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Design and Analysis of Computer Experiments WithBranching and Nested FactorsYing Hung, V. Roshan Joseph and Shreyes N. MelkoteDepartment of Statistics, Rutgers, The State University of New Jersey, Piscataway, NJ08854School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta,GA 30332School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332

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To cite this article: Ying Hung, V. Roshan Joseph and Shreyes N. Melkote (2009): Design and Analysis of ComputerExperiments With Branching and Nested Factors, Technometrics, 51:4, 354-365

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Design and Analysis of Computer ExperimentsWith Branching and Nested Factors

Ying HUNG

Department of StatisticsRutgers, The State University of New Jersey

Piscataway, NJ 08854([email protected])

V. Roshan JOSEPH

School of Industrial and Systems EngineeringGeorgia Institute of Technology

Atlanta, GA 30332([email protected])

Shreyes N. MELKOTE

School of Mechanical EngineeringGeorgia Institute of Technology

Atlanta, GA 30332([email protected])

In many experiments, some of the factors exist only within the level of another factor. Such factors areoften called nested factors. A factor within which other factors are nested is called a branching factor.Suppose, for example, that we want to experiment with two processing methods. The factors involved inthese two methods can be different. Thus in this experiment, the processing method is a branching factor,and the other factors are nested within the branching factor. The design and analysis of experiments withbranching and nested factors are challenging and have not received much attention in the literature. Mo-tivated by a computer experiment in a machining process, we have developed optimal Latin hypercubedesigns and kriging methods that can accommodate branching and nested factors. Through the applica-tion of the proposed methods, optimal machining conditions and tool edge geometry are attained, whichresulted in a remarkable improvement in the machining process.

KEY WORDS: Finite element model; Kriging; Latin hypercube design.

1. INTRODUCTION

Nested factors are those factors that exist only within thelevel of another factor. A factor within which other factors arenested is called a branching factor. Suppose, for example, thatwe want to experiment with two surface preparation methods inprinted circuit board (PCB) manufacturing: mechanical scrub-bing and chemical treatment. Mechanical scrubbing can be op-timized by changing the pressure of the scrub, and chemicaltreatment can be optimized by changing the micro-etch rate.Here the surface preparation method is the branching factor,and pressure and micro-etch rate are the nested factors. Whendesigning an experiment, the two nested factors (pressure andmicro-etch rate) are assigned to the same column in the designmatrix. If we choose two levels for the pressure and micro-etchrate, then the experimental design will appear like that shownin Table 1. Because nested factors can differ with respect to thelevel of branching factor, designing and analyzing experimentswith such factors is not trivial.

Taguchi (1987, p. 280) proposed an innovative approachto designing experiments with branching and nested fac-tors. He called nested factors “pseudo-factors” and the result-ing designs “pseudo-factor designs.” Phadke (1989, p. 168)called these “branching designs.” The core idea is to carefullyassign branching and nested factors to the columns of orthog-onal arrays using linear graphs in such a way that their inter-actions can be estimated. The interactions between branchingand nested factors are important, because the nested factorsdiffer with respect to the levels of the branching factors, andthus their effects can change depending on the level of the

branching factors. Consider the PCB experiment, for example.As shown in Figure 1(a), the surface quality of the PCB mayincrease with increasing pressure but may decrease with in-creasing micro-etch rate. There is strong interaction betweenthe branching and the nested factors. The interaction effectcould be reduced if the effects of the nested factors were knownbefore the experiment; for example, if the two pressure lev-els are interchanged, then the interaction becomes small [seeFigure 1(b)]. In general, however, the effects of the nested fac-tors are not known before the experiment, and thus this cannotbe done. Furthermore, any factor correlated with a significantbranching-by-nested interaction will be misspecified. There-fore, we should design experiments that are capable of esti-mating the potentially large branching-by-nested interactions.Although Taguchi’s approach using orthogonal arrays and lin-ear graphs is very intuitive, it is not sufficiently general to applyto more complex situations, such as the design of computerexperiments.

The designs that we consider here differ from the so-called“nested designs” reported in the literature (see, e.g., Hicks andTurner 1999, p. 190; Montgomery 2004, p. 525), in whichthe nested factors are assumed to be similar (e.g., differentbatches of material nested within different suppliers). Because

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TECHNOMETRICS, NOVEMBER 2009, VOL. 51, NO. 4DOI 10.1198/TECH.2009.07097

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http://dx.doi.org/10.1198/TECH.2009.07097

COMPUTER EXPERIMENTS WITH BRANCHING FACTORS 355

Table 1. Experimental design for the PCB experiment

Run Method (branching factor) Nested factors

1 mechanical scrubbing pressure12 mechanical scrubbing pressure23 chemical treatment micro-etch rate14 chemical treatment micro-etch rate2

the nested factors are similar, the branching-by-nested interac-tions do not play a critical role, as they do in the present prob-lem. Moreover, in nested designs, the nested factors are usuallytreated as random effects, and the main focus is on estimat-ing the variance components, whereas in the present problem,the nested factors are treated as fixed effects, and the objectiveis to simultaneously identify the optimal settings of branching,nested, and other factors.

Our work is motivated by a computer experiment involvingbranching and nested factors in which the objective is to opti-mize a turning process for hardened bearing steel with a cBNcutting tool (see Figure 2). This process, commonly known ashard turning, is of considerable interest to bearing manufactur-ers as a potential replacement for the grinding process. Becausethe material being machined is very hard (hardness > 60 Rock-well C), the cutting tool is subjected to high forces, stresses, andtemperatures during the operation. In practice, the tool’s cuttingedge is shaped to withstand the severe conditions. Two com-monly used cutting edge shapes, hone and chamfer, are shownin Figure 3. Note that Figure 3 represents the idealized viewof the instantaneous cutting action in the cross-section “A–A”shown in Figure 2. These cutting edge shapes are intended tostrengthen the cutting edge to bear the large tool stresses gen-erated during cutting. The chamfer tool design can be changedusing two factors, chamfer length and chamfer angle, whereasthe hone design is fixed. In other words, the two factors lengthand angle are nested within the chamfer edge, and no factors arenested within the hone edge. In our terminology, the tool edgeis a branching factor. Thus, when the branching factor takes thelevel chamfer, two additional factors are present in the exper-iment; but when the branching factor takes the level hone, no

Figure 2. Schematic of the turning process. A–A is a perpendicularsection through the tool.

additional factors are present. A few other factors are commonto both of the tool edges, including the cutting edge radius, toolnose radius, and rake angle. In addition, the machining parame-ters, such as cutting speed, feed, and depth of cut, also do notdepend on the type of tool edge (see Figure 2). To distinguishthese factors from the branching and nested factors, we callthem “shared factors.” Table 2 lists all of the factors involvedin this experiment and their allowable ranges. The experimentscan be performed with computers using the commercially avail-able finite element software AdvantEdge.

Latin hypercube designs (LHDs) are commonly used incomputer experiments (McKay, Beckman, and Conover 1979).A desirable property of a LHD is its one-dimensional balance;that is, when an N-point design is projected onto any factor,there will be N different levels for that factor. Clearly, this can-not be satisfied for branching and nested factors. The branch-ing factor is usually a qualitative factor, and thus the numberof levels of a branching factor is fixed (i.e., it does not dependon the number of runs N). Moreover, the nested factors differfor different levels of the branching factor. Therefore, we need

Figure 1. Illustration of branching-by-nested interaction when the effects are unknown (a) and when the effects are known (b).

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356 YING HUNG, V. ROSHAN JOSEPH, AND SHREYES N. MELKOTE

Figure 3. Illustration of hone and chamfer tool edges.

a one-dimensional balance for the nested factors within eachlevel of the branching factor. As an example, consider an ex-periment with one branching factor z1, one nested factor v1,and two shared factors x1 and x2. Suppose that the branchingfactor has two levels and that we want to do the experimentin eight runs. Table 3 presents a possible design for this experi-ment. We can see that the shared factors have a one-dimensionalbalance, because they take eight different levels in the experi-ment. The nested factor has a one-dimensional balance withineach level of the branching factor. Note that v1 represents twodifferent factors, one when z1 = 1 and the other when z1 = 2;therefore, v1 = 1 in run 1 is not the same as v1 = 1 in run 5. Inthe next section we discuss some general strategies for design-ing such experiments using LHDs.

It is well known that not all LHDs are “good.” Most of theresearch in this area has focused on finding “good” LHDs basedon some optimal design criteria (see Iman and Conover 1982;Tang 1993; Owen 1994; Morris and Mitchell 1995; Tang 1998;Ye 1998; Ye, Li, and Sudjianto 2000; Jin, Chen, and Sudjianto2005; Joseph and Hung 2008). We need to extend those opti-mal design criteria for experiments with branching and nestedfactors. Take, for example, the case of maximin LHD proposedby Morris and Mitchell (1995), where the optimal design crite-rion is to maximize the intersite distance among the experimen-tal points (runs). Now with the branching and nested factors,the notion of “distance” does not exist for all factors. Branch-ing factors are qualitative and thus cannot be measured by dis-tances. Moreover, for nested factors, the notion of “distance”exists only if the corresponding levels of the branching factorare the same. Another major factor that makes the design of ex-periment different from the usual designs is the importance of

the interaction between branching and nested factors. As notedearlier, these interactions usually are not negligible; therefore,if any of the main effects is highly correlated with one of theseinteractions, then that main effect will be misspecified. Thusthe optimal design criteria should be modified to capture thebranching-by-nested interaction effects.

The remainder of the article is organized as follows. In Sec-tion 2 we discuss some general strategies for designing ex-periments with branching and nested factors and introduce theconcept of branching LHD. In Section 3 we discuss three dif-ferent criteria for finding an optimal branching LHD. We re-port an analysis of experiments with branching and nested fac-tors in Section 4, and illustrate the proposed methods usingthe hard turning experiment in Section 5. We end with someconcluding remarks and directions for future research in Sec-tion 6.

2. BRANCHING LATIN HYPERCUBE DESIGNS

In general, an LHD with N runs and p factors, denoted byLHD(N,p), can be generated using a random permutation of{1,2, . . . ,N} for each factor. This cannot be done if the exper-iment involves branching and nested factors, however, as dis-cussed earlier.

Consider a simple case in which the branching factor z1has only two levels and m1 factors are nested within eachlevel of the branching factor. Thus there are m1 nested factors(vz1

1 , . . . , vz1m1 ), where each of the nested factors represents two

different factors, depending on the two levels of the branchingfactor. In addition, there are t more shared quantitative factors.

Table 2. Factors and their ranges in the hard turning experiment

Type of factor Notation Factor Ranges

Branching factor z1 Cutting edge shape hone or chamfer

Nested factors v1|z1 = chamfer Angle (degree) 17 ∼ 20v2|z1 = chamfer Length (μm) 115 ∼ 140v1|z1 = hone None Nonev2|z1 = hone None None

Shared factors x1 Cutting edge radius (μm) 5 ∼ 25x2 Rake angle (degree) −15 ∼ −5x3 Tool nose radius (mm) 0.4 ∼ 1.6x4 Cutting speed (m/min) 120 ∼ 240x5 Feed (mm/rev) 0.05 ∼ 0.15x6 Depth of cut (mm) 0.1 ∼ 0.25


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Table 3. An example of a branching LHD

Run z1 v1 x1 x2

1 1 1 4 12 1 2 3 83 1 3 8 54 1 4 2 3

5 2 1 7 26 2 2 1 67 2 3 6 78 2 4 5 4

We discuss some strategies for constructing LHDs with branch-ing and nested factors.

A naive strategy is to choose an LHD for the nested andshared factors and repeat it under each level of the branchingfactor; that is, first choose an LHD(n0, m1 + t) that can ac-commodate the nested and shared factors, then repeat it forthe two levels of the branching factor. The resulting design,shown in Table 4, is easy to construct. Moreover, optimal LHDsfor the nested and shared factors can be readily chosen usingthe existing methods. In addition, because all of the combina-tions of nested and shared factors are repeated at each levelof the branching factor, the interactions involving the branch-ing factor can be estimated. This approach has two drawbacks,however. First, if the design matrix is projected onto one ofthose shared factors (x1, . . . , xt), then some replications occur;thus the design points are not spread out as uniformly as theycould be. Second, the run size of these designs can be quitelarge.

The foregoing problems with the naive strategy can be easilyovercome by using one LHD for all of the shared factors. Bydoing this, the design points are spread out more uniformly inthe experimental region, and the required run size is compara-bly smaller. As an example, for m1 = 3 and t = 5, the run sizeof the naive approach (2n0 in Table 4) should be at least 16 (be-cause n0 ≥ 8). This can be reduced to 6 (2n1 ≥ 6) by the newdesign illustrated in Table 5. Following the terminology usedby Phadke (1989, p. 168), we call a design with this structure abranching Latin hypercube design (BLHD).

Now consider a more general case with q branching factorsdenoted by z = (z1, . . . , zq). Assume that all of these factorsare qualitative by nature. For each branching factor zu, thereare ku levels, and under each of these different levels, thereare mu nested factors. Note that in general, the number of fac-tors nested under each level of a branching factor can differ;

Table 4. Illustration of the naive strategy

Run z1 vz11 · · · vz1

m1 ; x1 · · · xt

1 1...

... LHD(n0, m1 + t)n0 1

n0 + 1 2...

... LHD(n0, m1 + t)2n0 2

Table 5. Branching LHD with one branching factor

Run z1 vz11 · · · vz1

m1 x1 · · · xt

1 1...

... LHD(n1, m1)n1 1 LHD(2n1, t)n1 + 1 2...

... LHD(n1, m1)2n1 2

for example, in the hard turning experiment, two factors arenested under chamfer but no factors are nested under hone.Here, for notational simplicity, we assume that the number ofnested factors is the same (for a given branching factor) anddevelop the construction of the BLHD. Later we explain howthis can be extended to deal with unequal numbers of nestedfactors.

Denote the nested factors by vzu = (vzu1 , . . . , vzu

mu)′, 1 ≤ u ≤ q.

Again note that each nested factor corresponds to differentfactors, depending on the branching factor and its level; thatis why we use a superscript to denote the branching factorlevel. In addition to the branching and nested factors, thereare t shared quantitative factors, x = (x1, . . . , xt)

′. Let v =((vz1)′, . . . , (vzq)′)′, and w = (x′, z′,v′)′ represent all of the pfactors involved in the experiment, where p = t +q+∑q

u=1 mu.A N-run BLHD then can be denoted by W = (w1, . . . ,wN)′. Ingeneral, this consists of three parts. The first part is a designfor branching factors. Because branching factors are qualita-tive factors, we can choose an orthogonal array of appropri-ate size depending on the number of levels of each branch-ing factor. The second part comprises LHDs for the nestedfactors. Choose LHD(nu, mu) for the mu nested factors un-der the branching factor zu, u = 1,2, . . . ,q. The third part isan LHD(N, t) for all of the shared factors x. If there are ku

levels for branching factor zu, where 1 ≤ u ≤ q, then clearlyN = k1n1 = k2n2 = · · · = kqnq. Thus, we have one orthogonalarray for the branching factors, q LHDs for the nested factors,and one LHD for the shared factors. These designs can be as-sembled to obtain a BLHD.

As an example, consider the case with two branching factors(z1 and z2) each at two levels. There are m1 nested factors un-der z1 and m2 nested factors under z2. Furthermore, there aret shared factors. Table 6 illustrates a N-run BLHD for this ex-ample. The first part is a four-run orthogonal array for thosetwo branching factors. For the second part, we choose LHD(n1,m1) for the nested factors under z1. Similarly, LHD(n2, m2) ischosen for the nested factors under z2. This LHD is dividedinto two halves and distributed among the two levels of z2, asshown in the table. The third part consists of an LHD(N, t) forthe t shared factors.

As is the case with LHDs, not all BLHDs are good. Someoptimal design criteria are needed to ensure the best BLHD.We address this in the next section.


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Table 6. Branching LHD with two branching factors

Run z1 vz11 · · · vz1

m1 z2 vz21 · · · vz2

m2 x1 · · · xt

1 1 1 LHD(n2, m2) first halfLHD(n1, m1)... 1 2 LHD(n2, m2) first half LHD(N, t)... 2 1 LHD(n2, m2) second halfLHD(n1, m1)N 2 2 LHD(n2, m2) second half

3. OPTIMAL BRANCHING LATINHYPERCUBE DESIGNS

As discussed in Section 1, several approaches to findinga good LHD are available. Using one of those approaches togenerate q + 1 optimal LHDs for the nested and shared factorsand assemble them to obtain the BLHD may be considered suf-ficient. Such an assembly of optimal LHDs may not lead to anoptimal BLHD, however. Moreover, we need to make sure thatin a BLHD, the correlation between the branching-by-nested in-teraction and any other main effect is small. In this section wepropose three optimal design criteria for finding good BLHDs.

3.1 Maximin Branching Latin Hypercube Design

Morris and Mitchell (1995) proposed finding LHDs thatmaximize the minimum intersite distance. Let g and h betwo design points (or sites or runs). Consider the distancemeasure d(g,h) = {∑p

j=1 |gj − hj|ς }1/ς , in which ς = 1 andς = 2 correspond to the rectangular and Euclidean distances.For simplicity, we consider only the rectangular distance(ς = 1) hereinafter. For a given LHD, define a distance list(D1,D2, . . . ,DM) in which the elements are the distinct val-ues of the intersite distances, sorted from the smallest to thelargest. Let Ji be the number of pairs of design points in the de-sign separated by Di. Then a design is called a maximin designif it sequentially maximizes the Di’s and minimizes the Ji’sin the following order: D1, J1,D2, J2, . . . ,DM, JM . A scalar-valued function that can be used to rank competing designs insuch a way that the maximin design receives the highest rankingis given by

φλ =(

M∑i=1

JiD−λi

)1/λ

=(∑

g�=h

d(g,h)−λ

)1/λ

, (1)

where λ is a positive integer.Extension of the maximin criterion to BLHDs is not straight-

forward. In contrast to LHDs, in which all factors can be mea-sured by distances, BLHDs have branching factors that have nonotion of distance and nested factors for which the definition ofdistance depends on the corresponding branching factors. Be-cause of the different roles of factors, instead of calculatingall of the pairwise distances over all factors, we need to con-sider branching and nested factors separately from shared fac-tors. First, note that for BLHDs, not all factors are divided intothe same number of levels as in LHDs; there are ku levels forbranching factor zu, nu levels for nested factors vzu , and N lev-els for x. Therefore, before calculating the distances, the designmatrix should be scaled to (−1,1).

Start with a simple case in which q = 1 and m1 = 1 and thusthere are t + 2 factors in the experiment. Assume that the last

two factors are the branching factor z1 and the nested factor vz11 .

We need to define two types of intersite distances. The first typeof distance focuses on all of the shared factors. It is the dis-tance projection onto the t-dimensional space (x), which can bedefined by dx(g,h) = ∑t

j=1 |gj − hj|, where g = (g1, . . . ,gt+2)

and h = (h1, . . . ,ht+2) are (t + 2)-dimensional design points.There are a total of (N

2 ) distances. The second type of dis-tance takes into account the branching and nested factors byconsidering distances within each level of the branching fac-tors. The objective here is to spread out the design points foreach level of branching factors. To do this, for each level z1,i

of the branching factor, where 1 ≤ i ≤ k1, distances are calcu-lated based on x and vz1

1 . Define the second type of distance bydB(g,h) = ∑t

l=1 |gl − hl| + |gt+2 − ht+2|. We can easily obtaindB(g,h) = dv1(g,h)+dx(g,h), where dv1(g,h) = |gt+2 −ht+2|.The second type of intersite distances are calculated only forthose within the same level of the branching factor (gt+1 =ht+1 = z1,i, for some 1 ≤ i ≤ k1), and thus there are

(n12

)k1 of

them.Note that the first type of distance measure consists of t di-

mensions, whereas the second type consists of t+1 dimensions.After standardizing with respect to their dimensions, we canextend the maximin distance criterion to BLHDs by defining adistance list based on these (N

2 ) + (n12

)k1 standardized intersite

distances. Furthermore, as in (1), the scalar-valued function canbe defined as

φλ =(∑

g�=h

[t

dx(g,h)

]λ

+k1∑

i=1

∑gt+1=ht+1=z1,i

[1 + t

dv1(g,h) + dx(g,h)

]λ)1/λ

, (2)

where∑

gt+1=ht+1=z1,idv1(g,h) is the sum of

(n12

)pairwise dis-

tances, in which g and h have the same level of branchingfactor and

∑g�=h dx(g,h) = ∑k1

i=1

∑gt+1=ht+1=z1,i

dx(g,h) +∑gt+1 �=ht+1

dx(g,h).

To illustrate this idea, consider the simple example givenin Table 3. Assume that the branching factor z1 is qualita-tive. The optimal design found by the modified maximin cri-terion (2) (with λ = 15) is x1 = {1,5,6,4,7,3,2,8}, x2 ={4,8,1,5,6,2,7,3}, and z1 and v1 remain the same as in Ta-ble 3. This maximin BLHD is plotted in Figure 4, with the “×”srepresenting the design points with z1 = 1 and solid points rep-resenting those with z1 = 2. The first part in (2) tries to max-imize the intersite distances in the space of x1 and x2 [Fig-ure 4(a)], in which the “×” points and solid points are not dis-


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Figure 4. Maximin BLHD. “×” represents z1 = 1, and the solid points represent z1 = 2.

tinguished. In contrast, the second part tries to maximize theintersite distances in the space of x1, x2, and v1 [Figures 4(b)and (c)]. Moreover, because these distances are calculated onlywithin the same level of the branching factor, the intersite dis-tances among the “×” points and among the solid points aremaximized.

If only the first part in (2) were used as the criterion, thenthe optimal design would be space-filling only over the sharedfactors. The design points can be structured with respect to thebranching and nested factors. This can be seen in Figure 5. Itis clear that although the design points are distributed evenlythroughout the (x1, x2) space [Figure 5(a)], experiments forz1 = 1 concentrate on the lower level of x1, and experimentsfor z1 = 2 concentrate on the higher level of x1. Furthermore,the nested factor v1 is highly correlated with x1. This clearlydemonstrated the importance of the new criterion in (2). Alsonote that the two designs for the shared factors [Figure 4(a)and Figure 5(a)] are isomorphic; however, these two isomor-phic designs are not equally good when the branching factor isincluded. The new criterion can clearly identify that Figure 4(a)gives a better design than Figure 5(a).

Now consider the general case with q branching factors z andcorresponding nested factors v. Assume that for each branchingfactor zu, there are ku levels, denoted by zu,i, 1 ≤ i ≤ ku. Let gand h denote design points with p dimensions, and let δu =(t + ∑u

l=1(ml−1 + 1) + 1), such that the δuth factor is the uthbranching factor and the [δu + 1]th to [δu + mu]th factors arethe corresponding nested factors. As an extension of (2), the

maximin criterion for BLHDs can be written as

φλ =(∑

g�=h

[t

dx(g,h)

]λ

+q∑

u=1

ku∑i=1

∑gδu =hδu =zu,i

[mu + t

dvu(g,h) + dx(g,h)

]λ)1/λ

, (3)

where dx(g,h) = ∑tj=1 |gj − hj| and the distance measure

for the uth branching factor is denoted by dvu(g,h) =∑δu+mul=δu+1 |gl − hl|. This criterion is general and includes some

interesting special cases.

Case 1. If q = 0 and mu = 0 for all u, then this will leadto the standard LHD(N, t). Up to a constant, the maximin crite-rion (3) will be the same as (1) proposed by Morris and Mitchell(1995).

Case 2. If there is no nested factor corresponding to thebranching factors (i.e., mu = 0 for all u), then this can be con-sidered an experiment with q qualitative factors z and t quanti-tative factors x. As a special case of (3), the maximin criterionfor experimental design with quantitative and qualitative factorscan be written as

φλ =(∑

g�=h

[t

dx(g,h)

]λ

+q∑

u=1

ku∑i=1

∑gδu=hδu =zu,i

[t

dx(g,h)

]λ)1/λ

.

(4)

Case 3. Another special case is when t = 0. In this situation,experiments have branching factors and nested factors but no

Figure 5. Maximin for shared factors. “×” represents z1 = 1, and solid points represent z1 = 2.


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shared factors. Because dx(g,h) = 0, for all g and h, (3) can besimplified to

φλ =( q∑

u=1

ku∑i=1

∑gδu=hδu =zu,i

[mu

dv(g,h)

]λ)1/λ

.

3.2 Minimum Correlation BranchingLatin Hypercube Design

Along with space-filling, another important issue in experi-mental designs is how to construct them such that the signif-icant factors can be correctly identified. For LHDs, this canbe achieved by minimizing the pairwise correlation among fac-tors (Iman and Conover 1982; Owen 1994; Tang 1998). Owen(1994) proposed a performance measure ρ2 for evaluating thegoodness of an LHD with respect to pairwise correlations. ForLHD(N, t),

ρ2 =∑t

i=2∑i−1

j=1 ρ2ij

t(t − 1)/2, (5)

where ρij is the linear correlation between columns i and j.In contrast to LHDs, in BLHDs orthogonality among main

effects is not sufficient—considering the branching-by-nestedinteractions is equally important. Thus we propose a modi-fied correlation criterion that minimizes the correlations amongthe main effects of all factors, as well as those between themain effect of a shared factor and a branching-by-nested in-teraction effect. We first enlarge the BLHDs by including two-factor interactions that represent the branching-by-nested in-teractions. There are mu such interactions for each branchingfactor zu, where mu is the number of nested factors; there-fore, the total number of branching-by-nested interactions is s,where s = ∑q

u=1 mu. Thus the new criterion for BLHDs is givenby

ρ2 =∑p

i=2

∑i−1j=1 ρ2

ij + ∑ti=1

∑sj=1 ρ2

ij

(p(p − 1)/2) + st, (6)

where ρ2ij is the linear correlation between columns i and j in

the design and ρ2ij is the linear correlation between xi and the

jth branching-by-nested interaction.Consider the example used in Table 3. The optimal design

that minimizes ρ2 in (6) is x1 = (2,8,5,3,4,7,1,6), x2 =(2,8,5,3,6,1,7,4), and z1 and v1 remain the same as in Ta-ble 3. This design is plotted in Figure 6.

3.3 Orthogonal-Maximin BranchingLatin Hypercube Design

Maximizing minimum intersite distances does not ensureminimizing pairwise correlations, and vice versa. Therefore,Joseph and Hung (2008) proposed a multiobjective criterionfor LHD that combines the maximin distance and the mini-mum correlation criteria. This criterion becomes even moreimportant in the case of BLHDs, because it is important toensure small correlations between the shared factors and thebranching-by-nested interactions along with ensuring goodspace-filling properties. To extend the result of Joseph andHung (2008) to BLHDs, we should scale φλ and ρ2 to thesame range, so that some meaningful weights can be assignedin the multiobjective function. The following result gives thelower and upper bounds for φλ, which can be used for scalingit to [0,1]. Here we consider only the case of a single branch-ing factor; the results can be extended to include more than onebranching factor, but then the expressions become more com-plicated.

Proposition 1. If there is only one branching factor, thenφλ,L ≤ φλ ≤ φλ,U, where

φλ,L = 3

[ k1∑i=1

∑gδ1 =hδ1 =z1,i

21/(p+1)(m1 + t)p/(p+1)

+∑

gδ1 �=hδ1

tp/(p+1)

](λ+1)/λ

×[

t(N2 − 1) +k1∑

i=1

m1(n21 − 1)

]−1

and

φλ,U = N

2

[ k1∑i=1

n1−1∑j=1

(n1 − j)(t + m1)λ

jλ(t + k1m1)λ+

N−1∑j=1

N − j

jλ

]1/λ

.

Thus the multiobjective criterion is to minimize

ψλ = wρ2 + (1 − w)φλ − φλ,L

φλ,U − φλ,L. (7)

We usually take w = 0.5 and call the design that minimizes thiscriterion orthogonal-maximin BLHD.

Figure 6. Minimum correlation BLHD. “×” represents z1 = 1, and the solid points represent z1 = 2.


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Figure 7. Orthogonal-maximin BLHD. “×” represents z1 = 1, and solid points represent z1 = 2.

The design matrix used in Table 3 is an orthogonal-maximinBLHD. This matrix is plotted in Figure 7. The optimal designsfound by the forgoing three criteria (plotted in Figures 4, 6,and 7) are compared in Table 7. It can be seen that the corre-lation between the shared factor and the branching-by-nestedinteraction (denoted by INT) is 0 in the case of minimum corre-lation design; however, the points are much closer compared tothe maximin BLHD. The orthogonal-maximin BLHD providesa good compromise between these two designs.

Because of the combinatorial nature of the optimizationproblem, finding the optimal BLHD for large dimensions is adifficult task. Several methods for finding the optimal LHD, in-cluding simulated annealing (Morris and Mitchell 1995), thecolumnwise-pairwise algorithm (Ye, Li, and Sudjianto 2000),the enhanced stochastic evolutionary algorithm (Jin, Chen, andSudjianto 2005), and modified simulated annealing (Joseph andHung 2008), have been proposed. These methods can be easilyadapted for finding the optimal BLHD as well. A C++ codebased on the algorithm of Joseph and Hung (2008) is availablefrom the authors on request.

4. KRIGING WITH BRANCHING ANDNESTED FACTORS

In this section we explain how branching and nested factorscan be incorporated in kriging (Sacks et al. 1989). Althoughsimilar extensions can be made on other methods, such as lin-ear regression and spline methods, here we focus on krigingbecause of its popularity in computer experiments (Santner,Williams, and Notz 2003, p. 86; Fang, Li, and Sudjianto 2006,p. 28). We note that if branching and nested factors are encoun-tered in a physical experiment, then regression models and ex-perimental designs based on orthogonal arrays should be con-sidered.

The ordinary kriging model is given by Y(w) = μ + Z(w),where Z(w) is a weakly stationary stochastic process with

mean 0 and covariance function σ 2ψ and w ∈ Rp. The corre-

lation function is defined as cor{Y(w1),Y(w2)} = ψ(w1,w2).Usually a product correlation structure is assumed for the cor-relation function. Consider the example in Table 3. The correla-tion function between two points w1 = (x11, x12, z11, vz11

11 ) andw2 = (x21, x22, z21, vz21

21 ) can be described as a product of cor-relation functions of each factor (ψi) as follows: In the usualcases, a common correlation function is chosen for each factor;however, this cannot be done in the present problem, becauseof the different types of factors.

For the shared factors, a Gaussian correlation function maybe used (Santner et al. 2003, p. 36),

ψi(x1i, x2i) = exp{−αi(x1i − x2i)2},

whereas for a branching factor, an isotropic correlation functionmay be used (Joseph and Delaney 2007; Qian, Wu, and Wu2008),

ξ1(z11, z21) = exp{−θ1I[z11 �=z21]

}.

Here αi and θ1 are correlation parameters, and IA is an indicatorfunction that takes value 1 when A is true and 0 otherwise.

A new correlation function needs to be developed for nestedfactors. Assume that these are quantitative factors. We cannotuse the Gaussian correlation function here, because a nestedfactor represents different factors depending on the level of thebranching factor. Therefore, using one correlation parameter fora given nested factor is not reasonable. Instead, they should bedifferent, depending on the level of branching factor. For theexample in Table 3, the correlation function for vz1

1 can be de-fined as follows. If two points have the same level in the branch-ing factor (e.g., z11 = z21 = 1), then the correlation functionwill be exp{−γ1(v

z1111 − vz21

21 )2}. Similarly, if z11 = z21= 2, thenthe correlation function will be exp{−γ2(v

z1111 − vz21

21 )2}. If thetwo points do not have the same level in the branching factor(i.e., z11 �= z21), then the correlation should be determined by

Table 7. Comparison of different BLHDs

Maximin distance Minimum correlation Orthogonal-maximin

φλ 2.36 4.35 2.37D1(J1) 1

2 (12) 14 (3) 1

2 (13)ρ2 0.0287415 0.000283 0.01445(cor(x1, INT), cor(x2, INT)) (0.195, 0) (0, 0) (−0.098, 0)


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the branching factor, not the nested factor. Thus in this case,the correlation function for the nested factor will be equal to 1.The correlation function for the nested factor can be succinctlydefined as

1(vz1111 , vz21

21 ) = exp

{−

2∑j=1

γj(vz1111 − vz21

21 )2I[z11=z21=j]

}. (8)

We can easily extend this to a more general situation. As-sume that there are q branching factors z1, . . . , zq, and thatfor each branching factor zu, there are mu nested factors. Alsoassume that x1 = (x11, . . . , x1t)

′, z1 = (z11, . . . , z1q)′, vzu

1 =(vzu

11, . . . , vzu1 mu

)′, and v1 = ((vz11 )′, . . . , (vzq

1 )′). Similarly, x2 =(x21, . . . , x2t)

′, z2 = (z21, . . . , z2q)′, vzu

2 = (vzu21, . . . , vzu

2mu)′, and

v2 = ((vz12 )′, . . . , (vzq

2 )′). Given any two design points w1 =(x1, z1,v1) and w2 = (x2, z2,v2), the correlation function canbe written as

cor(Y(w1),Y(w2))

=(

t∏i=1

ψi(x1,x2)

)

×q∏

u=1

[ξu(z1, z2)

( mu∏j=1

u(vz1u1j , vz2u

2j )

)], (9)

where ψi(x1,x2) = exp{−αi(x1i −x2i)2} is the correlation func-

tion for the shared factors, ξu(z1, z2) = exp{−θuI[z1u �=z2u]} is thecorrelation function for the branching factors, and

u(vz1u1i , vz2u

2i ) = exp

{−

ku∑j=1

γuij(vz1u1i − vz2u

2i )2I[z1u=z2u=zu,j]

}

(10)

is the correlation function for the nested factors. Note that zu,j,1 ≤ j ≤ ku are the ku levels for each branching factor zu. Thuswe obtain

cor(Y(w1),Y(w2))

= exp

{−

t∑i=1

αi(x1i − x2i)2

−q∑

u=1

[θuI[z1u �=z2u] +

mu∑i=1

ku∑j=1

γuij(vz1u1i − vz2u

2i )2

× I[z1u=z2u=zu,j]

]}.

(11)

Denote the correlation parameters by � = (α′, θ ′,γ ′), whereα = (α1, . . . , αt)

′, θ = (θ1, . . . , θq)′, and γ = (γ111, . . . ,

γqmqkq)′. These parameters can be estimated from the data to

obtain the ordinary kriging predictor, as we explain using anexample in the next section.

5. HARD TURNING EXPERIMENT

The objective of our experiment is to optimize a hard turningprocess with respect to cutting forces. Hard turning is a metal

cutting process that produces machined parts out of hard ma-terials with good dimensional accuracy, surface finish, and sur-face integrity. Minimizing cutting forces will help reduce powerrequirements, elastic distortion of the workpiece, and tool wear;thereby reducing manufacturing costs and improving the qual-ity of the machined part.

Nine factors are selected for experimentation, including onebranching factor, two nested factors, and six shared factors.The factors and their ranges are given in Table 2. A 30-runorthogonal-maximin BLHD is generated by using the modi-fied simulated annealing algorithm proposed by Joseph andHung (2008). The optimal design matrix is given in Table 8.The branching factor (cutting edge shape, z1), is labeled “1”for chamfer or “2” for hone. Two nested factors (v1 and v2) arenested within the cutting edge shape. Recall that if the cuttingedge is chamfer, then v1 represents chamfer angle and v2 repre-sents chamfer land length; otherwise, there is no factor.

The experiments are performed using the highly sophis-ticated finite-element–based machining simulation programAdvantEdge. This software models the underlying physicsof metal cutting as a thermomechanical plastic deformationprocess and captures various material and geometric nonlin-earities of the process. (The theoretical basis of the simulationmodel is provided in Marusich and Ortiz 1995.) The simula-tions are computationally intensive and require hours of run-ning time (about 12–24 hours) to produce a single output.The simulation outputs are deterministic and incorporate allof the factors listed in Table 2. The software produces variousresponses, including temperature, residual stresses, and forces.A finite-element mesh and temperature distribution is shown inFigure 8. In this work, we chose to analyze only the resultantcutting force (y). The data are given in Table 8.

In this example, the number of nested factors is not the samefor different levels of the branching factor. In the discussionin Section 3, we assumed these to be the same, for simplicityof notation. Here we explain how the criteria can be modifiedto deal with unequal numbers of nested factors. This is eas-ily done. First, consider the maximin criterion φλ. In (3), usem1 = 2 for the first 15 runs, and use m1 = 0 for the last 15 runs.Now consider ρ2 in (6). The pairwise correlations involving anested factor should be calculated using the first 15 runs. More-over, because there are no factors at a branching factor levelof 2, we do not need to consider the branching-by-nested inter-action.

Because the cutting forces are positive, we first apply alog transformation before fitting the ordinary kriging model.We also normalize all of the factor settings in Table 8 to[−1,1]. The 30 design points after normalization are de-noted by {w1, . . . ,w30}, where for all 1 ≤ j ≤ 30, wj =(xj1, . . . , xj6, zj1, v

zj1j1 , v

zj1j2 )′, zj1 = −1 represents the chamfer

edge and zj1 = 1 represents the hone edge. The parameters inthe kriging model can be estimated as (Santner, Williams, andNotz 2003, p. 66)

� = arg min�

N log σ 2 + log |�|,

μ = (1′�−11)−11′�−1y,

σ 2 = 1

N(y − μ1)′�−1(y − μ1),


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Table 8. Orthogonal-maximin BLHD and data for the hard turning experiment

Run z1 v1 v2 x1 x2 x3 x4 x5 x6 y

1 1 1 6 15 23 7 9 18 10 162.12 1 2 11 25 3 25 14 25 19 284.93 1 3 3 4 20 18 18 5 26 160.34 1 4 14 9 6 6 27 7 17 121.15 1 5 8 16 8 21 2 2 1 104.66 1 6 1 17 10 5 25 19 25 241.97 1 7 12 29 26 15 5 14 12 195.48 1 8 5 26 16 30 22 15 6 159.59 1 9 15 7 13 26 7 11 27 241.6

10 1 10 10 1 29 20 23 6 5 88.3311 1 11 2 20 21 27 10 20 29 320.412 1 12 7 8 11 14 4 29 21 218.813 1 13 13 22 9 1 24 27 9 193.514 1 14 4 10 2 24 28 13 13 198.615 1 15 9 28 25 13 17 3 28 155.116 2 19 5 9 1 8 20 164.417 2 14 28 17 6 21 24 323.618 2 6 17 4 16 12 4 109.119 2 11 1 12 15 4 8 115.420 2 27 22 8 30 24 16 254.821 2 21 14 23 19 10 22 217.022 2 23 18 22 12 28 3 243.723 2 3 27 3 3 26 14 131.524 2 13 15 19 29 16 30 258.725 2 24 12 2 11 1 18 109.326 2 18 24 28 8 17 2 174.827 2 12 30 11 26 9 11 157.028 2 2 4 16 13 30 15 133.129 2 30 7 10 20 23 7 210.130 2 5 19 29 21 22 23 273.3

where 1 is a vector of 1’s with length 30, y = (y1, . . . , y30)′, and

� is a 30 × 30 matrix whose njth element is

exp

{−

6∑i=1

αi(xni − xji)2 − θ1I[zn1 �=zj1]

− γ111(vzn1n1 − v

zj1j1 )I[zn1=zj1=−1]

− γ121(vzn1n2 − v

zj1j2 )I[zn1=zj1=−1]

}.

We obtain α = (0.091,0.014,0.027,0.009,0.944,1.082)′,θ = θ1 = 0.127, γ = (γ111, γ121)

′ = (0.140,0.008)′, μ =5.124. Note that we do not need to estimate γ112 and γ122 inthis example, because there no factors are nested within thehone edge. Thus the ordinary kriging predictor is given by (see,e.g., Joseph 2006)

y(w) = 5.124 + ψ(w)′�−1(y − 5.1241), (12)

where w = (x1, . . . , x6, z1, vz11 , vz1

2 )′ ∈ [−1,1]9 and ψ(w) is avector of length 30 with the jth element

exp

{−

6∑i=1

αi(xi − xji)2 − θ1I[z1 �=zj1]

− γ111(vz11 − v

zj1j1 )I[z1=zj1=−1]

− γ121(vz12 − v

zj1j2 )I[z1=zj1=−1]

}.

To explore the effects of the factors, we applied the sensi-tivity analysis technique on the ordinary kriging predictor (seeWelch et al. 1992). The main effects plot, given in Figure 9(a),shows that the cutting edge radius (x1), feed (x5), depth of cut(x6), and chamfer angle (v1) have significant effects on the cut-ting force. A significant interaction between cutting edge radius

Figure 8. Finite-element mesh and temperature distribution..


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Figure 9. (a) Main-effects plot. (b) Interaction between x1 and x6.

and depth of cut also can be seen [Figure 9(b)]. The depth of cuthas a positive effect on the cutting forces, but this effect is moresignificant when the cutting edge radius is smaller. This can beexplained in physical terms as follows. For a small cutting edgeradius, an increase in the depth of cut produces an increase inmaterial deformation through shear, and thus a more significanteffect on the force. For larger cutting edge radius values, thecontribution of ploughing of material around the cutting edgeto the cutting force is more pronounced, and consequently, anincrease in depth of cut does not produce as significant a changein the cutting force.

The optimal setting of the factors can be found by min-imizing the ordinary kriging predictor in (12). We obtain(x1, x2, x3, x4, x5, x6, z1, v1, v2) = (−1.00, −0.76, 0.69, 0.70,−0.66, −0.70, −1, 0.05, 0.16). In their original scales, the op-timal setting for the shared factors is (x1, x2, x3, x4, x5, x6) =(5,−13.80,1.41,222,0.067,0.123), and the optimal cuttingedge geometry is chamfer, with angle of 18.74 degrees anda length of 128.13 microns. The resulting cutting force pre-dicted under this setting is 81 N, much smaller than the forcesobserved in the experiment. We also performed a new exper-iment at the optimal setting and obtained a resultant force of79 N, confirming the validity of the optimal setting obtainedwith our model.

6. CONCLUSIONS

Surprisingly, the design and analysis of experiments withbranching and nested factors have received scant attention inthe literature. One possible reason for this could be that the ex-periments can be performed in two stages, with the first stageinvolving an experiment with the branching factors and sharedfactors. Because there are no nested factors, this experiment canbe designed using existing methods. The data can be analyzedto determine the optimum level of the branching factor. Then a

second stage of the experiment can be performed using only thenested factors under the optimum level of the branching factor.The design of this experiment also can be easily obtained usingexisting methods. Although this two-stage approach is quite in-tuitive, the final results may not be optimal. This is because adifferent level of the branching factor may be the true optimum,but it cannot be identified in the first stage of the experiment,because the nested factors under that branching level are notset at their optimal levels. This problem can be avoided by us-ing branching designs, which determines the optimal settings ofthe branching factors, nested factors, and shared factors simul-taneously.

Taguchi (1987, p. 280) and Phadke (1989, p. 168) have re-ported several case studies on experiments using branching de-signs; however, their approaches are not sufficiently generalfor more complex experiments, such as computer experiments.Moreover, the optimality properties of their approaches usingorthogonal arrays are not known. In this article, we proposedBLHDs that are suitable for computer experiments involvingbranching and nested factors and discussed the optimal choiceof such designs. We applied our proposed approach to the opti-mization of a machining process.

Although our primary focus here is on LHDs, some issuesregarding the use of orthogonal arrays and their applications inphysical experiments are important as well. Research on thesetopics is currently underway and will be reported elsewhere.

APPENDIX: PROOF OF PROPOSITION 1

If one branching factor (q = 1) is present, then a BLHD willinclude k1 small LHDs (n1, m1) for the k1 levels of the branch-ing factor and a LHD(N, t) for the shared factors. φλ can be


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written as

φλ =(∑

g�=h

[t

dx(g,h)

]λ

+k1∑

i=1

∑gδ1 =hδ1 z1,i

[m1 + t

dv1(g,h) + dx(g,h)

]λ)1/λ

.

As shown by Joseph and Hung (2008), for a given LHD(N, t),the average intersite distance (rectangular measure) is N(N2 −1)t/6, which is a constant. With these constraints, finding alower bound for φλ can be formulated as a constraint minimiza-tion problem:

minφλ

subject to∑

gδ1=hδ1 =z1,i

dv1(g,h) = m1n1(n21 − 1)

6,

1 ≤ i ≤ k1, (13)

∑g�=h

dx(g,h) = tN(N2 − 1)

6,

where∑

gδ1=hδ1 =z1,idv1(g,h) is the sum of intersite distances

for those smaller LHDs (n1,m1) and∑

g�=h dx(g,h) is the sumof intersite distances for LHD(N,t). Because

φλ ≥ φ∗λ =

( ∑gδ1 �=hδ1

[t

dx(g,h)

]λ

+k1∑

i=1

∑gδ1=hδ1 =z1,i

2

[m1 + t

dv1(g,h) + dx(g,h)

]λ)1/λ

,

the lower bound can be found by minimizing φ∗λ with the same

constraints as in (13). Thus the lower bound can be obtained us-ing the Lagrange multiplier method. For the upper bound of φλ,the result for the BLHD is a simple extension of that for theLHD and thus can be proved by the same argument used byJoseph and Hung (2008).

ACKNOWLEDGMENTS

This research was supported by U.S. National Science Foun-dation grants CMMI-0654369 and CMMI-0620259. The au-thors thank the editor, an associate editor, and four referees fortheir helpful comments and suggestions.

[Received June 2007. Revised June 2008.]

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