Risk-Calibrated Process Capability Approval with Finite Samples

Process capability indices such as $C_{pk}$ are widely used in manufacturing to support supplier qualification, pilot-build release, and production approval. In practice, approval decisions are often based on deterministic threshold rules of the form $\widehat{C}_{pk} \ge C_0$. Because $\widehat{C}_{pk}$ is estimated from finite samples, however, such decisions are inherently stochastic, especially when the true capability lies near the approval threshold. This paper develops a risk-calibrated decision framework for process capability approval that explicitly accounts for estimation uncertainty and asymmetric operational loss. Capability approval is formulated as a binary statistical decision problem, leading to a rule of the form $\widehat{C}_{pk} \ge C_0 + k\,SE(\widehat{C}_{pk})$, where the calibration constant $k$ is determined either by a tolerable failure probability or by a false-accept/false-reject cost ratio. The resulting formulation unifies several commonly used procedures, including deterministic thresholding, lower confidence bound rules, and probability-based approval rules, and naturally extends them to cost-sensitive decision rules derived from asymmetric operational loss. Simulation experiments and an industrial case study show that risk calibration primarily affects near-threshold decisions, improves approval stability, and can substantially reduce expected operational loss when false acceptance is more costly than false rejection.