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Fractional Box-Behnken Designs

Fractional Box-Behnken Designs. David J. Edwards Virginia Commonwealth University Robert W. Mee The University of Tennessee Fall Technical Conference October 8-9, 2009. Response Surface Methods (RSM). Sequential RSM Strategy (Box and Wilson (1951))

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Fractional Box-Behnken Designs

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  1. Fractional Box-Behnken Designs David J. Edwards Virginia Commonwealth University Robert W. Mee The University of Tennessee Fall Technical Conference October 8-9, 2009

  2. Response Surface Methods (RSM) • Sequential RSM Strategy (Box and Wilson (1951)) • 1. Initial small design to estimate linear main effects • 2. Exploration along path of steepest ascent • 3. Repeat step 1 in a new optimal location • 4. Augment to complete a 2nd-order design (add axial runs and center point runs) • 5. Optimization based on fitted second-order model

  3. One-Step RSM • Cheng and Wu (2001) (CW) introduce the idea of response surface exploration using only one 3-level design • Factor Screening from Design • Begin with 3-level regular or nonregular design in some number of factors • CW suggests a main-effects only analysis • Interactions are not considered until after screening • Assumes strong effect heredity

  4. One-Step RSM (cont.) • Projection and Optimization from same Design • Project 3-level design onto the factors identified as important when screening • Projected design must be a second-order design (termed eligible) in order to explore the response surface • That is, we want to estimate • CW and Xu, Cheng, and Wu (XCW) provides a set of optimal designs in terms of the proportion of eligible projections and D-efficiency

  5. Example 1 – PVC Insulation • Nine factors in 27-runs: 39-6 • Screening step identifies factors A, B, C, D, and G • There is no eligible projected design of four or five of these factors in the 39-6 • CW finds an eligible projection in the three most significant factors

  6. Example 2 – XCW Recommended OA • 3-level, 18-run Orthogonal Array (OA) • Response is simulated from the following ‘true’ model • Y=10xD + 9xE + 8xG + 5xDG -6xAD -6xAG + 6xD2 • Fit the following polynomial model: • Factors D, E, and G stand out as active • Project the OA onto D, E, and G (this is eligible) • Fit a second-order model in those three factors

  7. Example (cont.) Clearly, we have missed the opportunity to explore the large interactions involving factor A. True model: Y=10xD + 9xE + 8xG + 5xDG -6xAD -6xAG + 6xD2

  8. What have we learned? • Two problems with CW’s method: • Initial screening stage does not entertain the possibility of two-factor interactions – so we miss out on potentially important effects • Projection onto the factors of interest does not always yield a second-order design • Let’s consider a different initial design

  9. Box-Behnken Designs (BBDs) • Popular 3-level designs for estimating second-order models in spherical regions • Possess a simple construction method by combining two-level factorial design with incomplete block designs (IBDs) • For 5 or more factors, BBDs contain many more runs than required to fit a second-order model • Thus, it is somewhat unusual to see a BBD with more than 5 factors used in practice

  10. Box-Behnken Design Construction Each set of 8 runs forms a 23. 7*8+n0 runs

  11. Fractional Box-Behnken Designs 23-1 23-1 23-1 23-1 23-1 23-1 23-1 Each set of 8 runs forms a 23. 7*8+n0 runs Each set of 4 runs forms a 23-1 7*4 + n0 runs

  12. Fractional Box-Behnken Designs (FBBDs) • Motivated by a D-optimal design search for three-level designs with smaller run sizes than BBDs • Use Bayesian D-optimality when searching for designs with fewer runs than required to estimate a second-order model • Spherical design region –wider ranges • Constructed by combining IBDs with fractions of two-level designs composing a BBD • Can either estimate the second-order model or has a simpler aliasing structure than recommended OA’s

  13. Aliasing of 29-run FBBD (7-factors)

  14. Aliasing of XCW 27-run OA (7-factors)

  15. Proposed FBBDs

  16. One-Half Fractions of BBDs • t=5 (k=2): • Each resolution I subset is paired with another “opposite” subset (e.g. I=A with I=-A) • Estimates full second-order model • t=6,7,9 (k=3): • 23-1 resolution III subsets • Has fewer runs than a second-order model, yet possesses a simple aliasing structure to exploit for analysis; more later • t=9, 10, 11, 12, 13 (k=4): • 24-1 resolution III subsets • Each factor may appear at most 3 times among the t generators in order to estimate the second-order model (see example, next slide) • Maximum D-efficiency is achieved when each generator involves factor pairs associated with l=2 (when based on a partially balanced IBD)

  17. One-Half Fraction of BBD Example • t=10

  18. Three-Quarter Fractions of BBDs • Needed for t=4 (k=2),6,7,9 (k=3) in order to estimate second-order model • Exclude runs with defining relation of the form I=P=-Q=-PQ from full 2k • Maximum D-efficiency: • t=4: See example • t=6: See example • t=7,9: use any three-quarter fraction, differences of at most 1% among D-efficiencies

  19. Three-Quarter Fractions (cont.) • Example • t=4

  20. Three-Quarter Fractions (cont.) • Example • t=6 Note: AD, BE, and CF appear together in two blocks (i.e. l1=2)

  21. Design Comparisons • FBBDs either outperform or are comparable to the 18 and 27-run OAs as well as D-optimal designs of the same run size in terms of the number of eligible projections and D-efficiency • Simulation results indicate that FBBDs are able to better identify important factors using the main effects only analysis strategy vs. the OAs proposed by CW and XCW • Since, Box-Behnken designs are often D-optimal for second-order models in a spherical region, it makes sense that fractions of these designs perform well

  22. Analysis Strategy (when we cannot estimate the 2nd order model) • What do we assume? • Effect Hierarchy • Effect Sparsity (not necessarily factor sparsity) • Weak Effect Heredity • Structure of FBBDs allows us to obtain r=3,4 independent estimates of each factor’s linear main effect

  23. Analysis Strategy (Simulated Data) • 9-factor, 49-run FBBD, r=4 • Simulate from: • Aliasing:

  24. Analysis Graphic

  25. Analysis Example (cont.) • Estimate a PSE based on deviations from median • Compare Uj’s to critical values • Interactions aliased with B, E, and J are most likely at a=0.05 • NOTE: Using the 27-run OA of XCW, no linear or quadratic effects were detected at 10% using the CW analysis strategy • Now project onto A, B, C, E, F, G, H (which is eligible)

  26. Real Data Example • Experiment conducted to optimize a very-large-scale-integrated (VLSI) process, device, and circuit design • They use a 50-run, 6 factor Box-Behnken design • Authors identify all terms in a second-order model involving factors D and E to be important • We use only half their Box-Behnken design (FBBD with 25-runs and 6-factors)

  27. VLSI Data Graphic

  28. VLSI Data Example • Interactions partially aliased with A and D are likely • Same effects stand out in graphic • All interactions satisfy weak effect heredity

  29. VLSI Data – Second-order Model • Project onto factors A, B, D, E • R2 = 0.9977 • We identify the same effects as the authors

  30. Comments Regarding FBBDs • Provide a useful alternative to the OAs of CW and XCW • More real case studies involving Box-Behnken designs would be useful • What properties to FBBDs of smaller run sizes have? • Develop critical values for analysis based on center point replication

  31. Final Comments • Benefits of FBBDs • 1. Simple to construct • More intuitive than optimal designs • Software is not required • 2. Augmentation is straightforward • Resolving analysis ambiguity or increasing precision can be done by adding runs to one or more fractions • Or one may complete the runs required for a full Box-Behnken design

  32. Thank you! Any Questions?

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