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.
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