Process and Measurement System Capability Analysis

1
Chapter 7

• Process and Measurement System Capability Analysis

2
7-1. Introduction

• Process capability refers to the uniformity of
  the process.
• Variability in the process is a measure of the
  uniformity of output.
• Two types of variability
• Natural or inherent variability (instantaneous)
• Variability over time
• Assume that a process involves a quality
  characteristic that follows a normal distribution
  with mean ?, and standard deviation, ?. The upper
  and lower natural tolerance limits of the process
  are
• UNTL ? 3?
• LNTL ? - 3?

3
7-1. Introduction

• Process capability analysis is an engineering
  study to estimate process capability.
• In a product characterization study, the
  distribution of the quality characteristic is
  estimated.

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7-1. Introduction

• Major uses of data from a process capability
  analysis
• Predicting how well the process will hold the
  tolerances.
• Assisting product developers/designers in
  selecting or modifying a process.
• Assisting in Establishing an interval between
  sampling for process monitoring.
• Specifying performance requirements for new
  equipment.
• Selecting between competing vendors.
• Planning the sequence of production processes
  when there is an interactive effect of processes
  on tolerances
• Reducing the variability in a manufacturing
  process.

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7-1. Introduction

• Techniques used in process capability analysis
• Histograms or probability plots
• Control Charts
• Designed Experiments

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7-2. Process Capability Analysis Using a
Histogram or a Probability Plot

• 7-2.1 Using a Histogram
• The histogram along with the sample mean and
  sample standard deviation provides information
  about process capability.
• The process capability can be estimated as
• The shape of the histogram can be determined
  (such as if it follows a normal distribution)
• Histograms provide immediate, visual impression
  of process performance.

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7-2.2 Probability Plotting

• Probability plotting is useful for
• Determining the shape of the distribution
• Determining the center of the distribution
• Determining the spread of the distribution.
• Recall normal probability plots (Chapter 2)
• The mean of the distribution is given by the 50th
  percentile
• The standard deviation is estimated by
• 84th percentile 50th percentile

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7-2.2 Probability Plotting

• Cautions in the use of normal probability plots
• If the data do not come from the assumed
  distribution, inferences about process capability
  drawn from the plot may be in error.
• Probability plotting is not an objective
  procedure (two analysts may arrive at different
  conclusions).

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7-3. Process Capability Ratios

• 7-3.1 Use and Interpretation of Cp
• Recall
• where LSL and USL are the lower and upper
  specification limits, respectively.

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7-3.1 Use and Interpretation of Cp

• The estimate of Cp is given by
• Where the estimate can be calculated using
  the sample standard deviation, S, or

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7-3.1 Use and Interpretation of Cp

• Piston ring diameter in Example 5-1
• The estimate of Cp is

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7-3.1 Use and Interpretation of Cp

• One-Sided Specifications
• These indices are used for upper specification
  and lower specification limits, respectively

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7-3.1 Use and Interpretation of Cp

• Assumptions
• The quantities presented here (Cp, Cpu, Clu) have
  some very critical assumptions
• The quality characteristic has a normal
  distribution.
• The process is in statistical control
• In the case of two-sided specifications, the
  process mean is centered between the lower and
  upper specification limits.
• If any of these assumptions are violated, the
  resulting quantities may be in error.

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7-3.2 Process Capability Ratio an
Off-Center Process

• Cp does not take into account where the process
  mean is located relative to the specifications.
• A process capability ratio that does take into
  account centering is Cpk defined as
• Cpk min(Cpu, Cpl)

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7-3.3 Normality and the Process
Capability Ratio

• The normal distribution of the process output is
  an important assumption.
• If the distribution is nonnormal, Luceno (1996)
  introduced the index, Cpc, defined as

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7-3.3 Normality and the Process
Capability Ratio

• A capability ratio involving quartiles of the
  process distribution is given by
• In the case of the normal distribution Cp(q)
  reduces to Cp

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7-3.4 More About Process Centering

• Cpk should not be used alone as an measure of
  process centering.
• Cpk depends inversely on ? and becomes large as ?
  approaches zero. (That is, a large value of Cpk
  does not necessarily reveal anything about the
  location of the mean in the interval (LSL, USL)

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7-3.4 More About Process Centering

• An improved capability ratio to measure process
  centering is Cpm.
• where ? is the squre root of expected
  squared deviation from target T ½(USLLSL),

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7-3.4 More About Process Centering

• Cpm can be rewritten another way
• where

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7-3.4 More About Process Centering

• A logical estimate of Cpm is
• where

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7-3.4 More About Process Centering

• Example 7-3. Consider two processes A and B.
• For process A
• since process A is centered.
• For process B

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7-3.4 More About Process Centering

• A third generation process capability ratio,
  proposed by Pearn et. al. (1992) is
• Cpkm has increased sensitivity to departures of
  the process mean from the desired target.

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7-3.5 Confidence Intervals and Tests on Process
Capability Ratios

• Cp
• Cp is a point estimate for the true Cp, and
  subject to variability. A 100(1-?) percent
  confidence interval on Cp is

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7-3.5 Confidence Intervals and Tests on
Process Capability Ratios

• Example 7-4. USL 62, LSL 38, n 20,
• S 1.75, The process mean is centered. The
  point estimate of Cp is
• 95 confidence interval on Cp is

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7-3.5 Confidence Intervals and Tests on Process
Capability Ratios

• Cpk
• Cpk is a point estimate for the true Cpk, and
  subject to variability. An approximate 100(1-?)
  percent confidence interval on Cpk is

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7-3.5 Confidence Intervals and Tests on
Process Capability Ratios

• Example 7-5. n 20, Cpk 1.33. An approximate
  95
• confidence interval on Cpk is
• The result is a very wide confidence interval
  ranging from below unity (bad) up to 1.67 (good).
  Very little has really been learned about actual
  process capability (small sample, n 20.)

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7-3.5 Confidence Intervals and Tests on Process
Capability Ratios

• Cpc
• Cpc is a point estimate for the true Cpc, and
  subject to variability. An approximate 100(1-?)
  percent confidence interval on Cpc is
• where

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7-3.5 Confidence Intervals and Tests on
Process Capability Ratios

• Example 7-5. n 20, Cpk 1.33. An approximate
  95
• confidence interval on Cpk is
• The result is a very wide confidence interval
  ranging from below unity (bad) up to 1.67 (good).
  Very little has really been learned from this
  result, (small sample, n 20.)

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7-3.5 Confidence Intervals and Tests on Process
Capability Ratios

• Testing Hypotheses about PCRs
• May be common practice in industry to require a
  supplier to demonstrate process capability.
• Demonstrate Cp meets or exceeds some particular
  target value, Cp0.
• This problem can be formulated using hypothesis
  testing procedures

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7-3.5 Confidence Intervals and Tests on Process
Capability Ratios

• Testing Hypotheses about PCRs
• The hypotheses may be stated as
• H0 Cp ? Cp0 (process is not capable)
• H0 Cp ? Cp0 (process is capable)
• We would like to reject Ho
• Table 7-5 provides sample sizes and critical
  values for testing H0 Cp Cp0

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7-3.5 Confidence Intervals and Tests on Process
Capability Ratios

• Example 7-6
• H0 Cp 1.33
• H1 Cp gt 1.33
• High probability of detecting if process
  capability is below 1.33, say 0.90. Giving
  Cp(Low) 1.33
• High probability of detecting if process
  capability exceeds 1.66, say 0.90. Giving
  Cp(High) 1.66
• ? ? 0.10.
• Determine the sample size and critical value, C,
  from Table 7-5.

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7-3.5 Confidence Intervals and Tests on Process
Capability Ratios

• Example 7-6
• Compute the ratio Cp(High)/Cp(Low)
• Enter Table 7-5, panel (a) (since ? ? 0.10).
  The sample size is found to be n 70 and
  C/Cp(Low) 1.10
• Calculate C

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7-3.5 Confidence Intervals and Tests on Process
Capability Ratios

• Example 7-6
• Interpretation
• To demonstrate capability, the supplier must take
  a sample of n 70 parts, and the sample process
  capability ratio must exceed 1.46.

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7-4. Process Capability Analysis Using a
Control Chart

• If a process exhibits statistical control, then
  the process capability analysis can be conducted.
  
• A process can exhibit statistical control, but
  may not be capable.
• PCRs can be calculated using the process mean and
  process standard deviation estimates.

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7-5. Process Capability Analysis Designed
Experiments

• Systematic approach to varying the variables
  believed to be influential on the process.
  (Factors that are necessary for the development
  of a product).
• Designed experiments can determine the sources of
  variability in the process.

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7-6. Gage and Measurement System
Capability Studies

• 7-6.1 Control Charts and Tabular Methods
• Two portions of total variability
• product variability which is that variability
  that is inherent to the product itself
• gage variability or measurement variability which
  is the variability due to measurement error

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7-6.1 Control Charts and Tabular Methods

• and R Charts
• The variability seen on the chart can be
  interpreted as that due to the ability of the
  gage to distinguish between units of the product
• The variability seen on the R chart can be
  interpreted as the variability due to operator.

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7-6.1 Control Charts and Tabular Methods

• Precision to Tolerance (P/T) Ratio
• An estimate of the standard deviation for
  measurement error is
• The P/T ratio is

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7-6.1 Control Charts and Tabular Methods

• Total variability can be estimated using the
  sample variance. An estimate of product
  variability can be found using

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7-6.1 Control Charts and Tabular Methods

• Percentage of Product Characteristic Variability
• A statistic for process variability that does not
  depend on the specifications limits is the
  percentage of product characteristic variability

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7-6.1 Control Charts and Tabular Methods

• Gage RR Studies
• Gage repeatability and reproducibility (RR)
  studies involve breaking the total gage
  variability into two portions
• repeatability which is the basic inherent
  precision of the gage
• reproducibility is the variability due to
  different operators using the gage.

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7-6.1 Control Charts and Tabular Methods

• Gage RR Studies
• Gage variability can be broken down as
• More than one operator (or different conditions)
  would be needed to conduct the gage RR study.

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7-6.1 Control Charts and Tabular Methods

• Statistics for Gage RR Studies (The Tabular
  Method)
• Say there are p operators in the study
• The standard deviation due to repeatability can
  be found as
• where
• and d2 is based on the of observations per
  part per operator.

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7-6.1 Control Charts and Tabular Methods

• Statistics for Gage RR Studies (the Tabular
  Method)
• The standard deviation for reproducibility is
  given as
• where
• d2 is based on the number of operators, p

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7-6.2 Methods Based on Analysis of
Variance

• The analysis of variance (Chapter 3) can be
  extended to analyze the data from an experiment
  and to estimate the appropriate components of
  gage variability.
• For illustration, assume there are a parts and b
  operators, each operator measures every part n
  times.

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7-6.2 Methods Based on Analysis of
Variance

• The measurements, yijk, could be represented by
  the model
• where i part, j operator, k measurement.

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7-6.2 Methods Based on Analysis of
Variance

• The variance of any observation can be given by
• are the variance components.

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7-6.2 Methods Based on Analysis of
Variance

• Estimating the variance components can be
  accomplished using the following formulas