Bayesian Group Chain Sampling Plan for Poisson Distribution with Gamma Prior

Abstract

Acceptance sampling is a statistical quality control technique that consists of procedures for sentencing one or more incoming lots of finished products. Acceptance or rejection is based on the inspection of sampled products drawn randomly from the lot. The theory of previous acceptance sampling was built upon the assumption that the process from which the lots are produced is stable and the process fraction nonconforming is a constant. Process variability is inevitable due to random fluctuations, which may inadvertently lead to quality variation. As an alternative to traditional sampling plans, Bayesian approach can be used by considering prior information of the process. Using different combinations of design parameters, this study introduces a Bayesian group chain sampling plan (BGChSP). For the first time in group chain sampling plan, the probability of lot acceptance is derived by using Poisson distribution to estimate an average number of defectives. Gamma distribution is used as a prior distribution with Poisson distribution. Taking into account both consumer's and producer's risks, this research considers two quality regions namely, probabilistic quality region (PQR) and indifference quality region (IQR). By minimizing consumer's and producer's risks, BGChSP can be used to minimize the average number of defective products in industry.

Designing Bayesian Two-Sided Group Chain Sampling Plan for Gamma Prior Distribution

“…When developing the procedures, L(p) can be calculated for the chain acceptance sampling plans, with the assumption that the underlying distribution for the plan is following either binomial or Poisson distribution [1,[11][12][13][14][15]. For the average number of defectives, this paper considers Poisson distribution, such that:…”

“…where the shape parameter s > 0, shape and the rate parameter t > 0 with mean l ¼ s t under the proposed sampling plan. For the average probability of lot acceptance, the general expression used in Bayesian [12] is:…”

“…For comparison purposes, BTSGChSP is compared with the existing BGChSP [12] for the same values of design parameters. For the specified design parameters, s = 2, r = 3 and i = j = 2, the average number of defectives is plotted against the average probability of acceptance in Fig.…”

“…They considered binomial distribution and estimate the average probability of acceptance for average proportion of defective. Later, the plan was extended for an average number of defectives by using Poisson distribution and gamma distribution as a prior distribution [12].…”

“…Based on [9] and [12] plans, this study concerns the development of a Bayesian two-sided group chain sampling plan (BTSGChSP) that considers preceding as well as succeeding lots. Poisson distribution function is used to estimate the probability of lot acceptance based on conforming and non-conforming products and gamma distribution is used as a suitable prior for Poisson distribution.…”