This study addresses optimisation challenges in scheduling automatic guided vehicles (AGVs) for express distribution centres. A comprehensive model is developed that simultaneously considers makespan, AGV usage, and AGV recharging frequency to identify the optimal scheduling strategy and determine an effective recharging threshold. To address this, an enhanced adaptive genetic algorithm called LS-AGA (L-value and S-value based Adaptive Genetic Algorithm) is proposed. The LS-AGA employs a Logistic chaotic map to create an initial population. Fitness value factors and fitness entropy are incorporated to calculate individual L-values for selection, crossover, and mutation, maintaining a balance between fitness value and population diversity. The SoftMax function is introduced to map the L-values and fitness values into the corresponding probabilities, subsequently calculating individual S-values to optimise crossover and mutation rates. The addition of a catastrophe operator further enhances optimisation. Numerical and validation experiments demonstrate that LS-AGA outperforms existing improved genetic algorithms in solving the proposed model.