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![Page 1: 1 Process-oriented SPC and Capability Calculations Russell R. Barton, Smeal College of Business : rbarton@psu.edu, 814-863-7289rbarton@psu.edu Enrique.](https://reader035.fdocuments.in/reader035/viewer/2022062407/56649d035503460f949d5fec/html5/thumbnails/1.jpg)
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Process-oriented SPC and Capability Calculations
Russell R. Barton, Smeal College of Business : [email protected], 814-863-7289Enrique del Castillo, Earnest Foster, Amanda Schmitt
The Harold and Inge Marcus Department of Industrial and Manufacturing EngineeringPenn State
George RungerIndustrial Engineering
Arizona State
Collaboration with Jeff Tew and Lynn Truss, GM Enterprise Systems Lab, David Drain and John Fowler, Intel and Arizona State University, and graduate students at PSU and ASU
Process-oriented Representation of Multivariate Quality Data
Applications
Process-oriented SPC
Process-oriented Capability
Research Activities
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Process-oriented Representation of
Multivariate Quality Data
Define the set of n measured deviations from nominal to be a multivariate quality vector Y.
Suppose that n different patterns of interest for n different process causes, say a1, a2, ... , an.
If the process-oriented basis vectors a1, a2, ... , an are independent then they provide an alternative basis (or subspace if fewer than n)
Y = z1a1 + z2a2 + ... + znan.
A = [a1|a1| …|an]
z = A-1y or z = (A'A)-1A’y
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Applications : Chip Capacitor Manufacturing
Silver
Clay
PRESSURE
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Silver Square Printing: Registration (position) Critical
SilverSquares
ClaySubstrate
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Process-oriented Representation for Chip
Capacitors: Printing Registration Errors
actual
target
i ii
iv iii
2.1
1.4
1.7
3.9
1.6
-2.8
1.8
1.7
1.7
3.9
x =
h
v
h
v
h
v
h
v
i
ii
iii
iv
}
}
}
}
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Process-oriented Representation: Determining
the Basis (characteristic signatures)
Misregister
variationsin flat thickness locating fences
screen
stretch
rotationheight
verticalmisplacement
horizontalmisplacement
slurryinhomogeneity
variations insheet pull speed
frametwist
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10000000
standard basis
process-orientedbasis
uniform errors rotation uniformstretch/shrink
differentialstretch/shrink
a = e =i i
01000000
00100000
00010000
00001000
00000100
00000010
00000001
10101010
a =i
01010101
111
-1-1-1-11
1-111
-11
-1-1
-101010
-10
01010
-10
-1
10
-1010
-10
010
-1010
-1
i = 1 i = 2 i = 3 i = 4 i = 5 i = 6 i = 7 i = 8
diagonalstretch/shrink
Process-oriented Representation: Standard vs
Process-oriented Basis
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Process-oriented Representation
Standard Representation of y = (0, 1, 2, -1, 0, -1, -2, 1)'
POBREP Representation of y = (0, 1, 2, -1, 0, -1, -2, 1)’ is z = (0, 0, 1, 0, 1, 0, 0, 0)’
uniform errors rotation uniformstretch/shrink
differentialstretch/shrink
diagonalstretch/shrink
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Applications : Solder Paste Deposition
Drops of solder paste
Location for Processor Chip
Gonzalez-Barreto Example:
•52 leads per side
•208 solder drop volume measurements in quality vector
•5 process-oriented basis elements:
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Applications : Aircraft Stringer Drilling
A Drilled Stringer
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Case 1: the common cause variation is not related to the characteristic patterns:
Y = Az + , ~ N(0,)
Y = ,
= (A’A)-1A’y. Case 2: the common cause variation is due solely to process-oriented basis elements:
Y = AZ, Z ~ N(0,z)
Case 3: Of course, many situations might fall between these two cases, giving:
Y = AZ + , ~ N(0,), Z ~ N(0,z)
Y = Az,A’ +
= (A’Y-1A)-1A’Y
-1y.
Process-oriented SPC
olsz
wlsz
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1. SPC using or : separate charts for each.
2. SPC using T2 or U2 applied to or : a single chart, diagnosis requires extra steps, but still more effective than T2 applied to y’s.
Example: Case 3, Solder Paste Volume, Strategy 1, z >> , Var(Z5) small
Z’s vs Principal Components, EWMA Chart
52 elements rather than 208 (software difficulties with Princ. Comp.)
Process-oriented SPC: Strategies
olsz wlsz
olsz wlsz
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Process-oriented SPC: Strategies
ScenarioARL
POBREPARL Princ.
Comp.P-value
Shift z1 9 11.5 .301
Shift z1,z4 10 13 .039
Shift z5 24 68 .000
Drift z1 19 45 .000
Drift z1, z4 16 29 .010
Drift z5 13 60 .000
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Three univariate indices:
Cp = (USL-LSL)/6σ
22 )(6 T
LSLUSLCpm
3
,3
minLSLUSL
Cpk
Process-oriented Capability
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Process Capability and Multivariate
Capability Indices
Taam et al.: Assumed elliptical specifications
Shahriari et al.: Presented three numbers that describe
multivariate capability
Chen: A general approach allowing rectangular or
elliptical specifications and non-normal distributions
Wierda: Direct computation of percentage conforming
approach
(Taam et. al (1993), Shahriari et. al (1995), Chen (1994),Wierda (1992))
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Wierda (1993) approach to the multivariate index:
Multivariate index proposed that uses p-dimensional rectangular specification area.
Minimum expected or potentially attainable proportion of non-conformance items approach.
Original “proportion conforming” definition of capability indices is explicitly preserved
= probability of producing a good part
)(3
1 1 θMCpk
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is a bivariate “reliability” capability measure
gives multivariate proportion conforming: Integrate over bivariate normal density for the dependent case
Independent case: = 12
Wierda multivariate capability index :
x2
x1
USL1LSL1
LSL2
USL2
1
2
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Multivariate Process-oriented Capability Example
Chip capacitor: z = A-1y (Eight z’s per part)
x rectangular specifications LSL < x < USL also apply to Az (since x = Az, LSL < Az < USL )
Often, covariance matrix z will have zero non-diagonal elements—independent causes
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Multivariate Process-oriented Capability : Six Scenarios
Scenarios for computing Z matrix capability
Variances for Z
1. Base 1 (Z = 0) (12, .052, .052, .052, .052, .052, .052, .052)
2. Base 1 with z1 mean shift =.5 (12, .052, .052, .052, .052, .052, .052, .052)
3. Base 1 with z1 variance increase (1.52,.052, .052, .052, .052, .052,.052, .052)
4. Base 2 (Z = 0) (12, 12, 12, .052, .052, .052, .052, .052)
5. Base 2 with z1 mean shift =.5 (12, 12, 12, .052, .052, .052, .052, .052)
6. Base 2 with z1 variance increase (1.52, 12, 12, .052, .052, .052, .052, .052)
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Errors in Capability Estimates
Scenario Actual yield
Estimated yield
Based on Y
Estimated yield
Z
1. .94
.67 .91
2. .91
.63 .88
3.
.79 .50 .84
4.
.54 .28 .59
5. .51 .28 .57
6. .42 .21 .44
Multivariate Capability Errors without POBREP (Z values)
Based on
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Process-oriented SPC and Capability Calculations
Multivariate Capability and SPC - difficult to interpret
Process-Oriented Multivariate SPC/Capability Vectors
interpretable
practical (can be calculated with adequate precision) in many cases
efficient
Acknowledgments: NSF DDM-9700330, DMI-0084909 , GM Enterprise Systems Lab
Conclusions
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Wierda (1993) multivariate indexdetails:
Compute when quality variables independent:
Compute when quality variables dependent
( known):
np is MVN density
is covariance matrix L and U are vectors of
specifications
1
1 1
1
1 11 ΦΦ
s
XLSL
s
XUSL
p
p p
p
p pp
s
XLSL
s
XUSLθ ΦΦ
p ... ˆ 21).ˆ (1/3Φˆ 1 pkCM
dyn
1n,X|yn
U][L,
p )(ˆ
means of vector a is X