Statistical Thinking to Improve Quality

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 | Comments (0) | Send to a Friend

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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.

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

After collecting data on key measures, the next step is to analyze process stability based on that data.  First we define a stable process or one that is in-control.   Shewhart (1931, p. 6) states: “..a phenomenon will be said to be controlled when, through the use of past experience, we can predict, at least within limits, how the phenomenon may be expected to vary in the future.”   More recently, Montgomery (2005, p. 148) states: “In any production process, …., a certain amount of inherent or natural variability will always exist.  …. this natural variability is often called a ‘a stable system of chance causes.’  A process that is operating with only chance causes of variation present is said to be in statistical control.  … We refer to those sources of variability that are not part of the chance cause pattern as ‘assignable causes.’  A process that is operating in the presence of assignable causes is said to be out of control.”   Montgomery references Shewhart for the terminology chance and assignable causes.   He states that many now use the terminology common cause rather than chance cause and special cause rather than assignable cause.  

An important characteristic of a stable or in-control process is that it is predictable.   This comes from Shewhart’s definition.  That is, one can predict future behavior from past behavior.   Breyfogle (2003, p. 1109) and Wheeler (1993, p. 124 and 128) state that an in-control process is predicable whereas a process that is not in-control is unpredictable.   This means that statistical methods such as t tests and ANOVA are inappropriate for unstable processes.

The definitions stated above immediately suggest methods for identifying whether a process is in-control.   They include run charts and SPC control charts.   A run chart is a time plot of quality characteristic and a control chart is a run chart with control limits.   Using the points on these charts that signal lack of control, we can conduct investigations to determine what caused these points to be different.  Two previous postings that do that are:

• Rosin yield run chart on March 21.   The team found that their cause was a drop in air pressure.
• Customer complaint process posted on January 30.   The investigation identified the high defect rate in October, 91 was due to a supplier using the wrong material.

Two major reasons for assessing stability and removing assignable causes prior to addressing common-cause variation are:

• Detecting and removing assignable causes is easier than reducing the variation due to common causes.   Remove the low hanging fruit first.
• The analysis approach to removing assignable causes is different than reducing common-cause variation.   Common causes affect variation in all data points for a stable process.   When searching for the root cause of variation in specific observations due to an assignable cause, one can focus on the differences between these observations and the others.

Consider the possibility of wasting effort when a process is in-control (stable) but some results do not meet targets.   Managers could pressure staff to find the cause of specific results not meeting targets.    That is, managers could direct staff to find causes for specific undesirable outcomes when the variation is present in all outcomes.

References
1.     Breyfogle, Forrest W. (2003). Implementing Six Sigma: Smarter Solutions Using Statistical Methods, Second Edition, John Wiley & Sons, Inc.
2.     Britz, G. C., D. W. Emerling, et al. (2000). Improving Performance Through Statistical Thinking. Milwaukee, WI, ASQ Quality Press.
3.     Hoerl, R. and R. D. Snee (2002). Statistical Thinking - Improving Business Performance. Pacific Grove, CA, Duxbury.
4.     Montgomery, Douglas C. (2005). Introduction to Statistical Quality Control, John Wiley & Sons, Inc.
5.     Shewhart, Walter A. (1931). Economic Control of Quality of Manufactured Product, D. Van Nostrand Company, Inc.
6.     Wheeler, Donald J. (1993). Understanding Variation: The Key to Managing Chaos, SPC Press, Inc.

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This posting discusses the second step in the Hoerl-Snee Process Improvement Strategy.   Refer to the figure in the previous posting for an overview of the process.    Use Britz et al (2000) and Hoerl and Snee (2002) as references.

After understanding and documenting the process, the next step is to collect data on key process and output measures.  These key measures can include the overall process performance measure(s) and measures derived from the inputs and outputs of each process step.

For example, the Ricoh team in the Resin Output Variation example, March 21 posting, was concerned with the product yields being greater than theoretical expectations so they collected yield-ratio data.   In the Pease Industry example, posted on March 4, the company team wanted to improve quality of their residential entry doors so they collected defect-rate data from their customers.  In the automotive door frame example, posted on February 21, the manufacturer wanted to improve the quality of critical dimensions on the welded door frame.   They collected data from incoming material and after each processing step, i.e., roll mill, bender and saw.  These data consisted mainly of dimensional measurements.  

The process may be a sequence of steps required to perform a task with a cycle-time principal performance measure.    For example, the process might be the activities required to fill a prescription in a hospital.    For each order submitted, the data might include the submittal time, the arrival time at each processing step, the actual step processing time, the completion time for each processing step, and the drug prescribed.   In addition, one would need the number of servers at each processing step.  

Breyfogle (2003) on page 10 introduces several terms that are useful in identifying important process variables.   A Key Process Output Variable (KPOV) in an important output for a process.   Another name for this variable is a Critical to Quality Characteristic (CTQ).   Key Process Input Variables (KPIVs) are process inputs that affect the KPOVs.  

One might ask: how do we select the data to collect?  We use a combination of the output of the previous step, Understand the Process, and existing process knowledge.   Also, a previous iteration of the Process Improvement Strategy (see the posting on April 4) may have identified some KPIVs.   The author has found in his consulting experience that manufacturers of process equipment may have important information regarding the sensitivity of their equipment to process variables.  Also, do not forget the internet.   A search may reveal research reports indicating the sensitivity of equipment to process variables.    

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.
3.     Breyfogle, Forrest W. (2003). Implementing Six Sigma: Smarter Solutions Using Statistical Methods, Second Edition, John Wiley & Sons, Inc.

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This posting discusses the first step in the Hoerl-Snee Process Improvement Strategy.   Refer to the figure in the previous posting for an overview of the process.    Use Britz et al (2000) and Hoerl and Snee (2002) as references.   

The first step is to develop a common understanding of the process by recording and documenting it.   In the Monthly-Billing-Cycle Time example, posted on 1/21/2008, key participants did not have the same understanding of the principal process steps.  In the Ricoh Resin example, Hoerl-Snee Example posted on 3/21/2008, the team created a flowchart of the process which is the usual method for documenting the process.   We start by documenting the process as it is currently performed.   The flowchart serves as a reference and it facilitates communication.   Sometimes the flowchart is called a process map.

The flowchart is particularly important when on can not visually observe the process flow.   Many administrative or service processes have this property.   The Service-Time example, Service Time Flowchart posted on 2/18/2008, may have had this property.

The flowchart may immediately suggest areas for improvement.   Some steps in the flowchart may represent non-value added activity.    For example, customers waiting in  the Automatic Call Distribution (ACD) System Queue depicted in the service time example represent non-value added activity.

A flowchart or process map may have a number of variations.   The Ricoh resin example flowchart is a high-level flowchart.   Hoerl and Snee (2002) describe other more detailed flowcharts on pages 195-199.   Our flowchart of the automotive door frame example, Flowchart and Process Map posted on 2/21/2008, displayed key quality characteristics for the process steps.

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.

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This posting describes the Hoerl-Snee Process-Improvement Strategy.   This strategy was originally described in Hoerl-Snee (1995), and it also appears in Britz et (2000) and Hoerl-Snee (2002).   Prior to implementing the Process Improvement Strategy, one should define the scope and objectives for the improvement effort.  The following figure displays a flowchart of the improvement strategy steps and lists some example tools to perform the corresponding steps. 

The figure does not show the entire process improvement flow.    Eliminating special causes involves the Problem Solving Strategy.   Future postings will describe this strategy. 

Note that the resin output variation case study clearly illustrated the above features of the process improvement strategy.  Improvement occurred in sequential cycles involving planning, implementing and collecting data.   Also, the first improvement action by the resin team was to determine whether special causes were present and then to correct them.   After that they moved on to reduce the variation contributed by common causes.

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The overall process had two weighing processes.  The first was an in-process manual method, and the second method was a final, automatic scale.   The manual method had individuals reading a line on a scale.  They observed that individuals of different heights read the line from different viewpoints.  Thus, they produced different readings.   The team changed the presentation of the line so people of different heights had the same view point.   This change reduced in-process measurement variation.

Next the team investigated the automatic scale and found significant measurement errors.   They reduced these errors by:

1.     Redesigning the scales protective cover.

2.     Establishing procedures for checking the alignment on a periodic basis.

The following figure presents a control chart showing the results for this project.  The difference between the final upper and lower control limits was less than the team objective of ± 5 kg.  However, the resulting average was 4292 kg which is less than the original target of 4300 kg.   Given the reduction in output variability, management regarded the results as more than adequate.   The improvement also resulted in reduction in the variation of resin viscosity.   This verified the team’s motivation to reduce variation of finished product quality by reducing the output quantity variation. To maintain the results, the team created procedure manuals and established a schedule adjusting the automatic weighing process.

The overall improvement process consisted of four Plan-Do-Check-Act (PDCA) cycles.   This posting describes the last one, the previous post describes two of them and the posting on March 21 (Hoerl-Snee Example) describes the first one.   That one focused on finding and correcting special causes.   This process is different than that suggested by a serial DMAIC process.   Our next posting will present the Hoerl-Snee process improvement strategy which has an overall PDCA approach.

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.

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1. The procedure for dividing the resin after phase B (potentially the cause of difference between line A and B).
   
2. The solvent feed ratio.
   
3. The weighing process, i.e., final (automatic) and in-process (manual).
   

After attacking the first potential cause, the team found that some resin remained in the reaction tank after sending the materials to the two lines.  That meant that line B had less input and therefore less output.   After changing the dividing procedure, the team found no significant difference between the outputs of the two lines.

The output quantities still had too much variation.   The team turned to the second potential cause, i.e., the solvent feed ratio.  The following figure shows a scatter plot indicating that increasing solvent feed ratio is correlated with increasing output.   In calculating the regression line the team regarded the high output occurring at a feed ratio slightly less than 1, as an outlier.   This correlation violated the team’s knowledge of the underlying process.   They investigated the measurement of the feed ratio, and they found that the ratio measurement was affected by the length of time the solvent was in the tank.   They changed the procedure to insure that the solvent had stabilized prior to measurement.   They collected more data to measure the impact of this change and found less variation in the measured feed ratio and no correlation between the measured feed ratio and the output quantity.

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.
   

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