Robust Parameter Design

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

644 Chapter 15 2 k Factorial Experiments and Fractions

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 The goal of robust parameter design is to choose the levels of the control vari- Parameter Design ables (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 vari- ables 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 2 2 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

B (depth) Shallow Medium Deep

A (wall thickness)

Case Study 15.3: Solder Process Optimization: In an experiment described in Understanding Industrial Designed Experiments by Schmidt and Launsby (1991; see the Bibli- ography), 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 2 k Factorial Experiments and Fractions

Table 15.19: Control Factors for Case Study 15.3

A, solder pot temperature ( ◦ F)

B, conveyor speed (ft/min)

C, flux density

D, preheat temperature

E, wave height (in.)

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 (solder- ing 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 2 Both the control array (inner array) and the noise array (outer array) were

chosen to be fractional factorials, the former a 1 4 of a 2 5 and the latter a 1 2 of a 2 3 . The crossed array and the response values are shown in Table 15.21. The first three columns of the inner array represent a 2 3 . 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 2 3 . 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

Inner Array

Outer Array

A B C D E (1) a*b* a*c*

b*c* y ¯ s y

104 145.50 47.35 Solder Pot Temperature

Flux Density 250

Mean, y

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 σ 2 y 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 s y .

It turns out that temperature and flux density are the most important factors in Case Study 15.3. They seem to influence both s y 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 2 k 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 x 1 and flux density x 2 , linear regression on the response (number of errors per million joints) produces

ˆ y = 197.125 − 27.5x 1 + 57.875x 2 .

To find the most robust levels of temperature and flux density, it is impor- tant 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 s 2 =γ 0 +γ 1 (x 1 )+γ 2 (x 2 ).

This modeling process produces the following result:

ln s 2 2 = 6.6975 − 0.7458x 1 + 0.6150x 2 .

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 s 2 rather than s 2 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 ap- proach 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

s ⫽ 57.4 y⫽ ^ 260 s ⫽ 48.2 y⫽ ^ 240

s ⫽ 40.4 y⫽ ^ 220

s ⫽ 34.0 ^ y⫽ 0.0 200

s ⫽ 28.5 ^ y⫽ 180

, Flux Density

x 2 ^ 160 s ⫽ 23.9 y⫽

^ y⫽ 140 s ⫽ 14.2 y⫽ ^ 120

x 1 , Temperature

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 vari- ance 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

650 Chapter 15 2 k Factorial Experiments and Fractions

Table 15.22: Experimental Data in a Crossed Array Inner Array

Outer Array

A B C = −1

C = +1 Response Mean

15.21. One must understand that while A and B are held constant in the process

C follows a probability distribution during the process. Given this information, it becomes clear that A = −1 and B = +1 are levels that produce smaller values for

the process variance, while A = +1 and B = −1 give larger values. Thus, we say that A = −1 and B = +1 are robust values, i.e., insensitive to inevitable changes in the noise variable C during the process.

20 A=+1 20 B=−1

− 1 C +1 (a) AC interaction plot.

− 1 C +1

(b) BC interaction plot.

Figure 15.21: Interaction plots for the data in Table 15.22.

In the above example, we say that both A and B are dispersion effects (i.e. both factors impact the process variance). In addition, both factors are location effects since the mean of y changes as both factors move from −1 to +1.

Analysis Involving the Model Containing Both Control and Noise Variables

While it has been emphasized that noise variables are not constant during the working of the process, analysis that results in desirable or even optimal condi- tions on the control variables is best accomplished through an experiment in which both control and noise variables are fixed effects. Thus, both main effects in the control and noise variables and all the important control-by-noise interactions can

be evaluated. This model in x and z, often called a response model, can both

15.12 Robust Parameter Design 651 directly and indirectly provide useful information regarding the process. The re-

sponse model is actually a response surface model in vector x and vector z, where x contains control variables and z the noise variables. Certain operations allow models to be generated for the process mean and variance much as in Case Study

15.4. Details are supplied in Myers, Montgomery, and Anderson-Cook (2009); we will illustrate with a very simple example. Consider the data of Table 15.22 on page 650 with control variables A and B and noise variable C. There are eight

experimental runs in a 2 2 × 2, or 2 3 , factorial. Thus, the response model can be written

y(x, z) = β 0 +β 1 x 1 +β 2 x 2 +β 3 z+β 12 x 1 x 2 +β 1z x 1 z+β 2z x 2 z + ǫ. We will not include the three-factor interaction in the regression model. A, B, and

C in Table 15.22 are represented by x 1 ,x 2 , and z, respectively, in the model. We assume that the error term ǫ has the usual independence and constant variance properties.

The Mean and Variance Response Surfaces

The process mean and variance response surfaces are best understood by consid- ering the expectation and variance of z across the process. We assume that the noise variable C [denoted by z in y(x, z)] is continuous with mean 0 and variance

σ 2 z . The process mean and variance models may be viewed as

E z [y(x, z)] = β 0 +β 1 x 1 +β 2 x 2 +β 12 x 1 x 2 ,

Var z [y(x, z)] = σ 2 +σ 2 z (β 3 +β 1z x 1 +β 2z x 2 ) 2 =σ 2 +σ 2 z l 2 x , where l x is the slope ∂y(x,z) ∂z in the direction of z. As we indicated earlier, note how

the interactions of factors A and B with the noise variable C are key components of the process variance.

Though we have already analyzed the current example through plots in Figure

15.21, which displayed the role of AB and AC interactions, it is instructive to look at the analysis in light of E z [y(x, z)] and Var z [y(x, z)] above. In this example, the reader can easily verify the estimate b 1z for β 1z is 15/8 while the estimate b 2z for

β 2z is −15/8. The coefficient b 3 = 25/8. Thus, the condition x 1 = +1 and x 2 = −1 results in a process variance estimate of

Var 2 2 2 z 2 [y(x, z)] = σ +σ z (b 3 +b 1z x 1 +b 2z x 2 )

whereas for x 1 = −1 and x 2 = 1, we have

2 Var z [y(x, z)] = σ 2 +σ 2 z 2 (b 3 +b 1z x 1 +b 2z x 2 )

8 Thus, for the most desirable (robust) condition of x 1 = −1 and x 2 = 1, the

estimated process variance due to the noise variable C (or z) is (25/64)σ 2 z . The

652 Chapter 15 2 k Factorial Experiments and Fractions most undesirable condition, the condition of maximum process variance (i.e., x 1 =

+1 and x 2 = −1), produces an estimated process variance of (3025/64)σ 2 z . As far as the mean response is concerned, Figure 15.21 indicates that if maximum

response is desired x 1 = +1 and x 2 = −1 produce the best result. y

x 2 =+1 x 1 =+1

x 2 =−1 x 1 =−1

−1 z +1 −1 +1 (a) x 1 z interaction plot.

(b) x 2 z interaction plot.

Figure 15.22: Interaction plots for the data in Exercise 15.31.

Exercises

15.31 Consider an example in which there are two 0 and variance σ z 2 . Of concern is the variance of the control variables x 1 and x 2 and a single noise variable shrinkage response in the process itself. In the analysis z. The goal is to determine the levels of x 1 and x 2 that of Figure 15.7, it is clear that mold temperature, injec- are robust to changes in z, i.e., levels of x 1 and x 2 that tion velocity, and the interaction between the two are

minimize the variance produced in the response y as z the only important factors. moves between −1 and +1. The variables x 1 and x 2 are (a) Can the setting on velocity be used to create some at two levels, −1 and +1, in the experiment. The data

type of control on the process variance in shrinkage produce the plots in Figure 15.22 above. Note that

which arises due to the inability to control temper- x 1 and x 2 interact with the noise variable z. What

ature? Explain.

settings on x 1 and x 2 (−1 or +1 for each) result in minimum variance in y? Explain.

(b) Using parameter estimates from Figure 15.7, give an estimate of the following models:

15.32 Consider the following 2 3 factorial with control

(i)mean shrinkage across the distribution of tem-

variables x 1 and x 2 and noise variable z. Can x 1 and

perature;

x 2 be chosen at levels for which Var(y) is minimized? Explain why or why not.

(ii)shrinkage variance as a function of σ 2 z . (c) Use the estimated variance model to determine the z = −1

z = +1

level of velocity that minimizes the shrinkage vari-

x 2 = −1 x 2 = +1 x 2 = −1 x 2 = +1

ance.

x 1 = −1 4 6 8 10 (d) Use the mean shrinkage model to determine what x 1 = +1

1 3 3 5 value of velocity minimizes mean shrinkage. (e) Are your results above consistent with your anal-

15.33 Consider Case Study 15.1 involving the injec- ysis from the interaction plot in Figure 15.6? Ex- tion molding data. Suppose mold temperature is dif-

plain.

ficult to control and thus it can be assumed that in the process it follows a normal distribution with mean

15.34 In Case Study 15.2 involving the coal cleans-

Review Exercises 653 ing data, the percent solids in the process system is where the levels are

known to vary uncontrollably during the process and x 1 , percent solids: 8, 12 is viewed as a noise factor with mean 0 and variance σ 2 z . The response, cleansing efficiency, has a mean and

x 2 , flow rate: 150, 250 gal/min variance that change behavior during the process. Use

x 3 , pH: 5, 6

only significant terms in the following parts. Center and scale the variables to design units. Also (a) Use the estimates in Figure 15.9 to develop the pro- conduct a test for lack of fit, and comment concerning cess mean efficiency and variance models.

the adequacy of the linear regression model. (b) What factor (or factors) might be controlled at cer-

tain levels to control or otherwise minimize the pro- 15.36 A 2 5 factorial plan is used to build a regres- cess variance?

sion model containing first-order coefficients and model (c) What conditions of factors B and C within the de- terms for all two-factor interactions. Duplicate runs are sign region maximize the estimated mean?

made for each factor. Outline the analysis-of-variance (d) What level of C would you suggest for minimization table, showing degrees of freedom for regression, lack of fit, and pure error. of process variance when B = 1? When B = −1? 15.37 Consider the 1 16 of the 2 7 factorial discussed in

15.35 Use the coal cleansing data of Exercise 15.2 on Section 15.9. List the additional 11 defining contrasts. page 609 to fit a model of the type

15.38 Construct a Plackett-Burman design for 10

E(Y ) = β 0 +β 1 x 1 +β 2 x 2 +β 3 x 3 ,

variables containing 24 experimental runs.

Review Exercises

15.39 A Plackett-Burman design was used to study 15.40 A large petroleum company in the Southwest the rheological properties of high-molecular-weight regularly conducts experiments to test additives to copolymers. Two levels of each of six variables were drilling fluids. Plastic viscosity is a rheological mea- fixed in the experiment. The viscosity of the poly- sure reflecting the thickness of the fluid. Various poly- mer is the response. The data were analyzed by the mers are added to the fluid to increase viscosity. The Statistics Consulting Center at Virginia Tech for per- following is a data set in which two polymers are used sonnel in the Chemical Engineering Department at the at two levels each and the viscosity measured. The University. The variables are as follows: hard block concentration of the polymers is indicated as “low” or

chemistry x 1 , nitrogen flow rate x 2 , heat-up time x 3 , “high.” Conduct an analysis of the 2 2 factorial ex- percent compression x 4 , scans (high and low) x 5 , per- periment. Test for effects for the two polymers and cent strain x 6 . The data are presented here.

interaction.

Polymer 1 1 1 −1

Obs. x 1 x 2 x 3 x 4 x 5 x 6 y

15.41 A 2 2 factorial experiment is analyzed by the

Statistics Consulting Center at Virginia Tech. The 7 −1

client is a member of the Department of Housing, Inte- 8 −1 −1

rior Design, and Resource Management. The client is 10 1 −1 −1

−1 interested in comparing cold start to preheating ovens 11 −1

−1 in terms of total energy delivered to the product. In ad- dition, convection is being compared to regular mode.

Four experimental runs are made at each of the four Build a regression equation relating viscosity to the factor combinations. Following are the data from the

levels of the six variables. Conduct t-tests for all main experiment: effects. Recommend factors that should be retained for future studies and those that should not. Use the resid- ual mean square (5 degrees of freedom) as a measure of experimental error.

654 Chapter 15 2 k Factorial Experiments and Fractions Preheat

ponent 1. Assume that no interaction exists among the Convection 618 619.3 575 573.7

Cold

six factors.

Intensity Mode

Combination Ratio Total 1 abef

2.2480 Do an analysis of variance to study main effects and

1.8570 interaction. Draw conclusions.

2.3270 15.42 In the study “The Use of Regression Analy-

4 ace

1.8830 sis for Correcting Matrix Effects in the X-Ray Fluo-

5 bde

1.8078 rescence Analysis of Pyrotechnic Compositions,” pub-

6 abcd

2.1424 lished in the Proceedings of the Tenth Conference on

7 adf

1.9122 the Design of Experiments in Army Research Devel-

8 bcf

opment and Testing, ARO-D Report 65-3 (1965), an experiment was conducted in which the concentrations

15.43 Use Table 15.16 to construct a 16-run design of four components of a propellant mixture and the with 8 factors that is resolution IV. weights of fine and coarse particles in the slurry were each allowed to vary. Factors A, B, C, and D, each

15.44 Verify that your design in Review Exercise at two levels, represent the concentrations of the four

15.43 is indeed resolution IV. components, and factors E and F , also at two levels, represent the weights of the fine and coarse particles

15.45 Construct a design that contains 9 design present in the slurry. The goal of the analysis was points, is orthogonal, contains 12 total runs and 3 de-

to determine if the X-ray intensity ratios associated grees of freedom for replication error, and allows for a with component 1 of the propellant were significantly lack-of-fit test for pure quadratic curvature. influenced by varying the concentrations of the vari- ous components and the weights of the particles in the

15.46 Consider a design which is a 2 3−1

with 2 cen- 8 fraction of a 2 factorial experiment was ter runs. Consider ¯ y f as the average response at the

mixture. A 1 6

III

used, with the defining contrasts being ADE, BCE, design parameter and ¯ y 0 as the average response at the and ACF . The data shown here represent the total of design center. Suppose the true regression model is a pair of intensity readings. The pooled mean square error with 8 degrees of free-

E(Y ) = β 0 +β 1 x 1 +β 2 x 2 +β 3 x 3 dom is given by 0.02005. Analyze the data using a

+β 11 x 2 x 1 2 +β 22 2 +β 33 x 2 . 0.05 level of significance to determine if the concentra-

tions of the components and the weights of the fine and (a) Give (and verify) E(¯ y f −¯ y 0 ). coarse particles present in the slurry have a significant influence on the intensity ratios associated with com- (b) Explain what you have learned from the result in (a).