  We investigate ultrafast phonon dynamics in the Bi$_{1-x}$Sb$_{x}$ alloy
system for various compositions $x$ using a reflective femtosecond pump-probe
technique. The coherent optical phonons corresponding to the A$_{1g}$ local
vibrational modes of Bi-Bi, Bi-Sb, and Sb-Sb are generated and observed in the
time domain with a few picoseconds dephasing time. The frequencies of the
coherent optical phonons were found to change as the Sb composition $x$ was
varied, and more importantly, the relaxation time of those phonon modes was
dramatically reduced for $x$ values in the range 0.5--0.8. We argue that the
phonon relaxation dynamics are not simply governed by alloy scattering, but are
significantly modified by anharmonic phonon-phonon scattering with implied
minor contributions from electron-phonon scattering in a Weyl-semimetal phase.
