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High-quality processes with very low defect rates are usually modeled using a Poisson distribution. Considering any correlation between the number of defects on a unit and the fraction of nonconforming units, a bi-attribute process is introduced and the relationship between the two features is discussed in this research. A generalization of the Poisson distribution which is called  k -inflated Poisson (KIP) distribution is derived to model these types of processes. Based on the KIP distribution, a two-parameter distribution, a sampling method is suggested for inspection of a high-quality process. The joint distribution of the bi-attribute high-quality process is also introduced and the correlation between its attributes is discussed. The results of numerical examples provided some perspective to the model and supported the theoretical findings that there is no significant correlation between the two features. Then a charting procedure is suggested for monitoring the two parameters of the KIP distribution.

Monitoring a bi-attribute high-quality process using a mixture probability distribution