General Comments on Control Charts

16.1 General Comments on Control Charts

  A central message throughout this book has been the pervasiveness of naturally occurring variation associated with any characteristic or attribute of different indi- viduals or objects. In a manufacturing context, no matter how carefully machines are calibrated, environmental factors are controlled, materials and other inputs are monitored, and workers are trained, diameter will vary from bolt to bolt, some plastic sheets will be stronger than others, some circuit boards will be defective whereas others are not, and so on. We might think of such natural random variation as uncontrollable background noise.

  There are, however, other sources of variation that may have a pernicious impact on the quality of items produced by some process. Such variation may be attributable to contaminated material, incorrect machine settings, unusual tool wear, and the like. These sources of variation have been termed assignable causes in the quality control literature. Control charts provide a mechanism for recognizing situations where assignable causes may be adversely affecting product quality. Once

  a chart indicates an out-of-control situation, an investigation can be launched to identify causes and take corrective action.

  A basic element of control charting is that samples have been selected from the process of interest at a sequence of time points. Depending on the aspect of the process under investigation, some statistic, such as the sample mean or sample proportion of defective items, is chosen. The value of this statistic is then calculated for each sample in turn. A traditional control chart then results from plotting these calculated values over time, as illustrated in Figure 16.1.

  Value of quality statistic

  UCL Upper control limit

  Center line

  LCL Lower control limit

  Time 1 2 3 4 5 ...

  Figure 16.1

  A prototypical control chart

  16.1 General Comments on Control Charts

  Notice that in addition to the plotted points themselves, the chart has a center line and two control limits. The basis for the choice of a center line is sometimes a target value or design specification, for example, a desired value of the bearing diam- eter. In other cases, the height of the center line is estimated from the data. If the points on the chart all lie between the two control limits, the process is deemed to be in control. That is, the process is believed to be operating in a stable fashion reflect- ing only natural random variation. An out-of-control “signal” occurs whenever a plotted point falls outside the limits. This is assumed to be attributable to some assignable cause, and a search for such causes commences. The limits are designed so that an in-control process generates very few false alarms, whereas a process not in control quickly gives rise to a point outside the limits.

  There is a strong analogy between the logic of control charting and our previ- ous work in hypothesis testing. The null hypothesis here is that the process is in con- trol. When an in-control process yields a point outside the control limits (an out-of-control signal), a type I error has occurred. On the other hand, a type II error results when an out-of-control process produces a point inside the control limits. Appropriate choice of sample size and control limits (the latter corresponding to specifying a rejection region in hypothesis testing) will make the associated error probabilities suitably small.

  We emphasize that “in control” is not synonymous with “meets design speci- fications or tolerance.” The extent of natural variation may be such that the percent- age of items not conforming to specification is much higher than can be tolerated. In such cases, a major restructuring of the process will be necessary to improve process capability. An in-control process is simply one whose behavior with respect to vari- ation is stable over time, showing no indications of unusual extraneous causes.

  Software for control charting is now widely available. The journal Quality Progress contains many advertisements for statistical quality control computer pack- ages. In addition, SAS and Minitab, among other general-purpose packages, have attractive quality control capabilities.

  EXERCISES Section 16.1 (1–3)

  1. A control chart for thickness of rolled-steel sheets is based on

  which the probability of observing at least one outside the

  an upper control limit of .0520 in. and a lower limit of .0475 in.

  control limits exceeds .10?

  The first ten values of the quality statistic (in this case , the X 4. A cork intended for use in a wine bottle is considered accept-

  sample mean thickness of n55 sample sheets) are .0506,

  able if its diameter is between 2.9 cm and 3.1 cm (so the

  .0493, .0502, .0501, .0512, .0498, .0485, .0500, .0505, and

  lower specification limit is and LSL 5 2.9 the upper specifi-

  .0483. Construct the initial part of the quality control chart, and

  cation limit is ). USL 5 3.1

  comment on its appearance.

  a. If cork diameter is a normally distributed variable with

  2. Refer to Exercise 1 and suppose the ten most recent values of

  mean value 3.04 cm and standard deviation .02 cm, what

  the quality statistic are .0493, .0485, .0490, .0503, .0492,

  is the probability that a randomly selected cork will con-

  .0486, .0495, .0494, .0493, and .0488. Construct the relevant

  form to specification?

  portion of the corresponding control chart, and comment on

  b. If instead the mean value is 3.00 and the standard devia-

  its appearance.

  tion is .05, is the probability of conforming to specifica-

  3. Suppose a control chart is constructed so that the probability

  tion smaller or larger than it was in (a)?

  of a point falling outside the control limits when the process

  5. If a process variable is normally distributed, in the long run

  is actually in control is .002. What is the probability that ten

  virtually all observed values should be between m 2 3s and

  successive points (based on independently selected samples)

  m 1 3s , giving a process spread of 6s.

  will be within the control limits? What is the probability that

  a. With LSL and USL denoting the lower and upper specifi-

  25 successive points will all lie within the control limits?

  cation limits, one commonly used process capability

  What is the smallest number of successive points plotted for

  index is . C p 5 (USL 2 LSL)6s The value C p 51

  CHAPTER 16 Quality Control Methods

  indicates a process that is only marginally capable of

  C pk

  5 min 5(USL 2 m)3s, (m 2 LSL)3s6

  meeting specifications. Ideally, C p should exceed 1.33 (a “very good” process). Calculate the value of C

  p for each

  Calculate the value of C

  pk for each of the cork-production

  of the cork production processes described in the previous

  processes described in the previous exercise, and com-

  exercise, and comment.

  ment. [Note: In practice, m and s have to be estimated

  b. The C index described in (a) does not take into account

  from process data; we show how to do this in Section 16.2]

  p

  process location. A capability measure that does involve

  c. How do C p and C pk compare, and when are they equal?

  the process mean is