Process Control, or Chaos?

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Quality Engineering, 12(1), 1-6 (1999-2000)

PROCESS CONTROL, OR CHAOS? Bruno-Marie BCchard, Raynald Gauvin, and Nico Pelletier Dkpartement de gtnie mkcanique Universitk de Sherbrooke Sherbrooke, Qutbec, Canada, J l K 2R1

Key Words Statistical process control; Control chart; Trends; Trend analysis rules; Western Electric; Non-Normal distributions; Chaos; Fractals; Fractal geometry; Fractal dimension.

Introduction Statistical Process Control (SPC) techniques are widely used to monitor the variability of important quality characteristics in order to identify and react to undesirable process behavior before out-of-tolerance products are manufactured. However, the use of traditional trend analysis rules (1) is often inadequate for real industrial situations when distributions are not perfectly Normal, or when variability is not evenly distributed between and within samples (2). A possible solution for solving this problem consists of elaborating a new trend analysis technique based on a completely different approach. The latest developments in the theories on chaos and fractals, especially the fractal dimension, introduce promising new tools for this purpose.

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that most of these rules require the measurements to be normally distributed to function properly. Even slight departures from this ideal situation often cause false alarms (2). In these cases, a possible solution is to use samples of size larger than 4 to benefit from the Central Limit Theorem (3). This does not solve all situations, however. For example, in several industrial situations, measurements are taken from batch processes, where each sample represents a different lot. In these cases, more variation sources influence the measurements between lots than within lots, and the use of traditional techniques for establishing control limits and analyzing trends also cause frequent false alarms. A partial solution to these situtations consists of using universal control limits instead of traditional control limits (2), but the traditional trend analysis rules remain inadequate. All of these false alarms provoke unneeded process interruptions for the search of a nonexistent root cause for the problem and can even induce inappropriate process adjustments. In these conditions, SPC becomes misleading and ineffective, and the mistrust and discouragement felt by operators often cause the abandonment of its use.

Chaos and Fractal Dimension Weaknesses of Traditional Trend Analysis Rules . In real industrial situations, the use of control charts with traditional trend analysis rules often poses problems of frequent false alarms. This phenomenon is partly due to the fact

Copyright 8 1999 by Marcel Dekker, Inc.

The theories on chaos and fractals introduce several new mathematical concepts (4). Among them, the l'ractal dimension D quantifies the irregularity of any curve or surface. A straight line, a plane, and a volume have dimensions of 1.2, and 3, respectively. A rough curve covers part of a plane, so

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B~CHARD,GAUVIN, AND PELLETlER

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D = 2, - lim log(C0 - Ci) its dimension lies between 1 and 2. The rougher the curve, the closer its climension gets to 2. When the dimension is not an integer, it is called fractal. The same concept applies to a rough surface, which has a fractal dimension between 2 and 3. Figure 1 illustrates curves with different fractal dimensions: D = 1 .OO (lower curve), D = 1.05, D = 1.50, and D = 1.90 (upper curve).

Elaboration of the New Trend Analysis Technique The new trend analysis technique consists of calculating the fractal dimension of a portion of the curve generated by the measurements of the quality characteristic observed, and comparing the result with the fractal dimensions obtained when the process is considered to behave in an undisturbed satisfactory fashion. This approach of comparing the process behavior to its usual or natural behavior is radically different than the traditional trend analysis rules, which consist of comparing the process behavior to a statistically ideal situation based on a random variable drawn from a Normal distribution with evenly distributed variability. Because several methods exist to calculate the fractal dimension, it is imperative to choose methods that can reveal changes in the central tendency (mean) and in the vrtriabilily (standard deviation) of the quality characteristic measurements.

Detection of Changes in the Central Tendency (Mean) To detect changes in the mean, the autocorrelation method (5) is chosen for the calculation of the fractal dimensions. This method gives the best results (sensitivity and accuracy) in the different simulations performed. It is based on the autocomelation value of the quality characteristic measurements. The fractal dimension D is given by applying the following equations to the centered-reduced measurements x (mean of 0 and standard deviation of I):

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where slope, is the slope of the least squares straight line that best fits the graph of log(C, - Ci) versus log i, i being the sequential number of the measurement xi considered. For the trend analysis technique to be dynamic, the calculation'of the fractal dimension uses a limited number of past measurements. The number of measurements to use depends on the amplitude of the change in the mean that must be detected. The empirical equation

determines the number of measurements n, verhs the change in the mean to be detected A x , expressed in number of standard deviations a (6).The fractal dimension thus allows one to track the process behavior over time by observing the evolution of the results calculated on the curve generated by a constant number of measurements. An analysis of the frequency spectrum of the fractal di:mensions,using the fast Fourier transform (FFT) (7), further optimizes the detection of changes in the mean. Indeed, such changes particularly affect the frequencies close to the frequency associated to the number of measurements used for the calculation of the fractal dimensions. The sum of the amplitudes of the two closest frequencies can be plotted on a graph to allow a second glance at the process, which increases the reliability of the technique. For the examples in this article, the control limits on all charts based on fractal dimensions have been set to the extreme values obtained during the first half of the graph, where the process is considered to behave in an undisturbed satisfactory fashion. This rudimentary method for establishing the control limits for the trend analysis charts could be refined to aim at a specific type 1 or type 2 error, as is the case for all basic control charts. '

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PROCESS CONTROL,OR CHAOS?

Detection of Changes in the Variability (Standard Deviation) Changes in the standard deviation are another important tendency that must be detected by SPC. The method chosen for calculating fractal dimensions is the sausage dimension method proposed by Minkowski (8). It consists of centering circles of variable radii on each point of the structure being analyzed. The fractal dimension is linked to the total area of the sausage defined by all the circles versus their radii. The relation

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where Au is the minimum change in the standard deviation to be detected, expressed in number of standard deviations u (6). Again, an analysis of the frequency spectrum of the fractal dimensions, using the FFT, is performed to improve the detection of changes in the standard deviation. In this case, the lower frequencies react to variations in the fractal dimension. Hence, the sum of the amplitudes corresponding to the two lowest frequencies may be plotted on a graph to increase the certainty of the diagnostic.

Application Example of the New Technique in an Ideal Situation A numerical example illustrates the response of the technique developed using fractal dimensions when changes in the mean and standard deviation occur. Figure 2 shows 1200 measurements normally distributed with a mean of 500 and a standard deviation of 10. A change in the mean of - 12 is simulated between measurements number 601 and 800, and the standard deviation is increased by 60% between measurements number 1001 and 1200. This example shows that the new trend analysis technique

Figure 2.

Application example of the new technique.

allows a quick and obvious detection of the change in the mean with graphs b and c. As mentioned earlier, the analysis of the frequency spectrum of the fractal dimensions increases the reliability of the technique. The change in the standard deviation is also detected very efficiently by graphs d and e. Again, the analysis of the frequency spectrum of the fractal dimensions provides more certainty in !he diagnostic. Globally, the new trend analysis technique gives excellent results in this ideal situation and would allow efficient control of industrial processes showing similar behaviors.

Comparison of Techniques for a Realistic Nonideal Situation The following example is based on measurements distributed normally, with an arbitrary mean of 0 and a standard deviation of 1. To simulate the between-lot supplementary variation of the batch effect, the measurements from each sample are modified by adding a value taken at random from

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BGCHARD, GAUVIN, AND PELLETIER

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another Normal distribution of mean 0 and standard deviation of 1. Again, measurements number 1-600 represent the process considered to behave in an undisturbed satisfactory fashion. All control limits calculations are based on this first half of the graph. To simulate a change in the mean, a value of 1 is subtracted from each measurement between numbers 601 and 800. Also, the standard deviation is multiplied by 1.5 between measurements number 1001 and 1200. Figure 3 presents the results obtained with an ordinary X and Sx control chart (9), with an P control chart using universal control limits (2), and with the new trend analysis technique. The control limits for the ordinary X and Sx control chart are established using the following equations:

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In this example, the use of an ordinary Y and Sx control chart with traditional trend analysis rules is inappropriatedue to the high false alarms rate, as shown on graphs b and c of Fig. 3. Effectively, almost all plotted points of all zones cause alarms, especially on the f chart. The I chart with universal control limits, on graph d, eliminates most false alarms, but the detection of the simulated sudden changes is poor. The new trend analysis technique is illustrated on graphs e-h of Fig. 3. It causes no false alarms and detects the change in the standard deviation accurately, but it is slow to detect the change in the mean. As illustrated by this simple example, the new technique eliminates false alarms in nonideal situations and provides a better detection of changes in the standard deviation. Given the negative impact of false alarms (lost production time, inappropriate process adustments, mistrust and abandonment of control charting, etc.), the new trend analysis technique seems advantageous. Also, considering that a better detection of changes in the standard deviation helps to concentrate the efforts of searching for new root causes of variations only when they really exist, the new technique appears to lead to better process improvement. This is particularly true for the simulated batch process, although the new trend analysis technique also gives excellent results for the ideal situation of the first example.

Case Study

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With the collaboration of General Electric Canada, Aircraft Engines Division, a case study has been conducted to verify the pelformance of the new trend analysis technique in a real industrial situation. A batch precision grinding process has been selected for the study, because all previous attempts to apply different SPC techniques have been abandoned because of their poor ability to detect problems and/ or the important false alarm rates. Over a typical period of production, two key characteristics have been measured for 576 parts (3 per batch for 192 batches). The measurements have been analyzed using an individuals and moving-range control chart, an ordinary E and Sx control chart (9), an f control chart using universal control limits (2), and the new trend analysis technique. For every out-of-control alarm, a root cause investigation was conducted. In all situations, the new trend analysis technique was.

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PROCESS CONTROL, OR CHAOS?

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preferred. Every out-of-control situation explained by the finding of a root cause has been detected, and false alarm rates have been reduced by more than 90% compared to using traditional trend analysis rules with the individuals and moving-range control chart, or with the ordinary F and S, control chart. When the same control charts were used without the traditional trend analysis rules, they typically did not detect 70% of the problems, while still causing six times more false alarms than the new trend analysis technique. The performance of the F control chart using universal control limits was also interesting, with 30% less false alarms than the ordinary f control chart, and with only a slight drop of detection performance. In this case study, however, the improved false alarm rates were still too high to recommend its application.. When considered globally, the control charts using the new trend analysis technique clearly reveal problematic zones with almost all of the out-of-control alarms concentrated in these areas. This is singularly different than all other control charts, using traditional trend analysis rules or not, where out-of-control alarms are spread all over, causing root cause investigations repeatedly with little success. Figure 4 illustrates'this difference for a typical situation where only five problems were found after investigation. The ease and accuracy of the diagnostic are therefore significantly improved by using the new trend analysis technique. Surpris-

ingly, in most cases, the diagnostic was not delayed compared to traditional methods: An alarm was even signaled earlier in some cases. Also, the case study showed that the more complicated analysis using FFT did not significantly improve the diagnostic.

Conclusion The latest developments in the theories on chaos and fractals introduce several promising concepcs for industrial applications of SPC. Capable of measuring the irregularity of a curve, the fractal dimension can be used to improve the analysis of trends on control charts. The new trend analysis technique presented in this article can efficiently detect changes in the mean and standard deviation of the measurements of a quality characteristic and is especially preferable to the traditional trend analysis rules in situiitions such as typical batch processes. This was proved by an extensive case study conducted on two key characteristics generated by a batch precision grinding process at General Electric Canada, Aircraft Engines Division. It is obvious that the complexity of the new technique dictates the use of a computer to perform the necessary calculations. Once programmed, however, the new technique poses no difficulty of use in an industrial environment. This breakthrough in the development of trend analysis techniques is &e first exploration of these possibilities, and refinements of the new technique will certainly improve its performance further, Finally, this research also opens the door to the control of industrial processes with chaotic behavior, notably by using other concepts such as the Feigenbaum diagram.

Acknowledgments This research has been made possible by the financial support of Bombardier Inc. (industrial research grant) and the UniversitC de Sherbrooke (institutional grant), and by the collaboration of General Electric Canada, Aircraft Engines Division.

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References

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Western Electric, Statistical Quality Control Handbook. Western Electric Corporation, Indianapolis, IN, 1956. 2. BCchard, B.-M., Universal Control Limits forF-Charts,Quality 1:

and Its Applications, Penshaw Press, 1993, pp. 559-604. 3. Montgomery, D. C., Introduction to Sratistical Quality Control, 2nd ed., John Wiley & Sons, New York, 1991.

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B ~ C H A R DGAUVJN, , AND PELLETIER 4.

Peitgen, Jiirgens, Saupe, Maletsky, Perciante, and Yunke,

Fractals for rhe Classroom, Strategic Activities Volumes One and Two, Springer-Verlag, New York, 1993. 5 . Falconer, K., Fractal Geometry, Mathemtical Foundations and Applications, John Wiley & Sons, New York, 1990. 6. Pelletier, N., Application de la thJorie du chaos et des fractales au conrr6le statistique des prockdks, Mtmoire de maitrise es

sciences appliquks, Vol. JV, UniversitC de Sherbrooke, Sherbrooke. Quebec, 1995. 7. Bendat, J. S. and Piersol, A. G., Random Data: Analysis and Measurement Procedures, 2nd ed., John Wiley & Sons, New York, 1986. 8. Gouyet, J.-F., Physique et structuresfractales, Masson, Paris. 1992. and Larose, P., Initiation au gknie-qualitP, 2nd 9. Rfchard, B.-M. ed., Les tditions Ptdagogik, st-glie-d'orford, Qutbec, 1996: About the Authors: Bruno-Marie Btchard, P. Eng., M.A.Sc.,

is the professor responsible for education and research in quality engineering at the Dtpartement de gtnie mkcanique, Universitt de Sherbrooke, Sherbrooke, Qutbec, Canada. He

is a Professional Engineer, with 8 years of industrial experience. Most of his research is made in collaboration with the industry and focuses on process control and monitoring, characterization of natural variability, quality control, accep.tame sampling, quality assurance, and reliability. Professor BCchard participates in different advisory committees (Bureau de normalisation du QuCbec, Semaine de la qualitt) and is the Vice-President for Strategic Alliances of the Association qutbtcoise de la qualitt and member of the Board for the Mouvement qutbtcois de la qualitC. Raynald Gauvin, P.Eng., Ph.D., is a professor in materials at the Dtpartment de g6nie mtcanique, Universitt de Sherbrooke, Sherbrooke, Qutbec, Canada. He is a Professional Engineer and his research domain is the application of the theories on chaos and fractals to the study of materials. Nico Pelletier, P.Eng., M.A.Sc., is the Quality Manager for H. Fontaine, Magog, Qutbec, Canada. He is a Professional Engineer and studied the topic of this article for his research.