  The ratio of the $B^0_s$ and $B^0$ fragmentation fractions, $f_s/f_d$, in
proton-proton collisions at the LHC, is obtained as a function of $B$-meson
transverse momentum and collision centre-of-mass energy from the combined
analysis of different $B$-decay channels measured by the LHCb experiment. The
results are described by a linear function of the meson transverse momentum, or
with a function inspired by Tsallis statistics. Precise measurements of the
branching fractions of the $B^0_s \to J/\psi \phi$ and $B^0_s \to D^-_s \pi^+$
decays are performed, reducing their uncertainty by about a factor of two with
respect to previous world averages. Numerous $B^0_s$ decay branching fractions,
measured at the LHCb experiment, are also updated using the new values of
$f_s/f_d$ and branching fractions of normalisation channels. These results
reduce a major source of systematic uncertainty in several searches for new
physics performed through measurements of $B^0_s$ branching fractions.
