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 present the results of the analysis of the experiments specified in the 9/18/2008 posting (Part 2).  

The following figures display graphically the relative significance of the six factors, i.e., A, B, C, D, AB and AC.   The figures show the average response at the factor low (-1) and high (+1) values.    Factors B and C are not nearly as significant as factors A and D since the average responses of B and C are nearly the same at their low and high values.  That is, a change in the factor levels for factors B and C has little effect on the response.  Also, the interaction factor AC is more significant than the interaction factor AB.  

We can test the significance of the factors using an Analysis of Variance (ANOVA).   Refer to Montgomery, Peck and Vining (2006).   Let SS_T be the total sum of squares.   That is:

where Y_i is the response on experiment i and ybar is the average response over the 8 experiments.   That is, SS_T is the sum of the 8 squared deviations between the experiment responses and the average response.   The value of ybar is 49.582, and the value of SS_T is 118.151.   Then we partition SS_T into a sum of squares due to the estimated effects (SS_R) and a sum of squared deviations from the estimated effects (SS_RES).  That is, SS_T = SS_R + SS_RES.  The value of SS_R is the same as a sum of squares due to an estimated regression function when we have a two-level experiment.   Consider the contribution of factor A to SS_R.    The posting on 9/18/2008 gives the estimated effect of factor A to be -6.067.  That is the difference between the average of the responses at the low values of factor A and the high values of factor A.    Thus the estimated average response at the high values of factor A is ybar - 6.067/2 = 46.5485.  Similarly, the estimated average response at the low values of factor A is  ybar + 6.067/2  = 52.6155.   The deviation between the mean response and the effect of A conditioned on whether A is high or low is 6.067/2.   Since we have 8 experiments, the contribution of factor A to SS_R is 8*(6.067/2)² = 73.60788.   For factor D and the interaction effect AC, the corresponding contributions to SS_R are 18.67308 and 11.38575.   Thus, SS_R is 103.6667.   The value of SS_RES is SS_T – SS_R = 14.48432.  We can test whether these three factors are statistically significant using the F statistic.    The F statistic assumes that the individual responses have a normal distribution.   The F statistic is:

where df_R = degrees of freedom for SS_R = 3 (the number of factors),
df_RES = degrees of freedom for SS_RES = 8-1-3 = 4 (we loose one degree of freedom due to estimating the mean and 3 due to estimating the 3 factor effects.
We can tell whether this value of F is statistically significant by calculating its PValue.    The PValue is the probability of obtaining this value of F, i.e., 9.543, or higher by chance when the factor effects have at true value of zero.   The PValue for this F is .027.    Usually, we regard a PValue as statistically significant when it is less than .05.   Thus the factors A, D and AC are statistically significant.   If we attempt to add a forth factor, i.e., AB,  the PValue becomes .0625; thus, we do not include AB. 

Higher values of the response S/N are desirable.   Thus, the low value of factor A (feed rate of .0008 mm/Revolution) and the low value of factor D (wheel grade of A54) are preferred.  Since the low value (-1) of the interaction effect AC is preferred, we select the high value of factor C which is a work speed of 360 RPM.   For the insignificant factor, the team chose its low value ( a wheel speed of 2200 RPM).

The posting on 2/28/2008 reports that the preferred factor levels specified above improved the process performance index (P_pk) from .49 to 1.25.   This is based on a sample of 40 parts.   The posting on 5/1/2008 defines the process capability index C_pk.   Process capability indices assume the process is stable.   When we have insufficient evidence the process is stable, we call the capability index a performance index and use the same equation.   

References

1. Montgomery, Douglas C., Elizabeth Peck, Geoffrey Vining (2006). Introduction to Linear Regression Analysis, John Wiley & Sons, p26.

Posted by Gordon M. Clark at 10:01 PM | Permalink | Comments (0)

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	53.4692	+1	+1	+1
2	50.9704	-1	-1	+1
3	49.0298	-1	+1	-1
4	56.991	+1	-1	-1
5	49.0298	+1	+1	+1
6	46.1079	-1	-1	+1
7	46.1079	-1	+1	-1
8	44.9483	+1	-1	-1
Effect		3.056	-0.345	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 | Permalink | Comments (0)

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	53.4692
2	-1	-1	+1	+1	50.9704
3	-1	+1	-1	+1	49.0298
4	-1	+1	+1	-1	56.991
5	+1	-1	-1	-1	49.0298
6	+1	-1	+1	+1	46.1079
7	+1	+1	-1	+1	46.1079
8	+1	+1	+1	-1	44.9483
Effect	-6.067	-0.625	0.345	-3.056

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	-1.416	-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 | Permalink | Comments (0)

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 | Permalink | Comments (0)

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 | Permalink | Comments (0)

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 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)

This posting continues the discussion of the seventh step, Study Cause and Effect, of the Hoerl-Snee Process Improvement Strategy.  Tools that might be used in this step that were not summarized in the previous posting are: 

• Experimental Design.  A systematic planned variation of input factors for an actual process.   The experimenter observes the effect of these variations on important quality characteristics.   The 1/30/2008 posting mentions the use of designed experiments by an OEM manufacturer to determine an improved raw material composition.  The 2/28/2008 posting discusses the effort by a company to reduce the rejection rate at one of its machine shops.   Based on a Cause & Effect diagram, project members selected four factors for further analysis based on designed experiments.   These factors were Feed Rate, Wheel Speed, Work Speed, and Wheel Grade.   Analysis of the experimental results identified “optimum” levels for the four factors.
• Model Building.  One could construct a model of a process that predicts quality performance based on input variables.   The 2/18/2008 posting describes the actions of a Midwest manufacturing firm to reduce time delays experienced by customers contacting their order processing center.  They constructed a simulation model of the order-taking process.  Using the simulation model they determined the staffing level of customer service representatives by the hour of a work day to meet time-delay objectives.  Why don’t software companies use simulation models to specify technical support personnel requirements?

Subsequent postings will illustrate the use of experimental design and model building to Study Cause and Effect.

Posted by Gordon M. Clark at 03:57 PM | Permalink | Comments (0)

This posting discusses the seventh step, Study Cause and Effect, 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.

The previous step analyzed common-cause variation to identify the source (s) of variation.   If the previous step did not identify the source or if knowing the source does not reveal the root cause, we proceed to study cause and effect.  

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

• Scatter plot.   A plot of a quality characteristic versus a potential explanatory variable.   See the plot in the 3/28/2008 posting showing the effect of solvent feed ratio on output weight.
• Cause & Effect Diagram.  A diagram portraying the potential causes of an effect.  See the diagram in the 2/28/2008 posting showing the potential causes of rejections at the grinding operations.  Frequently, the Cause & Effect diagram summarizes the results of a brainstorming session.   However, some improvement efforts will use data to substantiate the cause and effect diagram.
• Box Plot.   Box Plots depict the relationship between a discrete variable, such as location on a part, and the distribution of continuous variable, such as a dimension.
• Multi-Vari Charts.   Multi-Vari charts display variations in categories that aid in identifying causes.
• Interrelationship Digraphs.   Teams construct cause and effect relationships from a list of issues.

The next posting will summarize additional tools for this step.   Subsequent postings will give examples of Box Plots, Multi-Vari Charts and Interrelationship Digraphs.

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 02:31 PM | Permalink | Comments (0)