A Cumulative Score Control Scheme

Abstract

The x-chart is one of the most widely used techniques in process control. When a point is outside control limits, an out-of-control signal is indicated. The x-chart can also indicate an out-of-control condition even though no single point is outside the control limits, if the pattern of the plotted points exhibits non-random or systematic behaviour such as shift, trend and stratification (West Electric, 1956). The cumulative sum (CUSUM) scheme was introduced by Page (1954) to detect a shift in the process mean and is more effective in detecting relatively small shifts than the x-chart. However, diagnosis of patterns other than shift on the CUSUM scheme is difficult, although not impossible, because the sequence of plotted points is not uncorrelated. Munford (1980) originally proposed cumulative score schemes in which a score of 1, 1 or 0 is assigned to each sample according to whether the sample mean lies below, above or within the range from k to k, assuming that the process mean is 0 and the standard deviation of the sample mean is 1. Scores are then accumulated and an out-of-control signal will be indicated if the cumulative score reaches a critical value. Lewis (1981) further discussed the determination of parameters of Munford's schemes. In this paper we propose an extended cumulative score control scheme to detect a shift in the process mean. The scheme divides the x-chart into zones of width a,,. For various values of the sample mean, assign various scores. This is illustrated in Fig. 1. For detecting an increase in the mean, the cumulative score scheme with some critical value h and reference value k is performed. When the accumulated score reaches h, a corrective signal is indicated. In the two-sided case for detecting either an increase or a decrease in the mean, a second cumulative score is added to detect a decrease.