  Phase diagrams supported by density functional theory methods can be crucial
for designing high-entropy alloys that are subset of multi-principal$-$element
alloys. We present phase and property analysis of quinary
(MoW)$_{x}$Zr$_{y}$(TaTi)$_{1-x-y}$ refractory high-entropy alloys from
combined Calculation of Phase Diagram (CALPHAD) and density-functional theory
results, supplemented by molecular dynamics simulations. Both CALPHAD and
density-functional theory analysis of phase stability indicates a Mo-W-rich
region of this quinary has a stable single-phase body-centered-cubic structure.
We report first quinary composition from Mo$-$W$-$Ta$-$Ti$-$Zr family of alloy
with pseudo-elastic behavior, i.e., hysteresis in stress$-$strain. Our analysis
shows that only Mo$-$W$-$rich compositions of Mo$-$W$-$Ta$-$Ti$-$Zr, i.e.,
Mo$+$W$\ge$ 85 at.%, show reproducible hysteresis in stress-strain responsible
for pseudo-elastic behavior. The (MoW)$_{85}$Zr$_{7.5}$(TaTi)$_{7.5}$ was
down-selected based on temperature-dependent phase diagram analysis and
molecular dynamics simulations predicted elastic behavior that reveals twinning
assisted pseudoelastic behavior. While mostly unexplored in body-centered-cubic
crystals, twinning is a fundamental deformation mechanism that competes against
dislocation slip in crystalline solids. This alloy shows identical cyclic
deformation characteristics during uniaxial $\lt$100$\gt$ loading, i.e., the
pseudoelasticity is isotropic in loading direction. Additionally, a temperature
increase from 77 to 1500 K enhances the elastic strain recovery in load-unload
cycles, offering possibly control to tune the pseudoelastic behavior.
