Based On The Complement Set, The MA Mixed Level 4~12~n Design Is Constructed

Experiment is an important method for people to know and understand nature,which has wide application in industry,agriculture,business and so on.In a lot of cases,the full experiment cannot be carried out because of the time,the outlay and other experiment conditions cannot be fulfilled.So fractional factorial design has been the focus of attention for researchers all the time whose purpose is to make the experiment economic and efficient by carrying out a part of the full design.And the results of two level fractional factorial design are the most abundant.But in many situations,there are not just two levels of each factor which can influence the experiment.So the mixed-level experiment is also significant.Naturaly how to get the optimal mixed-level design has become a valuable problem.The minimum aberration(MA)criterion is commonly used to compare the optimality of the designs.There are many results about the construction of the MA mixed-level design.Wu and Zhang(1993)studied the MA criterion for the mixed-level 412n and 422n designs,and constructed MA 412n design and 422n design of 16 and 32 runs.Mukerjee and Wu(2001)used the theory of finite geometry projection to study the relationship between the 412n design and its complementary design and constructed the MA 412n design through the complementary design.Ai and Zhang(2005)investigated the relationship between factorial design theory and coding theory.It can also identify and construct the mixed-level design through complementary design.The condition of using these two methods is that the com-plementary design is small,i.e.,n is large,otherwise the method is not so economic and efficient.The defining word which consists of only two-level factors is called type 0 defining word.MA criterion of type 0 is defined that type 0 defining word is the most important when some words have same wordlengh.In this paper,a new method for constructing type 0 MA mixed-level 412n design is proposed,and some rules of constructing the MA 412n design of type 0 are given.The new method and method of complementary design have some relation,but the new method developes the method of complementary design and is more applicable.Let P,Pw-1 denote the finite projective geometry PG(k-1,2)and the(w-1)-flat of P,respectively.Let N be the number of experiment treatment combinations.A regular MA 2(n+3)-(n+3-k)(N/2-3?n?2k-1)design corresponds to a set T in P.which satisfies T=T U Pw-1,Pw-1=T U T,where T is the complementary design of T,and T is the complementary set of the complementary design in Pw-1.Taking three two-level columns from T and replacing them with one four-level column,a 412n design D(T0,T')is obtained.Correspondingly,let T'=T' ? P'w-1,P'w-1=T'?T,where T' and T' represent the complementary sets of mixed-level design in Pw-1 and the two level factor part of the design,respectively.It is proved that the relationship of the wordlengh patterns between the design and the complementary set is As,j(T')=As,j(T')+constant,where As,j stands for the number of defining word of type j whose word length is s,s=3,4,j=0,1.This relationship shows that the design D(T0,T')and D(T0,T')have the same type 0 MA property.So the construction of MA design D(T0,T')is equivalent to the construction of the MA design D(T0,T').The MA designs D(T0,T')with the factor number g?4 in T' are given,and the MA designs D(T0,T')with larger n is further obtained.This solves the problem that the design is not easy to construct when the factor number is larger in the complementary design.Furthermore,if a MA 412n design is known,the MA design with larger n can be constructed simply and quickly by using the new method.By the way,a new design with multiplied runs can be obtained by using the new method repeatedly.Examples are given at the end of the paper.