1. 1 Process-oriented SPC and Capability Calculations Russell R. Barton, Smeal College of Business : rbarton@psu.edu, 814-863-7289rbarton@psu.edu Enrique del Castillo, Earnest Foster, Amanda Schmitt The Harold and Inge Marcus Department of Industrial and Manufacturing Engineering Penn State George Runger Industrial 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. 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 a 1, a 2,..., a n. If the process-oriented basis vectors a 1, a 2,..., a n are independent then they provide an alternative basis (or subspace if fewer than n) Y = z 1 a 1 + z 2 a 2 +... + z n a n. A = [a 1 |a 1 | …|a n ] z = A -1 y or z = (A'A) -1 A ’ y

3. 3 Applications : Chip Capacitor Manufacturing

4. 4 Silver Square Printing: Registration (position) Critical

5. 5 Process-oriented Representation for Chip Capacitors: Printing Registration Errors

6. 6 Process-oriented Representation: Determining the Basis (characteristic signatures)

7. 7 1 0 0 0 0 0 0 0 standard basis process-oriented basis uniform errorsrotation uniform stretch/shrink differential stretch/shrink a =e = ii 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 a = i 0 1 0 1 0 1 0 1 1 1 1 1 1 1 1 1 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 i = 1i = 2i = 3i = 4i = 5i = 6i = 7i = 8 diagonal stretch/shrink Process-oriented Representation: Standard vs Process-oriented Basis

8. 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 errorsrotation uniform stretch/shrink differential stretch/shrink diagonal stretch/shrink

9. 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. 10 Applications : Aircraft Stringer Drilling

11. 11 Case 1: the common cause variation is not related to the characteristic patterns: Y = Az + ,  ~ N(0,   )  Y =  , = (A ’ A) -1 A ’ 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 = A  z,A ’ +   = (A ’  Y -1 A) -1 A ’  Y -1 y. Process-oriented SPC

12. 12 1. SPC using or : separate charts for each. 2. SPC using T 2 or U 2 applied to or : a single chart, diagnosis requires extra steps, but still more effective than T 2 applied to y’s. Example: Case 3, Solder Paste Volume, Strategy 1,  z >>  , Var(Z 5 ) small Z’s vs Principal Components, EWMA Chart 52 elements rather than 208 (software difficulties with Princ. Comp.) Process-oriented SPC: Strategies

13. 13 Process-oriented SPC: Strategies Scenario ARL POBREP ARL Princ. Comp. P-value Shift z 1 911.5.301 Shift z 1,z 4 1013.039 Shift z 5 2468.000 Drift z 1 1945.000 Drift z 1, z 4 1629.010 Drift z 5 1360.000

14. 14 Three univariate indices: Cp = (USL - LSL)/6σ Process-oriented Capability

15. 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. 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

17. 17  is a bivariate “reliability” capability measure  gives multivariate proportion conforming: Integrate over bivariate normal density for the dependent case Independent case:  =  1  2 Wierda multivariate capability index : x2x2 x1x1 USL 1 LSL 1 LSL 2 USL 2 11 22

18. 18 Multivariate Process-oriented Capability Example Chip capacitor: z = A -1 y (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. 19 Multivariate Process-oriented Capability : Six Scenarios Scenarios for computing Z matrix capability Variances for  Z 1. Base 1 (  Z = 0) (1 2,.05 2,.05 2,.05 2,.05 2,.05 2,.05 2,.05 2 ) 2. Base 1 with z 1 mean shift =.5 (1 2,.05 2,.05 2,.05 2,.05 2,.05 2,.05 2,.05 2 ) 3. Base 1 with z 1 variance increase (1.5 2,.05 2,.05 2,.05 2,.05 2,.05 2,.05 2,.05 2 ) 4. Base 2 (  Z = 0) (1 2, 1 2, 1 2,.05 2,.05 2,.05 2,.05 2,.05 2 ) 5. Base 2 with z 1 mean shift =.5 (1 2, 1 2, 1 2,.05 2,.05 2,.05 2,.05 2,.05 2 ) 6. Base 2 with z 1 variance increase (1.5 2, 1 2, 1 2,.05 2,.05 2,.05 2,.05 2,.05 2 )

20. 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. 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

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27. 27 Wierda (1993) multivariate index details:  Compute  when quality variables independent:  Compute  when quality variables dependent (  known):  n p is MVN density   is covariance matrix  L and U are vectors of specifications 