  N-body dark matter simulations of structure formation in the $\Lambda$CDM
model predict a population of subhalos within Galactic halos that have higher
central densities than inferred for satellites of the Milky Way, a tension
known as the `too big to fail' problem. Proposed solutions include baryonic
effects, a smaller mass for the Milky Way halo, and warm dark matter. We test
these three possibilities using a semi-analytic model of galaxy formation to
generate luminosity functions for Milky Way halo-analogue satellite
populations, the results of which are then coupled to the Jiang & van den Bosch
model of subhalo stripping to predict the subhalo $V_\mathrm{max}$ functions
for the 10 brightest satellites. We find that selecting the brightest
satellites (as opposed to the most massive) and modelling the expulsion of gas
by supernovae at early times increases the likelihood of generating the
observed Milky Way satellite $V_\mathrm{max}$ function. The preferred halo mass
is $6\times10^{11}M_{\odot}$, which has a 14 percent probability to host a
$V_\mathrm{max}$ function like that of the Milky Way satellites. This
probability is reduced to 8 percent for a $1.0\times10^{12}M_{\odot}$ halo and
to 3 percent for a $1.4\times10^{12}M_{\odot}$ halo. We conclude that the Milky
Way satellite $V_\mathrm{max}$ function is compatible with a CDM cosmology, as
previously found by Sawala et al. using hydrodynamic simulations. Sterile
neutrino-warm dark matter models achieve a higher degree of agreement with the
observations, with a maximum 35 percent chance of generating the observed Milky
Way satellite $V_\mathrm{max}$ function. However, more work is required to
check that the semi-analytic stripping model is calibrated correctly in the
sterile neutrino cosmology, and to check if our sterile neutrino models produce
sufficient numbers of faint satellites.
