Statistical Thinking to Improve Quality

This posting continues the grinding process case study (Gigo, 2008) that illustrates the use of design and analysis of experiments to reduce common-cause variation.  We examine the properties of the experimental design reported by Gijo.   The examination illustrates the potential for aliasing in an experimental design and shows how it can bias the results.   The experimental design described by Gijo uses an orthogonal array which Taguchi recommended.   We contrast the properties of that design with a standard fractional factorial.

The 9/15/2008 posting initiated the design of experiments portion of the case study.  The primary purpose of the experimental design was to reduce the variation in the outer diameter produced by a grinding operation.   That posting reports that the team was primarily interested in estimating the following effects:

A – Feed Rate
B – Wheel Speed
C – Work Speed
D – Wheel Grade
AB – Interaction between A and B
AC - Interaction between A and C

Gijo states that the experimental design was developed using an L₈ orthogonal array.  He references Phadke (1989) for use of orthogonal arrays to construct designs.   Taguchi made extensive use of orthogonal arrays in constructing robust designs.  Hicks and Turner (1999, p381) give a table for using an L₈ orthogonal array to construct a design with the desired properties.   That is, we do not want the A, B, C, D, AB, and AC effects aliased with each other.   Two effects that have the same estimator are aliased.  The previous posting on September 15 gives the design and estimates of the factor effects.   Clearly the design meets the desired criterion since the factor effect estimates are all different.

However, consider the estimates of the of the BC, BD and CD interaction effects shown in the following table.

Experiment	Response (S/N)	Wheel Speed X Work Speed (BC)	Wheel Speed X Wheel Grade (BD)	Work Speed X Wheel Grade (CD)
1	53.46787	+1	+1	+1
2	50.9691	-1	-1	+1
3	49.0309	-1	+1	-1
4	56.9897	+1	-1	-1
5	49.0309	+1	+1	+1
6	46.10834	-1	-1	+1
7	46.10834	-1	+1	-1
8	44.9485	+1	-1	-1
Effect		3.055	-0.344	0.625

Note that the BC interaction effect is exactly equal to the negative of the D effect, the BD interaction effect is equal to the negative of the C effect and the CD interaction effect equals the negative of the B effect.  That is true because the sequences of +1 and -1s in the BC, BD and CD columns are precisely the negatives of those in the D, C and B columns.    With this design, the BC and D effects are aliased.   That is, if the BC effect is not zero, then our estimate of the D effect is affected by the BC effect.   Similarly, the BD effect estimate is aliased with the C effect, and the CD effect is aliased with the B effect.  Then this design provides no information on whether the BC, BD and CD interaction effects are negligible.   Also, this design can give a biased estimate of the D effect if the BC interaction defect is significant.

Montgomery (2005, p. 288) gives a standard one-half fraction of the 2⁴ factorial design.   Call it the 2^4-1 design.  This design uses 8 experiments and has four factors.   The properties of this design are:
·        Estimates of the main effects are not aliased with any two-factor interactions.
·        Estimates of the main effects are aliased with three factor interactions.
·        Every two factor interaction is aliased with another two factor interaction.   That is AB=CD, AC=BD and BC=AD.

The 2^4-1 design might be superior to the one described by Gijo.   Estimates of the A, B, C and D effects are not aliased with any two factor interaction.  Also, estimates of the AB and AC effects are not aliased with a main effect.

The next posting will present results from the experimental design.

References

1. Gijo, E. V. (2005). "Improving Process Capability of Manufacturing Process by Application of Statistical Techniques." Quality Engineering 17(2): 309-315.
2. Hicks, Charles R. and Kenneth V. Turner Jr. (1999).  Fundamental Concepts in the Design of Experiments, Oxford University Press.
3. Montgomery, Douglas C. (2005). Design and Analysis of Experiments, 6^th Edition, John Wiley & Sons, Inc.
4. Phadke, Madhav S. (1989).  Quality Engineering Using Robust Design, Prentice Hall.

Posted by Gordon M. Clark at 08:51 PM | Comments (0) | Send to a Friend

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This posting continues the grinding process case study (Gigo, 2008) that illustrates the use of design and analysis of experiments to reduce common-cause variation.  The 9/15/2008 posting initiated the design of experiments portion of the case study.

The response variable was a measure of the variability of the outer diameter of the machined components.   One could use the estimated variance, i.e. s², for each set of experimental conditions.   That is, one would replicate the experiment for each set of experimental conditions and estimate s².  Gijo chose to use -10*ln(s²).   He lets the symbol S/N represent the -10*ln(s²).  Could S/N mean that the response is a Taguchi signal-to-noise ratio?   Montgomery (2005, p. 469) discourages the use of signal-to-noise ratios.   He states that a more effective approach is to model the mean and variance separately.   Hunter (1987) comes to the same conclusion.   Gijo does not justify the use of S/N other than a reference to the 3^rd edition of Montgomery’s book.

A response variable that has a constant variance over the set of experimental conditions facilitates regression analyses of the results.   Montgomery (2005, p. 83) recommends the use of the logarithmic transformation when the standard deviation of the response is proportional to its mean.   Let’s proceed by assuming the team used S/N since they wanted to estimate the contribution of the selected factors to the variance of the outer diameter and the standard deviation was roughly proportional to the mean.

The following table gives the experimental design and the observed response for each experiment.   The team replicated the experiment twice for each set of experimental conditions.   From the two observed outer diameters, they calculated a variance estimate, i.e., s², and from that computed the response value S/N.  The -1 and +1 symbols represent the lower and higher levels of the respective factors. 

Experiment	Feed Rate (A)	Wheel Speed (B)	Work Speed (C)	Wheel Grade (D)	Response (S/N)
1	-1	-1	-1	-1	53.46787
2	-1	-1	+1	+1	50.9691
3	-1	+1	-1	+1	49.0309
4	-1	+1	+1	-1	56.9897
5	+1	-1	-1	-1	49.0309
6	+1	-1	+1	+1	46.10834
7	+1	+1	-1	+1	46.10834
8	+1	+1	+1	-1	44.9485
Effect	-6.065	-0.625	0.344	-3.055

Montgomery (2005, p208) shows how to calculate the average factor effects using the -1 and +1 coding.  For a single factor effect, we sum the products of the factor coding times the experiment response over all experiments.   Then we divide the sum by the number of -1, +1 pairs.   In this experiment, the number of pairs is 4.   The last row in the above table shows the estimated factor effects.   For an interaction effect, we multiply the experiment coding for each factor to get a coding for the interaction effect.

Experiment	Feed Rate X Wheel Speed AB	Feed Rate X Work Speed AC
1	+1	+1
2	+1	-1
3	-1	+1
4	-1	-1
5	-1	-1
6	-1	+1
7	+1	-1
8	+1	+1
Effect	-1.417	-2.386

Notice that the estimated AB and AC interaction effects are larger than the single factor B and C effects.

The next posting will examine the properties of the experimental design.

References

1. Gijo, E. V. (2005). "Improving Process Capability of Manufacturing Process by Application of Statistical Techniques." Quality Engineering 17(2): 309-315.
2. Hunter, J. S. (1987). "Signal-to-Noise Ratio Debated." Quality Progress 20(5): 7-9.
3. Montgomery, Douglas C. (2005). Design and Analysis of Experiments, 6^th Edition, John Wiley & Sons, Inc.

Posted by Gordon M. Clark at 09:13 PM | Comments (0) | Send to a Friend

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This posting describes a grinding process case study to illustrate the use of design and analysis of experiments to study cause and effect and reduce common-cause variation.  We continue the case study reported by Gijo (2005) in the 2/28/2005 posting.   That posting describes the construction of a cause-and-effect diagram by a team in an engineering organization identify potential causes of low grinding machine capability.  The team selected four factors for further analysis based on designed experiments.   These factors were Feed Rate, Wheel Speed, Work Speed, and Wheel Grade.  The team chose to perform experiments using two levels for each factor.   The following table shows the levels and factors selected for experimentation.  The levels with an * were existing operating levels.

Factor	Code	Low Level (-1)	High Level (+1)	Unit
Feed rate	A	.0008*	.0010	Mm/Rev
Wheel speed	B	2200	2450*	RPM
Work speed	C	250*	360	RPM
Wheel grade	D	A54	A60*	-

 

Experimental design terminology defines the effect of a factor as the change in the response produced by a change in the level of the factor.   Assume that the response in this experiment is the variance of the outer diameter measurements.   For example, if increasing the feed rate from .0008 to .0010 mm/revolution increases the variance of the outer diameter by .003 mm² then the feed-rate (factor A) effect is .003 mm².  When the difference in response to a factor level change is not the same at all levels of another factor, an interaction effect exists between the factors.   The factor A effect might be .003 mm² when the wheel speed is 2200 rpm and .005 mm² when the wheel speed (factor B) is 2400 rpm, then an interaction effect exists between factors A and B.   The magnitude of the interaction effect is the average difference between the two A effects.   Thus the AxB interaction effect is (.005-.003)/2 = .001 mm².

The team selected an experimental design the enables them to estimate the effects of the four factors in the above table.   They also wanted to estimate two interaction effects: 1. (AxB) between Feed Rate and Wheel Speed (AxB) and 2. (AxC) between Feed Rate and Work Speed.  The linear graph shown below depicts the effects the experimental design must be capable of estimating.  That is, the A, B, C and D effects, the AxB and AxC interaction effects and the error variance.

The next posting will describe the experimental design. 

References

1. Gijo, E. V. (2005). "Improving Process Capability of Manufacturing Process by Application of Statistical Techniques." Quality Engineering 17(2): 309-315.

Posted by Gordon M. Clark at 09:09 PM | Comments (0) | Send to a Friend

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This posting illustrates the use of model building to study cause and effect and reduce common-cause variation.  One approach to model building is to build a model such as a regression model based on either results from an experimental design or observed process data.  Another approach illustrated in this posting is to construct a simulation model based on the system flow chart or process map.    One application of a simulation model is to predict flow times or service times for complex systems.   In service or health system applications customer service or wait times could be useful quality measures.   One uses the simulation model by varying input variables such as the number of servers to predict their effect on customer service times.

Davies (2007) describes a case study involving the treatment of minor injuries and medical problems in an emergency department in England.   Receptionists route arriving patients with minor injuries or medical conditions are routed to the “Minors” department.   The standard processing procedure has receptionists in the Minors department assign patients to a queue for triage nurses who assess the patient condition and needs.   Then the triage nurse routes the patients to a doctor or nurse for treatment.   The nurses are qualified to assess and treat minor injuries but not to handle minor medical conditions which are handled by doctors.   These nurses are Emergency Nurse Practitioners (EPNs).  Call this procedure “See” and “Treat”.   The UK national health service recommended that emergency departments skip the triage nurse step.   The health service recommended that receptionists route patients to a doctor or ENP for diagnosis and treatment.  Call this procedure “See & Treat”.   The intent was to reduce patient system time by eliminating a step and its associated queuing time.   The following figure depicts the “See & Treat” patient flow.

Davies describes a simulation model for comparing the two procedures.   This model represents the processing of individual patients, their waiting times, and individual task processing times.   Inputs to the model would include distributions for task times, distributions for times between patient arrivals, and the numbers of doctors and EPNs.  The following figure presents some of the simulation results.   The new procedure “See & Treat” that eliminates the triage step gives the lowest system time.

References

1. Davies, R. (2007). "See and Treat" or "See" and "Treat" in an Emergency Department. 2007 Winter Simulation Conference. Washington, DC.
   

 

Posted by Gordon M. Clark at 06:54 PM | Comments (0) | Send to a Friend

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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 | Comments (0) | Send to a Friend

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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 | Comments (0) | Send to a Friend

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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 | Comments (0) | Send to a Friend

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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 | Comments (1) | Send to a Friend

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