10 Other Two-Level Resolution III Designs; The Plackett-Burman Designs
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15.11 Introduction to Response Surface Methodology
639
a resolution III design with k variables (k = 1, 2, . . . , N ). The basic lines are as
follows:
N = 12
N = 16
+ + − + + + − − − + −
+ + + + − + − + + − − + − − −
N = 20
N = 24
+ + − − + + + + − + − + − − − − + + −
+ + + + + − + − + + − − + + − − + − + − − − −
Example 15.7: Construct a two-level screening design with 6 variables containing 12 design points.
Solution : Begin with the basic line in the initial column. The second column is formed by
bringing the bottom entry of the ﬁrst column to the top of the second column
and repeating the ﬁrst column. The third column is formed in the same fashion,
using entries in the second column. When there is a suﬃcient number of columns,
simply ﬁll in the last row with negative signs. The resulting design is as
follows:
⎡ x1
+
⎢ +
⎢
⎢ −
⎢
⎢ +
⎢
⎢ +
⎢
⎢ +
⎢
⎢ −
⎢
⎢ −
⎢
⎢ −
⎢
⎢ +
⎢
⎣ −
−
x2
−
+
+
−
+
+
+
−
−
−
+
−
x3
+
−
+
+
−
+
+
+
−
−
−
−
x4
−
+
−
+
+
−
+
+
+
−
−
−
x5
−
−
+
−
+
+
−
+
+
+
−
−
x6 ⎤
−
− ⎥
⎥
− ⎥
⎥
+ ⎥
⎥
− ⎥
⎥
+ ⎥
⎥
+ ⎥
⎥
− ⎥
⎥
+ ⎥
⎥
+ ⎥
⎥
+ ⎦
−
The Plackett-Burman designs are popular in industry for screening situations.
Because they are resolution III designs, all linear eﬀects are orthogonal. For any
sample size, the user has available a design for k = 2, 3, . . . , N − 1 variables.
The alias structure for the Plackett-Burman design is very complicated, and
thus the user cannot construct the design with complete control over the alias
structure, as in the case of 2k or 2k−p designs. However, in the case of regression
models, the Plackett-Burman design can accommodate interactions (although they
will not be orthogonal) when suﬃcient degrees of freedom are available.
15.11
Introduction to Response Surface Methodology
In Case Study 15.2, a regression model was ﬁtted to a set of data with the speciﬁc
goal of ﬁnding conditions on those design variables that optimize (maximize) the
cleansing eﬃciency of coal. The model contained three linear main eﬀects, three
two-factor interaction terms, and one three-factor interaction term. The model response was the cleansing eﬃciency, and the optimum conditions on x1 , x2 , and x3
640
Chapter 15 2k Factorial Experiments and Fractions
were found by using the signs and the magnitude of the model coeﬃcients. In this
example, a two-level design was employed for process improvement or process optimization. In many areas of science and engineering, the application is expanded
to involve more complicated models and designs, and this collection of techniques
is called response surface methodology (RSM). It encompasses both graphical and analytical approaches. The term response surface is derived from the
appearance of the multidimensional surface of constant estimated response from a
second-order model, i.e., a model with ﬁrst- and second-order terms. An example
will follow.
The Second-Order Response Surface Model
In many industrial examples of process optimization, a second-order response surface model is used. For the case of, say, k = 2 process variables, or design variables,
and a single response y, the model is given by
y = β0 + β1 x1 + β2 x2 + β11 x21 + β22 x22 + β12 x1 x2 + .
Here we have k = 2 ﬁrst-order terms, two pure second-order, or quadratic, terms,
and one interaction term given by β12 x1 x2 . The terms x1 and x2 are taken to be
in the familiar ±1 coded form. The term designates the usual model error. In
general, for k design variables the model will contain 1 + k + k + k2 model terms,
and hence the experimental design must contain at least a like number of design
points. In addition, the quadratic terms require that the design variables be ﬁxed
in the design with at least three levels. The resulting design is referred to as a
second-order design. Illustrations will follow.
The following central composite design (CCD) and example is taken from
Myers, Montgomery, and Anderson-Cook (2009). Perhaps the most popular class
of second-order designs is the class of central composite designs. The example given
in Table 15.17 involves a chemical process in which reaction temperature, ξ1 , and
reactant concentration, ξ2 , are shown at their natural levels. They also appear in
coded form. There are ﬁve levels of each of the two factors. In addition, we have
the order in which the observations on x1 and x2 were run. The column on the
right gives values of the response y, the percent conversion of the process. The ﬁrst
four design points represent the familiar factorial points at levels ±1. The next
four points are called axial points. They are followed by the center runs that were
discussed and illustrated earlier in this chapter. Thus, the ﬁve levels of each of the
two factors are −1, +1, −1.414, +1.414, and 0. A clear picture of the geometry of
the central composite design for this k = 2 example appears in Figure 15.16. This
ﬁgure illustrates the source of the term√axial points.These four points are on the
factor axes at an axial distance of α = 2 = 1.414 from the design center. In fact,
for this particular
CCD, the perimeter points, axial and factorial, are all at the
√
distance 2 from the design center, and as a result we have eight equally spaced
points on a circle plus four replications at the design center.
Example 15.8: Response Surface Analysis: An analysis of the data in the two-variable example
may involve the ﬁtting of a second-order response function. The resulting response
surface can be used analytically or graphically to determine the impact that x1
15.11 Introduction to Response Surface Methodology
641
Table 15.17: Central Composite Design for Example 15.8
+2
30
+1
25
0
ξ2, Concentraion (%)
x2
Temperature (◦ C) Concentration (%)
Observation Run
ξ1
ξ2
x1
x2
15
200
4
1
−1
−1
15
250
12
2
1
−1
25
200
11
3
−1
1
25
250
5
4
1
1
20
189.65
6
5
−1.414
0
20
260.35
7
6
1.414
0
12.93
225
1
7
0
−1.414
27.07
225
3
8
0
1.414
20
225
8
9
0
0
20
225
10
10
0
0
20
225
9
11
0
0
20
225
2
12
0
0
y
43
78
69
73
48
78
65
74
76
79
83
81
20
−1
15
−2
10
175
200
225
ξ1 , Temperature ( C)
250
275
−2
−1
0
x1
+1
+2
Figure 15.16: Central composite design for Example 15.8.
and x2 have on percent conversion of the process. The coeﬃcients in the response
function are determined by the method of least squares developed in Chapter 12
and illustrated throughout this chapter. The resulting second-order response model
is given in the coded variables as
yˆ = 79.75 + 10.18x1 + 4.22x2 − 8.50x21 − 5.25x22 − 7.75x1 x2 ,
whereas in the natural variables it is given by
yˆ = −1080.22 + 7.7671ξ1 + 23.1932ξ2 − 0.0136ξ12 − 0.2100ξ22 − 0.0620ξ1 ξ2 .
Since the current example contains only two design variables, the most illumi-
Chapter 15 2k Factorial Experiments and Fractions
642
nating approach to determining the nature of the response surface in the design
region is through two- or three-dimensional graphics. It is of interest to determine
what levels of temperature x1 and concentration x2 produce a desirable estimated
percent conversion, yˆ. The estimated response function above was plotted in three
dimensions, and the resulting response surface is shown in Figure 15.17. The height
of the surface is yˆ in percent. It is readily seen from this ﬁgure why the term response surface is employed. In cases where only two design variables are used,
two-dimensional contour plotting can be useful. Thus, make note of Figure 15.18.
Contours of constant estimated conversion are seen as slices from the response surface. Note that the viewer of either ﬁgure can readily observe which coordinates
of temperature and concentration produce the largest estimated percent conversion. In the plots, the coordinates are given in both coded units and natural units.
Notice that the largest estimated conversion is at approximately 240◦ C and 20%
concentration. The maximum estimated (or predicted) response at that location
is 82.47%.
Concentration
82.47
60.45
38.43
16.41
1.414
27.07
24.71
ξ2 (
22.36
con
20.00
c
ent
1
17.64
rati
on,
15.29
%)
0
12.93
x
2
−1
−1.414
213.2
201.4
189.7
260.4
248.6
236.8
)
225.0
, °C
ξ1
ture
0
a
per
(tem
1
1.414
x1
−1
−1.414
Figure 15.17: Plot for the response surface prediction conversion for Example 15.8.
Other Comments Concerning Response Surface Analysis
The book by Myers, Montgomery, and Anderson-Cook (2009) provides a great
deal of information concerning both design and analysis of RSM. The graphical
illustration we have used here can be augmented by analytical results that provide
information about the nature of the response surface inside the design region.
15.12 Robust Parameter Design
643
1.414
27.07
65
75
1
22.36
2
(Concentration, ξ %)
24.71
x2
0
20.00
17.64
40
70
201.4
55
12.93
189.7
65
45
25
−1.414
60
50
15.29
35
1
80
213.2
225.0
236.8
248.6
260.4
1
1.414
ξ 1 (Temperature,°C)
−1.414
−1
0
x1
Figure 15.18: Contour plot of predicted conversion for Example 15.8.
Other computations can be used to determine whether the location of the optimum
conditions is, in fact, inside or remote from the experimental design region. There
are many important considerations when one is required to determine appropriate
conditions for future operation of a process.
Other material in Myers, Montgomery, and Anderson-Cook (2009) deals with
further experimental design issues. For example, the CCD, while the most generally
useful design, is not the only class of design used in RSM. Many others are discussed
in the aforementioned text. Also, the CCD discussed here is a special case in which
k = 2. The more general k > 2 case is discussed in Myers, Montgomery, and
Anderson-Cook (2009).
15.12
Robust Parameter Design
In this chapter, we have emphasized the notion of using design of experiments
(DOE) to learn about engineering and scientiﬁc processes. In the case where the
process involves a product, DOE can be used to provide product improvement or
quality improvement. As we pointed out in Chapter 1, much importance has been
attached to the use of statistical methods in product improvement. An important
aspect of this quality improvement eﬀort that surfaced in the 1980s and continued
through the 1990s is to design quality into processes and products at the research
stage or the process design stage. One often requires DOE in the development of
processes that have the following properties:
1. Insensitive (robust) to environmental conditions
Chapter 15 2k Factorial Experiments and Fractions
644
2. Insensitive (robust) to factors diﬃcult to control
3. Provide minimum variation in performance
The methods used to attain the desirable characteristics in 1, 2, and 3 are a part
of what is referred to as robust parameter design, or RPD (see Taguchi, 1991;
Taguchi and Wu, 1985; and Kackar, 1985, in the Bibliography). The term design
in this context refers to the design of the process or system; parameter refers to
the parameters in the system. These are what we have been calling factors or
variables.
It is very clear that goals 1, 2, and 3 above are quite noble. For example,
a petroleum engineer may have a ﬁne gasoline blend that performs quite well as
long as conditions are ideal and stable. However, the performance may deteriorate
because of changes in environmental conditions, such as type of driver, weather
conditions, type of engine, and so forth. A scientist at a food company may have
a cake mix that is quite good unless the user does not exactly follow directions on
the box, directions that deal with oven temperature, baking time, and so forth. A
product or process whose performance is consistent when exposed to these changing
environmental conditions is called a robust product or robust process. (See
Myers, Montgomery, and Anderson-Cook, 2009, in the Bibliography.)
Control and Noise Variables
Taguchi (1991) emphasized the notion of using two classes of design variables in a
study involving RPD: control factors and noise factors.
Deﬁnition 15.2: Control factors are variables that can be controlled both in the experiment and
in the process. Noise factors are variables that may or may not be controlled
in the experiment but cannot be controlled in the process (or not controlled well
in the process).
An important approach is to use control variables and noise variables in the
same experiment as ﬁxed eﬀects. Orthogonal designs or orthogonal arrays are
popular designs to use in this eﬀort.
Goal of Robust
Parameter Design
The goal of robust parameter design is to choose the levels of the control variables (i.e., the design of the process) that are most robust (insensitive) to changes
in the noise variables.
It should be noted that changes in the noise variables actually imply changes during
the process, changes in the ﬁeld, changes in the environment, changes in handling
or usage by the consumer, and so forth.
The Product Array
One approach to the design of experiments involving both control and noise variables is to use an experimental plan that calls for an orthogonal design for both
the control and the noise variables separately. The complete experiment, then, is
merely the product or crossing of these two orthogonal designs. The following is a
simple example of a product array with two control and two noise variables.
15.12 Robust Parameter Design
645
Example 15.9: In the article “The Taguchi Approach to Parameter Design” in Quality Progress,
December 1987, D. M. Byrne and S. Taguchi discuss an interesting example in
which a method is sought for attaching an electrometric connector to a nylon
tube so as to deliver the pull-oﬀ performance required for an automotive engine
application. The objective is to ﬁnd controllable conditions that maximize pull-oﬀ
force. Among the controllable variables are A, connector wall thickness, and B,
insertion depth. During routine operation there are several variables that cannot
be controlled, although they will be controlled during the experiment. Among
them are C, conditioning time, and D, conditioning temperature. Three levels are
taken for each control variable and two for each noise variable. As a result, the
crossed array is as follows. The control array is a 3 × 3 array, and the noise
array is a familiar 22 factorial with (1), c, d, and cd representing the four factor
combinations. The purpose of the noise factor is to create the kind of variability
in the response, pull-oﬀ force, that might be expected in day-to-day operation with
the process. The design is shown in Table 15.18.
Table 15.18: Design for Example 15.9
Thin
Medium
A (wall thickness)
Thick
Shallow
(1)
c
d
cd
(1)
c
d
cd
(1)
c
d
cd
B (depth)
Medium
(1)
c
d
cd
(1)
c
d
cd
(1)
c
d
cd
Deep
(1)
c
d
cd
(1)
c
d
cd
(1)
c
d
cd
Case Study 15.3: Solder Process Optimization: In an experiment described in Understanding
Industrial Designed Experiments by Schmidt and Launsby (1991; see the Bibliography), solder process optimization is accomplished by a printed circuit-board
assembly plant. Parts are inserted either manually or automatically into a bare
board with a circuit printed on it. After the parts are inserted, the board is put
through a wave solder machine, which is used to connect all the parts into the
circuit. Boards are placed on a conveyor and taken through a series of steps. They
are bathed in a ﬂux mixture to remove oxide. To minimize warpage, they are
preheated before the solder is applied. Soldering takes place as the boards move
across the wave of solder. The object of the experiment is to minimize the number
of solder defects per million joints. The control factors and levels are as given in
Table 15.19.
646
Chapter 15 2k Factorial Experiments and Fractions
Table 15.19: Control Factors for Case Study 15.3
Factor
A, solder pot temperature (◦ F)
B, conveyor speed (ft/min)
C, ﬂux density
D, preheat temperature
E, wave height (in.)
(−1)
480
7.2
0.9◦
150
0.5
(+1)
510
10
1.0◦
200
0.6
These factors are easy to control at the experimental level but are more formidable
at the plant or process level.
Noise Factors: Tolerances on Control Factors
Often in processes such as this one, the natural noise factors are tolerances on the
control factors. For example, in the actual on-line process, solder pot temperature
and conveyor-belt speed are diﬃcult to control. It is known that the control of
temperature is within ±5◦ F and the control of conveyor-belt speed is within ±0.2
ft/min. It is certainly conceivable that variability in the product response (soldering performance) is increased because of an inability to control these two factors
at some nominal levels. The third noise factor is the type of assembly involved.
In practice, one of two types of assemblies will be used. Thus, we have the noise
factors given in Table 15.20.
Table 15.20: Noise Factors for Case Study 15.3
Factor
A*, solder pot temperature tolerance (◦ F)
(deviation from nominal)
B*, conveyor speed tolerance (ft/min)
(deviation from ideal)
C*, assembly type
(−1)
−5
(+1)
+5
−0.2
+0.2
1
2
Both the control array (inner array) and the noise array (outer array) were
chosen to be fractional factorials, the former a 14 of a 25 and the latter a 12 of a 23 .
The crossed array and the response values are shown in Table 15.21. The ﬁrst three
columns of the inner array represent a 23 . The fourth and ﬁfth columns are formed
by D = −AC and E = −BC. Thus, the deﬁning interactions for the inner array
are ACD, BCE, and ABDE. The outer array is a standard resolution III fraction
of a 23 . Notice that each inner array point contains runs from the outer array.
Thus, four response values are observed at each combination of the control array.
Figure 15.19 displays plots which reveal the eﬀect of temperature and density on
the mean response.
15.12 Robust Parameter Design
647
Table 15.21: Crossed Arrays and Response Values for Case Study 15.3
A
1
1
1
1
−1
−1
−1
−1
Inner Array
B C D
1 −1
1
1
1 −1
1 −1
−1
1
−1 −1
1
1
1
1 −1 −1
1
1
−1
−1 −1 −1
E
−1
1
1
−1
−1
1
1
−1
(1)
194
136
185
47
295
234
328
186
a*b*
197
136
261
125
216
159
326
187
Outer Array
a*c* b*c*
275
193
136
132
264
264
42
127
293
204
157
231
322
247
104
105
Solder Pot Temperature
250
Mean, y
Mean, y
sy
40.20
2.00
39.03
47.11
48.75
43.04
39.25
47.35
Flux Density
250
185
120
y¯
214.75
135.00
243.50
85.25
252.00
195.25
305.75
145.50
Low
High
(−1)
(+1)
185
120
Low
High
(−1)
(+1)
Figure 15.19: Plot showing the inﬂuence of factors on the mean response.
Simultaneous Analysis of Process Mean and Variance
In most examples using RPD, the analyst is interested in ﬁnding conditions on
the control variables that give suitable values for the mean response y¯. However,
varying the noise variables produces information on the process variance σy2 that
might be anticipated in the process. Obviously a robust product is one for which
the process is consistent and thus has a small process variance. RPD may involve
the simultaneous analysis of y¯ and sy .
It turns out that temperature and ﬂux density are the most important factors
in Case Study 15.3. They seem to inﬂuence both sy and y¯. Fortunately, high
temperature and low ﬂux density are preferable for both. From Figure 15.19, the
“optimum” conditions are
solder temperature = 510◦ F,
ﬂux density = 0.9◦ .
648
Chapter 15 2k Factorial Experiments and Fractions
Alternative Approaches to Robust Parameter Design
One approach suggested by many is to model the sample mean and sample variance
separately. Separate modeling often helps the experimenter to obtain a better
understanding of the process involved. In the following example, we illustrate this
approach with the solder process experiment.
Case Study 15.4: Consider the data set of Case Study 15.3. An alternative approach is to ﬁt separate
models for the mean y¯ and the sample standard deviation. Suppose that we use the
usual +1 and −1 coding for the control factors. Based on the apparent importance
of solder pot temperature x1 and ﬂux density x2 , linear regression on the response
(number of errors per million joints) produces
yˆ = 197.125 − 27.5x1 + 57.875x2 .
To ﬁnd the most robust levels of temperature and ﬂux density, it is important to procure a compromise between the mean response and variability, which
requires a modeling of the variability. An important tool in this regard is the log
transformation (see Bartlett and Kendall, 1946, or Carroll and Ruppert, 1988):
ln s2 = γ0 + γ1 (x1 ) + γ2 (x2 ).
This modeling process produces the following result:
ln s2 = 6.6975 − 0.7458x1 + 0.6150x2 .
The log linear model ﬁnds extensive use for modeling sample variance, since the
log transformation on the sample variance lends itself to use of the method of
least squares. This results from the fact that normality and homogeneous variance
assumptions are often quite good when one uses ln s2 rather than s2 as the model
response.
The analysis that is important to the scientist or engineer makes use of the
two models simultaneously. A graphical approach can be very useful. Figure 15.20
shows simple plots of the mean and standard deviation models simultaneously. As
one would expect, the location of temperature and ﬂux density that minimizes
the mean number of errors is the same as that which minimizes variability, namely
high temperature and low ﬂux density. The graphical multiple response surface approach allows the user to see tradeoﬀs between process mean and process variability.
For this example, the engineer may be dissatisﬁed with the extreme conditions in
solder temperature and ﬂux density. The ﬁgure oﬀers estimates of how much is
lost as one moves away from the optimum mean and variability conditions to any
intermediate conditions.
In Case Study 15.4, values for control variables were chosen that gave desirable
conditions for both the mean and the variance of the process. The mean and
variance were taken across the distribution of noise variables in the process and
were modeled separately, and appropriate conditions were found through a dual
response surface approach. Since Case Study 15.4 involved two models (mean and
variance), this can be viewed as a dual response surface analysis. Fortunately,
in this example the same conditions on the two relevant control variables, solder
15.12 Robust Parameter Design
649
1.0
.4
sϭ
x2, Flux Density
0.5
57
0
^y ϭ 26
sϭ
.2
48
0
^y ϭ 24
.4
sϭ
40
0
^y ϭ 22
sϭ
0.0
.0
34 ^ 200
yϭ
.5
28
180
ϭ
^
s
yϭ
.9
sϭ
Ϫ0.5
23
0
^y ϭ 16
.1
sϭ
20
0
^y ϭ 14
.2
Ϫ1.0
Ϫ1.0
sϭ
Ϫ0.5
0.0
x1, Temperature
0.5
14
0
^y ϭ 12
1.0
Figure 15.20: Mean and standard deviation for Case Study 15.4.
temperature and ﬂux density, were optimal for both the process mean and the
variance. Much of the time in practice some type of compromise between the
mean and variance would need to be invoked.
The approach illustrated in Case Study 15.4 involves ﬁnding optimal process
conditions when the data used are from a product array (or crossed array) type of
experimental design. Often, using the product array, a cross between two designs,
can be very costly. However, the development of dual response surface models, i.e.,
a model for the mean and a model for the variance, can be accomplished without
a product array. A design that involves both control and noise variables is often
called a combined array. This type of design and the resulting analysis can be used
to determine what conditions on the control variables are most robust (insensitive)
to variation in the noise variables. This can be viewed as tantamount to ﬁnding
control levels that minimize the process variance produced by movement in the
noise variables.
The Role of the Control-by-Noise Interaction
The structure of the process variance is greatly determined by the nature of the
control-by-noise interaction. The nature of the nonhomogeneity of process variance is a function of which control variables interact with which noise variables.
Speciﬁcally, as we will illustrate, those control variables that interact with one or
more noise variables can be the object of the analysis. For example, let us consider
an illustration used in Myers, Montgomery, and Anderson-Cook (2009) involving
two control variables and a single noise variable with the data given in Table 15.22.
A and B are control variables and C is a noise variable.
One can illustrate the interactions AC and BC with plots, as given in Figure