A unified approach to adaptive Shewhart control charts

A completely adaptive (CA) X‾$\bar{X}$ chart, that is, an X‾$\bar{X}$ chart in which sampling interval, sample size, control limits, and warning limits are all adaptive and switch between two values, is explored. The exact expressions for the statistical and operational performance measures for this chart are derived. Obviously, these expressions are directly applicable to all the X‾$\bar{X}$ charts in which any one or more of the design parameters are adaptive and switch between two values, as those are particular cases of a CA X‾$\bar{X}$ chart. Thus, a CA X‾$\bar{X}$ chart provides a unified approach to explore all those charts. The simultaneous evaluation of all such charts through extensive numerical comparisons of their performances accentuated how each of the design parameters affects the chart performances when it is made adaptive. Also, the comparisons facilitated to determine the optimal adaptive X‾$\bar{X}$ charts for different situations in the sense of having the best overall performance. Investigation of a statistical design for a CA X‾$\bar{X}$ chart supported the inferences of the numerical comparisons.