MJA Rev 8/26/2005 5:50:00 PM
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 (23) 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
MJA Rev 8/26/2005 5:50:00 PM
Table 8-4: FCD done on DOE Golfer
Std
Blk
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
A:
Length
(inches)
5.5
11.5
5.5
11.5
5.5
11.5
5.5
11.5
8.5
8.5
8.5
8.5
8.5
5.5
11.5
8.5
8.5
8.5
8.5
8.5
8.5
B:
Angle
(deg)
20
20
60
60
20
20
60
60
40
40
40
40
40
40
40
20
60
40
40
40
40
C:
Weight
(washers)
0
0
0
0
2
2
2
2
1
1
1
1
1
1
1
1
1
0
2
1
1
Distance
(inches)
2.2
17.3
16.5
82.3
8.2
23.2
37.0
115.3
40.0
37.0
38.5
34.5
39.5
16.0
55.0
14.0
71.0
31.0
41.5
35.5
36.0
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.
Figure 8-2.1: Choosing central composite design (CCD)
Further down this screen for Blocks select 2.
Prob 8-2 p. 2
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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 8-2.4.
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 point s for block 2
(added to the 6 axial or ‘star’ points) for a total of 21 runs as shown in Figure 8-2.5.
Prob 8-2 p. 3
MJA Rev 8/26/2005 5:50:00 PM
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.dx7” 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 8-2.8 (below), if you like, but
note that run orders may vary due to randomization.
Prob 8-2 p. 4
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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
MJA Rev 8/26/2005 5:50:00 PM
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 8-2.10: Standard error contours
Prob 8-2 p. 6
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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 8-2.11: 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 8-2.12: Fit summary
Prob 8-2 p. 7
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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 8-2.13: Model specification
Press ahead to the ANOVA to accept the full quadratic as suggested by default.
Figure 8-2.14: Analysis of variance (ANOVA)
Prob 8-2 p. 8
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Notice that B2 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 8-2.15: 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 8-2.16: 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
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Figure 8-2.16: 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 8-2.17: 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.dx7” or the like. This saves the model you
created and optimization criteria you entered.
Prob 8-2 p. 10
MJA Rev 8/26/2005 5:50:00 PM
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 8-2.18: Changing type of point back to display blocks
Now click the button to the left of the first row in Block 2.
Figure 8-2.19: Selecting a row
Then while pressing the Shift key click the button to the left of the last row in Block 2.
Figure 8-2.20: Highlighting a block of rows
You should now see all of Block 2 highlighted as shown in Figure 8-2.20. 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
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Figure 8-2.21: Ignoring a block of runs
The software will now warn you about losing all the information from Block 2.
Figure 8-2.22: 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 8-2.23: Editing information on the response
Click the option to Analyze as factorial.
Figure 8-2.24: 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
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Figure 8-2.25: 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.
Figure 8-2.26: Applying square root transformation
Then press ahead to Effects.
Figure 8-2.27: 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
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Figure 8-2.28: Choosing effects
Press ANOVA to see the analysis of variance for this factorial model. Note that the
curvature is highly-significant.
Figure 8-2.29: 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