This study is threefold objective. First, a nonlinear one-dimensional steady-state simulation model for hydrogen production from proton exchange membrane water electrolyzer (PEMWE) is established. The model comprehensively addresses activation, diffusion overpotentials and ohmic potential drop, and particularly incorporating Nafion membrane properties in the presence of liquid water. The model is enhanced by analyzing the hydrogen crossover, introducing corresponding diffusion equations to evaluate Faraday efficiency. The model accurately quantifies voltage efficiency, hydrogen production rate and specific energy consumption, and it is validated with published experimental data found in the scientific literature. Second, a variance-based global sensitivity analysis (GSA) on the model, is employed as a metric to assess the impact of operating conditions and material characteristics on three objective functions: the hydrogen production rate; the voltage efficiency and the specific energy consumption. Third, to support the GSA analysis, genetic algorithm (GA) technique is performed to maximize hydrogen production rate while minimizing specific energy consumption. In the course of this novel investigation, it was discerned that the number of cells, the cross-sectional area of the electrolyzer, the current density and temperature all exerted a significant impact on the three objective functions. This combined approach GSA-GA provides a thorough exploration of parameter influences and an efficient optimization strategy for enhancing the model's performance. These findings provide valuable insights that can guide analysis of the performance and the optimization of the PEMWE system.