The quasiparticle (QP) band structures of both strainless and strained monolayer MoS${}_2$ are investigated using more accurate many-body perturbation $\mathit{GW}$ theory and maximally localized Wannier functions (MLWFs) approach. By solving the Bethe-Salpeter equation (BSE) including excitonic effects on top of the partially self-consistent $\mathit{GW}$ ${}_0$ (sc$\mathit{GW}$ ${}_0$) calculation, the predicted optical gap magnitude is in good agreement with available experimental data. With increasing strain, the exciton binding energy is nearly unchanged, while optical gap is reduced significantly. The sc$\mathit{GW}$ ${}_0$ and BSE calculations are also performed on monolayer WS${}_2$, similar characteristics are predicted and WS${}_2$ possesses the lightest effective mass at the same strain among monolayers Mo(S,Se) and W(S,Se). Our results also show that the electron effective mass decreases as the tensile strain increases, resulting in an enhanced carrier mobility. The present calculation results suggest a viable route to tune the electronic properties of monolayer transition-metal dichalcogenides (TMDs) using strain engineering for potential applications in high performance electronic devices.