response surface methods and the use of noise variables

Response Surface Methods and the Use of Noise Variables

RAYMOND H. MYERS Virginia Polytechnic Institute and State University, Blacksburg, VA 24061

YOON KIM Humboldt State University, Arcata, CA 95521

KRISTI L. GRIFFITHS Eli Lilly and Company, Indianapolis, IN 46285

The ideas of robust parameter design focus on reducing product or process variability that is transmitted by noise variables. These variables are difficult to control in the process or are not constant across different levels of consumer usage. In this paper we develop and illustrate the use of response surface methods that are extended to cover modeling of the process mean and variance.

Considerable attention has been placed on the response surface for the process variance. A methodology is given that allows for a confidence region on the location on the control factors of minimum process variance. This is the location where the process variance is no larger than the experimental error variance. The mean and variance response surfaces can also be combined to produce prediction limits on a future response and one-sided tolerance intervals.

Introduction

'1T'mE modern approach to product improvement II and optimization has borrowed considerably

from principles of Genichi Taguchi. Articles by Taguchi (1986, 1987), Taguchi and Wu (1980), and many others highlight an approach that puts emphasis on product variability in the statistical modeling scheme. They address the notion that products and processes lack high quality because of inconsistency in performance. This inconsistency is often produced by factors that are uncontrollable in the design of the product or process (i.e., tolerances on design factors, environmental factors, or factors that are a function of usage by the customers). As a result, in recent

Dr. Myers is Professor Emeritus in the Department of Statistics.

Dr. Kim is Assistant Professor in the Department of Mathematics and Statistics.

Dr. Griffiths is a Senior Statistician in Statistical and

Mathematical Sciences.

Vol. 29, No. 4, October 1997 429

years, attention has been placed on the choice of a product design that is said to be resistant (robust) to these environmental or noise variables. This notion motivated the rather uncommon interest in industry and academia in so called "parameter design" issues. (For a panel discussion on the topic see Nair (1992).)

Recently, much of the work in parameter design has dealt with alternative methodology in design and analysis when both standard (or control) variables and noise variables are used in the same experiment and in the same model. This has often been referred to as the response model approach and has led to interesting contributions by Montgomery (1991), Shoemaker, Tsui, and Wu (1991), Welch, Yu, Kang, and Sacks (1990), and Box and Jones (1990). These authors point out the economical advantage gained by designing experiments with a combined array approach as opposed to the crossed array advocated by Taguchi and discussed by Kackar (1985). It has been suggested that the response model approach avoids unnecessary biases that appear in main effect estimates due to certain interactions that are ignored when the experimental design is highly fractionated.

Journal of Quality Technology

430 RAYMOND H. MYERS, YOON KIM, AND KRISTI L. GRIFFITHS

These biases are likely to occur in the design and analysis approach often taken by Taguchi.

Since the underpinnings of parameter design revolve around experimental design, determination of optimum conditions, and a better understanding of the process, it is quite natural that response surface methods (RSM) would be suggested as an approach. The criteria on which choices of levels of the control variables are made should focus on variability in the process response. Lucas (1994) and Myers, Khuri, and Vining (1992) discuss the development of response surfaces for the process mean and variance. These response surfaces are generated from the response model with control and noise variables as fixed effects.

In this paper we develop some general ideas that deal with modeling the process mean and variance and how these models can be used in the determination of robust conditions and product consistency. It is natural for the engineer or scientist to focus on the variance model as a function of control variables in order to shed light on the workings of the process. Indeed the practitioner may wish to search for conditions of minimum process variance. Myers and Montgomery (1995) deal with the idea of performing process optimization through stationary point and ridge analysis on the process variance or on the slopes of the response model in the direction of the noise variables. The connection between these slopes and the process variance is exploited. This relates directly to the work of Shoemaker, Tsui, and Wu (1991) and Shoemaker and Tsui (1993) in which minimization of the process variance is approached through the use of plots revealing the nature of the interaction between control and noise variables. In the present paper, the concepts underlying the use of interaction plots and dual response modeling are extended to provide the practitioner properties of predicted response. These properties take sampling variability in both the mean and variance response surfaces into account. That allows the computation of prediction intervals and tolerance intervals on the process performance measure. These intervals serve as techniques for combining the process mean and variance. These criteria nicely characterize a process and serve as natural performance measures for response surface optimization.

If the analysis deals with the use of a model for the process variance, questions that relate to the quality of the model naturally arise. We begin by reviewing a method that allows a confidence region to be placed


on the location in the control factors where the process variance is minimized. This region allows the user to gain a sense of the precision of the process variance model and to determine the amount of flexibility that exists on the choice of levels to obtain a robust product. However, the choice of a robust product cannot focus on variance alone. The engineer can gain important insight about the process if he or she considers the process mean and variance simultaneously. Tradeoffs often exist in the selection of optimum conditions. An approach that brings the two response surfaces together can be very useful.

Consider the example in Byrne and Taguchi (1987) in which the engineer seeks a method to assemble an elastomeric connector to a nylon tube that would deliver the required pull-off performance for use in an automotive engine application. The objective is to maximize the pull-off force. Variance is transmitted due to variation in the noise variables: conditioning time, conditioning temperature, and conditioning relative humidity. An important component of the robust product analysis may be to estimate the levels of the control variables that minimize this variance and to gain some notion of the sampling variability of these estimated levels. The control factors include interference, connector wall thickness, and insertion depth. In addition, there will be considerable interest in determining the control levels that, in some sense, optimize the capability of the final product. The use of prediction and tolerance intervals as tools for evaluating the process capability and affording a single number optimization criterion for pull-off force can be very informative.

Some Response Surface Methods Involving Process Variance

The reduction of variability is an important objective of robust design experiments. Shoemaker and Tsui (1992) point out that the careful introduction of noise factors into the experiment is very important. As in Welch, et al (1990) and others, we will assume that the control variables (x) and noise variables (z) are fixed effects in the same model. That is, noise factors can be controlled in a laboratory or on the pilot plant level. This would certainly apply to the food industry (oven temperature, baking time, amount of milk, etc.) , petroleum industry (types of automobile, road conditions) , chemical industry (tolerances in temperature, pressure, etc.) , and many others. We will also assume in what follows that either a crossed array (see Kackar (1985) and Phadke

Vol. 29, No. 4, October 1997

RESPONSE SURFACE METHODS AND THE USE OF NOISE VARIABLES 431

(1989)) or a combined array is used. In the combined array x and z are in the same design array. As we indicated in the introduction, this allows for overall economy in the design of the experiment. One can take the view that there are three pertinent response surface models: (i) that which results from the response model; (ii) that which describes the process mean and is derived from the response model; and, (iii) that which describes the process variance and is also derived from the response model.

An RSM Model For Process Variance

The response model can take many forms, but we begin with the model described by Box and Jones (1990) and discussed in Lucas (1994) and Myers, Khuri, and Vining (1992), namely

y = (30 + x'(3 + x'Bx + z', + x' Az + E. (1) We allow x' = [Xl , X2, ... , xrx ] , z' = [Zl ' Z2, . . . , zrz ] and assume the Ei are independently and identically distributed (iid) N(O, a;). The model accounts for a second order response surface in the control variables ({30 + x' (3 + x'Bx), linear main effect terms in the noise variables (z',), and the very important control by noise interaction terms (x' Az). It should be emphasized that variations in the form of the model in equation (1) may be necessary. As in the case of previous researchers, it is our intention to use it as a basis from which to work. It is simple and allows for ease in illustration of the results. It certainly does not rule out discrete control variables (i.e., control variables which take on integer or nominal levels). Any response surface approach to data analysis will allow discrete control variables. It does not rule out the use of a subset of the model terms, as in the case where a two level design is used. Additional model terms may be required in the noise variables (e.g., quadratic terms or interactions). In what follows the results can easily be adjusted to account for these model alterations.

Clearly the existence of control by noise interactions makes the process variance a function of the levels of the control variables. That is, the elements of A, the 6ij, produce the dispersion effects created by the noise variables. This, of course, renders a model for the process variance, an important component of the robust parameter design problem. Figure 1 provides an illustration. In (a) of Figure 1 the obvious lack of interaction between X and Z suggests that the process variance contribution from x is the same at both the + and - levels. As a result, x cannot be used to control the process variance. However, in (b)


of Figure 1, the interaction between x and z implies that a larger contribution of process variability on y from the noise variable z occurs at the + level of x.

Let us now return directly to the model of equation (1). Assume that the process encounters random z's. Also assume that all noise variables are continuous and, in accordance with design level centering and scaling,

E(z) = 0

Var(z) = V

where the diagonal elements of V contain the variances

(j = 1, 2, ... , r z ) .

If enough is known about the process, estimates of the variances of the noise variables could be used. Plant data and empirical information about how the consumer handles the product are sources from which estimates can be found. In many cases noise variables are uncorrelated in the process. In much of the development that follows, knowledge of the elements of V is not necessary. Though we often simplify results by assuming V is diagonal, the methodology can easily be extended to the case where nonzero covariances are present. In cases where techniques depend on knowledge of the variances, little is known about robustness to poor estimation.

Using expectation and variance operations taken over z on equation (1) we obtain estimates of the mean and variance response surfaces as

and

J1:z[Y(x)] = bo + x'b + x'Bx (2)

2[y(x)] = (;y + a'x) ' V (;y + a'x) + 2;. (3) Here b, ;y, B, and a contain regression coefficients from the fitted model of equation (1). The value 2; is the error mean square in the fitted model. Clearly the use of equations (2) and (3) via an optimization

(a)

________ x+

________ x. Zhlgh

(b)

llow

FIGURE 1. Control by Noise Interaction.


432 RAYMOND H. MYERS. YOON KIM. AND KRISTI L. GRIFFITHS

type of approach is useful. This will be addressed in a later section. First, however, we shall briefly review the use of response surface methods to accommodate the estimated process variance model of equation (3).

Response Surface Slopes and Minimization of Process Variance

Much has been written about the importance of slopes of a response surface in the factor directions. (See, for example, Ott and Mendenhall (1972), Atkinson (1970), Hader and Park (1978), Myers and Lahoda (1975), and many others.) In the robust parameter design problem (RPD) the slopes of the model of equation (1) in the direction of the noise variables contain the important information in the process variance. This is certainly clear from the plots in Figure 1. In the "no interaction" situation, (constant slope) the process variance generated from the random noise variable is constant, whereas in the "interaction" case the changing slopes allows a[y(x)] to be controlled with choice of x. Consider that equation (3) can be written as (see Myers, Khuri, and Vining (1992) and Myers and Montgomery (1995))

a[y(x) ] = l'(x)Vl(x) + a;

where l(x) = :y + Li'x = 8fj/8z, the vector of slopes of the "response model" in the direction of the noise variables. The slopes are linear functions of the control variables and thus lead to useful and easy manipulation for determining desirable operating conditions on x. In practice, of course, minimum process variance is estimated by setting l(x) = O. A confidence region on the location of minimum process variance can be generated. We know that if Xo represents the conditions where the slopes of the true response surface are simultaneously zero (i.e., l(xo) = 0) then l(xa) rv N(O, Var(l(xo))) , and

The term Va;: (l(x)) = C(xo)a; contains no random variables and is a function of the experimental design. The term dtE represents the error degrees of freedom for the fitted response model of equation (1). For designs in which the noise main effects and control by noise interaction terms are all mutually orthogonal C(xo) will be a diagonal matrix. A confidence region on Xo is developed by making use of


Myers and Montgomery (1995),

Pr { l'(xo) [C(xo)r11(xo) < F } = 1 a. (4) ......... 2 - o:,rz ,dfE TZ{}"f

Here Fa,rz,dfE, is the upper ath percent point of the F distribution. Values of Xo that satisfy the inequality in the brackets of equation (4) define a (1 a) 100% confidence region on the location of minimum process variance.

Further Comments On the Structure and Interpretation of the Confidence Region

The C(xo) matrix quantifies the properties of the estimated slopes, the 0(x), j = 1, 2, ... , TZ. The nature of the matrix has a profound impact on the quality of the estimated optimum conditions (i.e., the estimated location of the minimum process variance). It does not contain values solely depending on the experimental design but is a function of location, Xo, in the control factor space. A close look at C(xo) sheds considerable light on the RSM that deals with the process variance.

Consider Bjj, (j = 1, 2, ... , TZ ) , to be the variance covariance matrix of the relevant coefficients in 0(x), namely

(j = 1, 2, ... ,Tz )

(\xj

If we assume that the design used possesses the orthogonality property discussed earlier, then

C(xo) = diag{x*Bjjxn

where Bjj is a diagonal matrix and xC; = [1, X1,a, X2,O, . . . , xrx,o]. The variances of noise main effect coefficients will be equal, (Var(;:Yj) / {}"2 = vd, as will the variances of Jij, (Var(Jij)/{}"2 = V2)' As a result the confidence region reduces to choices of x for which

rz 12 L lj (xo) j=l

-:-::---c--------=,...,- :::: Fa.rz ,dfE a;Tz [V1 + V2p2(xO)] (5)

where p2 (xo) is the squared distance that the point X1,O, X2,O, . . . , xrx,o is away from the design origin in the control variables. In equation (5) the roles of the slopes and their variance, V1 + V2p2(xO), are even more evident. Small variances of the slopes lead to

Vol. 29. No. 4. October 1997


tight confidence regions for the conditions of minimum process variance. Note that the variance of a slope becomes larger as one moves further away from the design origin.

This confidence region may be utilized to determine how much flexibility the researcher has in the choice of robust conditions. However, the researcher must be cautious in interpretation. A narrow confidence region suggests that Xo is estimated well. A relatively large confidence region may reveal a flat variance response surface near the estimated minimum variance location, suggesting that there are many locations in the control factors for which the slopes do not differ significantly from zero. However, a large region may result from poor estimation of interaction coefficients and thus large variances of slopes. In this case, Xo is being estimated poorly. Quality of fit of the response model and the standard errors of interaction coefficients play a huge role. For a better understanding of the process variance system, the confidence region should be supplemented with a graphical depiction of the variance or standard deviation response surface. The reader should be reminded that when rx, the number of control variables, exceeds r z, the minimum process variance conditions will be a line or a plane. An example with interpretation will be discussed in detail later.

Confidence Statement on Process Variance

To this point we have concentrated our efforts on producing response surfaces for the process mean and variance and finding favorable conditions on the control variables with regards to the process variance. Before we turn our attention to the analysis that combines the two response surfaces, we will first discuss a confidence statement which utilizes the process variance. The reader should view this as being analogous to a confidence statement on mean response using regression analysis, a useful notion in standard RSM as it would be in any regression analysis where predicted response is important. Here, however, the confidence statement is on the variance that is encountered in the process. As in standard RSM, the confidence statement is conditional on x, a location in design space.

For illustration, consider the case where the noise variables are continuous and uncorrelated, the model errors are iid N(O, aD, and the design allows for the orthogonality assumed in previous sections. The


process variance is given by l'(x)l(x) + u; and

l'(x)l(x) , 2 c(x)F(r dfE A) CTErZ z, ,

where F' is a noncentral F variate with noncentrality parameter A = [l'(x)l(x)]/c(x)u;, c(x) = Vl + V2p2(x) is the variance of a slope, and p2(X) is the squared distance from the design origin. As a result, for a fixed x and hence c(x), an approximate (1 - 00)100% confidence interval on l'(x)l(x)/u; is attained from the inequality

, l'(x)l(x) F < < F' (6) (rz,dfE,A,1-a/2) - u;rzc(x) - (rz,dfE,A,a/2).

One can approximate the confidence interval at a fixed x by first computing AU and AL, the large and small values of A respectively that satisfy the inequality in (6). Here, of course, percent points of the noncentral F are vital. Approximate confidence bounds on the process variance can be obtained as &;[1 + AUC(X)] and &;[1 + ALC(X)]. One should keep in mind that the complexion of this confidence interval will depend a great deal on the location x. Note as always the role of the variance of the control by noise interaction coefficients. This variance is quite prominent in c(x) and it is clear that as V2 becomes smaller the confidence interval becomes larger.

In many cases an exact confidence interval on the ratio of process variance to u; will be more useful to the experimenter. Thus the values 1 + AUC(X) and 1 + ALC(X) will be the appropriate bounds. Indeed, in many cases the engineer or scientist may not have a feeling for what a large process variance is in the context of product development. However, he or she may feel comfortable working with the ratio of process variance to u;. For example, if conditions in the control variables are found for which the ratio is l.25, this implies that variability induced by troublesome noise variables is 25% larger than the "baseline variance" experienced from model error or the inability to reproduce a response value with all factors held constant. Obviously, if the ratio is close to 1, one would feel confident in the success of the parameter design experiment, at least as far as variability is concerned. This ratio, then, becomes a measure of efficiency of the parameter design effort.

Use of the Mean and Variance Response Surfaces Simultaneously

We have dealt thus far with response surface analysis of the process variance which is generated from



the inability of the scientist to control noise variables in the process. Many times the process variance is the major response surface of interest. At times one may find that the process mean may be easy to bring to target or, indeed, in a case of a maximization or minimization problem one is able to bring the process mean to an acceptable level. However, there certainly are situations where it is important to deal with the mean and variance response surfaces simultaneously. In a purely RSM approach, the dual response procedure suggested by Vining and Myers (1990) may be helpful. Here, one finds conditions on x which minimize l'(x)V l(x) + &; subject to f1;z[y(x)] being held at some acceptable level. However, there are other procedures which combine the information from the two response surfaces. These procedures may be more appealing in quality areas in which attention is paid to process specifications and specification limits. The criteria used in process optimization should address the distribution of response values in the process. One approach is the use of prediction limits on a future response. Prediction intervals are, perhaps, the most commonly used type of intervals in regression analysis for determining adequate operating conditions. These limits can be extended to the process optimization arena. Another approach provides estimates of quantiles of the distribution of response values through estimated tolerance intervals. These intervals are certainly helpful and may, in fact, be the most useful technique when dealing with both response surfaces. If the robust parameter experiment produces estimates of "extremes" in the distribution of product responses, an engineer or manager may gain important insight regarding process capability.

Prediction Limits On A Future Response

Prediction intervals play an important role when an engineer desires probabilistic bounds on a new response at fixed conditions on the control factors. These intervals have a familiar interpretation and are used frequently in the multiple linear regression setting. The goal of the engineer or scientist is to find the limits around fj(xo) on Yj(xo), a single future observation, corresponding to a probability level of (1 - a) at fixed process conditions. The development of these intervals is somewhat more complicated in this setting. The details surrounding the theoretical manipulations necessary to compute the prediction limits can be found in Appendices A and B. Appendix A provides the theoretical groundwork for the development of prediction intervals found in


Appendix B. The approximate probability statement for determining the prediction limits can be written

Pr {f1;(xo) - k&yp.(xo) :


It is important to understand that both ILz[Y(xo)] and &z[y(xo)] depend on the location, Xo, in the design space. Of course, one seeks k for which

Pr [pr {y! :2: ILz[Y(xo)]- k&z[Y(xo)]} :2: p

]

ilz,az[y(xo)] Yf

:2: 1 - , (7)

where &z[y(xo)] = jVa;z[Y(xo)] is the estimated process standard deviation at a particular setting on the control factors, Xo. Note that a similar statement can be written for the upper one-sided tolerance limit. Here y! is a future observation. The probability statement in (7) can be modified to form a standard normal variate.

Pr [pr { y! - Ez(y) ilz,az[y(xo)] Yf jl'(xo)l(xo) + (J

> ILz - Ez(y) - k&z[y(xo)] }

> p] - jl'(xo)l(xo) + (J -

:2:1 -, (8)

where ILz[Y(xo)] = /30 + xsi3 + xSBxo is the estimated process mean and the estimated standard

deviation is given by &z[y(xo)] = ylhxo) l(xo) + &. Denote the standard normal percentage point as K p' Then (8) can be rewritten as

Pr [ILZ - Ez(y) - k&z[y(xo)] :S: Kp] p.z,az[Y(xo)] jl'(xo)l(xo) + (J;

:2:1 -"

Further modification yields

Pr [ ILz - Ez(y) ilz,az[y(xo) (JEjxG(X'X)lxO

< Kpjl'(xo)l(xo) + (J + k&z[y(Xo)]] - (JEjxG(X'X)lXO :2: 1 - "

Pr [Z < A + k&z[y(xo)] ] p.z,az[y(xo)) - (JEjxo'(X'X)lxO

:2:.1-,

Z- A k Pr ilz,az[y(xo)) -c==========

= < -r= l'( )l( ) 2 . Ixo' (X'X)lxO Xo Xo + (JE V (J

:2:1-" (9)


We must now determine the distribution of the denominator of the left hand side of (9). The approximate distribution was derived in Appendix A. It is given by

where

and

v* = (dv + (illEdfE) f

2 . d2v + (dfEJ dfE

Using this approximation, (9) can be rewritten as

ilz,a;[(xo)] [ --e--x- :S: -jrx=C: G::;::: (X'="'="X=C= )=l;= X= o

]

:2:1 - ,

which becomes

(10)

Now, we will turn the left hand side into a noncentral t variate by writing (10) as

p" ,";[;('0)] r-)-;-.> -jr=x=;=;c;:e::==;=lX=" j :2:1 -"

And, finally,

Pr [(* A :2: m*] :2: 1 -, ilz,az [y(xo)) ,

or equivalently,

where

Pr [ t * .\ :S: m *] :S: , ilz,o-z[Y(xo)] ,

-kvez;* m * = -r==;=:==::=:===;== jxG(X'X)lXO

(11 )

The probability statement in (11 ) allows us to calculate k through the following procedure:


RAYMOND H. MYERS, YOON KIM, AND KRISTI L. GRIFFITHS

(i) Estimate all model terms in the response-model as well as u;.

(ii) Select probability levels p and f. (iii) Select a setting of the control factors, xo,

and calculate x(X'X)-1 Xo, f'Lz[y(xo)], and &z[y(xo)].

(iv) Estimate A and calculate e and v*. (v) Using the estimate of A, find the percentage

point of the noncentral t variate, found in (11) and equate it to

m* -kVeV*

(vi) Solve for k. (vii) The lower tolerance limit at Xo is then

f'Lz[y(xo)] - k&z[y(xo)]. Note that the lower one-sided tolerance interval

is used primarily when the engineer is interested in maximizing the response. If a minimum response is of interest an upper one-sided tolerance interval would be more appropriate. This interval can be created in a fashion similar to that derived above. Adjustments can be easily made.

This procedure is an illustration of one method for approximating lower tailed tolerance limits using both the model for the process mean and the process variance. However, there are certainly other approaches available that can be adjusted for computing tolerance intervals in the scenario discussed here. One can actually avoid the chi-square approximation of a noncentral chi-square variate by using numerical integration procedures to solve the probability statement of (9). Details are given in Myers and Kim (1992). It is our experience that the difference in the estimated k values obtained with and without the chi square approximation is not substantial. For additional information on tolerance intervals see Hahn and Meeker (1991).

Example

The purpose of this example is to provide the potential user a graphical illustration of the meanvariance tradeoff along with the confidence region on the location of minimum variance. In addition, prediction limits and tolerance intervals are used to compare competing coordinates in the control variables.

The example involves a chemical data set taken from Montgomery (1997). A factorial experiment was conducted by a firm in the United States on a


pilot plant level to study the factors thought to influence the filtration rate of a chemical bonding substance. Four factors were varied; pressure (xd, concentration of formaldehyde (X2), stirring rate (X3), and temperature (z). The process engineer is interested in maximizing the filtration rate. However, when the process is in the production stage the temperature is difficult to control. As a result, it is of interest to deal with the variation transmitted by fluctuations of temperature in the process. Thus, temperature is treated as a noise variable. All factors are varied at 1 level in a 24 factorial array. The 1 level associated with temperature is assumed to be at u z , representing temperature variability in the process. Pressure (Xl) was found to have no impact on filtration rate. In fact, the response model is given by (Montgomery (1997))

fj = 70.0625 + 1O.8125z + 4.9375x2 + 7.3125x3 - 9.0625x2Z + 8.3125x3z - 0.5625x2X3

with R2 = 0.9668 and &E = 4.5954. From the response model it is apparent that the variability transmitted by fluctuations in the noise factor can be controlled through proper choice of formaldehyde concentration and stirring rate. The estimated mean model is given by

f'L(X2' X3) = 70.0625 + 4.9375x2 + 7.3125x3 - 0.5625x2x3

and the slope in the z-direction is estimated by

l(X2' X3) = 10.8125 - 9.0625x2 + 8.3125x3

and thus the estimate of process variance is given by

&[y(x)] = (10.8125 - 9.0625x2 + 8.3125x;3)2 +(4.5954)2. (12)

Figure 2 gives contours of constant mean filtration rate along with the locus of points l(x2' X3) = 0, depicting a line of minimum estimated process variance. In order to capture some notion of sampling variability, the 95% confidence region around the line is shaded. Figure 3 gives the superimposed contours of constant mean and standard deviations. The tradeoff between the mean and standard deviation is apparent.

The researcher may be tempted to recommend X2 = 1, X3 = 1 where a mean estimated filtration rate exceeding 80.52 gal/hr is experienced. However, using equation (12), the estimated process standard deviation at these conditions is 10.8 gal/hr The "minimum variance line" (i.e., the location Xo where l(xo) = 0) produces a standard deviation in filtration rate



X3

0.5

0.0

-0.5

62.15

58.47

-0.5 0.0 X2

73.17

0.5

,, (1,0.817)

(1,-0.21)

1.0

FIGURE 2. Contour Plot of the Mean Filtration Rate and the Line of Minimum Process Variance Along with its 95% Confidence Region.

of 4.5954 gal/hr. However, the estimated mean can be, at best, slightly over 73 gal/hr when l(xo) = O. If the process mean can be increased with other variables (e.g., "tuning factors" that do not impact variance) then the coordinates (X2 = 1, X3 = -0.2) are certainly attractive for minimization of process variance. However, if the tuning factors do not exist, then one must deal with the mean-variance tradeoff.

The standard deviation plot and the confidence region in Figure 2 suggest a reasonable amount of flexibility in achieving minimum variance. In this regard, coordinates indicated in the figure would seem to be of interest, Figure 4 represents lower 95% onesided prediction limits, This graphical illustration indicates the need to operate the chemical process at the high concentration of formaldehyde (X2)' There is some flexibility in the operating level of stirring rate (X3), Similarly, Figure 5 depicts lower tolerance limits on filtration rates for p = 0.95 and r = 0.05. Thus we have values that are exceeded 95% of the time with probability 0.95. This illustration also indicates operating the process at the high level of X2 with some flexibility around the X3 = 0 setting.

Combining the information from the various RSM


procedures indicates operating the chemical process at the coordinates X2 = 1 and -0.4 X3 O. If one operates in the region it can be said with 0,95 probability that 95% of the process values will exceed a filtration rate of about 60 gal/hr. These locations appear to produce the best mean-variance tradeoff. Note that the location of maximum estimated response results in a relatively large process variance and thus is not as desirable as other coordinates that are inside the confidence region. The most desirable location on the line l(xo) = 0 is (X2 = 1.0, X3 = -0.21) and, even though the estimated mean is as low as 73.583 gal/hr, the prediction limits and tolerance limits indicate that these are promising process conditions.

Summary

This paper is concerned with the details associated with response surface analysis for situations where noise variables are involved. The use of slopes or partial derivatives with respect to the noise variables are highlighted in the analysis which utilizes the response surface on the process variance. The use of methodology that addresses the simultaneous use of the process mean and variance response surfaces is

5.50 -'';- "'. 62.15

" " '"

58.47 5.50 " " '" -1.0 ,--_ ___ -"'-'L..., ____ ----''--..L. __ -1 1.0 0.5 0.0

X2

0.5 1.0

FIGURE 3. Contour Plot of Both the Mean Filtration Rate and the Process Standard Deviation.



5194

6408

'0,5

.1. -h-.,...,-r'--r-o."...,....,..J.,-,-.,--,-.,...,-r-r;-,-,....,....,...,.,..---,-,...-,-,c-rr-rT"T""r"---,-rl .1.0 0,5 0,0

Xl

0,\ 1.0

FIGURE 4. Contour Plot of Lower 95% One-Sided Pre-diction Limits.

addressed. The use of prediction limits and tolerance intervals are discussed as a simple methodology for comparing competing locations in the design region of the control variables.

X3 1.Or-.,.........--,.........-----r--------..,..----,

0,5

'4689

53.22

59.55

'0,5

.1. 0 h-,....,...,+-:-..,-,-,.-.,--.f.-,-,....,....,.---,-,.-.,--"....,....,....,...,-,-..,.-,-,.,.,....,....,.-r;-..,.-,-m .1.0 '0,5 0,0

Xl

0,5 1.0

FIGURE 5. Contour Plot of O.95-Content Lower 95% One-Sided Tolerance Limits.


Appendix A

The distribution of l"'(x)l(x) can be determined through the use of the distributional properties of l(x). It can be shown that I(x) rv N(l(x), a;C). Thus, lj(x) rv N(lj(x), a;cj) where Cj = Var(lj (x)), apart from a;. Then

rz I'(x)I(x) = LlJ(x)

j=l

(AI)

where >'j = l; (x) / a; Cj. The noncentral distribution of equation (AI) can be approximated by a linear combination of central chi-squares:

where Pj = (1 + 2>'j)/(1 + >'j) and Vj = (1 +>'j)2/(1 + 2>'j). This approximation is due to Patnaik (1949). We can further approximate this linear combination of central chi-squares with

I'(x)l(x) rv a;dx

where

This approximation is due to Welch (1956). Thus, we have an approximate distribution of I' (x)I(x). It is also true that

2 a; 2 a E rv dfE XdfE'

So we can approximate l' (x )l( x) + 0-; by

2 {d 2 1 2 } 2 2 aE XV+dfEXdfE rvaEexV* where



and

v* =

Therefore, the approximate distribution is

Appendix B The development of prediction limits in this setting

iti somewhat more complicated than that found in a multiple regression setting. In order to develop the bounds, the distribution of l' (x)l(x) must first be determined.

Recall the approximate distribution of l' (x)l(x) derived in Appendix A,

where

It is also true that

2 (J"; 2 (J"E rv dfE XdfE'

By combining the distributions of I' (x )l( x) and &; using the method attributed to Patnaik (1949) we can also determine the distribution of the estimated variance of [Yj(xo) - t?(xo)]. It should be clear that, approximately,

&_p, = [1'(xo)l(xo) + &;(1 + x(X'X)-lxO)]

where

rv (J";aX4 (B1)


and

(dv + (dill (1 + xo(X'X)-lXO)) dfE f V4 = ---'-------'----------'--;;-----'----d2v + (dill ( 1 + xo(X'X)-lXO)) 2 dfE

The probability statement for determining the prediction limits can be written

Pr {t?(xo) - k&y_p,(xo) :::; Yj(xo) :::; t?(xo) + k&y_p,(xo)}

=l - a and,

1 Yj(XO) - t?(xo) ) Pr -k :::; -(J" E-----c'l=' (=xo=r=!(=xo=) =+=l=+== X== ( =X ='X=)=-=l x=o :::; k

= 1- a. (B2) Denote

rv N (O, 1) , then equation (B2) can be rewritten as

1 Z(J"E 1 '(X;;(X) + 1 + x(X'X)-lxO ) Pr -k ::; -(J"-E ---"-rl =, (==)!=( X=)= + =l=+=X = (=X=' =X =) =-=l x=o=- ::; k (J"E

= 1 - a. (B3)

Now, the distribution of '(xo)l(xo) +&;(1 + xo(X'X)-lxO)], found in equation (B1) , can be applied to the statement in equation (B3), yielding

Pr 1-k :::; _Z--'-----l'_( X_;; _l ( X_)_+---= l +==-X_ ( _X _'X_)_- _l x _O :::; k) JaX4 = 1- a.

Further manipulations yield

Pr { -m ::; tV4 :::; m} = 1 - a where



and

In order to determine the estimated prediction limits, k must be calculated using the procedure found in the text.

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Key Words: Parameter Design, Prediction Limits, Process Variability, Response Surface Methodology, Robust Design, Tolerance Limits.


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