1. Lecture 17 Today: Start Chapter 9 Next day: More of Chapter 9

2. Example Chemical Engineer is interested in maximizing yield of a process 2 variables influence process yield: –reaction time (x 1 ) and reaction temperature (x 2 ) Current operating conditions have reaction time at 35 minutes and temperature at 155 o F, which give a yield of about 40% Best operating conditions may be far from current conditions

3. Response Surface Methods Often the goal of experimentation is to optimize (say maximize) a system response When there are only a few quantitative factors, response surface methodology can be use for understanding the relationship between the response and input factors Experimentation strategy is sequential

4. Models Relationship between input variables and response: – If the expected response is denoted E(y)=f(x 1, x 2,…, x k ), then f(x 1, x 2,…, x k ) is called the response surface

5. Models First order model to approximate f: – Second order model to approximate f: – Typically a lower order polynomial is used to approximate the local response surface

6. General Idea If there are many factors to consider, perform a screening design first (e.g., 2 k-p fractional factorial design) to screen out unimportant factors Response surface methodology involves experimentation, modeling, data analysis and optimization First run a sequence of small experiment designs to fit a first order model to identify the experiment region that is near or contains the optimum Next an experiment design is performed to estimate the second order model close to the optimum Designs to estimate the response surface are called response surface designs

7. General Idea Design that estimates a first order model is called a first order design Design that estimates a second order model is called a second order design Analysis is performed using regression

8. Coding the Variables It is convenient to code the variables (as we have done so far) For example consider a factor with 3 equally spaced levels Let z i be the mid-point of the levels The 3 levels are: Z: z i -c, z i, z i +c Transformation: 3 coded levels: X: -1, 0, 1

9. First Order Designs Will run a series of first order designs until near the optimum When there is substantial curvature, first order model becomes ineffective for approximating the surface How do we know if there is curvature?

10. First Order Designs Must have more than 2 level factors to check for curvature Solution is to add experiment trials at the center of the experimental region (e.g., x=(0,0,0,0…0) ) Designs combine factorials and center points

11. Example Engineer decide that reaction time should be investigated in the area of the operating conditions –time = 35 minutes (z 1 =30 or 40) –temperature = 155 o F (z 2 =150 or 160) Will include center points at the current operating conditions Coded variables:

12. Example Design and responses:

13. Curvature Check Have n f trials at the factorial design points (e.g., -1 and +1 combinations) Have n c trials at the center point Motivation for test: Test:

14. Example Fit regression line to data, include a quadratic term for curvature check

15. Method of Steepest Ascent (climbing the hill) Linear effects for first order model estimated by least squares: Take partial derivative with respect to each variable: Direction of steepest ascent:

16. Method of Steepest Ascent Several experiment trials are taken along the line from the center point of the design, in the direction of the steepest ascent until no further increase is observed The location where the maximum has occurred is the center point of the next first order design Design should have n c trials at the center point If curvature is detected, augment the design with additional trials so that the second order model can be estimated

17. Method of Steepest Ascent