Process capability indices (PCIs) have been widely used to analyze capability of the process that measures how the customer expectations have been conformed. Two of the well-known PCIs, named indices C-pm, and C-pmk have been developed to consider customers' ideal value that called target value (T). Although, these indices have similar features of the well-known indices C-p and C-pk, one of the most important differences is to consider T. In real case problems, we need to add some uncertainties related with human's evaluations into process capability analysis (PCA). One of the uncertainty modelling methods called neutrosophic sets (NSs), have an important role in modeling uncertainty based on incomplete and inconsistent information. For this aim, the PCIs have been designed by using NSs to manage the uncertainties of systems and to increase sensitiveness, flexibility and to obtain more detailed results of PCA in this paper. For this aim, the indices C-pm, and C-pmk have been performed and re-designed by using single valued neutrosophic numbers for the first time in the literature. Additionally, specification limits (SLs) have been re-considered by using NSs. The neutrosophic state of the SLs provide us to have more knowledge about the process and easily applied for engineering problems that includes uncertainty. Finally, the neutrosophic process capability indices (NPCIs) ((C) over tilde )over dot(pm) and ((C) over tilde )over dot(pmk) have been obtained and the main formulas of them have been produced. Additionally, the proposed ((C) over tilde )over dot(pm) and ((C) over tilde )over dot(pmk )have been applied on real case studies from manufacturing industry. The obtained results show that the indices ((C) over tilde )over dot(pm) and ((C) over tilde )over dot(pmk) include more informative and flexible results to evaluate capability of process.