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This paper divides into two parts. Let $(X,\omega)$ be a compact Hermitianmanifold. Firstly, if the Hermitian metric $\omega$ satisfies the assumptionthat $\partial\overline{\partial}\omega^k=0$ for all $k$, we generalize thevolume of the cohomology class in the K\"{a}hler setting to the Hermitiansetting, and prove that the volume is always finite and theGrauert-Riemenschneider type criterion holds true, which is a partial answer toa conjecture posed by Boucksom. Secondly, we observe that if the anticanonicalbundle $K^{-1}_X$ is nef, then for any $\varepsilon>0$, there is a smoothfunction $\phi_\varepsilon$ on $X$ such that$\omega_\varepsilon:=\omega+i\partial\overline{\partial}\phi_\varepsilon>0$ andRicci$(\omega_\varepsilon)\geq-\varepsilon\omega_\varepsilon$. Furthermore, if$\omega$ satisfies the assumption as above, we prove that for aHarder-Narasimhan filtration of $T_X$ with respect to $\omega$, the slopes$\mu_\omega(\mathcal{F}_i/\mathcal{F}_{i-1})\geq 0$ for all $i$, whichgeneralizes a result of Cao which plays a very important role in his studyingof the structures of K\"{a}hler manifolds.

On the volume of a pseudo-effective class and semi-positive properties of the Harder-Narasimhan filtration on a compact Hermitian manifold