Process Variation and Capability Assessment
Competitive companies understand that satisfying customers' needs and running a successful operation requires a system that is dependable, predictable and consistent. Many of todays quality management philosophies, such as Six Sigma, focus on reducing variations and scrap. A modern view of quality defines it as the inverse of variability. Therefore, measuring variability and finding ways to reduce it is important for the success of a company. This article presents a Weibull analysis approach to modeling production data for a process in order to assess its deviation and capabilities. This approach augments the Six Sigma-type calculations, which typically rely on the normality assumption.
We have discussed in past issues how process availability and throughput can be studied through the modeling of the process (for example, see Issue 28s Reliability Basics article). This "in-depth" way of assessing a process yield can be complemented by a "higher view" approach that relies on using actual periodical production data of the system. This is a quick and useful way to assess the "health" of a process and it is a stepping stone for getting the process as close to the target as possible.
The next figure compares production volumes of two processes, a healthy one and a troubled one.
Figure 1 Two Different Processes
In Six Sigma, variations are typically modeled with the bell-curve normal distribution. Oftentimes, processes and products are not "well-behaved" and their distributions are typically skewed. The skewness is generally more apparent when the variations are significant. Therefore, Six Sigma calculations have been criticized for relying heavily on the normality assumption.
In this article, we will focus on the Weibull distribution as our distribution model because of its flexibility and generally skewed pdf shape, as shown in Figure 2; however, other types of distributions (such as the lognormal, Gumbel and logistic distributions) might be appropriate depending on the types of variation.
Figure 2 Normal and Weibull pdf Plots Comparison
The following items are some of the key characteristics to look at when studying the probability plots of the process. These characteristics can help you discover signs of an unhealthy process.
Figure 3 Process Showing Significant Production Losses
Figure 4 Probability Plot for Processes with Different Variations Level
Figure 5 pdf Plot for Processes with Different Variations Level
Figure 6 Process Showing Mixed Variations Behavior
Figure 7 Process Showing Mixed Variations Behavior Modeled with a Mixed Weibull Distribution
Figure 8 Process Sample Production Data
The above graph shows that the process is off-target. It also does not follow the normal distribution, but rather a Weibull distribution with β=26.01. This indicates that the process suffers from significant variations.
Metrics such as Cp, Cpk and Cpm are common Six Sigma metrics used to assess the capabilities of a process. A great deal of importance is often assigned to these metrics, but they can be flawed because they assume that a process follows the normal distribution. Although some remedies have been suggested, such as using data transformation methods or modifying the way the process capability measures are calculated , these remedies remain unrobust and may complicate the analysis. We suggest using probability calculations, as they are straightforward and do not violate the assumptions of the chosen distribution. For this example, the percentage of daily productions that met the goal of 975 tons to 1025 tons can be found using the following probability calculations:
P(975<p<1025) = P(p<1025) P(p<975)
where p is a random variable that describes daily production. Calculating P(p<a) in Weibull++ means calculating the probability of failure by a. P(p<1025) and P(p<975) are found using Weibull++s QCP as follows.
Therefore, P(975<p<1025) = P(p<1025) P(p<975) = 0.998654 0.834599 = 0.164055 or 16.4055%. In other words, 83.5945% of the days in the sample did not meet the production requirements.
It is important to convert your statistical analysis into money figures, which will have more impact on management and are more likely to motivate actions. Think about defects cost, scrapping cost, rework cost, penalties, lost revenue, customer dissatisfaction, etc. For this example, the manufacturer is penalized by its customer $1000 for every day of insufficient production and $500 for every day of excessive production that requires storage. For a one year period, the penalties amount to:
Penalties = 365.[P(p<975).$1000 + P(p>1025).$500]
Which is equivalent to:
Penalties = 365.[P(p<975).$1000 + (1-P(p<1025)).$500]
Penalties = 365.[0.834599.$1000 + (1-0.998654).$500]= $304,874.3
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