Chapter 7 Process Capability

Introduction A “capable” process is one for which the distributions of the process characteristics do lie almost entirely within the engineering tolerances or customer’s needs. Process capability indices –Simple –Careful in use and interpretation Two phases in a process capability study –Determining how to data are to be collected, and then collecting the data –Selecting 1 or more indices and performing the computations

7.1 Data Acquisition for Capability Indices Data must come from an in-control process The sample must be representative of the population The sample size must be large enough –To assess the extent of the non-normality –To allow a non-normal distribution to be fit to the data Process capability indices <> Process Performance indices

7.2 Process Capability Indices Should be easy to compute Should not be undermined by slight-to-moderate departures from normality

(7.1)

(7.2)

7.3 Estimating the Parameters in Process Capability Indices

7.4 Distributional Assumption for Capability Indices It is assumed that the observations have come from a normal distribution. A normal distribution is also assumed when the capability indices are used.

7.5 Confidence Intervals for Process Capability Indices Unless the sample size was large, it is desirable to also report a confidence interval for the index. Lower confidence bound is more appropriate than a 2- sided confidence interval. It is assumed that individual observations are used in computing the parameter estimates (7.5.1~7.5.4)

(7.3)

7.5.5 Confidence Intervals Computed Using Data in Subgroups

7.5.6 Nonparametric Capability Indices and Confidence Limits Some quality characteristics such as diameter, roundness, mold dimensions, and customer waiting time will be non- normal, and flatness, runout, and % contamination will have skewed distributions. Process capability indices are not robust to non-normality in the individual observations. 4 approaches for non-normal distributions: –Robust capability index –Fit a distribution to a set of data and use percentiles in an index –Transform the data to approximate normal –Resample from the n sampled observations

Robust Capability Indices

Capability Indices Based on Fitted Distributions

Data Transformation Data can be transformed so the transformed data will be approximately normally distributed. Lognormal data

Capability Indices Computed Using Resampling Methods Resampling methods have been used to approximate sampling distributions when no assumption is made of the distribution of the random variable. Bootstrapping is one type of resampling. –Naïve bootstrap: keep the original sample size, resample with replacement The standard bootstrap methods can not be relied on to produce a lower confidence limit for a capability index.

7.6 Asymmetric Bilateral Tolerances (7.4)

7.6.1 Example ABC USL62 LSL50  5662  111 T59 CpCp 222 C pk 020 C pk ’000 C pm C pmk % def

7.7 Capability Indices that are a Function of % Non-conforming

7.11 Process Capability Indices vs. Process Performance Indices Process Capability IndicesProcess Performance Indices

7.13 Software for Process Capability Indices Minitab: Process Capability IndicesProcess Performance Indices