Transcript TM 720 Lecture10: Short Run SPC & Gage R & R

TM 720 - Lecture 10
Short Run SPC and Gage
Reproducibility &
Repeatability
7/16/2015
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Assignment:


Reading:
•
Finish Chapters 7 and 9
•
•
Sections 7.4 – 7.8
Sections 9 – 9.2
Assignment:
•
•
•
Access Excel Template for Individuals Control Charts:
•
•
Download Assignment 7 for practice
Use the data on the HW7 Excel sheet to do the charting, verify
the control limits by hand calculations
Solutions for 6 and 7 will post on Thursday
Review for Exam II
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Review

Shewhart Control charts
•
Are for sample data from an approximate Normal distribution
•
•
•

Three lines appear on all Shewhart Control Charts
•
UCL, CL, LCL
Two charts are used:
•
•
X-bar for testing for change in location
R or s-chart for testing for change in spread
We check the charts using 4 Western Electric rules
Attributes Control charts
•
Are for Discrete distribution data
•
•
•
•
Use p- and np-charts for tracking defective units
Use c- and u-charts for tracking defects on units
Use p- and u-charts for variable sample sizes
Use np- and c-charts with constant sample sizes
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Short Run SPC

Many products are made in smaller quantities than are
practical to control with traditional SPC
•
•
•
In order to have enough observations for statistical control
to work, batches of parts may be grouped together onto a
control chart
This usually requires a transformation of the variable on
the control chart, and a logical grouping of the part
numbers (different parts) to be plotted.
A single chart or set of charts may cover several different
part types
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DNOM Charts

Deviation from Nominal
•
Variable computed is the difference between the
measured part and the target dimension
xi  M i  Tp
where:
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Mi is the measured value of the ith part
Tp is the target dimension for all of part number p
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DNOM Charts

The computed variable (xi)
is part of a sub-sample of
size n
•
•

xi is normally distributed
n is held constant for all
part numbers in the chart
group.
Charted variables are x
and R, just as in a
traditional Shewhart control
chart, and control limits are
computed as such, too:
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UCL  x  A2 R  A2 R
CL  x  0
LCL  x  A2 R   A2 R
UCL  D4 R
CL  R
LCL  D3 R
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DNOM Charts

Usage:
•
•
•
A vertical dashed line is used to mark the charts at the point
at which the part numbers change from one part type to the
next in the group
The variation among each of the part types in the group
should be similar (hypothesis test!)
Often times, the Tp is the nominal target value for the
process for that part type
•
•
Allows the use of the chart when only a single-sided
specification is given
If no target value is specified, the historical average (x) may be
used in its’ place
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Standardized Control Charts

If the variation among the part types
within a logical group are not similar, the
variable may be standardized
• This is similar to the way that we converted
from any normally distributed variable to a
standard normal distribution:
• Express the measured variable in terms of how
many units of spread it is away from the central
location of the distribution
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Standardized Charts – x and R

Standardized Range:
•
Plotted variable is
Ri
R 
Rj
s
i
where:
Ri is the range of measure values for the ith
sub-sample of this part type j
Rj is the average range for this jth part type
UCL  D4
CL  R j
LCL  D3
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Standardized Charts – x and R

Standardized x:
•
Plotted variable for the sample is
s
i
x 
where:
M i  Tj
Rj
Mi is the mean of the original measured values for
this sub-sample of the current part type (j)
Tj is the target or nominal value for this jth part type
UCL   A2
CL  0
LCL   A2
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Standardized Charts – x and R

Usage:
•
•
Two options for finding Rj:
• Prior History
• Estimate from target σ:
Examples:
 d2 
R j  σ  
 c4 
• Parts from same machine
•
with similar dimensions
Part families – similar part
tolerances from similar
setups and equipment
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Standardized Charts – Attributes

Standardized zi for Proportion Defective:
•
Plotted variable is
zi 
•
pi  p
p(1  p)
n
Control Limits:
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UCL  3
CL  0
LCL  3
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Standardized Charts – Attributes

Standardized zi for Number Defective:
•
Plotted variable is
zi 
•
npi  n p
n p(1  p)
Control Limits:
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UCL  3
CL  0
LCL  3
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Standardized Charts – Attributes

Standardized zi for Count of Defects:
•
Plotted variable is
zi 
•
ci  c
c
Control Limits:
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UCL  3
CL  0
LCL  3
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Standardized Charts – Attributes

Standardized zi for Defects per Inspection Unit:
•
Plotted variable is
zi 
•
ui  u
u
n
Control Limits:
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UCL  3
CL  0
LCL  3
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Gage Capability Studies

Ensuring an adequate gage and inspection system capability
is an important consideration!

In any problem involving measurement the observed
variability in product due to two sources:

•
Product variability - σ2product
•
Gage variability - σ2gage
i.e., measurement error
Total observed variance in product:
σ2total = σ2product + σ2gage (system)
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e.g. Assessing Gage Capability

Following data were taken by one
operator during gage capability study.
Measurement
Part #
1
2
3
4
5
6
…
17
18
19
20
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1
21
24
20
27
19
23
…
20
19
25
19
2
20
23
21
27
18
21
…
20
21
26
19
x-bar R
20.5
1
23.5
1
20.5
1
27
0
18.5
1
22
2
…
…
20
0
20
2
25.5
1
19
0
x  22.3
R  1.0
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e.g. Assessing Gage Capability
Cont'd

Estimate standard deviation of measurement error:
 gage  R d2  1.0 1.128  0.8865

Dist. of measurement error is usually well approximated by
the Normal, therefore
•
Estimate gage capability:
6ˆ gage  6  0.8865  5.32
•
That is, individual measurements expected to vary as much as
3 gage  3  0.8865  2.67
owing to gage error.
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Precision-to-Tolerance
(P/T) Ratio


Common practice to compare gage capability with the width
of the specifications
In gage capability, the specification width is called the
tolerance band
•
(not to be confused with natural tolerance limits, NTLs)
P T

USL  LSL
Specs for above example: 32.5 ± 27.5
PT

6ˆ gage
6ˆ gage
USL  LSL

6  0.8865
55
 0.0967
Rule of Thumb:
•
P/T  0.1  Adequate gage capability
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Estimating Variance Components of Total
Observed Variability

Estimate total variance:
2
ˆ total
 S2
2
1 n
S 
xi  x 


n  1 i 1
2


1 n
2
   xi  22.3  10.05
19 i 1
Compute an estimate of product variance
Since :
2
2
2
 total   product   gage
ˆ
2
product
 ˆ
2
total
 ˆ
2
gage
 10.05   0.8865  9.26
2
ˆ product  9.26  3.04
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Gage Std Dev Can Be Expressed as % of
Product Standard Deviation

Gage standard deviation as percentage
of product standard deviation :
ˆ gage
0.8865
100% 
100%  29.2%
ˆ product
3.04

This is often a more meaningful
expression, because it does not depend
on the width of the specification limits
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Using x and R-Charts for a Gage
Capability Study

On x chart for measurements:
•
•
Expect to see many out-of-control points
x chart has different meaning than for process control
•
shows the ability of the gage to discriminate between units
(discriminating power of instrument)
Why? Because estimate of σx used for control limits is
based only on measurement error, i.e.:  x   gage  R d 2
X-bar Chart for Measurements
30
X-bar

UCL = 24.18
28
CTR = 22.30
26
LCL = 20.42
24
22
20
18
0
4
8
12
16
20
Subgroup
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Using x and R-Charts for a Gage
Capability Study

On R-chart for measurements:
•
•
R-chart directly shows magnitude of measurement error
Values represent differences between measurements made by same
operator on same unit using the same instrument
Range Chart for Measurements
4
UCL = 3.27
CTR = 1.00
Range
3
LCL = 0.00
2
1
0
0
4
8
12
16
20
Subgroup

Interpretation of chart:
•
•
In-control: operator has no difficulty making consistent measurements
Out-of-control: operator has difficulty making consistent measurements
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Repeatability & Reproducibility:
Gage R & R Study

If more than one operator used in study then measurement
(gage) error has two components of variance:
σ2total = σ2product + σ2gage
σ2reproducibility +
σ2repeatability


Repeatability:
•
σ2repeatability - Variance due to measuring instrument
Reproducibility:
•
σ2reproducibility - Variance due to different operators
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Ex. Gage R & R Study


20 parts, 3 operators, each operator measures each part twice
Operator i
xi
Ri
1
22.30
1.00
2
22.28
1.25
3
22.10
1.20
Estimate repeatability (measurement error):
R
1
3
R

1
3
1.00  1.25  1.20   1.15
1
 R2  R3 
ˆ repeatabiltiy 
•
R
1.15

 1.0195
d 2 1.128
Use d2 for n = 2 since each range uses 2 repeat measures
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Ex. Gage R & R Study Cont'd

Estimate reproducibility:
•
Differences in xi  operator bias since all operators measured
same parts
xmax  max  x1 , x2 , x3   22.30
xmin  min  x1 , x2 , x3   22.10
Rx  xmax  xmin  0.20
ˆ reproducibility
•
Rx
0.20


 0.1181
d 2 1.693
Use d2 for n = 3 since Rx is from sample of size 3
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Ex. Gage R & R Study Cont'd

Total Gage variability:
2
2
2
ˆ gage
 ˆ repeatability
 ˆ reproducibility
2
ˆ gage
 1.0195   0.1181  1.0533
2

2
Gage standard deviation (measurement error):
ˆ gage  1.0533  1.0263


P/T Ratio:
•
Specs: USL = 60, LSL = 5
•
Would like P/T < 0.1!
6ˆ gage
6 1.0263
P


 0.1120
T USL  LSL
60  5
Note: P T  0.1120  0.1
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Comparison of Gage Capability
Examples
σ2 repeatability
σ2 reproducibility
Single
operator
Three
operators

1.0195
0.1181
σ2 product
P/T
0.8865
0.0967
1.0263
0.1120
Gage capability is not as good when we account
for both reproducibility and repeatability
•
•
Train operators to reduce σ2reproducability from 0.1181
Since σ2repeatability = 1.0195 (largest component), direct
effort toward finding another inspection device.
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Gage Capability Based on
Analysis of Variance

A gage R & R study is actually a designed
experiment

Therefore ANOVA can be used to analyze the
data from an experiment and to estimate the
appropriate components of gage variability

Assume there are:
•
•
•
a parts
b operators
each operator measures every part n times
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The measurements, yijk, are
represented by a model
yijk  m  t i  b j  tb ij  e ijk


where
•
•
•
•
•
constant m – overall measurement mean
r.v. ti – effect from part differences
r.v. bj – effect from operator differences
r.v. tbij – joint effect of parts & operator differences
r.v. eijk – error from measuring instrument
with
•
•
•
i = part (i = 1, …, a)
j = operator (j = 1, …, b)
k = measurement (k = 1, …, n)
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The Variance Components for the Gage
R&R Study Using the Model

The variance of an observation yijk is
Var  yijk         
2
t

2
b
2
tb
So:
•  t2
•  b2
•  tb2
2

•
e
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is the variance from parts
is the variance from operators
is the joint variance from parts & operators
is the variance from measuring instrument
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2
e
Repeatability & Reproducibility
2
2
2
2


Var  yijk    t   b   tb   e
Reproducibility
(Operators)
Repeatability
(Measuring Device)
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Gage R&R – ANOVA Method StatGraphics
Output
ANOVA Table
Source
Sum Squares
Df Mean Square F-Ratio P-Value
-------------------------------------------------------Oper
0.95
2
0.475
Part
957.758
19
50.4083
Oper*Part
128.717
38
3.38728
3.42
0.0000
Residual
59.5
60
0.991667
-------------------------------------------------------Total
1146.92 119
Operator variable: Operator
Part variable: Part
Trial variable: Trial
Measurement variable: Measurement
3 operators
20 parts 2 trials
Estimated Estimated
Percent
Sigma
Variance of Total
-----------------------------------------------Repeatability
0.995825
0.991667
45.29
Reproducibility
0.0
0.0
0.00
Interaction
1.09444
1.19781
54.71
-----------------------------------------------R & R
1.47969
2.18947
100.00
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6ˆ gage
P

T
USL  LSL
6 1.4797 

 0.1614
60  5
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Comparison of Gage Capability
Examples
σ2repeatability
σ2reproducibility
Single
operator
σ2gage
P/T
0.8865
0.0967
Three
operators
(Tabular
Method)
1.0195
0.1181
1.0263
0.1120
Three
operators
(ANOVA
Method)
0.9958
1.0944
1.4797
0.1614
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Questions & Issues

Topics for Exam II:
•
•
•
•
•
Shewhart Continuous Variable Control Charts
•
•
•
X-bar and R; X-bar and S-charts
Control Limits from samples or standards using table
Western Electric Rules
Shewhart-Like Discrete Variable Control Charts
•
•
P, NP, C, U-charts
Defectives vs. Defects; Variable or Constant Sample Sizes
Control Charts for Individual Measurements
•
X and Moving Range; Moving Average, EWMA, CUSUM
Short Run Statistical Process Control
•
DNOM and Standardized charts (continuous / discrete)
Gage Repeatability and Reproducibility
•
Control Chart Method – only!
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