🤖 AI Summary
This study addresses the risk of misclassification in process capability assessment under limited sample sizes when using fixed thresholds such as Cpk ≥ 1.33, due to the stochastic nature of estimators near decision boundaries. The work establishes, for the first time, a probabilistic framework for Cpk-based threshold decisions and employs local asymptotic theory to demonstrate that when the true process capability deviates from the threshold by an order of O(n⁻¹/²), the decision probability converges to a non-degenerate limit governed by the signal-to-noise ratio. Through Monte Carlo simulations, resampling techniques, and empirical analysis across 880 manufacturing dimensions, the study quantifies substantial decision instability around commonly used thresholds, thereby providing both theoretical justification and practical tools for evaluating borderline release risks in engineering practice.
📝 Abstract
Process capability indices such as $C_{pk}$ are widely used in manufacturing quality control to support supplier qualification and product release decisions based on fixed acceptance thresholds (e.g., $C_{pk} \geq 1.33$). In practice, these decisions rely on sample-based estimates computed from moderate sample sizes ($n \approx$ 20-50), yet the stochastic nature of the estimator is often overlooked when interpreting threshold compliance. This study establishes a local asymptotic characterization of decision behavior when the true process capability lies near a fixed threshold. Under standard regularity conditions, if the true capability equals the threshold, the acceptance probability converges to 0.5 as sample size increases, implying that a fixed $C_{pk}$ gate embeds an inherent boundary decision risk even under ideal distributional assumptions. When the true capability deviates from the threshold by $O(n^{-1/2})$, the decision probability converges to a non-degenerate limit governed by a scaled signal-to-noise ratio. Monte Carlo simulations and an empirical study on 880 manufacturing dimensions demonstrate substantial resampling-based decision instability near the commonly used 1.33 criterion. These findings provide a probabilistic interpretation of threshold-based capability decisions and quantitative guidance for assessing boundary-induced release risk in engineering practice.