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 $6times10^{11}M_{odot}$, 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 $1.0times10^{12}M_{odot}$ halo and to 3 percent for a $1.4times10^{12}M_{odot}$ 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.
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