Optimal Defects-Per-Unit Acceptance Sampling Plans Using Truncated Prior Distributions

Optimal sampling inspection plans for defects per unit with fixed acceptance numbers and limiting quality levels are developed to provide the appropriate protection to customers when the number of nonconformities per sampled item follows a Poisson distribution. The best inspection scheme assures the customer, who has to judge the quality of the submitted material, that a supplier's lot is released only when there is conclusive evidence that it is satisfactory. The underlying integer nonlinear programming problem is formulated and solved in the frequentist setting, and a practically exact approximation to the minimum sample size is presented. Because there is often no reason to assume that the process average is constant, the classical perspective is then extended to those situations in which there is substantial prior information on the supplier's process. A family of generalized truncated gamma models and several restricted maximum entropy distributions satisfying typical constraints are adopted to describe the stochastic fluctuations in the process average. Optimal defects-per-unit acceptance plans are determined by solving the corresponding constrained minimization problems. Lower and upper bounds on the required sample size are deduced in closed-forms. A general procedure based on Taylor series expansions of the operating characteristic function around the mean quality level of the rejectable lots is proposed to derive an explicit, accurate, easily computable approximation to the smallest sample size that provides the required average customer protection. This procedure greatly simplifies the determination of optimal plans from defect or failure count data and prior knowledge, and also requires little prior information, namely the prior mean and variance of the quality level of the rejectable lots, which could be estimated from past data and expert opinions. The suggested methodology is applied to the manufacturing of paper and glass for illustrative purposes. Our approach allows the practitioners to incorporate into the quality analysis a reduced parameter space for the process average. Furthermore, the proposed sampling plans are reasonably insensitive to small disturbances in the prior knowledge on the process average, and the effective use of the available information on the supplier's process provides a more realistic assessment of the actual customer protection, as well as considerable savings in testing time and sample size.