Three different algorithms combining the methodology of Gibbs energy minimization (GEM) and the iteration function method with an extended phase-check procedure have been developed. The computational strategies devised in this study attempt to balance algorithm reliability, simplicity, generalizability, and computational efficiency. As a result, different numerical methods and improved initialization procedures have been implemented. The efficacy of the three algorithms is highly dependent on the multiphase system encountered. Specifically, the reliability, versatility and computational efficiency of the different algorithms vary significantly. A number of challenging case studies were successfully used to demonstrate the viability of the proposed algorithms and the initialization procedures. The results of this study indicate that (a) the GEM algorithm should be applied in demanding process design tasks and (b) the accelerated GEM algorithms should prove effective in dealing with systems involving a large number of components.