This paper offers an approach to process capability risk assessment based upon estimating the impact on the fraction nonconforming of a shift in the process mean, variance, or specification limits. The methodology generates a sensitivity measure for process capability that the practitioner may find valuable because it provides an understanding of what the impact of certain changes might have on the fraction nonconforming. Also, this approach has the advantage of providing a more intuitive risk measure making it much easier to understand and communicate to others.

Introduction

Process capability can be defined as the ability of a process to produce products or services that meet the specified requirements [ASQ, 1983] [Duncan, 1986]. But, how can this ability be measured? A reasonable approach might be to try to estimate the probability that the product or service falls within the acceptance region defined by the specification. There are three common methods for generating this estimate:

1. Empirical: Based on direct observation of the number conforming divided by the total number of units produced from the process in a given time interval.

2. Parametric: Based on the assumption that the observed values come from some theoretical distribution. This assumption might be given credibility because the nature of the process may a priori give rise to the theoretical distribution. However, it can also be validated by goodness-of-fit tests. This top-down approach is the classical method used by most practitioners to assess process capability [Flaig, 1996] [Somerville, 1997].

3. Modeling: Based on curve-fitting techniques such as polynomial regression or Johnson curves. This is a bottom-up approach [Pyzdek, 1992] [Farnum, 1996][Chou, 1998].

When analyzing process capability the practitioner seeks to understand the performance of the process and the effect of variation, parameter targeting, and specification changes on product conformance. The risk associated with producing a product or service is the cost of failure (both internal and external) [Bernstein, 1996]. A common belief is that product risk is binary. That is, a product is either good or bad. However, a more realistic appraisal suggests that cost usually increase as the performance characteristic deviates from the specification target. One approach to estimating this risk is the quadratic loss function popularized by Taguchi. Taguchi also suggests a deviation from target based process capability metric -- Cpm [Chan, 1988]. One problem with Cpm is that it has no intuitive interpretation making it hard to explain to practitioners. The second problem is that establishing a relationship between Cpm and the fraction nonconforming requires the application of restrictive assumptions.

The Johnson curve approach will be used in this paper to fit the observed data distribution and then to evaluate the effect of changes in the process parameters and/or specifications on the fraction nonconforming (i.e., tail probabilities) that determine process capability.

Methodology

To determine the effect of a shift in mean on the fraction nonconforming requires that the practitioner either have a great deal of data (in which case, an empirical estimate can be derived) or a model must be used. To make this methodology as simple as possible, a Johnson-curve-fitting approach is recommended. Using a Johnson curve to model the observed distribution allows the practitioner to use the Normal probability distribution to estimate the tail probabilities and the effect on nonconformance rates of changes in the mean or variance.

The procedure for estimating the effect of a mean shift is outlined below:

1. The current left and right tail probabilities are determined.

2. The d shift effect ratio is found by taking the difference of the areas before and after the shift divided by the magnitude of the shift (i.e., Darea /Dx).

3. The shift sensitivity is the limit of the ratio as Dx goes to zero.

Figure 1. Original and Shifted Distribution

The methodology will be illustrated in the example below:

Consider the following distribution that arose as part of a real-world study at a semiconductor equipment manufacturer. Five hundred sixty-nine readings were taken and the distribution formed by these data is presented in Figure 2.

Figure 2. Observed and Normal Distribution

The distribution has a mean of 12.499 and a standard deviation of 1.422; it is roughly symmetrical and highly peaked (as indicated by a kurtosis of 7.42). This distribution is also non-Normal as can be seen by comparing it to the superimposed Normal distribution and this is confirmed by the Shapiro-Wilk’s normality test statistic of .927. If the upper specification limit (USL) = 17.5 and the lower specification limit (LSL) = 10, then the fraction nonconforming expressed in defectives per million (DPM) can be estimated based on the various distribution assumptions. This analysis is given below in the following subsections.

Observed

The amount of product falling outside the specification limits

Based on the observed data is given below:

Proportion of units above the USL = 12,000 DPM

Proportion of units below the LSL = 14,000 DPM

Total fraction nonconforming = 26,000 DPM

Normal

Assuming the observed data are normally distributed, the amount of product falling outside the specification limits is given below:

Proportion of units above the USL = 0 DPM

Proportion of units below the LSL = 40,000 DPM

Total fraction nonconforming = 40,000 DPM

Johnson

Using a Johnson curve to approximate the observed distribution yields the following estimates for the amount of product falling outside the specification limits:

Proportion of units above the USL = 0 DPM

Proportion of units below the LSL = 23,000 DPM

Total fraction nonconforming = 23,000 DPM

However, this paper are not so much concerned with the various estimates of the nonconformance rate as it is with the effect of changes in the distribution parameters and/or specifications on the fraction nonconforming estimates. This is covered in the next section.

Sensitivity Analysis

To demonstrate the effect of a change in the process mean on the tail probabilities, a shift of one-half standard deviation (s/2) will be applied to the semiconductor data presented earlier. (see Table 1)

Table 1. The Effect of a Mean Shift on Tail Probabilities

No Shift

Left Tail Right Tail Total

Observed 14,000 12,000 26,000

Johnson 23,000 0 23,000

Normal 40,000 0 40,000

Right Shift (m₁ = m₀ + s/2)

Left Tail Right Tail Total

Observed 7,000 14,000 21,000

Johnson 7,000 1,000 9,000

Normal 12,000 1,000 14,000

Left Shift (m₁ = m₀ – s/2)

Left Tail Right Tail Total

Observed 67,000 7,000 74,000

Johnson 68,000 0 68,000

Normal 105,000 0 105,000

Further analysis shows that a +s/1.5 shift would result in an estimated nonconformance rate of 10,000 for a Normal approximation and 7,000 for a Johnson. However, the goal is not to find the shift that minimizes the fraction nonconforming but rather to assess the effect of mean shifts on the fraction nonconforming. In this case, the Johnson curve sensitivities for the s/2 shift on the left tail (LT) and right tail (RT) would be computed as follows:

Right Shift

LT(s/2) = (F(0) - F(s/2))/(0 - s/2) = (23,000 - 7,000)/(0 - .711) = - 23,000 DPM/unit x

RT(s/2) = (F(0) - F(s/2))/(0 - s/2) = (0 - 1,000)/(0 - .711) = 1,000 DPM/unit x

Net Avg. Rate of Change = LT + RT = -23,000 + 1,000 = - 22,000 DPM/unit x

Right Shift Net Effect = 7,000 - 23,000 + 1,000 - 0 = - 15,000 DPM

Left Shift

LT (-s/2) = - (F(0) - F(-s/2))/(0 + s/2) = -(23,000 - 68,000)/(0 + .711) = 63,000 DPM/unit x

RT (-s/2) = - (F(0) - F(-s/2))/(0 + s/2) = -(0 - 0)/(0 + .711) = 0 DPM/unit x

Net Avg. Rate of Change = LT + RT = 63,000 + 0 = 63,000 DPM/unit x

Left Shift Net Effect = 68,000 - 23,000 + 0 - 0 = 45,000 DPM

The above figures represent the average rate of change of the area in the tails of the distribution for a given shift in the mean. It can be seen that the distribution is more sensitive to left shifts than it is to right shifts and that the effect of the s/2 right shift reduces the DPM, whereas the left shift can be expected to increase it.

The result can be expressed symbolically for the left tail (LT) and the distribution sensitivity on the left (SL) measured at the lower specification limit (LSL) as follows:

Extending the Method

As can be seen from the above analysis, the shift sensitivities are measures of the average effect of a shift. In other words, they are measures of DF/Dx, where F is the cumulative density function (cdf) of f(x). Therefore, the left and right sensitivity measures could be defined as follows:

Let A_L be the area under the Johnson curve to the left of the LSL, then

is the standard Normal curve, then the Sensitivity on the Left (SL) is

If j(x) is roughly Normal then z = (x – m)/s, which implies dz/dx = 1/s, but in general for

Johnson curves where the k_i are given in the appendix, so

for SU type distributions and

for SB type distributions

Let A_R be the area under the curve to the right of the USL, then

then the Sensitivity on the Right (SR) is

but where the k_i are given in the appendix, so

for SU type distributions and

for SB type distributions

If the observed distribution is approximately Normal, then Net Sensitivity (NS) is given by:

This formula establishes the functional relationship between net sensitivity (NS), and the mean (m), standard deviation (s), upper specification limit (USL), and lower specification limit (LSL) of the process. For example, the true distribution of the semiconductor data is unknown but if we assume that it is approximately Normal, then the left and right sensitivities derived from the Normal curve f(x) are

SL = f(z_L)/s = f(10)/s = .05459/1.4219 defectives/unit x = 38,000 DPM/unit x

SR = –f(z_R)/s = –f(17.5)/s = –.00139/1.4219 defectives/unit x = -1,000 DPM/unit x

and the net sensitivity is:

NS = SL + SR = 37,000 DPM/unit x

This analysis indicates that a “small” shift in the mean will increase (for a left shift) or decrease (for a right shift) the process nonconformance level at a rate of 37,000 DPM per unit change in x. Additionally, the analysis suggests that the process capability is 38 times more sensitive to a change in the LSL than it is to a change in the USL based on the ratio SL/SR.

Sensitivity Model

If the observed process distribution can be approximated by a Normal curve, then the net sensitivity model allows the practitioner to understand the relationship between sensitivity, and the process parameters and specifications. That is, it establishes the functional relationship between net sensitivity, and the process mean, standard deviation, upper and lower specification limits based on the equation

The contour plot of NS for the semiconductor data is given in Figure 3 below:

LSL Target USL

Figure 3. Contour Plot of NS(m, s)

The net sensitivity function has two singularities: +¥ at m = LSL, s = 0; and -¥ when m = USL, s = 0. The practitioner must try to avoid these points. Remember, there are three goals in process optimization: (1) move the mean toward the target (i.e., m ® T); (2) reduce variation (i.e., s ® 0); (3) generate a robust process by minimizing sensitivity (i.e., NS ® 0). Sometimes, these goals cannot be achieved simultaneously and the practitioner must choose, as far as is possible, an acceptable operating region.

In addition, the rate of change in net sensitivity given a change in the standard deviation can be evaluated using the partial derivative of NS with respect to s, which is given by:

For the example data, m = 12.5, USL = 17.5, and LSL = 10, Figure 4 shows the rate of change in NS with respect to the standard deviation.

Figure 4. Rate of Change in Net Sensitivity with Respect to s

This curve tells us that net sensitivity is changing most rapidly when s is about 1.25.

Summary

Sensitivity analysis provides a way of assessing the possible impact of changes in the process distribution parameters or specification limits on process capability. However, the practitioner must exercise caution and good sense. The fact that a theoretical shift in the process mean generates a more capable process does not imply that a real shift will generate the same results. This is one of the common fallacies of capability analysis fostered by calling Cp the “potential” process capability, thus, suggesting that this potential can be realized by simply adjusting the mean. In some cases, this might be true but not in all cases. For example, if the mean and variance are not independent, then shifting the mean in the direction that would presumably reduce nonconformances might actually increase them.

Not withstanding the above concern, it is advisable for the practitioner to generate a trend chart (to assess stability), a histogram (to see what the process looks like), a measure of process capability (i.e., fraction nonconforming), a measure of sensitivity (i.e., NS), and the left and right sensitivities (SL and SR). The sensitivity measures can be used to both assess risk and, as an indicator of the direction in which to move the process mean or shift specifications for maximum improvement and minimum risk.

Appendix

The probability density functions (pdf’s) used in this paper are listed below:

Normal

Johnson

Unbounded (SU type)

Bounded (SB type)

Lognormal (SL type)

References

American Society for Quality, Statistics Division: Glossary and Tables for Statistical Quality Control, 2nd ed., ASQ, Milwaukee, WI.

Bernstein, P. L. (1996). The New Religion of Risk Management. Harvard Business Review, March-April 1996.

Chan, L. K. (1988). A New Measure of Process Capability: Cpm. Journal of Quality Technology, Vol. 20, No. 3, pg. 162-175.

Chou, Y-M., Polansky, A. M. and Mason, R. L. (1998). Transforming Non-Normal Data to Normality in Statistical Process Control. Quality Engineering, Journal of Quality Technology, Vol. 30, No. 2, pg. 133-141.

Duncan, A. J. (1986). Quality Control and Industrial Statistics. Irwin, Homewood, IL.

Farnum, N. R. (1996). Using Johnson Curves to Describe Non-normal Process Data. Quality Engineering, Marcel Dekker, Vol. 9, No. 2, pg. 329-336.

Flaig, J. J. (1996). A New Approach to Process Capability Analysis. Quality Engineering, Marcel Dekker, Vol. 9, No. 2, pg. 205-212.

Pyzdek, T. (1992). Process Capability Analysis Using Personal Computers. Quality Engineering, Marcel Dekker, Vol. 4, No. 3, pg. 419-440.

Somerville, S. E. and Montgomery, D. C. (1997). Process Capability Indices and Non-normal Distributions. Quality Engineering, Marcel Dekker, Vol. 9, No. 2, pg. 305-316.

Additional Reading:

Bissell, A. F. (1990). How Reliable is Your Capability Index. Applied Statistics, Vol. 39,

pg. 331-340.

Boyles, R. (1991). The Taguchi Capability Index. Journal of Quality Technology, Vol. 23, No. 1, pg. 17-26.

Chou, Y. M., Owen, D. B., and Borrego, S. A. (1990). Lower Confidence Limits on Process Capability Indices. Journal of Quality Technology, Vol. 22, No. 3, pg. 223-229.

Electronic Industries Association (1990): EIA Interim Standard IS-32: Assessment of Quality Levels in PPM Using Variables Test Data. Washington, DC.

Electronic Industries Association (1990) Bulletin QB6: Guidelines on the Use and Application of Cpk. Washington, DC.

Gunter, B. (1989). The Use and Abuse of Cpk. Quality Progress, January through September.

Gunter, B. (1991). Process Capability Studies. Quality Progress, Vol. 24, No. 8, pg. 123-132.

Kane, V. E. (1986). Process Capability Indices. Journal of Quality Technology, Vol. 18, No. 1, pg. 41-52.

Kotz, S. and Johnson, N. L. (1993). Process Capability Indices. New York, Chapman and Hall, 1993.

Kushler, R. H. and Hurley, P. (1992). Confidence Bounds for Capability Indices. Journal of Quality Technology, Vol. 24, No. 4, pg. 188-195.

Lewis, S. S. (1991). Process Capability Estimates from Small Samples. Quality Engineering, Marcel Dekker, Vol. 3, No. 3, pg. 381-394.

McNeese, W. H., and Klein, R. A. (1992). Measurement Systems, Sampling, and Process Capability. Quality Engineering, Marcel Dekker, Vol. 4, No. 1, pg. 21-40.

Pearn, W. L., Kotz, S. and Johnson, N. L. (1992). Distributional and Inferential Properties of Process Capability Indices. Journal of Quality Technology, Vol. 24, No. 4, pg. 216-231.

Price, B. and Price, K. (1993). A Methodology to Estimate the Sampling Variability of the Capability Index Cpk. Quality Engineering, Vol. 5, No. 4, pg. 527-544.

Rodriguez, R. N. (1992). Recent Developments in Process Capability Analysis. Journal of Quality Technology, Vol. 24, No. 4, pg. 176-187.

Runger, G. C. (1993). Designing Process Capability Studies. Quality Progress, Vol. 26, No. 7, pg. 31-34.

Key Words

Process capability; Sensitivity analysis; Fraction nonconforming

Authors Biography

John J. Flaig is managing director of Applied Technology a consulting and software publishing company. Dr. Flaig’s special interests are in statistical process control, process capability analysis, supplier management, design of experiments, and process optimization. He holds a doctorate in engineering and technology management from Southern California University, a master’s degree in mathematics from the University of California, and a bachelor’s degree in mathematics and economics from California State Polytechnic University. He is a senior member of the ASQ.

Editor:

Paper Reference Number: 1659

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