Research ArticleGeneral Minimum Lower-Order Confounding Designs withMulti-Block Variables

Yuna Zhao

School of Mathematics and Statistics Shandong Normal University Jinan 250358 China

Correspondence should be addressed to Yuna Zhao yunazhao0504163com

Received 7 February 2021 Revised 31 March 2021 Accepted 8 April 2021 Published 21 April 2021

Academic Editor Gengxin Sun

Copyright copy 2021 Yuna Zhao is is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Blocking the inhomogeneous units of experiments into groups is an efficient way to reduce the influence of systematic sources onthe estimations of treatment effects In practice there are two types of blocking problems One considers only a single blockvariable and the other considers multi-block variablese present paper considers the blocking problem of multi-block variableseoretical results and systematical construction methods of optimal blocked 2nminus m designs with (N4) + 1le nle 5N16 aredeveloped under the prevalent general minimum lower-order confounding (GMC) criterion where N 2nminus m

1 Introduction

e regular 2nminus m factorial experiment has played an im-portant role in engineering manufacturing industry agri-culture and medicine and so on It allows efficient andeconomic experimentation to estimate treatment effectsWhen the size of the experimental units is large the in-homogeneity will cause unwanted variance to the estima-tions of treatment effects To reduce such bad influence acrucial way is to partition the experimental units into blocks

ere are two kinds of blocking problems as pointed outin [1] One is called the single block variable problem whichconsiders only a single block variable and the other is calledthe multi-block variable problem which considers two ormore block variables In the last decades choosing optimalblocked 2nminus m designs with a single block variable has beenwell investigated for example the authors in [2ndash9] studiedthe blocked 2nminus m designs under different minimum aber-ration criteria Chen et al and Zhao et al [10 11] exploredthe blocked 2nminus m designs under the clear effects criterionZhang and Mukerjee Zhao et al and Zhao and Zhao[12ndash14] explored the constructions of the blocked 2nminus m

designs under the general minimum lower-order con-founding (GMC) criterion proposed in [15] and Zhao et al

[16ndash18] gave construction methods of the blocked 2nminus m

designs under another GMC criterion proposed in [19]Compared to the large body of work on the blocking

problem of single block variable the studies on choosingoptimal blocked 2nminus m designs with multi-block variables arerelatively rare However it has been recognized that theblocking problem of multi-block variables can arise quitenaturally in many practical situations For example in theagricultural context Bisgaard [1] pointed that when designsare laid out in rectangular schemes both row and columninhomogeneity effects probably exist in the soil Anotherexample of multi-block variables is from [20] Consideringthe comparison of two gasoline additives by testing them ontwo cars with two drivers over two days the ldquocarsrdquo ldquodriversrdquoand ldquodaysrdquo are three block variables which should be takeninto account when performing experiments

Under the clear effects criterion Zhao and Zhao [21]proposed an algorithm for finding optimal blocked 2nminus m

designs with multi-block variables Under the minimumaberration criterion Zhao and Zhao [22] developed somerules for constructing optimal blocked 2nminus m designs withmulti-block variables Zhang et al [23] extended the ideaof GMC criterion to the case of multi-block variableproblem and developed the blocked GMC criterion called

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 5548102 11 pageshttpsdoiorg10115520215548102

the B2-GMC criterion Inheriting the advantage of theGMC criterion the B2-GMC designs are particularlypreferable when some prior information on the impor-tance ordering of treatment effects is present By com-puter search Zhang et al [23] tabulated some B2-GMCdeigns with small n and N where N 2nminus m When n or N

is large computer search becomes computationally ex-pensive Zhao et al [24] and Zhao and Zhao [25] com-pleted the constructions of B2-GMC designs with5N16 + 1le nleN minus 1 is paper aims at providing the-ories and systematical construction methods of theB2-GMC designs with N4 + 1le nle 5N16

e rest of the paper is organized as follows Section 2reviews doubling theory and B2-GMC criterion Section 3provides theoretical results and construction methods ofB2-GMC designs Section 4 gives concluding remarks Someuseful lemmas are deferred to Appendix

2 Preliminaries

21 Doubling eory Let X (x1 xt) be a matrix withentries 1 or minus 1 Denote J0 (1 1)prime and J1 (1 minus 1)prime where primedenotes transpose Define

D(X) J0 J1( 1113857otimes x1 J0 J1( 1113857otimes xt( 1113857 (1)

as a double of X where otimes is the Kronecker product LetDq(X) denote the design obtained by repeatedly doublingXq times ie Dq(X) D(Dqminus 1(X)) When X 1 we writeDq(1) (I2q 12q 22q 12q22q 12q22q32q q2q ) wherethe subscript 2q means the dimension of a column I2q is acolumn of 1rsquos

12qprime (1 1 minus 1 minus 1)

22qprime (1 1 minus 1 minus 1 1 1 minus 1 minus 1)

⋮

q2qprime (1 minus 1 1 minus 1 1 minus 1 1 minus 1)

(2)

are q independent columns and the other columns are thecomponent-wise products of some of these q independentcolumns For example 12q22q is the component-wiseproduct of the columns 12q and 22q Write Dq(1)

(I2q Dq(middot)) then Dq(middot) is just the regular two-level satu-rated design with columns arranged in Yates order As somesubdesigns in Dq(middot) denote Hq

0 empty Hq1 12q Hq

r

(Hqrminus 1 r2q r2qHq

rminus 1) for r 2 3 q where empty denotes theempty set the superscript q of Hq

r refers to that Hqr is a

subdesign of Dq(middot) and r2qHqrminus 1 (r2qd2q d2q isin Hq

rminus 1) ieHq

r consists of the first 2r minus 1 columns of Dq(middot) EspeciallyHq

q Dq(middot)

22 B2-GMC Criterion Before introducing the B2-GMCcriterion we first review some principles in the multi-blockvariable problem Let b1 b2 bs denote the s block var-iables which cause the inhomogeneity of the experimentalunits Suppose that the block variable bj partitions the N(

2nminus m) experimental units into 2lj blocks then lj indepen-dent columns are needed to carry out this blocking plan

Denote Sj as the set of the lj independent columns related tothe block variable bj e block columns should follow thefollowing rules

(i) e lj block columns in Sj(j 1 2 s) are in-dependent of each other

(ii) A block column from Sj is not necessarily inde-pendent of the block columns from Si with jne i

In this paper we focus on the case where each blockvariable is at two levels ie lj 1

e effect hierarchy principle for blocked designs withmulti-block variables is as follows (see [23])

(i) e lower-order treatment factorial effects are morelikely to be important than the higher-order onesand the treatment factorial effects of the same orderare equally likely to be important

(ii) e lower-order block factorial effects are morelikely to be important than the higher-order onesand the block factorial effects of the same order areequally likely to be important

(iii) All the interactions between treatment factors andblock factors are negligible

Since each variable or factor is assigned to one column ofthe design matrix when an experiment is carried out we donot differentiate the variable factor and column Based onthe effect hierarchy principle and weak assumption that theeffects involving three or more factors are usually not im-portant and negligible Zhang et al [23] proposed theB2-GMC criterion which pays attention to only the con-founding among main treatment effects and the two-factorinteractions of treatment factors (2firsquos for short) For thesame reason a common assumption in blocking problem isthat only the main effects of block variables and the in-teractions of any two block factors are potentially significantand if a treatment effect is confounded with a potentiallysignificant block effect the treatment effect cannot be es-timated us the confounding between the main effects oftreatment factors and any potentially significant block effectis not allowed

Denote D (Dt Db) as a 2nminus m 2s design where Dt

consists of n treatment factors corresponding to a regular2nminus m design and Db consists of s block factors each of whichcan partition the 2nminus m runs into 2 blocks Denote 1 C

(p)

2 (D)

as the number of main treatment effects which are aliasedwith p2firsquos but not with any potentially significant blockeffects where p 0 1 2 P and P n(n minus 1)2 Simi-larly 2 C

(p)

2 (D) denotes the number of 2firsquos which are aliasedwith the other p2firsquos but not with any potentially significantblock effects where p 0 1 P Denote

1 C2(D)

1 C

(0)

2 (D)1 C

(1)

2 (D) 1 C

(p)

2 (D)1113874 1113875

2 C2(D)

2 C

(0)

2 (D)2 C

(1)

2 (D) 2 C

(p)

2 (D)1113874 1113875

(3)

C(D)

1 C2(D)

2 C2(D)1113874 1113875 (4)

2 Mathematical Problems in Engineering

A 2nminus m 2s blocked design D (Dt Db) is called aB2-GMC design if it sequentially maximizes (4) Let1 C

(p)

2 (Dt) be the number of main effects which are aliasedwith p 2firsquos ofDt and 2 C

(p)

2 (Dt) be the number of 2firsquos whichare aliased with the other p 2firsquos of Dt Let

1 C2 Dt( 1113857

1 C

(0)

2 Dt( 11138571 C

(1)

2 Dt( 1113857 1 C

(p)

2 Dt( 11138571113874 1113875

2 C2 Dt( 1113857

2 C

(0)

2 Dt( 11138572 C

(1)

2 Dt( 1113857 2 C

(p)

2 Dt( 11138571113874 1113875

C Dt( 1113857

1 C2 Dt( 1113857

2 C2 Dt( 11138571113874 1113875

(5)

A 2nminus m design Dt is called a GMC design if Dt se-quentially maximizes (5)

Let q n minus m Constructing a B2-GMC design is tochoose Dt and Db from Dq(middot) such that (4) is sequentiallymaximized In the following without causing confusions weomit the subscript of a column and the superscript of adesign when they are taken from Dq(middot) For example we useaHr andHq instead of a2q Hq

r andHqq respectively Denote

U Db( 1113857 γ isin Hq γ isin Db or γ abwith a b isin Db1113966 1113967 (6)

and then U(Db) consists of all the potentially significantblock effects As previously stated the confounding betweenmain treatment effects and potentially significant block ef-fects is not allowed is requires Dt capU(Db) empty and thus1 C

(p)

2 (D) 1 C

(p)

2 (Dt) for p 0 1 2 PFor Dt sub Hq and γ isin Hq define

B2 Dt γ( 1113857 d1d2( 1113857 d1 d2 isin Dt d1d2 γ1113864 1113865 (7)

where denotes the cardinality of a set and d1d2 standsfor the two-factor interaction of d1 and d2 us B2(Dt γ) isthe number of 2firsquos of Dt appearing in the alias set thatcontains γ

Isomorphism introduced by Tang and Wu [26] is auseful concept which helps narrow down the search of theoptimal blocked designs here An isomorphism ϕ is a one-to-one mapping fromHq toHq such that ϕ(xy) ϕ(x)ϕ(y) forevery x ne y isin Hq e 2nminus m designs Dt and Dlowastt are iso-morphic if there exists an isomorphism ϕ that mapsDt ontoDlowastt e 2nminus m 2s designs D (Dt Db) and Dlowast (Dlowastt Dlowastb )

are isomorphic if there exists an isomorphism ϕ that mapsDt onto Dlowastt and Db onto Dlowastb

3 Constructions of B2-GMC Designs

31 B2-GMC 2nminus m 2s Designs with n (N4) + 1 A designis of MaxC2 (see [27]) if it has resolution IV and maximumnumber of clear 2firsquos where a resolution R design has noc-factor interaction confounded with any other interactioninvolving less than R minus c factors (see [28]) and a 2fi is calledclear if it is not aliased with any main treatment effect andother 2firsquos Cheng and Zhang [29] showed that a 2nminus m designwith n N4 + 1 is a MaxC2 design if and only if it is a GMCdesign ey also pointed out that up to isomorphism theGMC 2nminus m design with n N4 + 1 can be uniquelyexpressed as SN4+1 (q minus 1 q q(q minus 1)Hqminus 2) It is easy toobtain that

B2 SN4+1 γ( 1113857

1 for γ isin (q minus 1)Hqminus 2 qHqminus 2 q(q minus 1)1113966 1113967

N

8minus 1 for γ isin Hqminus 2

0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)

erefore

1 C

(p)

2 SN4+1( 1113857

N

4+ 1 forp 0

0 otherwise

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(9)

2 C

(p)

2 SN4+1( 1113857

2N

4minus 11113874 1113875 + 1 forp 0

N

4minus 11113874 1113875

N

8minus 11113874 1113875 forp

N

8minus 2

0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(10)

Lemma 1 is a straightforward extension of Lemma A1in Appendix introduced from [24 25]

Lemma 1 Suppose Db is any s-projection of Hr γ γHr1113864 1113865

with 2k le sle 2k+1 minus 1 for some k(0le kle r minus 1) where γ isindependent of the columns of Hr We have

(i) If Db cap γ γHr1113864 1113865 empty then U(Db)capHr1113864 1113865ge 2k+1 minus 1and the equality holds when Db has k + 1 indepen-dent columns

(ii) If Db cap γ γHr1113864 1113865neempty then U(Db)capHr1113864 1113865ge 2k minus 1and the equality holds when Db has k + 1 indepen-dent columns

(iii) If Db cap γ γHr1113864 1113865neempty then U(Db)cap γ γHr1113864 11138651113864 1113865ge 2k

and the equality holds when Db has k + 1 indepen-dent columns

(iv) If Db sub Hk+1 then U(Db) Hk+1(v) If Db sub Hk γ γHk1113864 1113865 then U(Db) Hk γ γHk1113864 1113865

Lemma 2 provides a necessary condition for a 2nminus m 2s

design D (Dt Db) with n N4 + 1 to be a B2-GMCdesign

Lemma 2 Suppose D (Dt Db) is a 2nminus m 2s design withn N4 + 1 and 2k le sle 2k+1 minus 1 for some k(0le kle q minus 3)then D (Dt Db) is a B2-GMC design only if Dt SN4+1

Proof Let 1113957D (SN4+11113957Db) be a 2nminus m 2s design and

1113957Db isin Hqminus 2 From (8)ndash(10) we can obtain

1 C2(

1113957D) N

4+ 1 0 01113874 1113875 (11)

which is sequentially maximized by 1113957D and 2 C

(0)

2 ( 1113957D)

2(N4 minus 1) + 1 N2 minus 1 Suppose that any D (SN4+1

Db) is not a B2-GMC 2nminus m 2s design for n N4 + 1 en

Mathematical Problems in Engineering 3

by the definition of B2-GMC criterion there should be a2nminus m 2s design Dlowast (Dlowastt Dlowastb ) outperforming1113957D (SN4+1

1113957Db) in terms of (4) is leads to

1 C2 Dlowast( 1113857

N

4+ 1 0 01113874 1113875 (12)

and 2 C

(0)

2 (Dlowast)ge (N2) minus 1 Clearly

1 C2 Dlowastt( 1113857

1 C2 Dlowast( 1113857

N

4+ 1 0 01113874 1113875 (13)

and 2 C

(0)

2 (Dlowastt )ge 2 C(0)

2 (Dlowast)geN2 minus 1 noting that Dlowastt is theunblocked part of Dlowast and Dlowastt has n N4 + 1 columns

In fact the inequality in the formula 2 C(0)

2 (Dlowastt )geN2 minus

1 is not valid Recall that the MaxC2 design SN4+1 has thelargest number N2 minus 1 of clear 2firsquos among all the 2nminus m

designs with n N4 + 1 ereforeDlowastt has at most N2 minus 1clear 2firsquos ie

2 C(0)

2 (Dlowastt ) N2 minus 1 Wu and Wu [30]showed that a 2nminus m design with n N4 + 1 is a MaxC2design if and only if this design has N2 minus 1 clear 2firsquos isobtains that Dlowastt SN4+1 up to isomorphism

With Lemma 2eorem 1 provides the constructions ofB2-GMC designs with n N4 + 1

Theorem 1 SupposeD (Dt Db) is a 2nminus m 2s design withn N4 + 1 and 2k le sle 2k+1 minus 1 for some k(0le kle q minus 3)then D (Dt Db) is a B2-GMC design if Dt SN4+1 andDb is any s-projection of Hk+1

Proof By Lemma 2 if D (Dt Db) is a B2-GMC 2nminus m 2s

design with n N4 + 1 then Dt SN4+1 up to isomor-phism us

1 C2(D) (N4 + 1 0 0) which is se-quentially maximized by D (SN4+1 Db) Let u1

U(Db)cap (q minus 1)Hqminus 2 qHqminus 2 q(q minus 1)1113966 11139671113966 1113967 and u2 U(Db)capHqminus 21113966 1113967 then from (10) we have

2 C

(p)

2 (D)

2N

4minus 11113874 1113875 + 1 minus u1 forp 0

N

4minus 1 minus u21113874 1113875

N

8minus 11113874 1113875 forp

N

8minus 2

0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)

If D is a B2-GMC design then D must sequentiallyminimize (u1 u2) ere are two different ways to chooseDb

from HqSN4+1

(i) Db cap (q minus 1)Hqminus 2 qHqminus 2 q(q minus 1)1113966 1113967 empty(ii) Db cap (q minus 1)Hqminus 2 qHqminus 2 q(q minus 1)1113966 1113967neempty

It is not hard to verify thatDb in case (ii) results in u1 gt 0while Db in case (i) gives u1 0 erefore ifD (SN4+1 Db) is a B2-GMC design then Db must be ofcase (i) Recall thatDb cap SN4+1 empty and we haveDb sub Hqminus 2By Lemma 1 (i) if Db sub Hqminus 2 then u2 U(Db)capHqminus 21113966 1113967ge 2k+1 minus 1 and the equality holds when

Db sub Hk+1 up to isomorphism is completes theproof

e following example illustrates the constructionmethod in eorem 1

Example 1 Consider the construction of theB2-GMC29minus 4 25design Here n 9 m 4 q 5 N 32 and s 5 whichleads to k 2 According to eorem 1 let Dt (4 5 45H3)

and Db be any 5-projection of H3 say Db (1 2 12 3 13)en D (Dt Db) is a B2-GMC 29minus 4 25 design

32 B2-GMC Designs with N4 + 1lt nle 5N16 To se-quentially maximize (4) the first part 1 C2(D) of (4) shouldbe first maximized Recall that 1 C2(D)

1 C2(Dt) If Dt hasresolution IV then

1 C2(D) must be maximized Accordingto [31] when N4 + 1lt nle 5N16 theDt with resolution IVmust be an n-projection of some second-order saturated(SOS) designs In the following we first review the conceptof SOS design

A 2nminus m design is called an SOS design if all of its degrees offreedom can be used to estimate only themain treatment effectsand 2firsquos In terms of coding theory and projective geometryDavydov and Tombak [32] showed that given N only the SOSdesigns of N4 + 1 N(2wminus 2 + 1)2w with wge 4 and N2factors exist Block and Mee [31] further showed that an SOSdesign of N(2wminus 2 + 1)2w factors can be obtained by doublingsome smaller SOS 2(2wminus 2+1)minus (2wminus 2+1minus w) designs q minus w timesZhang and Cheng [33] showed that the SOS design of N2factors can be uniquely represented by SN2 (q qHqminus 1) up toisomorphism Let

L(w) Dqminus w

(Y) Y is an SOS 2 2wminus 2+1( )minus 2wminus 2+1minus w( ) design1113882 1113883

(15)

denote the collection of all the SOS designs obtained bydoubling some SOS 2(2wminus 2+1)minus (2wminus 2+1minus w) designs q minus w timesEspecially in L(w) we denote the SOS design obtained bydoubling the MaxC2 2(2wminus 2+1)minus (2wminus 2+1minus w) designΦ(w) ((w minus 1)2w w2w w2w (w minus 1)2wHw

wminus 2) as SN(2wminus 2+1)2w With a little algebra it is easy to verify that

Dqminus w I2w( 1113857 I2q Hq

qminus w1113872 1113873 IHqminus w1113872 1113873

Dqminus w 12w( 1113857 (q minus w + 1)2q I2q Hq

qminus w1113872 1113873

(q minus w + 1) IHqminus w1113872 1113873

Dqminus w 22w( 1113857 (q minus w + 2)2q I2q Hq

qminus w1113872 1113873

(q minus w + 2) IHqminus w1113872 1113873

⋮

Dqminus w

(w minus 1)2w( 1113857 (q minus 1)2q I2q Hqqminus w1113872 1113873 (q minus 1) IHqminus w1113872 1113873

Dqminus w w2w( 1113857 q2q I2q Hq

qminus w1113872 1113873 q IHqminus w1113872 1113873

(16)

where I2qminus w otimes 12w (q minus w + 1)2q I2qminus w otimes 22w (q minus w+ 2)2q

I2qminus w otimes (w minus 1)2w (q minus 1)2q and I2qminus w otimesw2w q2q

en

4 Mathematical Problems in Engineering

Dqminus w

(w minus 1)2ww2wHwwminus 2( 1113857 q(q minus 1) Hqminus 2Hqminus w1113872 1113873

(17)

erefore SN(2wminus 2+1)2w can be expressed as

SN 2wminus 2+1( )2w Dqminus w

(Φ(w))

(q minus 1) (q minus 1)Hqminus w q qHqminus w q(q minus 1)1113872

Hqminus 2Hqminus w1113872 11138731113873

(18)

e following lemma provides a necessary condition fora 2nminus m 2s design D (Dt Db) with N4 + 1lt nle 5N16 tobe a B2-GMC design

Lemma 3 Suppose D (Dt Db) is a 2nminus m 2s design withN(2wminus 1 + 1)2w+1 lt nleN(2wminus 2 + 1)2w for wge 4 and2k le sle 2k+1 minus 1 for some k(0le kle q minus 3) thenD (Dt Db) is a B2-GMC design only if Dt sub SN(2wminus 2+1)2w

Proof As discussed in the first paragraph of this section ifD (Dt Db) is a B2-GMC 2nminus m 2s design withN(2wminus 1 + 1)2w+1 lt nleN(2wminus 2 + 1)2w then Dt must be ann-projection of some SOS designs in L(v) with 4le vlew orSN2

Let P1 be an n-projection of SN2 According toeorem3 in [33] we obtain

2 C

(p)

2 P1( 1113857

0 forplt n minusN

4minus 1

N

4n minus

N

41113874 1113875 forp n minus

N

4minus 1

0 for n minusN

4minus 1ltplt

N

8minus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(19)

Let Xv (a1 a2 a2vminus 2+1) be any SOS2(2vminus 2+1)minus (2vminus 2+1minus v) design but not a MaxC2 design thenDqminus v(Xv) isinL(v) Denote pi as the number of clear 2firsquos ofXv involving ai for i 1 2 2vminus 2 + 1 and c as the totalnumber of clear 2firsquos of Xv Let Pv be an n-projection ofDqminus v(Xv) and Pv Dqminus v(Xv)Pv en Pv N

(2vminus 2 + 1)2v minus nleN2v Applying equations (23) and (24)in the proof of eorem 31 of [29] among all the n-pro-jections of Dqminus v(Xv) Pv sequentially maximizes 2 C2 in (5)only if Pv sub Dqminus v(aj) where aj is the column such thatpj max p1 p2 p2vminus 2+11113864 1113865 For a Pv with Pv sub Dqminus v(aj)according to equation (24) in the proof of eorem 31 of[29] it is obtained that

2 C

(p)

2 Pv( 1113857

0 for 0leplt n minusN

4minus 1

pjN

2v (n minus N4) forp n minusN

4minus 1

0 for n minusN

4minus 1ltplt

N

2v minus 1

c minus pj1113872 1113873N

2v1113874 1113875

2 forp

N

2v minus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)

For a MaxC22(2vminus 2+1)minus (2vminus 2+1minus v) design it has 2vminus 1 minus 1clear 2firsquos and pj 2vminus 2 Since Xv is not a MaxC2 design wehave

pj 2vminus 2

c minus pj lt 2vminus 1

minus 2vminus 2minus 1

(21)

or

pj lt 2vminus 2

clt 2vminus 1minus 1

(22)

Let Qv be an n-projection of SN(2vminus 2+1)2v obtained bydoubling Φ(v) the MaxC22(2vminus 2+1)minus (2vminus 2+1minus v) design asmentioned above With a column permutation rewriteDq(middot) as

Dq

RC(middot) Hqminus vHqminus 2Hqminus v (q minus 1) Hqminus 2Hqminus v1113872 11138731113872

q Hqminus 2Hqminus v1113872 1113873 q(q minus 1) IHqminus v1113872 1113873 SN 2vminus 2+1( )2v 1113873

(23)

in a re-changed Yates order Write Φ(v) asΦ(v) (s1 s2 s2vminus 2+1) where s1 (v minus 1)2v s2 v2v and(s3 s2vminus 2+1) v2v (v minus 1)2vHv

vminus 2 Applying equations (23)and (24) in the proof of eorem 31 of [29] among all then-projections of SN(2vminus 2+1)2v Qv sequentially maximizes 2 C2in (3) only ifQv sub Dqminus v(si) with i 1 or 2 In the followingwe investigate 2 C

(p)

2 (Qv) with Qv sub Dqminus v(s1) for which thefollowing analysis and final conclusion are the same as that

for 2 C

(p)

2 (Qv) with Qv sub Dqminus v(s2) For Qv withQv sub Dqminus v(s1) from equations (18) and (23) it is obtainedthat

Mathematical Problems in Engineering 5

B2 Qv γ( 1113857

0 for γ isin SN 2vminus 2+1( )2v

n minusN

4 for γ isin q Hqminus 2Hqminus v1113872 1113873 q(q minus 1) IHqminus v1113872 11138731113966 1113967

N

2v for γ isin (q minus 1) Hqminus 2Hqminus v1113872 1113873

N

8minus

N

2v for γ isin Hqminus 2Hqminus v

B2 Qv γ( 1113857 +N

8 for γ isin Hqminus v

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

where the second equality can be easily verified by notingthat

q Hqminus 2Hqminus v1113872 1113873 q(q minus 1) IHqminus v1113872 11138731113966 1113967

Dqminus v s1s2( 1113857 D

qminus v s1s3( 1113857 Dqminus v s1s2vminus 2+1( 11138571113864 1113865

(25)

erefore

2 C

(p)

2 Qv( 1113857

0 for 0leplt n minusN

4minus 1

N

4(n minus N4) forp n minus

N

4minus 1

0 for n minusN

4minus 1ltplt

N

2v minus 1

N

2v N4 minus N2v( 1113857 forp

N

2v minus 1

0 forN

2v minus 1ltpltN

8minus

N

2v minus 1

N

4minus

N

2v1113874 1113875N

8minus

N

2v1113874 1113875 forp N

8minus

N

2v minus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)

Denote Dlowast (Dlowastt Dlowastb ) where Dlowastt sub SN(2wminus 2+1)2w SN(2wminus 2+1)2w Dlowastt sub Dqminus w(s1) and Dlowastb sub Hk+1 Since kle q minus 3

we have U(Dlowastb ) sub Hqminus 2 erefore from (24) and (26) weobtain

2 C

(p)

2 Dlowast( 1113857

0 for 0leplt n minusN

4minus 1

N

4(n minus N4) forp n minus

N

4minus 1

0 for n minusN

4minus 1ltplt

N

2w minus 1

N

2w N4 minus N2w( 1113857 forp

N

2w minus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

6 Mathematical Problems in Engineering

Comparing (27) with (19) (20) for 4le vlew and (26) for4le vlew minus 1 it is obtained that ifD (Dt Db) is a B2-GMCdesign D should not be worse than Dlowast in terms of (3)erefore if D (Dt Db) is a B2-GMC design Dt shouldbe an n-projection of SN(2wminus 2+1)2w is completes theproof

e lemma below is an extension of Lemma A2 inAppendix introduced from [34]

Lemma 4 Suppose Dt consists of the last nN(2wminus 1 + 1)2w+1 lt nleN(2wminus 2 + 1)2w columns ofSN(2wminus 2+1)2w then B2(Dt γ1)geB2(Dt γ2) if γ1 is ahead of γ2in Hqminus w

Proof Suppose O consists of the last n minus N4 columns ofDqminus w((w minus 1)2w ) en Dt O SN(2wminus 2+1)2w Dqminus w1113966

(w minus 1) From (16) and (18) for any γ isin Hqminus wB2(Dt γ) B2(O γ) + N8 Straightforwardly fromLemma A2 it is obtained that B2(O γ1)geB2(O γ2) if γ1 isahead of γ2 in Hqminus w is completes the proof

With Lemmas 3 and 4 eorem 2 provides the con-struction methods of B2-GMC 2nminus m 2s designs withN4 + 1lt nle 5N16

Theorem 2 Let D (Dt Db) be a 2nminus m 2s design withN(2wminus 1 + 1)2w+1 lt nleN(2wminus 2 + 1)2w for wge 4 and2k le sle 2k+1 minus 1 for some 0le kle q minus 3 Suppose 2r leN

(2wminus 2 + 1)2w minus nle 2r+1 minus 1 for some 0le rle q minus w minus 1 thenD is a B2-GMC design if Dt consists of the last n columns ofSN(2wminus 2+1)2w and

(a) Db is any s-projection of Hk q minus 1 (q minus 1)Hk1113864 1113865 whenkle rle q minus w minus 1

(b) Db is any s-projection of Hk+1 whenr + 1le kle q minus w minus 1

(c) Db is any s-projection of Hqminus 2 whenq minus w minus 1lt kle q minus 3

Proof By Lemma 3 and its proof if Dlowast (Dlowastt Dlowastb ) is aB2-GMC 2nminus m 2s design there should be Dlowastt sub SN(2wminus 2+1)2w

and Dlowastt SN(2wminus 2+1)2w Dlowastt sub Dqminus w((w minus 1)2w ) Substitutingv with w in (24) we obtain

B2 Dlowastt γ( 1113857

0 for γ isin S N 2wminus 2+1( )2w( )

n minusN

4 for γ isin q Hqminus 2Hqminus w1113872 1113873 q(q minus 1) IHqminus w1113872 11138731113966 1113967

N

2w for γ isin (q minus 1) Hqminus 2Hqminus w1113872 1113873

N

8minus

N

2w for γ isin Hqminus 2Hqminus w

B2 Dlowastt γ) +

N

8 for γ isin Hqminus w1113874

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)

Let

u1 U Dlowastb( 1113857cap q Hqminus 2Hqminus w1113872 1113873 q(q minus 1) IHqminus w1113872 11138731113966 11139671113966 1113967

u2 U Dlowastb( 1113857cap (q minus 1) Hqminus 2Hqminus w1113872 11138731113966 1113967

u3 U Dlowastb( 1113857cap Hqminus 2Hqminus w1113872 11138731113966 1113967

(29)

and then

Mathematical Problems in Engineering 7

2 C

(p)

2 Dlowast( 1113857

n minusN

41113874 1113875

N

4minus u11113874 1113875 forp n minus

N

4minus 1

N

2w

N

4minus

N

2w minus u21113874 1113875 forp N

2w minus 1

N

8minus

N

2w1113874 1113875N

4minus

N

2w minus u31113874 1113875 forp N

8minus

N

2w minus 1

f Dlowastt p( 1113857 forpgeN

8minus 1

0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(30)

where f(Dlowastt p) 2 C

(p)

2 (Dlowastt ) minus (p + 1) γ γ isin Hqminus1113966

wcapU(Dlowastb ) B2(Dlowastt γ) p + 1 erefore Dlowast sequentiallymaximizes 2 C2 among all the possible 2nminus m 2s designs onlyif Dlowast sequentially minimizes

u1 u2 u3( 1113857 (31)

For ease of presenting let A q(Hqminus 2Hqminus w)1113966

q(q minus 1)(IHqminus w) B (q minus 1)(Hqminus 2Hqminus w) and C

Hqminus 2Hqminus wFor (a) when kle q minus w minus 1 there are two different ways

to choose Dlowastb from Hq (i) Dlowastb cap ABC empty and (ii)Dlowastb cap ABC neempty Recall that if Dlowast (Dlowastt Dlowastb ) is aB2-GMC design then Dlowastt sub SN(2wminus 2+1)2w with D

lowastt

SN(2wminus 2+1)2w Dlowastt sub Dqminus w((w minus 1)2w ) Case (i) implies thatDlowastb sub Hqminus w D

lowastt1113966 1113967 and thus Dlowastb sub Hqminus w q minus 1 (q minus 1)1113966

Hqminus w erefore choosing Dlowastb according to case (i) resultsin u1 u2 u3 0 If Dlowastb is chosen according to case (ii)then U(Dlowastb )cap ABC neempty which results in u1 gt 0 u2 gt 0 oru3 gt 0 Clearly if Dlowast is a B2-GMC design Dlowastb should be

chosen as (i) In the following we consider onlyDlowastb cap ABC empty ie Dlowastb sub Hqminus w (q minus 1) (q minus 1)Hqminus w1113966 1113967

For D (Dt Db) in (a) according to (v) in Lemma 1 ifDb sub Hk (q minus 1) (q minus 1)Hk1113864 1113865 with kle q minus w minus 1 thenU(Db) Hk (q minus 1) (q minus 1)Hk1113864 1113865 Denote b as the lastcolumn of Hk in Yates order and B2(Dt b) b thenbgeN8 According to Lemma 4 for γ isin Hqminus w if γ is ahead ofb in Hqminus w then B2(Dt γ)ge b by (18) For anyγ isin (q minus 1) (q minus 1)Hk1113864 1113865 we have B2(Dt γ) 0 For any γsuch that B2(Dt γ)lt b we have γ isin HqHk erefore forple b minus 2

2 C

(p)

2 (D) (p + 1) γ isin Hq γ notin U Db( 1113857 B2 Dt γ( 1113857 p + 11113966 1113967

(p + 1) γ isin Hq B2 Dt γ( 1113857 p + 11113966 1113967

2 C

(p)

2 Dt( 1113857

(32)

and for pge b

2 C

(p)

2 (D) (p + 1) γ isin Hq γ notin U Db( 1113857 B2 Dt γ( 1113857 p + 11113966 1113967

(p + 1) γ isin Hq B2 Dt γ( 1113857 p + 11113966 1113967 minus (p + 1) γ isin Hk B2 Dt γ( 1113857 p + 11113864 1113865

0

(33)

From (33) it is obtained that

2 C

(0)

2 (D) 2 C

(N8minus 1)

2 (D)2 C

(N8)

2 (D) 2 C

(bminus 2)

2 (D)1113874 1113875

2 C

(0)

2 Dt( 1113857 2 C

(N8minus 1)

2 Dt( 11138572 C

(N8)

2 Dt( 1113857 2 C

(bminus 2)

2 Dt( 11138571113874 1113875

(34)

Note that Dt is a GMC design [29] then D maximizes

2 C

(0)

2 (middot) 2 C

(N8minus 1)

2 (middot)2 C

(N8)

2 (middot) 2 C

(bminus 2)

2 (middot)1113874 1113875

(35)

among all the possible 2nminus m 2s designsSupposeD is not aB2-GMC design then there should be

aDlowast (Dlowastt Dlowastb ) which outperformsD (Dt Db) in termsof (3)is implies that there exists some p1 ge b minus 1 such that

8 Mathematical Problems in Engineering

2 C

(0)

2 Dlowast( 1113857 2 C

(N8minus 1)

2 Dlowast( 11138572 C

(N8)

2 Dlowast( 1113857 2 C

(bminus 2)

2 Dlowast( 11138571113874 1113875

2 C

(0)

2 (D) 2 C

(N8minus 1)

2 (D)2 C

(N8)

2 (D) 2 C

(bminus 2)

2 (D)1113874 1113875

(36)

2 C

p1minus 1( )2 Dlowast( 1113857gt 2 C

p1minus 1( )2 (D) (37)

Recalling the definitions of 2 C(p)

2 (D) and B2(Dt γ) wehave

1113944

P

p0

2 C

(p)

2 (D)

(p + 1) 1113944

P

p0 γ isin Hq γ notin U Db( 1113857 B2 Dt γ( 1113857 p + 11113966 1113967

1113944P

p0 γ isin HqSN 2wminus 2+1( )2w1113872 1113873 B2 Dt γ( 1113857 p + 11113966 1113967

minus 1113944P

p0 γ isin U Db( 1113857 B2 Dt γ( 1113857 p + 11113864 1113865

HqSN 2wminus 2+1( )2w1113872 1113873 minus Hk

2qminus

N 2wminus 2+ 11113872 1113873

2w minus 2k

(38)

where the second and third equalities are due to B2(Dt γ)

0 for any γ isin SN(2wminus 2+1)2w Similarly

1113944

P

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1) 1113944

P

p0 γ isin Hq γ notin U Dlowastb( 1113857 B2 Dlowastt γ( 1113857 p + 11113966 1113967

1113944P

p0 γ isin HqSN 2wminus 2+1( )2w1113872 1113873 B2 Dlowastt γ( 1113857 p + 11113966 1113967

minus 1113944P

p0 γ isin U Dlowastb( 1113857 B2 Dlowastt γ( 1113857 p + 11113864 1113865

HqSN 2wminus 2+1( )2w1113872 1113873 minus U Dlowastb( 1113857

2qminus

N 2wminus 2+ 11113872 1113873

2w minus U Dlowastb( 1113857

(39)

From (33) and (38) we have 1113944

P

p0

2 C

(p)

2 (D)

(p + 1) 1113944

bminus 2

p0

2 C

(p)

2 (D)

(p + 1)+

2 C

(bminus 1)

2 (D)

b

2qminus

N 2wminus 2+ 11113872 1113873

2w minus 2k

(40)

By (36) and (37) it is obtained that

Mathematical Problems in Engineering 9

1113944

P

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1)ge 1113944

bminus 2

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1)+

2 C

(bminus 1)

2 Dlowast( 1113857

b

gt 1113944bminus 2

p0

2 C

(p)

2 (D)

(p + 1)+

2 C

(bminus 1)

2 (D)

b

(41)

en according to (39)ndash(41) we obtain U(Dlowastb )lt 2k minus 1which contradicts Lemma 1 (i) and (ii)

For (b) similar to (a) when kle q minus w minus 1 there are twodifferent ways to choose Dlowastb from Hq (i) Dlowastb cap ABC emptyand (ii) Dlowastb cap ABC neempty With a similar argument to (a)case (i) results in u1 u2 u3 0 while case (ii) results inu1 gt 0 u2 gt 0 or u3 gt 0 erefore if Dlowast is a B2-GMC designDlowastb should be chosen according to case (i) ieDlowastb sub Hqminus w (q minus 1) (q minus 1)Hqminus w1113966 1113967 When choosing Dlowastbaccording to (i) there should be Dlowastb cap q minus 11113864

(q minus 1)Hqminus w empty ie Dlowastb sub Hqminus w otherwise Dlowastt capU

(Dlowastb )neempty noting that U(Dlowastb )cap (q minus 1 (q minus 1)Hqminus w)1113966 1113967

ge 2k gt 2r+1 minus 1 as shown in Lemma 1 (iii) e remainder ofthe proof is similar to that of (a)

For (c) when kgt q minus w minus 1 there are two different waysto choose Dlowastb from Hq (i) Dlowastb cap BC empty and (ii)Dlowastb cap BC neempty With a similar argument to (a) it is nothard to verify that case (ii) results in u1 gt 0 or u2 gt 0 whilecase (i) results in u1 u2 0 erefore if Dlowast is a B2-GMCdesign Dlowastb should be an s-projection of Hqminus 2 BC eremainder of the proof is similar to that of (a)

Remark 1 eorem 2 shows that when constructing theB2-GMC designs with N4 + 1lt nle 5N16 we can firstpartition the range (N4 + 1 5N16) into q minus 4 sequentialsubranges as (N(2wminus 1 + 1)2w+1 N(2wminus 2 + 1)2w) with w

4 5 q minus 1 and then obtain the B2-GMC designsaccording to eorem 2

In the following an example is provided to illustrate theconstruction method in eorem 2 and Remark 1

Example 2 Consider the constructions of B2-GMC219minus 13 2s designs for s 1 3 4 Here n 19 m 13 q 6and N 64 e values of the parameters N and n satisfyN4 + 1lt nle 5N16 ie n isin (17 20] From Remark 1 wepartition the range (17 20] into two subranges (17 18]

because w 5 and (18 20] because w 4 Since19 isin (18 20] according to eorem 2 Dt should be the last19 columns of S5N16 whereS5N16 Dqminus 4(Φ(4)) (5 5H2 6 6H2 56(H4H2)) from(18) erefore Dt (5H2 6 6H2 56(H4H2)) Note that5N16 minus n 20 minus 19 1 then r 0 Next we chooseDb fors 1 3 4

For s 1 we have k 0 According to eorem 2 (a)Db 5 as H0 empty

For s 3 we have k 1 According toeorem 2 (b)Db

should be a 3-projection of H2 ie Db H2 (1 2 12)For s 4 we have k 2 According toeorem 2 (c)Db

should be a 4-projection of H4 Without loss of generalitywe choose Db (1 2 12 3)

4 Concluding Remarks

e regular 2nminus m designs have wide applications in engi-neering manufacturing industry agriculture and medicineand so on When the size of experimental units is large theinhomogeneity of experimental units results in unwantedvariances of estimations of treatment effects An essentialway to solve this problem is to partition the experimentalunits into blocks

ere are two kinds of blocking problems as pointed outin [1] One is called the single block variable problem whichconsiders only a single block variable and the other is calledmulti-block variable problem which considers two or moreblock variables As stated in Section 1 experiments whichinvolve multi-block variables are more widely concerned inpractice than those which involve only a single block var-iable However due to the complexity of multi-block var-iable problem the studies on constructing optimal designswith multi-block variables are relatively rare

In this paper we aim at exploring the theories andconstructions of optimal blocked 2nminus m designs with multi-block variables e prevalent B2-GMC criterion is adoptedis criterion is preferable when there is some priorknowledge on the importance ordering of the treatmenteffects e systematical construction methods of theB2-GMC 2nminus m 2s designs with N4 + 1le nle 5N16 aredevelopede constructionmethods are concise and easy toimplement as indicated by the examples provided

Appendix

Lemma A1 is some result from Lemmas A2 A4 and A5 in[21] and Lemma 1 in [25]

Lemma A1 Suppose O is any s-projection of Hq with2k le sle 2k+1 minus 1 and kle q minus 2

(i) If Ocap q qHqminus 11113966 1113967 empty then U(O)capHqminus 11113966 1113967

ge 2k+1 minus 1 and when O has k + 1 independent col-umns the equality holds

(ii) If Ocap q qHqminus 11113966 1113967neempty then U(O)capHqminus 11113966 1113967

ge 2k minus 1 and when O has k + 1 independent col-umns the equality holds

(iii) If Ocap q qHqminus 11113966 1113967neempty then U(O)cap q qHqminus 11113966 11139671113966 1113967

ge 2k and when O has k + 1 independent columnsthe equality holds

(iv) If O sub Hk+1 then U(O) Hk+1(v) If O sub Hk q qHk1113864 1113865 then U(O) Hk q qHk1113864 1113865

Lemma A2 is some result of Lemma 1 in [34]

Lemma A2 Suppose O consists of the last n columns ofq qHqminus 11113966 1113967 then B2(O γ1)geB2(O γ2) if γ1 is ahead of γ2 inHqminus 1

Data Availability

No data were used to support this study

10 Mathematical Problems in Engineering

Conflicts of Interest

e author declares that there are no conflicts of interest

Acknowledgments

is study was supported by the National Natural ScienceFoundation of China (grant no 11801331)

References

[1] S Bisgaard ldquoA note on the definition of resolution for blocked2 k-p designsrdquo Technometrics vol 36 no 3 pp 308ndash3111994

[2] R R Sitter J Chen and M Feder ldquoFractional resolution andminimum aberration in blocked 2 n-k designsrdquo Techno-metrics vol 39 no 4 pp 382ndash390 1997

[3] H Chen and C S Cheng ldquoeory of optimal blocking ofsnminus mdesignsrdquo Annals of Statistics vol 27 no 6pp 1948ndash1973 1999

[4] R Zhang and D Park ldquoOptimal blocking of two-level frac-tional factorial designsrdquo Journal of Statistical Planning andInference vol 91 no 1 pp 107ndash121 2000

[5] S W Cheng and C F J Wu ldquoChoice of optimal blockingschemes in two-level and three-level designsrdquo Technometricsvol 44 no 3 pp 269ndash277 2002

[6] H Xu ldquoBlocked regular fractional factorial designs withminimum aberrationrdquo Annals of Statistics vol 8 no 5pp 2534ndash2553 2006

[7] H Xu and R W Mee ldquoMinimum aberration blockingschemes for 128-run designsrdquo Journal of Statistical Planningand Inference vol 140 no 11 pp 3213ndash3229 2010

[8] S Zhao P Li and R Karunamuni ldquoBlocked two-level regularfactorial designs with weak minimum aberrationrdquo Bio-metrika vol 100 no 1 pp 249ndash253 2013

[9] S-L Zhao and P-F Li ldquoConstruction of minimum aberrationblocked two-level regular factorial designsrdquo Communicationsin Statistics - eory and Methods vol 45 no 17 pp 5028ndash5036 2016

[10] B-J Chen P-F Li M-Q Liu and R-C Zhang ldquoSome resultson blocked regular 2-level fractional factorial designs withclear effectsrdquo Journal of Statistical Planning and Inferencevol 136 no 12 pp 4436ndash4449 2006

[11] S Zhao P Li and M-Q Liu ldquoOn blocked resolution IVdesigns containing clear two-factor interactionsrdquo Journal ofComplexity vol 29 no 5 pp 389ndash395 2013

[12] R C Zhang and RMukerjee ldquoGeneral minimum lower-orderconfounding in block designs using complementary setsrdquoStatistica Sinica vol 19 no 4 pp 1787ndash1802 2009

[13] S Zhao D K J Lin and P Li ldquoA note on the construction ofblocked two-level designs with general minimum lower orderconfoundingrdquo Journal of Statistical Planning and Inferencevol 172 pp 16ndash22 2016

[14] S-L Zhao and Q-Q Zhao ldquoSome results on constructingtwo-level block designs with general minimum lower orderconfoundingrdquo Communications in Statistics-eory andMethods vol 47 no 9 pp 2227ndash2237 2018

[15] R C Zhang P Li S L Zhao and M Y Ai ldquoA generalminimum lower-order confounding criterion for two-levelregular designsrdquo Statistica Sinica vol 18 no 4 pp 1689ndash1705 2008

[16] S Zhao P Li R Zhang and R Karunamuni ldquoConstruction ofblocked two-level regular designs with general minimum

lower order confoundingrdquo Journal of Statistical Planning andInference vol 143 no 6 pp 1082ndash1090 2013

[17] Y Zhao S Zhao and M-Q Liu ldquoA theory on constructingblocked two-level designs with general minimum lower orderconfoundingrdquo Frontiers of Mathematics in China vol 11no 1 pp 207ndash235 2016

[18] S-L Zhao and Q Sun ldquoOn constructing general minimumlower order confounding two-level block designsrdquo Commu-nications in Statistics-eory and Methods vol 46 no 3pp 1261ndash1274 2017

[19] J Wei P Li and R Zhang ldquoBlocked two-level regular designswith general minimum lower order confoundingrdquo Journal ofStatistical eory and Practice vol 8 no 1 pp 46ndash65 2014

[20] C F J Wu and M S Hamada Experiments PlanningAnalysis and Optimization Wiley Hoboken NJ USA 2edition 2009

[21] Q Zhao and S Zhao ldquoSome results on two-level regulardesigns with multi block variables containing clear effectsrdquoStatistical Papers vol 60 no 5 pp 1569ndash1582 2019

[22] S Zhao and Q Zhao ldquoMinimum aberration blocked designswith multiple block variablesrdquo Metrika vol 84 no 2pp 121ndash140 2021

[23] R Zhang P Li and J Wei ldquoOptimal two-level regular designswith multi block variablesrdquo Journal of Statistical eory andPractice vol 5 no 1 pp 161ndash178 2011

[24] Y N Zhao S L Zhao and M Q Liu ldquoOn constructingoptimal two-level designs with multi block variablesrdquo Journalof Systems Science and Complexity vol 31 pp 1ndash14 2018

[25] Y N Zhao and S L Zhao ldquoConstruction of optimal blockeddesigns with multi block variablesrdquoAIMSmathematics vol 6no 6 pp 6293ndash6308 2021

[26] B Tang and C F J Wu ldquoCharacterization of minimumaberration 2nminus kdesigns in terms of their complementarydesignsrdquo Annals of Statistics vol 25 no 6 pp 2549ndash25591996

[27] C F J Wu and Y Chen ldquoA graph-aided method for planningtwo-level experiments when certain interactions are impor-tantrdquo Technometrics vol 34 no 2 pp 162ndash175 1992

[28] G E P Box and J S Hunter ldquoe 2kminus pfractional factorialdesigns I and IIrdquo Technometrics vol 3 no 4 pp 311ndash3511961

[29] Y Cheng and R Zhang ldquoOn construction of general mini-mum lower order confounding 2nminus m designs with (N4)+

1le nle 9N32rdquo Journal of Statistical Planning and Inferencevol 140 no 9 pp 2384ndash2394 2010

[30] H Wu and C F J Wu ldquoClear two-factor interactions andminimum aberrationrdquo Annals of Statistics vol 30 no 5pp 1496ndash1511 2002

[31] R M Block and R W Mee ldquoSecond order saturated reso-lution IV designsrdquo Journal of Statistical eory and Appli-cations vol 2 no 2 pp 96ndash112 2003

[32] A A Davydov and L M Tombak ldquoQuasi-perfect linear bi-nary codes with distance 4 and complete caps in projectivegeometryrdquo Problems of Information Transmission vol 25no 4 pp 265ndash275 1990

[33] R Zhang and Y Cheng ldquoGeneral minimum lower orderconfounding designs an overview and a construction theoryrdquoJournal of Statistical Planning and Inference vol 140 no 7pp 1719ndash1730 2010

[34] Q Zhou N Balakrishnan and R Zhang ldquoe factor aliasedeffect number pattern and its application in experimentalplanningrdquo Canadian Journal of Statistics vol 41 no 3pp 540ndash555 2013

Mathematical Problems in Engineering 11

A 2nminus m 2s blocked design D (Dt Db) is called aB2-GMC design if it sequentially maximizes (4) Let1 C

(p)

2 (Dt) be the number of main effects which are aliasedwith p 2firsquos ofDt and 2 C

(p)

2 (Dt) be the number of 2firsquos whichare aliased with the other p 2firsquos of Dt Let

1 C2 Dt( 1113857

1 C

(0)

2 Dt( 11138571 C

(1)

2 Dt( 1113857 1 C

(p)

2 Dt( 11138571113874 1113875

2 C2 Dt( 1113857

2 C

(0)

2 Dt( 11138572 C

(1)

2 Dt( 1113857 2 C

(p)

2 Dt( 11138571113874 1113875

C Dt( 1113857

1 C2 Dt( 1113857

2 C2 Dt( 11138571113874 1113875

(5)

A 2nminus m design Dt is called a GMC design if Dt se-quentially maximizes (5)

Let q n minus m Constructing a B2-GMC design is tochoose Dt and Db from Dq(middot) such that (4) is sequentiallymaximized In the following without causing confusions weomit the subscript of a column and the superscript of adesign when they are taken from Dq(middot) For example we useaHr andHq instead of a2q Hq

r andHqq respectively Denote

U Db( 1113857 γ isin Hq γ isin Db or γ abwith a b isin Db1113966 1113967 (6)

and then U(Db) consists of all the potentially significantblock effects As previously stated the confounding betweenmain treatment effects and potentially significant block ef-fects is not allowed is requires Dt capU(Db) empty and thus1 C

(p)

2 (D) 1 C

(p)

2 (Dt) for p 0 1 2 PFor Dt sub Hq and γ isin Hq define

B2 Dt γ( 1113857 d1d2( 1113857 d1 d2 isin Dt d1d2 γ1113864 1113865 (7)

where denotes the cardinality of a set and d1d2 standsfor the two-factor interaction of d1 and d2 us B2(Dt γ) isthe number of 2firsquos of Dt appearing in the alias set thatcontains γ

Isomorphism introduced by Tang and Wu [26] is auseful concept which helps narrow down the search of theoptimal blocked designs here An isomorphism ϕ is a one-to-one mapping fromHq toHq such that ϕ(xy) ϕ(x)ϕ(y) forevery x ne y isin Hq e 2nminus m designs Dt and Dlowastt are iso-morphic if there exists an isomorphism ϕ that mapsDt ontoDlowastt e 2nminus m 2s designs D (Dt Db) and Dlowast (Dlowastt Dlowastb )

are isomorphic if there exists an isomorphism ϕ that mapsDt onto Dlowastt and Db onto Dlowastb

3 Constructions of B2-GMC Designs

31 B2-GMC 2nminus m 2s Designs with n (N4) + 1 A designis of MaxC2 (see [27]) if it has resolution IV and maximumnumber of clear 2firsquos where a resolution R design has noc-factor interaction confounded with any other interactioninvolving less than R minus c factors (see [28]) and a 2fi is calledclear if it is not aliased with any main treatment effect andother 2firsquos Cheng and Zhang [29] showed that a 2nminus m designwith n N4 + 1 is a MaxC2 design if and only if it is a GMCdesign ey also pointed out that up to isomorphism theGMC 2nminus m design with n N4 + 1 can be uniquelyexpressed as SN4+1 (q minus 1 q q(q minus 1)Hqminus 2) It is easy toobtain that

B2 SN4+1 γ( 1113857

1 for γ isin (q minus 1)Hqminus 2 qHqminus 2 q(q minus 1)1113966 1113967

N

8minus 1 for γ isin Hqminus 2

0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)

erefore

1 C

(p)

2 SN4+1( 1113857

N

4+ 1 forp 0

0 otherwise

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(9)

2 C

(p)

2 SN4+1( 1113857

2N

4minus 11113874 1113875 + 1 forp 0

N

4minus 11113874 1113875

N

8minus 11113874 1113875 forp

N

8minus 2

0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(10)

Lemma 1 is a straightforward extension of Lemma A1in Appendix introduced from [24 25]

Lemma 1 Suppose Db is any s-projection of Hr γ γHr1113864 1113865

with 2k le sle 2k+1 minus 1 for some k(0le kle r minus 1) where γ isindependent of the columns of Hr We have

(i) If Db cap γ γHr1113864 1113865 empty then U(Db)capHr1113864 1113865ge 2k+1 minus 1and the equality holds when Db has k + 1 indepen-dent columns

(ii) If Db cap γ γHr1113864 1113865neempty then U(Db)capHr1113864 1113865ge 2k minus 1and the equality holds when Db has k + 1 indepen-dent columns

(iii) If Db cap γ γHr1113864 1113865neempty then U(Db)cap γ γHr1113864 11138651113864 1113865ge 2k

and the equality holds when Db has k + 1 indepen-dent columns

(iv) If Db sub Hk+1 then U(Db) Hk+1(v) If Db sub Hk γ γHk1113864 1113865 then U(Db) Hk γ γHk1113864 1113865

Lemma 2 provides a necessary condition for a 2nminus m 2s

design D (Dt Db) with n N4 + 1 to be a B2-GMCdesign

Lemma 2 Suppose D (Dt Db) is a 2nminus m 2s design withn N4 + 1 and 2k le sle 2k+1 minus 1 for some k(0le kle q minus 3)then D (Dt Db) is a B2-GMC design only if Dt SN4+1

Proof Let 1113957D (SN4+11113957Db) be a 2nminus m 2s design and

1113957Db isin Hqminus 2 From (8)ndash(10) we can obtain

1 C2(

1113957D) N

4+ 1 0 01113874 1113875 (11)

which is sequentially maximized by 1113957D and 2 C

(0)

2 ( 1113957D)

2(N4 minus 1) + 1 N2 minus 1 Suppose that any D (SN4+1

Db) is not a B2-GMC 2nminus m 2s design for n N4 + 1 en

Mathematical Problems in Engineering 3

by the definition of B2-GMC criterion there should be a2nminus m 2s design Dlowast (Dlowastt Dlowastb ) outperforming1113957D (SN4+1

1113957Db) in terms of (4) is leads to

1 C2 Dlowast( 1113857

N

4+ 1 0 01113874 1113875 (12)

and 2 C

(0)

2 (Dlowast)ge (N2) minus 1 Clearly

1 C2 Dlowastt( 1113857

1 C2 Dlowast( 1113857

N

4+ 1 0 01113874 1113875 (13)

and 2 C

(0)

2 (Dlowastt )ge 2 C(0)

2 (Dlowast)geN2 minus 1 noting that Dlowastt is theunblocked part of Dlowast and Dlowastt has n N4 + 1 columns

In fact the inequality in the formula 2 C(0)

2 (Dlowastt )geN2 minus

1 is not valid Recall that the MaxC2 design SN4+1 has thelargest number N2 minus 1 of clear 2firsquos among all the 2nminus m

designs with n N4 + 1 ereforeDlowastt has at most N2 minus 1clear 2firsquos ie

2 C(0)

2 (Dlowastt ) N2 minus 1 Wu and Wu [30]showed that a 2nminus m design with n N4 + 1 is a MaxC2design if and only if this design has N2 minus 1 clear 2firsquos isobtains that Dlowastt SN4+1 up to isomorphism

With Lemma 2eorem 1 provides the constructions ofB2-GMC designs with n N4 + 1

Theorem 1 SupposeD (Dt Db) is a 2nminus m 2s design withn N4 + 1 and 2k le sle 2k+1 minus 1 for some k(0le kle q minus 3)then D (Dt Db) is a B2-GMC design if Dt SN4+1 andDb is any s-projection of Hk+1

Proof By Lemma 2 if D (Dt Db) is a B2-GMC 2nminus m 2s

design with n N4 + 1 then Dt SN4+1 up to isomor-phism us

1 C2(D) (N4 + 1 0 0) which is se-quentially maximized by D (SN4+1 Db) Let u1

U(Db)cap (q minus 1)Hqminus 2 qHqminus 2 q(q minus 1)1113966 11139671113966 1113967 and u2 U(Db)capHqminus 21113966 1113967 then from (10) we have

2 C

(p)

2 (D)

2N

4minus 11113874 1113875 + 1 minus u1 forp 0

N

4minus 1 minus u21113874 1113875

N

8minus 11113874 1113875 forp

N

8minus 2

0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)

If D is a B2-GMC design then D must sequentiallyminimize (u1 u2) ere are two different ways to chooseDb

from HqSN4+1

(i) Db cap (q minus 1)Hqminus 2 qHqminus 2 q(q minus 1)1113966 1113967 empty(ii) Db cap (q minus 1)Hqminus 2 qHqminus 2 q(q minus 1)1113966 1113967neempty

It is not hard to verify thatDb in case (ii) results in u1 gt 0while Db in case (i) gives u1 0 erefore ifD (SN4+1 Db) is a B2-GMC design then Db must be ofcase (i) Recall thatDb cap SN4+1 empty and we haveDb sub Hqminus 2By Lemma 1 (i) if Db sub Hqminus 2 then u2 U(Db)capHqminus 21113966 1113967ge 2k+1 minus 1 and the equality holds when

Db sub Hk+1 up to isomorphism is completes theproof

e following example illustrates the constructionmethod in eorem 1

Example 1 Consider the construction of theB2-GMC29minus 4 25design Here n 9 m 4 q 5 N 32 and s 5 whichleads to k 2 According to eorem 1 let Dt (4 5 45H3)

and Db be any 5-projection of H3 say Db (1 2 12 3 13)en D (Dt Db) is a B2-GMC 29minus 4 25 design

32 B2-GMC Designs with N4 + 1lt nle 5N16 To se-quentially maximize (4) the first part 1 C2(D) of (4) shouldbe first maximized Recall that 1 C2(D)

1 C2(Dt) If Dt hasresolution IV then

1 C2(D) must be maximized Accordingto [31] when N4 + 1lt nle 5N16 theDt with resolution IVmust be an n-projection of some second-order saturated(SOS) designs In the following we first review the conceptof SOS design

A 2nminus m design is called an SOS design if all of its degrees offreedom can be used to estimate only themain treatment effectsand 2firsquos In terms of coding theory and projective geometryDavydov and Tombak [32] showed that given N only the SOSdesigns of N4 + 1 N(2wminus 2 + 1)2w with wge 4 and N2factors exist Block and Mee [31] further showed that an SOSdesign of N(2wminus 2 + 1)2w factors can be obtained by doublingsome smaller SOS 2(2wminus 2+1)minus (2wminus 2+1minus w) designs q minus w timesZhang and Cheng [33] showed that the SOS design of N2factors can be uniquely represented by SN2 (q qHqminus 1) up toisomorphism Let

L(w) Dqminus w

(Y) Y is an SOS 2 2wminus 2+1( )minus 2wminus 2+1minus w( ) design1113882 1113883

(15)

denote the collection of all the SOS designs obtained bydoubling some SOS 2(2wminus 2+1)minus (2wminus 2+1minus w) designs q minus w timesEspecially in L(w) we denote the SOS design obtained bydoubling the MaxC2 2(2wminus 2+1)minus (2wminus 2+1minus w) designΦ(w) ((w minus 1)2w w2w w2w (w minus 1)2wHw

wminus 2) as SN(2wminus 2+1)2w With a little algebra it is easy to verify that

Dqminus w I2w( 1113857 I2q Hq

qminus w1113872 1113873 IHqminus w1113872 1113873

Dqminus w 12w( 1113857 (q minus w + 1)2q I2q Hq

qminus w1113872 1113873

(q minus w + 1) IHqminus w1113872 1113873

Dqminus w 22w( 1113857 (q minus w + 2)2q I2q Hq

qminus w1113872 1113873

(q minus w + 2) IHqminus w1113872 1113873

⋮

Dqminus w

(w minus 1)2w( 1113857 (q minus 1)2q I2q Hqqminus w1113872 1113873 (q minus 1) IHqminus w1113872 1113873

Dqminus w w2w( 1113857 q2q I2q Hq

qminus w1113872 1113873 q IHqminus w1113872 1113873

(16)

where I2qminus w otimes 12w (q minus w + 1)2q I2qminus w otimes 22w (q minus w+ 2)2q

I2qminus w otimes (w minus 1)2w (q minus 1)2q and I2qminus w otimesw2w q2q

en

4 Mathematical Problems in Engineering

Dqminus w

(w minus 1)2ww2wHwwminus 2( 1113857 q(q minus 1) Hqminus 2Hqminus w1113872 1113873

(17)

erefore SN(2wminus 2+1)2w can be expressed as

SN 2wminus 2+1( )2w Dqminus w

(Φ(w))

(q minus 1) (q minus 1)Hqminus w q qHqminus w q(q minus 1)1113872

Hqminus 2Hqminus w1113872 11138731113873

(18)

e following lemma provides a necessary condition fora 2nminus m 2s design D (Dt Db) with N4 + 1lt nle 5N16 tobe a B2-GMC design

Lemma 3 Suppose D (Dt Db) is a 2nminus m 2s design withN(2wminus 1 + 1)2w+1 lt nleN(2wminus 2 + 1)2w for wge 4 and2k le sle 2k+1 minus 1 for some k(0le kle q minus 3) thenD (Dt Db) is a B2-GMC design only if Dt sub SN(2wminus 2+1)2w

Proof As discussed in the first paragraph of this section ifD (Dt Db) is a B2-GMC 2nminus m 2s design withN(2wminus 1 + 1)2w+1 lt nleN(2wminus 2 + 1)2w then Dt must be ann-projection of some SOS designs in L(v) with 4le vlew orSN2

Let P1 be an n-projection of SN2 According toeorem3 in [33] we obtain

2 C

(p)

2 P1( 1113857

0 forplt n minusN

4minus 1

N

4n minus

N

41113874 1113875 forp n minus

N

4minus 1

0 for n minusN

4minus 1ltplt

N

8minus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(19)

Let Xv (a1 a2 a2vminus 2+1) be any SOS2(2vminus 2+1)minus (2vminus 2+1minus v) design but not a MaxC2 design thenDqminus v(Xv) isinL(v) Denote pi as the number of clear 2firsquos ofXv involving ai for i 1 2 2vminus 2 + 1 and c as the totalnumber of clear 2firsquos of Xv Let Pv be an n-projection ofDqminus v(Xv) and Pv Dqminus v(Xv)Pv en Pv N

(2vminus 2 + 1)2v minus nleN2v Applying equations (23) and (24)in the proof of eorem 31 of [29] among all the n-pro-jections of Dqminus v(Xv) Pv sequentially maximizes 2 C2 in (5)only if Pv sub Dqminus v(aj) where aj is the column such thatpj max p1 p2 p2vminus 2+11113864 1113865 For a Pv with Pv sub Dqminus v(aj)according to equation (24) in the proof of eorem 31 of[29] it is obtained that

2 C

(p)

2 Pv( 1113857

0 for 0leplt n minusN

4minus 1

pjN

2v (n minus N4) forp n minusN

4minus 1

0 for n minusN

4minus 1ltplt

N

2v minus 1

c minus pj1113872 1113873N

2v1113874 1113875

2 forp

N

2v minus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)

For a MaxC22(2vminus 2+1)minus (2vminus 2+1minus v) design it has 2vminus 1 minus 1clear 2firsquos and pj 2vminus 2 Since Xv is not a MaxC2 design wehave

pj 2vminus 2

c minus pj lt 2vminus 1

minus 2vminus 2minus 1

(21)

or

pj lt 2vminus 2

clt 2vminus 1minus 1

(22)

Let Qv be an n-projection of SN(2vminus 2+1)2v obtained bydoubling Φ(v) the MaxC22(2vminus 2+1)minus (2vminus 2+1minus v) design asmentioned above With a column permutation rewriteDq(middot) as

Dq

RC(middot) Hqminus vHqminus 2Hqminus v (q minus 1) Hqminus 2Hqminus v1113872 11138731113872

q Hqminus 2Hqminus v1113872 1113873 q(q minus 1) IHqminus v1113872 1113873 SN 2vminus 2+1( )2v 1113873

(23)

in a re-changed Yates order Write Φ(v) asΦ(v) (s1 s2 s2vminus 2+1) where s1 (v minus 1)2v s2 v2v and(s3 s2vminus 2+1) v2v (v minus 1)2vHv

vminus 2 Applying equations (23)and (24) in the proof of eorem 31 of [29] among all then-projections of SN(2vminus 2+1)2v Qv sequentially maximizes 2 C2in (3) only ifQv sub Dqminus v(si) with i 1 or 2 In the followingwe investigate 2 C

(p)

2 (Qv) with Qv sub Dqminus v(s1) for which thefollowing analysis and final conclusion are the same as that

for 2 C

(p)

2 (Qv) with Qv sub Dqminus v(s2) For Qv withQv sub Dqminus v(s1) from equations (18) and (23) it is obtainedthat

Mathematical Problems in Engineering 5

B2 Qv γ( 1113857

0 for γ isin SN 2vminus 2+1( )2v

n minusN

4 for γ isin q Hqminus 2Hqminus v1113872 1113873 q(q minus 1) IHqminus v1113872 11138731113966 1113967

N

2v for γ isin (q minus 1) Hqminus 2Hqminus v1113872 1113873

N

8minus

N

2v for γ isin Hqminus 2Hqminus v

B2 Qv γ( 1113857 +N

8 for γ isin Hqminus v

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

where the second equality can be easily verified by notingthat

q Hqminus 2Hqminus v1113872 1113873 q(q minus 1) IHqminus v1113872 11138731113966 1113967

Dqminus v s1s2( 1113857 D

qminus v s1s3( 1113857 Dqminus v s1s2vminus 2+1( 11138571113864 1113865

(25)

erefore

2 C

(p)

2 Qv( 1113857

0 for 0leplt n minusN

4minus 1

N

4(n minus N4) forp n minus

N

4minus 1

0 for n minusN

4minus 1ltplt

N

2v minus 1

N

2v N4 minus N2v( 1113857 forp

N

2v minus 1

0 forN

2v minus 1ltpltN

8minus

N

2v minus 1

N

4minus

N

2v1113874 1113875N

8minus

N

2v1113874 1113875 forp N

8minus

N

2v minus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)

Denote Dlowast (Dlowastt Dlowastb ) where Dlowastt sub SN(2wminus 2+1)2w SN(2wminus 2+1)2w Dlowastt sub Dqminus w(s1) and Dlowastb sub Hk+1 Since kle q minus 3

we have U(Dlowastb ) sub Hqminus 2 erefore from (24) and (26) weobtain

2 C

(p)

2 Dlowast( 1113857

0 for 0leplt n minusN

4minus 1

N

4(n minus N4) forp n minus

N

4minus 1

0 for n minusN

4minus 1ltplt

N

2w minus 1

N

2w N4 minus N2w( 1113857 forp

N

2w minus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

6 Mathematical Problems in Engineering

Comparing (27) with (19) (20) for 4le vlew and (26) for4le vlew minus 1 it is obtained that ifD (Dt Db) is a B2-GMCdesign D should not be worse than Dlowast in terms of (3)erefore if D (Dt Db) is a B2-GMC design Dt shouldbe an n-projection of SN(2wminus 2+1)2w is completes theproof

e lemma below is an extension of Lemma A2 inAppendix introduced from [34]

Lemma 4 Suppose Dt consists of the last nN(2wminus 1 + 1)2w+1 lt nleN(2wminus 2 + 1)2w columns ofSN(2wminus 2+1)2w then B2(Dt γ1)geB2(Dt γ2) if γ1 is ahead of γ2in Hqminus w

Proof Suppose O consists of the last n minus N4 columns ofDqminus w((w minus 1)2w ) en Dt O SN(2wminus 2+1)2w Dqminus w1113966

(w minus 1) From (16) and (18) for any γ isin Hqminus wB2(Dt γ) B2(O γ) + N8 Straightforwardly fromLemma A2 it is obtained that B2(O γ1)geB2(O γ2) if γ1 isahead of γ2 in Hqminus w is completes the proof

With Lemmas 3 and 4 eorem 2 provides the con-struction methods of B2-GMC 2nminus m 2s designs withN4 + 1lt nle 5N16

Theorem 2 Let D (Dt Db) be a 2nminus m 2s design withN(2wminus 1 + 1)2w+1 lt nleN(2wminus 2 + 1)2w for wge 4 and2k le sle 2k+1 minus 1 for some 0le kle q minus 3 Suppose 2r leN

(2wminus 2 + 1)2w minus nle 2r+1 minus 1 for some 0le rle q minus w minus 1 thenD is a B2-GMC design if Dt consists of the last n columns ofSN(2wminus 2+1)2w and

(a) Db is any s-projection of Hk q minus 1 (q minus 1)Hk1113864 1113865 whenkle rle q minus w minus 1

(b) Db is any s-projection of Hk+1 whenr + 1le kle q minus w minus 1

(c) Db is any s-projection of Hqminus 2 whenq minus w minus 1lt kle q minus 3

Proof By Lemma 3 and its proof if Dlowast (Dlowastt Dlowastb ) is aB2-GMC 2nminus m 2s design there should be Dlowastt sub SN(2wminus 2+1)2w

and Dlowastt SN(2wminus 2+1)2w Dlowastt sub Dqminus w((w minus 1)2w ) Substitutingv with w in (24) we obtain

B2 Dlowastt γ( 1113857

0 for γ isin S N 2wminus 2+1( )2w( )

n minusN

4 for γ isin q Hqminus 2Hqminus w1113872 1113873 q(q minus 1) IHqminus w1113872 11138731113966 1113967

N

2w for γ isin (q minus 1) Hqminus 2Hqminus w1113872 1113873

N

8minus

N

2w for γ isin Hqminus 2Hqminus w

B2 Dlowastt γ) +

N

8 for γ isin Hqminus w1113874

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)

Let

u1 U Dlowastb( 1113857cap q Hqminus 2Hqminus w1113872 1113873 q(q minus 1) IHqminus w1113872 11138731113966 11139671113966 1113967

u2 U Dlowastb( 1113857cap (q minus 1) Hqminus 2Hqminus w1113872 11138731113966 1113967

u3 U Dlowastb( 1113857cap Hqminus 2Hqminus w1113872 11138731113966 1113967

(29)

and then

Mathematical Problems in Engineering 7

2 C

(p)

2 Dlowast( 1113857

n minusN

41113874 1113875

N

4minus u11113874 1113875 forp n minus

N

4minus 1

N

2w

N

4minus

N

2w minus u21113874 1113875 forp N

2w minus 1

N

8minus

N

2w1113874 1113875N

4minus

N

2w minus u31113874 1113875 forp N

8minus

N

2w minus 1

f Dlowastt p( 1113857 forpgeN

8minus 1

0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(30)

where f(Dlowastt p) 2 C

(p)

2 (Dlowastt ) minus (p + 1) γ γ isin Hqminus1113966

wcapU(Dlowastb ) B2(Dlowastt γ) p + 1 erefore Dlowast sequentiallymaximizes 2 C2 among all the possible 2nminus m 2s designs onlyif Dlowast sequentially minimizes

u1 u2 u3( 1113857 (31)

For ease of presenting let A q(Hqminus 2Hqminus w)1113966

q(q minus 1)(IHqminus w) B (q minus 1)(Hqminus 2Hqminus w) and C

Hqminus 2Hqminus wFor (a) when kle q minus w minus 1 there are two different ways

to choose Dlowastb from Hq (i) Dlowastb cap ABC empty and (ii)Dlowastb cap ABC neempty Recall that if Dlowast (Dlowastt Dlowastb ) is aB2-GMC design then Dlowastt sub SN(2wminus 2+1)2w with D

lowastt

SN(2wminus 2+1)2w Dlowastt sub Dqminus w((w minus 1)2w ) Case (i) implies thatDlowastb sub Hqminus w D

lowastt1113966 1113967 and thus Dlowastb sub Hqminus w q minus 1 (q minus 1)1113966

Hqminus w erefore choosing Dlowastb according to case (i) resultsin u1 u2 u3 0 If Dlowastb is chosen according to case (ii)then U(Dlowastb )cap ABC neempty which results in u1 gt 0 u2 gt 0 oru3 gt 0 Clearly if Dlowast is a B2-GMC design Dlowastb should be

chosen as (i) In the following we consider onlyDlowastb cap ABC empty ie Dlowastb sub Hqminus w (q minus 1) (q minus 1)Hqminus w1113966 1113967

For D (Dt Db) in (a) according to (v) in Lemma 1 ifDb sub Hk (q minus 1) (q minus 1)Hk1113864 1113865 with kle q minus w minus 1 thenU(Db) Hk (q minus 1) (q minus 1)Hk1113864 1113865 Denote b as the lastcolumn of Hk in Yates order and B2(Dt b) b thenbgeN8 According to Lemma 4 for γ isin Hqminus w if γ is ahead ofb in Hqminus w then B2(Dt γ)ge b by (18) For anyγ isin (q minus 1) (q minus 1)Hk1113864 1113865 we have B2(Dt γ) 0 For any γsuch that B2(Dt γ)lt b we have γ isin HqHk erefore forple b minus 2

2 C

(p)

2 (D) (p + 1) γ isin Hq γ notin U Db( 1113857 B2 Dt γ( 1113857 p + 11113966 1113967

(p + 1) γ isin Hq B2 Dt γ( 1113857 p + 11113966 1113967

2 C

(p)

2 Dt( 1113857

(32)

and for pge b

2 C

(p)

2 (D) (p + 1) γ isin Hq γ notin U Db( 1113857 B2 Dt γ( 1113857 p + 11113966 1113967

(p + 1) γ isin Hq B2 Dt γ( 1113857 p + 11113966 1113967 minus (p + 1) γ isin Hk B2 Dt γ( 1113857 p + 11113864 1113865

0

(33)

From (33) it is obtained that

2 C

(0)

2 (D) 2 C

(N8minus 1)

2 (D)2 C

(N8)

2 (D) 2 C

(bminus 2)

2 (D)1113874 1113875

2 C

(0)

2 Dt( 1113857 2 C

(N8minus 1)

2 Dt( 11138572 C

(N8)

2 Dt( 1113857 2 C

(bminus 2)

2 Dt( 11138571113874 1113875

(34)

Note that Dt is a GMC design [29] then D maximizes

2 C

(0)

2 (middot) 2 C

(N8minus 1)

2 (middot)2 C

(N8)

2 (middot) 2 C

(bminus 2)

2 (middot)1113874 1113875

(35)

among all the possible 2nminus m 2s designsSupposeD is not aB2-GMC design then there should be

aDlowast (Dlowastt Dlowastb ) which outperformsD (Dt Db) in termsof (3)is implies that there exists some p1 ge b minus 1 such that

8 Mathematical Problems in Engineering

2 C

(0)

2 Dlowast( 1113857 2 C

(N8minus 1)

2 Dlowast( 11138572 C

(N8)

2 Dlowast( 1113857 2 C

(bminus 2)

2 Dlowast( 11138571113874 1113875

2 C

(0)

2 (D) 2 C

(N8minus 1)

2 (D)2 C

(N8)

2 (D) 2 C

(bminus 2)

2 (D)1113874 1113875

(36)

2 C

p1minus 1( )2 Dlowast( 1113857gt 2 C

p1minus 1( )2 (D) (37)

Recalling the definitions of 2 C(p)

2 (D) and B2(Dt γ) wehave

1113944

P

p0

2 C

(p)

2 (D)

(p + 1) 1113944

P

p0 γ isin Hq γ notin U Db( 1113857 B2 Dt γ( 1113857 p + 11113966 1113967

1113944P

p0 γ isin HqSN 2wminus 2+1( )2w1113872 1113873 B2 Dt γ( 1113857 p + 11113966 1113967

minus 1113944P

p0 γ isin U Db( 1113857 B2 Dt γ( 1113857 p + 11113864 1113865

HqSN 2wminus 2+1( )2w1113872 1113873 minus Hk

2qminus

N 2wminus 2+ 11113872 1113873

2w minus 2k

(38)

where the second and third equalities are due to B2(Dt γ)

0 for any γ isin SN(2wminus 2+1)2w Similarly

1113944

P

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1) 1113944

P

p0 γ isin Hq γ notin U Dlowastb( 1113857 B2 Dlowastt γ( 1113857 p + 11113966 1113967

1113944P

p0 γ isin HqSN 2wminus 2+1( )2w1113872 1113873 B2 Dlowastt γ( 1113857 p + 11113966 1113967

minus 1113944P

p0 γ isin U Dlowastb( 1113857 B2 Dlowastt γ( 1113857 p + 11113864 1113865

HqSN 2wminus 2+1( )2w1113872 1113873 minus U Dlowastb( 1113857

2qminus

N 2wminus 2+ 11113872 1113873

2w minus U Dlowastb( 1113857

(39)

From (33) and (38) we have 1113944

P

p0

2 C

(p)

2 (D)

(p + 1) 1113944

bminus 2

p0

2 C

(p)

2 (D)

(p + 1)+

2 C

(bminus 1)

2 (D)

b

2qminus

N 2wminus 2+ 11113872 1113873

2w minus 2k

(40)

By (36) and (37) it is obtained that

Mathematical Problems in Engineering 9

1113944

P

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1)ge 1113944

bminus 2

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1)+

2 C

(bminus 1)

2 Dlowast( 1113857

b

gt 1113944bminus 2

p0

2 C

(p)

2 (D)

(p + 1)+

2 C

(bminus 1)

2 (D)

b

(41)

en according to (39)ndash(41) we obtain U(Dlowastb )lt 2k minus 1which contradicts Lemma 1 (i) and (ii)

For (b) similar to (a) when kle q minus w minus 1 there are twodifferent ways to choose Dlowastb from Hq (i) Dlowastb cap ABC emptyand (ii) Dlowastb cap ABC neempty With a similar argument to (a)case (i) results in u1 u2 u3 0 while case (ii) results inu1 gt 0 u2 gt 0 or u3 gt 0 erefore if Dlowast is a B2-GMC designDlowastb should be chosen according to case (i) ieDlowastb sub Hqminus w (q minus 1) (q minus 1)Hqminus w1113966 1113967 When choosing Dlowastbaccording to (i) there should be Dlowastb cap q minus 11113864

(q minus 1)Hqminus w empty ie Dlowastb sub Hqminus w otherwise Dlowastt capU

(Dlowastb )neempty noting that U(Dlowastb )cap (q minus 1 (q minus 1)Hqminus w)1113966 1113967

ge 2k gt 2r+1 minus 1 as shown in Lemma 1 (iii) e remainder ofthe proof is similar to that of (a)

For (c) when kgt q minus w minus 1 there are two different waysto choose Dlowastb from Hq (i) Dlowastb cap BC empty and (ii)Dlowastb cap BC neempty With a similar argument to (a) it is nothard to verify that case (ii) results in u1 gt 0 or u2 gt 0 whilecase (i) results in u1 u2 0 erefore if Dlowast is a B2-GMCdesign Dlowastb should be an s-projection of Hqminus 2 BC eremainder of the proof is similar to that of (a)

Remark 1 eorem 2 shows that when constructing theB2-GMC designs with N4 + 1lt nle 5N16 we can firstpartition the range (N4 + 1 5N16) into q minus 4 sequentialsubranges as (N(2wminus 1 + 1)2w+1 N(2wminus 2 + 1)2w) with w

4 5 q minus 1 and then obtain the B2-GMC designsaccording to eorem 2

In the following an example is provided to illustrate theconstruction method in eorem 2 and Remark 1

Example 2 Consider the constructions of B2-GMC219minus 13 2s designs for s 1 3 4 Here n 19 m 13 q 6and N 64 e values of the parameters N and n satisfyN4 + 1lt nle 5N16 ie n isin (17 20] From Remark 1 wepartition the range (17 20] into two subranges (17 18]

because w 5 and (18 20] because w 4 Since19 isin (18 20] according to eorem 2 Dt should be the last19 columns of S5N16 whereS5N16 Dqminus 4(Φ(4)) (5 5H2 6 6H2 56(H4H2)) from(18) erefore Dt (5H2 6 6H2 56(H4H2)) Note that5N16 minus n 20 minus 19 1 then r 0 Next we chooseDb fors 1 3 4

For s 1 we have k 0 According to eorem 2 (a)Db 5 as H0 empty

For s 3 we have k 1 According toeorem 2 (b)Db

should be a 3-projection of H2 ie Db H2 (1 2 12)For s 4 we have k 2 According toeorem 2 (c)Db

should be a 4-projection of H4 Without loss of generalitywe choose Db (1 2 12 3)

4 Concluding Remarks

e regular 2nminus m designs have wide applications in engi-neering manufacturing industry agriculture and medicineand so on When the size of experimental units is large theinhomogeneity of experimental units results in unwantedvariances of estimations of treatment effects An essentialway to solve this problem is to partition the experimentalunits into blocks

ere are two kinds of blocking problems as pointed outin [1] One is called the single block variable problem whichconsiders only a single block variable and the other is calledmulti-block variable problem which considers two or moreblock variables As stated in Section 1 experiments whichinvolve multi-block variables are more widely concerned inpractice than those which involve only a single block var-iable However due to the complexity of multi-block var-iable problem the studies on constructing optimal designswith multi-block variables are relatively rare

In this paper we aim at exploring the theories andconstructions of optimal blocked 2nminus m designs with multi-block variables e prevalent B2-GMC criterion is adoptedis criterion is preferable when there is some priorknowledge on the importance ordering of the treatmenteffects e systematical construction methods of theB2-GMC 2nminus m 2s designs with N4 + 1le nle 5N16 aredevelopede constructionmethods are concise and easy toimplement as indicated by the examples provided

Appendix

Lemma A1 is some result from Lemmas A2 A4 and A5 in[21] and Lemma 1 in [25]

Lemma A1 Suppose O is any s-projection of Hq with2k le sle 2k+1 minus 1 and kle q minus 2

(i) If Ocap q qHqminus 11113966 1113967 empty then U(O)capHqminus 11113966 1113967

ge 2k+1 minus 1 and when O has k + 1 independent col-umns the equality holds

(ii) If Ocap q qHqminus 11113966 1113967neempty then U(O)capHqminus 11113966 1113967

ge 2k minus 1 and when O has k + 1 independent col-umns the equality holds

(iii) If Ocap q qHqminus 11113966 1113967neempty then U(O)cap q qHqminus 11113966 11139671113966 1113967

ge 2k and when O has k + 1 independent columnsthe equality holds

(iv) If O sub Hk+1 then U(O) Hk+1(v) If O sub Hk q qHk1113864 1113865 then U(O) Hk q qHk1113864 1113865

Lemma A2 is some result of Lemma 1 in [34]

Lemma A2 Suppose O consists of the last n columns ofq qHqminus 11113966 1113967 then B2(O γ1)geB2(O γ2) if γ1 is ahead of γ2 inHqminus 1

Data Availability

No data were used to support this study

10 Mathematical Problems in Engineering

Conflicts of Interest

e author declares that there are no conflicts of interest

Acknowledgments

is study was supported by the National Natural ScienceFoundation of China (grant no 11801331)

References

[1] S Bisgaard ldquoA note on the definition of resolution for blocked2 k-p designsrdquo Technometrics vol 36 no 3 pp 308ndash3111994

[2] R R Sitter J Chen and M Feder ldquoFractional resolution andminimum aberration in blocked 2 n-k designsrdquo Techno-metrics vol 39 no 4 pp 382ndash390 1997

[3] H Chen and C S Cheng ldquoeory of optimal blocking ofsnminus mdesignsrdquo Annals of Statistics vol 27 no 6pp 1948ndash1973 1999

[4] R Zhang and D Park ldquoOptimal blocking of two-level frac-tional factorial designsrdquo Journal of Statistical Planning andInference vol 91 no 1 pp 107ndash121 2000

[5] S W Cheng and C F J Wu ldquoChoice of optimal blockingschemes in two-level and three-level designsrdquo Technometricsvol 44 no 3 pp 269ndash277 2002

[6] H Xu ldquoBlocked regular fractional factorial designs withminimum aberrationrdquo Annals of Statistics vol 8 no 5pp 2534ndash2553 2006

[7] H Xu and R W Mee ldquoMinimum aberration blockingschemes for 128-run designsrdquo Journal of Statistical Planningand Inference vol 140 no 11 pp 3213ndash3229 2010

[8] S Zhao P Li and R Karunamuni ldquoBlocked two-level regularfactorial designs with weak minimum aberrationrdquo Bio-metrika vol 100 no 1 pp 249ndash253 2013

[9] S-L Zhao and P-F Li ldquoConstruction of minimum aberrationblocked two-level regular factorial designsrdquo Communicationsin Statistics - eory and Methods vol 45 no 17 pp 5028ndash5036 2016

[10] B-J Chen P-F Li M-Q Liu and R-C Zhang ldquoSome resultson blocked regular 2-level fractional factorial designs withclear effectsrdquo Journal of Statistical Planning and Inferencevol 136 no 12 pp 4436ndash4449 2006

[11] S Zhao P Li and M-Q Liu ldquoOn blocked resolution IVdesigns containing clear two-factor interactionsrdquo Journal ofComplexity vol 29 no 5 pp 389ndash395 2013

[12] R C Zhang and RMukerjee ldquoGeneral minimum lower-orderconfounding in block designs using complementary setsrdquoStatistica Sinica vol 19 no 4 pp 1787ndash1802 2009

[13] S Zhao D K J Lin and P Li ldquoA note on the construction ofblocked two-level designs with general minimum lower orderconfoundingrdquo Journal of Statistical Planning and Inferencevol 172 pp 16ndash22 2016

[14] S-L Zhao and Q-Q Zhao ldquoSome results on constructingtwo-level block designs with general minimum lower orderconfoundingrdquo Communications in Statistics-eory andMethods vol 47 no 9 pp 2227ndash2237 2018

[15] R C Zhang P Li S L Zhao and M Y Ai ldquoA generalminimum lower-order confounding criterion for two-levelregular designsrdquo Statistica Sinica vol 18 no 4 pp 1689ndash1705 2008

[16] S Zhao P Li R Zhang and R Karunamuni ldquoConstruction ofblocked two-level regular designs with general minimum

lower order confoundingrdquo Journal of Statistical Planning andInference vol 143 no 6 pp 1082ndash1090 2013

[17] Y Zhao S Zhao and M-Q Liu ldquoA theory on constructingblocked two-level designs with general minimum lower orderconfoundingrdquo Frontiers of Mathematics in China vol 11no 1 pp 207ndash235 2016

[18] S-L Zhao and Q Sun ldquoOn constructing general minimumlower order confounding two-level block designsrdquo Commu-nications in Statistics-eory and Methods vol 46 no 3pp 1261ndash1274 2017

[19] J Wei P Li and R Zhang ldquoBlocked two-level regular designswith general minimum lower order confoundingrdquo Journal ofStatistical eory and Practice vol 8 no 1 pp 46ndash65 2014

[20] C F J Wu and M S Hamada Experiments PlanningAnalysis and Optimization Wiley Hoboken NJ USA 2edition 2009

[21] Q Zhao and S Zhao ldquoSome results on two-level regulardesigns with multi block variables containing clear effectsrdquoStatistical Papers vol 60 no 5 pp 1569ndash1582 2019

[22] S Zhao and Q Zhao ldquoMinimum aberration blocked designswith multiple block variablesrdquo Metrika vol 84 no 2pp 121ndash140 2021

[23] R Zhang P Li and J Wei ldquoOptimal two-level regular designswith multi block variablesrdquo Journal of Statistical eory andPractice vol 5 no 1 pp 161ndash178 2011

[24] Y N Zhao S L Zhao and M Q Liu ldquoOn constructingoptimal two-level designs with multi block variablesrdquo Journalof Systems Science and Complexity vol 31 pp 1ndash14 2018

[25] Y N Zhao and S L Zhao ldquoConstruction of optimal blockeddesigns with multi block variablesrdquoAIMSmathematics vol 6no 6 pp 6293ndash6308 2021

[26] B Tang and C F J Wu ldquoCharacterization of minimumaberration 2nminus kdesigns in terms of their complementarydesignsrdquo Annals of Statistics vol 25 no 6 pp 2549ndash25591996

[27] C F J Wu and Y Chen ldquoA graph-aided method for planningtwo-level experiments when certain interactions are impor-tantrdquo Technometrics vol 34 no 2 pp 162ndash175 1992

[28] G E P Box and J S Hunter ldquoe 2kminus pfractional factorialdesigns I and IIrdquo Technometrics vol 3 no 4 pp 311ndash3511961

[29] Y Cheng and R Zhang ldquoOn construction of general mini-mum lower order confounding 2nminus m designs with (N4)+

1le nle 9N32rdquo Journal of Statistical Planning and Inferencevol 140 no 9 pp 2384ndash2394 2010

[30] H Wu and C F J Wu ldquoClear two-factor interactions andminimum aberrationrdquo Annals of Statistics vol 30 no 5pp 1496ndash1511 2002

[31] R M Block and R W Mee ldquoSecond order saturated reso-lution IV designsrdquo Journal of Statistical eory and Appli-cations vol 2 no 2 pp 96ndash112 2003

[32] A A Davydov and L M Tombak ldquoQuasi-perfect linear bi-nary codes with distance 4 and complete caps in projectivegeometryrdquo Problems of Information Transmission vol 25no 4 pp 265ndash275 1990

[33] R Zhang and Y Cheng ldquoGeneral minimum lower orderconfounding designs an overview and a construction theoryrdquoJournal of Statistical Planning and Inference vol 140 no 7pp 1719ndash1730 2010

[34] Q Zhou N Balakrishnan and R Zhang ldquoe factor aliasedeffect number pattern and its application in experimentalplanningrdquo Canadian Journal of Statistics vol 41 no 3pp 540ndash555 2013

Mathematical Problems in Engineering 11

by the definition of B2-GMC criterion there should be a2nminus m 2s design Dlowast (Dlowastt Dlowastb ) outperforming1113957D (SN4+1

1113957Db) in terms of (4) is leads to

1 C2 Dlowast( 1113857

N

4+ 1 0 01113874 1113875 (12)

and 2 C

(0)

2 (Dlowast)ge (N2) minus 1 Clearly

1 C2 Dlowastt( 1113857

1 C2 Dlowast( 1113857

N

4+ 1 0 01113874 1113875 (13)

and 2 C

(0)

2 (Dlowastt )ge 2 C(0)

2 (Dlowast)geN2 minus 1 noting that Dlowastt is theunblocked part of Dlowast and Dlowastt has n N4 + 1 columns

In fact the inequality in the formula 2 C(0)

2 (Dlowastt )geN2 minus

1 is not valid Recall that the MaxC2 design SN4+1 has thelargest number N2 minus 1 of clear 2firsquos among all the 2nminus m

designs with n N4 + 1 ereforeDlowastt has at most N2 minus 1clear 2firsquos ie

2 C(0)

2 (Dlowastt ) N2 minus 1 Wu and Wu [30]showed that a 2nminus m design with n N4 + 1 is a MaxC2design if and only if this design has N2 minus 1 clear 2firsquos isobtains that Dlowastt SN4+1 up to isomorphism

With Lemma 2eorem 1 provides the constructions ofB2-GMC designs with n N4 + 1

Theorem 1 SupposeD (Dt Db) is a 2nminus m 2s design withn N4 + 1 and 2k le sle 2k+1 minus 1 for some k(0le kle q minus 3)then D (Dt Db) is a B2-GMC design if Dt SN4+1 andDb is any s-projection of Hk+1

Proof By Lemma 2 if D (Dt Db) is a B2-GMC 2nminus m 2s

design with n N4 + 1 then Dt SN4+1 up to isomor-phism us

1 C2(D) (N4 + 1 0 0) which is se-quentially maximized by D (SN4+1 Db) Let u1

U(Db)cap (q minus 1)Hqminus 2 qHqminus 2 q(q minus 1)1113966 11139671113966 1113967 and u2 U(Db)capHqminus 21113966 1113967 then from (10) we have

2 C

(p)

2 (D)

2N

4minus 11113874 1113875 + 1 minus u1 forp 0

N

4minus 1 minus u21113874 1113875

N

8minus 11113874 1113875 forp

N

8minus 2

0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)

If D is a B2-GMC design then D must sequentiallyminimize (u1 u2) ere are two different ways to chooseDb

from HqSN4+1

(i) Db cap (q minus 1)Hqminus 2 qHqminus 2 q(q minus 1)1113966 1113967 empty(ii) Db cap (q minus 1)Hqminus 2 qHqminus 2 q(q minus 1)1113966 1113967neempty

It is not hard to verify thatDb in case (ii) results in u1 gt 0while Db in case (i) gives u1 0 erefore ifD (SN4+1 Db) is a B2-GMC design then Db must be ofcase (i) Recall thatDb cap SN4+1 empty and we haveDb sub Hqminus 2By Lemma 1 (i) if Db sub Hqminus 2 then u2 U(Db)capHqminus 21113966 1113967ge 2k+1 minus 1 and the equality holds when

Db sub Hk+1 up to isomorphism is completes theproof

e following example illustrates the constructionmethod in eorem 1

Example 1 Consider the construction of theB2-GMC29minus 4 25design Here n 9 m 4 q 5 N 32 and s 5 whichleads to k 2 According to eorem 1 let Dt (4 5 45H3)

and Db be any 5-projection of H3 say Db (1 2 12 3 13)en D (Dt Db) is a B2-GMC 29minus 4 25 design

32 B2-GMC Designs with N4 + 1lt nle 5N16 To se-quentially maximize (4) the first part 1 C2(D) of (4) shouldbe first maximized Recall that 1 C2(D)

1 C2(Dt) If Dt hasresolution IV then

1 C2(D) must be maximized Accordingto [31] when N4 + 1lt nle 5N16 theDt with resolution IVmust be an n-projection of some second-order saturated(SOS) designs In the following we first review the conceptof SOS design

A 2nminus m design is called an SOS design if all of its degrees offreedom can be used to estimate only themain treatment effectsand 2firsquos In terms of coding theory and projective geometryDavydov and Tombak [32] showed that given N only the SOSdesigns of N4 + 1 N(2wminus 2 + 1)2w with wge 4 and N2factors exist Block and Mee [31] further showed that an SOSdesign of N(2wminus 2 + 1)2w factors can be obtained by doublingsome smaller SOS 2(2wminus 2+1)minus (2wminus 2+1minus w) designs q minus w timesZhang and Cheng [33] showed that the SOS design of N2factors can be uniquely represented by SN2 (q qHqminus 1) up toisomorphism Let

L(w) Dqminus w

(Y) Y is an SOS 2 2wminus 2+1( )minus 2wminus 2+1minus w( ) design1113882 1113883

(15)

denote the collection of all the SOS designs obtained bydoubling some SOS 2(2wminus 2+1)minus (2wminus 2+1minus w) designs q minus w timesEspecially in L(w) we denote the SOS design obtained bydoubling the MaxC2 2(2wminus 2+1)minus (2wminus 2+1minus w) designΦ(w) ((w minus 1)2w w2w w2w (w minus 1)2wHw

wminus 2) as SN(2wminus 2+1)2w With a little algebra it is easy to verify that

Dqminus w I2w( 1113857 I2q Hq

qminus w1113872 1113873 IHqminus w1113872 1113873

Dqminus w 12w( 1113857 (q minus w + 1)2q I2q Hq

qminus w1113872 1113873

(q minus w + 1) IHqminus w1113872 1113873

Dqminus w 22w( 1113857 (q minus w + 2)2q I2q Hq

qminus w1113872 1113873

(q minus w + 2) IHqminus w1113872 1113873

⋮

Dqminus w

(w minus 1)2w( 1113857 (q minus 1)2q I2q Hqqminus w1113872 1113873 (q minus 1) IHqminus w1113872 1113873

Dqminus w w2w( 1113857 q2q I2q Hq

qminus w1113872 1113873 q IHqminus w1113872 1113873

(16)

where I2qminus w otimes 12w (q minus w + 1)2q I2qminus w otimes 22w (q minus w+ 2)2q

I2qminus w otimes (w minus 1)2w (q minus 1)2q and I2qminus w otimesw2w q2q

en

4 Mathematical Problems in Engineering

Dqminus w

(w minus 1)2ww2wHwwminus 2( 1113857 q(q minus 1) Hqminus 2Hqminus w1113872 1113873

(17)

erefore SN(2wminus 2+1)2w can be expressed as

SN 2wminus 2+1( )2w Dqminus w

(Φ(w))

(q minus 1) (q minus 1)Hqminus w q qHqminus w q(q minus 1)1113872

Hqminus 2Hqminus w1113872 11138731113873

(18)

e following lemma provides a necessary condition fora 2nminus m 2s design D (Dt Db) with N4 + 1lt nle 5N16 tobe a B2-GMC design

Lemma 3 Suppose D (Dt Db) is a 2nminus m 2s design withN(2wminus 1 + 1)2w+1 lt nleN(2wminus 2 + 1)2w for wge 4 and2k le sle 2k+1 minus 1 for some k(0le kle q minus 3) thenD (Dt Db) is a B2-GMC design only if Dt sub SN(2wminus 2+1)2w

Proof As discussed in the first paragraph of this section ifD (Dt Db) is a B2-GMC 2nminus m 2s design withN(2wminus 1 + 1)2w+1 lt nleN(2wminus 2 + 1)2w then Dt must be ann-projection of some SOS designs in L(v) with 4le vlew orSN2

Let P1 be an n-projection of SN2 According toeorem3 in [33] we obtain

2 C

(p)

2 P1( 1113857

0 forplt n minusN

4minus 1

N

4n minus

N

41113874 1113875 forp n minus

N

4minus 1

0 for n minusN

4minus 1ltplt

N

8minus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(19)

Let Xv (a1 a2 a2vminus 2+1) be any SOS2(2vminus 2+1)minus (2vminus 2+1minus v) design but not a MaxC2 design thenDqminus v(Xv) isinL(v) Denote pi as the number of clear 2firsquos ofXv involving ai for i 1 2 2vminus 2 + 1 and c as the totalnumber of clear 2firsquos of Xv Let Pv be an n-projection ofDqminus v(Xv) and Pv Dqminus v(Xv)Pv en Pv N

(2vminus 2 + 1)2v minus nleN2v Applying equations (23) and (24)in the proof of eorem 31 of [29] among all the n-pro-jections of Dqminus v(Xv) Pv sequentially maximizes 2 C2 in (5)only if Pv sub Dqminus v(aj) where aj is the column such thatpj max p1 p2 p2vminus 2+11113864 1113865 For a Pv with Pv sub Dqminus v(aj)according to equation (24) in the proof of eorem 31 of[29] it is obtained that

2 C

(p)

2 Pv( 1113857

0 for 0leplt n minusN

4minus 1

pjN

2v (n minus N4) forp n minusN

4minus 1

0 for n minusN

4minus 1ltplt

N

2v minus 1

c minus pj1113872 1113873N

2v1113874 1113875

2 forp

N

2v minus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)

For a MaxC22(2vminus 2+1)minus (2vminus 2+1minus v) design it has 2vminus 1 minus 1clear 2firsquos and pj 2vminus 2 Since Xv is not a MaxC2 design wehave

pj 2vminus 2

c minus pj lt 2vminus 1

minus 2vminus 2minus 1

(21)

or

pj lt 2vminus 2

clt 2vminus 1minus 1

(22)

Let Qv be an n-projection of SN(2vminus 2+1)2v obtained bydoubling Φ(v) the MaxC22(2vminus 2+1)minus (2vminus 2+1minus v) design asmentioned above With a column permutation rewriteDq(middot) as

Dq

RC(middot) Hqminus vHqminus 2Hqminus v (q minus 1) Hqminus 2Hqminus v1113872 11138731113872

q Hqminus 2Hqminus v1113872 1113873 q(q minus 1) IHqminus v1113872 1113873 SN 2vminus 2+1( )2v 1113873

(23)

in a re-changed Yates order Write Φ(v) asΦ(v) (s1 s2 s2vminus 2+1) where s1 (v minus 1)2v s2 v2v and(s3 s2vminus 2+1) v2v (v minus 1)2vHv

vminus 2 Applying equations (23)and (24) in the proof of eorem 31 of [29] among all then-projections of SN(2vminus 2+1)2v Qv sequentially maximizes 2 C2in (3) only ifQv sub Dqminus v(si) with i 1 or 2 In the followingwe investigate 2 C

(p)

2 (Qv) with Qv sub Dqminus v(s1) for which thefollowing analysis and final conclusion are the same as that

for 2 C

(p)

2 (Qv) with Qv sub Dqminus v(s2) For Qv withQv sub Dqminus v(s1) from equations (18) and (23) it is obtainedthat

Mathematical Problems in Engineering 5

B2 Qv γ( 1113857

0 for γ isin SN 2vminus 2+1( )2v

n minusN

4 for γ isin q Hqminus 2Hqminus v1113872 1113873 q(q minus 1) IHqminus v1113872 11138731113966 1113967

N

2v for γ isin (q minus 1) Hqminus 2Hqminus v1113872 1113873

N

8minus

N

2v for γ isin Hqminus 2Hqminus v

B2 Qv γ( 1113857 +N

8 for γ isin Hqminus v

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

where the second equality can be easily verified by notingthat

q Hqminus 2Hqminus v1113872 1113873 q(q minus 1) IHqminus v1113872 11138731113966 1113967

Dqminus v s1s2( 1113857 D

qminus v s1s3( 1113857 Dqminus v s1s2vminus 2+1( 11138571113864 1113865

(25)

erefore

2 C

(p)

2 Qv( 1113857

0 for 0leplt n minusN

4minus 1

N

4(n minus N4) forp n minus

N

4minus 1

0 for n minusN

4minus 1ltplt

N

2v minus 1

N

2v N4 minus N2v( 1113857 forp

N

2v minus 1

0 forN

2v minus 1ltpltN

8minus

N

2v minus 1

N

4minus

N

2v1113874 1113875N

8minus

N

2v1113874 1113875 forp N

8minus

N

2v minus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)

Denote Dlowast (Dlowastt Dlowastb ) where Dlowastt sub SN(2wminus 2+1)2w SN(2wminus 2+1)2w Dlowastt sub Dqminus w(s1) and Dlowastb sub Hk+1 Since kle q minus 3

we have U(Dlowastb ) sub Hqminus 2 erefore from (24) and (26) weobtain

2 C

(p)

2 Dlowast( 1113857

0 for 0leplt n minusN

4minus 1

N

4(n minus N4) forp n minus

N

4minus 1

0 for n minusN

4minus 1ltplt

N

2w minus 1

N

2w N4 minus N2w( 1113857 forp

N

2w minus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

6 Mathematical Problems in Engineering

Comparing (27) with (19) (20) for 4le vlew and (26) for4le vlew minus 1 it is obtained that ifD (Dt Db) is a B2-GMCdesign D should not be worse than Dlowast in terms of (3)erefore if D (Dt Db) is a B2-GMC design Dt shouldbe an n-projection of SN(2wminus 2+1)2w is completes theproof

e lemma below is an extension of Lemma A2 inAppendix introduced from [34]

Lemma 4 Suppose Dt consists of the last nN(2wminus 1 + 1)2w+1 lt nleN(2wminus 2 + 1)2w columns ofSN(2wminus 2+1)2w then B2(Dt γ1)geB2(Dt γ2) if γ1 is ahead of γ2in Hqminus w

Proof Suppose O consists of the last n minus N4 columns ofDqminus w((w minus 1)2w ) en Dt O SN(2wminus 2+1)2w Dqminus w1113966

(w minus 1) From (16) and (18) for any γ isin Hqminus wB2(Dt γ) B2(O γ) + N8 Straightforwardly fromLemma A2 it is obtained that B2(O γ1)geB2(O γ2) if γ1 isahead of γ2 in Hqminus w is completes the proof

With Lemmas 3 and 4 eorem 2 provides the con-struction methods of B2-GMC 2nminus m 2s designs withN4 + 1lt nle 5N16

Theorem 2 Let D (Dt Db) be a 2nminus m 2s design withN(2wminus 1 + 1)2w+1 lt nleN(2wminus 2 + 1)2w for wge 4 and2k le sle 2k+1 minus 1 for some 0le kle q minus 3 Suppose 2r leN

(2wminus 2 + 1)2w minus nle 2r+1 minus 1 for some 0le rle q minus w minus 1 thenD is a B2-GMC design if Dt consists of the last n columns ofSN(2wminus 2+1)2w and

(a) Db is any s-projection of Hk q minus 1 (q minus 1)Hk1113864 1113865 whenkle rle q minus w minus 1

(b) Db is any s-projection of Hk+1 whenr + 1le kle q minus w minus 1

(c) Db is any s-projection of Hqminus 2 whenq minus w minus 1lt kle q minus 3

Proof By Lemma 3 and its proof if Dlowast (Dlowastt Dlowastb ) is aB2-GMC 2nminus m 2s design there should be Dlowastt sub SN(2wminus 2+1)2w

and Dlowastt SN(2wminus 2+1)2w Dlowastt sub Dqminus w((w minus 1)2w ) Substitutingv with w in (24) we obtain

B2 Dlowastt γ( 1113857

0 for γ isin S N 2wminus 2+1( )2w( )

n minusN

4 for γ isin q Hqminus 2Hqminus w1113872 1113873 q(q minus 1) IHqminus w1113872 11138731113966 1113967

N

2w for γ isin (q minus 1) Hqminus 2Hqminus w1113872 1113873

N

8minus

N

2w for γ isin Hqminus 2Hqminus w

B2 Dlowastt γ) +

N

8 for γ isin Hqminus w1113874

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)

Let

u1 U Dlowastb( 1113857cap q Hqminus 2Hqminus w1113872 1113873 q(q minus 1) IHqminus w1113872 11138731113966 11139671113966 1113967

u2 U Dlowastb( 1113857cap (q minus 1) Hqminus 2Hqminus w1113872 11138731113966 1113967

u3 U Dlowastb( 1113857cap Hqminus 2Hqminus w1113872 11138731113966 1113967

(29)

and then

Mathematical Problems in Engineering 7

2 C

(p)

2 Dlowast( 1113857

n minusN

41113874 1113875

N

4minus u11113874 1113875 forp n minus

N

4minus 1

N

2w

N

4minus

N

2w minus u21113874 1113875 forp N

2w minus 1

N

8minus

N

2w1113874 1113875N

4minus

N

2w minus u31113874 1113875 forp N

8minus

N

2w minus 1

f Dlowastt p( 1113857 forpgeN

8minus 1

0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(30)

where f(Dlowastt p) 2 C

(p)

2 (Dlowastt ) minus (p + 1) γ γ isin Hqminus1113966

wcapU(Dlowastb ) B2(Dlowastt γ) p + 1 erefore Dlowast sequentiallymaximizes 2 C2 among all the possible 2nminus m 2s designs onlyif Dlowast sequentially minimizes

u1 u2 u3( 1113857 (31)

For ease of presenting let A q(Hqminus 2Hqminus w)1113966

q(q minus 1)(IHqminus w) B (q minus 1)(Hqminus 2Hqminus w) and C

Hqminus 2Hqminus wFor (a) when kle q minus w minus 1 there are two different ways

to choose Dlowastb from Hq (i) Dlowastb cap ABC empty and (ii)Dlowastb cap ABC neempty Recall that if Dlowast (Dlowastt Dlowastb ) is aB2-GMC design then Dlowastt sub SN(2wminus 2+1)2w with D

lowastt

SN(2wminus 2+1)2w Dlowastt sub Dqminus w((w minus 1)2w ) Case (i) implies thatDlowastb sub Hqminus w D

lowastt1113966 1113967 and thus Dlowastb sub Hqminus w q minus 1 (q minus 1)1113966

Hqminus w erefore choosing Dlowastb according to case (i) resultsin u1 u2 u3 0 If Dlowastb is chosen according to case (ii)then U(Dlowastb )cap ABC neempty which results in u1 gt 0 u2 gt 0 oru3 gt 0 Clearly if Dlowast is a B2-GMC design Dlowastb should be

chosen as (i) In the following we consider onlyDlowastb cap ABC empty ie Dlowastb sub Hqminus w (q minus 1) (q minus 1)Hqminus w1113966 1113967

For D (Dt Db) in (a) according to (v) in Lemma 1 ifDb sub Hk (q minus 1) (q minus 1)Hk1113864 1113865 with kle q minus w minus 1 thenU(Db) Hk (q minus 1) (q minus 1)Hk1113864 1113865 Denote b as the lastcolumn of Hk in Yates order and B2(Dt b) b thenbgeN8 According to Lemma 4 for γ isin Hqminus w if γ is ahead ofb in Hqminus w then B2(Dt γ)ge b by (18) For anyγ isin (q minus 1) (q minus 1)Hk1113864 1113865 we have B2(Dt γ) 0 For any γsuch that B2(Dt γ)lt b we have γ isin HqHk erefore forple b minus 2

2 C

(p)

2 (D) (p + 1) γ isin Hq γ notin U Db( 1113857 B2 Dt γ( 1113857 p + 11113966 1113967

(p + 1) γ isin Hq B2 Dt γ( 1113857 p + 11113966 1113967

2 C

(p)

2 Dt( 1113857

(32)

and for pge b

2 C

(p)

2 (D) (p + 1) γ isin Hq γ notin U Db( 1113857 B2 Dt γ( 1113857 p + 11113966 1113967

(p + 1) γ isin Hq B2 Dt γ( 1113857 p + 11113966 1113967 minus (p + 1) γ isin Hk B2 Dt γ( 1113857 p + 11113864 1113865

0

(33)

From (33) it is obtained that

2 C

(0)

2 (D) 2 C

(N8minus 1)

2 (D)2 C

(N8)

2 (D) 2 C

(bminus 2)

2 (D)1113874 1113875

2 C

(0)

2 Dt( 1113857 2 C

(N8minus 1)

2 Dt( 11138572 C

(N8)

2 Dt( 1113857 2 C

(bminus 2)

2 Dt( 11138571113874 1113875

(34)

Note that Dt is a GMC design [29] then D maximizes

2 C

(0)

2 (middot) 2 C

(N8minus 1)

2 (middot)2 C

(N8)

2 (middot) 2 C

(bminus 2)

2 (middot)1113874 1113875

(35)

among all the possible 2nminus m 2s designsSupposeD is not aB2-GMC design then there should be

aDlowast (Dlowastt Dlowastb ) which outperformsD (Dt Db) in termsof (3)is implies that there exists some p1 ge b minus 1 such that

8 Mathematical Problems in Engineering

2 C

(0)

2 Dlowast( 1113857 2 C

(N8minus 1)

2 Dlowast( 11138572 C

(N8)

2 Dlowast( 1113857 2 C

(bminus 2)

2 Dlowast( 11138571113874 1113875

2 C

(0)

2 (D) 2 C

(N8minus 1)

2 (D)2 C

(N8)

2 (D) 2 C

(bminus 2)

2 (D)1113874 1113875

(36)

2 C

p1minus 1( )2 Dlowast( 1113857gt 2 C

p1minus 1( )2 (D) (37)

Recalling the definitions of 2 C(p)

2 (D) and B2(Dt γ) wehave

1113944

P

p0

2 C

(p)

2 (D)

(p + 1) 1113944

P

p0 γ isin Hq γ notin U Db( 1113857 B2 Dt γ( 1113857 p + 11113966 1113967

1113944P

p0 γ isin HqSN 2wminus 2+1( )2w1113872 1113873 B2 Dt γ( 1113857 p + 11113966 1113967

minus 1113944P

p0 γ isin U Db( 1113857 B2 Dt γ( 1113857 p + 11113864 1113865

HqSN 2wminus 2+1( )2w1113872 1113873 minus Hk

2qminus

N 2wminus 2+ 11113872 1113873

2w minus 2k

(38)

where the second and third equalities are due to B2(Dt γ)

0 for any γ isin SN(2wminus 2+1)2w Similarly

1113944

P

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1) 1113944

P

p0 γ isin Hq γ notin U Dlowastb( 1113857 B2 Dlowastt γ( 1113857 p + 11113966 1113967

1113944P

p0 γ isin HqSN 2wminus 2+1( )2w1113872 1113873 B2 Dlowastt γ( 1113857 p + 11113966 1113967

minus 1113944P

p0 γ isin U Dlowastb( 1113857 B2 Dlowastt γ( 1113857 p + 11113864 1113865

HqSN 2wminus 2+1( )2w1113872 1113873 minus U Dlowastb( 1113857

2qminus

N 2wminus 2+ 11113872 1113873

2w minus U Dlowastb( 1113857

(39)

From (33) and (38) we have 1113944

P

p0

2 C

(p)

2 (D)

(p + 1) 1113944

bminus 2

p0

2 C

(p)

2 (D)

(p + 1)+

2 C

(bminus 1)

2 (D)

b

2qminus

N 2wminus 2+ 11113872 1113873

2w minus 2k

(40)

By (36) and (37) it is obtained that

Mathematical Problems in Engineering 9

1113944

P

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1)ge 1113944

bminus 2

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1)+

2 C

(bminus 1)

2 Dlowast( 1113857

b

gt 1113944bminus 2

p0

2 C

(p)

2 (D)

(p + 1)+

2 C

(bminus 1)

2 (D)

b

(41)

en according to (39)ndash(41) we obtain U(Dlowastb )lt 2k minus 1which contradicts Lemma 1 (i) and (ii)

For (b) similar to (a) when kle q minus w minus 1 there are twodifferent ways to choose Dlowastb from Hq (i) Dlowastb cap ABC emptyand (ii) Dlowastb cap ABC neempty With a similar argument to (a)case (i) results in u1 u2 u3 0 while case (ii) results inu1 gt 0 u2 gt 0 or u3 gt 0 erefore if Dlowast is a B2-GMC designDlowastb should be chosen according to case (i) ieDlowastb sub Hqminus w (q minus 1) (q minus 1)Hqminus w1113966 1113967 When choosing Dlowastbaccording to (i) there should be Dlowastb cap q minus 11113864

(q minus 1)Hqminus w empty ie Dlowastb sub Hqminus w otherwise Dlowastt capU

(Dlowastb )neempty noting that U(Dlowastb )cap (q minus 1 (q minus 1)Hqminus w)1113966 1113967

ge 2k gt 2r+1 minus 1 as shown in Lemma 1 (iii) e remainder ofthe proof is similar to that of (a)

For (c) when kgt q minus w minus 1 there are two different waysto choose Dlowastb from Hq (i) Dlowastb cap BC empty and (ii)Dlowastb cap BC neempty With a similar argument to (a) it is nothard to verify that case (ii) results in u1 gt 0 or u2 gt 0 whilecase (i) results in u1 u2 0 erefore if Dlowast is a B2-GMCdesign Dlowastb should be an s-projection of Hqminus 2 BC eremainder of the proof is similar to that of (a)

Remark 1 eorem 2 shows that when constructing theB2-GMC designs with N4 + 1lt nle 5N16 we can firstpartition the range (N4 + 1 5N16) into q minus 4 sequentialsubranges as (N(2wminus 1 + 1)2w+1 N(2wminus 2 + 1)2w) with w

4 5 q minus 1 and then obtain the B2-GMC designsaccording to eorem 2

In the following an example is provided to illustrate theconstruction method in eorem 2 and Remark 1

Example 2 Consider the constructions of B2-GMC219minus 13 2s designs for s 1 3 4 Here n 19 m 13 q 6and N 64 e values of the parameters N and n satisfyN4 + 1lt nle 5N16 ie n isin (17 20] From Remark 1 wepartition the range (17 20] into two subranges (17 18]

because w 5 and (18 20] because w 4 Since19 isin (18 20] according to eorem 2 Dt should be the last19 columns of S5N16 whereS5N16 Dqminus 4(Φ(4)) (5 5H2 6 6H2 56(H4H2)) from(18) erefore Dt (5H2 6 6H2 56(H4H2)) Note that5N16 minus n 20 minus 19 1 then r 0 Next we chooseDb fors 1 3 4

For s 1 we have k 0 According to eorem 2 (a)Db 5 as H0 empty

For s 3 we have k 1 According toeorem 2 (b)Db

should be a 3-projection of H2 ie Db H2 (1 2 12)For s 4 we have k 2 According toeorem 2 (c)Db

should be a 4-projection of H4 Without loss of generalitywe choose Db (1 2 12 3)

4 Concluding Remarks

e regular 2nminus m designs have wide applications in engi-neering manufacturing industry agriculture and medicineand so on When the size of experimental units is large theinhomogeneity of experimental units results in unwantedvariances of estimations of treatment effects An essentialway to solve this problem is to partition the experimentalunits into blocks

ere are two kinds of blocking problems as pointed outin [1] One is called the single block variable problem whichconsiders only a single block variable and the other is calledmulti-block variable problem which considers two or moreblock variables As stated in Section 1 experiments whichinvolve multi-block variables are more widely concerned inpractice than those which involve only a single block var-iable However due to the complexity of multi-block var-iable problem the studies on constructing optimal designswith multi-block variables are relatively rare

In this paper we aim at exploring the theories andconstructions of optimal blocked 2nminus m designs with multi-block variables e prevalent B2-GMC criterion is adoptedis criterion is preferable when there is some priorknowledge on the importance ordering of the treatmenteffects e systematical construction methods of theB2-GMC 2nminus m 2s designs with N4 + 1le nle 5N16 aredevelopede constructionmethods are concise and easy toimplement as indicated by the examples provided

Appendix

Lemma A1 is some result from Lemmas A2 A4 and A5 in[21] and Lemma 1 in [25]

Lemma A1 Suppose O is any s-projection of Hq with2k le sle 2k+1 minus 1 and kle q minus 2

(i) If Ocap q qHqminus 11113966 1113967 empty then U(O)capHqminus 11113966 1113967

ge 2k+1 minus 1 and when O has k + 1 independent col-umns the equality holds

(ii) If Ocap q qHqminus 11113966 1113967neempty then U(O)capHqminus 11113966 1113967

ge 2k minus 1 and when O has k + 1 independent col-umns the equality holds

(iii) If Ocap q qHqminus 11113966 1113967neempty then U(O)cap q qHqminus 11113966 11139671113966 1113967

ge 2k and when O has k + 1 independent columnsthe equality holds

(iv) If O sub Hk+1 then U(O) Hk+1(v) If O sub Hk q qHk1113864 1113865 then U(O) Hk q qHk1113864 1113865

Lemma A2 is some result of Lemma 1 in [34]

Lemma A2 Suppose O consists of the last n columns ofq qHqminus 11113966 1113967 then B2(O γ1)geB2(O γ2) if γ1 is ahead of γ2 inHqminus 1

Data Availability

No data were used to support this study

10 Mathematical Problems in Engineering

Conflicts of Interest

e author declares that there are no conflicts of interest

Acknowledgments

is study was supported by the National Natural ScienceFoundation of China (grant no 11801331)

References

[1] S Bisgaard ldquoA note on the definition of resolution for blocked2 k-p designsrdquo Technometrics vol 36 no 3 pp 308ndash3111994

[2] R R Sitter J Chen and M Feder ldquoFractional resolution andminimum aberration in blocked 2 n-k designsrdquo Techno-metrics vol 39 no 4 pp 382ndash390 1997

[3] H Chen and C S Cheng ldquoeory of optimal blocking ofsnminus mdesignsrdquo Annals of Statistics vol 27 no 6pp 1948ndash1973 1999

[4] R Zhang and D Park ldquoOptimal blocking of two-level frac-tional factorial designsrdquo Journal of Statistical Planning andInference vol 91 no 1 pp 107ndash121 2000

[5] S W Cheng and C F J Wu ldquoChoice of optimal blockingschemes in two-level and three-level designsrdquo Technometricsvol 44 no 3 pp 269ndash277 2002

[6] H Xu ldquoBlocked regular fractional factorial designs withminimum aberrationrdquo Annals of Statistics vol 8 no 5pp 2534ndash2553 2006

[7] H Xu and R W Mee ldquoMinimum aberration blockingschemes for 128-run designsrdquo Journal of Statistical Planningand Inference vol 140 no 11 pp 3213ndash3229 2010

[8] S Zhao P Li and R Karunamuni ldquoBlocked two-level regularfactorial designs with weak minimum aberrationrdquo Bio-metrika vol 100 no 1 pp 249ndash253 2013

[9] S-L Zhao and P-F Li ldquoConstruction of minimum aberrationblocked two-level regular factorial designsrdquo Communicationsin Statistics - eory and Methods vol 45 no 17 pp 5028ndash5036 2016

[10] B-J Chen P-F Li M-Q Liu and R-C Zhang ldquoSome resultson blocked regular 2-level fractional factorial designs withclear effectsrdquo Journal of Statistical Planning and Inferencevol 136 no 12 pp 4436ndash4449 2006

[11] S Zhao P Li and M-Q Liu ldquoOn blocked resolution IVdesigns containing clear two-factor interactionsrdquo Journal ofComplexity vol 29 no 5 pp 389ndash395 2013

[12] R C Zhang and RMukerjee ldquoGeneral minimum lower-orderconfounding in block designs using complementary setsrdquoStatistica Sinica vol 19 no 4 pp 1787ndash1802 2009

[13] S Zhao D K J Lin and P Li ldquoA note on the construction ofblocked two-level designs with general minimum lower orderconfoundingrdquo Journal of Statistical Planning and Inferencevol 172 pp 16ndash22 2016

[14] S-L Zhao and Q-Q Zhao ldquoSome results on constructingtwo-level block designs with general minimum lower orderconfoundingrdquo Communications in Statistics-eory andMethods vol 47 no 9 pp 2227ndash2237 2018

[15] R C Zhang P Li S L Zhao and M Y Ai ldquoA generalminimum lower-order confounding criterion for two-levelregular designsrdquo Statistica Sinica vol 18 no 4 pp 1689ndash1705 2008

[16] S Zhao P Li R Zhang and R Karunamuni ldquoConstruction ofblocked two-level regular designs with general minimum

lower order confoundingrdquo Journal of Statistical Planning andInference vol 143 no 6 pp 1082ndash1090 2013

[17] Y Zhao S Zhao and M-Q Liu ldquoA theory on constructingblocked two-level designs with general minimum lower orderconfoundingrdquo Frontiers of Mathematics in China vol 11no 1 pp 207ndash235 2016

[18] S-L Zhao and Q Sun ldquoOn constructing general minimumlower order confounding two-level block designsrdquo Commu-nications in Statistics-eory and Methods vol 46 no 3pp 1261ndash1274 2017

[19] J Wei P Li and R Zhang ldquoBlocked two-level regular designswith general minimum lower order confoundingrdquo Journal ofStatistical eory and Practice vol 8 no 1 pp 46ndash65 2014

[20] C F J Wu and M S Hamada Experiments PlanningAnalysis and Optimization Wiley Hoboken NJ USA 2edition 2009

[21] Q Zhao and S Zhao ldquoSome results on two-level regulardesigns with multi block variables containing clear effectsrdquoStatistical Papers vol 60 no 5 pp 1569ndash1582 2019

[22] S Zhao and Q Zhao ldquoMinimum aberration blocked designswith multiple block variablesrdquo Metrika vol 84 no 2pp 121ndash140 2021

[23] R Zhang P Li and J Wei ldquoOptimal two-level regular designswith multi block variablesrdquo Journal of Statistical eory andPractice vol 5 no 1 pp 161ndash178 2011

[24] Y N Zhao S L Zhao and M Q Liu ldquoOn constructingoptimal two-level designs with multi block variablesrdquo Journalof Systems Science and Complexity vol 31 pp 1ndash14 2018

[25] Y N Zhao and S L Zhao ldquoConstruction of optimal blockeddesigns with multi block variablesrdquoAIMSmathematics vol 6no 6 pp 6293ndash6308 2021

[26] B Tang and C F J Wu ldquoCharacterization of minimumaberration 2nminus kdesigns in terms of their complementarydesignsrdquo Annals of Statistics vol 25 no 6 pp 2549ndash25591996

[27] C F J Wu and Y Chen ldquoA graph-aided method for planningtwo-level experiments when certain interactions are impor-tantrdquo Technometrics vol 34 no 2 pp 162ndash175 1992

[28] G E P Box and J S Hunter ldquoe 2kminus pfractional factorialdesigns I and IIrdquo Technometrics vol 3 no 4 pp 311ndash3511961

[29] Y Cheng and R Zhang ldquoOn construction of general mini-mum lower order confounding 2nminus m designs with (N4)+

1le nle 9N32rdquo Journal of Statistical Planning and Inferencevol 140 no 9 pp 2384ndash2394 2010

[30] H Wu and C F J Wu ldquoClear two-factor interactions andminimum aberrationrdquo Annals of Statistics vol 30 no 5pp 1496ndash1511 2002

[31] R M Block and R W Mee ldquoSecond order saturated reso-lution IV designsrdquo Journal of Statistical eory and Appli-cations vol 2 no 2 pp 96ndash112 2003

[32] A A Davydov and L M Tombak ldquoQuasi-perfect linear bi-nary codes with distance 4 and complete caps in projectivegeometryrdquo Problems of Information Transmission vol 25no 4 pp 265ndash275 1990

[33] R Zhang and Y Cheng ldquoGeneral minimum lower orderconfounding designs an overview and a construction theoryrdquoJournal of Statistical Planning and Inference vol 140 no 7pp 1719ndash1730 2010

[34] Q Zhou N Balakrishnan and R Zhang ldquoe factor aliasedeffect number pattern and its application in experimentalplanningrdquo Canadian Journal of Statistics vol 41 no 3pp 540ndash555 2013

Mathematical Problems in Engineering 11

Dqminus w

(w minus 1)2ww2wHwwminus 2( 1113857 q(q minus 1) Hqminus 2Hqminus w1113872 1113873

(17)

erefore SN(2wminus 2+1)2w can be expressed as

SN 2wminus 2+1( )2w Dqminus w

(Φ(w))

(q minus 1) (q minus 1)Hqminus w q qHqminus w q(q minus 1)1113872

Hqminus 2Hqminus w1113872 11138731113873

(18)

e following lemma provides a necessary condition fora 2nminus m 2s design D (Dt Db) with N4 + 1lt nle 5N16 tobe a B2-GMC design

Lemma 3 Suppose D (Dt Db) is a 2nminus m 2s design withN(2wminus 1 + 1)2w+1 lt nleN(2wminus 2 + 1)2w for wge 4 and2k le sle 2k+1 minus 1 for some k(0le kle q minus 3) thenD (Dt Db) is a B2-GMC design only if Dt sub SN(2wminus 2+1)2w

Proof As discussed in the first paragraph of this section ifD (Dt Db) is a B2-GMC 2nminus m 2s design withN(2wminus 1 + 1)2w+1 lt nleN(2wminus 2 + 1)2w then Dt must be ann-projection of some SOS designs in L(v) with 4le vlew orSN2

Let P1 be an n-projection of SN2 According toeorem3 in [33] we obtain

2 C

(p)

2 P1( 1113857

0 forplt n minusN

4minus 1

N

4n minus

N

41113874 1113875 forp n minus

N

4minus 1

0 for n minusN

4minus 1ltplt

N

8minus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(19)

Let Xv (a1 a2 a2vminus 2+1) be any SOS2(2vminus 2+1)minus (2vminus 2+1minus v) design but not a MaxC2 design thenDqminus v(Xv) isinL(v) Denote pi as the number of clear 2firsquos ofXv involving ai for i 1 2 2vminus 2 + 1 and c as the totalnumber of clear 2firsquos of Xv Let Pv be an n-projection ofDqminus v(Xv) and Pv Dqminus v(Xv)Pv en Pv N

(2vminus 2 + 1)2v minus nleN2v Applying equations (23) and (24)in the proof of eorem 31 of [29] among all the n-pro-jections of Dqminus v(Xv) Pv sequentially maximizes 2 C2 in (5)only if Pv sub Dqminus v(aj) where aj is the column such thatpj max p1 p2 p2vminus 2+11113864 1113865 For a Pv with Pv sub Dqminus v(aj)according to equation (24) in the proof of eorem 31 of[29] it is obtained that

2 C

(p)

2 Pv( 1113857

0 for 0leplt n minusN

4minus 1

pjN

2v (n minus N4) forp n minusN

4minus 1

0 for n minusN

4minus 1ltplt

N

2v minus 1

c minus pj1113872 1113873N

2v1113874 1113875

2 forp

N

2v minus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)

For a MaxC22(2vminus 2+1)minus (2vminus 2+1minus v) design it has 2vminus 1 minus 1clear 2firsquos and pj 2vminus 2 Since Xv is not a MaxC2 design wehave

pj 2vminus 2

c minus pj lt 2vminus 1

minus 2vminus 2minus 1

(21)

or

pj lt 2vminus 2

clt 2vminus 1minus 1

(22)

Let Qv be an n-projection of SN(2vminus 2+1)2v obtained bydoubling Φ(v) the MaxC22(2vminus 2+1)minus (2vminus 2+1minus v) design asmentioned above With a column permutation rewriteDq(middot) as

Dq

RC(middot) Hqminus vHqminus 2Hqminus v (q minus 1) Hqminus 2Hqminus v1113872 11138731113872

q Hqminus 2Hqminus v1113872 1113873 q(q minus 1) IHqminus v1113872 1113873 SN 2vminus 2+1( )2v 1113873

(23)

in a re-changed Yates order Write Φ(v) asΦ(v) (s1 s2 s2vminus 2+1) where s1 (v minus 1)2v s2 v2v and(s3 s2vminus 2+1) v2v (v minus 1)2vHv

vminus 2 Applying equations (23)and (24) in the proof of eorem 31 of [29] among all then-projections of SN(2vminus 2+1)2v Qv sequentially maximizes 2 C2in (3) only ifQv sub Dqminus v(si) with i 1 or 2 In the followingwe investigate 2 C

(p)

2 (Qv) with Qv sub Dqminus v(s1) for which thefollowing analysis and final conclusion are the same as that

for 2 C

(p)

2 (Qv) with Qv sub Dqminus v(s2) For Qv withQv sub Dqminus v(s1) from equations (18) and (23) it is obtainedthat

Mathematical Problems in Engineering 5

B2 Qv γ( 1113857

0 for γ isin SN 2vminus 2+1( )2v

n minusN

4 for γ isin q Hqminus 2Hqminus v1113872 1113873 q(q minus 1) IHqminus v1113872 11138731113966 1113967

N

2v for γ isin (q minus 1) Hqminus 2Hqminus v1113872 1113873

N

8minus

N

2v for γ isin Hqminus 2Hqminus v

B2 Qv γ( 1113857 +N

8 for γ isin Hqminus v

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

where the second equality can be easily verified by notingthat

q Hqminus 2Hqminus v1113872 1113873 q(q minus 1) IHqminus v1113872 11138731113966 1113967

Dqminus v s1s2( 1113857 D

qminus v s1s3( 1113857 Dqminus v s1s2vminus 2+1( 11138571113864 1113865

(25)

erefore

2 C

(p)

2 Qv( 1113857

0 for 0leplt n minusN

4minus 1

N

4(n minus N4) forp n minus

N

4minus 1

0 for n minusN

4minus 1ltplt

N

2v minus 1

N

2v N4 minus N2v( 1113857 forp

N

2v minus 1

0 forN

2v minus 1ltpltN

8minus

N

2v minus 1

N

4minus

N

2v1113874 1113875N

8minus

N

2v1113874 1113875 forp N

8minus

N

2v minus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)

Denote Dlowast (Dlowastt Dlowastb ) where Dlowastt sub SN(2wminus 2+1)2w SN(2wminus 2+1)2w Dlowastt sub Dqminus w(s1) and Dlowastb sub Hk+1 Since kle q minus 3

we have U(Dlowastb ) sub Hqminus 2 erefore from (24) and (26) weobtain

2 C

(p)

2 Dlowast( 1113857

0 for 0leplt n minusN

4minus 1

N

4(n minus N4) forp n minus

N

4minus 1

0 for n minusN

4minus 1ltplt

N

2w minus 1

N

2w N4 minus N2w( 1113857 forp

N

2w minus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

6 Mathematical Problems in Engineering

Comparing (27) with (19) (20) for 4le vlew and (26) for4le vlew minus 1 it is obtained that ifD (Dt Db) is a B2-GMCdesign D should not be worse than Dlowast in terms of (3)erefore if D (Dt Db) is a B2-GMC design Dt shouldbe an n-projection of SN(2wminus 2+1)2w is completes theproof

e lemma below is an extension of Lemma A2 inAppendix introduced from [34]

Lemma 4 Suppose Dt consists of the last nN(2wminus 1 + 1)2w+1 lt nleN(2wminus 2 + 1)2w columns ofSN(2wminus 2+1)2w then B2(Dt γ1)geB2(Dt γ2) if γ1 is ahead of γ2in Hqminus w

Proof Suppose O consists of the last n minus N4 columns ofDqminus w((w minus 1)2w ) en Dt O SN(2wminus 2+1)2w Dqminus w1113966

(w minus 1) From (16) and (18) for any γ isin Hqminus wB2(Dt γ) B2(O γ) + N8 Straightforwardly fromLemma A2 it is obtained that B2(O γ1)geB2(O γ2) if γ1 isahead of γ2 in Hqminus w is completes the proof

With Lemmas 3 and 4 eorem 2 provides the con-struction methods of B2-GMC 2nminus m 2s designs withN4 + 1lt nle 5N16

Theorem 2 Let D (Dt Db) be a 2nminus m 2s design withN(2wminus 1 + 1)2w+1 lt nleN(2wminus 2 + 1)2w for wge 4 and2k le sle 2k+1 minus 1 for some 0le kle q minus 3 Suppose 2r leN

(2wminus 2 + 1)2w minus nle 2r+1 minus 1 for some 0le rle q minus w minus 1 thenD is a B2-GMC design if Dt consists of the last n columns ofSN(2wminus 2+1)2w and

(a) Db is any s-projection of Hk q minus 1 (q minus 1)Hk1113864 1113865 whenkle rle q minus w minus 1

(b) Db is any s-projection of Hk+1 whenr + 1le kle q minus w minus 1

(c) Db is any s-projection of Hqminus 2 whenq minus w minus 1lt kle q minus 3

Proof By Lemma 3 and its proof if Dlowast (Dlowastt Dlowastb ) is aB2-GMC 2nminus m 2s design there should be Dlowastt sub SN(2wminus 2+1)2w

and Dlowastt SN(2wminus 2+1)2w Dlowastt sub Dqminus w((w minus 1)2w ) Substitutingv with w in (24) we obtain

B2 Dlowastt γ( 1113857

0 for γ isin S N 2wminus 2+1( )2w( )

n minusN

4 for γ isin q Hqminus 2Hqminus w1113872 1113873 q(q minus 1) IHqminus w1113872 11138731113966 1113967

N

2w for γ isin (q minus 1) Hqminus 2Hqminus w1113872 1113873

N

8minus

N

2w for γ isin Hqminus 2Hqminus w

B2 Dlowastt γ) +

N

8 for γ isin Hqminus w1113874

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)

Let

u1 U Dlowastb( 1113857cap q Hqminus 2Hqminus w1113872 1113873 q(q minus 1) IHqminus w1113872 11138731113966 11139671113966 1113967

u2 U Dlowastb( 1113857cap (q minus 1) Hqminus 2Hqminus w1113872 11138731113966 1113967

u3 U Dlowastb( 1113857cap Hqminus 2Hqminus w1113872 11138731113966 1113967

(29)

and then

Mathematical Problems in Engineering 7

2 C

(p)

2 Dlowast( 1113857

n minusN

41113874 1113875

N

4minus u11113874 1113875 forp n minus

N

4minus 1

N

2w

N

4minus

N

2w minus u21113874 1113875 forp N

2w minus 1

N

8minus

N

2w1113874 1113875N

4minus

N

2w minus u31113874 1113875 forp N

8minus

N

2w minus 1

f Dlowastt p( 1113857 forpgeN

8minus 1

0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(30)

where f(Dlowastt p) 2 C

(p)

2 (Dlowastt ) minus (p + 1) γ γ isin Hqminus1113966

wcapU(Dlowastb ) B2(Dlowastt γ) p + 1 erefore Dlowast sequentiallymaximizes 2 C2 among all the possible 2nminus m 2s designs onlyif Dlowast sequentially minimizes

u1 u2 u3( 1113857 (31)

For ease of presenting let A q(Hqminus 2Hqminus w)1113966

q(q minus 1)(IHqminus w) B (q minus 1)(Hqminus 2Hqminus w) and C

Hqminus 2Hqminus wFor (a) when kle q minus w minus 1 there are two different ways

to choose Dlowastb from Hq (i) Dlowastb cap ABC empty and (ii)Dlowastb cap ABC neempty Recall that if Dlowast (Dlowastt Dlowastb ) is aB2-GMC design then Dlowastt sub SN(2wminus 2+1)2w with D

lowastt

SN(2wminus 2+1)2w Dlowastt sub Dqminus w((w minus 1)2w ) Case (i) implies thatDlowastb sub Hqminus w D

lowastt1113966 1113967 and thus Dlowastb sub Hqminus w q minus 1 (q minus 1)1113966

Hqminus w erefore choosing Dlowastb according to case (i) resultsin u1 u2 u3 0 If Dlowastb is chosen according to case (ii)then U(Dlowastb )cap ABC neempty which results in u1 gt 0 u2 gt 0 oru3 gt 0 Clearly if Dlowast is a B2-GMC design Dlowastb should be

chosen as (i) In the following we consider onlyDlowastb cap ABC empty ie Dlowastb sub Hqminus w (q minus 1) (q minus 1)Hqminus w1113966 1113967

For D (Dt Db) in (a) according to (v) in Lemma 1 ifDb sub Hk (q minus 1) (q minus 1)Hk1113864 1113865 with kle q minus w minus 1 thenU(Db) Hk (q minus 1) (q minus 1)Hk1113864 1113865 Denote b as the lastcolumn of Hk in Yates order and B2(Dt b) b thenbgeN8 According to Lemma 4 for γ isin Hqminus w if γ is ahead ofb in Hqminus w then B2(Dt γ)ge b by (18) For anyγ isin (q minus 1) (q minus 1)Hk1113864 1113865 we have B2(Dt γ) 0 For any γsuch that B2(Dt γ)lt b we have γ isin HqHk erefore forple b minus 2

2 C

(p)

2 (D) (p + 1) γ isin Hq γ notin U Db( 1113857 B2 Dt γ( 1113857 p + 11113966 1113967

(p + 1) γ isin Hq B2 Dt γ( 1113857 p + 11113966 1113967

2 C

(p)

2 Dt( 1113857

(32)

and for pge b

2 C

(p)

2 (D) (p + 1) γ isin Hq γ notin U Db( 1113857 B2 Dt γ( 1113857 p + 11113966 1113967

(p + 1) γ isin Hq B2 Dt γ( 1113857 p + 11113966 1113967 minus (p + 1) γ isin Hk B2 Dt γ( 1113857 p + 11113864 1113865

0

(33)

From (33) it is obtained that

2 C

(0)

2 (D) 2 C

(N8minus 1)

2 (D)2 C

(N8)

2 (D) 2 C

(bminus 2)

2 (D)1113874 1113875

2 C

(0)

2 Dt( 1113857 2 C

(N8minus 1)

2 Dt( 11138572 C

(N8)

2 Dt( 1113857 2 C

(bminus 2)

2 Dt( 11138571113874 1113875

(34)

Note that Dt is a GMC design [29] then D maximizes

2 C

(0)

2 (middot) 2 C

(N8minus 1)

2 (middot)2 C

(N8)

2 (middot) 2 C

(bminus 2)

2 (middot)1113874 1113875

(35)

among all the possible 2nminus m 2s designsSupposeD is not aB2-GMC design then there should be

aDlowast (Dlowastt Dlowastb ) which outperformsD (Dt Db) in termsof (3)is implies that there exists some p1 ge b minus 1 such that

8 Mathematical Problems in Engineering

2 C

(0)

2 Dlowast( 1113857 2 C

(N8minus 1)

2 Dlowast( 11138572 C

(N8)

2 Dlowast( 1113857 2 C

(bminus 2)

2 Dlowast( 11138571113874 1113875

2 C

(0)

2 (D) 2 C

(N8minus 1)

2 (D)2 C

(N8)

2 (D) 2 C

(bminus 2)

2 (D)1113874 1113875

(36)

2 C

p1minus 1( )2 Dlowast( 1113857gt 2 C

p1minus 1( )2 (D) (37)

Recalling the definitions of 2 C(p)

2 (D) and B2(Dt γ) wehave

1113944

P

p0

2 C

(p)

2 (D)

(p + 1) 1113944

P

p0 γ isin Hq γ notin U Db( 1113857 B2 Dt γ( 1113857 p + 11113966 1113967

1113944P

p0 γ isin HqSN 2wminus 2+1( )2w1113872 1113873 B2 Dt γ( 1113857 p + 11113966 1113967

minus 1113944P

p0 γ isin U Db( 1113857 B2 Dt γ( 1113857 p + 11113864 1113865

HqSN 2wminus 2+1( )2w1113872 1113873 minus Hk

2qminus

N 2wminus 2+ 11113872 1113873

2w minus 2k

(38)

where the second and third equalities are due to B2(Dt γ)

0 for any γ isin SN(2wminus 2+1)2w Similarly

1113944

P

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1) 1113944

P

p0 γ isin Hq γ notin U Dlowastb( 1113857 B2 Dlowastt γ( 1113857 p + 11113966 1113967

1113944P

p0 γ isin HqSN 2wminus 2+1( )2w1113872 1113873 B2 Dlowastt γ( 1113857 p + 11113966 1113967

minus 1113944P

p0 γ isin U Dlowastb( 1113857 B2 Dlowastt γ( 1113857 p + 11113864 1113865

HqSN 2wminus 2+1( )2w1113872 1113873 minus U Dlowastb( 1113857

2qminus

N 2wminus 2+ 11113872 1113873

2w minus U Dlowastb( 1113857

(39)

From (33) and (38) we have 1113944

P

p0

2 C

(p)

2 (D)

(p + 1) 1113944

bminus 2

p0

2 C

(p)

2 (D)

(p + 1)+

2 C

(bminus 1)

2 (D)

b

2qminus

N 2wminus 2+ 11113872 1113873

2w minus 2k

(40)

By (36) and (37) it is obtained that

Mathematical Problems in Engineering 9

1113944

P

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1)ge 1113944

bminus 2

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1)+

2 C

(bminus 1)

2 Dlowast( 1113857

b

gt 1113944bminus 2

p0

2 C

(p)

2 (D)

(p + 1)+

2 C

(bminus 1)

2 (D)

b

(41)

en according to (39)ndash(41) we obtain U(Dlowastb )lt 2k minus 1which contradicts Lemma 1 (i) and (ii)

For (b) similar to (a) when kle q minus w minus 1 there are twodifferent ways to choose Dlowastb from Hq (i) Dlowastb cap ABC emptyand (ii) Dlowastb cap ABC neempty With a similar argument to (a)case (i) results in u1 u2 u3 0 while case (ii) results inu1 gt 0 u2 gt 0 or u3 gt 0 erefore if Dlowast is a B2-GMC designDlowastb should be chosen according to case (i) ieDlowastb sub Hqminus w (q minus 1) (q minus 1)Hqminus w1113966 1113967 When choosing Dlowastbaccording to (i) there should be Dlowastb cap q minus 11113864

(q minus 1)Hqminus w empty ie Dlowastb sub Hqminus w otherwise Dlowastt capU

(Dlowastb )neempty noting that U(Dlowastb )cap (q minus 1 (q minus 1)Hqminus w)1113966 1113967

ge 2k gt 2r+1 minus 1 as shown in Lemma 1 (iii) e remainder ofthe proof is similar to that of (a)

For (c) when kgt q minus w minus 1 there are two different waysto choose Dlowastb from Hq (i) Dlowastb cap BC empty and (ii)Dlowastb cap BC neempty With a similar argument to (a) it is nothard to verify that case (ii) results in u1 gt 0 or u2 gt 0 whilecase (i) results in u1 u2 0 erefore if Dlowast is a B2-GMCdesign Dlowastb should be an s-projection of Hqminus 2 BC eremainder of the proof is similar to that of (a)

Remark 1 eorem 2 shows that when constructing theB2-GMC designs with N4 + 1lt nle 5N16 we can firstpartition the range (N4 + 1 5N16) into q minus 4 sequentialsubranges as (N(2wminus 1 + 1)2w+1 N(2wminus 2 + 1)2w) with w

4 5 q minus 1 and then obtain the B2-GMC designsaccording to eorem 2

In the following an example is provided to illustrate theconstruction method in eorem 2 and Remark 1

Example 2 Consider the constructions of B2-GMC219minus 13 2s designs for s 1 3 4 Here n 19 m 13 q 6and N 64 e values of the parameters N and n satisfyN4 + 1lt nle 5N16 ie n isin (17 20] From Remark 1 wepartition the range (17 20] into two subranges (17 18]

because w 5 and (18 20] because w 4 Since19 isin (18 20] according to eorem 2 Dt should be the last19 columns of S5N16 whereS5N16 Dqminus 4(Φ(4)) (5 5H2 6 6H2 56(H4H2)) from(18) erefore Dt (5H2 6 6H2 56(H4H2)) Note that5N16 minus n 20 minus 19 1 then r 0 Next we chooseDb fors 1 3 4

For s 1 we have k 0 According to eorem 2 (a)Db 5 as H0 empty

For s 3 we have k 1 According toeorem 2 (b)Db

should be a 3-projection of H2 ie Db H2 (1 2 12)For s 4 we have k 2 According toeorem 2 (c)Db

should be a 4-projection of H4 Without loss of generalitywe choose Db (1 2 12 3)

4 Concluding Remarks

e regular 2nminus m designs have wide applications in engi-neering manufacturing industry agriculture and medicineand so on When the size of experimental units is large theinhomogeneity of experimental units results in unwantedvariances of estimations of treatment effects An essentialway to solve this problem is to partition the experimentalunits into blocks

ere are two kinds of blocking problems as pointed outin [1] One is called the single block variable problem whichconsiders only a single block variable and the other is calledmulti-block variable problem which considers two or moreblock variables As stated in Section 1 experiments whichinvolve multi-block variables are more widely concerned inpractice than those which involve only a single block var-iable However due to the complexity of multi-block var-iable problem the studies on constructing optimal designswith multi-block variables are relatively rare

In this paper we aim at exploring the theories andconstructions of optimal blocked 2nminus m designs with multi-block variables e prevalent B2-GMC criterion is adoptedis criterion is preferable when there is some priorknowledge on the importance ordering of the treatmenteffects e systematical construction methods of theB2-GMC 2nminus m 2s designs with N4 + 1le nle 5N16 aredevelopede constructionmethods are concise and easy toimplement as indicated by the examples provided

Appendix

Lemma A1 is some result from Lemmas A2 A4 and A5 in[21] and Lemma 1 in [25]

Lemma A1 Suppose O is any s-projection of Hq with2k le sle 2k+1 minus 1 and kle q minus 2

(i) If Ocap q qHqminus 11113966 1113967 empty then U(O)capHqminus 11113966 1113967

ge 2k+1 minus 1 and when O has k + 1 independent col-umns the equality holds

(ii) If Ocap q qHqminus 11113966 1113967neempty then U(O)capHqminus 11113966 1113967

ge 2k minus 1 and when O has k + 1 independent col-umns the equality holds

(iii) If Ocap q qHqminus 11113966 1113967neempty then U(O)cap q qHqminus 11113966 11139671113966 1113967

ge 2k and when O has k + 1 independent columnsthe equality holds

(iv) If O sub Hk+1 then U(O) Hk+1(v) If O sub Hk q qHk1113864 1113865 then U(O) Hk q qHk1113864 1113865

Lemma A2 is some result of Lemma 1 in [34]

Lemma A2 Suppose O consists of the last n columns ofq qHqminus 11113966 1113967 then B2(O γ1)geB2(O γ2) if γ1 is ahead of γ2 inHqminus 1

Data Availability

No data were used to support this study

10 Mathematical Problems in Engineering

Conflicts of Interest

e author declares that there are no conflicts of interest

Acknowledgments

is study was supported by the National Natural ScienceFoundation of China (grant no 11801331)

References

[1] S Bisgaard ldquoA note on the definition of resolution for blocked2 k-p designsrdquo Technometrics vol 36 no 3 pp 308ndash3111994

[2] R R Sitter J Chen and M Feder ldquoFractional resolution andminimum aberration in blocked 2 n-k designsrdquo Techno-metrics vol 39 no 4 pp 382ndash390 1997

[3] H Chen and C S Cheng ldquoeory of optimal blocking ofsnminus mdesignsrdquo Annals of Statistics vol 27 no 6pp 1948ndash1973 1999

[4] R Zhang and D Park ldquoOptimal blocking of two-level frac-tional factorial designsrdquo Journal of Statistical Planning andInference vol 91 no 1 pp 107ndash121 2000

[5] S W Cheng and C F J Wu ldquoChoice of optimal blockingschemes in two-level and three-level designsrdquo Technometricsvol 44 no 3 pp 269ndash277 2002

[6] H Xu ldquoBlocked regular fractional factorial designs withminimum aberrationrdquo Annals of Statistics vol 8 no 5pp 2534ndash2553 2006

[7] H Xu and R W Mee ldquoMinimum aberration blockingschemes for 128-run designsrdquo Journal of Statistical Planningand Inference vol 140 no 11 pp 3213ndash3229 2010

[8] S Zhao P Li and R Karunamuni ldquoBlocked two-level regularfactorial designs with weak minimum aberrationrdquo Bio-metrika vol 100 no 1 pp 249ndash253 2013

[9] S-L Zhao and P-F Li ldquoConstruction of minimum aberrationblocked two-level regular factorial designsrdquo Communicationsin Statistics - eory and Methods vol 45 no 17 pp 5028ndash5036 2016

[10] B-J Chen P-F Li M-Q Liu and R-C Zhang ldquoSome resultson blocked regular 2-level fractional factorial designs withclear effectsrdquo Journal of Statistical Planning and Inferencevol 136 no 12 pp 4436ndash4449 2006

[11] S Zhao P Li and M-Q Liu ldquoOn blocked resolution IVdesigns containing clear two-factor interactionsrdquo Journal ofComplexity vol 29 no 5 pp 389ndash395 2013

[12] R C Zhang and RMukerjee ldquoGeneral minimum lower-orderconfounding in block designs using complementary setsrdquoStatistica Sinica vol 19 no 4 pp 1787ndash1802 2009

[13] S Zhao D K J Lin and P Li ldquoA note on the construction ofblocked two-level designs with general minimum lower orderconfoundingrdquo Journal of Statistical Planning and Inferencevol 172 pp 16ndash22 2016

[14] S-L Zhao and Q-Q Zhao ldquoSome results on constructingtwo-level block designs with general minimum lower orderconfoundingrdquo Communications in Statistics-eory andMethods vol 47 no 9 pp 2227ndash2237 2018

[15] R C Zhang P Li S L Zhao and M Y Ai ldquoA generalminimum lower-order confounding criterion for two-levelregular designsrdquo Statistica Sinica vol 18 no 4 pp 1689ndash1705 2008

[16] S Zhao P Li R Zhang and R Karunamuni ldquoConstruction ofblocked two-level regular designs with general minimum

lower order confoundingrdquo Journal of Statistical Planning andInference vol 143 no 6 pp 1082ndash1090 2013

[17] Y Zhao S Zhao and M-Q Liu ldquoA theory on constructingblocked two-level designs with general minimum lower orderconfoundingrdquo Frontiers of Mathematics in China vol 11no 1 pp 207ndash235 2016

[18] S-L Zhao and Q Sun ldquoOn constructing general minimumlower order confounding two-level block designsrdquo Commu-nications in Statistics-eory and Methods vol 46 no 3pp 1261ndash1274 2017

[19] J Wei P Li and R Zhang ldquoBlocked two-level regular designswith general minimum lower order confoundingrdquo Journal ofStatistical eory and Practice vol 8 no 1 pp 46ndash65 2014

[20] C F J Wu and M S Hamada Experiments PlanningAnalysis and Optimization Wiley Hoboken NJ USA 2edition 2009

[21] Q Zhao and S Zhao ldquoSome results on two-level regulardesigns with multi block variables containing clear effectsrdquoStatistical Papers vol 60 no 5 pp 1569ndash1582 2019

[22] S Zhao and Q Zhao ldquoMinimum aberration blocked designswith multiple block variablesrdquo Metrika vol 84 no 2pp 121ndash140 2021

[23] R Zhang P Li and J Wei ldquoOptimal two-level regular designswith multi block variablesrdquo Journal of Statistical eory andPractice vol 5 no 1 pp 161ndash178 2011

[24] Y N Zhao S L Zhao and M Q Liu ldquoOn constructingoptimal two-level designs with multi block variablesrdquo Journalof Systems Science and Complexity vol 31 pp 1ndash14 2018

[25] Y N Zhao and S L Zhao ldquoConstruction of optimal blockeddesigns with multi block variablesrdquoAIMSmathematics vol 6no 6 pp 6293ndash6308 2021

[26] B Tang and C F J Wu ldquoCharacterization of minimumaberration 2nminus kdesigns in terms of their complementarydesignsrdquo Annals of Statistics vol 25 no 6 pp 2549ndash25591996

[27] C F J Wu and Y Chen ldquoA graph-aided method for planningtwo-level experiments when certain interactions are impor-tantrdquo Technometrics vol 34 no 2 pp 162ndash175 1992

[28] G E P Box and J S Hunter ldquoe 2kminus pfractional factorialdesigns I and IIrdquo Technometrics vol 3 no 4 pp 311ndash3511961

[29] Y Cheng and R Zhang ldquoOn construction of general mini-mum lower order confounding 2nminus m designs with (N4)+

1le nle 9N32rdquo Journal of Statistical Planning and Inferencevol 140 no 9 pp 2384ndash2394 2010

[30] H Wu and C F J Wu ldquoClear two-factor interactions andminimum aberrationrdquo Annals of Statistics vol 30 no 5pp 1496ndash1511 2002

[31] R M Block and R W Mee ldquoSecond order saturated reso-lution IV designsrdquo Journal of Statistical eory and Appli-cations vol 2 no 2 pp 96ndash112 2003

[32] A A Davydov and L M Tombak ldquoQuasi-perfect linear bi-nary codes with distance 4 and complete caps in projectivegeometryrdquo Problems of Information Transmission vol 25no 4 pp 265ndash275 1990

[33] R Zhang and Y Cheng ldquoGeneral minimum lower orderconfounding designs an overview and a construction theoryrdquoJournal of Statistical Planning and Inference vol 140 no 7pp 1719ndash1730 2010

[34] Q Zhou N Balakrishnan and R Zhang ldquoe factor aliasedeffect number pattern and its application in experimentalplanningrdquo Canadian Journal of Statistics vol 41 no 3pp 540ndash555 2013

Mathematical Problems in Engineering 11

B2 Qv γ( 1113857

0 for γ isin SN 2vminus 2+1( )2v

n minusN

4 for γ isin q Hqminus 2Hqminus v1113872 1113873 q(q minus 1) IHqminus v1113872 11138731113966 1113967

N

2v for γ isin (q minus 1) Hqminus 2Hqminus v1113872 1113873

N

8minus

N

2v for γ isin Hqminus 2Hqminus v

B2 Qv γ( 1113857 +N

8 for γ isin Hqminus v

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

where the second equality can be easily verified by notingthat

q Hqminus 2Hqminus v1113872 1113873 q(q minus 1) IHqminus v1113872 11138731113966 1113967

Dqminus v s1s2( 1113857 D

qminus v s1s3( 1113857 Dqminus v s1s2vminus 2+1( 11138571113864 1113865

(25)

erefore

2 C

(p)

2 Qv( 1113857

0 for 0leplt n minusN

4minus 1

N

4(n minus N4) forp n minus

N

4minus 1

0 for n minusN

4minus 1ltplt

N

2v minus 1

N

2v N4 minus N2v( 1113857 forp

N

2v minus 1

0 forN

2v minus 1ltpltN

8minus

N

2v minus 1

N

4minus

N

2v1113874 1113875N

8minus

N

2v1113874 1113875 forp N

8minus

N

2v minus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)

Denote Dlowast (Dlowastt Dlowastb ) where Dlowastt sub SN(2wminus 2+1)2w SN(2wminus 2+1)2w Dlowastt sub Dqminus w(s1) and Dlowastb sub Hk+1 Since kle q minus 3

we have U(Dlowastb ) sub Hqminus 2 erefore from (24) and (26) weobtain

2 C

(p)

2 Dlowast( 1113857

0 for 0leplt n minusN

4minus 1

N

4(n minus N4) forp n minus

N

4minus 1

0 for n minusN

4minus 1ltplt

N

2w minus 1

N

2w N4 minus N2w( 1113857 forp

N

2w minus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

6 Mathematical Problems in Engineering

Comparing (27) with (19) (20) for 4le vlew and (26) for4le vlew minus 1 it is obtained that ifD (Dt Db) is a B2-GMCdesign D should not be worse than Dlowast in terms of (3)erefore if D (Dt Db) is a B2-GMC design Dt shouldbe an n-projection of SN(2wminus 2+1)2w is completes theproof

e lemma below is an extension of Lemma A2 inAppendix introduced from [34]

Lemma 4 Suppose Dt consists of the last nN(2wminus 1 + 1)2w+1 lt nleN(2wminus 2 + 1)2w columns ofSN(2wminus 2+1)2w then B2(Dt γ1)geB2(Dt γ2) if γ1 is ahead of γ2in Hqminus w

Proof Suppose O consists of the last n minus N4 columns ofDqminus w((w minus 1)2w ) en Dt O SN(2wminus 2+1)2w Dqminus w1113966

(w minus 1) From (16) and (18) for any γ isin Hqminus wB2(Dt γ) B2(O γ) + N8 Straightforwardly fromLemma A2 it is obtained that B2(O γ1)geB2(O γ2) if γ1 isahead of γ2 in Hqminus w is completes the proof

With Lemmas 3 and 4 eorem 2 provides the con-struction methods of B2-GMC 2nminus m 2s designs withN4 + 1lt nle 5N16

Theorem 2 Let D (Dt Db) be a 2nminus m 2s design withN(2wminus 1 + 1)2w+1 lt nleN(2wminus 2 + 1)2w for wge 4 and2k le sle 2k+1 minus 1 for some 0le kle q minus 3 Suppose 2r leN

(2wminus 2 + 1)2w minus nle 2r+1 minus 1 for some 0le rle q minus w minus 1 thenD is a B2-GMC design if Dt consists of the last n columns ofSN(2wminus 2+1)2w and

(a) Db is any s-projection of Hk q minus 1 (q minus 1)Hk1113864 1113865 whenkle rle q minus w minus 1

(b) Db is any s-projection of Hk+1 whenr + 1le kle q minus w minus 1

(c) Db is any s-projection of Hqminus 2 whenq minus w minus 1lt kle q minus 3

Proof By Lemma 3 and its proof if Dlowast (Dlowastt Dlowastb ) is aB2-GMC 2nminus m 2s design there should be Dlowastt sub SN(2wminus 2+1)2w

and Dlowastt SN(2wminus 2+1)2w Dlowastt sub Dqminus w((w minus 1)2w ) Substitutingv with w in (24) we obtain

B2 Dlowastt γ( 1113857

0 for γ isin S N 2wminus 2+1( )2w( )

n minusN

4 for γ isin q Hqminus 2Hqminus w1113872 1113873 q(q minus 1) IHqminus w1113872 11138731113966 1113967

N

2w for γ isin (q minus 1) Hqminus 2Hqminus w1113872 1113873

N

8minus

N

2w for γ isin Hqminus 2Hqminus w

B2 Dlowastt γ) +

N

8 for γ isin Hqminus w1113874

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)

Let

u1 U Dlowastb( 1113857cap q Hqminus 2Hqminus w1113872 1113873 q(q minus 1) IHqminus w1113872 11138731113966 11139671113966 1113967

u2 U Dlowastb( 1113857cap (q minus 1) Hqminus 2Hqminus w1113872 11138731113966 1113967

u3 U Dlowastb( 1113857cap Hqminus 2Hqminus w1113872 11138731113966 1113967

(29)

and then

Mathematical Problems in Engineering 7

2 C

(p)

2 Dlowast( 1113857

n minusN

41113874 1113875

N

4minus u11113874 1113875 forp n minus

N

4minus 1

N

2w

N

4minus

N

2w minus u21113874 1113875 forp N

2w minus 1

N

8minus

N

2w1113874 1113875N

4minus

N

2w minus u31113874 1113875 forp N

8minus

N

2w minus 1

f Dlowastt p( 1113857 forpgeN

8minus 1

0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(30)

where f(Dlowastt p) 2 C

(p)

2 (Dlowastt ) minus (p + 1) γ γ isin Hqminus1113966

wcapU(Dlowastb ) B2(Dlowastt γ) p + 1 erefore Dlowast sequentiallymaximizes 2 C2 among all the possible 2nminus m 2s designs onlyif Dlowast sequentially minimizes

u1 u2 u3( 1113857 (31)

For ease of presenting let A q(Hqminus 2Hqminus w)1113966

q(q minus 1)(IHqminus w) B (q minus 1)(Hqminus 2Hqminus w) and C

Hqminus 2Hqminus wFor (a) when kle q minus w minus 1 there are two different ways

to choose Dlowastb from Hq (i) Dlowastb cap ABC empty and (ii)Dlowastb cap ABC neempty Recall that if Dlowast (Dlowastt Dlowastb ) is aB2-GMC design then Dlowastt sub SN(2wminus 2+1)2w with D

lowastt

SN(2wminus 2+1)2w Dlowastt sub Dqminus w((w minus 1)2w ) Case (i) implies thatDlowastb sub Hqminus w D

lowastt1113966 1113967 and thus Dlowastb sub Hqminus w q minus 1 (q minus 1)1113966

Hqminus w erefore choosing Dlowastb according to case (i) resultsin u1 u2 u3 0 If Dlowastb is chosen according to case (ii)then U(Dlowastb )cap ABC neempty which results in u1 gt 0 u2 gt 0 oru3 gt 0 Clearly if Dlowast is a B2-GMC design Dlowastb should be

chosen as (i) In the following we consider onlyDlowastb cap ABC empty ie Dlowastb sub Hqminus w (q minus 1) (q minus 1)Hqminus w1113966 1113967

For D (Dt Db) in (a) according to (v) in Lemma 1 ifDb sub Hk (q minus 1) (q minus 1)Hk1113864 1113865 with kle q minus w minus 1 thenU(Db) Hk (q minus 1) (q minus 1)Hk1113864 1113865 Denote b as the lastcolumn of Hk in Yates order and B2(Dt b) b thenbgeN8 According to Lemma 4 for γ isin Hqminus w if γ is ahead ofb in Hqminus w then B2(Dt γ)ge b by (18) For anyγ isin (q minus 1) (q minus 1)Hk1113864 1113865 we have B2(Dt γ) 0 For any γsuch that B2(Dt γ)lt b we have γ isin HqHk erefore forple b minus 2

2 C

(p)

2 (D) (p + 1) γ isin Hq γ notin U Db( 1113857 B2 Dt γ( 1113857 p + 11113966 1113967

(p + 1) γ isin Hq B2 Dt γ( 1113857 p + 11113966 1113967

2 C

(p)

2 Dt( 1113857

(32)

and for pge b

2 C

(p)

2 (D) (p + 1) γ isin Hq γ notin U Db( 1113857 B2 Dt γ( 1113857 p + 11113966 1113967

(p + 1) γ isin Hq B2 Dt γ( 1113857 p + 11113966 1113967 minus (p + 1) γ isin Hk B2 Dt γ( 1113857 p + 11113864 1113865

0

(33)

From (33) it is obtained that

2 C

(0)

2 (D) 2 C

(N8minus 1)

2 (D)2 C

(N8)

2 (D) 2 C

(bminus 2)

2 (D)1113874 1113875

2 C

(0)

2 Dt( 1113857 2 C

(N8minus 1)

2 Dt( 11138572 C

(N8)

2 Dt( 1113857 2 C

(bminus 2)

2 Dt( 11138571113874 1113875

(34)

Note that Dt is a GMC design [29] then D maximizes

2 C

(0)

2 (middot) 2 C

(N8minus 1)

2 (middot)2 C

(N8)

2 (middot) 2 C

(bminus 2)

2 (middot)1113874 1113875

(35)

among all the possible 2nminus m 2s designsSupposeD is not aB2-GMC design then there should be

aDlowast (Dlowastt Dlowastb ) which outperformsD (Dt Db) in termsof (3)is implies that there exists some p1 ge b minus 1 such that

8 Mathematical Problems in Engineering

2 C

(0)

2 Dlowast( 1113857 2 C

(N8minus 1)

2 Dlowast( 11138572 C

(N8)

2 Dlowast( 1113857 2 C

(bminus 2)

2 Dlowast( 11138571113874 1113875

2 C

(0)

2 (D) 2 C

(N8minus 1)

2 (D)2 C

(N8)

2 (D) 2 C

(bminus 2)

2 (D)1113874 1113875

(36)

2 C

p1minus 1( )2 Dlowast( 1113857gt 2 C

p1minus 1( )2 (D) (37)

Recalling the definitions of 2 C(p)

2 (D) and B2(Dt γ) wehave

1113944

P

p0

2 C

(p)

2 (D)

(p + 1) 1113944

P

p0 γ isin Hq γ notin U Db( 1113857 B2 Dt γ( 1113857 p + 11113966 1113967

1113944P

p0 γ isin HqSN 2wminus 2+1( )2w1113872 1113873 B2 Dt γ( 1113857 p + 11113966 1113967

minus 1113944P

p0 γ isin U Db( 1113857 B2 Dt γ( 1113857 p + 11113864 1113865

HqSN 2wminus 2+1( )2w1113872 1113873 minus Hk

2qminus

N 2wminus 2+ 11113872 1113873

2w minus 2k

(38)

where the second and third equalities are due to B2(Dt γ)

0 for any γ isin SN(2wminus 2+1)2w Similarly

1113944

P

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1) 1113944

P

p0 γ isin Hq γ notin U Dlowastb( 1113857 B2 Dlowastt γ( 1113857 p + 11113966 1113967

1113944P

p0 γ isin HqSN 2wminus 2+1( )2w1113872 1113873 B2 Dlowastt γ( 1113857 p + 11113966 1113967

minus 1113944P

p0 γ isin U Dlowastb( 1113857 B2 Dlowastt γ( 1113857 p + 11113864 1113865

HqSN 2wminus 2+1( )2w1113872 1113873 minus U Dlowastb( 1113857

2qminus

N 2wminus 2+ 11113872 1113873

2w minus U Dlowastb( 1113857

(39)

From (33) and (38) we have 1113944

P

p0

2 C

(p)

2 (D)

(p + 1) 1113944

bminus 2

p0

2 C

(p)

2 (D)

(p + 1)+

2 C

(bminus 1)

2 (D)

b

2qminus

N 2wminus 2+ 11113872 1113873

2w minus 2k

(40)

By (36) and (37) it is obtained that

Mathematical Problems in Engineering 9

1113944

P

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1)ge 1113944

bminus 2

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1)+

2 C

(bminus 1)

2 Dlowast( 1113857

b

gt 1113944bminus 2

p0

2 C

(p)

2 (D)

(p + 1)+

2 C

(bminus 1)

2 (D)

b

(41)

en according to (39)ndash(41) we obtain U(Dlowastb )lt 2k minus 1which contradicts Lemma 1 (i) and (ii)

For (b) similar to (a) when kle q minus w minus 1 there are twodifferent ways to choose Dlowastb from Hq (i) Dlowastb cap ABC emptyand (ii) Dlowastb cap ABC neempty With a similar argument to (a)case (i) results in u1 u2 u3 0 while case (ii) results inu1 gt 0 u2 gt 0 or u3 gt 0 erefore if Dlowast is a B2-GMC designDlowastb should be chosen according to case (i) ieDlowastb sub Hqminus w (q minus 1) (q minus 1)Hqminus w1113966 1113967 When choosing Dlowastbaccording to (i) there should be Dlowastb cap q minus 11113864

(q minus 1)Hqminus w empty ie Dlowastb sub Hqminus w otherwise Dlowastt capU

(Dlowastb )neempty noting that U(Dlowastb )cap (q minus 1 (q minus 1)Hqminus w)1113966 1113967

ge 2k gt 2r+1 minus 1 as shown in Lemma 1 (iii) e remainder ofthe proof is similar to that of (a)

For (c) when kgt q minus w minus 1 there are two different waysto choose Dlowastb from Hq (i) Dlowastb cap BC empty and (ii)Dlowastb cap BC neempty With a similar argument to (a) it is nothard to verify that case (ii) results in u1 gt 0 or u2 gt 0 whilecase (i) results in u1 u2 0 erefore if Dlowast is a B2-GMCdesign Dlowastb should be an s-projection of Hqminus 2 BC eremainder of the proof is similar to that of (a)

Remark 1 eorem 2 shows that when constructing theB2-GMC designs with N4 + 1lt nle 5N16 we can firstpartition the range (N4 + 1 5N16) into q minus 4 sequentialsubranges as (N(2wminus 1 + 1)2w+1 N(2wminus 2 + 1)2w) with w

4 5 q minus 1 and then obtain the B2-GMC designsaccording to eorem 2

In the following an example is provided to illustrate theconstruction method in eorem 2 and Remark 1

Example 2 Consider the constructions of B2-GMC219minus 13 2s designs for s 1 3 4 Here n 19 m 13 q 6and N 64 e values of the parameters N and n satisfyN4 + 1lt nle 5N16 ie n isin (17 20] From Remark 1 wepartition the range (17 20] into two subranges (17 18]

because w 5 and (18 20] because w 4 Since19 isin (18 20] according to eorem 2 Dt should be the last19 columns of S5N16 whereS5N16 Dqminus 4(Φ(4)) (5 5H2 6 6H2 56(H4H2)) from(18) erefore Dt (5H2 6 6H2 56(H4H2)) Note that5N16 minus n 20 minus 19 1 then r 0 Next we chooseDb fors 1 3 4

For s 1 we have k 0 According to eorem 2 (a)Db 5 as H0 empty

For s 3 we have k 1 According toeorem 2 (b)Db

should be a 3-projection of H2 ie Db H2 (1 2 12)For s 4 we have k 2 According toeorem 2 (c)Db

should be a 4-projection of H4 Without loss of generalitywe choose Db (1 2 12 3)

4 Concluding Remarks

e regular 2nminus m designs have wide applications in engi-neering manufacturing industry agriculture and medicineand so on When the size of experimental units is large theinhomogeneity of experimental units results in unwantedvariances of estimations of treatment effects An essentialway to solve this problem is to partition the experimentalunits into blocks

ere are two kinds of blocking problems as pointed outin [1] One is called the single block variable problem whichconsiders only a single block variable and the other is calledmulti-block variable problem which considers two or moreblock variables As stated in Section 1 experiments whichinvolve multi-block variables are more widely concerned inpractice than those which involve only a single block var-iable However due to the complexity of multi-block var-iable problem the studies on constructing optimal designswith multi-block variables are relatively rare

In this paper we aim at exploring the theories andconstructions of optimal blocked 2nminus m designs with multi-block variables e prevalent B2-GMC criterion is adoptedis criterion is preferable when there is some priorknowledge on the importance ordering of the treatmenteffects e systematical construction methods of theB2-GMC 2nminus m 2s designs with N4 + 1le nle 5N16 aredevelopede constructionmethods are concise and easy toimplement as indicated by the examples provided

Appendix

Lemma A1 is some result from Lemmas A2 A4 and A5 in[21] and Lemma 1 in [25]

Lemma A1 Suppose O is any s-projection of Hq with2k le sle 2k+1 minus 1 and kle q minus 2

(i) If Ocap q qHqminus 11113966 1113967 empty then U(O)capHqminus 11113966 1113967

ge 2k+1 minus 1 and when O has k + 1 independent col-umns the equality holds

(ii) If Ocap q qHqminus 11113966 1113967neempty then U(O)capHqminus 11113966 1113967

ge 2k minus 1 and when O has k + 1 independent col-umns the equality holds

(iii) If Ocap q qHqminus 11113966 1113967neempty then U(O)cap q qHqminus 11113966 11139671113966 1113967

ge 2k and when O has k + 1 independent columnsthe equality holds

(iv) If O sub Hk+1 then U(O) Hk+1(v) If O sub Hk q qHk1113864 1113865 then U(O) Hk q qHk1113864 1113865

Lemma A2 is some result of Lemma 1 in [34]

Lemma A2 Suppose O consists of the last n columns ofq qHqminus 11113966 1113967 then B2(O γ1)geB2(O γ2) if γ1 is ahead of γ2 inHqminus 1

Data Availability

No data were used to support this study

10 Mathematical Problems in Engineering

Conflicts of Interest

e author declares that there are no conflicts of interest

Acknowledgments

is study was supported by the National Natural ScienceFoundation of China (grant no 11801331)

References

[1] S Bisgaard ldquoA note on the definition of resolution for blocked2 k-p designsrdquo Technometrics vol 36 no 3 pp 308ndash3111994

[2] R R Sitter J Chen and M Feder ldquoFractional resolution andminimum aberration in blocked 2 n-k designsrdquo Techno-metrics vol 39 no 4 pp 382ndash390 1997

[3] H Chen and C S Cheng ldquoeory of optimal blocking ofsnminus mdesignsrdquo Annals of Statistics vol 27 no 6pp 1948ndash1973 1999

[4] R Zhang and D Park ldquoOptimal blocking of two-level frac-tional factorial designsrdquo Journal of Statistical Planning andInference vol 91 no 1 pp 107ndash121 2000

[5] S W Cheng and C F J Wu ldquoChoice of optimal blockingschemes in two-level and three-level designsrdquo Technometricsvol 44 no 3 pp 269ndash277 2002

[6] H Xu ldquoBlocked regular fractional factorial designs withminimum aberrationrdquo Annals of Statistics vol 8 no 5pp 2534ndash2553 2006

[7] H Xu and R W Mee ldquoMinimum aberration blockingschemes for 128-run designsrdquo Journal of Statistical Planningand Inference vol 140 no 11 pp 3213ndash3229 2010

[8] S Zhao P Li and R Karunamuni ldquoBlocked two-level regularfactorial designs with weak minimum aberrationrdquo Bio-metrika vol 100 no 1 pp 249ndash253 2013

[9] S-L Zhao and P-F Li ldquoConstruction of minimum aberrationblocked two-level regular factorial designsrdquo Communicationsin Statistics - eory and Methods vol 45 no 17 pp 5028ndash5036 2016

[10] B-J Chen P-F Li M-Q Liu and R-C Zhang ldquoSome resultson blocked regular 2-level fractional factorial designs withclear effectsrdquo Journal of Statistical Planning and Inferencevol 136 no 12 pp 4436ndash4449 2006

[11] S Zhao P Li and M-Q Liu ldquoOn blocked resolution IVdesigns containing clear two-factor interactionsrdquo Journal ofComplexity vol 29 no 5 pp 389ndash395 2013

[12] R C Zhang and RMukerjee ldquoGeneral minimum lower-orderconfounding in block designs using complementary setsrdquoStatistica Sinica vol 19 no 4 pp 1787ndash1802 2009

[13] S Zhao D K J Lin and P Li ldquoA note on the construction ofblocked two-level designs with general minimum lower orderconfoundingrdquo Journal of Statistical Planning and Inferencevol 172 pp 16ndash22 2016

[14] S-L Zhao and Q-Q Zhao ldquoSome results on constructingtwo-level block designs with general minimum lower orderconfoundingrdquo Communications in Statistics-eory andMethods vol 47 no 9 pp 2227ndash2237 2018

[15] R C Zhang P Li S L Zhao and M Y Ai ldquoA generalminimum lower-order confounding criterion for two-levelregular designsrdquo Statistica Sinica vol 18 no 4 pp 1689ndash1705 2008

[16] S Zhao P Li R Zhang and R Karunamuni ldquoConstruction ofblocked two-level regular designs with general minimum

lower order confoundingrdquo Journal of Statistical Planning andInference vol 143 no 6 pp 1082ndash1090 2013

[17] Y Zhao S Zhao and M-Q Liu ldquoA theory on constructingblocked two-level designs with general minimum lower orderconfoundingrdquo Frontiers of Mathematics in China vol 11no 1 pp 207ndash235 2016

[18] S-L Zhao and Q Sun ldquoOn constructing general minimumlower order confounding two-level block designsrdquo Commu-nications in Statistics-eory and Methods vol 46 no 3pp 1261ndash1274 2017

[19] J Wei P Li and R Zhang ldquoBlocked two-level regular designswith general minimum lower order confoundingrdquo Journal ofStatistical eory and Practice vol 8 no 1 pp 46ndash65 2014

[20] C F J Wu and M S Hamada Experiments PlanningAnalysis and Optimization Wiley Hoboken NJ USA 2edition 2009

[21] Q Zhao and S Zhao ldquoSome results on two-level regulardesigns with multi block variables containing clear effectsrdquoStatistical Papers vol 60 no 5 pp 1569ndash1582 2019

[22] S Zhao and Q Zhao ldquoMinimum aberration blocked designswith multiple block variablesrdquo Metrika vol 84 no 2pp 121ndash140 2021

[23] R Zhang P Li and J Wei ldquoOptimal two-level regular designswith multi block variablesrdquo Journal of Statistical eory andPractice vol 5 no 1 pp 161ndash178 2011

[24] Y N Zhao S L Zhao and M Q Liu ldquoOn constructingoptimal two-level designs with multi block variablesrdquo Journalof Systems Science and Complexity vol 31 pp 1ndash14 2018

[25] Y N Zhao and S L Zhao ldquoConstruction of optimal blockeddesigns with multi block variablesrdquoAIMSmathematics vol 6no 6 pp 6293ndash6308 2021

[26] B Tang and C F J Wu ldquoCharacterization of minimumaberration 2nminus kdesigns in terms of their complementarydesignsrdquo Annals of Statistics vol 25 no 6 pp 2549ndash25591996

[27] C F J Wu and Y Chen ldquoA graph-aided method for planningtwo-level experiments when certain interactions are impor-tantrdquo Technometrics vol 34 no 2 pp 162ndash175 1992

[28] G E P Box and J S Hunter ldquoe 2kminus pfractional factorialdesigns I and IIrdquo Technometrics vol 3 no 4 pp 311ndash3511961

[29] Y Cheng and R Zhang ldquoOn construction of general mini-mum lower order confounding 2nminus m designs with (N4)+

1le nle 9N32rdquo Journal of Statistical Planning and Inferencevol 140 no 9 pp 2384ndash2394 2010

[30] H Wu and C F J Wu ldquoClear two-factor interactions andminimum aberrationrdquo Annals of Statistics vol 30 no 5pp 1496ndash1511 2002

[31] R M Block and R W Mee ldquoSecond order saturated reso-lution IV designsrdquo Journal of Statistical eory and Appli-cations vol 2 no 2 pp 96ndash112 2003

[32] A A Davydov and L M Tombak ldquoQuasi-perfect linear bi-nary codes with distance 4 and complete caps in projectivegeometryrdquo Problems of Information Transmission vol 25no 4 pp 265ndash275 1990

[33] R Zhang and Y Cheng ldquoGeneral minimum lower orderconfounding designs an overview and a construction theoryrdquoJournal of Statistical Planning and Inference vol 140 no 7pp 1719ndash1730 2010

[34] Q Zhou N Balakrishnan and R Zhang ldquoe factor aliasedeffect number pattern and its application in experimentalplanningrdquo Canadian Journal of Statistics vol 41 no 3pp 540ndash555 2013

Mathematical Problems in Engineering 11

Comparing (27) with (19) (20) for 4le vlew and (26) for4le vlew minus 1 it is obtained that ifD (Dt Db) is a B2-GMCdesign D should not be worse than Dlowast in terms of (3)erefore if D (Dt Db) is a B2-GMC design Dt shouldbe an n-projection of SN(2wminus 2+1)2w is completes theproof

e lemma below is an extension of Lemma A2 inAppendix introduced from [34]

Lemma 4 Suppose Dt consists of the last nN(2wminus 1 + 1)2w+1 lt nleN(2wminus 2 + 1)2w columns ofSN(2wminus 2+1)2w then B2(Dt γ1)geB2(Dt γ2) if γ1 is ahead of γ2in Hqminus w

Proof Suppose O consists of the last n minus N4 columns ofDqminus w((w minus 1)2w ) en Dt O SN(2wminus 2+1)2w Dqminus w1113966

(w minus 1) From (16) and (18) for any γ isin Hqminus wB2(Dt γ) B2(O γ) + N8 Straightforwardly fromLemma A2 it is obtained that B2(O γ1)geB2(O γ2) if γ1 isahead of γ2 in Hqminus w is completes the proof

With Lemmas 3 and 4 eorem 2 provides the con-struction methods of B2-GMC 2nminus m 2s designs withN4 + 1lt nle 5N16

Theorem 2 Let D (Dt Db) be a 2nminus m 2s design withN(2wminus 1 + 1)2w+1 lt nleN(2wminus 2 + 1)2w for wge 4 and2k le sle 2k+1 minus 1 for some 0le kle q minus 3 Suppose 2r leN

(2wminus 2 + 1)2w minus nle 2r+1 minus 1 for some 0le rle q minus w minus 1 thenD is a B2-GMC design if Dt consists of the last n columns ofSN(2wminus 2+1)2w and

(a) Db is any s-projection of Hk q minus 1 (q minus 1)Hk1113864 1113865 whenkle rle q minus w minus 1

(b) Db is any s-projection of Hk+1 whenr + 1le kle q minus w minus 1

(c) Db is any s-projection of Hqminus 2 whenq minus w minus 1lt kle q minus 3

Proof By Lemma 3 and its proof if Dlowast (Dlowastt Dlowastb ) is aB2-GMC 2nminus m 2s design there should be Dlowastt sub SN(2wminus 2+1)2w

and Dlowastt SN(2wminus 2+1)2w Dlowastt sub Dqminus w((w minus 1)2w ) Substitutingv with w in (24) we obtain

B2 Dlowastt γ( 1113857

0 for γ isin S N 2wminus 2+1( )2w( )

n minusN

4 for γ isin q Hqminus 2Hqminus w1113872 1113873 q(q minus 1) IHqminus w1113872 11138731113966 1113967

N

2w for γ isin (q minus 1) Hqminus 2Hqminus w1113872 1113873

N

8minus

N

2w for γ isin Hqminus 2Hqminus w

B2 Dlowastt γ) +

N

8 for γ isin Hqminus w1113874

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)

Let

u1 U Dlowastb( 1113857cap q Hqminus 2Hqminus w1113872 1113873 q(q minus 1) IHqminus w1113872 11138731113966 11139671113966 1113967

u2 U Dlowastb( 1113857cap (q minus 1) Hqminus 2Hqminus w1113872 11138731113966 1113967

u3 U Dlowastb( 1113857cap Hqminus 2Hqminus w1113872 11138731113966 1113967

(29)

and then

Mathematical Problems in Engineering 7

2 C

(p)

2 Dlowast( 1113857

n minusN

41113874 1113875

N

4minus u11113874 1113875 forp n minus

N

4minus 1

N

2w

N

4minus

N

2w minus u21113874 1113875 forp N

2w minus 1

N

8minus

N

2w1113874 1113875N

4minus

N

2w minus u31113874 1113875 forp N

8minus

N

2w minus 1

f Dlowastt p( 1113857 forpgeN

8minus 1

0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(30)

where f(Dlowastt p) 2 C

(p)

2 (Dlowastt ) minus (p + 1) γ γ isin Hqminus1113966

wcapU(Dlowastb ) B2(Dlowastt γ) p + 1 erefore Dlowast sequentiallymaximizes 2 C2 among all the possible 2nminus m 2s designs onlyif Dlowast sequentially minimizes

u1 u2 u3( 1113857 (31)

For ease of presenting let A q(Hqminus 2Hqminus w)1113966

q(q minus 1)(IHqminus w) B (q minus 1)(Hqminus 2Hqminus w) and C

Hqminus 2Hqminus wFor (a) when kle q minus w minus 1 there are two different ways

to choose Dlowastb from Hq (i) Dlowastb cap ABC empty and (ii)Dlowastb cap ABC neempty Recall that if Dlowast (Dlowastt Dlowastb ) is aB2-GMC design then Dlowastt sub SN(2wminus 2+1)2w with D

lowastt

SN(2wminus 2+1)2w Dlowastt sub Dqminus w((w minus 1)2w ) Case (i) implies thatDlowastb sub Hqminus w D

lowastt1113966 1113967 and thus Dlowastb sub Hqminus w q minus 1 (q minus 1)1113966

Hqminus w erefore choosing Dlowastb according to case (i) resultsin u1 u2 u3 0 If Dlowastb is chosen according to case (ii)then U(Dlowastb )cap ABC neempty which results in u1 gt 0 u2 gt 0 oru3 gt 0 Clearly if Dlowast is a B2-GMC design Dlowastb should be

chosen as (i) In the following we consider onlyDlowastb cap ABC empty ie Dlowastb sub Hqminus w (q minus 1) (q minus 1)Hqminus w1113966 1113967

For D (Dt Db) in (a) according to (v) in Lemma 1 ifDb sub Hk (q minus 1) (q minus 1)Hk1113864 1113865 with kle q minus w minus 1 thenU(Db) Hk (q minus 1) (q minus 1)Hk1113864 1113865 Denote b as the lastcolumn of Hk in Yates order and B2(Dt b) b thenbgeN8 According to Lemma 4 for γ isin Hqminus w if γ is ahead ofb in Hqminus w then B2(Dt γ)ge b by (18) For anyγ isin (q minus 1) (q minus 1)Hk1113864 1113865 we have B2(Dt γ) 0 For any γsuch that B2(Dt γ)lt b we have γ isin HqHk erefore forple b minus 2

2 C

(p)

2 (D) (p + 1) γ isin Hq γ notin U Db( 1113857 B2 Dt γ( 1113857 p + 11113966 1113967

(p + 1) γ isin Hq B2 Dt γ( 1113857 p + 11113966 1113967

2 C

(p)

2 Dt( 1113857

(32)

and for pge b

2 C

(p)

2 (D) (p + 1) γ isin Hq γ notin U Db( 1113857 B2 Dt γ( 1113857 p + 11113966 1113967

(p + 1) γ isin Hq B2 Dt γ( 1113857 p + 11113966 1113967 minus (p + 1) γ isin Hk B2 Dt γ( 1113857 p + 11113864 1113865

0

(33)

From (33) it is obtained that

2 C

(0)

2 (D) 2 C

(N8minus 1)

2 (D)2 C

(N8)

2 (D) 2 C

(bminus 2)

2 (D)1113874 1113875

2 C

(0)

2 Dt( 1113857 2 C

(N8minus 1)

2 Dt( 11138572 C

(N8)

2 Dt( 1113857 2 C

(bminus 2)

2 Dt( 11138571113874 1113875

(34)

Note that Dt is a GMC design [29] then D maximizes

2 C

(0)

2 (middot) 2 C

(N8minus 1)

2 (middot)2 C

(N8)

2 (middot) 2 C

(bminus 2)

2 (middot)1113874 1113875

(35)

among all the possible 2nminus m 2s designsSupposeD is not aB2-GMC design then there should be

aDlowast (Dlowastt Dlowastb ) which outperformsD (Dt Db) in termsof (3)is implies that there exists some p1 ge b minus 1 such that

8 Mathematical Problems in Engineering

2 C

(0)

2 Dlowast( 1113857 2 C

(N8minus 1)

2 Dlowast( 11138572 C

(N8)

2 Dlowast( 1113857 2 C

(bminus 2)

2 Dlowast( 11138571113874 1113875

2 C

(0)

2 (D) 2 C

(N8minus 1)

2 (D)2 C

(N8)

2 (D) 2 C

(bminus 2)

2 (D)1113874 1113875

(36)

2 C

p1minus 1( )2 Dlowast( 1113857gt 2 C

p1minus 1( )2 (D) (37)

Recalling the definitions of 2 C(p)

2 (D) and B2(Dt γ) wehave

1113944

P

p0

2 C

(p)

2 (D)

(p + 1) 1113944

P

p0 γ isin Hq γ notin U Db( 1113857 B2 Dt γ( 1113857 p + 11113966 1113967

1113944P

p0 γ isin HqSN 2wminus 2+1( )2w1113872 1113873 B2 Dt γ( 1113857 p + 11113966 1113967

minus 1113944P

p0 γ isin U Db( 1113857 B2 Dt γ( 1113857 p + 11113864 1113865

HqSN 2wminus 2+1( )2w1113872 1113873 minus Hk

2qminus

N 2wminus 2+ 11113872 1113873

2w minus 2k

(38)

where the second and third equalities are due to B2(Dt γ)

0 for any γ isin SN(2wminus 2+1)2w Similarly

1113944

P

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1) 1113944

P

p0 γ isin Hq γ notin U Dlowastb( 1113857 B2 Dlowastt γ( 1113857 p + 11113966 1113967

1113944P

p0 γ isin HqSN 2wminus 2+1( )2w1113872 1113873 B2 Dlowastt γ( 1113857 p + 11113966 1113967

minus 1113944P

p0 γ isin U Dlowastb( 1113857 B2 Dlowastt γ( 1113857 p + 11113864 1113865

HqSN 2wminus 2+1( )2w1113872 1113873 minus U Dlowastb( 1113857

2qminus

N 2wminus 2+ 11113872 1113873

2w minus U Dlowastb( 1113857

(39)

From (33) and (38) we have 1113944

P

p0

2 C

(p)

2 (D)

(p + 1) 1113944

bminus 2

p0

2 C

(p)

2 (D)

(p + 1)+

2 C

(bminus 1)

2 (D)

b

2qminus

N 2wminus 2+ 11113872 1113873

2w minus 2k

(40)

By (36) and (37) it is obtained that

Mathematical Problems in Engineering 9

1113944

P

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1)ge 1113944

bminus 2

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1)+

2 C

(bminus 1)

2 Dlowast( 1113857

b

gt 1113944bminus 2

p0

2 C

(p)

2 (D)

(p + 1)+

2 C

(bminus 1)

2 (D)

b

(41)

en according to (39)ndash(41) we obtain U(Dlowastb )lt 2k minus 1which contradicts Lemma 1 (i) and (ii)

For (b) similar to (a) when kle q minus w minus 1 there are twodifferent ways to choose Dlowastb from Hq (i) Dlowastb cap ABC emptyand (ii) Dlowastb cap ABC neempty With a similar argument to (a)case (i) results in u1 u2 u3 0 while case (ii) results inu1 gt 0 u2 gt 0 or u3 gt 0 erefore if Dlowast is a B2-GMC designDlowastb should be chosen according to case (i) ieDlowastb sub Hqminus w (q minus 1) (q minus 1)Hqminus w1113966 1113967 When choosing Dlowastbaccording to (i) there should be Dlowastb cap q minus 11113864

(q minus 1)Hqminus w empty ie Dlowastb sub Hqminus w otherwise Dlowastt capU

(Dlowastb )neempty noting that U(Dlowastb )cap (q minus 1 (q minus 1)Hqminus w)1113966 1113967

ge 2k gt 2r+1 minus 1 as shown in Lemma 1 (iii) e remainder ofthe proof is similar to that of (a)

For (c) when kgt q minus w minus 1 there are two different waysto choose Dlowastb from Hq (i) Dlowastb cap BC empty and (ii)Dlowastb cap BC neempty With a similar argument to (a) it is nothard to verify that case (ii) results in u1 gt 0 or u2 gt 0 whilecase (i) results in u1 u2 0 erefore if Dlowast is a B2-GMCdesign Dlowastb should be an s-projection of Hqminus 2 BC eremainder of the proof is similar to that of (a)

Remark 1 eorem 2 shows that when constructing theB2-GMC designs with N4 + 1lt nle 5N16 we can firstpartition the range (N4 + 1 5N16) into q minus 4 sequentialsubranges as (N(2wminus 1 + 1)2w+1 N(2wminus 2 + 1)2w) with w

4 5 q minus 1 and then obtain the B2-GMC designsaccording to eorem 2

In the following an example is provided to illustrate theconstruction method in eorem 2 and Remark 1

Example 2 Consider the constructions of B2-GMC219minus 13 2s designs for s 1 3 4 Here n 19 m 13 q 6and N 64 e values of the parameters N and n satisfyN4 + 1lt nle 5N16 ie n isin (17 20] From Remark 1 wepartition the range (17 20] into two subranges (17 18]

because w 5 and (18 20] because w 4 Since19 isin (18 20] according to eorem 2 Dt should be the last19 columns of S5N16 whereS5N16 Dqminus 4(Φ(4)) (5 5H2 6 6H2 56(H4H2)) from(18) erefore Dt (5H2 6 6H2 56(H4H2)) Note that5N16 minus n 20 minus 19 1 then r 0 Next we chooseDb fors 1 3 4

For s 1 we have k 0 According to eorem 2 (a)Db 5 as H0 empty

For s 3 we have k 1 According toeorem 2 (b)Db

should be a 3-projection of H2 ie Db H2 (1 2 12)For s 4 we have k 2 According toeorem 2 (c)Db

should be a 4-projection of H4 Without loss of generalitywe choose Db (1 2 12 3)

4 Concluding Remarks

e regular 2nminus m designs have wide applications in engi-neering manufacturing industry agriculture and medicineand so on When the size of experimental units is large theinhomogeneity of experimental units results in unwantedvariances of estimations of treatment effects An essentialway to solve this problem is to partition the experimentalunits into blocks

ere are two kinds of blocking problems as pointed outin [1] One is called the single block variable problem whichconsiders only a single block variable and the other is calledmulti-block variable problem which considers two or moreblock variables As stated in Section 1 experiments whichinvolve multi-block variables are more widely concerned inpractice than those which involve only a single block var-iable However due to the complexity of multi-block var-iable problem the studies on constructing optimal designswith multi-block variables are relatively rare

In this paper we aim at exploring the theories andconstructions of optimal blocked 2nminus m designs with multi-block variables e prevalent B2-GMC criterion is adoptedis criterion is preferable when there is some priorknowledge on the importance ordering of the treatmenteffects e systematical construction methods of theB2-GMC 2nminus m 2s designs with N4 + 1le nle 5N16 aredevelopede constructionmethods are concise and easy toimplement as indicated by the examples provided

Appendix

Lemma A1 is some result from Lemmas A2 A4 and A5 in[21] and Lemma 1 in [25]

Lemma A1 Suppose O is any s-projection of Hq with2k le sle 2k+1 minus 1 and kle q minus 2

(i) If Ocap q qHqminus 11113966 1113967 empty then U(O)capHqminus 11113966 1113967

ge 2k+1 minus 1 and when O has k + 1 independent col-umns the equality holds

(ii) If Ocap q qHqminus 11113966 1113967neempty then U(O)capHqminus 11113966 1113967

ge 2k minus 1 and when O has k + 1 independent col-umns the equality holds

(iii) If Ocap q qHqminus 11113966 1113967neempty then U(O)cap q qHqminus 11113966 11139671113966 1113967

ge 2k and when O has k + 1 independent columnsthe equality holds

(iv) If O sub Hk+1 then U(O) Hk+1(v) If O sub Hk q qHk1113864 1113865 then U(O) Hk q qHk1113864 1113865

Lemma A2 is some result of Lemma 1 in [34]

Lemma A2 Suppose O consists of the last n columns ofq qHqminus 11113966 1113967 then B2(O γ1)geB2(O γ2) if γ1 is ahead of γ2 inHqminus 1

Data Availability

No data were used to support this study

10 Mathematical Problems in Engineering

Conflicts of Interest

e author declares that there are no conflicts of interest

Acknowledgments

is study was supported by the National Natural ScienceFoundation of China (grant no 11801331)

References

[1] S Bisgaard ldquoA note on the definition of resolution for blocked2 k-p designsrdquo Technometrics vol 36 no 3 pp 308ndash3111994

[2] R R Sitter J Chen and M Feder ldquoFractional resolution andminimum aberration in blocked 2 n-k designsrdquo Techno-metrics vol 39 no 4 pp 382ndash390 1997

[3] H Chen and C S Cheng ldquoeory of optimal blocking ofsnminus mdesignsrdquo Annals of Statistics vol 27 no 6pp 1948ndash1973 1999

[4] R Zhang and D Park ldquoOptimal blocking of two-level frac-tional factorial designsrdquo Journal of Statistical Planning andInference vol 91 no 1 pp 107ndash121 2000

[5] S W Cheng and C F J Wu ldquoChoice of optimal blockingschemes in two-level and three-level designsrdquo Technometricsvol 44 no 3 pp 269ndash277 2002

[6] H Xu ldquoBlocked regular fractional factorial designs withminimum aberrationrdquo Annals of Statistics vol 8 no 5pp 2534ndash2553 2006

[7] H Xu and R W Mee ldquoMinimum aberration blockingschemes for 128-run designsrdquo Journal of Statistical Planningand Inference vol 140 no 11 pp 3213ndash3229 2010

[8] S Zhao P Li and R Karunamuni ldquoBlocked two-level regularfactorial designs with weak minimum aberrationrdquo Bio-metrika vol 100 no 1 pp 249ndash253 2013

[9] S-L Zhao and P-F Li ldquoConstruction of minimum aberrationblocked two-level regular factorial designsrdquo Communicationsin Statistics - eory and Methods vol 45 no 17 pp 5028ndash5036 2016

[10] B-J Chen P-F Li M-Q Liu and R-C Zhang ldquoSome resultson blocked regular 2-level fractional factorial designs withclear effectsrdquo Journal of Statistical Planning and Inferencevol 136 no 12 pp 4436ndash4449 2006

[11] S Zhao P Li and M-Q Liu ldquoOn blocked resolution IVdesigns containing clear two-factor interactionsrdquo Journal ofComplexity vol 29 no 5 pp 389ndash395 2013

[12] R C Zhang and RMukerjee ldquoGeneral minimum lower-orderconfounding in block designs using complementary setsrdquoStatistica Sinica vol 19 no 4 pp 1787ndash1802 2009

[13] S Zhao D K J Lin and P Li ldquoA note on the construction ofblocked two-level designs with general minimum lower orderconfoundingrdquo Journal of Statistical Planning and Inferencevol 172 pp 16ndash22 2016

[14] S-L Zhao and Q-Q Zhao ldquoSome results on constructingtwo-level block designs with general minimum lower orderconfoundingrdquo Communications in Statistics-eory andMethods vol 47 no 9 pp 2227ndash2237 2018

[15] R C Zhang P Li S L Zhao and M Y Ai ldquoA generalminimum lower-order confounding criterion for two-levelregular designsrdquo Statistica Sinica vol 18 no 4 pp 1689ndash1705 2008

[16] S Zhao P Li R Zhang and R Karunamuni ldquoConstruction ofblocked two-level regular designs with general minimum

lower order confoundingrdquo Journal of Statistical Planning andInference vol 143 no 6 pp 1082ndash1090 2013

[17] Y Zhao S Zhao and M-Q Liu ldquoA theory on constructingblocked two-level designs with general minimum lower orderconfoundingrdquo Frontiers of Mathematics in China vol 11no 1 pp 207ndash235 2016

[18] S-L Zhao and Q Sun ldquoOn constructing general minimumlower order confounding two-level block designsrdquo Commu-nications in Statistics-eory and Methods vol 46 no 3pp 1261ndash1274 2017

[19] J Wei P Li and R Zhang ldquoBlocked two-level regular designswith general minimum lower order confoundingrdquo Journal ofStatistical eory and Practice vol 8 no 1 pp 46ndash65 2014

[20] C F J Wu and M S Hamada Experiments PlanningAnalysis and Optimization Wiley Hoboken NJ USA 2edition 2009

[21] Q Zhao and S Zhao ldquoSome results on two-level regulardesigns with multi block variables containing clear effectsrdquoStatistical Papers vol 60 no 5 pp 1569ndash1582 2019

[22] S Zhao and Q Zhao ldquoMinimum aberration blocked designswith multiple block variablesrdquo Metrika vol 84 no 2pp 121ndash140 2021

[23] R Zhang P Li and J Wei ldquoOptimal two-level regular designswith multi block variablesrdquo Journal of Statistical eory andPractice vol 5 no 1 pp 161ndash178 2011

[24] Y N Zhao S L Zhao and M Q Liu ldquoOn constructingoptimal two-level designs with multi block variablesrdquo Journalof Systems Science and Complexity vol 31 pp 1ndash14 2018

[25] Y N Zhao and S L Zhao ldquoConstruction of optimal blockeddesigns with multi block variablesrdquoAIMSmathematics vol 6no 6 pp 6293ndash6308 2021

[26] B Tang and C F J Wu ldquoCharacterization of minimumaberration 2nminus kdesigns in terms of their complementarydesignsrdquo Annals of Statistics vol 25 no 6 pp 2549ndash25591996

[27] C F J Wu and Y Chen ldquoA graph-aided method for planningtwo-level experiments when certain interactions are impor-tantrdquo Technometrics vol 34 no 2 pp 162ndash175 1992

[28] G E P Box and J S Hunter ldquoe 2kminus pfractional factorialdesigns I and IIrdquo Technometrics vol 3 no 4 pp 311ndash3511961

[29] Y Cheng and R Zhang ldquoOn construction of general mini-mum lower order confounding 2nminus m designs with (N4)+

1le nle 9N32rdquo Journal of Statistical Planning and Inferencevol 140 no 9 pp 2384ndash2394 2010

[30] H Wu and C F J Wu ldquoClear two-factor interactions andminimum aberrationrdquo Annals of Statistics vol 30 no 5pp 1496ndash1511 2002

[31] R M Block and R W Mee ldquoSecond order saturated reso-lution IV designsrdquo Journal of Statistical eory and Appli-cations vol 2 no 2 pp 96ndash112 2003

[32] A A Davydov and L M Tombak ldquoQuasi-perfect linear bi-nary codes with distance 4 and complete caps in projectivegeometryrdquo Problems of Information Transmission vol 25no 4 pp 265ndash275 1990

[33] R Zhang and Y Cheng ldquoGeneral minimum lower orderconfounding designs an overview and a construction theoryrdquoJournal of Statistical Planning and Inference vol 140 no 7pp 1719ndash1730 2010

[34] Q Zhou N Balakrishnan and R Zhang ldquoe factor aliasedeffect number pattern and its application in experimentalplanningrdquo Canadian Journal of Statistics vol 41 no 3pp 540ndash555 2013

Mathematical Problems in Engineering 11

2 C

(p)

2 Dlowast( 1113857

n minusN

41113874 1113875

N

4minus u11113874 1113875 forp n minus

N

4minus 1

N

2w

N

4minus

N

2w minus u21113874 1113875 forp N

2w minus 1

N

8minus

N

2w1113874 1113875N

4minus

N

2w minus u31113874 1113875 forp N

8minus

N

2w minus 1

f Dlowastt p( 1113857 forpgeN

8minus 1

0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(30)

where f(Dlowastt p) 2 C

(p)

2 (Dlowastt ) minus (p + 1) γ γ isin Hqminus1113966

wcapU(Dlowastb ) B2(Dlowastt γ) p + 1 erefore Dlowast sequentiallymaximizes 2 C2 among all the possible 2nminus m 2s designs onlyif Dlowast sequentially minimizes

u1 u2 u3( 1113857 (31)

For ease of presenting let A q(Hqminus 2Hqminus w)1113966

q(q minus 1)(IHqminus w) B (q minus 1)(Hqminus 2Hqminus w) and C

Hqminus 2Hqminus wFor (a) when kle q minus w minus 1 there are two different ways

to choose Dlowastb from Hq (i) Dlowastb cap ABC empty and (ii)Dlowastb cap ABC neempty Recall that if Dlowast (Dlowastt Dlowastb ) is aB2-GMC design then Dlowastt sub SN(2wminus 2+1)2w with D

lowastt

SN(2wminus 2+1)2w Dlowastt sub Dqminus w((w minus 1)2w ) Case (i) implies thatDlowastb sub Hqminus w D

lowastt1113966 1113967 and thus Dlowastb sub Hqminus w q minus 1 (q minus 1)1113966

Hqminus w erefore choosing Dlowastb according to case (i) resultsin u1 u2 u3 0 If Dlowastb is chosen according to case (ii)then U(Dlowastb )cap ABC neempty which results in u1 gt 0 u2 gt 0 oru3 gt 0 Clearly if Dlowast is a B2-GMC design Dlowastb should be

chosen as (i) In the following we consider onlyDlowastb cap ABC empty ie Dlowastb sub Hqminus w (q minus 1) (q minus 1)Hqminus w1113966 1113967

For D (Dt Db) in (a) according to (v) in Lemma 1 ifDb sub Hk (q minus 1) (q minus 1)Hk1113864 1113865 with kle q minus w minus 1 thenU(Db) Hk (q minus 1) (q minus 1)Hk1113864 1113865 Denote b as the lastcolumn of Hk in Yates order and B2(Dt b) b thenbgeN8 According to Lemma 4 for γ isin Hqminus w if γ is ahead ofb in Hqminus w then B2(Dt γ)ge b by (18) For anyγ isin (q minus 1) (q minus 1)Hk1113864 1113865 we have B2(Dt γ) 0 For any γsuch that B2(Dt γ)lt b we have γ isin HqHk erefore forple b minus 2

2 C

(p)

2 (D) (p + 1) γ isin Hq γ notin U Db( 1113857 B2 Dt γ( 1113857 p + 11113966 1113967

(p + 1) γ isin Hq B2 Dt γ( 1113857 p + 11113966 1113967

2 C

(p)

2 Dt( 1113857

(32)

and for pge b

2 C

(p)

2 (D) (p + 1) γ isin Hq γ notin U Db( 1113857 B2 Dt γ( 1113857 p + 11113966 1113967

(p + 1) γ isin Hq B2 Dt γ( 1113857 p + 11113966 1113967 minus (p + 1) γ isin Hk B2 Dt γ( 1113857 p + 11113864 1113865

0

(33)

From (33) it is obtained that

2 C

(0)

2 (D) 2 C

(N8minus 1)

2 (D)2 C

(N8)

2 (D) 2 C

(bminus 2)

2 (D)1113874 1113875

2 C

(0)

2 Dt( 1113857 2 C

(N8minus 1)

2 Dt( 11138572 C

(N8)

2 Dt( 1113857 2 C

(bminus 2)

2 Dt( 11138571113874 1113875

(34)

Note that Dt is a GMC design [29] then D maximizes

2 C

(0)

2 (middot) 2 C

(N8minus 1)

2 (middot)2 C

(N8)

2 (middot) 2 C

(bminus 2)

2 (middot)1113874 1113875

(35)

among all the possible 2nminus m 2s designsSupposeD is not aB2-GMC design then there should be

aDlowast (Dlowastt Dlowastb ) which outperformsD (Dt Db) in termsof (3)is implies that there exists some p1 ge b minus 1 such that

8 Mathematical Problems in Engineering

2 C

(0)

2 Dlowast( 1113857 2 C

(N8minus 1)

2 Dlowast( 11138572 C

(N8)

2 Dlowast( 1113857 2 C

(bminus 2)

2 Dlowast( 11138571113874 1113875

2 C

(0)

2 (D) 2 C

(N8minus 1)

2 (D)2 C

(N8)

2 (D) 2 C

(bminus 2)

2 (D)1113874 1113875

(36)

2 C

p1minus 1( )2 Dlowast( 1113857gt 2 C

p1minus 1( )2 (D) (37)

Recalling the definitions of 2 C(p)

2 (D) and B2(Dt γ) wehave

1113944

P

p0

2 C

(p)

2 (D)

(p + 1) 1113944

P

p0 γ isin Hq γ notin U Db( 1113857 B2 Dt γ( 1113857 p + 11113966 1113967

1113944P

p0 γ isin HqSN 2wminus 2+1( )2w1113872 1113873 B2 Dt γ( 1113857 p + 11113966 1113967

minus 1113944P

p0 γ isin U Db( 1113857 B2 Dt γ( 1113857 p + 11113864 1113865

HqSN 2wminus 2+1( )2w1113872 1113873 minus Hk

2qminus

N 2wminus 2+ 11113872 1113873

2w minus 2k

(38)

where the second and third equalities are due to B2(Dt γ)

0 for any γ isin SN(2wminus 2+1)2w Similarly

1113944

P

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1) 1113944

P

p0 γ isin Hq γ notin U Dlowastb( 1113857 B2 Dlowastt γ( 1113857 p + 11113966 1113967

1113944P

p0 γ isin HqSN 2wminus 2+1( )2w1113872 1113873 B2 Dlowastt γ( 1113857 p + 11113966 1113967

minus 1113944P

p0 γ isin U Dlowastb( 1113857 B2 Dlowastt γ( 1113857 p + 11113864 1113865

HqSN 2wminus 2+1( )2w1113872 1113873 minus U Dlowastb( 1113857

2qminus

N 2wminus 2+ 11113872 1113873

2w minus U Dlowastb( 1113857

(39)

From (33) and (38) we have 1113944

P

p0

2 C

(p)

2 (D)

(p + 1) 1113944

bminus 2

p0

2 C

(p)

2 (D)

(p + 1)+

2 C

(bminus 1)

2 (D)

b

2qminus

N 2wminus 2+ 11113872 1113873

2w minus 2k

(40)

By (36) and (37) it is obtained that

Mathematical Problems in Engineering 9

1113944

P

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1)ge 1113944

bminus 2

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1)+

2 C

(bminus 1)

2 Dlowast( 1113857

b

gt 1113944bminus 2

p0

2 C

(p)

2 (D)

(p + 1)+

2 C

(bminus 1)

2 (D)

b

(41)

en according to (39)ndash(41) we obtain U(Dlowastb )lt 2k minus 1which contradicts Lemma 1 (i) and (ii)

For (b) similar to (a) when kle q minus w minus 1 there are twodifferent ways to choose Dlowastb from Hq (i) Dlowastb cap ABC emptyand (ii) Dlowastb cap ABC neempty With a similar argument to (a)case (i) results in u1 u2 u3 0 while case (ii) results inu1 gt 0 u2 gt 0 or u3 gt 0 erefore if Dlowast is a B2-GMC designDlowastb should be chosen according to case (i) ieDlowastb sub Hqminus w (q minus 1) (q minus 1)Hqminus w1113966 1113967 When choosing Dlowastbaccording to (i) there should be Dlowastb cap q minus 11113864

(q minus 1)Hqminus w empty ie Dlowastb sub Hqminus w otherwise Dlowastt capU

(Dlowastb )neempty noting that U(Dlowastb )cap (q minus 1 (q minus 1)Hqminus w)1113966 1113967

ge 2k gt 2r+1 minus 1 as shown in Lemma 1 (iii) e remainder ofthe proof is similar to that of (a)

For (c) when kgt q minus w minus 1 there are two different waysto choose Dlowastb from Hq (i) Dlowastb cap BC empty and (ii)Dlowastb cap BC neempty With a similar argument to (a) it is nothard to verify that case (ii) results in u1 gt 0 or u2 gt 0 whilecase (i) results in u1 u2 0 erefore if Dlowast is a B2-GMCdesign Dlowastb should be an s-projection of Hqminus 2 BC eremainder of the proof is similar to that of (a)

Remark 1 eorem 2 shows that when constructing theB2-GMC designs with N4 + 1lt nle 5N16 we can firstpartition the range (N4 + 1 5N16) into q minus 4 sequentialsubranges as (N(2wminus 1 + 1)2w+1 N(2wminus 2 + 1)2w) with w

4 5 q minus 1 and then obtain the B2-GMC designsaccording to eorem 2

In the following an example is provided to illustrate theconstruction method in eorem 2 and Remark 1

Example 2 Consider the constructions of B2-GMC219minus 13 2s designs for s 1 3 4 Here n 19 m 13 q 6and N 64 e values of the parameters N and n satisfyN4 + 1lt nle 5N16 ie n isin (17 20] From Remark 1 wepartition the range (17 20] into two subranges (17 18]

because w 5 and (18 20] because w 4 Since19 isin (18 20] according to eorem 2 Dt should be the last19 columns of S5N16 whereS5N16 Dqminus 4(Φ(4)) (5 5H2 6 6H2 56(H4H2)) from(18) erefore Dt (5H2 6 6H2 56(H4H2)) Note that5N16 minus n 20 minus 19 1 then r 0 Next we chooseDb fors 1 3 4

For s 1 we have k 0 According to eorem 2 (a)Db 5 as H0 empty

For s 3 we have k 1 According toeorem 2 (b)Db

should be a 3-projection of H2 ie Db H2 (1 2 12)For s 4 we have k 2 According toeorem 2 (c)Db

should be a 4-projection of H4 Without loss of generalitywe choose Db (1 2 12 3)

4 Concluding Remarks

e regular 2nminus m designs have wide applications in engi-neering manufacturing industry agriculture and medicineand so on When the size of experimental units is large theinhomogeneity of experimental units results in unwantedvariances of estimations of treatment effects An essentialway to solve this problem is to partition the experimentalunits into blocks

ere are two kinds of blocking problems as pointed outin [1] One is called the single block variable problem whichconsiders only a single block variable and the other is calledmulti-block variable problem which considers two or moreblock variables As stated in Section 1 experiments whichinvolve multi-block variables are more widely concerned inpractice than those which involve only a single block var-iable However due to the complexity of multi-block var-iable problem the studies on constructing optimal designswith multi-block variables are relatively rare

In this paper we aim at exploring the theories andconstructions of optimal blocked 2nminus m designs with multi-block variables e prevalent B2-GMC criterion is adoptedis criterion is preferable when there is some priorknowledge on the importance ordering of the treatmenteffects e systematical construction methods of theB2-GMC 2nminus m 2s designs with N4 + 1le nle 5N16 aredevelopede constructionmethods are concise and easy toimplement as indicated by the examples provided

Appendix

Lemma A1 is some result from Lemmas A2 A4 and A5 in[21] and Lemma 1 in [25]

Lemma A1 Suppose O is any s-projection of Hq with2k le sle 2k+1 minus 1 and kle q minus 2

(i) If Ocap q qHqminus 11113966 1113967 empty then U(O)capHqminus 11113966 1113967

ge 2k+1 minus 1 and when O has k + 1 independent col-umns the equality holds

(ii) If Ocap q qHqminus 11113966 1113967neempty then U(O)capHqminus 11113966 1113967

ge 2k minus 1 and when O has k + 1 independent col-umns the equality holds

(iii) If Ocap q qHqminus 11113966 1113967neempty then U(O)cap q qHqminus 11113966 11139671113966 1113967

ge 2k and when O has k + 1 independent columnsthe equality holds

(iv) If O sub Hk+1 then U(O) Hk+1(v) If O sub Hk q qHk1113864 1113865 then U(O) Hk q qHk1113864 1113865

Lemma A2 is some result of Lemma 1 in [34]

Lemma A2 Suppose O consists of the last n columns ofq qHqminus 11113966 1113967 then B2(O γ1)geB2(O γ2) if γ1 is ahead of γ2 inHqminus 1

Data Availability

No data were used to support this study

10 Mathematical Problems in Engineering

Conflicts of Interest

e author declares that there are no conflicts of interest

Acknowledgments

is study was supported by the National Natural ScienceFoundation of China (grant no 11801331)

References

[1] S Bisgaard ldquoA note on the definition of resolution for blocked2 k-p designsrdquo Technometrics vol 36 no 3 pp 308ndash3111994

[2] R R Sitter J Chen and M Feder ldquoFractional resolution andminimum aberration in blocked 2 n-k designsrdquo Techno-metrics vol 39 no 4 pp 382ndash390 1997

[3] H Chen and C S Cheng ldquoeory of optimal blocking ofsnminus mdesignsrdquo Annals of Statistics vol 27 no 6pp 1948ndash1973 1999

[4] R Zhang and D Park ldquoOptimal blocking of two-level frac-tional factorial designsrdquo Journal of Statistical Planning andInference vol 91 no 1 pp 107ndash121 2000

[5] S W Cheng and C F J Wu ldquoChoice of optimal blockingschemes in two-level and three-level designsrdquo Technometricsvol 44 no 3 pp 269ndash277 2002

[6] H Xu ldquoBlocked regular fractional factorial designs withminimum aberrationrdquo Annals of Statistics vol 8 no 5pp 2534ndash2553 2006

[7] H Xu and R W Mee ldquoMinimum aberration blockingschemes for 128-run designsrdquo Journal of Statistical Planningand Inference vol 140 no 11 pp 3213ndash3229 2010

[8] S Zhao P Li and R Karunamuni ldquoBlocked two-level regularfactorial designs with weak minimum aberrationrdquo Bio-metrika vol 100 no 1 pp 249ndash253 2013

[9] S-L Zhao and P-F Li ldquoConstruction of minimum aberrationblocked two-level regular factorial designsrdquo Communicationsin Statistics - eory and Methods vol 45 no 17 pp 5028ndash5036 2016

[10] B-J Chen P-F Li M-Q Liu and R-C Zhang ldquoSome resultson blocked regular 2-level fractional factorial designs withclear effectsrdquo Journal of Statistical Planning and Inferencevol 136 no 12 pp 4436ndash4449 2006

[11] S Zhao P Li and M-Q Liu ldquoOn blocked resolution IVdesigns containing clear two-factor interactionsrdquo Journal ofComplexity vol 29 no 5 pp 389ndash395 2013

[12] R C Zhang and RMukerjee ldquoGeneral minimum lower-orderconfounding in block designs using complementary setsrdquoStatistica Sinica vol 19 no 4 pp 1787ndash1802 2009

[13] S Zhao D K J Lin and P Li ldquoA note on the construction ofblocked two-level designs with general minimum lower orderconfoundingrdquo Journal of Statistical Planning and Inferencevol 172 pp 16ndash22 2016

[14] S-L Zhao and Q-Q Zhao ldquoSome results on constructingtwo-level block designs with general minimum lower orderconfoundingrdquo Communications in Statistics-eory andMethods vol 47 no 9 pp 2227ndash2237 2018

[15] R C Zhang P Li S L Zhao and M Y Ai ldquoA generalminimum lower-order confounding criterion for two-levelregular designsrdquo Statistica Sinica vol 18 no 4 pp 1689ndash1705 2008

[16] S Zhao P Li R Zhang and R Karunamuni ldquoConstruction ofblocked two-level regular designs with general minimum

lower order confoundingrdquo Journal of Statistical Planning andInference vol 143 no 6 pp 1082ndash1090 2013

[17] Y Zhao S Zhao and M-Q Liu ldquoA theory on constructingblocked two-level designs with general minimum lower orderconfoundingrdquo Frontiers of Mathematics in China vol 11no 1 pp 207ndash235 2016

[18] S-L Zhao and Q Sun ldquoOn constructing general minimumlower order confounding two-level block designsrdquo Commu-nications in Statistics-eory and Methods vol 46 no 3pp 1261ndash1274 2017

[19] J Wei P Li and R Zhang ldquoBlocked two-level regular designswith general minimum lower order confoundingrdquo Journal ofStatistical eory and Practice vol 8 no 1 pp 46ndash65 2014

[20] C F J Wu and M S Hamada Experiments PlanningAnalysis and Optimization Wiley Hoboken NJ USA 2edition 2009

[21] Q Zhao and S Zhao ldquoSome results on two-level regulardesigns with multi block variables containing clear effectsrdquoStatistical Papers vol 60 no 5 pp 1569ndash1582 2019

[22] S Zhao and Q Zhao ldquoMinimum aberration blocked designswith multiple block variablesrdquo Metrika vol 84 no 2pp 121ndash140 2021

[23] R Zhang P Li and J Wei ldquoOptimal two-level regular designswith multi block variablesrdquo Journal of Statistical eory andPractice vol 5 no 1 pp 161ndash178 2011

[24] Y N Zhao S L Zhao and M Q Liu ldquoOn constructingoptimal two-level designs with multi block variablesrdquo Journalof Systems Science and Complexity vol 31 pp 1ndash14 2018

[25] Y N Zhao and S L Zhao ldquoConstruction of optimal blockeddesigns with multi block variablesrdquoAIMSmathematics vol 6no 6 pp 6293ndash6308 2021

[26] B Tang and C F J Wu ldquoCharacterization of minimumaberration 2nminus kdesigns in terms of their complementarydesignsrdquo Annals of Statistics vol 25 no 6 pp 2549ndash25591996

[27] C F J Wu and Y Chen ldquoA graph-aided method for planningtwo-level experiments when certain interactions are impor-tantrdquo Technometrics vol 34 no 2 pp 162ndash175 1992

[28] G E P Box and J S Hunter ldquoe 2kminus pfractional factorialdesigns I and IIrdquo Technometrics vol 3 no 4 pp 311ndash3511961

[29] Y Cheng and R Zhang ldquoOn construction of general mini-mum lower order confounding 2nminus m designs with (N4)+

1le nle 9N32rdquo Journal of Statistical Planning and Inferencevol 140 no 9 pp 2384ndash2394 2010

[30] H Wu and C F J Wu ldquoClear two-factor interactions andminimum aberrationrdquo Annals of Statistics vol 30 no 5pp 1496ndash1511 2002

[31] R M Block and R W Mee ldquoSecond order saturated reso-lution IV designsrdquo Journal of Statistical eory and Appli-cations vol 2 no 2 pp 96ndash112 2003

[32] A A Davydov and L M Tombak ldquoQuasi-perfect linear bi-nary codes with distance 4 and complete caps in projectivegeometryrdquo Problems of Information Transmission vol 25no 4 pp 265ndash275 1990

[33] R Zhang and Y Cheng ldquoGeneral minimum lower orderconfounding designs an overview and a construction theoryrdquoJournal of Statistical Planning and Inference vol 140 no 7pp 1719ndash1730 2010

[34] Q Zhou N Balakrishnan and R Zhang ldquoe factor aliasedeffect number pattern and its application in experimentalplanningrdquo Canadian Journal of Statistics vol 41 no 3pp 540ndash555 2013

Mathematical Problems in Engineering 11

2 C

(0)

2 Dlowast( 1113857 2 C

(N8minus 1)

2 Dlowast( 11138572 C

(N8)

2 Dlowast( 1113857 2 C

(bminus 2)

2 Dlowast( 11138571113874 1113875

2 C

(0)

2 (D) 2 C

(N8minus 1)

2 (D)2 C

(N8)

2 (D) 2 C

(bminus 2)

2 (D)1113874 1113875

(36)

2 C

p1minus 1( )2 Dlowast( 1113857gt 2 C

p1minus 1( )2 (D) (37)

Recalling the definitions of 2 C(p)

2 (D) and B2(Dt γ) wehave

1113944

P

p0

2 C

(p)

2 (D)

(p + 1) 1113944

P

p0 γ isin Hq γ notin U Db( 1113857 B2 Dt γ( 1113857 p + 11113966 1113967

1113944P

p0 γ isin HqSN 2wminus 2+1( )2w1113872 1113873 B2 Dt γ( 1113857 p + 11113966 1113967

minus 1113944P

p0 γ isin U Db( 1113857 B2 Dt γ( 1113857 p + 11113864 1113865

HqSN 2wminus 2+1( )2w1113872 1113873 minus Hk

2qminus

N 2wminus 2+ 11113872 1113873

2w minus 2k

(38)

where the second and third equalities are due to B2(Dt γ)

0 for any γ isin SN(2wminus 2+1)2w Similarly

1113944

P

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1) 1113944

P

p0 γ isin Hq γ notin U Dlowastb( 1113857 B2 Dlowastt γ( 1113857 p + 11113966 1113967

1113944P

p0 γ isin HqSN 2wminus 2+1( )2w1113872 1113873 B2 Dlowastt γ( 1113857 p + 11113966 1113967

minus 1113944P

p0 γ isin U Dlowastb( 1113857 B2 Dlowastt γ( 1113857 p + 11113864 1113865

HqSN 2wminus 2+1( )2w1113872 1113873 minus U Dlowastb( 1113857

2qminus

N 2wminus 2+ 11113872 1113873

2w minus U Dlowastb( 1113857

(39)

From (33) and (38) we have 1113944

P

p0

2 C

(p)

2 (D)

(p + 1) 1113944

bminus 2

p0

2 C

(p)

2 (D)

(p + 1)+

2 C

(bminus 1)

2 (D)

b

2qminus

N 2wminus 2+ 11113872 1113873

2w minus 2k

(40)

By (36) and (37) it is obtained that

Mathematical Problems in Engineering 9

1113944

P

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1)ge 1113944

bminus 2

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1)+

2 C

(bminus 1)

2 Dlowast( 1113857

b

gt 1113944bminus 2

p0

2 C

(p)

2 (D)

(p + 1)+

2 C

(bminus 1)

2 (D)

b

(41)

en according to (39)ndash(41) we obtain U(Dlowastb )lt 2k minus 1which contradicts Lemma 1 (i) and (ii)

For (b) similar to (a) when kle q minus w minus 1 there are twodifferent ways to choose Dlowastb from Hq (i) Dlowastb cap ABC emptyand (ii) Dlowastb cap ABC neempty With a similar argument to (a)case (i) results in u1 u2 u3 0 while case (ii) results inu1 gt 0 u2 gt 0 or u3 gt 0 erefore if Dlowast is a B2-GMC designDlowastb should be chosen according to case (i) ieDlowastb sub Hqminus w (q minus 1) (q minus 1)Hqminus w1113966 1113967 When choosing Dlowastbaccording to (i) there should be Dlowastb cap q minus 11113864

(q minus 1)Hqminus w empty ie Dlowastb sub Hqminus w otherwise Dlowastt capU

(Dlowastb )neempty noting that U(Dlowastb )cap (q minus 1 (q minus 1)Hqminus w)1113966 1113967

ge 2k gt 2r+1 minus 1 as shown in Lemma 1 (iii) e remainder ofthe proof is similar to that of (a)

For (c) when kgt q minus w minus 1 there are two different waysto choose Dlowastb from Hq (i) Dlowastb cap BC empty and (ii)Dlowastb cap BC neempty With a similar argument to (a) it is nothard to verify that case (ii) results in u1 gt 0 or u2 gt 0 whilecase (i) results in u1 u2 0 erefore if Dlowast is a B2-GMCdesign Dlowastb should be an s-projection of Hqminus 2 BC eremainder of the proof is similar to that of (a)

Remark 1 eorem 2 shows that when constructing theB2-GMC designs with N4 + 1lt nle 5N16 we can firstpartition the range (N4 + 1 5N16) into q minus 4 sequentialsubranges as (N(2wminus 1 + 1)2w+1 N(2wminus 2 + 1)2w) with w

4 5 q minus 1 and then obtain the B2-GMC designsaccording to eorem 2

In the following an example is provided to illustrate theconstruction method in eorem 2 and Remark 1

Example 2 Consider the constructions of B2-GMC219minus 13 2s designs for s 1 3 4 Here n 19 m 13 q 6and N 64 e values of the parameters N and n satisfyN4 + 1lt nle 5N16 ie n isin (17 20] From Remark 1 wepartition the range (17 20] into two subranges (17 18]

because w 5 and (18 20] because w 4 Since19 isin (18 20] according to eorem 2 Dt should be the last19 columns of S5N16 whereS5N16 Dqminus 4(Φ(4)) (5 5H2 6 6H2 56(H4H2)) from(18) erefore Dt (5H2 6 6H2 56(H4H2)) Note that5N16 minus n 20 minus 19 1 then r 0 Next we chooseDb fors 1 3 4

For s 1 we have k 0 According to eorem 2 (a)Db 5 as H0 empty

For s 3 we have k 1 According toeorem 2 (b)Db

should be a 3-projection of H2 ie Db H2 (1 2 12)For s 4 we have k 2 According toeorem 2 (c)Db

should be a 4-projection of H4 Without loss of generalitywe choose Db (1 2 12 3)

4 Concluding Remarks

e regular 2nminus m designs have wide applications in engi-neering manufacturing industry agriculture and medicineand so on When the size of experimental units is large theinhomogeneity of experimental units results in unwantedvariances of estimations of treatment effects An essentialway to solve this problem is to partition the experimentalunits into blocks

ere are two kinds of blocking problems as pointed outin [1] One is called the single block variable problem whichconsiders only a single block variable and the other is calledmulti-block variable problem which considers two or moreblock variables As stated in Section 1 experiments whichinvolve multi-block variables are more widely concerned inpractice than those which involve only a single block var-iable However due to the complexity of multi-block var-iable problem the studies on constructing optimal designswith multi-block variables are relatively rare

In this paper we aim at exploring the theories andconstructions of optimal blocked 2nminus m designs with multi-block variables e prevalent B2-GMC criterion is adoptedis criterion is preferable when there is some priorknowledge on the importance ordering of the treatmenteffects e systematical construction methods of theB2-GMC 2nminus m 2s designs with N4 + 1le nle 5N16 aredevelopede constructionmethods are concise and easy toimplement as indicated by the examples provided

Appendix

Lemma A1 is some result from Lemmas A2 A4 and A5 in[21] and Lemma 1 in [25]

Lemma A1 Suppose O is any s-projection of Hq with2k le sle 2k+1 minus 1 and kle q minus 2

(i) If Ocap q qHqminus 11113966 1113967 empty then U(O)capHqminus 11113966 1113967

ge 2k+1 minus 1 and when O has k + 1 independent col-umns the equality holds

(ii) If Ocap q qHqminus 11113966 1113967neempty then U(O)capHqminus 11113966 1113967

ge 2k minus 1 and when O has k + 1 independent col-umns the equality holds

(iii) If Ocap q qHqminus 11113966 1113967neempty then U(O)cap q qHqminus 11113966 11139671113966 1113967

ge 2k and when O has k + 1 independent columnsthe equality holds

(iv) If O sub Hk+1 then U(O) Hk+1(v) If O sub Hk q qHk1113864 1113865 then U(O) Hk q qHk1113864 1113865

Lemma A2 is some result of Lemma 1 in [34]

Lemma A2 Suppose O consists of the last n columns ofq qHqminus 11113966 1113967 then B2(O γ1)geB2(O γ2) if γ1 is ahead of γ2 inHqminus 1

Data Availability

No data were used to support this study

10 Mathematical Problems in Engineering

Conflicts of Interest

e author declares that there are no conflicts of interest

Acknowledgments

is study was supported by the National Natural ScienceFoundation of China (grant no 11801331)

References

[1] S Bisgaard ldquoA note on the definition of resolution for blocked2 k-p designsrdquo Technometrics vol 36 no 3 pp 308ndash3111994

[2] R R Sitter J Chen and M Feder ldquoFractional resolution andminimum aberration in blocked 2 n-k designsrdquo Techno-metrics vol 39 no 4 pp 382ndash390 1997

[3] H Chen and C S Cheng ldquoeory of optimal blocking ofsnminus mdesignsrdquo Annals of Statistics vol 27 no 6pp 1948ndash1973 1999

[4] R Zhang and D Park ldquoOptimal blocking of two-level frac-tional factorial designsrdquo Journal of Statistical Planning andInference vol 91 no 1 pp 107ndash121 2000

[5] S W Cheng and C F J Wu ldquoChoice of optimal blockingschemes in two-level and three-level designsrdquo Technometricsvol 44 no 3 pp 269ndash277 2002

[6] H Xu ldquoBlocked regular fractional factorial designs withminimum aberrationrdquo Annals of Statistics vol 8 no 5pp 2534ndash2553 2006

[7] H Xu and R W Mee ldquoMinimum aberration blockingschemes for 128-run designsrdquo Journal of Statistical Planningand Inference vol 140 no 11 pp 3213ndash3229 2010

[8] S Zhao P Li and R Karunamuni ldquoBlocked two-level regularfactorial designs with weak minimum aberrationrdquo Bio-metrika vol 100 no 1 pp 249ndash253 2013

[9] S-L Zhao and P-F Li ldquoConstruction of minimum aberrationblocked two-level regular factorial designsrdquo Communicationsin Statistics - eory and Methods vol 45 no 17 pp 5028ndash5036 2016

[10] B-J Chen P-F Li M-Q Liu and R-C Zhang ldquoSome resultson blocked regular 2-level fractional factorial designs withclear effectsrdquo Journal of Statistical Planning and Inferencevol 136 no 12 pp 4436ndash4449 2006

[11] S Zhao P Li and M-Q Liu ldquoOn blocked resolution IVdesigns containing clear two-factor interactionsrdquo Journal ofComplexity vol 29 no 5 pp 389ndash395 2013

[12] R C Zhang and RMukerjee ldquoGeneral minimum lower-orderconfounding in block designs using complementary setsrdquoStatistica Sinica vol 19 no 4 pp 1787ndash1802 2009

[13] S Zhao D K J Lin and P Li ldquoA note on the construction ofblocked two-level designs with general minimum lower orderconfoundingrdquo Journal of Statistical Planning and Inferencevol 172 pp 16ndash22 2016

[14] S-L Zhao and Q-Q Zhao ldquoSome results on constructingtwo-level block designs with general minimum lower orderconfoundingrdquo Communications in Statistics-eory andMethods vol 47 no 9 pp 2227ndash2237 2018

[15] R C Zhang P Li S L Zhao and M Y Ai ldquoA generalminimum lower-order confounding criterion for two-levelregular designsrdquo Statistica Sinica vol 18 no 4 pp 1689ndash1705 2008

[16] S Zhao P Li R Zhang and R Karunamuni ldquoConstruction ofblocked two-level regular designs with general minimum

lower order confoundingrdquo Journal of Statistical Planning andInference vol 143 no 6 pp 1082ndash1090 2013

[17] Y Zhao S Zhao and M-Q Liu ldquoA theory on constructingblocked two-level designs with general minimum lower orderconfoundingrdquo Frontiers of Mathematics in China vol 11no 1 pp 207ndash235 2016

[18] S-L Zhao and Q Sun ldquoOn constructing general minimumlower order confounding two-level block designsrdquo Commu-nications in Statistics-eory and Methods vol 46 no 3pp 1261ndash1274 2017

[19] J Wei P Li and R Zhang ldquoBlocked two-level regular designswith general minimum lower order confoundingrdquo Journal ofStatistical eory and Practice vol 8 no 1 pp 46ndash65 2014

[20] C F J Wu and M S Hamada Experiments PlanningAnalysis and Optimization Wiley Hoboken NJ USA 2edition 2009

[21] Q Zhao and S Zhao ldquoSome results on two-level regulardesigns with multi block variables containing clear effectsrdquoStatistical Papers vol 60 no 5 pp 1569ndash1582 2019

[22] S Zhao and Q Zhao ldquoMinimum aberration blocked designswith multiple block variablesrdquo Metrika vol 84 no 2pp 121ndash140 2021

[23] R Zhang P Li and J Wei ldquoOptimal two-level regular designswith multi block variablesrdquo Journal of Statistical eory andPractice vol 5 no 1 pp 161ndash178 2011

[24] Y N Zhao S L Zhao and M Q Liu ldquoOn constructingoptimal two-level designs with multi block variablesrdquo Journalof Systems Science and Complexity vol 31 pp 1ndash14 2018

[25] Y N Zhao and S L Zhao ldquoConstruction of optimal blockeddesigns with multi block variablesrdquoAIMSmathematics vol 6no 6 pp 6293ndash6308 2021

[26] B Tang and C F J Wu ldquoCharacterization of minimumaberration 2nminus kdesigns in terms of their complementarydesignsrdquo Annals of Statistics vol 25 no 6 pp 2549ndash25591996

[27] C F J Wu and Y Chen ldquoA graph-aided method for planningtwo-level experiments when certain interactions are impor-tantrdquo Technometrics vol 34 no 2 pp 162ndash175 1992

[28] G E P Box and J S Hunter ldquoe 2kminus pfractional factorialdesigns I and IIrdquo Technometrics vol 3 no 4 pp 311ndash3511961

[29] Y Cheng and R Zhang ldquoOn construction of general mini-mum lower order confounding 2nminus m designs with (N4)+

1le nle 9N32rdquo Journal of Statistical Planning and Inferencevol 140 no 9 pp 2384ndash2394 2010

[30] H Wu and C F J Wu ldquoClear two-factor interactions andminimum aberrationrdquo Annals of Statistics vol 30 no 5pp 1496ndash1511 2002

[31] R M Block and R W Mee ldquoSecond order saturated reso-lution IV designsrdquo Journal of Statistical eory and Appli-cations vol 2 no 2 pp 96ndash112 2003

[32] A A Davydov and L M Tombak ldquoQuasi-perfect linear bi-nary codes with distance 4 and complete caps in projectivegeometryrdquo Problems of Information Transmission vol 25no 4 pp 265ndash275 1990

[33] R Zhang and Y Cheng ldquoGeneral minimum lower orderconfounding designs an overview and a construction theoryrdquoJournal of Statistical Planning and Inference vol 140 no 7pp 1719ndash1730 2010

[34] Q Zhou N Balakrishnan and R Zhang ldquoe factor aliasedeffect number pattern and its application in experimentalplanningrdquo Canadian Journal of Statistics vol 41 no 3pp 540ndash555 2013

Mathematical Problems in Engineering 11

1113944

P

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1)ge 1113944

bminus 2

p0

2 C

(p)

2 Dlowast( 1113857

(p + 1)+

2 C

(bminus 1)

2 Dlowast( 1113857

b

gt 1113944bminus 2

p0

2 C

(p)

2 (D)

(p + 1)+

2 C

(bminus 1)

2 (D)

b

(41)

en according to (39)ndash(41) we obtain U(Dlowastb )lt 2k minus 1which contradicts Lemma 1 (i) and (ii)

For (b) similar to (a) when kle q minus w minus 1 there are twodifferent ways to choose Dlowastb from Hq (i) Dlowastb cap ABC emptyand (ii) Dlowastb cap ABC neempty With a similar argument to (a)case (i) results in u1 u2 u3 0 while case (ii) results inu1 gt 0 u2 gt 0 or u3 gt 0 erefore if Dlowast is a B2-GMC designDlowastb should be chosen according to case (i) ieDlowastb sub Hqminus w (q minus 1) (q minus 1)Hqminus w1113966 1113967 When choosing Dlowastbaccording to (i) there should be Dlowastb cap q minus 11113864

(q minus 1)Hqminus w empty ie Dlowastb sub Hqminus w otherwise Dlowastt capU

(Dlowastb )neempty noting that U(Dlowastb )cap (q minus 1 (q minus 1)Hqminus w)1113966 1113967

ge 2k gt 2r+1 minus 1 as shown in Lemma 1 (iii) e remainder ofthe proof is similar to that of (a)

For (c) when kgt q minus w minus 1 there are two different waysto choose Dlowastb from Hq (i) Dlowastb cap BC empty and (ii)Dlowastb cap BC neempty With a similar argument to (a) it is nothard to verify that case (ii) results in u1 gt 0 or u2 gt 0 whilecase (i) results in u1 u2 0 erefore if Dlowast is a B2-GMCdesign Dlowastb should be an s-projection of Hqminus 2 BC eremainder of the proof is similar to that of (a)

Remark 1 eorem 2 shows that when constructing theB2-GMC designs with N4 + 1lt nle 5N16 we can firstpartition the range (N4 + 1 5N16) into q minus 4 sequentialsubranges as (N(2wminus 1 + 1)2w+1 N(2wminus 2 + 1)2w) with w

4 5 q minus 1 and then obtain the B2-GMC designsaccording to eorem 2

In the following an example is provided to illustrate theconstruction method in eorem 2 and Remark 1

Example 2 Consider the constructions of B2-GMC219minus 13 2s designs for s 1 3 4 Here n 19 m 13 q 6and N 64 e values of the parameters N and n satisfyN4 + 1lt nle 5N16 ie n isin (17 20] From Remark 1 wepartition the range (17 20] into two subranges (17 18]

because w 5 and (18 20] because w 4 Since19 isin (18 20] according to eorem 2 Dt should be the last19 columns of S5N16 whereS5N16 Dqminus 4(Φ(4)) (5 5H2 6 6H2 56(H4H2)) from(18) erefore Dt (5H2 6 6H2 56(H4H2)) Note that5N16 minus n 20 minus 19 1 then r 0 Next we chooseDb fors 1 3 4

For s 1 we have k 0 According to eorem 2 (a)Db 5 as H0 empty

For s 3 we have k 1 According toeorem 2 (b)Db

should be a 3-projection of H2 ie Db H2 (1 2 12)For s 4 we have k 2 According toeorem 2 (c)Db

should be a 4-projection of H4 Without loss of generalitywe choose Db (1 2 12 3)

4 Concluding Remarks

e regular 2nminus m designs have wide applications in engi-neering manufacturing industry agriculture and medicineand so on When the size of experimental units is large theinhomogeneity of experimental units results in unwantedvariances of estimations of treatment effects An essentialway to solve this problem is to partition the experimentalunits into blocks

ere are two kinds of blocking problems as pointed outin [1] One is called the single block variable problem whichconsiders only a single block variable and the other is calledmulti-block variable problem which considers two or moreblock variables As stated in Section 1 experiments whichinvolve multi-block variables are more widely concerned inpractice than those which involve only a single block var-iable However due to the complexity of multi-block var-iable problem the studies on constructing optimal designswith multi-block variables are relatively rare

In this paper we aim at exploring the theories andconstructions of optimal blocked 2nminus m designs with multi-block variables e prevalent B2-GMC criterion is adoptedis criterion is preferable when there is some priorknowledge on the importance ordering of the treatmenteffects e systematical construction methods of theB2-GMC 2nminus m 2s designs with N4 + 1le nle 5N16 aredevelopede constructionmethods are concise and easy toimplement as indicated by the examples provided

Appendix

Lemma A1 is some result from Lemmas A2 A4 and A5 in[21] and Lemma 1 in [25]

Lemma A1 Suppose O is any s-projection of Hq with2k le sle 2k+1 minus 1 and kle q minus 2

(i) If Ocap q qHqminus 11113966 1113967 empty then U(O)capHqminus 11113966 1113967

ge 2k+1 minus 1 and when O has k + 1 independent col-umns the equality holds

(ii) If Ocap q qHqminus 11113966 1113967neempty then U(O)capHqminus 11113966 1113967

ge 2k minus 1 and when O has k + 1 independent col-umns the equality holds

(iii) If Ocap q qHqminus 11113966 1113967neempty then U(O)cap q qHqminus 11113966 11139671113966 1113967

ge 2k and when O has k + 1 independent columnsthe equality holds

(iv) If O sub Hk+1 then U(O) Hk+1(v) If O sub Hk q qHk1113864 1113865 then U(O) Hk q qHk1113864 1113865

Lemma A2 is some result of Lemma 1 in [34]

Lemma A2 Suppose O consists of the last n columns ofq qHqminus 11113966 1113967 then B2(O γ1)geB2(O γ2) if γ1 is ahead of γ2 inHqminus 1

Data Availability

No data were used to support this study

10 Mathematical Problems in Engineering

Conflicts of Interest

e author declares that there are no conflicts of interest

Acknowledgments

is study was supported by the National Natural ScienceFoundation of China (grant no 11801331)

References

[1] S Bisgaard ldquoA note on the definition of resolution for blocked2 k-p designsrdquo Technometrics vol 36 no 3 pp 308ndash3111994

[2] R R Sitter J Chen and M Feder ldquoFractional resolution andminimum aberration in blocked 2 n-k designsrdquo Techno-metrics vol 39 no 4 pp 382ndash390 1997

[3] H Chen and C S Cheng ldquoeory of optimal blocking ofsnminus mdesignsrdquo Annals of Statistics vol 27 no 6pp 1948ndash1973 1999

[4] R Zhang and D Park ldquoOptimal blocking of two-level frac-tional factorial designsrdquo Journal of Statistical Planning andInference vol 91 no 1 pp 107ndash121 2000

[5] S W Cheng and C F J Wu ldquoChoice of optimal blockingschemes in two-level and three-level designsrdquo Technometricsvol 44 no 3 pp 269ndash277 2002

[6] H Xu ldquoBlocked regular fractional factorial designs withminimum aberrationrdquo Annals of Statistics vol 8 no 5pp 2534ndash2553 2006

[7] H Xu and R W Mee ldquoMinimum aberration blockingschemes for 128-run designsrdquo Journal of Statistical Planningand Inference vol 140 no 11 pp 3213ndash3229 2010

[8] S Zhao P Li and R Karunamuni ldquoBlocked two-level regularfactorial designs with weak minimum aberrationrdquo Bio-metrika vol 100 no 1 pp 249ndash253 2013

[9] S-L Zhao and P-F Li ldquoConstruction of minimum aberrationblocked two-level regular factorial designsrdquo Communicationsin Statistics - eory and Methods vol 45 no 17 pp 5028ndash5036 2016

[10] B-J Chen P-F Li M-Q Liu and R-C Zhang ldquoSome resultson blocked regular 2-level fractional factorial designs withclear effectsrdquo Journal of Statistical Planning and Inferencevol 136 no 12 pp 4436ndash4449 2006

[11] S Zhao P Li and M-Q Liu ldquoOn blocked resolution IVdesigns containing clear two-factor interactionsrdquo Journal ofComplexity vol 29 no 5 pp 389ndash395 2013

[12] R C Zhang and RMukerjee ldquoGeneral minimum lower-orderconfounding in block designs using complementary setsrdquoStatistica Sinica vol 19 no 4 pp 1787ndash1802 2009

[13] S Zhao D K J Lin and P Li ldquoA note on the construction ofblocked two-level designs with general minimum lower orderconfoundingrdquo Journal of Statistical Planning and Inferencevol 172 pp 16ndash22 2016

[14] S-L Zhao and Q-Q Zhao ldquoSome results on constructingtwo-level block designs with general minimum lower orderconfoundingrdquo Communications in Statistics-eory andMethods vol 47 no 9 pp 2227ndash2237 2018

[15] R C Zhang P Li S L Zhao and M Y Ai ldquoA generalminimum lower-order confounding criterion for two-levelregular designsrdquo Statistica Sinica vol 18 no 4 pp 1689ndash1705 2008

[16] S Zhao P Li R Zhang and R Karunamuni ldquoConstruction ofblocked two-level regular designs with general minimum

lower order confoundingrdquo Journal of Statistical Planning andInference vol 143 no 6 pp 1082ndash1090 2013

[17] Y Zhao S Zhao and M-Q Liu ldquoA theory on constructingblocked two-level designs with general minimum lower orderconfoundingrdquo Frontiers of Mathematics in China vol 11no 1 pp 207ndash235 2016

[18] S-L Zhao and Q Sun ldquoOn constructing general minimumlower order confounding two-level block designsrdquo Commu-nications in Statistics-eory and Methods vol 46 no 3pp 1261ndash1274 2017

[19] J Wei P Li and R Zhang ldquoBlocked two-level regular designswith general minimum lower order confoundingrdquo Journal ofStatistical eory and Practice vol 8 no 1 pp 46ndash65 2014

[20] C F J Wu and M S Hamada Experiments PlanningAnalysis and Optimization Wiley Hoboken NJ USA 2edition 2009

[21] Q Zhao and S Zhao ldquoSome results on two-level regulardesigns with multi block variables containing clear effectsrdquoStatistical Papers vol 60 no 5 pp 1569ndash1582 2019

[22] S Zhao and Q Zhao ldquoMinimum aberration blocked designswith multiple block variablesrdquo Metrika vol 84 no 2pp 121ndash140 2021

[23] R Zhang P Li and J Wei ldquoOptimal two-level regular designswith multi block variablesrdquo Journal of Statistical eory andPractice vol 5 no 1 pp 161ndash178 2011

[24] Y N Zhao S L Zhao and M Q Liu ldquoOn constructingoptimal two-level designs with multi block variablesrdquo Journalof Systems Science and Complexity vol 31 pp 1ndash14 2018

[25] Y N Zhao and S L Zhao ldquoConstruction of optimal blockeddesigns with multi block variablesrdquoAIMSmathematics vol 6no 6 pp 6293ndash6308 2021

[26] B Tang and C F J Wu ldquoCharacterization of minimumaberration 2nminus kdesigns in terms of their complementarydesignsrdquo Annals of Statistics vol 25 no 6 pp 2549ndash25591996

[27] C F J Wu and Y Chen ldquoA graph-aided method for planningtwo-level experiments when certain interactions are impor-tantrdquo Technometrics vol 34 no 2 pp 162ndash175 1992

[28] G E P Box and J S Hunter ldquoe 2kminus pfractional factorialdesigns I and IIrdquo Technometrics vol 3 no 4 pp 311ndash3511961

[29] Y Cheng and R Zhang ldquoOn construction of general mini-mum lower order confounding 2nminus m designs with (N4)+

1le nle 9N32rdquo Journal of Statistical Planning and Inferencevol 140 no 9 pp 2384ndash2394 2010

[30] H Wu and C F J Wu ldquoClear two-factor interactions andminimum aberrationrdquo Annals of Statistics vol 30 no 5pp 1496ndash1511 2002

[31] R M Block and R W Mee ldquoSecond order saturated reso-lution IV designsrdquo Journal of Statistical eory and Appli-cations vol 2 no 2 pp 96ndash112 2003

[32] A A Davydov and L M Tombak ldquoQuasi-perfect linear bi-nary codes with distance 4 and complete caps in projectivegeometryrdquo Problems of Information Transmission vol 25no 4 pp 265ndash275 1990

[33] R Zhang and Y Cheng ldquoGeneral minimum lower orderconfounding designs an overview and a construction theoryrdquoJournal of Statistical Planning and Inference vol 140 no 7pp 1719ndash1730 2010

[34] Q Zhou N Balakrishnan and R Zhang ldquoe factor aliasedeffect number pattern and its application in experimentalplanningrdquo Canadian Journal of Statistics vol 41 no 3pp 540ndash555 2013

Mathematical Problems in Engineering 11

Conflicts of Interest

e author declares that there are no conflicts of interest

Acknowledgments

is study was supported by the National Natural ScienceFoundation of China (grant no 11801331)

References

[1] S Bisgaard ldquoA note on the definition of resolution for blocked2 k-p designsrdquo Technometrics vol 36 no 3 pp 308ndash3111994

[2] R R Sitter J Chen and M Feder ldquoFractional resolution andminimum aberration in blocked 2 n-k designsrdquo Techno-metrics vol 39 no 4 pp 382ndash390 1997

[3] H Chen and C S Cheng ldquoeory of optimal blocking ofsnminus mdesignsrdquo Annals of Statistics vol 27 no 6pp 1948ndash1973 1999

[4] R Zhang and D Park ldquoOptimal blocking of two-level frac-tional factorial designsrdquo Journal of Statistical Planning andInference vol 91 no 1 pp 107ndash121 2000

[5] S W Cheng and C F J Wu ldquoChoice of optimal blockingschemes in two-level and three-level designsrdquo Technometricsvol 44 no 3 pp 269ndash277 2002

[6] H Xu ldquoBlocked regular fractional factorial designs withminimum aberrationrdquo Annals of Statistics vol 8 no 5pp 2534ndash2553 2006

[7] H Xu and R W Mee ldquoMinimum aberration blockingschemes for 128-run designsrdquo Journal of Statistical Planningand Inference vol 140 no 11 pp 3213ndash3229 2010

[8] S Zhao P Li and R Karunamuni ldquoBlocked two-level regularfactorial designs with weak minimum aberrationrdquo Bio-metrika vol 100 no 1 pp 249ndash253 2013

[9] S-L Zhao and P-F Li ldquoConstruction of minimum aberrationblocked two-level regular factorial designsrdquo Communicationsin Statistics - eory and Methods vol 45 no 17 pp 5028ndash5036 2016

[10] B-J Chen P-F Li M-Q Liu and R-C Zhang ldquoSome resultson blocked regular 2-level fractional factorial designs withclear effectsrdquo Journal of Statistical Planning and Inferencevol 136 no 12 pp 4436ndash4449 2006

[11] S Zhao P Li and M-Q Liu ldquoOn blocked resolution IVdesigns containing clear two-factor interactionsrdquo Journal ofComplexity vol 29 no 5 pp 389ndash395 2013

[12] R C Zhang and RMukerjee ldquoGeneral minimum lower-orderconfounding in block designs using complementary setsrdquoStatistica Sinica vol 19 no 4 pp 1787ndash1802 2009

[13] S Zhao D K J Lin and P Li ldquoA note on the construction ofblocked two-level designs with general minimum lower orderconfoundingrdquo Journal of Statistical Planning and Inferencevol 172 pp 16ndash22 2016

[14] S-L Zhao and Q-Q Zhao ldquoSome results on constructingtwo-level block designs with general minimum lower orderconfoundingrdquo Communications in Statistics-eory andMethods vol 47 no 9 pp 2227ndash2237 2018

[15] R C Zhang P Li S L Zhao and M Y Ai ldquoA generalminimum lower-order confounding criterion for two-levelregular designsrdquo Statistica Sinica vol 18 no 4 pp 1689ndash1705 2008

[16] S Zhao P Li R Zhang and R Karunamuni ldquoConstruction ofblocked two-level regular designs with general minimum

lower order confoundingrdquo Journal of Statistical Planning andInference vol 143 no 6 pp 1082ndash1090 2013

[17] Y Zhao S Zhao and M-Q Liu ldquoA theory on constructingblocked two-level designs with general minimum lower orderconfoundingrdquo Frontiers of Mathematics in China vol 11no 1 pp 207ndash235 2016

[18] S-L Zhao and Q Sun ldquoOn constructing general minimumlower order confounding two-level block designsrdquo Commu-nications in Statistics-eory and Methods vol 46 no 3pp 1261ndash1274 2017

[19] J Wei P Li and R Zhang ldquoBlocked two-level regular designswith general minimum lower order confoundingrdquo Journal ofStatistical eory and Practice vol 8 no 1 pp 46ndash65 2014

[20] C F J Wu and M S Hamada Experiments PlanningAnalysis and Optimization Wiley Hoboken NJ USA 2edition 2009

[21] Q Zhao and S Zhao ldquoSome results on two-level regulardesigns with multi block variables containing clear effectsrdquoStatistical Papers vol 60 no 5 pp 1569ndash1582 2019

[22] S Zhao and Q Zhao ldquoMinimum aberration blocked designswith multiple block variablesrdquo Metrika vol 84 no 2pp 121ndash140 2021

[23] R Zhang P Li and J Wei ldquoOptimal two-level regular designswith multi block variablesrdquo Journal of Statistical eory andPractice vol 5 no 1 pp 161ndash178 2011

[24] Y N Zhao S L Zhao and M Q Liu ldquoOn constructingoptimal two-level designs with multi block variablesrdquo Journalof Systems Science and Complexity vol 31 pp 1ndash14 2018

[25] Y N Zhao and S L Zhao ldquoConstruction of optimal blockeddesigns with multi block variablesrdquoAIMSmathematics vol 6no 6 pp 6293ndash6308 2021

[26] B Tang and C F J Wu ldquoCharacterization of minimumaberration 2nminus kdesigns in terms of their complementarydesignsrdquo Annals of Statistics vol 25 no 6 pp 2549ndash25591996

[27] C F J Wu and Y Chen ldquoA graph-aided method for planningtwo-level experiments when certain interactions are impor-tantrdquo Technometrics vol 34 no 2 pp 162ndash175 1992

[28] G E P Box and J S Hunter ldquoe 2kminus pfractional factorialdesigns I and IIrdquo Technometrics vol 3 no 4 pp 311ndash3511961

[29] Y Cheng and R Zhang ldquoOn construction of general mini-mum lower order confounding 2nminus m designs with (N4)+

1le nle 9N32rdquo Journal of Statistical Planning and Inferencevol 140 no 9 pp 2384ndash2394 2010

[30] H Wu and C F J Wu ldquoClear two-factor interactions andminimum aberrationrdquo Annals of Statistics vol 30 no 5pp 1496ndash1511 2002

[31] R M Block and R W Mee ldquoSecond order saturated reso-lution IV designsrdquo Journal of Statistical eory and Appli-cations vol 2 no 2 pp 96ndash112 2003

[32] A A Davydov and L M Tombak ldquoQuasi-perfect linear bi-nary codes with distance 4 and complete caps in projectivegeometryrdquo Problems of Information Transmission vol 25no 4 pp 265ndash275 1990

[33] R Zhang and Y Cheng ldquoGeneral minimum lower orderconfounding designs an overview and a construction theoryrdquoJournal of Statistical Planning and Inference vol 140 no 7pp 1719ndash1730 2010

[34] Q Zhou N Balakrishnan and R Zhang ldquoe factor aliasedeffect number pattern and its application in experimentalplanningrdquo Canadian Journal of Statistics vol 41 no 3pp 540ndash555 2013

Mathematical Problems in Engineering 11