Response Surface Exercise

The Minitab macro we will use in class is available on the home page. If you call the macro with a given set of x-coordinates, the macro will evaluate an "unknown" function for you at those x-coordinates.

As an example, suppose you have the macro stored on subdirectory mtb on a diskette. If the design matrix is stored in columns c1 and c2 and you want to store the response in c3, you would enter the following command in the Session window:

%a:$\backslash$ mtb $\backslash$ rse c1 c2 c3

You will use the macro as a tool in the following experiment. Please save output and data so we can discuss results briefly in class on Thursday.

Experiment 1

The bond strength between an integrated circuit and a substrate is a function of holding pressure and holding time. In a discussion with the process engineer, a reasonable start value for holding pressure is thought to be 40 to 44 psi while a reasonable start value for holding time is 20 to 25 minutes. Use this information to design an initial screening experiment in the two factors. Use the fractional factorial pull-down menus in Minitab to generate the appropriate design matrix. Under Factors, change the Low and High factor levels as well as the factor Names accordingly. Call the rse macro to obtain your data. Fit a first-order model (You could use Fit Factorial Model or choose Fit RS Model... under the Stat menu). Is the first-order model appropriate? What is the gradient direction?

Choose a step-size for your search from the center of your initial design in the gradient direction (be generous here-I found I had to take alot of steps). Save the appropriate set of x-coordinates and use the macro rse once again to obtain data. At what point does the response decline? Choose a new set of design points for a first-order design. Analyze the first-order design. Are you ready to fit a central composite design or to continue your search? If you are ready for a central composite design, select Create RS Design and select factor levels. Under Storage, select Fits and Coefficients. We use fits to attempt a contour plot (not entirely satisfactory since the grid is irregular) and coefficients to check for a local maximum.

If you have Minitab select a central composite design for you, note that your original set of observations would be deleted (unless you went to the trouble of saving them then copying them back into the new set of design columns, etc...don't bother), so you're not really "augmenting" your old design. After you choose an appropriate central composite design, use the rse macro once again, select Fit RS Model... and fit a full quadratic model. What set of x values is optimal? Do they correspond to a local maximum? Comment on your residual analysis.

Here's some matrix math in Minitab. Copy the coefficients into a column for b (the linear coefficients) and two columns for B (the quadratic coefficients-remember to take 1/2 of the coefficient for the interaction). Use the copy command to copy them into a matrix. Suppose I entered the coefficients from the linear effects into c1 (this is the vector b) and the coefficients from the quadratic effects into c2 and c3 (this is the matrix B). To obtain the stationary point and check the eigenvalues of B, type

copy c2 c3 m1             # m1 is B 
invert m1 m2              # m2 is B-1 
mult m2 c1 c4             # c4 is b*B-1 
mult c4 -.5 c4            # c4 is xo 
eigen m2 c5 m6            # c5 contains eigenvalues, m6 contains eigenvectors 
print c4 c5