Process capability index (PCI) is a common and simple way to evaluate the process capability in the field of quality control. So far, most existing PCIs have been studied based on using the normality assumption, while only few PCIs have been proposed for specified non-normal distributions or based on robust methods. In practical applications, users may not be able to identify the exact distribution of quality characteristics or to collect a large amount of samples to implement robust methods. To overcome these two problems, new modified Clements' PCIs based on a model selection approach, named robust PCIs method, for the location-scale distribution (LSD) family are proposed to evaluate the process capability of a production process. The most suitable model in the robust PCIs method can be automatically selected by using a competitive procedure from a pool of candidate LSDs. Furthermore, the point estimation, interval inference and new hypothesis testing methods for the PCIs are also studied. A strategy is proposed to improve the manufacturing process. Monte Carlo simulation results show that the proposed robust PCIs method performs well as the distribution of quality characteristic is unable to be clearly identified from several LSD competitors. Finally, the proposed robust PCIs method is applied to two examples for illustration and an improvement strategy is provided to improve the manufacturing process of products.