Skip to main content
main-content

Über dieses Buch

Cumulative sum (CUSUM) control charting is a valuable tool for detecting and diagnosing persistent shifts in series of readings. It is used in traditional statistical process control (SPC) settings such as manufacturing, but is also effective in settings as diverse as personnel management, econometrics, and conventional data analysis. It is an essential tool for the quality professional. This book covers CUSUMs from an application-oriented viewpoint, while also providing the essential theoretical underpinning. It is accessible to anyone with a basic statistical training, and is aimed at quality practitioners, teachers and students of quality methodologies, and people interested in analysis of time-ordered data. The text is supported by a Web site containing CUSUM software and data sets. Douglas M. Hawkins is Chair of the Department of Applied Statistics, University of Minnesota. He is a Fellow of the American Statistical Association, a Member of the International Statistical Institute and a Senior member of the American Society for Quality Control. His work on multivariate CUSUMs won him the Ellis R. Ott Award for the best paper on quality published in 1993. He has been Associate Editor of Technometrics and Journal of the American Statistical Association. David H. Olwell is Associate Professor in the Department of Mathematical Sciences at the United States Military Academy. He is a member of the American Statistical Association, the American Society for Quality Control, and the Military Operations Research Society, where his work on applications of CUSUMs to managing sexual harassment was nominated for the 1998 Barchi prize. He is Editor of Mathematica

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract
Suppose you have some “process” whose ongoing quality you want to assess. You do this by making regular readings on some measurable property of the process. Some examples might be:
  • You run a sugar packaging plant where a continuous filler line fills paper bags with sugar. Each bag is supposed to contain 10 pounds of sugar. Although the inevitable random variability makes a constant weight in all bags impossible, you would like to check that there is no excessive variability from one bag to another, and that the average weight of all bags is correct. To achieve this, you take random samples of the production from each shift and weigh the sugar in each sampled bag accurately. These weights are your process measurements.
  • You run the emergency room in a hospital. There is some concern about the time taken with the paperwork admitting accident victims. To check this, you have an observer watch a random sample of incoming accident victims and see how long it takes to fill out the forms for each of them. It is clearly important to have a sample that represents all different times of day and different days of the week, so you ensure that your sample correctly represents all these different time periods.
Douglas M. Hawkins, David H. Olwell

2. CUSUM design

Abstract
In Chapter 1, we illustrated the major properties of the CUSUM: its descriptive properties, its ability to signal persistent shifts, even if these are quite modest, and its diagnosis using either the V-mask or the equivalent decision interval form. In this chapter, we look more closely at the design of the CUSUM — that is, at the choice of the parameters k and h that define the decision interval scheme or the equivalent V-mask.
Douglas M. Hawkins, David H. Olwell

3. More about normal data

Abstract
The most common application of CUSUMs is controlling the mean of a stream of process data modeled by the normal distribution. The last two chapters illustrated the CUSUM for the mean of normal data; this chapter gives more detail. We look at extensions to the basic method, covering Fast Initial Response CUSUMs, CUSUMs for variance shifts, weighted CUSUMs, and combined Shewhart/CUSUM charting. We also examine the effects of model departures on CUSUM schemes. We begin with a closer look at the mechanics of determining ARLs.
Douglas M. Hawkins, David H. Olwell

4. Other continuous distributions

Abstract
Until this chapter, we have looked almost exclusively at CUSUM schemes for the normal distribution. We now take a detailed look at two other continuous members of the exponential family of distributions (the gamma and the inverse Gaussian) and their CUSUMs. We find that the design of CUSUMs for these distributions differs from the normal distribution; in particular, and sometimes the optimal CUSUM involves CUSUMming a transformed variable.
Douglas M. Hawkins, David H. Olwell

5. Discrete data

Abstract
The data we have been considering so far were on the continuous scale. In classical SPC terminology they are “variables” measures. The other classical SPC data type is “attribute” data, originally counts of the number of good and of defective (or in current terminology “conforming” and “nonconforming ” items) in a sample. We draw the distinction a little differently: into the standard statistical distinction between continuous and discrete measurements.
Douglas M. Hawkins, David H. Olwell

6. Theoretical foundations of the CUSUM

Abstract
This chapter gives the technical underpinnings of the rest of the book. It may be used as a reference as necessary, and skipped without loss of continuity.
Douglas M. Hawkins, David H. Olwell

7. Calibration and short runs

Abstract
Up to now, we have not said much about the estimation of the in-control parameters, except to point out that large samples are necessary to estimate them adequately. To get some feeling for this, we try to quantify the impact of uncertainty in the parameter estimates in the case of a normal CUSUM. Suppose that the process stream is N (μ, σ2). To calibrate the CUSUM, we take a sample ofsize m and compute its mean X and standard deviation s, and substitute X for μ and s for σ. This means that our figure for the true mean is in error by the amount μ — X, and the figure for the true standard deviation is in error by the ratio σ /s.
Douglas M. Hawkins, David H. Olwell

8. Multivariate data

Abstract
Many processes involve multiple process measurements. These can be of several types, as the following examples illustrate.
  • Different properties may be measured on each unit produced. In manufacturing roller bearings, for example, we might measure the length, maximum diameter, and minimum diameter of each sampled bearing.
  • In process control, we often see a number of different but connected processes. The measurements made on the different processes may then be related to each other. For example, in semiconductor wafer fabrication, chips go through sequences of processing steps. The quality of a chip at one stage depends, not only on this most recent process step performed on it, but also on the outcomes of all the earlier processing steps. To look at the measures on one step in isolation from those on previous steps then is a bad idea. The product at some stage may be bad, not because of problems in that stage, but because of problems created in earlier stages. Not recognizing this possibility can lead to tinkering with a process that is working perfectly well, which wastes time and resources, and is also likely to make quality worse rather than better.
  • A related problem has some of the process steps located outside the operation being controlled. For example, in a coal washing plant, the yield and ash content of the washed coal are important measures of the quality of the product and the effectiveness of the washing plant. Both however are strongly affected by the quality of the raw coal entering the plant, and this is something outside the control of the washing plant. Still the plant should measure these properties of the incoming coal so that the yield and washed ash figures can be “adjusted for” the quality of the coal supplied. Making the adjustment leads to more sensitive SPC controls and to more accurate recognition of when there are and are not process problems.
Douglas M. Hawkins, David H. Olwell

9. Special topics

Abstract
We conclude this monograph by discussing three special topics.
Douglas M. Hawkins, David H. Olwell

10. Software

Abstract
Software for CUSUM design calculations is available from many sources. Yashchin (1986) describes an APL2 package with the capability of solving many design and ARL problems. At the time of this writing, this software is available on request from the author (yashchi@watson.ibm.com) and is scheduled to be put onto a Web site.
Douglas M. Hawkins, David H. Olwell

11. References

Abstract
Software for CUSUM design calculations is available from many sources. Yashchin (1986) describes an APL2 package with the capability of solving many design and ARL problems. At the time of this writing, this software is available on request from the author (yashchi@watson.ibm.com) and is scheduled to be put onto a Web site.
Douglas M. Hawkins, David H. Olwell

Backmatter

Weitere Informationen