  To resum large logarithms in multi-scale problems a generalization of $\MS$
is introduced allowing for as many renormalization scales as there are generic
scales in the problem. In the new \lq\lq minimal multi-scale subtraction
scheme'' standard perturbative boundary conditions become applicable. However,
the multi-loop beta functions depend on the various renormalization scale
ratios and a large logarithms resummation has to be performed on them. Using
these improved beta functions the \lq\lq partial'' renormalization group
equations corresponding to the renormalization point independence of physical
quantities allows one to resum the logarithms. As an application the leading
and next-to-leading order two-scale analysis of the effective potential in the
$O(N)$-symmetric $\phi^4$-theory is performed. This calculation indicates that
there is no stable vacuum in the broken phase of the theory for $1<N\leq4$.
