Acceptance or Rejection of Lots while Minimizing and Controlling Type I and Type II Errors

TL;DR

The paper presents a sequential Double Hypotheses Test (DHT) for Bernoulli trials to bound defect probability $p$ by comparing two distributions $p_0$ and $p_1$, thereby controlling Type I and Type II errors. It develops four practical implementations—Binomial, Poisson, Normal via Norm_N (Newton-Raphson) and Norm_I (intuitive)—to compute decision thresholds, and augments the framework with the Successive Failures Limit (SFL) from Renewal Theory for faster, staged decisions. It also introduces a fuzzy design to automatically select the most appropriate algorithm based on precision and runtime, and demonstrates the methods through examples and extensive discussion of trade-offs. The approach is applicable to manufacturing quality control, supplier assessments, and continuous monitoring of service systems, offering flexible, real-time decision capabilities even when the true defect rate is unknown.

Abstract

The double hypothesis test (DHT) is a test that allows controlling Type I (producer) and Type II (consumer) errors. It is possible to say whether the batch has a defect rate, p, between 1.5 and 2%, or between 2 and 5%, or between 5 and 10%, and so on, until finding a required value for this probability. Using the two probabilities side by side, the Type I error for the lower probability distribution and the Type II error for the higher probability distribution, both can be controlled and minimized. It can be applied in the development or manufacturing process of a batch of components, or in the case of purchasing from a supplier, when the percentage of defects (p) is unknown, considering the technology and/or process available to obtain them. The power of the test is amplified by the joint application of the Limit of Successive Failures (LSF) related to the Renewal Theory. To enable the choice of the most appropriate algorithm for each application. Four distributions are proposed for the Bernoulli event sequence, including their computational efforts: Binomial, Binomial approximated by Poisson, and Binomial approximated by Gaussian (with two variants). Fuzzy logic rules are also applied to facilitate decision-making.

Figures (16)

• Figure 1: Operating Characteristic - OC curve
• Figure 2: Double Hypothesis Test for Binomial Distribution - Bin Algorithm
• Figure 3: DHT Steps for Binomial Distribution = Bin Algorithm
• Figure 4: DHT Steps for Poisson Distribution - Poiss Algorithm
• Figure 5: DHT for Normal Newton Raphson Generalized Algorithm - Norm_N