To optimize the production scheduling of a flexible job shop, this paper, based on the NSGA-II algorithm, proposes an adaptive simulated annealing non-dominated sorting genetic algorithm II with enhanced elitism (ASA-NSGA-EE) that establishes a multi-objective flexible job shop scheduling model with the objective functions of minimizing the maximum completion time, processing cost, and carbon emissions generated from processing. The ASA-NSGA-EE algorithm adopts an adaptive crossover and mutation genetic strategy, which dynamically adjusts the crossover and mutation rates based on the evolutionary stage of the population, aiming to reduce the loss of optimal solutions. Additionally, it incorporates the simulated annealing algorithm to optimize the selection strategy by leveraging its cooling characteristics. Furthermore, it improves the elite strategy through incorporating elite selection criteria. Finally, by simulation experiments, the effectiveness of the improved NSGA-II algorithm is validated by comparing it with other algorithms.