1

Process-oriented SPC and Capability Calculations

Russell R. Barton, Smeal College of Business : rbarton@psu.edu, 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

2

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

3

Applications : Chip Capacitor Manufacturing

Silver

Clay

PRESSURE

4

Silver Square Printing: Registration (position) Critical

SilverSquares

ClaySubstrate

5

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

}

}

}

}

6

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

7

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

8

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

9

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:

10

Applications : Aircraft Stringer Drilling

A Drilled Stringer

11

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

12

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

13

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

14

Three univariate indices:

Cp = (USL-LSL)/6σ

22 )(6 T

LSLUSLCpm

3

,3

minLSLUSL

Cpk

Process-oriented Capability

15

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))

16

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

17

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

18

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

19

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)

20

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

21

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

22

23

24

25

26

27

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