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.

翻译：过程能力指数（如$C_{pk}$）在制造业中被广泛用于支持供应商资质认定、试生产放行和生产批准。在实践中，批准决策通常基于$\widehat{C}_{pk} \ge C_0$形式的确定性阈值规则。然而，由于$\widehat{C}_{pk}$是从有限样本中估计得出的，此类决策本质上是随机的，尤其是当真实能力接近批准阈值时。本文开发了一个用于过程能力批准的风险校准决策框架，该框架明确考虑了估计不确定性和非对称操作损失。能力批准被表述为一个二元统计决策问题，从而推导出$\widehat{C}_{pk} \ge C_0 + k\,SE(\widehat{C}_{pk})$形式的规则，其中校准常数$k$由可容忍的失效概率或误接受/误拒绝成本比决定。所得框架统一了多种常用程序，包括确定性阈值法、置信下限规则和基于概率的批准规则，并自然地将它们扩展到由非对称操作损失导出的成本敏感决策规则。仿真实验和工业案例研究表明，风险校准主要影响接近阈值的决策，提高了批准稳定性，并且当误接受比误拒绝成本更高时，能显著降低预期操作损失。