Process capability analysis (PCA) is an important statistical approach for measuring and analyzing the ability to meet specifications. This analysis has been generally applied by obtaining process capability indices (PCIs). The indices named C-p and C-pk are the most commonly used for this aim. Although PCIs are completely effective statistics to analyze process' capability, the complexity of the production processes based on uncertainty arising from human thinking, incomplete or vague information makes it difficult to analyze the process capability with precise values. When the process includes uncertain, complex, incomplete and inaccurate information, the capability of the process can be successfully analyzed by using the fuzzy set theory (FST). Neutrosophic sets (NSs), one of the fuzzy set extensions, have an ability to deliver more successful results for modeling uncertainty, since they contain the membership functions of truth, indeterminacy, and falsity definitions rather than an only membership function. This feature provides a strong advantage and important capability for modeling uncertainty. In this paper, PCA has been performed based on NSs for more effectively modeling uncertainties of the process. For this purpose, specification limits (SLs) have been reconsidered by using NSs and two of the well-known process capability indices (PCIs) named C-p and C-pk have been reformulated. Additionally, design and analysis of the indices Cp and Cpk are investigated based on NSs. Finally, the neutrosophic process capability indices (NPCIs) C-p (C-p ($) triple over dot($$) over tilde) and C-pk (C-pk ($) triple over dot($$) over tilde) have been derived for three cases that are created by defining SLs to model uncertainties. Additionally, the indices C-p ($) triple over dot($$) over tilde and C-pk ($) triple over dot($$) over tilde have also been applied and analyzed on some real case problems from automotive industry. The obtained results show that the NPCIs support the quality engineers to easily define SLs and obtain more flexible and realistic evaluations for PCA.