Product Robust Design and Process Robust Design

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Product Robust Design and Process Robust Design:Are They the Same? (No.)

Haim Shore * and Ram Arad

Department of Industrial Engineering, Faculty of Engineering Science, Ben-Gurion University

of the Negev, Beer-Sheva, Israel

ABSTRACT

Taguchi’s ideas of robust parameter design motivated the development of the dual-response approach, where both the mean and the variance of the quality response aremodeled in terms of the design parameters and noise factors. These are then used toidentify optimal settings that achieve the dual objective of optimizing the signal (themean) and minimizing variation. While much research has been published recentlywith regard to how to solve the dual-response problem (DRP), relatively little attentionhas been given to the unique characteristics of process robust design, like the existence

of systematic variation or intercorrelations among the process “controllable” variables.These properties indeed put process robust design in a category of its own (separatefrom product robust design). In this paper, we rst expound these unique properties anddevelop a general formulation of the DRP as it applies to process robust design. Wethen report on an implementation to an industrial process in a high-tech corporation.

Key Words : Dual response modeling; Parameter design; Robust design; SPC;Taguchi quality philosophy.

INTRODUCTION

Robust parameter design originated in Taguchi’squality philosophy, which called for a dual objective indesigning a new product or process: optimization of thesignal (usually the mean) and minimization of thevariability transmitted to the response via noise

(uncontrollable) factors. The robust parameter designproblem is to nd settings for the design variables that

will optimize some criterion that reects the above dualobjective. Thus, for a process that produces soap, we mayaspire to be close to a target value of pH, but we also wishthe variation around this target to be minimal. This maybe attained by setting the controllable variables at values

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DOI: 10.1081/QEN-120024007 0898-2112 (Print); 1532-4222 (Online)Copyright # 2003 by Marcel Dekker, Inc. www.dekker.com

*Correspondence: Haim Shore, Department of Industrial Engineering, Faculty of Engineering Science, Ben-Gurion University of theNegev, P.O.B. 653, Beer-Sheva 84105, Israel; E-mail: [email protected].

QUALITY ENGINEERING w

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that would cause various noise sources (like heterogen-eity of raw materials or uctuations in process variables)

to contribute minimally to the observed responsevariation.

Robust design is commonly addressed in theframework of response surface methodology (RSM).With this approach, response surfaces, expressed interms of the design factors and relevant noise factors, areconstructed for both the mean and the variance. Theseexpressions are then used to identify optimal settings forthe design variables so that the dual objective of anoptimal mean and minimum variance is achieved. Thisproblem is known in the literature as the dual-responseproblem (DRP). Henceforth we will introduce the DRPas it commonly appears in the literature.

Two general approaches, which reect differentexperimental design structure, exist to model the DRP. Ina cross-product array (similar to Taguchi’s inner andouter arrays), the design factors and the noise factorsare located in different arrays, so that each experi-mental combination of design factors provides noisereplicates that allow independent estimation of theassociated variability (via signal-to-noise ratios or, mostprobably, via direct variance estimates). This approach isreferred to as the “summary measure approach” (Chip-man, 1998). The variance response-surface is built byreferring to the variance estimates of the various

experimental conditions as response data that need tobe modeled.

Alternatively, a combined array may be used wherenoise factors are included in the experimental designtogether with design factors. From this design, a“response model” is estimated, and response surfacesderived for the mean and the variance. Obviously, aresponse model, which includes both design and noisefactors, may also be built from data arranged in a crossedarray. Since in a response model noise factors arecommonly assumed to be normally distributed, with zeromean and constant variance, any signicant design-noiseinteraction in the response model provides an opportu-nity for variance reduction via appropriate settings of thedesign factors. Thus, a response-surface for the mean isderived from the response model, which includes onlydesign factors, and similarly a response-surface for thevariance is derived, which comprises both the errorvariance and variance components associated with thenoise factors. Once appropriate response-surfaces havebeen constructed (estimated), the optimal settings for thecontrollable factors are identied.

Let us demonstrate this approach with a simpleexample. Suppose that there are three design factors, T 1 ,T 2 , and T 3 , all assuming deterministic nonnegativevalues, and two noise factors, expressed in deviations

around the respective means (namely, having zeromean), Z 1 and Z 2 . Suppose that from available data, the

following response model has been obtained:Y ¼ 6:7 þ 2:6(T 1) þ 0:7(T 2) 1:6(T 3) þ 0:22( T 1)( Z 1)

þ 4:6( Z 2) 0:88(T 1)( Z 1)( Z 2) þ 1 (1)

where 1 is the error, assumed to be independent of any of the noise variables, and having a zero mean and constantvariance s 1

2 .Taking expectation of Eq. (1), we obtain a response

surface for the mean:

E (Y ) ¼ 6:7 þ 2:6(T 1 ) þ 0:7(T 2) 1:6(T 3 ) (2)

Next, we derive the response surface for the variance.Since Z 2 appears both individually and in a product (with Z 1), we will use a well-known formula for the variance of Y , Var( Y ), based on conditional expectations:

Var( Y ) ¼ E ½Var( Y j Z 2) þ Var ½E (Y j Z 2) (3)

From Eq. (1) we have for the conditional variance andmean of Y , given Z 2:

Var( Y j Z 2) ¼ (0:22T 1 0:88T 1 Z 2)2Var( Z 1) þ s 21(4a)

E (Y j Z 2) ¼ 6:7 þ 2:6(T 1) þ 0:7(T 2)

1:6(T 3) þ 4:6( Z

2) (4b)

Taking expected value of (4a) and applying the varianceoperator to (4b), we obtain, on introducing back into (3):

Var( Y ) ¼ Var( Z 1)½(0 :22T 1Þ2 þ (0:88T 1)2E ( Z 22 )

þ (4:6Þ2Var( Z 2 ) þ s 21 (5)

Note that E ( Z 22) is the variance of Z 2 . Observing Eq. (5),

we realize that the right-hand side is a function of T 1 , andthe value of T 1 that minimizes V (Y ) is T 1 ¼ 0. While thisselected value for T 1 is optimal in relation to the responsevariability, it is not necessarily so in terms of the

optimality of the mean. For example, to maximize themean, the selected value of T 1 ¼ 0 is obviouslynonoptimal (refer to Eq. 2). Thus, a composite objectivefunction needs to be optimized, which takes into accountthe dual nature of the required design optimization.

A good summary of the dual response analysis maybe found in Myers and Montgomery (1995, Chapter 10).Some more recent developments may be found in thefollowing references: (Borror et al., 2002; Savage andSwan, 2001; Fan, 2000; Fogliatto and Albin, 2000;Myers, 1999; and discussants Kim and Lin, 1998; Viningand Bohn, 1998; Semple, 1997; Del Castillo et al., 1997;Copeland and Nelson, 1996; Vining and Myers, 1990)and references therein.

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As noted earlier, the above depiction and demon-stration of the DRP represents the most common

perception of this problem, as it is addressed in thereferences cited above and in others. Several featurescharacterize this formulation:

(a) There are essentially no real differences betweenproduct robust design and process robust design.

(b) “By design parameters we imply the mean of probability distributions or nominal values of deterministic variables” (Savage and Swan,2001).

(c) All observed variation in design parameters (if there is any) is random (no systematic variationand no correlations with external noise factors oramong design parameters).

It is apparent that the major focus of this prole is onhow design parameters are dened. As (b) above implies,the only possible difference between process and productdesign parameters is that occasionally process designparameters may experience random variation, which hasto be accounted for. How? Essentially, by determiningdesirable means for those design parameters thatexperience random variation. Thus, in Borror et al.(2002), a general response model is introduced where thedeterministic controllable variables appear in a full

quadratic model (namely, all linear, quadratic, andinteraction terms appear). However, noise factors appearonly as main effects or in controllable-factor by noise-factor interactions. This is obviously a restrictivedescription of the diversity of process variables, andtheir interrelations, that one may encounter in reality. Forexample, strong correlations among process factors arethe norm rather than the exception. Furthermore, designfactors will often be “noisy.” This is in sharp contrast todesign factors that one may realistically expect toencounter in product design. In the latter, design factorsare assumed to have little variation, which is accounted

for mainly by setting tolerances. However, this variationis assumed to be negligible relative to variationassociated with noise factors. Thus, the distinctionbetween deterministic (or approximately deterministic)“design” parameters and “noise” nondesign factors isquite clear. This separation between design and noisefactors, in terms of the above characterization, does notcarry over naturally and necessarily to process robustdesign.

All these imply that general denitions of designparameters for a process and a product should differsubstantially. Correspondingly, different formulations of the DRP that reect the differences in denitions shouldbe put forward. These, in turn, would most probably lead

to different solution procedures. Each of these topics willbe addressed in detail in the next two sections.

WHAT ARE DESIGN PARAMETERSIN ROBUST PARAMETER DESIGN?

Design parameters, by their very name, implydecision variables that may be incorporated in the designof a product or a process once their values aredetermined. While it is well known that with productdesign any design parameter has a tolerance, whichindicates that the design parameter is not reallydeterministic, the variation around the target value isusually negligible. Thus, we may justiably assume thatall design parameters in a product design are indeeddeterministic. Furthermore, all noise factors that mayinteract with the design parameters are external to theproduct design and therefore can be handled asindependent variation-transmission agents, the effect of which we wish to eliminate (or reduce) by robust design.

When it comes to process design, two new factorsare introduced, which present-day formulations of theDRP seem largely to ignore. We will discuss each in turn.

Systematic Variation

Unlike in product design, where design factors areassumed to be deterministic and noise factors may beassumed to have a characteristic stable distribution,process design variables may have built-in systematicvariation. What we mean by systematic variation is thatcertain process variables are forced to vary in apredetermined fashion that is dictated by the processbuild-up. Thus, as we shall see in the reported applicationbelow, uid density in a tank that provides liquid for acertain industrial process changes in a cyclic fashion, dueto injection of fresh concentrate of a certain component,

at predetermined time intervals, into the tank.Possible existence of systematic variation for aprocess variable also seems to be in contrast toconventional wisdom when SPC is concerned. Thebasic assumption of the conventional SPC methodologyis that for a process in a state of control, the processvariables have a characteristic stable distribution. Inreality, many process variables regularly experiencesystematic variation that does not conform to thistypology. Obviously, all process variables that change ina cyclic manner exhibit systematic variation. Conse-quently, application of conventional control chartsbecomes inadmissible. For such cases, special controlcharts need to be adopted; for example, control charts

Product and Process Robust Design 195



that monitor cyclic process variables (for one possiblesolution, refer to Box and Luceno (1997). In a similar

fashion, one cannot assume that design parametersdened for process robust design are necessarily randomvariables with distributions that have locally stableparameters (stable mean and stable standard deviation, inthe case of the normal distribution). The observedvariation in process variables often hides a component of systematic variation, most probably of a cyclic nature.

This leads to a very important conclusion: when itcomes to applying Taguchi’s ideas of robust design to aprocess, robustness to systematic variation, and not justto random variation, should be of prime concern. This isparticularly important since in cases where there is acomponent of systematic variation, this component willprobably be dominant in determining the total responsevariation. We may thus state the rst major differencebetween product and process robust designs:

. In product design, design factors are controllabledeterministic variables. Therefore, random vari-ation is transmitted to the response only viaexternal noise factors, which are distinguishableand separate from controllable factors.

. In process design, design controllable factors mayat the same time be noise factors, which transmitto the response both random and systematic

components of variation. Thus, for process robustdesign, design factors may be controllable only inthe sense that their long-term means (and lessoften also their long-term variance) may bedetermined.

These distinctions have important implications, as wewill discuss shortly.

Intercorrelations Among DesignParameters

Unlike product design parameters, which aredeterministic and therefore cannot be correlated, processdesign parameters may represent random variables thatare occasionally intercorrelated. This implies that inmodeling of the response mean, but particularly inmodeling the response variance, the usual assumption of statistical independence among design or noise factors(as in the example in the Introduction) cannot be made(we assume that the variables are normally distributedand therefore correlation and statistical dependence maybe used interchangeably). This makes the modeling of both the response mean and the response variancesomewhat more complex. For one thing, the modeler is

required to estimate not just means and variances of theprocess design and noise parameters but also their

intercorrelations. Additionally, the modeler is not free todetermine independently the means of all designparameters, because a high correlation between a pairof process variables implies that only the mean of one of these variables can independently be determined and set.Thus, the structure of the intercorrelation congurationdictates which of the design parameters can be freelydetermined (even if only the mean is decided) and whichwill result automatically from that decision, due to highintercorrelation with the determined design parameters.

Given this characterization of process designparameters, a modied procedure for the derivation of response surfaces for the response mean and the responsevariance needs to be developed. This will be done in thenext section, where a general model that includesintercorrelations is developed and expressions for theresponse mean and the response variance are derived.

MODELING OF THE RESPONSEMEAN AND THE RESPONSE

VARIANCE IN THE PRESENCE OFINTERCORRELATIONS

Suppose that as a result of an appropriatelycontrolled experiment, a response model has been built:

Y ¼ a 0 þ Sa iT i þ Sbi X i þ SS b ij Z 0i Z 0 jþ SS (cijT j) X i þ 1 (6a)

where 1 is the error with zero mean and variance s 1

2 , fT igis the set of process variables with small enough variationto be treated as deterministic, f X ig is the set of processvariables with either or both random and systematicvariation, and f Z 0 ig is the set of process random variablesthat appear in interaction terms with other randomvariables (note that the last two sets may be overlapping).

The f Z

0 ig variables are expressed in terms of deviationsfrom the present mean, namely, Z 0 i ¼ X i 2 m0 i, where

fm0 ig is the set of the means of the factors in theinteraction term at the time of deriving the regressionequation Eq. (6a). This form of presenting interactionterms obtained from regression analysis is standard incurrent statistical packages. However, the means of theprocess variables are decision variables, the values of which may be changed when applying an optimizationroutine. Therefore, we wish to express the interactionterms in the response model in terms of the deviationfrom the actual mean, whatever its value. Let us dene: Z i ¼ X i 2 mi or Z 0 i ¼ Z i þ di, where: di ¼ mi 2 m0 i, and

mi is the actual mean. Introducing back into Eq. (6a) we

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obtain:

Y ¼ a0 þ Sa

iT

i þ Sb

i X

i þ SS b

ij½ Z i Z j þ d j Z i þ d i Z j þ d id jþ SS (cijT j) Xi þ 1

¼ a 0 þ Sa iT i þ Sk i X i þ SS bij

½ Z i Z j þ d j Z i þ d i Z j þ SS b ij(d id j) þ 1 (6b)

where k i ¼ b i þ S ( j) (c ij T j). This equation will be used toderive an expression for the response mean.

Alternatively, let U ij ¼ Z i Z j. Then we may writeEq. (6b) in terms of f X ig and fU ijg only:

Y ¼ A0 þ SK i X i þ SS b ij U ij þ 1 (6c)

where A0 includes all the constants in the regressionequation, and the terms that include f Z ig were mergedinto terms that include f X ig so that Eq. (6c) is expressedin terms of f X ig and fU ijg only (hence the modieddenition of the coefcients of f X ig). This equation willbe used to derive an expression for the response variance.

We now wish to model the response mean and theresponse variance. Assume that the f X ig may beintercorrelated. While the model presented in Eq. (6a)reects the relationship between the process variables(deterministic and random) and the response, arelationship that may have been derived from controlled

experimental settings, in deriving the response mean andthe response variance we have to be tuned to the actualbehavior of the process. This implies that althoughEq. (6a) is based on designed experimental data, whereprocess variables have been changed in a preplannedmanner, data has now to be collected from the operatingprocess itself so that actual means, variances, andintercorrelations can be estimated. This will ensure thatwith the application of an optimization procedure andderivation of optimal settings for the design parameters,realistically optimal solutions are achieved. Henceforth itis assumed that estimates are based on data collectedfrom such an operating process.

Deriving the response surface for the mean, weobtain from Eq. (6b):

E (Y ) ¼ a 0 þ Sa iT i þ Sk imi þ Sb ijE ( Z i Z j)

þ SS b ij (d id j) (7)

Note, that we assume that the fmig are decision variables,the optimal values of which will be determined in theoptimization stage. Thus, we allow the means to appearas decision variables in the response surface for theresponse mean. By the denition of f Z ig:

E ( Z i Z j) ¼ Cov( X i , X j) ¼ r ij s ( X i)s ( X j) (8)

where Cov( X i, X j) is the covariance between X i and X i,s ( X ) is the standard deviation of X , and r ij is the

correlation between X i and X j. Introducing back into Eq.7 obtain:

E (Y ) ¼ a 0 þ Sa iT i þ Sk imi

þ Sb ij r ij s ( X i)s ( X j)

þ SS bij (d jd j) (9)

We learn that unlike the conventional introduction of theDRP, the mean may include terms that depend on theintercorrelations and the variances of the designparameters, f X ig. Henceforth we assume that thevariances and correlations that appear in the model’sresponse surfaces are independent of the means andtherefore have constant values. This assumption is self-evident under the normal scenario, if the process variableexperiences only random variation. If the normalscenario is not valid, and the variance depends on themean, then modeling of the variance in terms of the meanis needed. Once such a model is derived, the abovevariance components may be expressed in terms of therespective means. The latter would later serve as thedecision variables in the optimization procedure. If asystematic component of variation is also present, onecannot automatically assume that changing the targetmean for a process variable necessarily does not affect

also the systematic component of variation, hence thevariance. In the rest of this paper, however, we assumethat variances of individual process variables are notaffected by changes in the means.

To model the response variance, let us denote thesums that appear in the response model, Eq. (6c), by:

SX ¼ SK i X i ; SU ¼ SS b jk U jk

For two correlated random variables, V i and V j, thevariance of the weighted sum SV ¼ a iV i þ a jV j is:

Var( SV ) ¼ (a i)2Var( V i) þ (a j)2Var( V j)

þ 2(a i)(a j)Cov( V i , V j) (10)Furthermore, for two weighted sums of randomvariables:

Cov Sa i X i, Sb jY j ¼ SS a ib jCov( X i , Y j) (11)

(See Evans, 1992, p. 259.)We may write for the response variance, from Eqs.

(6c), (10), and (11):

Var( Y ) ¼ Var( SX ) þ Var( SU ) þ 2Cov( SX , SU ) þ s 21¼ Var( SX ) þ Var( SU ) þ 2S (i, j,k )(K i)(b jk )

r ( X i, U jk )½Var( X i)Var( U jk )1=2

þ s 21 (12)




where s 1

2 is the error variance from the response model(error is assumed to be independent of SX and SU ),

r ( X ,U ) is the correlation between X and U , andsummation is over all possible combinations of ( i, j, k ).

Let us now develop expressions for the variances of SX and SU . From the above formula for the weighted sumof correlated variables Eq. (10), we obtain:

Var( SX ) ¼ SK 2i Var( X i)

þ 2SS K iK jCov( X i , X j)

¼ SK 2i Var( X i) þ 2SS (K iK j)

r ( X i , X j)½Var( X i)Var( X j) 1=2

Var( SU ) ¼ Sb2ijVar( U ij) þ 2SS

(bij )(bg,h)Cov( U ij , U gh )

¼ Sb2ijVar( U ij) þ 2SS (b ij)(bg,h)

r (U ij , U gh )½Var( U ij )Var( U gh ) 1=2 (13)

These expressions contain terms that can all be estimatedfrom current data. However, apart from the values of thedeterministic variables, fT ig, which are embedded(hidden) in the coefcients fK ig, none of the means fmigappears explicitly in any of the variance components. We

will now develop expressions for Var( U ij ) and nd outwhether these depend on fmig.According to our basic assumptions, expounded

earlier, neither the correlations nor the variances of theprocess variables depend on the means. Therefore, therst expression in Eq. (13) (Var( SX )) is affected by thedeterministic design parameters, fT ig, and by the means,fmig, only via the values fK ig. The second expressionabove (Var( SU )) includes variances of interaction terms(fU ijg and fU gh g), which may be affected by changes inthe means. We will now develop a procedure to calculateVar( U ij ) and nd out whether this expression indeeddepends on the means, fmi, m jg.

First, assume that two variables, V i and V j, withmeans mi and m j, respectively, are independent. Then, itmay be easily shown that:

Var( V iV j) ¼ Var( V i)Var( V j) þ m2i Var( V j)

þ m2 j Var( V i) (14)

(See Stuart and Ord, 1987, p. 343, Exc. 10.23.)Since E ( Z i) ¼ 0 (for all i), we obtain from Eq. (14):

Var( U ij) ¼ Var( Z i Z j) ¼ Var( Z i)Var( Z j)

¼ Var( X i)Var( X j)

Conversely, when the variables Z i and Z j are notindependent, assume that they are from a bi-variate

normal distribution with correlation r ij ¼ r ( Z i, Z j)(which implies that Z i and Z j are not independent).Then, it may be shown (See Stuart and Ord 1987, p. 539,Exc. 16.8.) that:

Var( U ij) ¼ Var( X i)Var( X j)(1 þ r 2ij) (15)

We have thus shown that if we pursue the basic assump-tion that the variances of the process variables, in anoperating process, and their intercorrelations do notdepend on the means, then the terms that include Var( U ij )do not depend on the means either. This implies thatVar( Y ) depends on themeans only via thecoefcients fK ig.

However, Eq. 15 will still be needed in order tocalculate the correct values of the response variance.

If we cannot assume that the mean and the varianceof a process variable are unrelated, Eq. (15) cannot beused since it assumes that Z i and Z j are from a bi-normaldistribution. A general expression for Var( U ij ), whichdoes not require the normality assumption, can be foundin Stuart and Ord (1987, p. 343, Exc. 10.23).

Introducing from Eq. (15) into Eq. (13) and then intoEq. (12), we obtain the nal form of the expression forthe response variance when intercorrelations exist. It

may now be incorporated in the objective function for theoptimization procedure. Any of the objective functionsthat one may nd in the literature dealing with the DRPmay be appropriate, and we will select one of these (thequadratic loss function of Taguchi) in the reportedapplication below.

Two assumptions are implicit in the aboveformulation, and they should again be clearly stated:

1. We assume that changes in the means of theprocess design parameters do not affect theirvariances. This assumption is obviously true for

normal variables, when only random variation ispresent. When a process variable is not normal orwhen there is also systematic variation that canbe linked to the mean, the relationship betweenthe variance and the mean has to be established.This relationship may then be used to express thevariance in terms of the respective mean in allexpressions for the variances.

2. We assume that changes in the means of theprocess design parameters do not affect theintercorrelations among design parameters. Thisassumption again may be easily justied if amulti-variate normal distribution is assumed forthe set of design parameters f X ig.

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Both of the above assumptions allow us to refer tothe response surfaces obtained for the response mean and

the response variance as functions of the means only andlet the optimization procedure determine their optimalvalues.

Before we conclude this section with a description of an application, let us take stock of the types of estimates,based on data from an operating process, that arerequired in order to develop the response surfaces neededfor the design optimization. Assuming that the responsemodel Eq. (6a) has already been derived, we need thefollowing estimates:

1. For the response mean [refer to (Eq. 9)]:. Estimates of the standard deviations of f X ig,

and the correlations, fr ijg:2. For the response variance [refer to (Eq. 12)]:

. Estimate of the error variance associated withthe response model ( s 1

2).. Estimates of variances of f X ig. Additionally, for

variables in interaction terms, the correlationbetween the variables that share an interactionterm;

. Estimates of intercorrelations between inter-action terms, fr (U ij , U gh )g.

AN APPLICATION

Description of the Process

The application reported here has been conducted ina high-tech multinational corporation. Due to proprietaryconsiderations, the exact nature of the process, as well asthe process variables, have been concealed. However, thedata, the data analyses, the nal model and other relevantoutputs are provided here intact. The application focuseson a process, which transforms a liquid solution into anal product. The solution is prepared in a reservoir that

mixes together various ingredients. These ingredientsdetermine the properties of the solution. A pumptransfers the “ready” solution to a process device,where the nal product is manufactured. The requiredproperties of the process output (the output at the processdevice) serve as the responses, investigated in thisapplication. We will focus on a single most importantresponse that will be denoted Y . Extended studies thathave been conducted in the corporation established thatthe process response ( Y ) is affected by the solutionproperties A, B, and C . For that reason, the solution in thetank is being kept at specic desirable levels of A, B, andC . Controlling these solution properties is performed by aclosed-loop control system, which is based on unique

sensors for each of the solution properties. Whenever Aor B reach predened low limits, the control system

automatically adds the needed ingredients to the solutiontank. Another mechanism keeps C at the required level.A schematic drawing of the complete system is given inFig. 1.

As related earlier, the system is designed to controlthe solution properties in the tank at predeterminedlevels. If a high level of Y is needed, this results in highsolution consumption and the levels of A and B in thetank fall. Thus, more frequent injections of ingredientsare needed in order to keep the response at the requiredlevel. In general, large efforts are invested in stabilizingthe solution properties. Property C has relatively stablevariance, while A, and to a lesser degree B, have built-insystematic variation. The root of this variation isembedded in the nature of the process, which requiresinjections of the ingredients when the control loop sodetermines. Figure 2 demonstrates a typical chart of property A level over time.

The trend line represents property A, the trianglesare points at which a concentrated ingredient was addedthat has affected A, and the squares are points at whichanother ingredient was added to the tank that affectsproperty B (these procedures are conveniently termed“ingredient A add” and “ingredient B add,” respectively).Clearly, one can see very high correlation between

“ingredient A add” points and A peaks, which results in asawtooth plot. The slope of the A line, in the negative-slope regions, is relatively constant, which impliesconstant solution consumption. The chart demonstratesthe built-in systematic variation caused by constantsolution consumption, on the one hand, and short-timeinjections of the ingredients, on the other hand.

This research aimed at optimizing the solutionproperties in the sense that the response is optimizedaccording to the dual objective of being as near asfeasible to the target value (1.45) while minimizing thevariability in the response caused by variation in thesolution properties.

The Research Scheme

The research comprised four stages:

(I) Deriving the response model , based on datacollected in a designed experiment.

(II–III) Deriving expressions for the response meanand the response variance in terms of designfactors’ levels (for deterministic factors) or




in terms of design factors’ means (for vari-able factors). This stage was based on thederived response model (from stage I) andon data collected from an operating process.

(IV) Finding optimal settings for the solution properties . Once the response model hadbeen established, the DRP was chosen as theframework approach, and a constrainedoptimization routine applied to an objectivefunction that reects the dual goal of proximity to target plus minimum sensi-tivity to variation in the solution parameters.Taguchi’s loss function was used as theobjective function for the optimizationprocedure. We will now describe each of these phases (analyses and results) in detail.

(I) The Response ModelThe relationship between the response and the

solution properties was investigated by conducting a 3 k

full factorial experiment. The purpose of this experimentwas to derive a response model for Y (or any requiredtransformation thereof) in terms of the affecting solutionproperties. In a separate experiment, which we do notdiscuss in this paper, the effects of different uncontrol-lable factors on the response were examined in theframework of Taguchi’s crossed array design (usinginner and outer orthogonal arrays for design anduncontrollable factors, respectively).

Collecting data from the designed experiment andexamining various alternative models, including a Box-Cox transformation that aimed to stabilize the variance, itwas determined that the best response is log( Y ). The

Figure 2. Typical variation plot of ingredient A.

Figure 1. Schematic drawing of the research system.

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following response model was obtained:

log( Y ) ¼ SX þ SU þ 1 (16)

where 1 is the error, and:

SX ¼ 0:1132 þ (0:1437) A

(0:002633) B þ (0:01057) C

SU ¼ (0:0007776) (A 1:748) (B 111 :1)

(0:001354) (A 1:748) (C 30:08)

(0:00004017)( B 111 :1) (C 30:08) (17)

Based on a 3 k full factorial experimental design with 16replicates for each run and with three outlyingobservations that were excluded, the total number of observations used in the regression analysis is n ¼ 429.The resulting goodness-of-t statistics are an adjusted R-squared of 0.9871, model F -ratio value of 5442 (highlysignicant), and s 1 ¼ 0.01136. Thus, good represen-tation of the process dynamics has been captured by Eq.(16). Various diagnostic checks (like identication of outliers, normal probability plots for residuals, checkingserial correlations, using known statistics for comparisonbetween alternative models like AIC and C p) showed thatthis model conforms to the normal scenario, assumed inmultiple linear regression analysis, and that this is thebest model.

(II) The Response Mean

While the above response model has been extractedfrom a designed experiment, deriving response surfacesfor the response mean and the response variance requiresthat the covariance matrix from actual process data becalculated, so that variances, co-variances, and corre-lations from an operating process can be used in theexpressions for the response mean and the responsevariance. This will ensure that optimization is done withregard to data from an operating process and not from a

controlled experiment. Summary statistics based on 7273observations (an observation per second) are displayed inTable 1.

We realize that the means are nearly identical tothose calculated in the designed experiment, as seen inthe interaction terms of the response model, given inEq. (17). However, certain local instabilities in theprocess caused the variance-covariance matrix to varyappreciably from one time period to the next. Therefore,to identify periods when the process was relativelystable, and thus representative of normal operatingconditions, Mahalanobis distances were calculated foreach observation. A description of this measure is givenin the appendix (from JUMP w Help). The Mahalanobis

distance provides a multidimensional measure of thedistance of an observation from the vector of the meansso that an observation with large distance can be regardedas relatively unrepresentative of normal workingconditions. Observing a plot of Mahalanobis distances,a certain period was identied where the process was judged to be relatively stable, and observations from thisperiod (1000 observations, 1 per second) were used tocalculate the variance-covariance matrix and theassociated correlations. These are displayed in Tables 2and 3. We are now in a position to derive expressions forthe response mean and the response variance, based on

normal operating conditions of the process.First, for the response mean, we obtain from Eqs. (9),(16), and (17) (and the data in the tables):

E log( Y ) ¼ 0:1132 þ (0:1437) mA

(0:002633) mB þ (0:01057) mC

(0:0007776) r AB ½Var(A)Var(B) 1=2

(0:001354) r AC ½Var(A)Var(C) 1=2

(0:00004017) r BC ½Var(B)Var(C) 1=2

(0:0007776) (mA 1:748)( mB 111 :1)

(0:001354) (mA 1:748)( mC 30:08)

(0:00004017) ) (mB 111 :1)(mC 30:08)

(18)

where the last three products represent SS b ij (di d j) inEq. (9) and the variances and correlations are taken fromTables 2 and 3, respectively.

If we introduce in Eq. (18) the current means (fromTable 1, which is based on all 7273 observations) weobtain:

E log( Y ) ¼ 0:3898, or: E (Y ) ¼ exp E log( Y )

¼ 1:4767

Table 1. Summary statistics from anoperating process (data from about 121

minutes, n¼

7273).n 7273Mean( A) 1.7800Mean( B) 114.05Mean( C ) 30.070Variance( A) 0.0044573Variance( B) 4.5399Variance( C ) 0.097409Median( A) 1.7780Median( B) 114.01Median( C ) 30.100




Note that the equation used for E (Y ) is onlyapproximately true (to realize that, expand log( Y ) in aTaylor series around the mean and apply the expectation

operator to the rst two terms).

(III) The Response Variance

To model the response variance, we need todetermine rst which of the three solution propertiescan be uniquely determined (and thus be considered asdeterministic) and which have components of variation(random and / or systematic), which allow only determi-nation of optimal means. Observing the variances of thethree parameters (Table 2) relative to the current meanswe obtain the following coefcients of variation

(CV ¼ s / m):

CV( C ) ¼ (0:09224) 1=2

30 :07 ¼ 0:0101;

CV( B) ¼ (4:234) 1=2

114 :1¼ 0:0180;

CV( A) ¼ (0:001340) 1=2

1:780¼ 0:02056

It is obvious from the foregoing description of theprocess that, unlike A and B, the solution property C hasno systematic component of variation attached to it. This

is now reected in its relatively small coefcient of

variation. Therefore, it may be decided to treat C asdeterministic (a T i variable in the formulation of Eq. (6a),while A and B may be treated as random ( X i variables).

This decision may be corroborated by the extremely highcorrelations between interaction terms comprising C andthe other variable in the interaction term. For example,the correlation between the interaction term ( A)(C ) and Ais 0.936, which implies that ( A)(C ) is nearly proportionalto A, evidence of the relative stability of C (refer toTable 3). However, we have decided to treat C asrandom, too, and once the appropriate expressions arederived, there is no difculty to revoke this decision andtreat C as deterministic by simply introducing determi-nistic values for C (for example, inserting Var( C ) ¼ 0,and r (C , B) ¼ r (C , A) ¼ 0).

In accordance with the general formulation given forthe DRP when intercorrelations are present, we obtain forthe variances, Eqs. (12) and (13):

Varlog( Y ) ¼ Var( SX ) þ Var( SU )

þ 2Cov( SX , SU ) þ s 21 (19)

Var( SX ) ¼ (K A)2Var( A) þ (K B)2Var( B)

þ (K C )2Var( C )

þ 2(K A)(K B)Cov( A, B)

þ 2(K A)(K C )Cov( A, C )

þ 2(K B)(K C )Cov( B, C ) (20)

Table 2. Variance-covariance matrix (from an operating process, based on n ¼ 1000 observations).

Row A B C A B A C C B

A 0.001340 0.03402 0.004054 0.2112 0.04754 1.487 B 4.234 0.2803 11.15 1.513 159.6C 0.09224 0.9491 0.2825 18.95 A B 43.25 8.027 444.4 A C 1.927 77.75C B 6972

Table 3. Correlations among factors f X ig and interaction terms f X i X jg (from an operating process, basedon n ¼ 1000 observations).

A B C A B A C C B

A 1.0000 0.4515 0.3646 0.8772 0.9355 0.4863 B 1.0000 0.4485 0.8242 0.5296 0.9290C 1.0000 0.4752 0.6701 0.7474 A B 1.0000 0.8793 0.8093 A C 1.0000 0.6709C B 1.0000

Shore and Arad202



where:

K A ¼ 0:1437 (0:0007776) d B (0:001354) d C

K B ¼ 0:002633 (0:0007776) d A (0:00004017) d C

K C ¼ 0:01057 (0:001354) d A (0:00004017) d B

and:

d A ¼ m A 1:748; d B ¼ m B 111 :1; d C

¼ mC 30:08;

Var( SU ) ¼ (0:0007776) 2 Var(U AB ) þ (0:001354) 2

Var( U AC ) þ (0:00004017) 2Var( U BC )

þ 2( 0:0007776)( 0:001354)

Cov( U AB, U AC )

þ 2( 0:0007776)( 0:00004017)

Cov( U AB, U BC )

þ 2( 0:001354)( 0:00004017)

Cov( U AC , U BC ) (21)

and from Eq. (11):

Cov( SX , SU ) ¼ S (K i)(bkj)Cov( X i , U kj)

¼ K Ab ABCov( A, U AB)þ K Ab AC Cov( A, U AC )

þ K Ab BC Cov( A, U BC )

þ K Bb ABCov( B, U AB)

þ K Bb AC Cov( B, U AC )

þ K Bb BC Cov( B, U BC )

þ K C b ABCov( C , U AB)

þ K C b AC Cov( C , U AC )

þ K C b BC Cov( C , U BC ) (22)

Introducing from Tables 2 and 3 for variances andcorrelations associated with f X ig and introducing from

Table 4 correlations between f X ig and fU ijg, while usingEq. (15) to calculate variances for the interaction terms,Eq. (19) is expressed in terms of the means and knownparameters’ values. Note that the assumption of independence between the f X ig is not used in calculatingthe expression for the variance of log( Y ) [namely,Eq. (14) is not used], and we have used the sampleestimates of correlations and variances in order that Eq.(19) will provide an accurate estimate of the variance.

The variance of log( Y ) is now expressed in terms of the means of A, B, and C and is ready to be incorporatedin the objective function used in the optimizationprocedure.

(IV) Finding Optimal Settings for the SolutionProperties (Optimization)

The expected value of Taguchi’s quadratic lossfunction, L (.), is also the response mean squared error(multiplied by a constant), namely:

E L ½log( Y ) ¼ (k )fVar log ( Y ) þ ½E log( Y ) LT 2g

¼ (k )MSElog( Y ) (23)

where LT ¼ log(1.45) ¼ 0.3716 is the target value forlog( Y ), and k is a coefcient, determined by a given pointon the loss function curve. Since k is not part of theoptimization routine, we may assume, arbitrarily, k ¼ 1,so that MSE becomes the objective function that we wishto minimize, as done in earlier published reports (forexample, Copeland and Nelson, 1996). The objectivefunction (OF) is:

OF ¼ Var log ( Y ) þ ½E log( Y ) LT 2 ! Min. (24)

Table 4. Correlations among factors f X ig and interaction terms fU ijg (from an operating process, basedon n ¼ 1000 observations).

A B C U AB U BC U AC

A 1.0000 0.4515 0.3646 2 0.2812 2 0.1150 0.0880 B 1.0000 0.4485 2 0.4882 2 0.1139 2 0.1643C 1.0000 2 0.0762 2 0.1933 2 0.2924U AB 1.0000 0.3014 0.2648U BC 1.0000 0.4963U

AC 1.0000




The following technological constraints wereimposed on the desirable means:

1:0 m A 2:5 (about + 0:75 around

the current mean)

104 m B 124 (about + 10 around

the current mean)

20 mC 40 (about + 10 around

the current mean) (25)

Applying rst an unconstrained minimization routine

(using MATHEMATICAw

), we nd out that the optimalvalues for the means of the process variables were withinacceptable range, and therefore no constraints wereadded to the optimization. The global optimal solutionand optimal solutions for certain prespecied C valuesare given in Table 5.

Below the table are values associated with thecurrent operating process. We realize that these valuesare far from being optimal. Figures 3–6 provide three-dimensional plots and contour plots for C ¼ 30 (currentvalue) and for C ¼ 37 (global optimum). It is apparentthat the means of the process variables may be allowed tochange over a large range of values without losing muchof the optimality of the process.

CONCLUSIONS

The purpose of this paper was to demonstrate thatsome of the basic assumptions, which are routine inmodels that one can nd in the published literaturedealing with robust design, do not apply in general toprocess robust design. Specically, ignoring systematicvariation and, most importantly, intercorrelations amongprocess variables may result in wrong modeling of boththe response mean and the response variance. This can

lead to the implementation of supposedly optimalsolutions that are not really so. Observing the expressionsfor the response mean and the response variance,developed above for the reported application (withintercorrelations accounted for), and comparing these tocorresponding expressions developed under the inde-pendence assumption may be constructive in demon-strating the enormous effect that intercorrelations mayhave on both the response mean and the responsevariance. The distinction made here between productrobust design and process robust design should nd moreemphasis in publications that justiably advocate theadoption of robust design concepts.

APPENDIX: MAHALANOBISDISTANCE

An outlier distance plot shows the Mahalanobisdistance of each point from the multivariate mean(centroid). The Mahalanobis distance takes into accountthe correlation structure of the data as well as theindividual scales. For each value, the distance is denotedd i and is computed as:

d i ¼ ½(Y i Y)0

S 1

(Y i Y)1=2

where Y i, is the data for the ith observation (a vector), Yis the vector of means, and S is the estimated covariancematrix for the data. The reference line drawn on aMahalanobis distance plot is computed as the square rootof “F nvars,” where “nvars” is the number of variablesand F is the 0.95 quantile of the F distribution, with nvarsdegrees of freedom in the nominator and ( n-nvars-1)degrees of freedom in the denominator. If a variable is anexact linear combination of other variables, then thecorrelation matrix is singular and the row and the columnfor that variable are zeroed out. The generalized inversethat results is still valid for forming the distances.

Table 5. Results of unconstrained optimization.

Scenario ( C ) A B Objective function Response variance Response mean dev.

OptimalC ¼ 37.21

1.810 145.5 0.0001604 0.0001603 2 0.0002939

C ¼ 25 2.445 134.5 0.0001648 0.0001648 0.00005609C ¼ 30 (Current) 2.095 134.3 0.0001630 0.0001630 0.00005780C ¼ 35 1.71738 134.6 0.0001612 0.0001612 0.00006939

Notes : Current values: A ¼ 1.75; B ¼ 111.1; C ¼ 30.OF at current operating conditions; OF ¼ 0.000498618; Var[log( Y )] ¼ 0.000167243; E [log( Y )] 2 0.3716 ¼ 0.01820.

Shore and Arad204



Multivariate distances are useful for spotting outliersin many dimensions. However, if the variables are highlycorrelated in a multivariate sense, then a point can beseen as an outlier in multivariate space without looking

unusual along any subset of dimensions. Said anotherway, when data are correlated, it is possible for a point tobe unremarkable when seen along one or two axes butstill be an outlier by violating the correlation.

Figure 4. Contour plot of OF at C ¼ 30.

Figure 3. Three-dimensional plot of the OF at C ¼ 30.




Statistical Warning: This outlier distance isnot particularly robust in the sense that outlyingpoints themselves can distort the estimate of theco-variances and means in such a way that outliers are

disguised. You might want to use an alternativeapproach where distances are computed with a jackknifemethod. The alternate distance for each observationuses estimates of the mean, standard deviation, and

Figure 6. Contour plot of the OF at C ¼ 37.

Figure 5. Three-dimensional plot of the OF at C ¼ 37.

Shore and Arad206



correlation matrix that do not include the observationitself.

ACKNOWLEDGMENTS

JUMP is a registered trademark of SAS. MATHE-MATICA is a registered trademark of WolframResearch.

ABOUT THE AUTHORS

Haim Shore is a professor in the Department of Industrial Engineering and Management, Ben-GurionUniversity of the Negev, Israel. He is an Associate Editorof Communications in Statistics and serves on theEditorial Boards of IIE Transactions (Quality and Reliability Engineering ) and Quality Engineering . In2002–03 he was a visiting professor at the Department of Mathematics and Statistics, McMaster University,Hamilton, Canada. He is a senior member of ASQ andof IIE.

Ram Arad is currently a testing specialist engineer ina high-tech rm. He received his M.Sc. in industrial

engineering and his B.Sc. in mechanical engineering,both at Ben-Gurion University of the Negev, Israel. Thereported application was the subject of his master thesis.

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