To improve the adaptability of solar refrigeration systems to different heat sources, a single-double-effect LiBr−H₂O absorption refrigeration system (ARS) driven by solar energy was designed and analyzed. The system was optimized using a multi-objective optimization method based on Sobol sensitivity analysis to enhance solar energy efficiency and reduce costs. The model of the solar single-double-effect LiBr−H₂O ARS was developed, and the continuous operation characteristics of the system in different configurations were simulated and compared. The results show that the average cooling time of the system without auxiliary heat source is approximately 8.5 h per day, and the double-effect mode (DEM) generates about 11 kW of cooling capacity during continuous operation for one week under the designated conditions, and the system with adding auxiliary heat source meet the requirements of daily cooling time, the solar fraction (SF) of the system reaches 59.29%. The collector area has a greater effect on SF, while the flowrate of the hot water circulating pump and the volume of storage tank have little effect on SF. The optimized SF increases by 3.22% and the levelized cost decreases by 10.18%. Moreover, compared with the solar single-effect LiBr−H₂O ARS, the SF of the system is increased by 15.51% and 17.42% respectively after optimization.