  Multi-principal-element alloys, including high-entropy alloys, experience
segregation or partially-ordering as they are cooled to lower temperatures. For
Ti$_{0.25}$CrFeNiAl$_{x}$, experiments suggest a partially-ordered B2 phase,
whereas CALculation of PHAse Diagrams (CALPHAD) predicts a region of
L2$_{1}$+B2 coexistence. We employ first-principles density-functional theory
(DFT) based electronic-structure approach to assess stability of phases of
alloys with arbitrary compositions and Bravais lattices (A1/A2/A3). In
addition, DFT-based linear-response theory has been utilized to predict
Warren-Cowley short-range order (SRO) in these alloys, which reveals
potentially competing long-range ordered phases. The resulting SRO is uniquely
analyzed using concentration-waves analysis for occupation probabilities in
partially-ordered states, which is then be assessed for phase stability by
direct DFT calculations. Our results are in good agreement with experiments and
CALPHAD in Al-poor regions ($x \le 0.75$) and with CALPHAD in Al-rich region
($0.75 \le {x} \le 1$), and they suggest more careful experiments in Al-rich
region are needed. Our DFT-based electronic-structure and SRO predictions
supported by concentration-wave analysis are shown to be a powerful method for
fast assessment of competing phases and their stability in
multi-principal-element alloys.
