Development of an Empirical Bayesian Single-Attribute Acceptance Sampling Plan for Food Quality using a Beta–Binomial Model

Abstract

Background: The variation in the quality characteristics of a manufactured product across different lots may arise from variation within lots (batches) or from variation between lots. Within-lot variation is addressed through statistical process control, whereas between-lot variation, which may arise from shifts in the process center, is more appropriately evaluated through acceptance sampling. Traditional acceptance sampling plans in food quality assurance rely primarily on frequentist frameworks that evaluate lots in isolation, thereby ignoring valuable historical process information and failing to adequately address uncertainty in the process center. In food manufacturing, where inspection is often destructive and costly, incorporating historical process information into acceptance decisions is essential. Bayesian methods provide an effective framework for integrating prior knowledge with current sample data. Objective: This study develops and evaluates a Bayesian single-attribute acceptance sampling plan using a Beta prior distribution to model the lot proportion nonconforming, p, and benchmarks its performance against classical frequentist baselines under realistic food safety constraints. Methods: The number of nonconforming items, Y , in a sample of size n is assumed to follow a Binomial distribution, Y ∼ Binomial(n, p), with a conjugate Beta prior, p ∼ Beta(α, β). Method of Moments estimation was applied to simulated continuous production records comprising K = 100 lots to calibrate an empirical baseline prior, Beta(2, 198). The pro- posed empirical plan was systematically benchmarked against three noninformative priors: Haldane, Beta(0.001, 0.001); Jeffreys, Beta(0.5, 0.5); and Uniform, Beta(1, 1), as well as three informative regimes: Low Compliance, Beta(2, 38); Conservative, Beta(5, 95); and Optimistic, Beta(1, 199). Plan performance was evaluated using Operating Characteristic (OC) curves, Bayes risk, Average Total Inspection (ATI), and prior sensitivity analysis. Results: The empirical Beta(2, 198) prior reduced the required sample size by 33.3% while maintaining Producer’s Risk (α0 ≤ 0.05 at AQL = 1%) and Consumer’s Risk (β0 ≤ 0.10 at LQL = 5.0%) within the specified boundaries. The OC curve of the empirical Bayesian plan closely matched the discrimination profile of the classical frequentist baseline while requiring substantially less inspection effort. Sensitivity analysis confirmed the robustness of the proposed plan to perturbations in the prior hyperparameters, while dynamic Bayesian updating provides a mechanism for maintaining consumer protection when process conditions change. Conclusion: The conjugate Beta–Binomial Bayesian framework improves sampling efficiency and decision-making by combining historical process stability with current inspection data. The proposed adaptive approach provides a mathe- matically sound framework for food quality assurance, potentially reducing laboratory inspection costs while maintaining specified food safety requirements.