  On the basis of the density functional calculations in combination with the
supercell approach, we report on a complete study of the influences of atomic
arrangement and Ni substitution for Al on the ground state structural and
magnetic properties for Fe$_2$Ni$_{1+x}$Al$_{1-x}$ Heusler alloys. We discuss
systematically the competition between five cubic Heusler-type structures
formed by shuffles of Fe and Ni atoms to reveal routes for improving the phase
stability and magnetic properties, in particular magnetocrystalline
anisotropy~(MAE). We predict that in case of Fe$_2$NiAl the ground state cubic
structure with alternated layers of Fe and Ni possesses the highest uniaxial
MAE which twice larger than that for the tetragonal L1$_0$ FeNi. The successive
Ni doping at Al sublattice leads to a change of ground state structure and to
reduce of the MAE. In addition, the phase stability against the decomposition
into the stable systems at finite-temperatures is discussed. All~Ni-rich
Fe$_2$Ni$_{1+x}$Al$_{1-x}$ are turned to be decomposed into a dual-phase
consisting of Fe$_2$NiAl and FeNi.
