  In the present study, the multiphase volume distribution problem, where there
can be an arbitrary number of phases, is addressed using a consistent and
conservative volume distribution algorithm. The proposed algorithm satisfies
the summation constraint, the conservation constraint, and the
\textit{consistency of reduction}. The first application of the volume
distribution algorithm is to determine the Lagrange multipliers in multiphase
Phase-Field models that enforce the mass conservation, and a multiphase
conservative Allen-Cahn model that satisfies the \textit{consistency of
reduction} is developed. A corresponding consistent and conservative numerical
scheme is developed for the model. The multiphase conservative Allen-Cahn model
has a better ability than the multiphase Cahn-Hilliard model to preserve
under-resolved structures. The second application is to develop a numerical
procedure, called the boundedness mapping, to map the order parameters,
obtained numerically from a multiphase model, into their physical interval, and
at the same time to preserve the physical properties of the order parameters.
Along with the consistent and conservative schemes for the multiphase
Phase-Field models, the numerical solutions of the order parameters are
reduction consistent, conservative, and bounded, which are theoretically
analyzed and numerically validated. Then, the multiphase Phase-Field models are
coupled with the momentum equation by satisfying the \textit{consistency of
mass conservation} and the \textit{consistency of mass and momentum transport},
thanks to the consistent formulation. It is demonstrated that the proposed
model and scheme converge to the sharp-interface solution and are capable of
capturing the complicated multiphase dynamics even when there is a large
density and/or viscosity ratio.
