Runs rules for bivariate Shewhart chart

Among the various statistical process control (SPC) charts, the Shewhart chart (Shewhart, Economic Control of Quality Improvement, Van Nostrand Reinhold, 1931) has been the most used control chart in practice due to its simplicity and explicit representation. In many practical situations, there are two or more related quality characteristics that must be monitored simultaneously. The Shewhart chart used at present is inefficient for monitoring multiple variables and lacks the capability to consider the relationships between variables. In this paper, we develop runs rules for the Shewhart chart to control bivariate data. The results show that the Shewhart chart with runs rules is comparable, and in some cases superior, to the T² control chart. We use the thickness data of an SiO₂ layer as an example to illustrate the methodology