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Minimum-Aberration Two-Level

Split-Plot Designs

Peng HUANG Dechang CHEN

Department of Statistics Department of Mathematics University of Rochester State University of New York at Buffalo

Rochester, NY 14627 Buffalo, NY 14214

Joseph 0. VOELKEL John D. Hromi Center for Quality

and Applied Statistics Rochester Institute of Technology Rochester, New York 14623-5604

(igvcqa@rit.edu)

The treatment-design portion of fractionated two-level split-plot designs is associated with a subset of the Zn-” fractional factorial designs. The concept of aberration is then extended to these split- plot designs to compare designs. Two methods are presented for constructing two-level minimum- aberration split-plot designs, along with examples. An extensive catalog of such designs is tabulated. Extensions to prime-level designs and relations to inner-outer arrays are also presented. KEY WORDS: Fractional factorial designs; Generating matrices; Inner-outer arrays; Word-length

patterns.

In multifactor experiments in which it is not practical to run all the factors in a completely random order, a split-plot design (e.g., Cochran and Cox 1957) is often used. Usually a split-plot design is obtained by combining two separate designs, one for the whole plot and one for the sub-plot. For fractional factorial designs, better split-plot designs can often be constructed for the sub-plot design by using all the factors involved in the sub-plot and some factors from the whole plot when creating the generators. This idea has long been known (e.g., Kempthorne 1952). No optimality criteria have been associated with such designs, however, and only small listings of such designs exist.

In this article, we explore the construction of optimal two-level split-plot designs. Section 1 provides an example that we use throughout. Section 2 establishes the relation between split-plot designs and fractional factorial designs. This relation is best seen through the generating matrices by which split-plot designs are well represented. In Section 3, we use this relation and the concept of minimum aberration for fractional factorial designs to study split-plot designs.

Sections 4 and 5 present two methods for constructing two-level minimum-aberration split-plot designs. Using the first method, one can use minimum-aberration fractional factorial designs to construct minimum-aberration split-plot designs. We provide an extensive list of such designs con- structed in this manner. This method cannot, however, gen- erate all minimum-aberration split-plot designs. Hence, in Section 5, we present a second method, using linear integer programming and some known properties on word-length patterns. To compare two split-plot designs, the aberration function is first defined so that comparing two split-plot

designs is equivalent to comparing the two values of the aberration function.

In Sections 6 and 7, we show how these ideas may be extended to prime-level designs and indicate the relation of these designs to the inner-outer array designs sometimes used in the area of robust design.

1. AN EXAMPLE

An experiment needed to be run to improve the proper- ties of a thin-film coating. A schematic for the process is shown in Figure 1. In this process, a set of substrates, typi- cally glass, were coated with numerous chemicals in a batch operation. The schematic shows the vessel in which the sub- strates (four shown in the schematic) would be coated, and 10 factors were to be studied. The first six, denoted by the letters A through F, could only be applied to all the sub- strates in a particular batch. These are called the whole-plot factors. The groups of four substrates form the experimen-

tal units for these factors and are called whole-plot units. After the batch operation, the substrates were removed from the vessel. The other four factors, denoted by P, Q, R, and S, could be varied for each individual substrate-these are called the sub-plot factors (or split-plot factors). These individual substrates form the experimental units for these factors and are called sub-plot units.

We shall use this example by considering two sets of designs. In the first set, we do not consider factors D, E, and F and so have only three whole-plot factors and four

@ 1998 American Statistical Association

and the American Society for Quality

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A Spatter

B 02

C Ion Clean D I-IF02

E Temperature F Operator

ea

P Substrate

Q Substrate

Purity

R Cleaning Agent S Cleaner

Figure 1. Schematic of a Thin-Film Process,

sub-plot factors for our designs. This is done primarily to provide simple examples. In the second set, we consider all

10 factors in our designs.

To fix ideas, say we ran the three whole-plot factors in a 2” design and completely randomized the run order of these eight combinations. For each of these combinations, say we were able to run 16 (not 4) substrates in each batch. Then we could arrange the four sub-plot factors in a 24 design and completely randomize these combinations to the 16 substrates. So there would be eight whole-plot units and 8 x 16 = 128 sub-plot units in the experiment.

2. GENERATING MATRICES AND SPLIT-PLOT FRACTIONAL FACTORIAL DESIGNS 2.1 Generating Matrices

The defining relation of a 2n-k fractional factorial design D can be represented by the generating matrix (Franklin

19841, and without loss of generality the generating matrix G can be written in the form

G = (I C), (1)

where I is the k x Ic identity matrix and C is a k x (n - k) matrix with its elements equal to 0 or 1, in which every row must contain at least one 1. For example, a 27-3 design with generating relation I = 1456 = 2457 = 3467 can be represented as i234567 1001110 G= 0101101 0011011 2.2 Split-Plot Designs

Now we apply this notation to two-level fractional fac- torial split-plot designs. Suppose that we have n1 factors in the whole plots with fractionation element kl and 122 factors in the subplots with fractionation element Icz. Then there

are 2nl-kl treatment combinations in the whole plots and 2(nl+n2)-(kl+kz) treatment combinations for the total split- plot design. Let G1 be the generating matrix of a 2nl-kl design D1 for the whole plots, so G1 = (11 Cl), where II is the kl x ICI identity matrix and Cl is a ICI x (n1 -ICI) matrix. Let Gz be the generating matrix of a 2n2-kz design D2 for the sub-plot. If the fractionation in the sub-plots depends only on the sub-plot factors, then GZ = (I2 C,), where IZ is the lea x k2 identity matrix, and CZ is a ICZ x (n2 - Ic2) matrix. The design matrix for the total split-plot design can be represented by the generating matrix

“=(“d Gq)

kl nl - h k2 m - h

=(

I1 Cl 01 02 kl 03 04 12 c2 > k.2 ’ (2)

where all the elements in 01, 02, 03, and 04 are 0. As an example, let us consider the thin-film example but only with the three whole-plot factors A-C, fractionation el- ement 1, and four sub-plot factors P-S with fractionation el- ement 2. (Our example is too small a design to normally be used, but it succinctly illustrates some key ideas.). Then we have a 23-1 design for the whole plots and a 24-2 design for the sub-plots. Let the 23-1 whole-plot design be D1 with generating relation I = ABC and the 24-2 design for the sub-plot be D2 with generating relation I = PR = QRS. Then the corresponding generating matrices for D1 and D2 are, respectively, G1 = (1 1 1) and so Gz = ( 1 0 1 0 1 0111’ ABCPQRS G= i 1110000 0001010. 0000111 1

The design matrix for the split-plot design is then

M= ABCPQRS 0 0 0 0 0 0 0’ 0001110 0000101 0001011 1010000 1011110 1010101 1011011 1100000 1101110 1100101 1101011 0110000 0111110 0110101 0 1 1 1 0 1 1

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316 PENG HUANG. DECHANG CHEN. AND JOSEPH 0. VOELKEL

Note that, although M is also the design matrix of a 27-3 design with the generating relation I = ABC = PR = QRS, this fractional factorial design is not fully equiva- lent to the preceding split-plot design. The reason is that the split-plot design consists of both a treatment design (here a 27-3) and an error-control design, the latter of which defines how the treatment combinations are ran- domized and blocked. [The italicized phrases are those used by Hinkelmann and Kempthorne (1994).]. We only consider the case in which the error-control design for the whole-plot units are completely randomized and the error-control design for the sub-plot units are completely randomized for each treatment combination in the whole- plot units. Because the sequel only discusses the treatment- design aspects of such split-plot designs, we will regard our split-plot restrictions of the treatment design as equiv- alent to these corresponding fractional factorial designs. In this treatment-design sense, the standard fractional factorial two-level design has generating matrix of form (l), but the standard fractional factorial two-level split-plot design has generating matrix of form (2).

The form (2) will now be extended in a useful way. To il- lustrate, let us reconsider our previous example. The design Di for the whole plots and the design D2 for the sub-plots were constructed separately-the generators in D1 were the words formed only from the letters (factors) A, B, and C in the whole plots, and the generators in D2 were the words formed only from the letters (factors) P, Q, R, and S in the sub-plots. If we instead fix the whole-plot design D1 as before but now modify the sub-plot design D2 as the one whose generating relation is I = APR = QRS, then this split-plot design has the generating matrix

ABCPQRS G= ( 1110000 1001010, 0000111 1

and this split-plot design can now be identified with the de- sign 27-3 having the generating relation I = ABC = APR = QRS, in the sense that their treatment designs are identical. Comparing the original split-plot design with the modified one, we find that the second design is better than the first one. For example, the second design is Resolution III (the defining relation is I = ABC = APR = QRS = BCPR = ABCQRS = APQS = BCPQS), but the first design is Res- olution II.

Here, letters from the whole plots are allowed to appear in the generators for subplots. In general, the generating matrix G of such a split-plot design, with n1 factors and kl fractionation in the whole plots and r12 factors and k2 fractionation in the sub-plots, will now have the form

h nl - kl kz n2 - k2

G = I1

BI

cl 01 g;

B2 I2 (3)

Here, 11, 12: Cl, C2: 01, and O2 are the matrices with the same structures as in (2); B1 and B2 are matrices with ele- ments 0 or 1; (It Cl), denoted by G1, represents the gener-

ating matrix of the whole-plot design; and (B1 B2 I2 C,) represents the generating matrix of the subplot design. The G in (3) is a natural generalization of the G in (2). These generalized split-plot designs (at two levels) can be formally identified as fractional factorial designs with the generating matrix of form (3).

The idea of using, for the sub-plots, one or more letters from the whole plots was mentioned by Kempthorne (1952) and Cochran and Cox (1957). Addelman (1964) presented a list of some split-plot designs generated in this manner, and these were reproduced by Daniel (1976). Rosenbaum (1996) also generated such designs but in a different context-see Section 7.

Note the asymmetry between the whole-plot and sub-plot factors in such designs. We can allow the generators of the sub-plot factors to include some whole-plot factors, but the split-plot nature of the design would be lost if we allowed the generators of the whole-plot factors to include some sub-plot factors.

For convenience, if G1 = (Ii Ci) is prescribed, which is natural in some cases, then a split-plot design with gener- ating matrix (3) will be said to be a 2(7~‘+7’2)-(~‘+ka)(G1) design. If G1 is not prescribed, then a split-plot design with generating matrix (3) will be said to be a 2(711+TL~)-(kl+ka) design.

We sometimes wish to consider designs of the form (3) in which there is no fractionation in the whole plots. In this case kl = 0, and G collapses down to (B2 I2 C,).

3. DEFINING MINIMUM-ABERRATION SPLIT-PLOT DESIGNS

3.1 Minimum Aberration and Word-Length Patterns Fries and Hunter (1980) introduced the notion of aber- ration as a finer method of comparing fractional factorial designs than resolution. See also Franklin (1984) Chen and Wu (1991), and Chen (1992). The concept of aberration can be illustrated by the following example. The two 27-2 de- signs with defining relations I = 1236 = 2457 = 134567 and I = 12346 = 12357 = 4567 are both Resolution IV de- signs, but the second one is said to have smaller aberration because it has fewer words of length 4.

The formal definition of minimum aberration needs the notion of word-length patterns. The number of letters in a word appearing in the defining relation of a fractional factorial design D is called the word length. Let A<(D) be the number of words of length i in the design D. The word-length pattern of the design D is defined as W(D) = (Al(D): A2(D), . , A,, +nz (D)). In this exam- ple the two designs have word-length patterns of (0, 0, 0, 2, 0, 1, 0) and (0, 0, 0, 1, 2, 0, 0). For the formal definition of a minimum-aberration fractional factorial design, we refer the reader to Chen (1992). This article only concerns itself with minimum-aberration split-plot designs, whose defini- tion will be given in Section 3.2. For brevity, we will often denote minimum aberration as MA.

3.2 Minimum Aberration Applied to Split-Plot Designs As discussed in Section 2.2, generalized split-plot designs

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are special kinds of fractional factorial designs, and thus the MA concept of fractional factorial designs can be applied. To construct a split-plot design of form (3), it is natural to first select a whole-plot design Dr with good properties. For example, the designs displayed in table 12.15 of Box, Hunter, and Hunter (1978) can be used because they are of minimum aberration. Then we need a sub-plot design Dz that will result in good properties for the entire split-plot design. Hence, a natural question is how the sub-plot design is obtained, given n spec$c whole-plot design. This ques- tion is equivalent to asking how to construct Br 1 Bz, CZ in (3), given Ci, so that the corresponding fractional factorial 2(nl+nz)-(kl+k-a) design is good under some criterion. The criterion we use is based on extending the idea of minimum aberration to split-plot designs.

Dejinition 1. Suppose that D’ and D” are two 2(711-tn2)--(kl+k2)(G1) designs. Let ‘r be the smallest % such that Ai f A,(D”). Then D’ is said to have less aberration than D” if AT(D’) < A,.(D”). If no such i exists, then D’ and D” are said to have equal aberration. A 2(n1+na)-(k1+kZ)(G1) design is said to be an MA 2(n1+712)-(k.1+k:,)(G1) design if no other 2(nl+nzi-(kl+k2) (Gr) design has less aberration.

Note the following:

All the designs 2(nl+rL2)-(k~+li2)(G1) already have Ici generators described by G1 = (I1 C,). Thus, finding an MA 2(7Ll+na)-(ki+ka)(GI) design requires a choice of only k:! generators. For this reason, Definition 1 can be regarded as a definition for restricted MA split-plot designs.

The class of 2(nl+n2)-(kl+li2)(Gr) designs for a given Gi are a subset of the class of 2n-k designs for n = n1 + n2 and k = k:r + ,&.

The whole-plot design corresponding to Gr is not re- quired to be MA.

In this definition, we have implicitly given the whole-plot design a priority over the sub-plot design because we ini- tially select the whole-plot design, usually according to some criterion such as MA. We now introduce another def- inition of MA split-plot designs, one that does not create such a whole-plot priority.

Definition 2. Suppose that D’ and D” are two 2(n1+7Q-(k1+bJ designs. Let T be the smallest i such

that Ai # AI( Then D’ is said to have less aberration than D” if A, (D’) < A, (D”). If no such i exists, then D’ and D” are said to have equal aber- ration. A 2(7L1+7La)--(liltk2) design is said to be an MA 2(7&l +n>)-(kl +k2) &sign if no other 2(n1 +w-(~I +b) design

has less aberration. Note the following:

1. There are only a finite number of 2(“Ll+71~)-(kltk~)

designs. Thus, an MA 2( nl+rl2)b(kl+k2) design for any

given nl; ~1~2; X-i. !Y;? always exists. Similarly, an MA ~(“I+~~“)-(~I+~~)(G~) design for any given Gr always exists.

MINIMUM-ABERRATION TWO-LEVEL SPLIT-PLOT DESIGNS

2. For the 2rL-k case, MA designs are unique (up to a relabeling of factor letters and run order) for k 5 4 (Chen 1992). But this is not true for 2(nl+Tr2)-(kl+li~) designs for kr + k2 I 4-see Section 4.3 for a counter- example.

3. If a design D is an MA 2(7L1+n?)--(h^l+kz) design and its whole-plot design has generating matrix Gr, then D is also an MA 2(711-t712)~(lilfka)(G1) design. Both of these definitions correspond to reasonable ways to create good split-plot designs and depend on the objectives of the experiment in which the design is used. In Section 4, we show how to create such MA designs.

4. FINDING MINIMUM-ABERRATION SPLIT-PLOT DESIGNS: METHOD I

We examine two methods to obtain MA split-plot de- signs. Method I finds such designs from available MA frac- tional factorial 2n-k designs. In addition to finding MA 2(711+n2)-(kl+kz) &signs, many MA 2(nl+nz)-(kl+kZ)(G1)

designs can be found with this method. Method II, which employs linear integer programming, can be used to find MA 2(nl+nz)-(kl+kz)(G1) or 2(7L1+na)F(kl+kz) designs and

will be discussed in Section 5. 4.1 Method I

For any n, the MA 271Pk. designs for k I 4 and k = 5 were given by Chen and Wu (1991) and Chen (1992), respectively. Moreover, for some ‘n and some k 2 6, the MA a”-” designs were given by Franklin (1984). (All of the MA fractional factorial designs with Resolution III or higher and most of the MA fractional factorial designs with Resolution II can be used to construct MA split-plot designs. In this article, we restrict our examples to the more useful designs of Resolution III or higher.)

For this purpose, we first want to get a simplified version of (3). Let gzJ be a typical element of G of (3), which lies in the ith row and jth column. If one nonzero element gi,i is in Br, then adding the jth row of G to the ith row with reduction modulo 2, to replace the ith row, will reduce the element gij to 0. This operation produces generators that are equivalent to the initial ones. Therefore, performing simi- lar row operations for all the nonzero elements of Br will reduce Br to Oa, a matrix with all elements equal to 0. The following matrix, which is also a generating matrix of the split-plot design corresponding to the G in (3), is then obtained:

kl n,l - ICI k.2 7~2 - kx

(

I1 Cl 01 02 kl

O3 Bs IZ C2 > k2

Switching some columns of the preceding matrix gives us the following matrix:

k1 k.2 121 - x:1 n,2 - k2 ( I1 01 Cl 02 ) h 03 12 B3 c2 k2 - (4)

which, of course, is a generating matrix of the same split- plot design except for relabeling of the factors. We will call

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318 PENG HUANG, DECHANG CHEN, AND JOSEPH 0. VOELKEL

this a standard form for the generating matrix G of a split- plot design, where, as before, each of the C matrices needs nonzero entries in each row.

4.2 A 2g-4 Example

This example shows how to obtain MA 2(nl+nz)-(kl+k2) designs from an MA 2”-4 design. The strategy here is to rearrange the matrix G of an MA 2+lc design to be of the form (4), whose two key features are as follows:

1. The left portion is the identity matrix of dimension k x k.

2. The upper right portion is a Icl x (n2 - &) matrix of OS.

For convenience, we will use the symbol [i] t) [j] to denote the elementary row operation that the positions of the ith row and jth row of a matrix G have been switched.

The following fractional factorial 2g-4 design with gen- erating relation I = 12346 = 12357 = 2458 = 3459 is an MA design (see Chen and Wu 1991) and has Resolution IV The generating matrix of this design is

123456789 G= i 111101000 111010100 010110010’ 001110001 1

Let us now switch columns to get

6 7 8 9 1 2 3

4

5 i 100011110 010011101 001001011 i = (I C). (5) 000100111

Thus, comparing (I C) with form (41, one gets kl = 1, Ic2 = 3, n1 - ICI = 4 (so nl = 4 + rFi = 51, and n2 - ka = 1 (so n2 = 1 + ka = 4). Therefore, the MA split-plot design represented by (I C) is easily stated-factors 1,2, . ,9; whole-plot design 2n1-k1 = 25-1 with factors 1, 2, 3, 4, and 6 and generating relation I = 12346; sub-plot design 2nz-kz = 24-3 with factors 5, 7, 8, and 9 and generat- ing relation I = 12357 = 2458 = 3459. By relabeling the factors, we get this MA split-plot design stated in a more standard form-factors A-E, P-S; whole-plot design 2nl-rcl = 25-1 with factors A-E and generating relation I = ABCDE; sub-plot design 2n2-k2 = 24-3 with factors P- S and generating relation I = ABCPQ = BDPR = CDPS. Note the power of this technique here-there are only two subplots for each of the 16 whole plots, with four sub- plot factors, yet the design is still MA and in particular Resolution IV.

A second MA split-plot design can be derived from the preceding 2’-* design. Observe that the column under fac- tor 1 in (5) has two 0 elements. We want to “move” them

to the upper right corner and so switch columns to get

6 7 8 9 2 3

4

5 1 ( 10 0 0 111 0 1 0 1 0 0 1 1 0 1 1 001010110 000101110 1 CL31 * [II; 141 tf PI) + 6 7 8 9 2 3

4

5 1 ( 001010110 000101110 100011101 010011011 i

and switch columns again to get

8 9 6 7 2 3

4

5 I ( 10 0 010 110 010001110 001011101 1 = G*. 000111011

Comparing G” with form (4), one has 1;i = 2; k2 = 2, n1 = 4 + ICI = 6, na = 1 + k2 = 3, and this MA split-plot design is obtained-factors 1: 2,. . .9; whole-plot design 2111-kl = 26p2 with factors 2, 3,4, 5, 8, and 9 and generating relation I = 2458 = 3459; sub-plot design 2712-kZ = 2”-” with fac- tors 1, 6, and 7 and generating relation I = 12346 = 12357. Relabeling gives six whole-plot factors A-F with I = ACDE = BCDF and subplot factors P, Q, and R, with I = ABCPQ = ABDPR.

In a similar way, a third MA design can be created by noting that the third row of C in (5) has two 0 elements that we can “move” to the upper right corner. By row and column switches, we arrive at

876924513 i 100011100 010010111 001011011' 000101101 i

We now have obtained a third MA split-plot design- whole-plot design 2n1-kl = 24-1 with factors 2, 4, 5 and 8 and generating relation I = 2458; sub-plot design 2na-kz = 25-3 with factors 1, 3, 6, 7, and 9 and generating relation I = 12357 = 12346 = 3459. Relabeling gives four whole-plot factors A-D with I = ABCD and sub-plot fac- tors P-T, with I = ACPQR = ABPQS = BCQT. Several other MA split-plot designs can be created-we discuss this in Section 4.3.

Note that all three of the preceding whole-plot designs are MA 2(7L1+n2)-(k1+k.a) designs, which, of course, are also MA 2(121+na)~(kl+kz)(G1) designs,

4.3 A Catalog of MA Designs

We now present a listing of MA 2(7’1+“p)--(k.l+li~)(G1) designs that we derived from Method I and indicate when such designs have the property that their whole-plot de-

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319

signs are also MA. We have tried to find all such designs We initially created these designs by using the generators for Zn-” = 16,32, and 64, with n = 5 to 15, k2 2 1, from MINITAB@. For each such resulting 2n-k design, we and Resolution at least III. These designs are listed in used SAS@ to calculate the word-length pattern and the

Tables l-3. generating matrix. We then examined the generating matrix

Consider the lo-factor thin-film example. Say we want as in Section 4.2.

to construct a 2(6+4)-(2+3) design. Referring to Table 2, we Of course, the generators for an MA 2n”.-k design are not find that such a design exists, with generators I = ABCE = unique. Initially, we felt that generators of the shortest pos- ABDF for the whole-plot factors and I = ABPQ = ACDPR sible word length would lead to more MA 2(nl+n~)-(kl+kz)

= BCDPS for the sub-plot factors. designs because such generators create more OS in G. We

Table 1. MA 2(nf+“Z)-(kf+kz) Designs in 16 Runs, 5 5 nr + n2 5 15, kz 2 1, Resolution > 111. # plots ny 16 5 16 3 16 2 16 6 16 4 16 3 16 2 16 7 16 4 16 3 16 2 16 8 16 4 16 2 16 9 16 5 16 4 16 3 16 2 16 10 16 6 16 5 16 4 16 3 16 2 16 10 16 6 16 3 16 11 16 6 16 5 16 4 16 3 16 3 16 2 16 12 16 6 16 4 16 3 16 13 16 7 16 6 16 5 16 3 16 14 16 7 16 6 16 3 16 15 16 7 16 3 - n2 2 3 2 3 4 3 4 5 4 6 4 7 6 8 9 6 7 8 IO 7 8 11 a 12 kl - 1 0 0 2 1 0 0 3 1 0 0 4 1 0 5 2 1 1 0 6 3 2 1 1 0 6 3 0 7 3 2 1 1 0 0 8 3 1 1 9 4 3 2 2 10 4 3 1 11 4 1 kz # whole plots - 1 8 1 4 1 8 2 8 2 4 2 8 3 8 3 4 3 8 4 4 3 8 4 8 4 4 5 4 3 8 4 8 5 8 5 4 6 4 3 8 6 8 4 a 5 8 6 8 6 4 7 8 7 4 5 8 7 8 7 4 5 8 6 8 7 8 7 4 6 8 7 8 9 4 7 8 10 4 # sub-

plots Word-length pattern/generators

2 4 2 2 4 2 2 4 2 4 2 2 4 4 2 2 2 4 4 2 2 2 2 2 4 2 4 2 2 4 2 2 2 4 2 2 4 2 4 00001 ABCPQ ABPQR 000300 ABCD, BCPQ ABPQ, BCPR APQR, BPQS 0007000 ABCD, BCPQ, ACPR ABPQ, BCPR, ACPS ABPR, BPQS, APQT 000140001

ABCD, BCPQ, ACPR, ABPS ABQR, ABPS, APQT, BPQU 0041480410

ABD, ACE, APQ, BCPR, ABCPS ABCD, APQ, BPR, CPS, ABCPT ABC, APR, AQS, BPQT, ABPQU PQR, APS, BPT, ABQU, ABQPV 008181688500

ABD, ACE, ABCF, APQ, BCPR, ABCPS ABCD, ABE, BCPQ, ACPR, ABPS, ABCPT ABCD, ABPQ, BCPR, ACPS, ABCPT, APU ABC, ABPR, BPQS, APQT, ABQU, ABPQV ABPR, ABQS, BPQT, APQU, ABPQV, APW

009161512731 0**

ABCD, ABE, ACF, ABPQ, ACPR, ABCPS ABPQ, ACPR, BCPS, APT, BPU, ABCPW 00122628242013400

ABCD, ABE, ACF, ABPQ, ACPR, BCPS, ABCPT ABCD, ABE, ABPQ, BCPR, ACPS, ABCPT, APU ABCD, ABPQ, BCPR, ACPS, ABCPT, APU, BPV ABC, ABPR, ABQS, APQT, BPQU, ABPQV, APW ABPQ, ACPR, BCPS, ABCPT, APU, BPV, CPW ABPR, APQS, BPQT, ABPQU, APV, BPW, QPX 0016394848483916001

ABCD, ABE, ACF, ABPQ, ACPR, BCPS, ABCPT, APU ABCD, ABPQ, ACPR, BCPS, ABCPT, APU, BP’/, CPW ABC, ABPR, ABQS, APQT, BPQU, ABPQV, APW, AQX 0022557296116874016610

ABCD, ABE, ACF, BCG, ABPQ, ACPR, BCPS, ABCPT, APU ABCD, ABE, ACF, ABPQ, ACPR, BCPS, ABCPT, APU, BPV ABCD, ABE, ABPQ, ACPR, BCPS, ABCPT, APU, BPV, CPW ABC, APR, AQS, BPT, BQU, PQV, ABPW, ABQX, APQY 0028771121682322031125628700

ABCD, ABE, ACF, BCG, ABPQ, ACPR, BCPS, ABCPT, APU, BPV ABCD, ABE, ACF, ABPQ, ACPR, BCPS, ABCPT, APU, BPV, CPW ABC, ABPR, ABQS, APQT, BPQU, ABPQV, APW, AQX, BPY, BQZ 003510516828043543528016810535001

ABCD, ABE, ACF, BCG, ABPQ, ACPR, BCPS, ABCPT, APU, BPV, CPW ABC, ABPR, ABQS, AQPT, BQPU, ABQPV, APW, AQX, BPY, BQZ, PQp

NOTE: (1) All designs except those dlscussed in (4) are MA 2(“’ +n2)--(k1 +@) designs that were dewed from MA 2”-k designs, where n = “3 + nz and k = kl + kz. (2) The word-length pattern for each of these designs appears on a line that Indicates its associated 2”-k design. For example, in Table 1 the 2(4+4’--(‘13) d esign has a word-length pattern of 0 0 0 14 0 0 0 1. the same as that of an MA 2’-’ design. (3) Some of these MA Z(“f +w)--(kl +‘,) designs ate. interestingly, not MA in the whole-plot factors. These are flagged as “*” (4) Occasionally, we found dwgns that did not correspond to MA 2”-” designs, but that appeared to be quite good. All these designs allowed for full factorial designs in the whole-plot factors and, for the 2(6+4)--(3+3) design. a nice design with fractionation m the whole-plot design. These designs are flagged as “**“.

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320 PENG HUANG, DECHANG CHEN, AND JOSEPH 0. VOELKEL

T-b/e 2, MA 2fnl+nzl-(ki+kz) D esigns in 32 Runs, 6 5 nl + nz < 15, kz > 1, Resolution 2 111

# plots - n7 n2 kj k2 32 6 32 4 32 3 32 2 32 7 32 5 32 4 32 4 32 3 32 2 32 8 32 6 32 5 32 4 32 4 32 3 32 2 32 9 32 6 32 5 32 4 32 4 32 3 32 2 32 10 32 6 32 5 32 4 32 3 32 2 32 10 32 4 32 11 32 7 32 6 32 5 32 4 32 3 32 2 32 11 32 4 32 12 32 8 32 7 32 6 32 5 32 4 32 3 32 2 32 12 32 4 32 13 32 8 32 7 32 6 32 4 32 3 32 2 32 14 32 8 32 7 32 4 32 3 32 2 32 15 32 8 32 4 # whole # sub-

plots plots Word-length pattern/generators

2 3 4 6 7 4 5 6 7 8 9 IO 8 5 6 7 9 10 11 6 7 10 11 12 7 11 1 0 0 0 2 1 1 0 0 0 3 2 1 1 0 0 0 4 2 1 1 0 0 0 5 2 1 1 0 0 5 0 8 3 2 1 I 0 0 6 0 7 4 3 2 1 1 0 0 7 0 8 4 3 2 1 0 0 9 4 3 1 0 0 10 4 1 1 16 2 1 8 4 1 4 8 1 16 2 1 8 4 2 16 2 2 8 4 2 4 8 1 16 2 2 16 2 2 8 4 3 16 2 3 8 4 3 4 8 2 16 2 3 16 2 3 8 4 4 16 2 4 8 4 4 4 8 3 16 2 4 16 2 4 8 4 5 8 4 5 4 8 5 16 2 3 16 2 4 16 2 5 16 2 5 8 4 6 8 4 6 4 8 6 16 2 3 16 2 4 16 2 5 16 2 6 16 2 6 8 4 7 8 4 7 4 8 7 16 2 4 16 2 5 16 2 6 16 2 7 8 4 8 8 4 8 4 8 5 16 2 6 16 2 8 8 4 9 8 4 9 4 8 6 16 2 9 8 4 000001 ABCDPQ ABCPQR ABPQRS 0001200 ABCDE, ABPQ ABCD, ABPQR ABCPQ, ADQR ABPQR, CPQS ABPQS, PQRT 00034000 ABCE, ABDF, BCDPQ ABCDE, ABPQ, ACPR ABCD, ABPR, BCPQS ABPQ, ACPR, BCDPS ABPR, ACPS, BCQPT ABPS, APQT, BPQRU 000680010

ABCE, ABDF, ABPQ, ACDPR ABCDE, ABPQ, ACPR, ADPS ABCD, ABPR, ABQS, ACPQT ABPQ, ACPR, ADPS, BCDPT ABPR, ACPS, AQPT, BCQPU ABPS, AQPT, ARPU, BQRPV 000101600500

ABCE, ABDF, ABPQ, ACDPR, BCDPS ABCDE, ABPQ, ACPR, ADPS, BCDPT ABCD, ABPR, ABQS, ACPQT, BCPQU APQR, BPQS, CPQT, ABCPU, ABCQV APQS, BPQT, PQRU, ABPRV, ABQRW 001101143110**

ABPQ, ACPR, ADPS, BCDPT, ABCDPU

0002502701001 0

ABCE, BCDF, ACDG, CDPQ, ADPR, BDPS ACDE, BCDF, ABPQ, BCPR, CDPS, ACPT ABCE, BCPQ, CDPR, ACPS, ADPT, BDPU * ABCD, ABPR, BPQS, BCPT, CPQU, APQV ABPR, BPQS, BCQT, ABQU, ACQV, CPQW ABPS, BPQT, BQRU, ABQV, AQRW, PQRX 00412181287200**

ABPQ, ACPR, ADPS, BCDPT, ABCPU, ABCDPV 000380520330400

ABDE, ABCF, BCDG, ACDH, ACPQ, ADPR, ABCDPS ABCE, ABDF, ACDG, ADPQ, ACPR, CDPS, ABCDPT ACDE, ABDF, ACPQ, ABPR, CDPS, ABCDPT, ADPU ABCE, ACPQ, ABPR, BDPS, CDPT, ABCDPU, BCPV * ABCD, ACPR, APQS, ABQT, CPQU, ABCPQV, ACRW ACQR, ACPS, ABPT, ABQU, CPQV, ABCPQW, APQX APRS, APQT, ABQU, ABRV, PQRW, ABPQRX, AQRY 006183024182110000**

ABPQ, ACPR, ADPS, BCDPT, ABCPU, ABDPV, ACDPW 00055096087016010

ACDE, BCDF, ABCG, ABDH, CDPQ, ABCDPR, ACPS, ADPT ABCE, ABDF, ACDG, ACPQ, BCPR, CDPS, ABPT, ADPU ABCE, ACDF, ACPQ, BCPR, CDPS, ABCDPT, ABPU, ADPV ABCD, BCPR, ABPS, BCQT, ABCPQU, ACPV, ABQW, ACQX CPQR, ACQS, AQPT, BCQU, ABCPQV, ACPW, BPQX, BCPY PQRS, AQRT, APQU, BQRV, ABPQRW, APRX, BPQY, BPRZ 00077016802030560700

ABCE, ABDF, ACDG, BCDH, ABPQ, ACPR, ADPS, BCPT, BDPU ABCE, ABDF, ACDG, ABPQ, ACPR, ADPS, BCPT, BDPU, CDPV ABCD, ABPR, ABQS, ACPT, ACQU, APQV, BCPW, BCQX, BPQY APQR, BPQS, CPQT, ABPU, ACPV, BCPW, ABQX, ACQY, BCQZ APQS, BPQT, PQRU, ABPV, APRW, BPRX, ABQY, AQRZ, BQRp 000105028004350168035000

ABCE, ABDF, ACDG, BCDH, ABPQ, ACPR, ADPS, BCPT, BDPU, CDPV ABCD, ABPR, ABQS, ACPT, ACQU, APQV, BCPW, BCQX, BPQY, CPQZ

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#

plots n7

321 Tab/e 3, MA 2(nl+nz)-(kr+k) D esigns in 64 Runs, 7 5 nl + n2 5 15, kp 2 1, Resolution > 111

# who/e # sub-

n2 kl k2 plots plots Word-length pattern/generators

64 7 64 5 64 4 64 3 64 2 84 8 64 6 64 5 64 5 64 4 64 3 64 2 64 9 64 7 64 6 64 5 64 5 64 4 64 4 64 3 64 2 64 10 64 6 64 6 64 6 64 5 64 5 64 4 64 3 64 2 64 11 64 8 64 7 64 6 64 5 64 5 64 4 64 4 64 3 64 2 64 12 64 8 64 7 64 5 64 5 64 4 64 4 64 3 64 2 64 13 64 9 64 7 64 5 64 5 64 4 64 4 64 3 64 2 64 14 64 12 64 9 64 7 64 7 64 5 64 5 64 4 64 4 64 3 64 2 4 5 7 7 8 8 9 IO 4 6 8 8 9 9 10 11 2 5 7 7 9 9 10 10 11 12 1 0 0 0 0 2 1 1 0 0 0 0 3 2 1 1 0 1 0 0 0 4 2 1 1 1 1 0 0 0 5 3 2 1 1 1 1 0 0 0 6 3 2 1 1 1 0 0 0 7 4 2 1 1 1 0 0 0 8 7 4 3 2 1 1 1 0 0 0 32 2 16 4 8 8 4 16 32 2 16 4 32 2 16 4 8 8 4 16 1 32 2 2 32 2 2 16 4 3 32 2 2 8 8 3 16 4 3 8 8 3 4 16 2 16 4 3 32 2 3 32 2 3 16 4 3 16 4 4 16 4 4 8 8 4 4 16 1 4 5 6 7 7 7 8 8 8 32 2 32 2 32 2 32 2 32 2 8 8 16 4 8 8 4 16 32 2 32 2 16 4 16 4 8 8 16 4 8 8 4 16 32 2 32 2 16 4 16 4 8 8 16 4 8 8 4 16 32 2 32 2 16 4 32 2 16 4 16 4 8 8 16 4 8 8 4 16 0000001 ABCDEPQ ABCDPQR ABCPQRS ABPQRST 00002100 ABCDF, ABEPQ * ABCDE, ABPQR ABCPQ, ADEPR ABCPR, ADPQS ABCPS, APQRT ABPQT, APRSU 000142000

ABCDF, ACDEG, ABEPQ ABCDF, ACEPQ, CDEPR * ABCDE, ACPQR, BCPQS ABCPQ, ADEPR, CDEPS ABCD, ABPQS, ACPRT ABCPR, ADPQS, CDPQT ABCPS, APQRT, CPQRU ABPQT, APRSU, PQRSV 0002840100

ABCF, BCDEG, DEPR, ABEPS ABCEF, BCEPQ, ACDPR, ABDPS * ABCF, EDPQ, ABEPR, BCDPS * ABCDE, ABPR, CDQS, ADPQT ABCE, DPQR, ABPQS, BCDPT * ABDQR, ABDPS, BCPQT, ACPQU ABQRS, ABPRT, BCPQU, ACPQV ABQST, ABPSU, BPQRV, APQRW 000414803200

ABEF, BCDEG, ACDEH, CDPQ, ABCPR CDEF, ABCDG, ABPQ, BDEPR, ADEPS ABEF, CDPQ, ABCPR, BDEPS, ADEPT * ABCDE, CDPR, ABQS, BDPQT, ADPQU ABCE, DPQR, ABPQS, BCDQT, ACDQU * ABCD, PQRS, ABPQT, BCQRU, ACQRV CDPR, ABCPS, ABQT, BDPQU, ADPQV CPQS, ABCPT, ABRU, BPQRV, APQRW PQRT, ABPQU, ABSV, BQRSW, AQRSX 00062416098000

ABCF, ABDEG, ACDEH, DEPQ, BCDPR, BCEPS ABCF, BCDEG, DEPQ, BCDPR, ABEPS, ACEPT ABCDE, CDQR, ABPS, ABCQT, ADPQU, BDPQV ABCE, DPQR, BCDPS, BCDQT, ABPQU, ACPQV * ABCD, PQRS, BCPQT, BCPRU, ABQRV, ACQRW CDQR, ABPS, ABCQT, ABDQU, ACDPV, BCDPW CQRS, ABPT, ABCQU, ABQRV, ACPRW, BCPRX QRTS, ABPU, ABQSV, ABQRW, APRSX, BPRSY 0001433161633140001

ABCF, ABDEG, ACDEH, BCDEJ, DEPQ, BCEPR, BCDPS * ABCF, BCDEG, DEPQ, BCDPR, ABEPS, ACEPT, BC EPU ABCDE, ABPR, CDQS, BDPQT, BCPQU, ACDPV, BCDPW ABCE, DPQR, BCDQS, BCDPT, ABPQU, ACPQV, BCPQW * ABCD, PQRS, BCPRT, BCPQU, ABQRV, ACQRW, BCQRX ABPR, CDQS, BCDPT, BCPQU, ADPQV, ABDQW, BDPQX ABPS, CQRT, BCPRU, BCPQV, APQRW, ABQRX, BPQRY ABPT, QRSU, BPRSV, BPQSW, APQRX, ABQRY, BPQRZ 00038175244335441201

ADEF, ABEG, CDEH, BDEJ, ABDK, BCEL, ABCDEM, BCDPQ BDEF, BCEG, CDEH, BCDJ, DEPQ, CEPR, ACDPS, BCDEPT ** BCDE, ACDF, ABDG, BCQR, ADQS, ACQT, ABCPU, ABCDQV BCEF, ACDEG, BEPQ, BCPR, DEPS, CEPT, CDPU, BCDEPV ABCDE, DPQR, BPQS, CDQT, BDQU, BDPV, BCQW, BCDPQX ABCE, ACQR, BCDS, BCQT, ABQU, CDQV, BDPQW, ABCDQX ** ABCD, ACQS, BCRT, BCQU, ABQV, CQRW, BPQRX, ABCQRY BPQR, BCQS, DPQT, CPQU, BCPV, CDQW, ACDPX, BCDPQY BPQS, BCQT, PQRU, CPQV, BCPW, CQRX, ACPRY, BCPQRZ BPQT, BQSU, PQRV, PQSW, BPSX, QRSY, APRSZ, BPQRSp

(continued)

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322 # plots 64 64 64 64 64 64 64 64 64 64 nl 15 11 10 7 6 5 4 4 3 2 n2 4 5 8 9 10 11 11 12 13 kl 9 6 5 3 2 0 1 0 0 0

PENG HUANG, DECHANG CHEN, AND JOSEPH 0. VOELKEL Table 3. (continued)

# whole # sub-

kz plots plots Word-length pattern/generators

0004901440219080019000

3 32 2 ABCF, ABDG, ABEH, ACDJ, ACEK, ADEL, ABPQ, ACPR, ADPS

4 32 2 ABCF, ABDG, ABEH, ACDJ, ACEK, ABPQ, ACPR, ADPS, AEPT *

6 16 4 ABCE, ABDF, ACDG, ABPR, ABQS, ACPT, ACQU, ADPV, ADQW

7 16 4 ABCE, ABDF, APQR, ABPS, ACPT, ADPU, ABQV, ACQW, ADQX

9 32 2 ABPQ, ACPR, ADPS, AEPT, BCPU, BDPV, BEPW, CDPX, CEPY

8 8 8 ABCD, ABPS, ABQT, ABRU, ACPV, ACQW, ACRX, APQY, APRZ

9 16 4 ABPR, ACPS, ADPT, APQU, BCPV, BDPW, BPQX, CDPY, CPQZ

9 8 8 ABPS, ACPT, APQU, APRV, BCPW, BPQX, BPRY, CPQZ, CPRp

9 4 16 ABPT, APQU, APRV, APSW, BPQX, BPRY, BPSZ, PQRp, PQSq

NOTE: See note to Table 1

found that some, but not all, of the generators in MINITAB were of this form, but we were not able to find a reference in which such shortest-word generators could be found.‘So we frequently created these ourselves. In addition, we later discovered that sometimes longer length words in genera- tors lead to designs that the shorter-word generators do not. For example. if a column in C has two OS in it, then it can be used for ki = 2 but not for Ici = 1. (Trying to use this for ICI = 1 results in a new split-plot factor that is con- founded only with whole-plot factors, a contradiction.) As an example, the 2(5+8)--(2+7) design, which we abbreviate to 5/8/2/7, came from one G, but the 3/10/2/7 design came from another G. Neither generating matrix contained both designs. So, our catalog is only as complete as the sets of generators we used. In many cases, we tried several sets of carefully chosen generators to create as many designs as possible; nevertheless, we are quite sure that our catalog, although extensive, is incomplete.

In trying to find alternative generators, we sometimes found designs that were almost MA but that generated use- ful designs we could not obtain from the MA designs them- selves. We have included some of these designs, so noted, in our listing. See, for example, the 4/6/O/5 design.

With this exception, all the other designs are, by neces- sity, MA 2(n1+n2)-(k1fkz) designs. These designs are not always MA in the whole-plot design, however. One exam- ple is the 5/6/l/5 design. This is Resolution IV, but the MA whole-plot design is Resolution V. A more interest- ing example is provided by the two 5/5/l/3 designs. One of these is MA in the whole-plot design, but the other is not. Because the overall word-length pattern is the same in both designs, however, which design is preferable must be decided on the objectives of the experiment, what interac- tions the researcher is more concerned about, and so on. Yet another example is provided by the two 6/4/l/3 designs. Neither of these whole-plot designs is MA, yet one of them is MA in the more restricted class of designs that can be gotten by selecting six factors in a 2s-1 design from the 10 factors of an MA 210-4 design.

To construct MA 2(nl+n2)-(kifk2) designs in which the whole-plot design is a full factorial (kl = 0), consider the matrix in (5) as an example. Remember for this case that this matrix must have the form G = (0s 12 Bs Ca) in (4). Suppose we want to construct a design in which nl = 4 so

that nz = 5 and Ica = 4. In that case, the Cz matrix is 4 x 1 and must correspond to one of the five right-most columns of (5). But each row of the Cz matrix is required to have at least one nonzero entry-in this case, all 1 ‘s-and because none of these five columns meets this requirement, no such design exists for this generating matrix. Next, let us try to construct a design with ni = 3 so that na = 6 and Ica = 4. In that case, the CZ matrix is 2 x 4 and must correspond to two of the five right-most columns of (5). If we select the two columns corresponding to factors 2 and 3, each row of the resulting Ca matrix has at least one nonzero entry. Relabeling factors 2 and 3 to P and Q and factors 1, 4, and 5 to ABC, we have a created a 2(3+6)-(o+4) design with whole-plot design a 23 in A, B, and C and sub-plot factors P-U with design given by I = ABPQR = ACPQS = PBCT = QBCU. In a similar way, a 2(2f7)--(0+4) can be constructed from (5).

We finally point out that MA 2(nl+nz)-(kl+kz) designs cannot always be derived from the associated MA 2n-k design. For example, any MA 2(nlfnz)--(kl+kz) design cre- ated from the MA 25-1 design is restricted to Ici = 0 be- cause the generating matrix consists of all 1s. See Table 1 for two such designs. Another way to see this is to note that the MA 25-1 design is Resolution V, but any associ- ated split-plot design with Ici > 0 must have resolution less than V.

Our catalog, containing 39 16-run designs, 54 32-run de- signs, and 70 64-run designs, has little in common with the listing of Addelman (1964). His 40 designs only include 7 in which the whole-plot design is fractionated. Of these, only 3 have a total of 16 to 64 runs; of these 3, 2 (Addel- man’s plans 19 and 21) correspond to designs in our tables. So, aside from designs in which the whole plots are run as a full factorial, our design listings only have 2 designs in common.

5. FINDING MINIMUM-ABERRATION SPLIT-PLOT DESIGNS: METHOD II

We now return to our original thin-film example, in which there are six whole-plot factors and four sub-plot factors. We wish to find an MA 2(6+4)--(3f2)(G1) design. Because the MA 21°p5 design is Resolution IV but the MA 26-3 is only Resolution III, such a design cannot be created by Method I. We use another method for such situations based on Chen (1992). This method might also be used to

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find designs that could have been, but were not, derived by Method I.

5.1 The Algorithm

We use an algorithm for constructing MA two-level split-plot designs for 2 (nl+nz)-(ti+kz) designs and also for 2(n1+n2)-(“l+“2)(G1) designs. To do so, we first look at some known properties of word-length patterns. As in the case of unrestricted fractional factorial designs, we can safely assume in the following that each of the n1 + n2 let- ters in a 2(nl+n2)-(kl+b) or 2(nl+nz)-(k~fk~)(G1) &sign must appear in the defining relation.

Given the word-length pattern (AI, A2,. . , A,,+,,) of a 2(n1+nz)-(k1+kz) or 2(n1+na)-(rcl-tkz)(G1) design, let n = ni + n2, and k = ICI + k2, and let q and T be the unique nonnegative integers such that n = q(2” - 1) + T with 0 I r < 2” - 1. Moreover, let R = min{ilAi # 0) and let m = [(R + 1)/a], where [x] denotes the greatest integer less than or equal to 5. Finally, let C denote the summation of i from 1 to n and let C’ denote such summation up to nl. Then (AI, Aa, . . , A,,+,,) must satisfy the following constraints: 1. C Ai = 2” - 1, C’ Ai > 2”’ - 1. 2. CAzl-l = 2”-l or CAzi-I = 0. 3. C iAi = n2k-1, C’iAi > n12k1-1 4. C i2Ai 2 2”- 2[n2 + q2(2k - 1) + iqr + r]. 5. Ci2A, is divisible by 2”-l. 6. C iA2i is divisible by 2k-3.

7. If A, = 1, then A,-i = Ai for i = 1,2, . . . , n - 1. 8. An-m+1

+ An--m+2

+. . +

A, I 1.

9. If there exists some j 2 n - m + 1 such that Aj = 1, then AzneRdj 5 AR.

10. A,-, 5 A2, + 1.

For references about the preceding constraints, see Bur- ton and Connor (1957), Fries and Hunter (1980), Franklin (1984), and Chen (1992).

Finally, if the design is a 2(nl+nz)-(kl+k2)(G1), and its word-length pattern of the whole-plot design G1 is

(al, a2,. . . , a,,), then we must also have

11. Aj >a,,j=1,2 ,..., nl;Aj >O,j=nl+l,nl+ 2,...,nl+nz.

Moreover, when Caai-1 > 0, then the first equality in constraint 2 clearly must hold.

Given the word-length pattern W = (AI, AZ, . , A,l+nz) ofadesign D = 2(nl+nz)-(kl+kz) [or 2(nl+nz)-(kl+kz)(G1)], we define the aberration function f(D) = f(w) = A, + A,-12k + . + A12cn-l)“, where n = n1 + n2 and k = k1 + k2. Let IV’ = (Ai,Ah,. . ,Akl+n2) be the word- length pattern of another design D’ that is 2(n1-tnz)-(k1+kz) [or 2(nl+n2)--(kl+kz) ( G1)]. Then the judgment about which of the two designs is better can be made through the aberration function. Precisely, we have the following theorem.

Theorem. (a) IV = IV’ if and only if f(D) = f(D’). (b) D has less aberration than D’ if and only if f(D) < f (D’) .

A proof of the theorem is given in Appendix A.

The algorithm follows. Its basis, integer linear program- ming, was used by Chen (1992) in his search for MA de- signs.

Step 1. Find a design Do = 2(nl+na)-(kl+kz) [or 2(“‘+“2)-(“l+k2)(G1)] such that Da is likely to have mini- mum aberration. Set B = 0.

Step 2. Let W = (Al,Az,. , A,) with n = n1 + n,2. Find the solution to the following (linear) integer program- ming problem:

minimizef (IV)

subject to constraints 1,2,3,4 (and 11 if G1 is given), and B < f(W) < f(Do). (6) If there is no solution, then Do has minimum aberration, so stop. If there is a solution, proceed to the next step.

Step 3. Check constraints 5 to 10 for the solution W to (6). If W does not satisfy all of them, or if the solution satisfies 5 to 10 but does not correspond to a design, then set B = f(w) and go back to step 2. If the solution W satisfies 5 to 10 and corresponds to a split-plot design (and, if Gi is given, has a whole-plot generating matrix G,), then this design has minimum aberration, so stop.

Note that, from constraint 2, if C u2i-l > 0, then it follows that CAzi-1 = 2’-l. If Cu2i-1 = 0, however, then 2 contains two possibilities-either CA2i-i = 2’-l or CAzi-1 = 0. In this case, constraint 2 can be ignored in (6) while running the integer programming problem and then can be checked later along with constraints 5 to 10.

In the preceding algorithm, the integer programming method is used to find the possible word-length pattern, which enables the aberration function to achieve its min- imum value. Thus, minimum aberration is obtained. [For more details on integer programming, see Walukiewicz (1991).] The potential difficulty in this method is that the constraints l-10 (and 11 if Gi is given) are necessary but not sufficient for W to correspond to a design. Thus, it may be difficult in practice to determine if a solution W that obeys these constraints does in facto correspond to a de- sign. And even if the solution does correspond to a design, it may well not correspond to a split-plot design (and if G1 is given, to a design with whole-plot generating matrix Gr). Our second example in Section 5.2 illustrates that these are real concerns.

5.2 Two Examples

Understanding of the algorithm will be aided by two ex- amples. For the first example, we use the seven-factor thin- film example in Section 2, in which the design we proposed had generating relation I = ABC = APR = QRS. This forms our initial split-plot design Da = 2(3t4)-(1+2)(G1), for which IV = (0, 0,3,2,1,1,0).

The solution to the integer programming problem (6) with B = 0 is IV = (O,O, 2,3,2,0,0). This IV satisfies constraints 5-10 and does in fact correspond to a design D = 2(3+4)-(1+2)(G1) with generating relation I = ABC

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324 PENG HUANG, DECHANG CHEN, AND JOSEPH 0. VOELKEL

= APR = BQRS. Therefore, this split-plot design D is an MA 2(3+4)-c1+2)(Gi) design. Note that this MA split-plot design D cannot be obtained by using Method I because the MA 27-3 design has Resolution IV, but this design can be no better than the Resolution III of the whole-plot design. For the second example, consider the thin-film example with 10 factors, 6 in the whole plots and 4 in the sub-plots. We want to construct an MA split-plot design in 32 runs, a 2(6+4)--(3+2) design. For the whole-plot design, we fix an MA 26-3 with generators I = ABD = ACE = BCF. The sub-plot contains the remaining four factors P, Q, R, and S. Once again, Method I cannot be used because our design is already Resolution III, but the MA 2l”-” design has Resolution IV. An intelligent guess for the two sub- plot generators is I = DPQR = ABCPS, which produces W(Do) = (0,0,4,8,8,4,4,3,0,0).

To run this problem without numeric overflow, the ob- jective function was modified to include only A3 to A7- f(w) = 1048576As +32768A4 + 1024A5 +32A6 +A7-and the constraints Al = 0 and A2 = 0 were added. (Because As-Al0 are not in the new objective function, it is possi- ble that the new objective function may generate additional solutions.)

The integer programming solution with B = 0 yields W = (0,0,4,3,12,11; O,O,O, l), but this does not meet constraint 7. Neither the second solution [found by setting B to the f(w) of the first solution&----W = (O,O, 4,4,11,10,01 1: l,O)-nor the third solution-IV = (O,O, 4: 4,11,10,1,0,0,1)-passed constraint 6. The fourth solution-w = (O,O, 4,4,12,8,0,3,0,0)-met all the con- straints 5-l 0, however.

Continuing on this way, even by the 50th integer pro- gramming solution, the solution corresponding to the ini- tial guess had not been achieved. Of these 50 solutions, 44 were rejected as designs by one or more of the con- straints 5-10, and one additional solution, which included words of length 8 and 9, was rejected by separate arguments (proof available from authors). This left 5 solutions-listed in Appendix B-that at this point might correspond to MA split-plot designs. By more carefully considering the split- plot nature of this design, however, an additional constraint, A4 > 8, was added to the integer programming problem- see Appendix C. With this additional constraint, all of the first 50 solutions were rejected. Restarting the integer pro- gramming problem, new solutions I to 6 were rejected by constraint 6, and the seventh solution corresponded to our initial guess. Thus, our initial guess led to a correct solution. This experience suggests that MA designs in the general two-level case will need constraints that directly reflect the split-plot nature of the design-for example, A4 > %--in addition to the current constraints created from 11. We have not examined how to create these additional constraints for the general case.

We note that replacing the original f(w) by an objec- tive function, say f*(w), that only includes terms A3 to AT, and then requiring Al = 0 and A2 = 0, has no prac- tical effect on the results. First, replacing AL and A2 in the objective function with the requirement that Al = 0

and A2 = 0 simply precludes factors whose levels do not change and whose main effects are confounded with each other. So these two constraints are needed for reasonable designs. Second, whether we use f(lv) or f*(r/l’) in the objective function does not affect the solution to A3 to A7 because of the lexicographic ordering of A3 to Alo-only the solutions to As to Aio may be affected. But these long word lengths are of no practical importance. Based on this reasoning, we may call the design resulting from use of f*(W) a near minimum-aberration design.

6. EXTENSION TO PRIME-LEVEL DESIGNS Most of the results presented in this article can be readily extended to the case of s-level split-plot designs, where s is a prime. The definition of MA sflpk fractional factorial designs was given by Franklin (1984) and Chen and Wu (1991). The obvious modification of the definition given in this article will lead to the definition of MA s-level split-plot designs. For any sn-k fractional factorial design, its gener- ating matrix can be written in the form G = (I C). where I is the k x k identity matrix and C is a X: x (71, -X:) matrix with

its elements from the Galois field G(s) = (0, 1 1 2.3, . s - I}, and that for s-level split-plot designs the elements of C,, C2; B1, B2, and B3 in the generating matrices (2), (3), and (4) belong to G(s). Therefore it is straightforward to use Method I. The aberration function now becomes f(D) = f(W) = A,, + A,_lsk + ... + Als(7’-1)/k, and the theorem stated in Section 5 is true in the general case. It appears, however, that enough conditions about the word- length patterns of s-level fractional factorial designs have not been found, so use of Method II will be especially diffi- cult to implement. In spite of this, we believe that minimum- aberration designs could often be generated by an intelligent user, even though this property would be dithcult to prove.

7. RELATION TO INNER-OUTER ARRAY DESIGNS Taguchi’s use of inner and outer arrays for robust de- sign (e.g., Phadke 1989) corresponds, from a treatment de- sign aspect, to our initial version of the fractional facto- rial split-plot design, where D1 for the whole plots and D2 for the sub-plots are constructed separately. Hence, from a minimum-aberration criterion, the designs in this article are better, or at least as good as, those proposed by Taguchi.

Minimum-aberration designs can also be more appeal- ing from a quality engineering perspective, for two reasons. First, our designs explore wider regions than Taguchi’s. For example, one weakness of Taguchi’s designs when the outer array is a 3”-” is that extreme settings of the factors will occur by design only for pairs of factors. Thus, a potentially important combination such as “high power, high temper- ature, and high humidity” can easily not get tested. Sec- ond, our design can explore more factors than allowed by Taguchi’s designs. For example, in Section 5 we constructed a 2(6+4)-(3+2) design. Taguchi’s designs would not allow “4 factors with 4 runs” in the outer array, but our design accomplishes this easily. Assume that we are willing to ig- nore interactions among the whole-plot factors. as Taguchi usually does, and that we are only interested in, at most, two-factor interactions between a whole-plot factor and a

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MINIMUM-ABERRATION TWO-LEVEL SPLIT-PLOT DESIGNS

325

sub-plot factor. (Such interactions, of which our design has 6 x 4 = 24, really form the basis of robust-design princi- ples.) Then the 31 df from our design include the six main effects of the whole-plot factors, the four main effects of the subplot factors, 12 individual whole-plot/sub-plot interac- tions, six pairs of such interactions, and three other terms. Such design construction can be an important addition to the field of robust design. Similar points were made by Box and Jones (1992) and Bisgaard (1992).

More recently, Rosenbaum (1996) referred to Taguchi’s treatment designs as direct product designs, corresponding to our (2), and to a generalization of these as compound designs, corresponding to our (3). His work, using the cri- terion of the strength of a design rather than minimum aberration, focused on generating compound designs whose strength properties are very well suited to robust-design ex- periments. These designs, using the terminology of resolu- tion, have main-plot (control-factor) designs of Resolution IV and overall Resolution IV.

8. CLOSING REMARKS

The collection of split-plot designs generated by the ma- trices of form (4) is an extension of the collection of com- monly used split-plot designs represented by the matrices of form (2). The representation of split-plot designs in the matrices of (4) clearly indicates that the treatment-design aspect of split-plot designs can be treated in the same way as in the case of fractional factorial designs. In particular, the concept of aberration can be applied, which allows us to compare designs in a natural way.

We have presented two definitions for MA split-plot de- signs. We have also presented two methods for construct- ing minimum-aberration split-plot designs. Method I can be done by hand quite easily, but Method II requires integer programming. Using Method I, we have listed an extensive, but likely incomplete, set of such designs for 16, 32, and 64 runs, such that 5 5 ~1~~ + n2 I 15, k2 2 1, and Resolution > III. Extensions to prime-level split-plot designs and the importance of extending these designs to the inner-outer array class of designs are also presented.

Some issues that remain to be addressed are (a) cre- ating a systematic way to generate all possible MA 2(7L1+11~)--(kl+k~) designs from a given MA 2”--li design (this includes various designs that make trade-offs between the whole-plot and the sub-plot word-length patterns); (b) cre- ating near MA 2(“1+~“)-(~1+~“) designs by first creating a given near MA 271--k designs, such as the 4/6/O/5 design; (c) characterizing constraints on word-length patterns that are generated by the split-plot nature of the design (as we saw from the 1 O-factor thin-film example, such a constraint was needed to show that the selected design was indeed MA); and (d) extending these ideas to prime-level, and es- pecially three-level, designs.

ACKNOWLEDGMENTS

We gratefully acknowledge the editor, associate editor, and two referees, whose insightful comments improved the earlier version of this article.

APPENDIX A: PROOF OF THEOREM

The proof is based on transforming the lexicographic or- dering of word-length patterns into an ordering of natural numbers.

For (a) the necessity is obvious. So let us consider the sufficiency. Suppose f(D) = f(D’); that is, (Al -A/,)2-" + (A2-A~)2-2k+...+(A,-A',)2Y7Lk =O,Wewanttoshow W = W’. It suffices to show AI = A',. In fact, if Al # A',, we have IAl - A',1 > 1. In this case,

2-’ < J(Al -A;)2-"1

= I(A2 - A;)2-2k + +(A, - A;,)2-""1 5 (2” - 1)2-2” + . + (2” - qpk

< (2” - 1)2-271 + 2-k + 2--2k + .) = 2-k.

This contradiction shows that Al = A',.

Now consider the necessity part of(b). If D has less aber- ration than D’, then there exists some m such that Al = Ai,A2 = A/2, . . . . A,,-1 = A',-l,A,r,, < AA. Therefore

UP’) - fP&-“”

n

=(A;, -A,)2-"'"+ 1 (A; -Ai)2-'"

i=WL+1

> 2-7nk -

izg+li2”:

- wik

> 2-nLk

_ c2k

_ 1)(2-(“+1)k

+ 2-(17L+2)k.

+ . .)

= 0.

For the sufficiency part of (b), suppose .1’(D) < f(D’). We want to show that D has less aberration than D’. If D and D’ have the same word-length pattern, then by (a), f(D) = f(D’), w lc 1s impossible. If D’ has less aber- h’ h ration, then, similar to the necessity part in (b), one has f(D’) < f(D), which is also impossible. Therefore, D must have less aberration than D’.

APPENDIX B: SOME ORIGINAL POSSIBLE SOLUTIONS TO THE DESIGN

Among the first 50 linear-programming solutions, the fol- lowing word-length patterns passed all the constraints 5-10: (0, 0, 4, 4, 12, 8, 0, 3, 0, OL (0, 0, 4, 5, 9, 10, 2, 0, 1, O), (0, 0, 4, 6, 8, 8, 4, 1, 0, 01, (0, 0, 4, 6, 12, 4, 0, 5, 0, Oh (0, 0, 4, 7, 8, 7, 4, 0, 0, l), (0, 0, 4, 7, 9, 6, 2, 2, 1, 0). It turns out that the last of these solutions, which includes words of length 8 and 9, is not consistent with the given design (proof available from the authors).

APPENDIX C: PROOF THAT THE DESIGN IN THE IO-FACTOR EXAMPLE MUST HAVE AT LEAST

EIGHT WORDS OF LENGTH 4

We need to add two generators to define sub-plot letters 9 and 10 for the sub-plot factors. To try to create a split-plot

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326 PENG HUANG, DECHANG CHEN, AND JOSEPH 0. VOELKEL

design of Resolution IV whose whole-plot features are fixed and given, the new generators are restricted as follows:

1. 2. 3. 4. 5. 6.

They must correspond to words of length 4 or more. They must include two or three sub-plot letters, with- out loss of generality. (If any included all four sub-plot letters, this could be reduced to an equivalent word with fewer sub-plot letters by reduction modulo 2.) They must include at least one letter from the whole- plot factors. The terms available from the whole plots are (1, 2, 3, 12, 13, 23, 123). By reduction to shortest-length words, where additional words of shortest length in a term are shown in parentheses, these are equivalent to { 1, 2, 3, 4, 5, 6, 16(+25 +34)}. Thus, one new generator, say 9, must be of the form 9 = ~~78, where w1 is from { 1, 2, 3, 4, 5, 6}, say 9 = 478, and the other must be of the form lo = 16~3, where s1 is either 7 or 8, say lo = 167. (The other alternative for 9, 9 = 1678, will not work because it forces JJ to be defined with only three letters in the generator.)

Now consider how many additional words of length 4 these two generators create in the defining relation. The first is 4789, and because 16 is confounded with 25 and 34, this adds 167lJ,257lJ, and 34710. The product of the first and fourth adds another, 38910. When we add in the three words of length 4 that are created only among whole-plot factors, we see that the best Resolution IV design must have at least eight words of length 4 in the defining relation.

(Received February 1997. Revised June 1998.1

REFERENCES

Addelman, S. (1964), “Some Two-Level Factorial Plans With Split-plot Confounding,” Technometrics, 6, 253-258.

Bisgaard, S. (1992) “The Design and Analysis of 2k--P x 24-r Inner and Outer Array Experiments,” Technical Report 90, University of Wisconsin-Madison, Center for Quality and Productivity Improvement. Box, G. E. P., Hunter, W. G., and Hunter, J. S. (1978), Statistics for Ex-

perimenters, New York: Wiley.

Box, G., and Jones, S. (1992), “Split-plot Designs for Robust Product Ex- perimentation,” Journal of Applied Statistics, 19, 3-26.

Burton, R. C., and Connor, W. S. (1957), “On the Identity Relationship for Fractional Replicates of the 2n Series,” The Annuls of Mathematical Statistics, 28, 762-767.

Chen, J. (1992), “Some Results on Zn-” Fractional Factorial Designs and Search for Minimum Aberration Designs,” The Annuls of Statistics, 20,

2124-2141.

Chen, J., and Wu, C. F. J. (1991) “Some Results on sn-’ Fractional Factorial Designs With Minimum Aberration or Optimal Moments,” The Annals of Statistics, 19, 1028-1041.

Cochran, W. G., and Cox, G. M. (1957), Experimental Designs, New York: Wiley.

Daniel, C. (1976), Applications of Statistics to Industrial Experimentation,

New York: Wiley.

Franklin, M. F. (1984), “Constructing Tables of Minimum Aberration P m-m Designs,” Technometrics, 26, 225-232.

Fries, A., and Hunter, W. G. (1980) “Minimum Aberration 2k-n Designs,”

Technometrics, 22, 601-608.

Hinkelmann, K., and Kempthorne, 0. (1994), Design and Anulysis of EC

periments. Volume I: Introduction to Experimental Design, New York: Wiley.

Kempthorne, 0. (1952), The Design and Analysis of Experiments, Hunt- ington, NY: Wiley.

Phadke, M. (1989), Quality Engineering Using Robust Design, Englewood Hills, NJ: Prentice-Hall.

Rosenbaum, P. R. (1996) “Some Useful Compound Dispersion Experi- ments in Quality Design,” Technometrics, 38, 354-364.

References

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