CUSUM Procedures

16.5 CUSUM Procedures

  A defect of the traditional chart is its inability to detect a relatively small change X in a process mean. This is largely a consequence of the fact that whether a process is judged out of control at a particular time depends only on the sample at that time, and not on the past history of the process. Cumulative sum (CUSUM) control charts and procedures have been designed to remedy this defect.

  There are two equivalent versions of a CUSUM procedure for a process mean, one graphical and the other computational. The computational version is used almost exclusively in practice, but the logic behind the procedure is most easily grasped by first considering the graphical form.

  16.5 CUSUM Procedures

  The V-Mask

  Let m 0 denote a target value or goal for the process mean, and define cumulative sums by

  S 1 5x 1 2m 0

  S 2 5(x 1 2m 0 )1(x 2 2m 0 )5 g (x i 2m 0 )

  i51

  l

  S l 5(x 1 2m 0 )1c1(x l 2m 0 )5 g (x i 2m 0 )

  i51

  (in the absence of a target value, is used in place of m 5 x 0 ). These cumulative sums

  are plotted over time. That is, at time l, we plot a point at height S l . At the current

  time point r, the plotted points are (1, S 1 ), (2, S 2 ), (3, S 3 ), c, (r, S r ) . Now a V-shaped “mask” is superimposed on the plot, as shown in Figure 16.7. The point 0, which lies a distance d behind the point at which the two arms of the mask intersect, is positioned at the current CUSUM point (r, S r ). At time r, the process is judged out of control if any of the plotted points lies outside the

  ⎩ ⎨ ⎧

  Figure 16.7 CUSUM plots: (a) successive points (I, S l ) in a CUSUM plot; (b) a V-mask with

  0 5 (r, S r ) ; (c) an in-control process; (d) an out-of-control process

  CHAPTER 16 Quality Control Methods

  V-mask—either above the upper arm or below the lower arm. When the process is in

  control, the x i ’s will vary around the target value m 0 , so successive S i ’s should vary

  around 0. Suppose, however, that at a certain time, the process mean shifts to a value

  larger than the target. From that point on, differences x i 2m 0 will tend to be posi-

  tive, so that successive S l ’s will increase and plotted points will drift upward. If a shift has occurred prior to the current time point r, there is a good chance that (r, S r ) will be substantially higher than some other points in the plot, in which case these other points will be below the lower arm of the mask. Similarly, a shift to a value smaller than the target will subsequently result in points above the upper arm of the mask.

  Any particular V-mask is determined by specifying the “lead distance”

  d and “half-angle” , or, equivalently, by specifying d and the length h of the u vertical line segment from 0 to the lower (or to the upper) arm of the mask. One method for deciding which mask to use involves specifying the size of a shift in the process mean that is of particular concern to an investigator. Then the parame-

  ters of the mask are chosen to give desired values of a and , the false-alarm b

  probability and the probability of not detecting the specified shift, respectively. An alternative method involves selecting the mask that yields specified values of the ARL (average run length) both for an in-control process and for a process in which the mean has shifted by a designated amount. After developing the compu- tational form of the CUSUM procedure, we will illustrate the second method of construction.

  Example 16.8

  A wood products company manufactures charcoal briquettes for barbecues. It pack- ages these briquettes in bags of various sizes, the largest of which is supposed to contain 40 lbs. Table 16.4 displays the weights of bags from 16 different samples, each of size n54 . The first 10 of these were drawn from a normal distribution with

  m5m 0 5 40 and s 5 .5 . Starting with the eleventh sample, the mean has shifted upward to m 5 40.3 .

  Table 16.4

  Observations, x’ s and Cumulative Sums for Example 16.8

  Sample Number

  Observations

  x

  g(x i 2 40) 1 40.77 39.95 40.86 39.21 40.20 .20

  16.5 CUSUM Procedures

  Figure 16.8 displays an chart with control limits X

  Figure 16.8 X control chart for the data of Example 16.8

  No point on the chart lies outside the control limits. This chart suggests a stable process for which the mean has remained on target.

  Figure 16.9 shows CUSUM plots with a particular V-mask superimposed. The plot in Figure 16.9(a) is for current time r 5 12 . All points in this plot lie inside the arms of the mask. However, the plot for r 5 13 displayed in Figure 16.9(b) gives an out-of-control signal. The point falling below the lower arm of the mask suggests an increase in the value of the process mean. The mask at r 5 16 is even more emphatic in its out-of-control message. This is in marked contrast to the chart. X

  CUSUM CUSUM

  Figure 16.9 CUSUM plots and V-masks for data of Example 16.8: (a) V-mask at time r 5 12 , process in control; (b) V-mask at time r 5 13 , out-of-control signal

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  CHAPTER 16 Quality Control Methods