In the 21st century, quality has become globally a major norm for success in both industrial and service sectors. People have started to realize the importance of quality and the consequences of bad quality. In the earlier stage, quality was totally inspection based. The actions were mostly reactive after the fault has occurred and any sort of corrective actions were mostly impossible. Then the idea of sampling came forward where random samples were taken for further statistical analysis to evaluate the capability of the process. Control chart is the most popular and widely used tool to analyze the performance of a process based on sample information. Process status (in control/out of control) can be easily understood and interpreted by simply observing the control chart. However, the rapid improvement of science and technology has led to enhancement of processes to such extent that traditional control charts are showing problems in performance or practical implementation. Based on the underlying distribution that the data follow, various control charts are developed. The control limits of traditional Shewhart type control charts are derived on the assumption that the distribution of the pattern is normal. However if the data is contaminated and the underlying process is not normal then the performance of the traditional control charts are highly affected. p-charts are used when the underlying distribution of the quality parameter follows binomial distribution. These charts are prepared by using normal approximation to binomial distribution to sample statistics. Nevertheless, in case of binomial distribution a prior estimate of p (fraction non-conforming in the population) must be available. When p is estimated from limited number of samples, the result becomes questionable and unreliable. During the development of p-control chart, Shewhart probably did not consider that the proportion of non-conforming items can take a very small value due to the rapid improvement of technology and sophisticated high precision machineries. For small p-values, binomial distribution is highly asymmetric, and the normal approximation becomes erroneous. Construction of p-charts using binomial assumption is unreliable in most cases where p is estimated from sample data. Binomial distribution assumes more precision than actually exists; it makes control limits more precise than they really are since the uncertainty in the value of p makes the limits of the control chart questionable. Thus, the control limits become narrower and may result in false detection when the process may still be in control. In this study, a modified p-chart is proposed based on hyperbinomial distribution when prior estimate of proportion nonconforming is unavailable and is estimated from limited sample information. The probability distribution of p is considered in determining the control limits. This reduces the rate of false detection significantly. The study also validates the use of hyperbinomial 3-σ control chart by comparing with the result obtained using the cumulative distribution function (CDF) of hyperbinomial distribution.

Gulam Kibria

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Application of Hyperbinomial Distribution in Developing a Modified p-chart