1. Process-Oriented Basis Representations (POBREP) for Multivariate SPC: Tracing Errors to their Source Russell R. Barton Penn State, Smeal College of Business Acknowledgments: J. McCool, D. Gonzalez-Barreto, E. Foster Multivariate Repeated Measurements - Motivation A Process-Oriented Approach Math Details Example If time: POBREP for Multivariate Capability Many products consist of multiple similar measurements, such as temperatures, thickness or registration errors at multiple locations. Under such conditions, it is possible to produce process diagnostics analyzing the multivariate process quality vector using a process-oriented basis. Many potential production problems have characteristic signatures that can be detected in the multivariate quality vector.

2. POBREP for Capturing Process Knowledge Chip Capacitor Manufacturing

3. POBREP for Capturing Process Knowledge Screen Printing of Silver Squares: Registration Errors Problematic

4. POBREP for Capturing Process Knowledge Measuring Registration Error

5. Define the set of n measured deviations from nominal to be a multivariate quality vector x. In this example suppose horizontal and vertical registration errors measured only for the pads at each corner of the sheet: x 1 is horizontal error at upper left pad, x 2 is vertical error at upper left pad, x 3 is horizontal error at upper right pad, etc. A littl math: notation for multivariate quality vector

6. Notation for multivariate quality vector Measuring Registration Error

7. A process-oriented approach The situation: a set of 8 misregistration numbers is hard to interpret SPC using Hotelling’s T 2 or principal components is complicated and not intuitive Ideally, the link to specific causes would be clear

8. Process-specific causes of misregister and characteristic signatures

9. Suppose have n different characteristic signatures 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 linearly independent (not linear combinations of each other) then they provide an alternative ‘basis’ for representing the x data as a linear combination of the signatures: x = z 1 a 1 + z 2 a 2 +... + z n a n. Mathematical notation: POBREP vector

10. x = z 1 a 1 + z 2 a 2 +... + z n a n. The z = (z 1, z 2,..., z n )' found by computing a matrix inverse and then doing a simple linear calculation: z = A -1 x A is the matrix consisting of the column vectors a 1, a 2,..., a n. These are the characteristic signatures. We call this basis { a 1, a 2,..., a n } a process-oriented basis. Thus POBREP: z is a process-oriented basis representation of the original data vector, x. Mathematical notation: POBREP vector

11. Screen printing example

12. 1 0 0 0 0 0 0 0 standard basis process-oriented basis uniform errorsrotation uniform stretch/shrink differential stretch/shrink e = i 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

13. Screen printing example z = A -1 x Inverse easy: use Excel or Octave (free MATLAB) A -1 =

14. Excel instructions: http://judgeg.myweb.port.ac.uk/SAME/Xmatinv.pdf Step 1 Highlight the block of cells for the inverse (for example if you are inverting a 3x3 matrix this should also be 3x3). In my illustration the matrix is in cells A1:C3 and the inverse is going to go in cells A6:C8. Step 2 In the top left hand cell of the new block (A6) type the following =MINVERSE( Step 3 Then use the mouse to paste over the cells where the matrix to invert is situated (i.e. A1:C3) Step 4 Enter the close bracket symbol ) Step 5 Press the following keys together Ctrl Shift Enter

15. Screen printing example z = A -1 x Inverse easy: use Excel or Octave (free MATLAB)

16. Screen Printing Example Standard Representation: x = (0, 1, 2, -1, 0, -1, -2, 1)'

17. z = (z 1, z 2,..., z n )‘ Let’s show the calculation of z 1. We will need the standard representation, x, and the first row of A -1 : x = (0, 1, 2, -1, 0, -1, -2, 1)‘ First row of A -1 : (0, 1/4, 0, 1/4, 0, 1/4, 0, 1/4) So the calculation is: z 1 = 0*0 + ¼*1 + 0*2 + ¼*-1 + 0*0 + ¼*-1 + 0*-2 + ¼*1 = 0 That means we observe NO HORIZONTAL SHIFT. KEY: once A -1 is constructed, this calculation is easy in Excel. The calculation

18. Screen Printing Example Standard Representation of x = (0, 1, 2, -1, 0, -1, -2, 1)' POBREP Representation of x = (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

19. x = z 1 a 1 + z 2 a 2 +... + z n a n. Using the process-oriented basis representation z, of the original vector x, diagnosis is possible: Potential causes are associated with patterns (a i ) having positive or negative coefficients (z i ) that are large in magnitude. These patterns are linked with one or more specific causes. Further, if the a i are scaled so that the maximum magnitude is 1, the z i value indicates the worst error magnitude introduced by this cause. (our example: one unit of error from rotation, one from horizontal stretch). The power of POBREP

20. POBREP for Data Reduction In many interesting cases, would like to keep full set of measurements (e.g. all misregistration errors) but have a relatively small set of signatures – giving both data reduction and cause connection. POPBREP facilitates this: instead of computing A- 1, solve x = Az by least squares

21. INFORMS Fall 200021 Fine Pitch Component with CCD =.020  and 208 leads Q: in this case is ((z 1 ), (z 2 ), …, (z 208 )) PRACTICAL?? POBREP for Data Reduction

22. Four Basis Elements for Fine Pitch Component Example a1a1 a2a2 a3a3 a4a4 POBREP for Data Reduction

23. Summary: POBREP Diagnosis Methodology Observed Error Patterns (x) Hypothesized or Observed Process Deviations Process Oriented Basis Matrix A A = [ a 1 | a 2 |.....…a n ] Process x = Az +  - solving the linear system (via least squares if A is not full rank) will provide a representation of the error vector in the basis matrix space: z i are coefficients for the a i......... Potential process causes are associated with patterns having large z i coefficients a 1 = z1z1 z2z2 z4z4 z3z3 znzn ……... 1010101010101010

24. Since causes are associated with signatures (a i ) having positive or negative coefficients (z i ) that are large in magnitude, multivariate SPC with POBREP can be more informative than univariate SPC. Univariate SPC: monitor for special cause variation, then separately investigate to find cause. Multivariate SPC: POBREP z coefficients give the cause! POBREP vector for SPC

25. POBREP for Diagnosis-based SPC: Charts for z Coefficients 05101520253035404550 -10 0 10 05 1520253035404550 -10 0 10 05 1520253035404550 -10 0 10 Basis # 2 Basis # 3 Basis # 1

26. Multivariate SPC – ‘usual’ methods (principal components, Hotellings T 2 ) difficult to interpret A Process-Oriented Multivariate Vector, z interpretable practical (can be calculated easily and with adequate precision) in many cases can induce independence between components Conclusions Process-Oriented Basis Representations (POBREP) for Multivariate SPC: Tracing Errors to their Source

28. POBREP and Multivariate Capability POBREP can address similar issues in multivariate capability. A brief overview…

29. Three widely accepted univariate indices: Cp = (USL - LSL)/6σ

30. 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))

31. Multivariate capability index literature review summary:

32. 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

33. 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 

34.  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 graphical aid: x2x2 x1x1 USL 1 LSL 1 LSL 2 USL 2 11 22

35. Current Limitations in Multivariate Capability: Estimating  x is difficult when there are many quality variables. Interpretation is difficult when one number represents the joint affect of many variables.

36. Multivariate Process-oriented Capability Method Example z = A -1 x (Eight z’s per part) Z = [z 1 | z 2 |…| z 100 ]. Using Z and the specification limits, capability can be computed Often, covariance matrix  z will have zero non- diagonal elements—independent causes

37. Multivariate Process-oriented Capability Method Example If the values 1 and –1 are in each column at least once, the full affect of basis elements is estimated x rectangular specifications LSL < x < USL may make less sense than specifications on z components – because of cause connection and scaling to match maximum x-deviation for a specific cause.

38. Using z values instead of x likely to yield independence Independent case:  =  1  2 POBREP multivariate capability index graphical aid: z2z2 z1z1 USL 1 LSL 1 LSL 2 USL 2 11 22

39. Multivariate Capability Indices using Process-Oriented Basis Representations M ultivariate Capability Indices - difficult to interpret A Process-Oriented Multivariate Capability Vector interpretable practical (can be calculated with adequate precision) in many cases can induce independence between components Conclusions