Design of control charts with m‐of‐m runs rules

This paper investigates economic–statistical properties of the X? charts supplemented with m‐of‐m runs rules. An out‐of‐control condition for the chart is either a point beyond a control limit or a run of m‐of‐m successive points beyond a warning limit. The sampling process is modeled by a Markov chain with 2m states. The steady‐state probability for each state and the average run length (ARL) from each state of the Markov chain are derived in explicit formulas. Then the stationary average run length (SALR) is derived so as to develop an economic–statistical model. Using this model, the design parameters are optimized by minimizing the cost function with constraints on the average time to signal (ATS). The X? chart supplemented with m‐of‐m runs rules is compared with the Shewhart X? chart in terms of the SARL and the cost function. Sensitivity of the design parameters with respect to the cost function is also analyzed. General guidelines for implementing the X? chart with m‐of‐m runs rules are presented from those observations. It should be emphasized that supplementing run rules may provide feasible and efficient solutions even if the sample size is limited, while the Shewhart X? chart may not. Copyright © 2009 John Wiley & Sons, Ltd.