Logic Programs with Ordered Disjunction (LPODs) are a well-known extension of logic programming used to express qualitative preferences via the operator of ordered disjunction. However, the semantics originally proposed for LPODs can yield counterintuitive preferred solutions in some cases. An alternative logical semantics for LPODs addresses these cases by introducing an additional truth value representing the failure to satisfy a preference. In this paper we present an implementation of this alternative many-valued semantics. We propose a transformation of an LPOD into an ASP program and reformulate the task of finding the many-valued preferred models as an answer set optimization task. Notably, our approach can be extended to also support standard disjunction for expressing preferences of equal importance. Finally, we compare our system with existing systems implementing the original semantics. The experimental results suggest that the alternative semantics are not only feasible to implement but also comparable in performance.