Statistical Process Control. Process (Machine) Capability

In section Statistical Process Control – Control Charts we investigated how wide a process is performing, what is its variation. In other words, we investigated whether there exist a control or not for that process.

Establishing that a process is under control, a normal question arises: on a long term, how measured data fit customer requirements? Here we face the specification limits, those values which are acceptable for the client. How we will do that? By using process capability studies. Please note that it makes no sense to investigate process capability as long as the process in not under statistical control. The existence of the control over the process is a fundamental assumption for process capability study.

Before performing process capability analyze, it's important to determine the normality of data. All studies of process capability rely on the assumption that data are normally distributed, because the estimation of nonconforming fraction from population is determined using standard normal tables. If analyzed data does not follow a normal distribution, there is the risk of overestimating or underestimating the proportion of nonconformities. After all, it's nothing wrong in overestimating the proportion of nonconformities. The worst scenario is the underestimation of it, resulting in more nonconformities that predicted. In other words, capability study indicates a capable process, when in fact is not capable.

Normality of data can be determined in several ways, described in many statistical books. Usual methods for determining the normality are Darling – Anderson test, chi-squared goodness-of-fit, kolmogorov-Smirnov (pretty similar to Darling-Anderson test), normal probability plot, and with a little more precaution, histograms. The basic of normality is the following: plotting the data on a normal probability graph, those data should fit a straight line. If plotted points do not follow the line, then you cannot assume that data is normally distributed. More investigation should be carried, in order to determine if data comes from a non-parametric distribution (binomial, Weibull, etc.). Where applicable, data may be transformed using a Box-Cox transformation, but this has to be done so that to make sense (square root of data, inverse of data). A coefficient of let's say 2.35 has no practical utility, being intended only for theoretical use.

Making as input data those from Control Charts, example one, the Darling-Anderson test value is 0.237 and the p value is 0.786. Assuming the alpha level of 0.05 (which is common practice), and observing that p value is greater than selected alpha level, we can conclude that data is normally distributed. A representation of data distribution is shown below.

Passing these two mandatory conditions for given example (process under control and normal distribution of data), we can determine the process capability. First of all, we must be aware of specification limits, imposed by the customer. For those precision tubes, lower specification limit (LSL) set by the client is 99.00 and upper specification limit (USL) is 101.00. What customer ideally wants is a tube with outer diameter of 100.

In a simplified method of computing capability indices, we will have to correct the standard deviation, because the number of observation is less than 60. If you are using Minitab or other equivalent software products, all computation is performed automatically, in a more complex manner. For given set of data, standard deviation (sigma) is 0.1991. Needed correction is sigma/c4, where c4 is 0.9823. Corrected sigma is 0.2027.

First capability index is Cp, which represent the potential performance of the process, by dividing the tolerance interval to natural process spread. The relation is Cp = (USL – LSL)/(6 sigmas). Remember that sigma is that which was corrected with c4 coefficient. The result is Cp = 1.64. As a conclusion, the process is potentially capable of performing at a level of 1.64.

The actual performance of the process is computed by selecting the minimum between CpU and CpL, which are capability indices which take in consideration the physical limits imposed. Cpu and CpL are determined as follow: CpU = (USL – mean)/(3 sigmas) and CpL = (mean – LSL)/(3 sigmas). Doing the computation is resulting CpU = 1.81 and CpL = 1.48. Minimum between these two values is 1.48, which means that Cpk = 1.48. This is the real (actual performance of the process).

From where comes the difference between potential and actual performance of the process? The answer is simple. Cp does not take in consideration the mean of the process, which is a real thing. In other words, Cp "believes" that the process is perfectly centered, while Cpk makes a difference when sensing the mean of the process in correlation with specification limits.

One more thing... The customer wants ideally to get those tubes with the outer diameter of 100. This is the target T. We can compute the capability indices for the target, using the formula Cpm = Cp/(1+((mean-T)**2)/(sigma**2)). Doing the computation is resulting Cpm = 1.48.

What is the significance of these indices? Indices are correlated with number of defects! The following table gives these correlations:

Cp	Capability
0.50	86.64%
0.62	93.50%
0.68	96.00%
0.75	97.50%
0.81	98.50%
0.86	99.00%
0.91	99.35%
1.00	99.73%
1.33	99.964%

If a capability index is 1.33, the capability of the process is 99.964%; is expected that 0.036% of the process to produce nonconforming products. This is equivalent of producing 36 nonconforming products within one million products produced. Many industries works with a value of capability index of 1.33 as a minimum requirement. Top companies are using minimum level set to 1.66 or even 2.

What would happen if customer requirements are tighter than those initially imposed? Let's assume that the client wants a specification range of 99.5 … 100.5. Redoing computations, results are: Cp = 0.82, Cpk = 0.66 and Cpm = 0.74. A drastic reduction in process performance is observed, resulting approximately 26,100 nonconforming products from a million produced. Analyzing the chart is observed that each tail of the distribution curve fall outside specification limits (more on the left tail, corresponding to lower specification limit). What would be the solution for complying with customer requirements? They will have to improve the process, in terms of narrowing the variation of data near the target specification. Yes, this will imply some costs, possible in terms of equipment, training, process technology, etc.

What initially appears to be a loss in business is turning to be a gold mine. As an example, if a tube is sold with a price of 15$, at a monthly production of 700,000 pieces and investment in improvements costs 200,000$, here is the saving:

• number of defective products for given production is 700,000 x 26,100 / 1,000,000 = 18,270, with a total cost of 274,050$;
• cost of investment: 200,000$;
• after the investment, in the first month, the profit is 74,050$;
• in next months, savings are turned to the equivalent in bad defects, prior the implementation of improvement.

Note: assumption is that investment in improvement will lead to zero defects. However, this is not possible all the time. Just think that investment in improvement will make a saving of 25%. On a month, this means 274,050 x 25% = 68,512$. On a year, savings are 822,144$. Clever stuff, yeah?

Entire previous discussion was for process capability, where, common practice imply the use of 6 sigmas. For machinery capability, common practice impose an 8 sigmas. The way of performing the studies is pretty similar to process capability.