The $C_{pk}$ index was originally created to measure the ability of the processes to produce products meeting specifications but, more recently, Rao and team proposed the use of the $C_{pk}$ index to control processes. In reality, they investigated the performance of the $C_{pk}$ chart with the GMDS (Generalized Multiple Dependent State Sampling) supplementary signaling rule. With the GMDS supplementary run rule, the $C_{pk}$ chart signals when a point falls in the action region or when a point falls in the warning region after a sequence of $m$ points, with less than $k$ , in the central region. The points on the $C_{pk}$ chart are the estimated values of the $C_{pk}$ index obtained with the mean and the variance of the samples. Rao and team obtained the ARLs of the GMDS $C_{pk}$ chart by simulation; reminding the ARL is the average number of samples the chart requires to signal a change in the process. This present work uses the Markov chain approach to obtain the ARLs, once this approach leads to exact results. The main conclusion is that the GMDS $\bar {X}$ chart is not only simpler to use than the GMDS $C_{pk}$ chart, but it is also more sensitive to process changes.