Finite-Sample Decision Instability in Threshold-Based Process Capability Approval

TL;DR

This paper analyzes the probabilistic behavior of process capability index decisions near thresholds, revealing inherent risks of decision instability due to sampling variability, especially around the commonly used 1.33 threshold.

Contribution

It provides a local asymptotic analysis of decision behavior at threshold boundaries, quantifies the instability risk, and offers practical guidance for manufacturing quality control.

Findings

Acceptance probability converges to 0.5 at the threshold as sample size increases.

Decision instability is significant near the 1.33 criterion in practice.

Monte Carlo simulations and empirical data confirm the theoretical insights.

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.