  In this paper, we introduce the notions of $\alpha$-Hermitian-Einstein metric
and $\alpha$-stability for $I_\pm$-holomorphic vector bundles on bi-Hermitian
manifolds. Moreover, we establish a Kobayashi-Hitchin correspondence for
$I_\pm$-holomorphic vector bundles on bi-Hermitian manifolds. Examples of such
vector bundles include generalized holomorphic bundles on generalized K\"ahler
manifolds. We also show that $\alpha$-stability of a vector bundle, in this
sense, can depend on the parameter $\alpha$.
