Statistical Thinking to Improve Quality

Histogram – Stratification.   The posting on 3/25/2008 describes statistical thinking by a team at Ricoh’s Numazu plant.   The plant makes raw material used as ingredients for copy machine toner.  The team wanted to reduce variation in output quantity which indicated a lack of control of the underlying process.   After removing a special cause, the team constructed a histogram of the output quantity.   The histogram clearly displayed excessive variation and two peaks.   The process flow chart showed a split after phase 2 into 2 separate lines, i.e., line A and line B.   Separate histograms for the two lines showed the output from line B was consistently lower that line A.  Constructing separate histograms for the two lines illustrates stratification by line.  Next, the team conducted a brainstorming session to formulate their collective thinking about the causes of excessive variation and the differences between the two lines.   They documented the results with a cause and effect diagram.   The brainstorming session and the construction of a cause and effect diagram illustrate step 7, Study Cause & Effect.

Stratification requires identifying a stratification factor, such as time of the day, and the partitioning of this factor into logical categories.   What tools may we use to aid in the selection of a stratification factor?    The team in the example above noticed two peaks in a histogram.   Breyfogle (2003) provides some guidance for this question.

1. On page 220, Breyfogle states that patterns on a control chart may suggest the need for stratification.   A sequence of points with small up and down variation relative to the control limits may suggest that the sequence of points comes from a single strata.   The opposite situation where a sequence of points that do not have values near the center line may indicate the combination of two strata.
   
2. On page 385, Breyfogle suggests dividing the data into categories based on posing basic questions such as who, what, when and where.
   

Disaggregation may be aided by constructing a process map such as the one used in the posting on 2/21/08.    The process map (Breyfogle, 2003, p. 103) is a flowchart with key process input variables listed for each step in the process.

1.     Breyfogle, F. W. (2003). Implementing Six Sigma. Hoboken, New Jersey, John Wiley & Sons, Inc.

Posted by Gordon M. Clark at 07:48 PM | Permalink | Comments (0)

This posting gives two additional examples illustrating the Analyze Common-Cause Variation step, step 6, in the Hoerl-Snee process improvement strategy.   Refer to the posting on 5/18/2008 for a description of this step.   Both examples include disaggregation as a tool.

·        Disaggregation – Stratification.  The posting on 2/18/2008 describes statistical thinking by a Midwest manufacturing firm to reduce waiting times by customers.   The company’s goal was to have 95% of incoming customer calls answered by a customer service representative in less than 2 minutes.   Based on a process flowchart, team collected service time data for each step in the process.   That is disaggregation.   The team also collected data for estimating the distribution of incoming calls by time of the day.   That is stratification by the time of day.  They used these data as inputs to a simulation of the call answering process.  They used the simulation construct staffing levels by the hour of the day.   The construction and use of the simulation illustrates step 7, Study Cause & Effect.
·        Disaggregation – Regression Analysis.  The posting on 2/21/2008 describes statistical thinking by a manufacturer of automotive door frames.  The purpose was to eliminate a problem meeting dimensional specifications of the finished product.   Shop floor personnel thought that variations in the incoming raw material characteristics caused the problem meeting dimensional specifications.  The team defined important quality characteristics for each step in the process.   They included quality characteristics of the incoming material.   The manufacturer collected data listing the important quality characteristics as well as the final part dimensions.    A regression analysis showed no effect by the incoming material characteristics.    Moreover, it identified several quality characteristics having a significant effect on finished product dimensions.    The regression analysis also showed that the left and right door frames had significantly different variation for two quality characteristics.   These results motivated corrective action and eliminated the need for rework.   In this example, the team did not need to employ step 7, Study Cause & Effect.

Posted by Gordon M. Clark at 12:17 PM | Permalink | Comments (0)

This posting gives two examples illustrating the Analyze Common-Cause Variation step, step 6, in the Hoerl-Snee process improvement strategy.   Refer to the previous posting for a description of this step.

·         Stratification – Pareto Chart.  The posting on 2/25/2008 describes statistical thinking by a company experiencing a high rejection rate in one of its machine shops.   In order to determine the root cause of these rejections they stratified by classifying the rejections with respect to machine type causing the rejections.   Then they created a Pareto Chart ranking the frequency of rejections by machine type.   They found that 60% of the rejections were due to grinding problems.   This finding did not give them the root cause of the rejections, but it allowed them to focus on grinding operations.  Their next step was to construct a cause and effect diagram and then to design experiments to determine improved grinding procedures.   This next step illustrates the implementation of the Study Cause & Effect step, step 7 in the Hoerl-Snee process improvement strategy.
·         Regression Analysis – Stratification.  The posting on 3/4/2008 describes statistical thinking by Pease Industries to reduce the defect rate of decorative glass inserts for a wooden entry door.   The prevailing opinion was that humidity and temperature variations in the mold department were the root cause.  The team collected data and did a regression analysis using temperature and humidity as independent variables and the number of defects as the dependent variable.   The result was no correlation between the independent variables and the number of defects.  They collected more data and stratified the data by part type, month of occurrence and day of week.   They were surprised by the result showing day of the week strongly affecting the defect rate.   A Chi-Square test showed the day of the week was statistically significant.   The next step was to construct a Cause-and-Effect diagram and do a Is-Is Not analysis.   This step illustrates the Study Cause and Effect step, step 7.

In both of the above examples, the use Cause-and-Effect diagrams, designed experiments and the Is-Is Not analysis required the previous results from the Analyze Cause and Effect steps.   One needs to identify the effects prior to studying the effects.

Posted by Gordon M. Clark at 05:51 PM | Permalink | Comments (0)

This posting discusses the sixth step, Analyze Common Cause Variation, of the Hoerl-Snee Process Improvement Strategy.   Refer to the figure in the April 4 posting for an overview of the process.  Use Britz et al (2000) and Hoerl and Snee (2002) as references.

Common-cause variation affects all of the data which distinguishes this step from the Address-Special-Causes step.  The purpose of the Analyze-Common-Cause-Variation step is to identify sources of variation.     Locating the sources of variation might also reveal its root cause without significant additional analysis.  On other occasions, knowing a source of common-cause variation might require further analysis to determine its root cause.   This additional analysis is performed in the next step, Study Cause and Effect.

Some of the tools we might use in this step are:

• Stratification.  Define a stratification factor such as the day of the week or machine.   Partition the factor into logical categories.  Compare the data for each category to highlight differences.
• Disaggregation.  Define quality measures for sub-processes or individual process steps.  Study the variation in the individual sub-processes.  How does it contribute to the overall process variation?
• Pareto Chart.  Classify defects into categories.  Highlight the categories having the most frequent occurrences.    
• Histogram.  Plot the distribution of quality measures.  One or more peaks might indicate the presence of categories that could be examined by stratification.
• Regression Analysis.   Existing opinion might suggest one or more input variables that influence the output quality measure.   A regression analysis might verify this opinion or indicate that these variables have negligible effect.

References

1. Britz, G. C., D. W. Emerling, et al. (2000). Improving Performance Through Statistical Thinking. Milwaukee, WI, ASQ Quality Press.
   
2. Hoerl, R. and R. D. Snee (2002). Statistical Thinking - Improving Business Performance. Pacific Grove, CA, Duxbury.
   

Posted by Gordon M. Clark at 02:49 PM | Permalink | Comments (0)

The following figures illustrate two problems with the C_pk index.
1.     In the first figure, two processes with identical C_pk values (1.5) have significantly different means and standard deviations.  Possibly changing the mean is easier to accomplish than the standard deviation.
2.     In the second figure, two processes with identical C_pk values (1.0) have different distributions.   One is normal and the other lognormal.  For the normal distribution, the probability of being below the lower spec limit is .00135, and the probability of exceeding the upper spec limit has the same value.   For the normal distribution, the total probability of not meeting the spec limit is .0027.   For the lognormal distribution, the probability of the quality measure being below the lower spec limit is approximately zero, while the probability of being greater than the upper spec limit is .007915.   For the lognormal, the probability of not meeting the spec limits is almost three times the corresponding value for the normal distribution.

For the above reasons and others, Breyfogle (2003) recommends the use of estimated parts per million (ppm) beyond specification limits rather than process capability estimates.

Due to sampling variability, Hare (2007) recommends estimating process capability indices using at least 100 values.

Reference

1. Hare, Lynne B. (2007).  “The Ubiquitous C_pk”, Quality Progress, pp. 72-73.
   
2. Breyfogle III, Forrest W. (2003). Implementing Six Sigma – Smarter Solutions^â Using Statistical Methods, John Wiley & Sons, Inc., pp 296-299.
   

Posted by Gordon M. Clark at 07:28 PM | Permalink | Comments (0)

This posting discusses the fourth and fifth steps in the Hoerl-Snee Process Improvement Strategy.   Refer to the figure in the April 4 posting for an overview of the process.    Use Britz et al (2000) and Hoerl and Snee (2002) as references.

The approach for addressing special causes is different than the Process Improvement Strategy.    Addressing special causes uses the Problem Solving Strategy which will be described in future postings.

The Evaluate Capability step compares process specifications (targets) and observed variation.   The motivation is to determine whether the process can consistently meet established specifications and/or goals.  

The histogram is an informative graphical method for assessing process capability.   The posting on March 25 showed three histograms displaying resin output variation and two of them gave upper and lower limits for the output quantities.   These histograms clearly showed excessive variation.   That is, output quantities were frequently less than the lower limit and greater than the upper limit.  One advantage of the histogram is that one does not have to assume a theoretical distribution to estimate the rate of non-conformances.   Also, the histogram shape may suggest a theoretical distribution.   For example a bell shaped histogram suggests a normal distribution.   If the histogram displays unexpected patterns, it may suggest corrective action.   For example, the resin output variation histogram showed two peaks suggesting difference between the two production lines.   Also, a histogram that is shifted towards a specification limit (upper or lower) suggests that centering the process mean may reduce non-conformances.

Another popular measure of process capability is a process capability index such as C_p or C_pk.   Let USL be the upper specification limit and LSL be the lower specification limit.   Then C_p = (USL-LSL)/(6*sigma) where sigma is the process standard deviation.  If the process quality characteristic has a normal distribution, then a C_p of 1.0 means that .27% of the items produced are non-conforming.   For a C_p of 1.33 the non-conforming percentage is .00636.   For one-sided specifications and calculation of C_pk, we define:

C_pu = (USL-mu)/(3*sigma) for the upper limit,

C_pl = (mu-LSL)/(3*sigma) for the lower limit,

C_pk = Min(C_pu, C_pl) where mu is the process mean.

If we think of three standard deviations as the process spread around its mean, then C_pk is the ratio between the allowable spread and the actual spread.   For short term performance, a C_pk of 2.0 is the target standard for a Six Sigma project.   In the past, C_pk of 1.33 had been required of suppliers in the automotive industry.

Important observations are:

• In order for C_p and C_pk to have any validity, the process must be stable.
• Both the Assess Stability and Evaluate Capability steps are important in estimating the amount of improvement needed for a project.
• Probability plots are another tool one can use in evaluating process capability.

The next posting will discuss problems in using process capability indices.

References
1.     Britz, G. C., D. W. Emerling, et al. (2000). Improving Performance Through Statistical Thinking. Milwaukee, WI, ASQ Quality Press.
2.     Hoerl, R. and R. D. Snee (2002). Statistical Thinking - Improving Business Performance. Pacific Grove, CA, Duxbury.

Posted by Gordon M. Clark at 09:45 AM | Permalink | Comments (0)