MJA Rev 10/17/2011 1:53:00 PM

Transcription

1 Problem 8-2 (as stated in RSM Simplified) Leonard Lye, Professor of Engineering and Applied Science at Memorial University of Newfoundland contributed the following case study. It is based on the DOE Golfer, a machine he invented to teach response surface methods. Figure 8-11a shows a student preparing a putt with the DOE Golfer. The weight of the club head can be adjusted by adding washers as shown in the close-up of the golfing machine on Figure 8-11b. Fig 8-12a: Student setting up golf machine Fig 8-12b: Golf machine club head Many factors can be varied in the DOE Golfer, including length of club, angle of swing, weight of club, and type of ball. Table 8-4 shows an experiment performed on the golfing machine by a group of students who studied three of these four factors. They ended up doing a face-centered central composite design (FCD) in two blocks. Notice that the first block is a full two-level factorial (2 3 ) with five center points. What did the students see from analysis of this block alone that led them to augment it with a second block of six face-centered star points (standard order rows 14 through 19) plus two more center points (20, 21)? (Hint: Do a square root transformation on the response.) Based on the predictive model generated from this RSM, what would be good setup for a six-foot (72 inch) putt with the DOE Golfer, assuming you can give or take 2 inches? Prob 8-2 p. 1

2 Table 8-4: FCD done on DOE Golfer Std Blk A: Length (inches) B: Angle (deg) C: Weight (washers) Distance (inches) Solution to Problem 8-2 If you choose to set this up from scratch using Design-Expert software, here s how to do it. Start by running the program and selecting File, New Design or click the blanksheet icon ( ) on the toolbar. Then click on the tab marked Response Surface. The default of Central Composite design is correct. Enter for Numeric Factors, 3. (Note: Your screen for this figure and the others that follow may differ somewhat due to revisions to the software.) Figure 8-2.1: Choosing central composite design (CCD) Further down this screen for Blocks select 2. Prob 8-2 p. 2

3 Figure 8-2.2: Choosing 2 blocks At the very bottom of the screen click the Options button, change the Center points in each factorial block to 5, press the Face Centered button and then OK. Figure 8-2.3: Specifying a face centered option for CCD Now you are ready to enter fields for the Name, Units and Alpha levels for each factor. Enter these as shown in the screen shot on Figure Figure 8-2.4: Factor specification Before moving on from this screen, check at the bottom to ensure you ve specified 5 center points for block 1 (added to the 8 factorial points) and 2 center points for block 2 (added to the 6 axial or star points) for a total of 21 runs as shown in Figure Prob 8-2 p. 3

4 Figure 8-2.5: Breakdown and number of points in CCD Continue to the screen for block names. In some cases, it would be worthwhile here to identify how runs will be blocked for example by specific lots or material, dates of manufacture or machine number. Figure 8-2.6: Block names For this problem, simply accept the defaults by pressing Continue. For the next screen offering options for Responses, enter for Name the Distance and the Units as inches. Figure 8-2.7: Entering the response names Press Continue to get the design layout in random run order. With only one response to enter, it will not be too hard to type in all the data (only 21 runs). However, feel free at this stage to open the file named 8-2 Prob - Golfing machine that we posted to the RSM Simplified website. To more easily compare what you see in this file (or the one you just created) with what s listed in Table 8-4, select View from the main menu and switch to Std Order. If you lack response data, enter it now. Read it off Figure (below), if you like, but note that run orders may vary due to randomization. Prob 8-2 p. 4

5 Figure 8-2.8: Design layout (run order may vary due to randomization) Let s get a feel for the layout of this face-centered central composite design (FCD) by first identifying points via a right-click option on the Block column-header. Select Display Point Type. Figure 8-2.9: Display point type Now you see that how the design has been broken into two blocks: factorial versus axial points, with some center points replicated in both. Prob 8-2 p. 5

6 Figure 8-2.9: Design points described Under the Design branch, click the Evaluation node and press the Graphs button. Now you see the cuboidal standard error contours characteristic of the FCD. Figure : Standard error contours Prob 8-2 p. 6

7 By default, Design-Expert displays factors A versus B with C set at its center value. This slice reveals the 7 center points (five in Block 1 plus two in Block 2) and axial points at the middle of the four edges. Press ahead to the Analysis branch of the software and select the node for Distance. Under Transformation choose Square root. Figure : Square root transformation Why the students who performed this experiment on DOE Golfer chose this transformation will become clear at the diagnostics stage of the analysis. For now, note at the bottom of the screen that the Ratio of max to min is 52. And that A ratio greater than 10 usually indicates a transformation is required. Click Fit Summary and observe that the program suggests the quadratic model. Figure : Fit summary Prob 8-2 p. 7

8 Scroll down this screen and check the lack of fit and R-squared statistics they all look good. Continue on to the Model specification. Figure : Model specification Press ahead to the ANOVA to accept the full quadratic as suggested by default. Figure : Analysis of variance (ANOVA) Prob 8-2 p. 8

9 Notice that B 2 and other model terms are not significant and thus could be eliminated, but let s not bother doing this. Scroll down this screen to review the post-anova statistics presented by the software. Then click the Diagnostics. Click through the series of plots offered on the Diagnostics Tool until you get to the Box Cox plot. Figure : Box-Cox plot recommending square root transformation Now you see why the students applied the square root transformation. Feel free at this stage to move on to the Model Graphs and look at the response surface. However, recall that in the problem statement you are asked to predict a good setup for a six-foot (72 inch) putt with the DOE Golfer, assuming you can give or take 2 inches, so without further ado, move on to the Optimization branch and select Numerical. Then click the Sqrt(Distance) response. Then change Display Options to Responses in Original Scale. Figure : Putting response back to original scale for optimization Now for Goal select a target of 72 with Limits at Lower of 70 and Upper at 74. Prob 8-2 p. 9

10 Figure : Targeting the putt from DOE Golfer Press ahead to Solutions and change the Solutions Tool to Ramps. As you will quickly realize by pressing through the solutions 1, 2 and beyond, there are many ways to vary the three process factors on DOE Golfer to make it putt the ball 72 inches. Figure : One solution to making a 72 inch putt (your results may differ) Problem 9-2 will ask that you consider which combination might make the putting most consistent by making the process more robust to variations in the input factors: club length, angle of pullback and the club weight. However, that must await your reading of Chapter 9 of RSM Simplified. Very Important Re-Save Your File to Preserve Modeling, Etc. To prepare for the follow-up problem noted above, do a File, Save as and change the File name to 8-2 Prob - Golf - analyzed or the like. This saves the model you created and optimization criteria you entered. Prob 8-2 p. 10

11 Postscript: Optional Post-Mortem As a postscript to this problem solution, let s guess at why the students working on the DOE Golfer decided to augment their first block of runs done with a simpler two-level factorial design. Go back to the Design branch and right-click on the Type column-header: Change it back to Display Blocks. Figure : Changing type of point back to display blocks Now click the button to the left of the first row in Block 2. Figure : Selecting a row Then while pressing the Shift key click the button to the left of the last row in Block 2. Figure : Highlighting a block of rows You should now see all of Block 2 highlighted as shown in Figure Now rightclick over the column of buttons to the left of the highlighted block and select Set Row Status, Ignore. Prob 8-2 p. 11

12 Figure : Ignoring a block of runs The software will now warn you about losing all the information from Block 2. Figure : Warning about ignored block Just press OK to ignore this warning. Now only Block 1, the two-level factorial core of the original FCD, remains available for analysis. Right-click the response column-header and select Edit Info. Figure : Editing information on the response Click the option to Analyze as factorial. Figure : Analyzing as a factorial The software will now warn you about changing from a polynomial (RSM) to the factorial model normally applied to a two-level design. Prob 8-2 p. 12

13 Figure : Warning about changing to the factorial model Just press OK to ignore this warning and OK the change in the Response screen. Then, under the Analysis branch click the Distance response node and for the Transform choose Square root. Then press ahead to Effects. Figure : Applying square root transformation Figure : Half-normal plot of effects from Block 1 (the factorial portion of FCD) Click the four largest effects the ones furthest right on the half-normal plot. (For details on how to make use of the half-normal plot for choosing two-level factorial effects, see DOE Simplified page 51.) Prob 8-2 p. 13

14 Figure : Choosing effects Press ANOVA to see the analysis of variance for this factorial model. Note that the curvature is highly-significant. Figure : Significant curvature revealed by ANOVA Evidently this prompted the students experimenting on DOE Golfer to add the second block of runs that created the FCD, thus making it a response surface (rather than factorial) design. Prob 8-2 p. 14