Bayesian economic cost model for a variable sampling plan for fraction defective and manufacturing process control.

Abstract

Acceptance sampling plans by variables are a basic quality control technique. These plans provide economical procedures to determine the acceptability of batches of product. Most of these plans are based on a single quality characteristic and are of the classical type. This work concentrates on Bayesian variable acceptance sampling plans for fraction defective. Both destructive and non-destructive sampling procedures are considered. A set of decision points are estimated and employed to make decisions about the inspected lots. Techniques to dispose of the rejected lots are provided. Components of the expected total cost relative to various decisions are estimated. The sample size required to obtain the expected optimum cost is found. An untrue assumption implicit in the measurement of the quality characteristic of items sampled is that the observed dimensions are error free. The distributions, means, and variances of a set of parameters for error free and error prone sampling is listed. Computer programs written in FORTRAN 77 are developed to compute the decision points and the costs for both destructive and nondestructive testing. Precise Bays estimate of the costs and other economic parameters involve the moments of the fraction defective p raised to the kᵗʰ power. Mathematical expressions for the conditional expectations of p|x and p|ẋ are derived and a computer program to estimate these moments is provided. Producing quality items with minimum cost requires keeping a production process under control. The quality characteristic X of each item produced is determined and the sample means are plotted on an Ẋ-control chart. A production process is assumed to start in control at time t = 0 with specific values of the mean and standard deviation. The occurrence of a single or multiple cause-failures shift the process mean outside the control limits. During the search for the causes of failure, the process is either allowed to continue in operation or shut down until the assignable cause or causes are discovered. The expected duration of time during which the process is shut down and the additional time to be taken to repair the process are considered. Computer programs are provided to estimate the optimal sample size, the interval between successive samples, the control limits, the probability of type I error, the power of the chart, and the average time the process operates in the presence of an assignable cause. The parameters estimated are employed to estimate the optimal loss-cost. The economic design of Ẋ -charts assumes one quality characteristic of interest. However a product quality in most industrial products and processes is characterized by more than one quality characteristic where the application of a Ẋ -control chart for each variable is inappropriate. In this work a Hotellings T² control chart is employed to handle cases of where products are tested relative to several quality characteristics.