Cumulative performance measures of X-bar charts for process control

Although X-bar charts have made significant contributions towards the pursuit of continuous quality improvement, the inability to predict chart performance over multiple subgroups has essentially restricted their functional capability. Performance measures on the multiple-subgroup level are developed for each of the four rules on the X-bar chart in terms of the probability of one or more violations over multiple subgroups, which is founded on the binomial distribution, rather than in terms of run length whose origin is the geometric distribution. The proposed methodology computes the joint probability of zero violations throughout process monitoring as the product of conditional probabilities along the path of no violations. An ergodic Markov chain is modeled to establish these conditional probabilities of no violation at a specified subgroup, given no violation at the previous subgroup. Once X-bar chart performance is known for process parameters (mean and variance), performance can be used to enhance the role of X-bar charts for a system: (a) as a measure of comparison with alternative charting methodologies; (b) as a tool to measure any improvement in chart performance as a result of control chart modifications; and (c) to provide diagnostic information that identifies special causes of variability and estimates process parameters given observed control chart performance. This type of system effectively represents a closed-loop feedback system and can be used for intelligent control of manufacturing processes.