When can I localise a model category by a set(or class) of morphisms, and how do I describe the localised model category ?

By 'localise' I mean to find a localisation functor $q_S : M \rightarrow M_S$
with the usual universality property: that is, $q_S(s)$ is a weak equivalence for any $s\in S$, and any functor $q:M \rightarrow M'$ with this property factors through $q_S$. Importantly, $M_S$ and $M$ may have *different* underlying categories and so it is not necessarily a Bousfield localisation.

The model category (defined here) I am interested in, is very degenerate: its underlying category is a partially ordered set (or class).