A Control Chart for the Joint Monitoring of Mean and Variance With Klein's Supplementary Rule

ABSTRACT Supplementary run rules have been employed since the 1950s to enhance the performance of control charts, such as the control chart, which aims to control the process mean, or control chart, which aims to control the process variance. One simple, effective, and popular suggestion is Klein's 2‐of‐2, which signals an out‐of‐control process only when two sequential sample averages are observed above the upper control limit or below the lower control limit. We propose to implement Klein's 2‐of‐2 for a more complex control chart, the joint control chart, which aims to simultaneously monitor the mean and the variance of a process. Compared with the traditional control chart, using this rule for the control chart reduces the average run length (ARL) and standard deviation of the run length (SDRL) for detecting small process shifts while maintaining high applicability owing to the ease of use in employing the 2‐of‐2 rule in practical environments. Klein's run rule was implemented in the control chart through a 15‐state Markov chain, and its performance was compared with the basic control chart as well as the exponentially weighted moving average (EWMA) chart, known to have excellent capabilities in detecting small process average shifts at the cost of a more complex control scheme. The newly proposed Klein control chart demonstrated superior performance compared to the traditional control chart and competitive performance even against the more complex joint EWMA control chart.