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 discusses the use of process capability indices in fifth step, Evaluate Capability, of the Hoerl-Snee Process Improvement Strategy.   Refer to the figure in the April 4 posting for an overview of the process.    See Hare (2007) or Breyfogle (2003) for references.

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.

Address Special Causes

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.

Evaluate Capability

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 (2)

The Cause and Effect Diagram graphically portrays the potential causes of an effect.   The causes are grouped into categories.   Common categories are manpower (personnel), materials, methods and machines.   When the diagram uses these specific categories we might call the diagram a 4M diagram.  Depending on the effect, the diagram might display other categories.   The diagram is also known as an Ishikawa diagram since Dr. Ishikawa devised its first use of the diagram.   Another name for the diagram is a Fishbone diagram because of its appearance.   Recording the results of a brainstorming session is a typical use for the diagram.   A project might use a brainstorming session to generate a list of potential causes of an effect or a quality problem.  

We will continue the case study reported by Gijo (2005) to illustrate the Cause and Effect diagram.   The previous post presented a Pareto chart for a machine shop showing that the grinding operations generated most of the rejections experienced by the shop. They estimated grinding machine capability based on a sample of 40 parts.   The estimated P_pk for this sample was .49.  This result verified the lack of grinding machine capability. 

The posting on 5/1/2008 defines the process capability indices C_p and C_pk.   Process capability indices assume the process is stable.   When we have lack evidence that the process is stable, we call the capability index a performance index and use the same equation.   The index P_pk is a process performance index.

Selected individuals participated in a brainstorming session to generate a set of potential causes of grinding machine rejections.    The following figure shows the resulting causes. 

After further study, 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.  The estimated P_pk  at the optimum factor levels was 1.25 based on a sample of 40 parts.   This showed significant improvement.

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 08:20 PM | Permalink | Comments (1)