Research On Several Types Of Fractional Factorial Designs Under The Clear Effect Criterion

Factorial experiment has played a fundamental role in many fields.Generally,suppose an experimental outcome is affected by n input variables.These input variables are called factors and the settings of these factors are called levels.An s1 x … x sn factorial experiment involves n(? 1)factors that appear at s1,...,sn(? 2)levels,respectively.In particular,if s1 = …= sn=s,it is called a symmetrical sn factorial design;otherwise it is called an asymmetrical factorial design or a mixed-level design.Any combination of the levels of the n factors is also called a treatment combination.Considering the costs of resources and time,the experimenters usually can not afford a full factorial experiment and only carry out a fraction of the treatment combi-nations.Such a fraction of the full factorial design is called a fractional factorial design.In fractional factorial experiments,one of the most important problems is using which optimality criterion to choose a "good" design.Up to now,the most popular criteria include maximum resolution,minimum aberration,clear effect,maximum estimation capacity and general minimum lower-order confounding criteria,etc.Clear effect(CE)criterion is an important criterion for selecting designs,which was first proposed by Wu and Chen([51]).Under the effect hierarchy principle,if the interactions involving three or more factors can be ignored,a clear main effect or two-factor interaction(2FI)can be estimated unbiasedly.There are mainly two aspects for the research of optimal designs under the CE criterion.One is the study of the conditions for designs to contain clear effects;the other is how to construct designs with more clear effects.In some experiments,if prior knowledge strongly suggests that some main effects and two-factor interactions(2FIs)are likely to be more important,the CE criterion is preferred in order to estimate the important effects as far as possible.The literature on the studies about the CE criterion are mainly focused on the two-level designs,mixed-level designs with two-and four-level factors,two-level blocked designs with single block variable,two-level and mixed-level split-plot designs.Howev-er,few people has studied the conditions for the general mixed-level designs to contain clear effects and the upper and lower bounds on the rmaximum number of clear effects.In addition,there is no literature on the study of the CE problem for blocked designs with multi block variables.Hence,this dissertation is devoted to the study of these designs under the CE criterion,and the dissertation is divided into seven chapters.In Chapter 1,we briefly introduce the background,the existing research work and some basic concepts.The mixed-level designs have many applications in physical experiment and in-dustrial experiment.Up to now,literature on the CE problem of mixed-level designs is mainly focused on designs containing two-and four-level factors,there is no much systematic research on the CE problem for mixed-level designs with general high level factors.We will do some research about this type of design in Chapter 2.Chap-ter 2 considers(2r)x 2n designs(with n two-level factors and one 2r-level factor)and(2r1)x(2r2)x 2n designs(with n two-level factors,one 2r1-level factor,and one 2r2-level factor).All the designs we considered have resolution ? or ?,and 2? runs.For(2r)x 2n designs,there are two types of two-factor interaction components(2FICs):those involving only two-level factors are called type 0,and those involving one com-ponent of the 2r-level factor and one of the two-level factors are called type 1.In Section 2.2,we mainly obtain the following results:if n ? 2?-1-(2r-1),there exist(2r)x 2n designs containing clear 2FICs of type 0 and type 1;and if n ? 2k-r-1,there exist(2r)x 2n designs with resolution IV which contain clear 2FICs;in particular,for n = 2k-r-1,if a(2r)x 2n design with resolution IV has clear 2FICs,then the clear 2FICs are all of type 1.For(2r1)x(2r2)x 2n designs,there are three types of 2FICs.Type 0 involves no high level factor,type 1 involves one component of one high level factor and one of the two-level factors,and type 2 involves both one of the components of the two high level factors.In Section 2.3,we prove that there exist(2r1)x(2r2)x 2n designs containing clear 2FICs if n ? 2k-1-(2r1-1 + 2r2-1)and k ? r1 + r2.And if n ? 2r2(2k-r1-r2-1)and ?>r1 +r2,there exist(2r1)x(2r2)x 2n designs with resolution IV which contain clear 2FICs.For n = 2r2(2k-r1-r2-1),if there exist clear 2FICs in(2r1)x(2r2)x 2n design with resolution IV,then each of the clear 2FICs is of type 1 or type 2.For a given number of runs,when there exist mixed-level designs containing clear 2FICs,we will hope to find the designs with more clear 2FICs.For 8 x 4 x 2n designs with resolution III and IV,Chapter 3 mainly gives the upper and lower bounds on the maximum number of clear 2FICs,and the lower bounds are given by constructing the designs containing more clear 2FICs.Table 3.1.1 compares the upper and lower bounds on the maximum number of clear 2FICs for 64-run 8 x 4 x 2n designs with resolution?.Let nj =2j + 2?-j-12,for n =n3 = 4 and n = n2 = 8,the upper bounds on the maximum number of clear 2FICs equal that of the lower bounds,respectively;and for n3<n? n2,the closer n is to n2,the smaller the difference between the upper bounds?u(6,n,?)and the lower bounds ??(6,n,?)is.Generally,our construction methods are satisfactory when nj+1? n ?nj and n is close to nj(j ? 2).For the 128-run 8 x 4 x 2n designs with resolution IV,Table 3.2.1 compares the maximum number of clear 2FICs,say ?(7,n,?),and the lower bounds ??(7,n,?),and shows that there is no much difference between ??(7,n,?)and(7,n,?).That is,the designs we have constructed are satisfactory.In some experimental situations,inhomogeneity of experimental units often exists.Blocking is the common method to deal with this kind of designs,which is also one of the fundamental principles in experimental designs.The study about the blocked designs was originally motivated by agricultural experiments,and has been widely used in industrial and medicine fields nowadays.Each blocked design has two kinds of factors:the treatment factors and the block factors.There are two kinds of blocking problems according to the number of block variables,one is called single block variable problem and the other is called multi block variables problem involving two or more block variables.Up to now,literature on the study of symmetric blocked designs with single block variable is reach.However,there is little literature on the asymmetric designs,i.e.the mixed-level blocked designs with single block variable.And few people has studied the blocked designs with multi block variables.Chapter 4 discusses the mixed-level blocked designs with single block variable.Suppose ? 2?-run(2r)x 2n design is grouped into 2? blocks by a block variable with 2?runs,denoted as(2r)x 2n:2? blocked design.This chapter mainly obtains the following two results:the necessary and sufficient conditions are n ? 2?-1-(2r-1)and ? + r ? ?for(2r)x 2n:2? blocked designs with resolution at least III to contain clear treatment 2FICs;and when ?? r,there exist resolution at least IV(2r)x 2n:2? blocked designs which contain clear treatment 2FICs if and only if ?-?>r and n ? 2?-?-2r.In Chapters 5 and 6,we study the theory for blocked designs with multi block variables under the CE criterion.We use the notation 2n-m:2? to denote the two-level blocked designs with ? block variables.In chapter 5,we first give the definitions of resolution and CE for the 2n-m:2? blocked designs.Then,we give some necessary and sufficient conditions on the existence of clear treatment main effects and clear treatment 2FIs for 2n-m:2? designs with resolution III,IV-and IV.Note that each alias set of the 2n-(n-?)design is corresponding to a column of H?,we partition all the 2?-1 columns in H? into four classes:the M-c1ass,C-class,UC-class and(?)-class.Based on these four classes,we propose an algorithm for constructing 2n-m?:2? designs with the maximum number of clear treatment 2FIs,and list some design tables.In Chapter 6,we extend the idea of CE to the mixed-level designs with multi block variables.A mixed-level blocked design with one 2r-level factor,n two-level factors and? block variables is denoted by a(2r)x 2n:2? design.For the(2r)x 2n:2? designs with resolution III,IV-and IV,we give some necessary and sufficient conditions on the existence of clear treatment main effects and clear treatment 2FICs.We also discuss how to find(2r)x 2n:2? designs with resolution at least III,which contain more clear treatment 2FICs.And for the 32-run 4 x 2n:2? designs,we construct the resolution III designs containing the most clear treatment 2FICs,and the designs with resolution at least IV-,which contain clear treatment 2FICs.Some concluding remarks and some questions worthy of further investigation are given in Chapter 7.