In this paper, we consider two related problems concerning homogeneous (α,β)-metrics. In the first part we consider homogeneous (α,β)-spaces of Douglas type. We prove that a homogeneous (α,β)-metric is a Douglas metric if and only if either F is a Berwald metric or F is a Douglas metric of Randers type. In the second part, we prove that if F is a homogeneous (α,β)-metric which is neither a Riemannian metric nor a Minkowski metric,
then F is locally projectively flat if and only if F is a locally projectively flat left invariant Randers metric on the hyperbolic space as a solvable Lie group. We also give all the explicit forms of the metric F in the second case. This result also provides new examples of locally projectively flat metrics which have not been described before in the literature, presenting some new insight in the study of the Hilbert's fourth problem.