AbstractAn atomic integral domain D is a half-factorial domain (HFD) if for any irreducible elements α₁,..., α_n, β₁,..., β_m of D with α₁... α_n = β₁ ...β_m, then n = m. In [3], Anderson, Anderson, and Zafrullah explore factorization problems in overrings of HFDs and ask whether a localization of a HFD is again a HFD. We construct an example of a Dedekind domain which is a HFD, but with a localization which is not a HFD. We also give an example of a Dedekind domain where each localization is a HFD, but with an overring which is not a HFD.