By curiosity, I would like to see an example of a model category with the underlying category locally presentable which is not accessible [in this sense](https://ncatlab.org/nlab/show/accessible+model+category) (and just in case: even by using Vopěnka's principle).



Example of non accessible model categories
I don't know whether set-theoretic hypotheses are necessary to answer this question. But if we assume the negation of Vopěnka's principle, here is an example: by Example 6.12 of Adamek-Rosicky *Locally presentable and accessible categories* the locally presentable category $\bf Gra$ of graphs has a reflective subcategory that is not accessible, and by Proposition 3.5 of [Salch](https://arxiv.org/abs/1501.00508) *The Bousfield localizations and colocalizations of the discrete model structure* this reflector is the fibrant replacement functor of a model structure on $\bf Gra$.


