The 𝑛𝑝-Chart with 3-𝜎 Limits and the ARL-Unbiased 𝑛𝑝-Chart Revisited

Abstract

In the statistical process control literature, counts of nonconforming items are frequently assumed to be independent and have a binomial distribution with parameters ( n , p ) (n,p) , where 𝑛 and 𝑝 represent the fixed sample size and the fraction nonconforming. In this paper, the traditional n ⁢ p np -chart with 3-𝜎 control limits is reexamined. We show that, even if its lower control limit is positive and we are dealing with a small target value p 0 p_{0} of the fraction nonconforming ( p ) (p) , this chart average run length (ARL) function achieves a maximum to the left of p 0 p_{0} . Moreover, the in-control ARL of this popular chart is also shown to vary considerably with the fixed sample size 𝑛. We also look closely at the ARL function of the ARL-unbiased n ⁢ p np -chart proposed by Morais [An ARL-unbiased n ⁢ p np -chart, Econ. Qual. Control 31 (2016), 1, 11–21], which attains a pre-specified maximum value in the in-control situation. This chart triggers a signal at sample 𝑡 with probability one if the observed number of nonconforming items, x t x_{t} , is beyond the lower and upper control limits (𝐿 and 𝑈), probability γ L \gamma_{L} (resp. γ U \gamma_{U} ) if x t x_{t} coincides with 𝐿 (resp. 𝑈). A graphical display for the ARL-unbiased n ⁢ p np -chart is proposed, taking advantage of the qcc package for the statistical software R. Furthermore, as far as we have investigated, its control limits can be obtained using three different search algorithms; their computation times are thoroughly compared.