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Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d>0$ withinfinite residue field. Let $M$ be a finitely generated proper $R$-submodule ofa free $R$-module $F$ with $\ell (F/M) < \infty$ and having rank $r$. In thisarticle, we study the fiber multiplicity $f_0(M)$ of the module $M$. We provethat if $(R,\mathfrak{m})$ is a two dimensional Cohen-Macaulay local ring, then$f_0(M)\le br_1(M)-br_0(M)+\ell (F/M)+\mu(M)-r$, where $br_i(M)$ denotes the$i^{th}$ Buchsbaum-Rim coefficient of $M$.

On the Vasconcelos inequality for the fiber multiplicity of modules