The Minimum Monotone Satisfying Assignment problem (MMSA) consists, given a monotone boolean formula ϕ, in searching for a minimum number of true variables such that ϕ is satisfied. A polynomial inapproximability ratio was given by Dinur et al. However, this ratio depends on a parameter that is not the size of the MMSA instance. It is instead the size of the problem from which the reduction is done. Consequently, it is hard to reuse this result to prove other hardness of approximability. In this paper, we deepen the previous work and prove two inapproximability ratio for MMSA depending on the size of the formula and the number of variables and we prove that MMSA cannot be polylogarithmically approximated.