Upload
cuthbert-lamb
View
220
Download
3
Tags:
Embed Size (px)
Citation preview
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
7.5.6.1 Robust Capability Indices
7.5.6.2 Capability Indices Based on Fitted Distributions
7.5.6.3 Data Transformation
• Data can be transformed so the transformed data will be approximately normally distributed.
• Lognormal data
7.5.6.4 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
A B CUSL 62 62 62LSL 50 50 50 50 56 62 1 1 1T 59 59 59
Cp 2 2 2
Cpk 0 2 0
Cpk’ 0 0 0
Cpm 0.2209 0.6325 0.6325
Cpmk 0 0.6325 0% def. 0.500000 0.000000 0.500000
7.7 Capability Indices that are a Function of % Non-conforming
7.11 Process Capability Indices vs. Process Performance Indices
7.13 Software for Process Capability Indices
• Minitab: