The effect of correlation on the economic design of warning limit X-bar charts

Abstract

Control charts are widely applied to monitor manufacturing processes. In 1962, Page presented a modified X-bar chart with warning limits, which includes an upper and lower warning band. In 1975, Gordon and Weindling presented a cost model for determining the five parameters of a warning limit X-bar chart: i.e., the sample size, the sampling interval between successive subgroups, the control limit coefficient, the warning limit coefficient and the significant run length. When designing control charts, one usually assumes that the measurements within a sample are independently distributed. However, this assumption may not be tenable in some specific production processes. Yang and Hancock presented a correlation model to describe the correlated data in a sample. In this paper, we study the effect of correlated data on the design of warning limit X-bar charts by combining Gordon and Weindling's cost model with Yang and Hancock's correlation model. Based on the study, it is observed that among the five parameters in the economic design, only the significant run length is affected by the correlated data. Highly correlated data or independent data result in a longer run in the warning band.