  In the present work, we study a cosmological model composed of a viscous dark
matter interacting with decaying vacuum energy in a spatially flat Universe. In
the first part, we find the analytical solution of different cosmological
parameters by assuming the physically viable forms of bulk viscosity and
decaying vacuum density with the interaction term. The second part is dedicated
to constrain the free parameters of the interacting viscous model with decaying
vacuum energy by employing latest observational data of $Pantheon+$, Cosmic
Chronometer and $f(z)\sigma_{8}(z)$. We find that the interacting model just
deviate very slightly from well-known concordance $\Lambda$CDM model and can
alleviate effectively the current $H_0$ tension between local measurement by
R21 and global measurement by Planck 2018, and the excess in the mass
fluctuation amplitude $\sigma_{8}$ essentially vanish in this context. We
report the Hubble constants as $H_0=72.150^{+0.989}_{-0.779}$, and $
72.202^{+0.796}_{-0.937}$ \;$km s^{-1} Mpc^{-1}$, deceleration parameters as
$q_0=-0.533 \pm 0.024$, and $-0.531 \pm 0.024$, and equation of state
parameters as $w_0=-0.689 \pm 0.016$, and $ -0.687 \pm 0.016$ for $\Lambda$CDM
and interacting models, respectively. It is found that the interacting model is
in good agreement with $\Lambda$CDM. Further, we discuss the amplitude of
matter power spectrum $\sigma_8$ and its associated parameter $S_8$ using
$f(z)\sigma_8(z)$ data. Finally, the information selection criterion and
Bayesian inference are discussed to distinguish the interacting model with
$\Lambda$CDM model.
