Statistical Thinking to Improve Quality

This posting describes the Multi-Vari Chart which is a tool for use in the seventh step, Study Cause and Effect, of the Hoerl-Snee Process Improvement Strategy.  The posting defines the chart and illustrates its use.

The Multi-Vari Chart graphically shows variation of a quality characteristic for multiple factors.   The purpose of the chart is to permit identification of the factor or factors having the greatest effect on variability.

Recall the example in the previous posting taken from Breyfogle (2003, page 389).  An injection molding process produced plastic cylindrical connectors.   The example included data from a sample of two parts collected hourly from four mold cavities for three hours consisting of measurements at three locations on the parts.  The three locations are bottom, middle, and top.  We want to display the variability by location, cavity and part.  The following figure shows averages over the three hours by location, cavity and part.   The figure shows that cavities 2,3 and 4 had larger diameters at the ends (top and bottom) while cavity 1 had a taper.   Thus, cavity and location have an interacting effect.

In this example, the Multi-Vari chart showed interactions among categories affecting variability.   In the previous posting, the Box Plot shows variation within a category, i.e., a cavit.

References

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

Posted by Gordon M. Clark at 08:57 PM | Permalink | Comments (0)

This posting describes the Box Plot (Box-and-whiskers plot) which is a tool for use in the seventh step, Study Cause and Effect, of the Hoerl-Snee Process Improvement Strategy.  The posting defines the plot and illustrates its use.   The Box Plot shows certain aspects of the distribution of data.  By classifying the data into categories, one can construct a Box Plot for each category and observe distributional differences among the categories.   These differences may reveal categories or factors that are increasing (or reducing) variability.

To illustrate the Box Plot, we refer to an example given by Breyfogle (2003, page 389).  An injection molding process produced plastic cylindrical connectors.   Breyfogle presents data from a sample of two parts collected hourly from four mold cavities for three hours consisting of measurements at three locations on the parts.   The Box Plot for the aggregated data appears below. 

The plot portrays key distribution characteristics as shown in the figure.  Twenty-five percent of the data are less than or equal to Q₁, half of the data are less than or equal to the median, and seventy-five percent of the data are less than or equal to Q₃.  The vertical lines are whiskers.   Call Q₁ the 25^th percentile, Q₃ the 75^th percentile, and the median the 50^th percentile. The lower whisker extends to the lower limit which is Q₁ – 1.5(Q₃ - Q₁), and the upper whisker extends to the upper limit which is Q₃ + 1.5(Q₃ - Q₁).   Values beyond the upper and lower limits are outliers and shown as asterisks (*).
 
The following figure illustrates the use of Box Plots to identify categories increasing variability and degrading quality.   Mold cavity 1 produces diameters greater than cavities 2, 3 and 4.  The 25^th percentile for mold cavity 1 diameters is greater than the 75^th percentiles for mold cavities 2,3 and 4.

References

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

Posted by Gordon M. Clark at 08:24 PM | Permalink | Comments (0)

This posting gives the background and source of the interrelationship digraph.   It differentiates this source from the ‘Seven major SPC Tools’ and the ‘Magnificent Seven’.

GOAL/QPC, an educational consulting company, noticed a new book proposing seven new QC tools.   This book (Mizuno, 1988) was eventually translated into English.  GOAL/QPC created the Memory Jogger Plus+ (Brassard, 1989) featuring these new tools.  They called these new tools the ‘Seven Management and Planning Tools’ to differentiate them from the ‘Seven Major SPC Tools’.   The Seven Management and Planning Tools are:

1. Affinity diagram
   
2. Interrelationship digraph
   
3. Tree diagram
   
4. Prioritization matrices
   
5. Matrix diagram
   
6. Process decision program chart (PDPC)
   
7. Activity network diagram
   

Montgomery (2005, page 148) identifies ‘Seven Major SPC Tools’.  He calls them the ‘Magnificent Seven’.  They are:

1. Histogram (3/25/2008 and 5/1/2008 postings) or stem-and-leaf plot
   
2. Check sheet
   
3. Pareto chart (2/25/2008 and 5/18/2008 postings)
   
4. Cause and effect diagram (2/28/2008 posting)
   
5. Defect concentration diagram
   
6. Scatter diagram (3/28/2008 posting)
   
7. Control chart (1/30/2008, 2/11/2008, 2/14/2008, and 4/1/2008 postings)
   

The implication is that we can perform SPC in most cases using these tools.

Earlier, Ishikawa (1985) identified ‘Seven Major TQM’ (Total Quality Management) tools.   They are:

1. Histogram
   
2. Flowchart
   
3. Pareto chart
   
4. Cause and effect diagram
   
5. Run charts and graphs
   
6. Scatter diagram
   
7. X-bar and R control charts
   

Ishikawa also felt that the above tools would support most TQM projects.

One could say that Montgomery replaced the ‘flowchart’ and ‘run charts and graphs’ with the ‘check sheet’ and ‘defect concentration diagram’.   Montgomery also generalized the X-bar and R control charts with all control charts.

References

1. Brassard, M. (1989). The Memory Jogger Plus+^â. Salem, NH, Goal/QPC.
2. Mizuno, S. (1988). Management for Quality Improvement: The Seven New QC Tools. Cambridge, Productivity Press.

Posted by Gordon M. Clark at 10:28 AM | Permalink | Comments (0)

This posting gives an example of an Interrelationship Digraph which is a tool for use in the seventh step, Study Cause and Effect, of the Hoerl-Snee Process Improvement Strategy.   The quality issue is the potential causes or factors contributing to late deliveries.   We take our example from Benbow and Kubiak (2005).   The interrelationship digraph appears below.

After constructing the interrelationship digraph we want to interpret its meaning.   What are the key factors or causes to investigate and improve?   Recall that we called the entries in the digraph concerns.  A concern with a high number of output arrows is a driver or key cause.  A key cause affects a large number of other items.  The above diagram shows the following key causes:

1. ‘Poor scheduling practices’ (6 outgoing arrows),
   
2. ‘Late order from customer’ (5 outgoing arrows), and
   
3. ‘Equipment breakdown (3 outgoing arrows).
   

A concern with a large number of input arrows is affected by a large number of other concerns.  Thus, it could be a source of a quality or performance metric.   ‘Poor scheduling of the trucker’ has 4 input arrows.   A measure of poor scheduling performance of the trucker could indicate the magnitude of system problems causing late delivery.

References:

1. Benbow, D. W. and T. M. Kubiak (2005). The Certified Six Sigma Black Belt Handbook. Milwaukee, Wisconsin, ASQ Quality Press.

Posted by Gordon M. Clark at 10:22 AM | Permalink | Comments (1)

This posting describes the Interrelationship Digraph which is a tool for use in the seventh step, Study Cause and Effect, of the Hoerl-Snee Process Improvement Strategy.  For example, assume that we start with a Cause & Effect diagram displaying potential causes of an effect or quality issue.   We want to determine which potential cause or causes are the key causes or drivers.

Form a team of knowledgeable individuals with respect to this quality issue.    The team will select a number, e.g., from six to twelve, of the potential causes from the Cause & Effect diagram.   Call these potential causes concerns.   The process for generating the Interrelationship Digraph will construct causal relationships among the concerns.   The word digraph is a combination of the two words diagram and graph.  The resulting digraph reflects the collective judgment of the team.

Benbow and Kubiak (2005, page 40) specify a procedure for constructing the digraph.   List the concerns on a sheet of easel paper or a whiteboard. Pick a pair of concerns.   Ask the team to specify whether the first concern influences the second, the second concern influences the first, or whether there is no influential relationship between the concerns.    If the team decides there is an influential relationship, draw an arrow from the most influential concern to the other concern.  Does the first concern influence the second more than the second concern influences the first?   If so, draw an arrow from the first concern to the second.  Repeat this assessment for all possible pairs.  A good way to proceed is to arrange the concerns in an approximate circular pattern.   Start with the concern in the 12 o’clock position and call it the first concern.   Compare it with the concern in the next clockwise position.  Then, move clockwise and select another concern to compare with the first concern.   Repeat this process until all possible combinations of concerns have been compared by the team.

The next posting will illustrate the construction of an interrelationship digraph.

Posted by Gordon M. Clark at 05:00 PM | Permalink | Comments (0)