  In this paper, we first propose the concepts of BiHom-$\Omega$-associative
algebras, BiHom-$\Omega$-dendriform algebras, BiHom-$\Omega$-pre-Lie algebras
and BiHom-$\Omega$-Lie algebras. We then obtain a new
BiHom-$\Omega$-associative (resp. Lie) algebra by defining a new multiplication
on a BiHom-$\Omega$-associative (resp. Lie) algebra with the Rota-Baxter family
of weight $\lambda$.In addition, we generalize the classical relationships of
associative algebras, pre-Lie algebras, dendriform algebras and Lie algebras to
the BiHom-$\Omega$ version.
