1. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 16 Quality Control Methods

2. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 16.1 General Comments on Control Charts

3. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Control Charts Control charts provide a mechanism for recognizing situations where assignable causes may be adversely affecting product quality. A basic element is that samples have been selected from the process of interest at a sequence of time points.

4. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 16.2 Control Charts for Process Location

5. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Let X be a rv and assume that for an in-control process, X has a normal distribution with mean and standard deviation Let denote the sample mean for a random sample of size n at a particular time point.

6. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Properties of We know 3. has a normal distribution.

7. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The Chart Suppose that at each of the time points 1,2,3…, a random sample of size n is available. Let denote the corresponding sample means. An chart results from plotting these over time and then drawing horizontal lines at

8. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The Chart LCL UCL Time Any point outside the control limits suggests that the process may have been out of control at that time.

9. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The Chart Based on Estimated Parameters sample standard deviations

10. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Control Limits Based on the Sample Standard Deviations

11. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Recomputing Control Limits Suppose that one of the points on the control chart falls outside the control limits. If an assignable cause can be found, it is recommended that new control limits be calculated after deleting the corresponding sample from the data set.

12. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Performance Characteristics of Control Charts 1. The use of 3-sigma limits makes it highly unlikely that an out-of-control signal will result from an in-control process. 2. When the process is in control, we expect to observe many samples before seeing one whose lies outside the control limits.

13. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Performance Characteristics of Control Charts 3. The chart is effective in detecting large process mean shifts but less effective at quickly identifying small shifts.

14. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Supplemental Rules Take corrective action whenever one of the following is satisfied: 1. 2 out of 3 successive points fall outside 2- sigma limits on the same side of the center. 2. 4 out of 5 successive points fall outside 1 sigma limits on the same side of the center. 3. 8 successive points fall on the same side of the center line.

15. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 16.5 CUSUM Procedures

16. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. CUSUM A defect of the traditional X chart is its inability to detect a relatively small change in a process mean. Cumulative Sum (CUSUM) control charts and procedures are designed to remedy this defect. There are two versions of a CUSUM – one graphical and the other computational.

17. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The V-Mask Let denote a target value or goal for the process mean and define cumulative sums by

18. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The V-Mask The cumulative sums are plotted over time. At time l, we plot a point at height S l. Now a V-shaped mask is superimposed on the plot. At a particular time, the process is judged to be out of control if any of the plotted points lies outside the V-mask.

19. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. CUSUM plot V-mask 0=(r,S r ) In-controlOut-of-control The V-Mask SrSr h d r

20. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Computational Form of the CUSUM Procedure Let d 0 = e 0 = 0, and calculate d 1,d 2,… and e 1,e 2,… recursively using l = 1,2,… If at current time r, either d r > h or e r > h, the process is judged to be out of control. k (slope of lower arm of the V) is customarily taken as

21. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Designing a CUSUM Procedure Let denote the size of a shift in that is to be quickly detected using a CUSUM procedure. Suppose a quality control practitioner specifies desired values of two average run lengths: 1. ARL when the process is in control 2. ARL when the process is out of control because the mean is shifted by

22. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Designing a CUSUM Procedure (continued) A chart, called a nomogram, can then be used to determine values of h and n that achieve the specified ARLs.

23. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Using the Kemp Nomogram 1. Locate the desired ARLs on the in- control and out-of-control scales. Connect these points with a line. 2. Note where the line crosses the scale, and solve for n using the equation Then round n up to the nearest integer.

24. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Using the Kemp Nomogram (continued) 3. Connect the point on the scale with the point on the in-control ARL scale using a second line, and note where this line crosses the scale. Then