Statistical Thinking to Improve Quality

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.

Posted by Gordon M. Clark at 09:45 AM | Permalink | Comments (2)