The process-parameter optimization of a wire electrical discharge machining (WEDM) process has been performed with an Inconel 825 alloy during taper profile machining. The input parameters have been considered at three different levels and the number of experiments to be performed was decided using Taguchi's Design of experiment (DOE). The output performances such as material removal rate (MRR), Surface roughness (SR) and taper error (TE) were considered simultaneously for the optimization process. The experimental results were analysed using Grey relational analysis (GRA) and the best experimental trial was identified based on ranking. The influence of the input parameters was analysed using the ANOVA technique. The regression equations to predict the output responses were formed using the experimental results and the correlation between the experimental and regression equation results.