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 recogniz- ing 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

  Figure 16.1 A prototypical control chart

  680 Chapter 16 Quality Control Methods

  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 diameter. 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 previous work in hypothesis testing. The null hypothesis here is that the process is in control. 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 will make the associated error probabilities suitably small.

  We emphasize that “in control” is not synonymous with “meets design specifi- cations or tolerance.” The extent of natural variation may be such that the percentage 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 pro- cess capability. An in-control process is simply one whose behavior with respect to variation 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 packages. In addition, SAS and Minitab, among other general-purpose packages, have attractive quality control capabilities.

  EXERCISES Section 16.1 (1–5)

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

  points plotted for which the probability of observing at least

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

  one outside the control limits exceeds .10?

  .0475 in. The first ten values of the quality statistic (in this

  4. A cork intended for use in a wine bottle is considered

  case X, the sample mean thickness of n 5 5 sample sheets)

  acceptable if its diameter is between 2.9 cm and 3.1 cm

  are .0506, .0493, .0502, .0501, .0512, .0498, .0485, .0500,

  (so the lower specification limit is LSL 5 2.9 and the

  .0505, and .0483. Construct the initial part of the quality

  upper specification limit is USL 5 3.1).

  control chart, and comment on its appearance.

  a. If cork diameter is a normally distributed variable

  2. Refer to Exercise 1 and suppose the ten most recent val-

  with mean value 3.04 cm and standard deviation

  ues of the quality statistic are .0493, .0485, .0490, .0503,

  .02 cm, what is the probability that a randomly

  .0492, .0486, .0495, .0494, .0493, and .0488. Construct

  selected cork will conform to specification?

  the relevant portion of the corresponding control chart,

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

  and comment on its appearance.

  deviation is .05, is the probability of conforming to

  3. Suppose a control chart is constructed so that the probabil-

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

  ity of a point falling outside the control limits when the

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

  process is actually in control is .002. What is the probability

  run virtually all observed values should be between

  that ten successive points (based on independently selected

  m2 3s and m 1 3s, giving a process spread of 6s.

  samples) will be within the control limits? What is the prob-

  a. With LSL and USL denoting the lower and upper

  ability that 25 successive points will all lie within the con-

  specification limits, one commonly used process capa-

  trol limits? What is the smallest number of successive

  bility index is C p 5 sUSL 2 LSLdy6s. The value

  16.2 Control Charts for process Location 681

  C p 5 1 indicates a process that is only marginally

  C pk 5 min { sUSL 2 mdy3s, sm 2 LSLdy3s}

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

  Calculate the value of C pk for each of the cork-

  value of C p for each of the cork production processes

  production processes described in the previous exer-

  described in the previous exercise, and comment.

  cise, and comment. [Note: In practice, m and s have

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

  to be estimated from process data; we show how to

  account process location. A capability measure that

  do this in Section 16.2]

  does involve the process mean is

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