Using Control Charts to Monitor
Process and Product Profiles
William H. Woodall and Dan J. Spitzner
Department of Statistics
Virginia Tech
Blacksburg, VA 24061-0439
( bwoodall@vt.edu
and dan.spitzner@vt.edu
)
Douglas C. Montgomery and Shilpa Gupta
Department of Industrial Engineering
Arizona State University
Tempe, AZ 85287-5906
( doug.montgomery@asu.edu
and shilpa.gupta@asu.edu
)
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Basic SPC assumptions:
1. Univariate quality characteristic or multivariate quality vector.
2. Some distributional assumptions, usually normality.
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ij
ij
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There are many examples and applications:
•
Aspartame function, Kang and Albin
(2000)
•
Tonnage stamping, Jin and Shi (2001)
•
Location data, Boeing (1998)
•
Calibration data, Mestek et al. (1994)
•
Vertical density profile data for wood boards, Walker and Wright (2002).
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Monitoring Profiles with Control
Charts
•
General Issues and Pitfalls
•
Linear Profiles
•
Nonlinear Profiles, Wavelets, and Splines
•
Relationships to Other Methods in SPC
•
Ideas for Further Research
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1. Phase I vs. Phase II
Each application and method applies to a particular phase. The goals and the methods of evaluating statistical performance vary by phase.
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Phase I – A set of historical data is available. Interest is on understanding process variation, assessing process stability, and estimating in-control process parameters.
Statistical performance measure: Probability of deciding process is unstable.
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Phase II – Using control limits estimated from Phase I with data as it is obtained successively over time.
Statistical performance measure:
Parameter (usually the mean) of the run length distribution.
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2. Principal Components and
Functional Data
Method of Jones and Rice (1992) is very useful in Phase I to understand profile variation.
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Profile monitoring is an application of functional data analysis, although only classical regression and multivariate ideas have been applied thus far.
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3.
Profile-to-Profile Common Cause
Variation
A basic issue in all applications is the extent to which variation between profiles should be incorporated into the control chart limits.
Pitfall #1: Failing to address this issue.
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4. The Control Chart Statistic(s)
Parametric model: Monitor each parameter with separate chart unless estimators are dependent, then use a T-squared chart.
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To form the T-squared statistics in
Phase I, one should use the estimator of the variance-covariance matrix proposed by Holmes and Mergen
(1993).
See Sullivan and Woodall (1996).
Pitfall #2: Pooling of all vectors in
Phase I.
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If a smoothing method is used, such as spline-fitting, then control charts based on metrics can be used to detect changes in observed profiles from a baseline profile.
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If only a few linear combinations of the Y-variables are monitored for each profile, then some shifts in profiles are undetectable.
Pitfall #3: Monitoring only a subset of principal components or wavelet coefficients determined from incontrol profiles.
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Analysis of Linear Profiles
Phase I: Mestek et al. (1994), Stover and Brill (1998), Kang and Albin
(2000), Kim et al. (2003), Mahmoud and Woodall (2003).
Phase II: Kang and Albin (2000),
Kim et al. (2003).
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Linear Calibration Applications
Croarkin and Varner (1982)
NIST/SEMATECH Engineering
Statistics Handbook www.itl.nist.gov/div898/handbook/
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Polynomial / Multiple Regression:
Jensen, Hui, and Ghare (1984)
Nonlinear Regression: Brill (2001),
Williams et al. (2003).
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Splines: Gardner et al. (1994),
Boeing (1998).
Wavelets: Jin and Shi (1999, 2001),
Lada et al. (2002), Sun et al. (2003).
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Relationships to Other SPC Methods
Multivariate SPC: Related, but dimensionality reduction is needed.
Regression-adjusted (or cause-selecting) charts:
[Hawkins (1991, 1993), Wade and Woodall (1993)]
Related, but more general and data-intensive.
Use of Trend Rules. Only artificially related.
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Much work is needed in profile monitoring. Only the linear profile case has been studied in any detail.
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•
Linear profile case with the X-variable random.
•
Use of generalized linear models.
•
Effect of estimation error.
•
Statistical evaluation of proposed methods.
•
Linear calibration monitoring.
•
Use of more powerful SPC methods.
•
Multiple response variables.
•
Comparisons of competing methods.
• … and many, many more.
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We strongly encourage work and research in the area of profile monitoring.
This framework opens SPC up to a much wider variety of statistical methods, models, and ideas.
It also greatly expands the variety of engineering applications.
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References
1. Ajmani, V. B. (2003). “Using EWMA Control Charts to Monitor Linear
Relationships in Semiconductor Manufacturing.” Paper to be presented at the 47 th
Annual Fall Technical Conference, El Paso, Texas.
2. Boeing Commercial Airplane Group, Materiel Division, Procurement Quality
Assurance Department (1998). Advanced Quality System Tools, AQS D1-9000-1,
The Boeing Company: Seattle, WA.
3. Brill, R. V. (2001). “A Case Study for Control Charting a Product Quality
Measure That is a Continuous Function Over Time”. Presentation at the 45 th
Annual Fall Technical Conference, Toronto, Ontario.
4. Croarkin, C., and Varner, R. (1982). “Measurement Assurance for Dimensional
Measurements on Integrated-Circuit Photomasks.” NBS Technical Note 1164, U.S.
Department of Commerce, Washington, D.C.
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5. Gardner, M. M., Lu, J. –C., Gyurcsik, R. S., Wortman, J. J., Hornung, B. E.,
Heinisch, H. H., Rying, E. A., Rao, S., Davis, J. C., and Mozumder, P. K. (1997).
“Equipment Fault Detection Using Spatial Signatures”. IEEE Transactions on
Components, Packaging, and Manufacturing Technology – Part C, 20, pp. 295-304.
6. Hawkins, D. M. (1991). “Multivariate Quality Control Based on Regression-
Adjusted Variables”. Technometrics 33, pp.61-75.
7. Hawkins, D. M. (1993). “Regression Adjustment for Variables in Multivariate
Quality Control”. Journal of Quality Technology 25, pp. 170-182.
8. Holmes, D. S., and Mergen, A. E. (1993). “Improving the Performance of the T 2
Control Chart”. Quality Engineering 5, pp. 619-625.
9. Jensen, D. R., Hui, Y. V., and Ghare, P.M. (1984). “Monitoring an Input-Output
Model for Production. I. The Control Charts”. Management Science 30, pp. 1197-
1206.
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10.
Jin, J., and Shi, J. (1999). “Feature-Preserving Data Compression of
Stamping Tonnage Information Using Wavelets”. Technometrics 41, pp. 327-339.
11. Jin, J., and Shi, J. (2001). “Automatic Feature Extraction of Waveform
Signals for In-Process Diagnostic Performance Improvement”. Journal of
Intelligent Manufacturing 12, pp. 257-268.
12. Jones, M. C., and Rice, J. A. (1992). “Displaying the Important Features of
Large Collections of Similar Curves”. American Statistician 46, pp. 140-145.
13.
Kang, L., and Albin, S. L. (2000). “On-Line Monitoring When the Process
Yields a Linear Profile”. Journal of Quality Technology 32, pp. 418-426.
14. Kim, K., Mahmoud, M. A., and Woodall, W. H. (2003). “On The Monitoring of Linear Profiles”. To appear in the Journal of Quality Technology.
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15. Lada, E. K., Lu, J. -C., and Wilson, J. R (2002). “A Wavelet-Based Procedure for Process Fault Detection”. IEEE Transactions on Semiconductor Manufacturing
15, pp. 79-90.
16. Mahmoud, M. A., and Woodall, W. H. (2003), “Phase I Monitoring of Linear
Profiles with Calibration Applications”, submitted to Technometrics.
17. Mestek, O., Pavlik, J., and Suchánek, M. (1994). “Multivariate Control Charts:
Control Charts for Calibration Curves”. Fresenius’ Journal of Analytical
Chemistry 350, pp. 344-351.
18. Stover, F. S., and Brill, R. V. (1998). “Statistical Quality Control Applied to Ion
Chromatography Calibrations”. Journal of Chromatography A 804, pp. 37-43.
19. Sullivan, J. H., and Woodall, W. H. (1996), “A Comparison of Multivariate
Quality Control Charts for Individual Observations”. Journal of Quality
Technology 28, pp. 398-408.
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20. Sun, B., Zhou, S., and Shi, J. (2003). “An SPC Monitoring System for Cycle-
Based Process Signals Using Wavelet Transform”. Unpublished manuscript.
21. Wade, M. R., and Woodall, W. H. (1993). “A Review and Analysis of Cause-
Selecting Control Charts”. Journal of Quality Technology 25, pp. 161-169.
22. Walker, E., and Wright, S. P. (2002). “Comparing Curves Using Additive
Models”. Journal of Quality Technology 34, pp. 118-129.
23. Williams, J. D., Woodall, W. H., and Birch, J. B. (2003). “Phase I Monitoring of Nonlinear Profiles”, paper to be presented at the 2003 Quality and Productivity
Research Conference, Yorktown Heights, New York.
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