The membrane process is a promising technology for CO2 removal and mitigation. Since the energy consumption and economy of membrane-based carbon capture systems (CCSs) are critical for their large-scale deployments, optimal design and operation of such systems are the primary aims of this study. To achieve these research goals, a numerical model based on the solution-diffusion mechanism for the multicomponent gas separation process with a hollow-fiber membrane module is developed using Aspen Custom Modeler. The model is employed to investigate the effects of important operating and design parameters. Multi-objective process optimization is then performed by linking Aspen Plus and MATLAB and using an evolutionary technique to determine the optimal operating and design conditions. Our results show that by increasing the CO2 concentration in the feed gas, the CO2 capture cost significantly decreases and CO2 removal improves, although the process energy requirement slightly increases. The best achievable tradeoffs between objective functions are generated and analyzed, which substantiate the significant potential for improving the sustainability of the process. The results show that at optimum design and operating conditions, CO2 capture cost and energy consumption of the process could be as low as 13.1 $/tCO2 and 61 MW, respectively. The results of this study provide valuable insights into membrane separation and can be used by decision-makers to achieve the optimal performance of the process for commercial development and deployment of the technology.