Process and Measurement System Capability Analysis

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Chapter 7

• Process and Measurement System Capability Analysis

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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?

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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
• Histograms provide immediate, visual impression
  of process performance

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Example 7-1

• Pgs. 353-354
• This procedure works if data are distributed
  normally

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Reasons for poor process capability

• See Fig. 7-3
• Poor process centering
• Assume that this can be corrected
• Excess process variability
• Harder to correct

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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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Example

• See Fig. 7-4
• First, mest 260 psi
• Then, sest 298 260 38 psi
• Can also use normal probability plot to estimate
  fallout
• If LSL 200 psi
• Then, from Fig 7-4, about 5 will be below that
  value

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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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Example 7-2

• Pg. 359

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Table 7-3

• Process fallout for one- and two-sided
  specifications

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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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Table 7-4

• Recommended minimum values of the PCR
• For example, a new process with two-sided
  specifications has a recommended Cp of 1.50
• This implies that process fallout would be 7 ppm
• Six s would result in a Cp of 2.0

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7-3.2 Process Capability Ratio on
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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Figure 7-7

• All of the panels in the figure have Cp 2.0
• But, when the process mean shifts, the capability
  of the process can change
• Note that s does not shift

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Figure 7-7, cont.

• For panel b, N(53, 22)
• Cpk min(Cpu, Cpl)
• Cpu (62-53)/3(2) 1.5
• Cpl (53-38)/3(2) 2.5
• Cpk 1.5

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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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Example

• USL 90, LSL 80
• So, T (90 80)/2 85
• (T target value)
• Let X 84
• Then, Cpc 1.33
• (Be careful with this resultI dont trust it!)

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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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Why does it reduce to Cp?

• In the case of the normal distribution
• x.00135 m 3s
• x.99865 m 3s

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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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Example

• Pgs. 373-375

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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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Example

• Machine that fills bottles with a soft-drink
  beverage
• Each machine has many filling heads that are
  independently adjusted
• Quality characteristic measured is syrup content
  in degrees brix
• Three possible causes of variabililty
• Machines, heads, analytical tests

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Example, cont.

• Variability is sB2 sM2 sH2 sA2
• Conduct an experiment
• Say the result is as shown in Fig. 7-12
• Head-to-head variability is large
• Improve the process by reducing this variance

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7-7 Setting spec limits on discrete components

• Setting specifications to insure that the final
  product meets specifications
• As discussed previously, if normally distributed
  variables are linked, the result is normally
  distributed with mean the sum of the individual
  means, and variance the sum of the individual
  variances

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Example 7-9

• Pgs. 388-389

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Example 7-10

• Pgs. 389-390

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Example 7-11

• Pgs. 391-392

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7-8 Estimating tolerance limits

• Confidence limits
• Provide an interval estimate of the parameters of
  a distribution
• Tolerance limits
• Indicate the limits between which we can expect
  to find a specified proportion of a population

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7-8.1 Tolerance limits based on the normal
distribution

• Suppose xN(m, s2), both unknown
• Take a sample of size n and compute xbar and S2
• Natural tolerance limits might be estimated
  using
• Xbar Za/2S
• Since xbar and S are only estimates, the interval
  may or may not always contain 100(1-a) of the
  distribution

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7-8.1 Tolerance limits based on the normal
distribution

• However, we may use a constant K such that in a
  large number of samples a fraction g of the
  intervals xbar KS will include at least
  100(1-a) of the distribution
• Values of K are tabulated in Appendix Table VII
• 2.01

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Example 7-13

• Pg. 396

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Assignment

• Work odd-numbered exercises on the topics covered
  in class

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End