In this paper, we study left φ-biflatness and left φ-biprojectivity of some Banach algebras, where φ is a non-zero multiplicative linear function. We show that if the Banach algebra A∗∗ is left φ-biprojective, then A is left φ-biflat. Using this tool we study left φ-biflatness of some matrix algebras. We also study left φ-biflatness and left φ-biprojectivity of the projective tensor product of some Banach algebras related to a locally compact group. We prove that for a locally compact group G, M(G) ⊗p A(G) is left φ ⊗ -biprojective if and only if G is finite. We show that M(G) ⊗p L1(G) is left φ ⊗ -biprojective if and only if G is compact
