1. Industrial Applications of Response Surface Methodolgy John Borkowski Montana State University Pattaya Conference on Statistics Pattaya, Thailand

2. Outline of the Presentation 1.The Experimentation Process 2.Screening Experiments 3.2 k Factorial Experiments 4.Optimization Experiments 5.Mixture Experiments 6.Final Comments

3. The Experimentation Process

4. Defining Experimental Objectives  Researchers often discover after running an experiment that the data are insufficient to meet objectives  The first and most important step in an experimental strategy is to clearly state the objectives of the experiment.  The objective is a precise answer to the question “What do you want to know when the experiment is complete?”

5. 2. Screening Experiments  The experimenter wants to determine which process variables are important from a list of potentially important variables.  Screening experiments are economical because a large number of factors can be studied in a small number of experimental runs.  The factors that are found to be important will be used in future experiments. That is, we have screened out the important factors from the list.

6. 2. Screening Experiments  Common screening experiments are 1.Plackett-Burman designs 2.Two-level full-factorial (2 k ) designs 3.Two-level fractional-factorial (2 k-p ) designs  Plackett-Burman designs allow you to study as many as k-1 factors in k points where k = 12, 20, 24… (k is a multiple of 4 but not a power of 2)

7. Example 1: Screening 6 Factors Response: Plastic Hardness Factor Levels Factors -1 +1 (X 1 ) Tension Control Manual Automatic (X 2 ) Machine #1 #2 (X 3 ) Throughput (liters/min) 10 20 (X 4 ) Mixing Single Double (X 5 ) Temperature 200 o 250 o (X 6 ) Moisture 20 % 30 %

8. Randomly assign 6 columns to the 6 factors and then randomize the run order

10. Analysis of the Screening Design Data Using SAS

12. 3.2 k Factorial Experiments  A 2 k factorial design is a k-factor design such that  The 2 k experimental runs are the 2 k combinations of + and – factor levels  Each factor has two levels coded + and -  The 2 k experimental runs may also be replicated

13. Example 2: 2 3 Design with 3 Replicates (Montgomery 2005)  An engineer is interested in the effects of – cutting speed (A), – tool geometry (B), – cutting angle (C) on the life (in hours) of a machine tool  Two levels of each factor were chosen  Three replicates of a 2 3 design were run  Run order was randomized

15. Analysis using SAS: Main Effects Model (r 2 =.50)

16. Analysis using SAS: Interaction Model (r 2 =.76)

17. Maximize Hours at A=-1 B=+1 C=+1

18. Unreplicated 2 k Designs  When the design is unreplicated (n=1 for each of the 2 k factor treatments), it is necessary to “pool” interaction terms to form an error term for hypothesis testing.  Three steps are recommended: 1.Estimate all effects for the full-factorial model. 2.Make a normal probability plot of these estimated effects. “Outlier” effects can be pooled together. 3.Run the ANOVA using the pooled error term.

19. Example 3: An Unreplicated 2 4 Design (Montgomery and Myers 2002)  An engineer studied four factors believed to affect the filtration rate (Y) of a chemical product: – temperature (A), – pressure (B), – concentration of formaldehyde (C) – stirring rate (D)  Two levels of each factor were chosen  An unreplicated 2 3 design were run  Run order was randomized

20. The Unreplicated 2 4 Design The Design and Data Step 1: Estimates

21. Step 2: Normal probability plot of effects (Minitab) A

22. Step 3: The ANOVA with a Pooled Error (r 2 =.97)

23. Main Effects Plots (using Minitab)

24. Interaction Plots (using Minitab)

25. 4. Optimization Experiments  The experimenter wants to model (fit a response surface) involving a response y which depends on process input variables  1,  2, …  k.  Because the exact functional relationship between y and  1,  2, …  k is unknown, a low order polynomial is used as an approximating function.  Before fitting a model,  1,  2, …  k are coded as x 1, x 2, …, x k. For example:  i = 100 150 200 x i = -1 0 +1

26. 4. Optimization Experiments The experimenter is interested in determining: 1. Values of the input variables  1,  2, …  k. that optimize the response y (known as the optimum operating conditions). 2. An operating region that satisfy operating specifications for y.  A common approximating function is the quadratic or second-order model:

27. Example 3: Approximating Functions  The experimental goal is to maximize process yield (y).  A two-factor 3 2 experiment with 2 replicates was run with: Temperature  1 : Uncoded Levels 100 o 150 o 200 o x 1 Coded Levels -1 0 +1 Process time (min)  2 : Uncoded Levels 6 8 10 x 2 Coded Levels -1 0 +1

28. True Function: y = e (.5x 1 – 1.5x 2 ) Fitted function (from SAS) 

29. Predicted Maximum Yield (y) at x 1 = +1, x 2 = -1 (or, Temperature = 200 o, Process Time = 6 minutes)

30. CONTOUR PLOTS TRUE FUNCTION QUADRATIC APPROX.

31. Central Composite Design Box-Behnken Design (CCD) (BBD) Factorial, axial, and Centers of edges and center points center points

32. Example 4: Central Composite Design (Myers 1976)  The experimenter wants to study the effects of sealing temperature (x 1 ) cooling bar temperature (x 2 ) polethylene additive (x 3 ) on the seal strength in grams per inch of breadwrapper stock (y).  The uncoded and coded variable levels are -  -1 0 1 . x 1 204.5 o 225 o 255 o 285 o 305.5 o x 2 39.9 o 46 o 55 o 64 o 70.1 o x 3.09%.5% 1.1% 1.7% 2.11%

33. Example 4: Central Composite Design

34. Canonical Analysis of Quadratic Model (using SAS)

35. Ridge Analysis of Quadratic Model (using SAS) Predicted Maximum at x 1 =-1.01 x 2 =0.26 x 3 =0.68

36. 5. Mixture Experiments  A mixture contains q components where x i is the proportion of the i th component (i=1,2,…, q)  Two constraints exist: 0 ≤ x i ≤ 1 and Σ x i = 1

37. Simplex Coordinate System

38. Mixture Experiment Models  Because the level of the final component can written as x q = 1 – (x 1 + x 2 + + x q-1 ) any response surface model used for independent factors can be reduced to a Scheffé model. Examples include:

39. Example of a 3-Component Mixture Design

40. Analysis of a 3-component Mixture Experiment

41. 4-Component Mixture Experiment with Component Level Constraints (McLean & Anderson 1966) Response: Flare Brightness

42. 6. Final Comments  Screening experiments  2 k and 2 k-p experiments  Optimization experiments  Mixture experiments  Other applications: Fractional factorial designs Path of steepest ascent (descent) Experiments with blocking Experiments with restrictions on randomization