This paper presents a hybrid genetic algorithm (GA) for the generalized time-window assignment problem (GTWAP), a complex artificial intelligence (AI) scheduling challenge that involves assigning agents to resources under strict temporal and capacity constraints. Our method integrates a problem specific heuristics and a repair mechanism to generate feasible and high quality solutions. We provide a mathematical formulation for GTWAP and introduce a new public benchmark set, using CPLEX to obtain exact solutions. Computational experiments demonstrate that our GA is highly competitive with CPLEX, often matching its performance. This effectiveness makes our method a practical and scalable AI-driven tool for complex scheduling in domains like logistics and healthcare.