CUSUM control charts

Appropriately designed, Cusum control charts give sensitive and instructive impressions on process changes. Cumulative stuns S = YT,= ( — 0) also contain information on actual as well as on previously obtained values. Therefore their display enables one to perceive earlier changes leading to OCS than by means of the chart of original values (see Woodward and Goldsmith [1964] Marshall [1977] Doerffel [1990]). 

The Cusum Control Chart is a very special chart from which a lot of information can be drawn. Cusum is the abbreviation for cumulative sum and means the sum of all differences from the target value. Every day the difference of the control analysis from the target value is added to the sum of all the previous ddferences. 

The control charts discussed earlier are very useful in the diagnostic aspects of quality process improvement. They can be used to stabilize a process by identifying out-of-control situations. After the process is stabilized and brought in control, further improvement of the process can be achieved by using some special control charts such as the cumulative sum (CUSUM) control chart and the exponentially weighted moving average (EWMA) control chart. These control charts can be used when small shifts in a process are of interest. 

CUSUM Control Chart A CUSUM chart provides an efficient way of detecting small shifts in the mean of a process (l/2 a), the chart is usually used.The CUSUM chart incorporates information contained in a sequence of sample points. It keeps track of the cumulative sum of the deviations between each sample point (a sample mean) and a target value. Unlike the x chart, which often bases its out-of-control decision on just the most recently collected sample, the CUSUM calculated for a sample point carries the history prior to that sample. For example, a sequence of sample points above the centerline can trigger an out-of-control signal although all of them stayed well below the UCLs of the x chart. 

Spreadsheet 4.2. Calculation of S+ and S for the CuSum control chart example of research octane number. 

NIST (2006), NIST/SEMATECH e-handbook of statistical methods—6.3.2.3. CuSum control charts, http //www.itl.nist.gov/div898/handbook/ (Accessed 22 January 2006). 

The relative performance of the Shewhart and CUSUM control charts is compared in Fig. 8-48 for a set of simulated data for the tensile strength of a resin. It is assumed that the tensile strength x is normally distributed with a mean of p = 70 M Pa and a standard deviation of a = 3 MPa. A single measurement is available at each sampling instant. A constant (a = 0.5a = 1.5) was added to x(k) for k >10 in order to evaluate each charts ability to detect a small process shift. The CUSUM chart was designed using K = 0.5a and H = 5a. 

The CUSUM control chart was originally developed using a graphical approach based on V masks. However, for computer calculations, it is more convenient to use an equivalent algebraic version that consists of two recursive equations... 

TABLE 8-7 Average Run Lengths for CUSUM Control Charts... 

FIG. 8-48 Comparison of Shewhart and CUSUM control charts for the resin example. (Source Seborg et al., Process Dynamics and Control, 2d ed., Wiley, New York, 2004.)... 

Two alternatives to the Shewhart control chart, which are more complicated to calculate but generally more effective to detect small shifts, are the Cumulative Sum (or Cusum) control chart and the Exponentially Weighted Moving Average (EWMA) control chart. These control charts will not be discussed here, but are described in standard references. ... 

Westgard JO, Groth T, Aronsson T, et al. Combined Shewhart-cusum control chart for improved quality control in chnical chemistry. Clin Chem 1977 23 1881-7. 

Figure 1.1 Examples of (a) Levey-Jennings and (b) cumulative sum (cusum) control charts using inter-assay quality control data from an alprazolam GC-MS assay. Since the cusum chart persents the cumulative sum of deviations from the mean, it is more sensitive to small biases that develop over time, whereas Levey-Jennings charts are most useful for detecting changes in the precision of the assay...

GC Runger, TR Willemain, and S Prabhu. Average run lengths for CUSUM control charts applied to residuals. Commun. Statist. -Theory Methods, 24(l) 273-282, 1995. 

Verify the accuracy of results obtained in a laboratory Monitor the performance of the method (e.g. cusum control charts) Calibrate equipment which requires a calibrant similar to the matrix (e.g. optical emission spectrometry. X-ray fluorescence spectrometry) Demonstrate equivalence between methods... 

FIG. 4. Cusum control chart calculated on the basis of the previously shown quality control data plotted in the Shewhart chart (Fig. 3). 

CUSUM control charts also are constructed for measures of variability such as the range or standard deviation (Ryan, 2000 Montgomery, 2009). 

Both simple methods and more complex methods have been presented in this article. While there are certain concerns and weaknesses for the simple methods such as lack of independence, those methods have proved adequate for many production operations over the years and those simple methods are recommended as an initial effort. The use of more sophisticated methods, even including moving average and CUSUM control charts, as well as DOE and the highly complex methods suggested for autocorrelated data, such as that found in nuclear and process industries, is based on a matter of need and level of technical understanding of what is to be accomplished. 

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