Comparing the performance of the Cpk and (X‐bar, S) sampling plans

When the Cpk sampling plan is in use, a sample of size (n) is taken from the lot, and the mean and the standard deviation of the sample observations are used to obtain the —the sample estimator of the Cpk index; if the is lower than a threshold (c0), then the lot is rejected, otherwise, the lot is accepted. In recent studies, the α and β risks are the risks of rejecting/accepting lots of items produced by in‐control/out‐of‐control processes with Cpks equal to (, that is, the two parameters (n, c0) of the Cpk sampling plans depend on the inputs (α, , βmax¸ ). When the design of the Cpk sampling plan is based on the inputs C0 and C1, the optimum sample size is always big and, excluding the cases where the magnitude of the mean shift is too small, the β risks associated to the combinations of mean shifts and variance increases (determined by the inputs C0 and C1) are always equal to the maximum allowed value βmax. This fact motivated us to compare the Cpk sampling plan with the (X‐bar, S) sampling plan, where the sample mean (X‐bar) and the sample standard deviation (S) are directly compared with thresholds. For most disturbances, the (X‐bar, S) sampling plan requires smaller samples to meet the condition of β ≤ βmax, that is, for a fixed sample size, it is always possible to find endless combinations of mean shifts and variance increases where the β risks of the (X‐bar, S) sampling plan are β = βmax, and the β risks of the Cpk sampling plan are β > βmax.