_{\Phi,\varphi}(\mathbb{R}^{n})$ is also obtained. In all the cases the
conditions for the boundedness are given in terms of supremal-type inequalities
on weights $\varphi(x,r)$, which do not assume any assumption on monotonicity
of $\varphi(x,r)$ on $r$.
