A Machine Learning Framework for Uncertainty-Calibrated Capability Decision under Finite Samples

Abstract: Process capability indices such as $C_{pk}$ are widely used for manufacturing decisions, yet are typically applied via deterministic thresholding of finite-sample estimates, ignoring uncertainty and leading to unstable outcomes near the capability boundary. This paper reformulates capability approval as a decision-risk calibration problem, quantifying the probability of misclassification under finite-sample variability. We propose an uncertainty-aware hybrid framework that combines a statistically grounded baseline with a data-driven residual learner, where the baseline provides an interpretable approximation of failure risk and the residual captures systematic deviations due to non-normality, measurement effects, and finite-sample uncertainty. A nested Monte Carlo procedure is introduced to approximate oracle decision risk under controlled synthetic settings, enabling direct evaluation of probabilistic calibration. Empirical results show that conventional approaches exhibit substantial miscalibration in near-threshold regimes, while the proposed framework provides a structured and uncertainty-aware representation of decision risk that remains stable under stricter leak-free evaluation. The framework is simple, compatible with existing capability metrics, and readily deployable in industrial analytics systems.