In this study, biogas production through anaerobic digestion of food waste was investigated by applying four factor-four level Taguchi design under experimental conditions such as solid concentration (5%–12.5% TS), pH (6–9), temperature (30–60 °C), and co-digestion of poultry waste (10%–40%). Multi-objective techniques such as grey relational analysis (GRA) and multi-variate principal component analysis (PCA) were used to determine the optimal level of process parameters. The interactive effect of process conditions on the biogas yield was studied with multi-response performance index. The obtained results were analyzed by analysis of variance and the percentage contributions of each parameter were determined. Optimum conditions for maximizing the biogas yield and volatile solid removal efficiency were determined using hybrid GRA-PCA technique and it was found to be solid concentration of 7.5% TS, pH of 7, temperature of 50 °C, and co-digestion of 30%. Supremacy of the chosen hybrid technique is confirmed from the confirmation experimental output.